Electronic structure and bonding in thermoelectric...

Electronic structure and bonding inthermoelectric skutterudites

Øystein Prytz

Thesis submitted in partial fulfillmentof the requirements for the degree of

Philosophiae Doctor

Department of PhysicsUniversity of Oslo

June 2007

Ich und Mich sind immer zu eifrig im Gesprache: wie ware es auszuhalten, wennes nicht einen Freund gabe?

- Friedrich Nietzsche, Also sprach Zarathustra.

Summary

The present work is a study of the electronic structure and bonding of the class of materials of-ten called skutterudites. These materials have received much attention during the past decade,largely because their thermoelectric properties are rather easily modified and improved. Mostinvestigations have been into synthesizing materials and measuring their thermoelectric proper-ties. There have also been several theoretical studies of their electronic structure. However, therehave been fewer experimental studies of the bonding of these materials. In this thesis, electronenergy-loss spectroscopy and x-ray photoelectron and Auger spectroscopy have been used to-gether with density functional calculations to study the electronic structure of skutterudites. Thiscombination of experimental and theoretical methods may be mutually beneficial, and the aim ofthis thesis is to bridge the gap between theoretical considerations and experimental investigationsof the bonding.

v

Acknowledgments

There is a saying due to John Donne, ’No man is an island’, but as this is a rather trite thing tosay when acknowledging help and support received, I’ll refrain from using it. It is nonethelesstrue. This thesis would never have come about without the constant interaction I have had witha lot of wonderful people. First and foremost my supervisors, Johan Taftø, Helmer Fjellvag, andTerje Finstad, deserve thanks for their scientific input and support. I would in particular like tothank Johan for excellent supervision and enthusiasm for my project, not to mention the endlessand fun discussions about everything between heaven and earth.

There are several other people with whom I have collaborated closely, their input and suggestionshave been invaluable. Ole Martin Løvvik and Krister Mangersnes have contributed greatly tothe scientific output with DFT calculations and discussion. Spyros Diplas has been central tothe XPS work, but his enthusiasm alone is worthy of mention! Ole Bjørn Karlsen has beenindispensable for the synthesis work, and Sissel Jørgensen deserves great thanks for help aroundthe lab. Furthermore Channing Ahn and Brent Fultz of Caltech have contributed with a lot ofinteresting discussions and input to much of the EELS work. Thank you all!

The people who have contributed scientifically to this thesis are important, but just as importantare all the people who have not! I am of course talking about friends and family. First I would liketo thank my mother and father. I don’t know how they did it, but they managed to stir my interestfor science and technology at quite an early age, encouraging me to stick with it throughout all my21 years of education. Furthermore I would like to thank Olav for friendly heckling, Klaus andIngvild for endless (no really, endless!) discussions of research and education policy, Annett, TorHelge, and Kanutte for being good friends, and Mariam, Hakon, and Tone for many wonderfuldinners and a very welcome diversion from a social life otherwise dominated by physicists.Finally I would like to give a big hug and thanks to Martin and Daniel for being my very bestfriends over many, many years.

Øystein PrytzJune 2007

vii

Preface

Work on this thesis started in the autumn of 2003, and has been funded by the University ofOslo through the FUNMAT@UiO programme. The major part of the research was carried outat the Department of Physics, under the supervision of professors Johan Taftø, Helmer Fjellvag,and Terje Finstad. The main focus of the work has been on studies of the electronic structureof skutterudites using electron energy-loss spectroscopy, density functional theory, and x-rayphotoelectron and Auger spectroscopy.

During the spring of 2006, I spent three months at the California Institute of Technology per-forming EELS experiments under the guidance of Channing C. Ahn and Brent Fultz. The workdone during my stay there is presented in Paper III of this thesis. A further month was spent atBrookhaven National Labs doing energy filtered electron diffraction experiments in collaborationwith Yimei Zhu and Lijun Wu; this is still work in progress and is not presented here.

In addition to the work done in direct relation to my Ph.D. project, I have participated in severalother projects that have resulted in publications that are not part of this thesis. These are listedbelow.

• Ø. Prytz and J. Taftø. Accurate determination of domain boundary orientation in LaNbO4.Acta Materialia 53, 297 (2005).

• U. Olsbye, A. Virnovskaia , Ø. Prytz et al. Mechanistic Insight in the Ethane Dehydro-genation reaction over Cr/Al2O3 catalysts. Catalysis Letters 103, 143 (2005).

• A. Virnovskaia, S. Jørgensen, J. Hafizovic, Ø. Prytz et al. In situ XPS investigation ofPt(Sn)/ Mg(Al)O catalysts during ethane dehydrogenation experiments. Surface Science601, 30 (2007).

• K. Mangersnes, O. M. Løvvik, and Ø. Prytz. Optimization of P-based skutterudites forthermoelectricity from first principles calculations. Submitted to Physical Review B (2007).

• Ø. Prytz, R. Sæterli, and J. Taftø. Comparison of the electronic structure of a thermoelec-tric skutterudite before and after adding rattlers: an electron energy loss study. Submittedto Micron (2007).

ix

Table of Contents

Summary v

Acknowledgments vii

Preface ix

Contents xi

1 Introduction 1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Thermoelectricity and its applications 52.1 The thermoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Simple classical model . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Evaluation of the Seebeck coefficient . . . . . . . . . . . . . . . . . . . 72.1.3 The figure of merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 State of the art thermoelectric materials and the search for new compositions . . . 122.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Heat pump effect: cooling and heating . . . . . . . . . . . . . . . . . . . 152.3.2 Power generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Skutterudites 233.1 Crystal structure and bonding in binary skutterudites . . . . . . . . . . . . . . . 233.2 Filled skutterudites and thermoelectric applications . . . . . . . . . . . . . . . . 28References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Methodology 354.1 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 X-ray photoelectron and Auger electron spectroscopy . . . . . . . . . . . . . . . 384.3 Electron Energy-Loss Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3.1 Core excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3.2 Collective excitations and the joint density of states . . . . . . . . . . . . 47

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

xi

xii TABLE OF CONTENTS

5 Overview of papers 55References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Paper I:O. M. Løvvik and Ø. PrytzDensity-functional band-structure calculations for La-, Y-, and Sc-filledCoP3-based skutterudite structures. 59

Paper II:Ø. Prytz, O. M. Løvvik, and J. TaftøComparison of theoretical and experimental dielectric functions:electron energy-loss spectroscopy and density-functional calculationson skutterudites. 67

Paper III:Ø. Prytz, J. Taftø, C. C. Ahn, and B. FultzTransition metal d-band occupancy in skutterudites studied byElectron Energy-Loss Spectroscopy. 77

Paper IV:S. Diplas, Ø. Prytz, O. B. Karlsen, J. F. Watts, and J. TaftøA quantitative study of valence electron transfer in the skutterudite compoundCoP3 by combining x-ray induced Auger and photoelectron spectroscopy. 85

List of Figures

2.1 Simple classical model of the thermoelectric effect . . . . . . . . . . . . . . . . 62.2 The figure of merit for several materials . . . . . . . . . . . . . . . . . . . . . . 142.3 Thermocouple in heat-pump mode . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Thermocouple in power generation mode . . . . . . . . . . . . . . . . . . . . . 192.5 Segmented thermocouple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1 The unit cell of the skutterudite structure . . . . . . . . . . . . . . . . . . . . . . 243.2 The total and projected density of states for CoP3, CoAs3, and CoSb3 . . . . . . 273.3 CoP3 band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4 DOS for various degrees of substitution . . . . . . . . . . . . . . . . . . . . . . 293.5 The thermal conductivity of CoSb3 and CeFe4Sb12 . . . . . . . . . . . . . . . . 30

4.1 XPS survey scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Production of a KL1L3 Auger electron . . . . . . . . . . . . . . . . . . . . . . . 414.3 The P 2p-KLL Auger parameter for several compounds . . . . . . . . . . . . . 434.4 Sketch of a magnetic prism spectrometer . . . . . . . . . . . . . . . . . . . . . . 444.5 Sketch of an EELS excitation process . . . . . . . . . . . . . . . . . . . . . . . 454.6 Transition metal L2,3 edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.7 The low loss region of Si and Co . . . . . . . . . . . . . . . . . . . . . . . . . . 504.8 The dielectric function of CoAs3 . . . . . . . . . . . . . . . . . . . . . . . . . . 51

xiii

Chapter 1

Introduction

The choice of thermoelectric materials as the topic of this Ph.D. study is largely motivated bythe challenges related to energy and environment. The past decade has seen increasing concernabout the world energy supply. Today, the world energy consumption is more than twenty timeshigher than in 1900. This increase has mainly occurred in the industrialized part of the world,and is seen as a necessity for a modern lifestyle. As the 3rd World economies grow and theirstandards of living increase, their energy use will nearly double by 2030 [1]. Indeed, the totalworld consumption is expected to increase by almost 60% from 2004 to 2030, and most of thisincrease will take place outside the OECD [1].

The burning of fossil fuels has for most of the 20th century accounted for a majority of the energysupply, contributing more than 80% of the consumption in 2004 [1]. This is the cause of majorenvironmental concern as evidence suggests that emission of large amounts of CO2 is causingheating of the atmosphere, in addition to the environmental damage caused by NOx and SOx

released in the burning of fossil fuels.

In light of the expected increase in energy consumption and growing environmental concerns,there is increased interest for alternative sources of energy and the technology for their use. Mostindustrialized countries have large programmes focused on developing and implementing new,environmentally friendly and renewable sources of energy. Although a handful of technologies,such as photovoltaics and biofuels, have received the most public attention, one can expect thata plethora of technologies will need to be implemented in all sectors of the economy.

1

2 Chapter 1. Introduction

However, the criteria for successful implementation of new energy technologies are not easilymet. To be deemed both politically and economically acceptable, the new technologies mustbe capable of supporting dramatic increases of living standards in the less developed world,moderate increases in living standards in the western world, and both of these with less damageto the environment and at lower costs than today’s petroleum based technology.

Dresselhaus and Thomas have reviewed some possible alternative sources for energy [2], andrecently the United Nations Energy Programme reported that investments in sustainable energyjumped by 40% from 2005 to 2006 [3]. The total world investments now exceed $70 billion, andthe growth is expected to continue into 2007. Although this is encouraging, one should not be toooptimistic about large scale exploitation of alternative sources of energy in the short term. Thetechnology and economy of many of these sources are still far from adequate, and fossil fuelsare expected to play a major role in supplying the world with the energy needed for decades tocome. It is therefore important to consider if already extant energy resources can be exploitedmore efficiently, and with less harm to the environment.

In the past decade, there has been a resurgence in interest in thermoelectric materials and de-vices [4]. In applications of thermoelectricity, the coupling between electric currents and heatflow are exploited. This allows a thermoelectric device to function either as a heat pump totransport heat when an electric current is applied, or as a generator when the device is placed ina temperature gradient. Use of thermoelectric devices may therefore facilitate a more efficientuse of energy, or at least the option of recuperating some of the energy lost to waste heat in e.g.industrial processes.

Extensive research efforts have gone into investigating these possibilities, and developing newmaterials for such applications [4, 5]. Although the prospects of using thermoelectric devices asa part of new and more efficient energy production and use are promising, there have so far beenfew applications in this regard. The main bottleneck is the efficiency of the existing materials,and this is the focus of intense research in the fields of physics, chemistry, and materials science.

In the present work, the electronic structure and bonding of the so-called skutterudite materialshave been studied. These materials have received much attention during the past decade, largelybecause their thermoelectric properties are rather easily modified and improved [6]. Most in-vestigations have been into synthesizing materials and measuring their thermoelectric properties.There have also been several theoretical studies of their electronic structure. However, there have

1.0. References 3

been fewer experimental studies of the bonding of these materials. In this thesis, electron energy-loss spectroscopy and x-ray photoelectron and Auger spectroscopy have been used together withdensity functional calculations to study the electronic structure. This combination of experimen-tal and theoretical methods may be mutually beneficial, and the aim of this thesis is to bridge thegap between theoretical considerations and experimental investigations of the bonding.

References

[1] International energy outlook 2007. Tech. Rep. DOE/EIA-0484(2007), Energy InformationAdministration/US Department of Energy, Washington DC (2007).

[2] Dresselhaus, M. S. & Thomas, I. L. Alternative energy technologies. Nature 414, 332(2001).

[3] Global trends in sustainable energy investment 2007. Tech. Rep. ISBN: 978-92-807-2859-0,DTI/0985/PA, United Nations Environment Programme (2007).

[4] Chen, G., Dresselhaus, M. S., Dresselhaus, G., Fleurial, J. P. & Caillat, T. Recent develop-ments in thermoelectric materials. International Materials Review 48, 45 (2003).

[5] Riffat, S. & Ma, X. Thermoelectrics: a review of present and potential applications. Applied

Thermal Engineering 23, 913 (2003).

[6] Uher, C. Skutterudites: Prospective novel thermoelectrics. Semiconductors and semimetals

69, 139 (2001).

Chapter 2

Thermoelectricity and its applications

In this chapter, a background on the thermoelectric effect and a simple model for the ther-mopower α of a material are presented. The figure of merit is introduced, and furthermore abrief review of the current status of thermoelectric materials is given. Finally, possible applica-tions of thermoelectric materials as heat pumps and generators of electrical power are discussed.

2.1 The thermoelectric effect

The discovery of the thermoelectric effect is often attributed to Thomas Johann Seebeck who in1821 observed that an electric current flows through a circuit made of two different materials,if the two junctions are kept at different temperatures. Another aspect of the same physicalphenomenon was observed somewhat later (1834) by Jean-Charles-Athanase Peltier. Workingwith similar circuits as Seebeck, he discovered that passing a current through the two materialscaused heat to be absorbed at one junction and expelled at the other.

2.1.1 Simple classical model

These effects, today known as the Seebeck and Peltier effects, are both caused by a couplingof the currents of heat and electrical charge. If one realizes that e.g. electrons are carriers of

5

6 Chapter 2. Thermoelectricity and its applications

e

e

e e

ee

e

e

e

e

e

e

e e

+

+

+

+

+

+

-

-

-

-

-

-

Heat current of electrons

T + ∆T T

Net +Q Net -Q

V + ∆V V

Figure 2.1: An intuitive model of the diffusion of electrons caused by a temperature gradient. Anelectrically isolated slab of material is heated on the left (∆T > 0) causing a net flux of electronstowards the right. Adapted from Lovell [1].

both heat and charge, the existence of such effects may not be too surprising. Indeed, a simpleclassical picture of an electrically isolated slab of material, such as that shown in figure 2.1, maysuffice as a first intuitive explanation of the phenomenon.

At finite temperatures, we can consider the electrons as particles which are continually in motionin random directions. If the two ends of the material are in thermal equilibrium (∆T = 0), theprobability of an electron making its way from one end to the other is the same as for an electrontraveling the opposite direction. Thus, there will be no net flux of electrons.

However, if one end is at a higher temperature than the other, the thermally induced motion ofthe electrons there will be more vigorous than on the cold side. If this is the case, there will be anet flux of charge from the hot to the cold side.

This process will continue until the induced electric potential between the two ends is largeenough to halt the net flow of electrons, and a stationary state is achieved. The Seebeck coef-ficient (or thermopower) of the material is then phenomenologically defined as the ratio of thepotential difference to the temperature difference:

α =∆V

∆T(2.1)

For a material where electrons are the dominant charge carrier, as in figure 2.1, if ∆T > 0 then∆V is a negative number and the Seebeck coefficient is also negative. Conversely, in this simpleclassical model the thermopower is positive if positively charged holes are the dominant carrier.

2.1. The thermoelectric effect 7

Although this model may be sufficient to gain a first intuitive understanding of the phenomenon,the transport properties of materials can not be fully understood classically. In the next section,a simple non-classical model for the thermopower of a material is presented.

2.1.2 Evaluation of the Seebeck coefficient

This derivation roughly follows that of Dugdale [2]. The perhaps easiest way to evaluate theSeebeck coefficient is though the related Peltier coefficient. In doing this, an electric field Ex

is applied in the x-direction of a material which is held at a constant temperature. The Peltiercoefficient π is given as the ratio between the heat current density relative to the electrical currentdensity, and is related to the Seebeck coefficient α through the Kelvin relation [3]:

π =heat current

electrical current=

jQx

jex

= Tα (2.2)

Under the assumption that all transport of heat and charge is in the x-direction and is solelydetermined by the flow of electrons—that is no heat or charge transport by holes, and no heat iscarried by phonons—the current densities of heat and electrical charge are given by:

jQx =

1

V

∑i

hivi(x) (2.3)

jex =

1

V

∑i

evi(x) (2.4)

Here e is the electron charge and vi(x) is the electron velocity in the x-direction, while the sumis over all electrons i and V the volume of the system. In order for a transport of heat to occur,an electron must be thermally excited at one end of the material, only to lose this energy in ade-excitation process after being moved some distance by the applied electrical field. The heattransported into a unit volume is given by the thermodynamic identity: dQ = TdS = dU−µdN

where dU is the change in internal energy, dS is the change in entropy, µ the chemical potential,and dN is the change in number of particles in the volume. The heat introduced by one electron


is then hi = εi − µ, where εi is the energy of the thermally excited electron.

Combining equations (2.2) and (2.3) with this definition of the carried heat, an expression for theSeebeck coefficient can be found:

α =π

T=

1

Te

∑i hivi(x)∑i vi(x)

=1

Te

∑i(εi − µ)vi(x)∑

i vi(x)(2.5)

Furthermore, the current density contributed by the i’th electron in the x-direction is defined asjei (x) = evi(x), giving:

α =1

Te

∑i(εi − µ)je

i (x)∑i j

ei (x)

(2.6)

It is often more convenient to deal with an integration over all energies instead of a sum overmany electrons. If the contribution to the electric current by electrons with energy between ε andε + dε is described as je

x(ε)dε, equation (2.6) can be written [2]:

α =1

Te

∫(ε− µ)je

x(ε)dε∫jex(ε)dε

(2.7)

The Seebeck coefficient can be further evaluated by examining jx(ε), which is the contribution tothe current density of electrons with with energy ε. The total current density is then the integralof this partial density over all energies. It can be shown [2] that this is given by:

jx =

∫jx(ε)dε =

∫−Exσx(ε)

df0(ε)

dεdε (2.8)

where Ex is the applied electric field, and σx(ε) is the conductivity of electrons with energy ε.Also, the equilibrium Fermi–Dirac distribution function f0(ε) has been introduced. The electricconductivity is to a large part determined by the number of available electrons for conduction andthe scattering of electrons in the material. Both of these are dependent on energy, and it is there-fore reasonable that the conductivity is a function of energy. Inserting this into equation (2.7)one arrives at:


α =1

Te

∫ df0(ε)dε

(ε− µ)σx(ε)dε∫ df0(ε)dε

σx(ε)dε(2.9)

which is a well known expression for the Seebeck coefficient [4, 2, 5].

It is important to remember that this model is limited: it assumes that there is no contributionfrom holes to the electrical current and transport of heat, and also that the there are no phononsthat carry heat. These are factors that can be incorporated into the model, but would complicatethe derived expression quite a bit. Furthermore, it is important to be aware that all factors ap-pearing in equation (2.9) are temperature dependent, and the expression is therefore only validfor situations where the heat transfer does not cause any changes in temperature.

Nevertheless, equation (2.9) gives a lot of important information on the thermopower. First,at energies far from the Fermi–level, f0 is a constant (either 1 or 0). Thus, the appearance ofderivative of the Fermi–Dirac function, df0(ε)

dε, shows that only electrons close to the Fermi-level

contribute. Furthermore, the function df0(ε)dε

(ε− µ) is anti-symmetric about the Fermi–level andintegrates to zero. Therefore, if non-zero values of α are to be obtained, the conductivity, σ(ε)

must be a non-symmetric function of energy. That is, the conductivity must vary for electrons ofdifferent energy. Indeed, it can be shown [2, 6] that the thermopower can be evaluated as

α =π2

3

(kBT )2

eT

1

σ(ε)

∂σ(ε)

∂ε

∣∣ε=εF

(2.10)

where the appearance of the term ∂σ(ε)∂ε

emphasizes this point. Such a variation of the electricconductivity can be achieved for example if the the scattering processes and the density of statesclose to the Fermi–level (available electrons) varies strongly as a function of energy.

2.1.3 The figure of merit

The macroscopic equations of charge and heat transfer are now examined [4]. Consider a samplewhere the two ends are held at constant, but different, temperatures, giving a temperature gradient∇T . This difference in temperature will cause a thermoelectric current J∇T

e = −σα∇T to flow


in the material. If an electric field E is applied, the total current density becomes:

Je = JEe + J∇T

e = σE − σα∇T (2.11)

with one contribution (J∇Te )arising from the temperature difference and, one from the electric

field (JEe ).

The thermal current density JQ can be analyzed in a similar manner. At zero electric field, but ina temperature gradient, the thermal current density will be given by J∇T

Q = −λ0∇T , where λ0 isthe thermal conductivity at zero electric field. If an electric field is applied, the induced currentof electrons will contribute to the thermal current through the term JE

Q = JEe π = JE

e αT , and thetotal heat current will be given by:

JQ = JEQ + J∇T

Q = JEe αT − λ0∇T (2.12)

There are two effects that have been ignored in these expressions. First, the internal resistance ofthe material will cause a heating effect I2R, which can be disregarded for small currents. Sec-ondly, the Thomson–effect is neglected, an approximation which is valid for small temperaturedifferences or if the Seebeck coefficient is independent of temperature [4, 3].

From equation (2.11) an expression for the electric current caused in an external field is achieved:

JEe = Je + α∇Tσ (2.13)

which is insert into equation (2.12) to arrive at:


JQ = (Je + α∇Tσ)αT − λ0∇T

= αJeT + α2σT∇T − λ0∇T

= αTJe + (α2σT − λ0)∇T

= αTJe − λ0(1− α2σ

λ0

T )∇T

= πJe − λ∇T (2.14)

The final line of equation (2.14) shows that the total heat current is the sum of two contributions:one which depends on the Peltier coefficient and the electric current, and one which varies withthe temperature gradient. In doing this, a new total thermal conductivity is defined that takes intoaccount the retarding effect of the thermoelectrically induced electric field: λ = λ0(1− α2σ

λ0T ) =

λ0 − α2σT . Rearranging this expression gives:

ZT =α2σ

λT =

λ0

λ− 1 (2.15)

where Z = α2σλ

. This relationship is of particular importance when investigating the thermoelec-tric effect in materials. For example, if heat is to be transferred along the temperature gradient,it is obvious from equation (2.14) that the highest amount of heat transfer for a given electricalcurrent is achieved for small values of λ. Equation (2.15) shows that this corresponds to largevalues of ZT .

Similarly, the maximum electric current for a given value of the temperature gradient can beinvestigated. Combining equations (2.14) and (2.12) gives:

JQ = πJe − λ∇T = σEαT − λ0∇T

αTJe = σEαT − λ0∇T + λ∇T

Je = σE − (λ0 − λ)

αT∇T (2.16)


Thus, at E = 0, the maximum current Jmaxe = − λ0

αT∇T is achieved with small values 1 of λ,

which corresponds to large values of ZT .

Because of its importance in determining how close the system is to an ’ideal’ thermoelectric ma-chine, ZT is often considered as the figure of merit for a thermoelectric material. Investigationsinto new thermoelectric materials often report this figure as their main result, and in section 2.2 abrief review of the state of current research into new materials is given. In general, commerciallyused materials today have a maximum figure of merit ZT ≈ 1.

Through similar arguments as those leading to the relation between ZT and the thermal con-ductivities in equation (2.15), one can obtain [7] the expression ZT = Vα/Vρ where Vα is theSeebeck voltage and Vρ is the voltage drop due to the ohmic resistance of the material. Althoughthis approach may allow a quick evaluation of a material in a single experiment, it is often prefer-able to separately measure the properties in equation (2.15), and thereby gain insight into theirindividual contributions to the figure of merit. This is particularly important when modifyingmaterials in an attempt to improve their thermoelectric performance.

2.2 State of the art thermoelectric materials and the searchfor new compositions

Since thermoelectric phenomena were first observed, there has periodically been great interestin investigating the thermoelectric (TE) effect in various materials. As shown in the previoussection, a ’good’ TE material needs to have a high figure of merit, that is, a high value for theSeebeck coefficient, good electric conductivity, and low thermal conductivity. Some materialsmay possess one or more of these characteristics, however, further improvements in the figure ofmerit may be achieved through careful modifications of the material.

One obvious way of improving the figure of merit is if the Seebeck coefficient can be increased.From equation (2.9) we see that this can be achieved by ensuring a large gradient in the electric

1The function |(λ0 − λ)| has a maximum value of λ0. This can be understood if one realizes that λ can not belarger than λ0 (from equation (2.15)). Furthermore, λ can only take positive values, since a negative value of thethermal conductivity would allow a spontaneous transport of heat from the cold end of the material to the hot side.This is not possible, as it would cause a net reduction of the entropy of the system, thereby violating the SecondLaw of Thermodynamics.

2.2. State of the art thermoelectric materials and the search for new compositions 13

conductivity, σ(ε), close to the Fermi–level. Unfortunately, tuning the Seebeck coefficient inthis manner will also affect the number of available conducting states, possibly giving a loweroverall electric conductivity, which in turn is detrimental to the TE performance of the material.Conversely, modifying the electric conductivity σ(ε) may have unwanted effects on the Seebeckcoefficient.

The coupling between thermal and electric conductivities must also be considered. There are twocontributions to the total thermal conductivity: the electron (λe) and lattice (λl) parts. Thus, thetotal thermal conductivity can then be written λ = λe + λl. The electron contribution is due toelectrons carrying heat, and can be described by the Wiedemann–Franz law. This empirical lawstates that the electron contribution to the thermal conductivity can be described by the relation:λe = L0σT , where L0 is approximately a constant [8], and σ is the electric conductivity. Thelattice contribution to the thermal conductivity is due to the heat carried by the lattice vibrationsthrough quantized oscillations called phonons.

In materials with high electrical conductivity, λe dominates, and the thermal conductivity is oftenconsidered to be described solely by the Wiedemann–Franz law. For these materials, increasingthe electric conductivity will cause a proportional increase in the thermal conductivity. Examin-ing equation (2.15), one notes that this defeats the goal of increasing the figure of merit. Indeed,for a metal, where the Wiedemann–Franz law accurately describes the total thermal conductivity,the figure of merit becomes ZT = α2

L0T and depends only on the value of the Seebeck coeffi-

cient. Although there in principle is no upper bound for the value of ZT , for practical purposesthe highest value will be achieved by striking a balance between several effects that may verywell counteract each other.

Roughly, one can say that there have been three periods of research into TE materials. Earlyfocus on the transport properties of metals revealed that while they have a high electric conduc-tivity, their Seebeck coefficients are usually too small and their thermal conductivity too largefor practical energy conversion or heat pump applications. After these initial investigations in-terest waned and there was little progress in developing new materials. It was only after theSecond World War that the advent of modern semiconductors gave a renewed interest in thermo-electric phenomena in materials. These materials have a much larger Seebeck coefficient thanmetals, and, through doping, can achieve reasonable values of the electric conductivity. Further-more, explorations of space created a demand for power sources which can operate unattendedand without maintenance for long periods of time, without any significant deterioration in the


Temperature (K)

ZT

Si0.8

Ge0.2

Alloy

PbTe Alloy

CoSb3

CeFe3.5

Co0.5

Sb12

Zn4Sb

3

Zn4Sb

3 Alloy

Bi2Te

3 Alloy

CsBi4Te

6

PbTeSeTe/PbTe

Quantum dots

Superlattices

Bi2Te

3/Se

2Te

3

Superlattices2.0

1.5

1.0

0.5

0.0

2.5

200 400 600 800 1000 1200 14000

Figure 2.2: The dimensionless figure of merit for several thermoelectric materials that have beeninvestigated the last decade. Adapted from [9].

amount of power supplied. As will be discussed in the next section, electric generators based onthermoelectric materials are prime candidates to fill such a niche. Up till the mid 1960’s, a hugenumber of binary semiconductors were investigated, and this burst of research resulted in manyof the thermoelectric materials that are used today. These include the Bi-Te alloys which have aZT ≈ 1 at room temperature, and the Si-Ge alloys which are suited to work at high temperatures.

The third bout of increased activity on thermoelectric materials started in the early 1990’s. Thereare two main features of the recent activity that should be mentioned: first, the exact tuningof the electronic properties has been greatly aided through modern quantitative theory, suchas calculations based on density functional theory (DFT). For example, through studies of thedensity of states and position of the Fermi–level it is possible to consider the above mentionedtradeoff between increasing the Seebeck coefficient and any concomitant reduction in electricconductivity.

Furthermore, researchers have endeavored to reduce the total thermal conductivity without at thesame time causing a reduction in the electric conductivity. This can be achieved by modificationsof the material that introduce defects or structural features that strongly scatter phonons, but havea smaller effect on the charge carriers. Thus, only the lattice part λl of the thermal conductivity

2.3. Applications 15

is affected, and changes in the electric conductivity are avoided. This can dramatically enhancethe figure of merit for a material. For example, Bi2Te3/Sb2Te3 super-lattices have been reportedto achieve a room temperature figure of merit up to 2.4, which is significantly higher than that ofsimilar bulk materials [10].

Figure 2.2 shows the figure of merit ZT as a function of temperature for some of the mostrecently investigated materials, as well as the ’old’ bulk semiconductors which are commerciallyavailable. It is important to note that any given material only achieves its maximum figure ofmerit for a certain temperature region. Outside this range, the figure of merit declines, or perhapsthe material decomposes at higher temperatures. Thus, for any application, the material has tobe selected based on the operating temperature of the device.

2.3 Applications

The coupling of electric and thermal currents in a material allows for several interesting appli-cations. By applying a current, one can achieve a heat pump effect giving heating or cooling,while naturally occurring temperature gradients or waste heat can be used to generate electric-ity. Thermoelectric materials have seen some success in applications where special conditionsdictate that conventional methods of heating/cooling or power generation cannot be utilized. Ingeneral, TE devices are of particular interest due to their compact and low weight nature. Inaddition, unlike most other technologies, TE devices have few or no moving parts, and the needfor mechanical maintenance is therefore usually very low. However, there have so far been onlylimited commercial success in consumer goods and large scale applications. This may changewith the increasing focus on environmentally friendly power use and generation, and as moreefficient TE materials are being developed.

2.3.1 Heat pump effect: cooling and heating

First, the use of TE materials as heat pumps is investigated. For most applications, two differentTE materials are combined in a TE device, or thermocouple. A sketch of such a device is shownin figure 2.3. The two materials used are e.g. semiconductors doped in such a way that one isan n-type conductor, while the other is p-type. The materials are connected electrically in series,


p n

Heating

Cooling

Active cooling/heating

I

Figure 2.3: Sketch of a thermoelectric device (thermocouple) operating as a heat pump. The twolegs are made of different materials, one with n-type conductivity and one with p-type. The legsare connected electrically in series and thermally in parallel. An external power source sets up anelectrical current though the device, thus causing a net flux of heat.

but thermally in parallel. Applying a voltage will set up a current as shown in the figure, drivingcharge carriers from one end of the device to the other. As the charge carriers are also carriers ofheat, this will cause a cooling of one end and heating of the other.

The rate of heat absorption at the cold end is given by

Q = α(Tm − ∆T

2)I − 1

2I2R− λ0∆T (2.17)

where Tm is the average temperature in the device and ∆T is the temperature difference. Thisexpression is analogous to equation (2.12), however, the Joule heating I2R caused by the electriccurrent and the internal ohmic resistance of the material is now also taken into account. It canbe assumed that half of this heat ends up at the cold end, thereby reducing the rate of cooling.The Thomson effect is neglected as it is very small compared to the effects under considerationhere. The cooling efficiency can be defined as the rate of heat transport for a given power input.The power input is given by W = α∆TI + I2R, where the first term is the power needed toovercome the Seebeck voltage arising from the temperature difference, while the second is thepower loss due to ohmic resistance in the device.


The cooling efficiency φ then becomes

φcooling =Q

W=

˙α(Tm − ∆T2

)I − 12I2R− λ0∆T

α∆TI + I2R(2.18)

In order to achieve maximum efficiency, there are two factors that need to be considered. First,the geometry of the thermocouple must be optimized so as to minimize power loss due to Jouleheating and losses from return heat conduction due to the temperature gradient. Second, thereis an optimum current for which the heat transport is maximum with regard to the heating I2R.For a thermocouple satisfying these conditions, it can be shown [3] that the cooling efficiency isgiven by

φcooling =Tm(

√1 + ZTm − 1)

∆T (√

1 + ZTm + 1)− 1

2(2.19)

Here Tm is the mean temperature across the thermocouple, while Z is the figure of merit of thedevice defined in a similar manner as in equation (2.15).

The highest cooling efficiency is achieved when the temperature difference between the twosides is relatively small, however, the overall efficiency of a TE cooling device is usually farlower than for gas-compression based technology. For example, a typical household refrigeratoroperates in a 20 temperature difference between room temperature (≈ 300 K) and somewhatabove freezing. With values of ZT ≈ 1, this will give an efficiency of approximately 1.4 for theTE device. In comparison, conventional compressor based refrigerators have an efficiency of 2–4under similar operating conditions [9]. For a TE refrigerator to achieve this level of efficiency,thermoelectric materials with a value of ZT around 3–4 would be needed.

Despite this rather low efficiency, TE refrigeration has become rather commonplace in specialapplications where traditional cooling technologies are impractical. The perhaps most visibleapplication to the general public is the use of portable TE picnic coolers intended for use in e.g.cars. Traditional cooling with compression/decompression of a cooling liquid is usually infea-sible for such applications due to volume and weight restrictions. However, this is only one ofmany applications that take advantage of the light weight and compactness of TE cooling tech-nology. Other advantages that often make TE cooling more attractive than the other alternatives


are localized and highly controllable cooling, noiseless operation, and very low maintenanceneeds.

In some cases, the reverse application of the heat-pump may be of more interest. Instead ofusing the TE device for cooling, one can consider the possibility of using the heat pump effect toachieve heating, for example to heat houses or offices by transporting heat from the outside to theinside. The physics of this process are very similar to that of the case for cooling, except that theJoule heat produced by the electric current is no longer a detrimental effect. Instead of reducingthe efficiency, the I2R heating now contributes to the desired result, meaning that the sign ofthe 1

2I2R term in equation (2.17) should be changed. The heating efficiency now becomes [3]

φheating = φcooling + 1 giving

φheating =Tm(

√1 + ZTm − 1)

∆T (√

1 + ZTm + 1)+

1

2(2.20)

Assuming a 20 temperature difference, and a room temperature of 295 K, the heating efficiencyis close to 3 for a TE device with ZTm ≈ 1. The efficiency is largest for small temperaturedifferences, meaning that if the outside temperature drops too much, the benefit of such a heatpump compared to conventional heating is reduced.

2.3.2 Power generation

An alternative application of the thermoelectric phenomenon is to take advantage of the volt-age caused by a temperature gradient. For example, thermoelectric devices are widely used asaccurate temperature sensors. However, in the following, the focus is on use of thermoelectricmaterials for power geneneration.

For power generation, thermoelectric materials are combined in a device in a similar manner asfor heating/cooling applications. The device is then placed in a temperature gradient which mayoriginate from for example solar or geothermal sources, or from industrial waste heat. The dif-ference in temperature causes an electric current to flow, thereby converting some of the thermalenergy to electric energy, see figure 2.4.

The conversion efficiency is the fraction of heat input that is converted to electricity, and can be


p n

Heat sink

Heat source

Power generation

I

Figure 2.4: Thermocouple in power generation mode

shown [3] to depend on the temperature difference and the figure of merit of the device throughthe relation

η =T1 − T2

T1

√1 + ZTm − 1√

1 + ZTm + T2/T1

(2.21)

where T1 and T2 are the hot and cold side side temperatures respectively. In contrast to thesituation where we wish to achieve heating or cooling, the maximum efficiency is now obtainedfor large temperature differences.

Thermoelectric generator based on this principle have been in use for several decades. Theperhaps most well known applications are as power sources for space probes launched by theUS and Soviet Union. This power source is particularly interesting for deep space probes thatoperate far from the Sun, and therefore cannot rely on photovoltaics for their power. Instead, heatis supplied by the decay of a radioactive isotope, and some of this heat is converted to electricityby the TE device. Several well known space missions, such as Voyager and Viking, have reliedon this technology for their power. In all, more than 50 space probes have used TE generators astheir main sources of power [11].

Terrestrial applications are not uncommon. In particular, thermoelectric generators are used forpowering electric equipment located in remote areas that do not have access to the electrical grid,


Figure 2.5: Example of a proposed segmented thermocouple in which different materials are useddepending on the temperature variation throughout the device [13]. The predicted conversion effi-ciency of this device is 15.5% with a hot side temperature of 975 K and cold side temperature of 300K.

or where frequent maintenance is impractical. These applications include powering meteorologi-cal equipment or other sensors, cathodic protection of pipelines, and telecommunication devices.The needed heat is usually supplied by burning of fossil fuels [12].

The conversion efficiency of these devices is usually in the range of 4–9% [12]. However, byincreasing the temperature difference, it would be possible to achieve a higher conversion rate.For example, from equation (2.21) one finds that a hot side temperature of 900 K and cold sidetemperature of 300 K would give a conversion efficiency of approximately 15% if the figure ofmerit is ZTm ≈ 1.

However, it is not a trivial matter to obtain such conversion rates. As seen in the previous sec-tion, each thermoelectric material has a certain temperature range over which it has favourableproperties. If a TE device is designed with materials suitable for the hot side temperature, thesematerials will most likely not be able to operate optimally at the cold end. In other words, ZT forany one material is far from optimal for the entire range of temperatures throughout the device.Thus, the efficiency cited above would probably not be possible with any single device/material.

Several possible solutions to this problem have been proposed. One possibility is to constructthe TE device from several materials which are suited to different temperatures. Closest to thehot side, one would use materials suited for high temperatures, while closer to the cold side other

2.3. References 21

materials would be used. An example of such a device is sketched in figure 2.5. The predictedconversion efficiency of this device is 15.5% with a hot side temperature of 975 K and cold sidetemperature of 300 K [13, 14], thus rivaling the efficiency of commercial photovoltaic cells.

In conclusion, thermoelectric materials have great potential as electric generators or heat pumps,especially if their figure of merit ZT can be raised above that of the currently available commer-cial materials. This is a field of intense and growing research activity world wide, although greatchallenges remain.

References

[1] Lovell, M. C., Avery, A. J. & Vernon, M. W. Physical Properties of Materials (Van Nos-trand Reinhold Company, New York, 1976).

[2] Dugdale, J. S. The Electrical Properties of Metals and Alloys. The Structure and Propertiesof Solids 5 (Edward Arnold, London, 1977).

[3] Goldsmid, H. J. Applications of thermoelectricity (Methuen & Co. Ltd., London, 1960).

[4] Mahan, G. D. Good Thermoelectrics. In Eherenreich, H. & Spaepen, F. (eds.) Solid State

Physics, vol. 51, 82–152 (Academic Press, San Diego, 1998).

[5] Cutler, M. & Mott, N. F. Observation of Anderson Localization in an Electron Gas. Physical

Review 181, 1336 (1969).

[6] Ashcroft, N. W. & Mermin, N. D. Solid State Physics (Thomson Learning, Saunders Col-lege, Philadelphia, 1976).

[7] Harman, T. C. Special Techniques for Measurement of Thermoelectric Properties. Journal

of Applied Physics 29, 1373 (1958).

[8] Kittel, C. Introduction to Solid State Physics (John Wiley & Sons, Inc., 1996), 7th edn.

[9] Chen, G., Dresselhaus, M. S., Dresselhaus, G., Fleurial, J. P. & Caillat, T. Recent develop-ments in thermoelectric materials. International Materials Review 48, 45 (2003).


[10] Venkatasubramanian, R., Siivola, E., Colpitts, T. & O’Quinn, B. Thin-flm thermoelectricdevices with high room-temperature fgures of merit. Nature 413, 597–602 (2001).

[11] Bennett, G. L. Space applications. In Rowe, D. M. (ed.) CRC Handbook of Thermo-

electrics, chap. 41, 515–537 (CRC Press, Boca Raton, 1995).

[12] McNaughton, A. G. Commercially available generators. In Rowe, D. M. (ed.) CRC Hand-

book of Thermoelectrics, chap. 36, 459–469 (CRC Press, Boca Raton, 1995).

[13] Caillat, T., Fleurial, J. P., Snyder, G. J. & Borshchevsky, A. Development of high efficiencysegmented thermoelectric unicouples. In Proceedings of the 20th International Conference

on Thermoelectrics, Beijing, 282. IEEE (IEEE catalog no. TH8589, 2001).

[14] Caillat, T. et al. A new high efficiency segmented thermoelectric unicouple. In Proceed-

ings of the 34th Intersociety Energy Conversion Engineering Conference, 2567 (Vancouver,1999).

Chapter 3

Skutterudites

In this chapter, materials with the skutterudite-like structure are presented. Their crystal struc-ture, bonding and physical properties are discussed. Particular emphasis is put on modificationsof the materials that lead to enhanced thermoelectric properties.

3.1 Crystal structure and bonding in binary skutterudites

The binary skutterudite compounds have the general chemical formula MX3, where M typicallyis one of the column 9 transition metal (Co, Rh, or Ir), and X is one of the elements P, As, orSb (often called pnicogen). The skutterudites belong to the cubic space group Im3 (no. 204),and their unit cell may be considered to consist of eight smaller cubes with metals atoms on thecorners. Six of these cubes are filled with mutually perpendicular (nearly square) rectangles madeup of the X atoms, while two of the cubes are left empty, see figure 3.1(a). The pnicogens arelocated slightly off center in a tetrahedron of two pnicogen and two metal atoms (figure 3.1(b)),while the metal atoms are octahedrally coordinated by pnicogens (figure 3.1(c)). Thus, eachpnicogen has two pnicogen and two metal nearest neighbours, while each metal has six pnicogennearest neighbours.

Using Wyckoff notation, the metal atoms are placed on the 8c sites, the X atoms are on the 24gsites and the voids are the 2a sites. The structure is fully defined by the cell parameter a, and two

23

24 Chapter 3. Skutterudites

(a) The skutterudite unit cell with the rectangu-lar pnicogen ’rings’ emphasized.

(b) The tetrahedral environment of the pnico-gens. Each pnicogen has two pnicogen and twotransition metal nearest neighbours.

(c) The octahedral environment of the transi-tion metal atoms.

(d) The filled skutterudite structure. Here thetransition metal has been fully substituted witha lower valent element (e.g. Fe) and a ’rattler’(in purple) introduced in the two large voids.

Figure 3.1: The unit cell of the skutterudite structure with the origin translated [1/4 1/4 1/4], andthe various structural features emphasized. The pnicogen atoms are shown in red, and the transitionmetals in yellow/gold.

3.1. Crystal structure and bonding in binary skutterudites 25

Table 3.1: Structural parameters of some binary skutterudites.

a (A) y zCoP3 ref. [1] 7.7112 0.34895 0.14513CoAs3 ref. [2] 8.195 0.3431 0.1503CoSb3 ref. [3] 9.0385 0.33537 0.15788RhP3 ref. [4] 7.9951 0.3547 0.1393IrP3 ref. [4] 8.0151 0.3540 0.1393

Table 3.2: Comparison of the covalent radii of Co and P, As, and Sb [6] with the observed bondlength in skutterudites [4].

Covalent radius r (pm) rCo + rX Observed Co-X bond length (pm)Co 125P 110 235 CoP3 222As 121 246 CoAs3 234Sb 141 266 CoSb3 252

positional parameters y and z for the pnicogens. Table 3.1 lists structural parameters for somebinary skutterudites.

The bonding of skutterudites is often considered to be covalent in nature. First, the difference inelectronegativity between the metal and pnicogen atoms is not large enough for the bonding to beconsidered ionic. As an example, the difference in electronegativity in an ionic compound suchas CoO is approximately 1.5 (Pauling units), while for the most extreme case in skutterudite(CoP3) the difference is only about 0.3 [5]. Thus, sharing of electrons in a covalent bond isexpected in stead of transfer of charge between the elements. Furthermore, the observed M–Xbond length is close to the sum of the elemental covalent radii. For example, the sum of covalentradii for Co and P is 235 pm, while the observed bond length in CoP3 is slightly less at 222pm, see table 3.2. This is comparable to the situation in e.g. GaAs. Thus covalent bonding isexpected, although a slight ionic character is possible.

The valence electron configuration of the pnicogen atoms is of the type ns2np3, giving fiveavailable electrons for bonding. As shown in figure 3.1(b) each pnicogen has two pnicogen andtwo metal nearest neighbours. In a covalent model of bonding, two pnicogen valence electronscontribute to bonding with the nearest pnicogens, while the remaining three form bonds with thetwo metal atoms. From the perspective of the metal atom, the octahedral environment gives rise


to d2sp3 hybrid metal orbitals with octahedral symmetry. The metal atom donates three electronstowards these bonds, while the remaining nine are supplied by the pnicogens [4, 7].

Turning to the electronic structure calculated by density functional theory (DFT), the variousskutterudite compounds show great similarities. As an example, the total density of states (DOS)and the symmetry projected local density of states for the three cobalt based skutterudites areshown in figure 3.2. The Co d states are split into states of t2g and eg symmetry due to the octa-hedral environment of the Co atoms, with the majority of the Co d states strongly peaked approx-imately 2-2.5 eV below the Fermi-level. The pnicogen p states are rather uniformly distributedabove and below the Fermi-level, hybridizing with the Co d states and contributing heavily tothe total DOS in the -7 to +5 eV region . The Co s and p states contribute only to a small degreeto the total DOS, while the pnicogen s states dominate in the region 9-15 eV below the Fermi-level. In CoP3, the Fermi-level is located at the very lower edge of the conduction band, while inCoAs3 and CoSb3 it is shifted slightly downwards towards the valence band, into a region wherethe density of states is very small. The presence of such a region with a low density of states istypical of the skutterudites [8, 9, 10], and is often referred to as a pseudo-gap. The calculatedpseudo-gap is approximately 1 eV wide in CoP3 and decreases in width in the skutterudites withthe heavier pnicogens and larger unit cells.

Although the density of states is very low in this region, it is generally not found to be zero. Thisbecomes seen most easily when the band structure is plotted. Figure 3.3 shows the band structureof CoP3 near the Fermi-level, and it is evident that the pseudo-gap is crossed by a single bandtouching the conduction band near the Γ point just above the Fermi-level. Thus, theory predictsthat CoP3 is a metal. However, this contradicts experimental evidence indicating semiconductingbehaviour of CoP3 [12]. This may perhaps be explained by the tendency of DFT calculations tounderestimate band gaps.

3.1. Crystal structure and bonding in binary skutterudites 27

0

20

40

CoP3

Total DOS

CoAs3

Total DOS

-15 -10 -5 0 5

CoSb3

Total DOS

0

1

2

3

DO

S

Co s

p

d

Co Co

-15 -10 -5 0 50

0.2

0.4 P

-15 -10 -5 0 5

E - EFermi

(eV)

As

-15 -10 -5 0 5

Sb

Figure 3.2: The total and projected density of states for CoP3, CoAs3, and CoSb3 [11].

Figure 3.3: CoP3 band structure showing that a single band crosses the pseudo-gap close to the Γpoint [13].


3.2 Filled skutterudites and thermoelectric applications

Several binary skutterudite compounds have relatively high values of the Seebeck coefficientand the electrical conductivity. For example, values as high as 630 µV/K have been reportedfor the Seebeck coefficient of CoSb3 [14], which together with reasonable electric conductivitymakes this compound interesting for thermoelectric applications. Unfortunately, the thermalconductivity of the skutterudites has been found to be too high, and only moderate values of thedimensionless figure of merit are usually achieved.

This changed, however, in 1994, when Slack and Tsoukala suggested that filling the the voids inthe structure could reduce the thermal conductivity of the material through scattering of thermalphonons [15]. It has been known for decades that ternary skutterudites can be made [16], whereadditional atoms are placed in the large voids as in figure 3.1(d). In the case of a completefilling of the structure, the general formula is RyM4X12 with y = 1. Here R is typically anelectropositive rare earth metal.

Complete filling is, however, generally not achieved in compounds made with Co or anothercolumn 9 transition metals. Filling fractions for these materials are typically well below y =

0.5, with the case of Ba filled Cobalt–Antimony skutterudites with y = 0.44 considered ananomaly [14]. For example, the maximum filling fraction for CayCo4Sb12 has been found to bey = 0.2 [17], and theoretical investigations suggest that the solubility of La, Y and Sc in CoP3 isbelow y = 0.06 [13].

In order to achieve high levels of filling while maintaining structural stability, it is often necessaryto perform a charge compensation, or doping, by substituting the transition metal with a lowervalent element. The argument here is that the extra valence electrons brought into the system bythe filling element are compensated for by reducing the number valence electrons contributed to-wards the metal–pnicogen bonding by the transition metal. Thus the octahedral metal–pnicogencomplex becomes electron deficient relative to the situation in the binary skutterudite, and chargeis transferred from the filling atom to compensate this deficiency. This allows large filling frac-tions, and even a complete filling of the structure [12, 16, 18].

Typical examples are the substitution of Co with Fe, which allows the complete filling of thevoids. As an example, tetravalent Ce can participate in a CeFe4P12 skutterudite. Here eachiron atom has one less valence electron than cobalt, and the Fe4P12 complex can be considered

3.2. Filled skutterudites and thermoelectric applications 29

−15 −10 −5 0 50

20

40

60

80

100

E (eV)

DO

S

LaFe8P

24

−15 −10 −5 0 50

20

40

60

80

100

E (eV)

DO

S

LaCo2Fe

6P

24

−15 −10 −5 0 50

20

40

60

80

100

E (eV)

DO

S

LaCo4Fe

4P

24

−15 −10 −5 0 50

20

40

60

80

100

E (eV)

DO

S

LaCo6Fe

2P

24

Figure 3.4: The calculated DOS for various degrees of substitution in a fully La-filled phosphorusskutterudite [19]. Increasing the amount of Co in the structure relative to Fe causes the Fermi–level tomove from the top of the valence band, across the pseudo-gap, and into the bottom of the conductionband.

to lack four electron compared to an analogous Co4P12 complex. However, this deficiency isexactly compensated for by the four valence electrons of Ce, and the chemical formula can bewritten Ce4+[Fe4P12]4−. Thus, whereas the metal–pnicogen bonding is usually considered to becovalent, cerium is assumed to be in a positively charged ionic state.

In addition to stabilizing the structure, substituting Fe for Co allows a tuning of the electronicstructure and transport properties of the material. Figure 3.4 shows the calculated density ofstates for a series of filled P-based skutterudites [19]. Here the voids are completely filled withLa, while the amount of substitution is varied. The general features of the DOS are very similarfor the various compositions. However, for the completely substituted La2Fe8P24 compound, theFermi-level is located at the top of the valence band, while with lower degrees of substitutionit moves across the pseudo-gap and into the conduction band. Thus, by varying the degreeof substitution the electronic properties can to some extent be controlled. This is importantsince it allows for keeping the already beneficial electronic properties relatively unchanged while


0

2

4

6

8

10Electronic contribution

Lattice contribution

CeFe4Sb

12CoSb

3

Therm

al con

ductivity (

W/K

m)

Figure 3.5: The thermal conductivity of CoSb3 and CeFe4Sb12 at 300 K. The lattice and electroniccontributions have been estimated using the Wiedemann-Franz law and a Lorenz number L0 = 2.44 ·10−8 WΩ

K2 . The data were adapted from refs. [22] and [24].

modifying the structure.

The effect of this filling on the thermal conductivity has been thoroughly demonstrated withmany different filling elements, typically the rare earth elements R=La, Ce, Nd, Sm, Eu, Thetc. Dramatic reductions in the thermal conductivity up to an order of magnitude have been ob-served [20, 21, 22, 18, 23]. For example, figure 3.5 shows the thermal conductivity of CoSb3

compared to that of CeFe4Sb12. By using the Wiedemann–Franz law, the total thermal conduc-tivity can be partitioned into a lattice and an electronic contribution. We see a dramatic reductionin the thermal conductivity of the filled and substituted compound compared to the unfilled one,with most of the reduction due to a reduction in the lattice contribution.

The rationale for this effect is that when the filler atoms are placed in the large voids in the struc-ture, they are very loosely bound to their neighbours giving a very high thermal displacementparameter. That is, they “rattle” in an oversized box. Thus they serve as very efficient scatterersof thermal phonons, thereby lowering the thermal conductivity of the material. In addition, it iswell documented that the maximum reduction in thermal conductivity is achieved for only a par-tial filling of the voids [22], indicating that scattering of phonons against randomly distributedpoint defects plays an important role. These point defects need not be the filling atoms, therehave also been several studies, both experimental and theoretical, into the effect of various tran-sition metal substitutions on the transport properties of the materials [25, 26, 27, 28]. Amongother things, these studies indicate that point defect scattering of phonons against the substitutionelements also play an important role in reducing the thermal conductivity of the crystal.

3.2. References 31

References

[1] Jeitschko, W. et al. Crystal structure and properties of some filled and unfilled skutterudites:GdFe4P12, SmFe4P12, NdFe4P12, Eu0.54Co4Sb12, Fe0.5Ni0.5P3, CoP3 and NiP3. Zeitschrift

fur Anorganische und Allgemeine Chemie 626, 1112 (2000).

[2] Mandel, N. & Donohue, J. The refinement of the crystal structure of skutterudite, CoAs3.Acta Crystallographica B 27, 2288 (1971).

[3] Schmidt, T., Kliche, G. & Lutz, H. D. Structure Refinement of Skutterudite-Type CobaltTriantimonide, CoSb3. Acta Crystallographica C 43, 1678 (1987).


69, 139 (2001).

[5] Atkins, P. W. Quanta – A Handbook of concepts (Oxford University Press, Oxford, 1991),2 edn.

[6] Shriver, D. F. & Atkins, P. W. Inorganic Chemistry (Oxford Univeristy Press, Oxford,1999), 3 edn.

[7] Dudkin, L. D. The chemical bond in semiconducting cobalt triantimonide. Soviet Physics

- Technical Physics 3, 216 (1958).

[8] Llunell, M., Alemany, P., Alvarez, S., Zhukov, V. P. & Vernes, A. Electronic structure ofskutterudite type phsophides. Physical Review B 53, 10605 (1996).

[9] Fornari, M. & Singh, D. J. Electronic structure and thermoelectric properties of phosphideskutterudites. Physical Review B 59, 9722 (1999).

[10] Koga, K., Akai, K., Oshiro, K. & Matsuura, M. Electronic structure and optical propertiesof binary skutterudite antimonides. Physical Review B 71, 155119 (2005).

[11] Prytz, Ø., Løvvik, O. M. & Taftø, J. Comparison of theoretical and experimental dielectricfunctions: Electron energy-loss spectroscopy and density-functional calculations on skut-terudites. Physical Review B 74, 245109 (2006).

[12] Watcharapasorn, A. et al. Preparation and thermoelectric properties of some phosphideskutterudite compounds. Journal of Applied Physics 86, 6213 (1999).


[13] Løvvik, O. M. & Prytz, Ø. Density-functional band-structure calculations for La-, Y-, andSc-filled CoP3 based skutterudite structures. Physical Review B 70, 195119 (2004).

[14] Chen, L. D. et al. Anomalous barium filling fraction and n-type thermoelectric performanceof BayCo4Sb12. Journal of Applied Physics 90, 1864 (2001).

[15] Slack, G. A. & Tsoukala, V. G. Some properties of semiconducting IrSb3. Journal of

Applied Physics 76, 1665 (1994).

[16] Jeitschko, W. & Braun, D. LaFe4P12 with Filled CoAs3-Type structure and IsotypicLanthanoid-Transition Metal Polyphosphides. Acta Crystallographica B 33, 3401 (1977).

[17] Puyet, M., Lenoir, B., Dauscher, A., Weisbecker, P. & Clarke, S. J. Synthesis and crystalstructure of CaxCo4Sb12 skutterudites. Journal of Solid State Chemistry 177, 2138 (2004).

[18] Sales, B. C., Mandrus, D., Chakoumakos, B. C., Keppens, V. & Thompson, J. R. Filledskutterudite antimonides: Electron crystals and phonom glasses. Physical Review B 56, 56(1997).

[19] Mangersnes, K. Density functional calculations of skutterudites for thermoelectric appli-

cations. Master’s thesis, Department of Physics, University of Oslo, Oslo, Norway (2006).

[20] Sales, B. C., Mandrus, D. & Williams, R. K. Filled skutterudite antimonides: A new classof thermoelectric materials. Science 272, 1325 (1996).

[21] Nolas, G. S., Slack, G. A., Morelli, D. T., Tritt, T. M. & Ehrlich, A. C. The effect of rare-earth filling on the lattice thermal conductivity of skutterudites. Journal of Applied Physics

79, 4002 (1996).

[22] Nolas, G. S., Cohn, J. L. & Slack, G. A. Effect of partial void filling on the lattice thermalconductivity of skutterudites. Physical Review B 58, 164 (1998).

[23] Grytsiv, A. et al. Structure and physical properties of the thermoelectric skutteruditesEuyFe4−xCoxSb12. Physical Review B 66, 094411 (2002).

[24] Morelli, D. T. & Meisner, G. P. Low temperature properties of the filled skutteruditeCeFe4Sb12. Journal of Applied Physics 77, 3777 (1995).

[25] Anno, H., Matsubara, K., Notohara, Y., Sakakibara, T. & Tashiro, H. Effects of doping onthe transport properties of CoSb3. Journal of Applied Physics 86, 3780 (1999).

3.2. References 33

[26] Yang, J., Meisner, G. P., Morelli, D. T. & Uher, C. Iron valence in skutterudites: Transportand magnetic properties of Co1−xFexSb3. Physical Review B 63, 014410 (2000).

[27] Yang, J. et al. Influence of electron-phonon interaction on the lattice thermal conductivityof Co1−xNixSb3. Physical Review B 65, 094115 (2002).

[28] Yang, J., Endres, M. G. & Meisner, G. P. Valence of Cr in skutterudites: Electrical transportand magnetic properties of Cr-doped CoSb3. Physical Review B 66, 014436 (2002).

Chapter 4

Methodology

In this chapter, density functional theory (DFT), X-ray photoelectron spectroscopy (XPS), andelectron energy-loss spectroscopy (EELS) are briefly reviewed.

4.1 Density functional theory

The field of solid state physics is concerned with examining the physical properties of matter insolid phases. Although physics is fundamentally an experimental science, the constant interac-tion between experiment and theory is an absolute necessity for making progress in understand-ing the basic properties of the systems under scrutiny. In solid state physics, one studies systemsthat in principle are well understood within the framework of already established quantum the-ory. Furthermore, many solids, such as crystals, are well ordered and can be succinctly describedmathematically. Although defects in the perfect crystal lattice play a major role in many materi-als’ properties, one would often initially be satisfied to consider idealized models with few or nodefects. All in all, calculations of the properties of a material should be within reach.

Unfortunately the reality of the matter proves to be rather more discouraging. Solving theSchrodinger equation quickly becomes intractable for any system of size. For a system of n

electrons and N nuclei, the Schrodinger equation depends on 3(n + N) variables, three spatialcoordinates for each electron and nucleus. For all practical models of a material one usually has

35

36 Chapter 4. Methodology

to consider some tens of atoms, which may easily entail more than a hundred electrons. Withsuch a large number of variables this becomes a completely insolvable problem. Some sim-plification is possible: applying the Born–Oppenheimer approximation allows us to considerednuclear and electronic degrees of freedom separately, and the nuclei can then be seen to supply astatic background potential in which the electrons move. Although this is a helpful approxima-tion, finding the many electron wave function incorporating the 3n electronic degrees of freedomis still an insurmountable problem in most cases [1].

One solution to this problem came as a result of two theorems proposed by Hohenberg and Kohn,which are now known as the Hohenberg–Kohn theorems. These theorems state [1, 2, 3] that i)

the ground state electron density ρ0(r) of a system uniquely determines the external potential,and thereby the Hamiltonian and wave function of the system, and ii) this ground state electrondensity can be obtained variationally: the density that gives the minimum system energy is theexact ground state density. In short, these theorems assert that all information available in thewave function is also available in the ground state electron density, and that this density can befound through energy minimization procedures.

The significance of these theorems becomes obvious when one considers that while the wavefunction itself is dependent on 3n variables, the electron density is a function of only three spatialvariables. The computational challenge of calculating the properties of a material is therebygreatly reduced. As long as one is able to find the proper functional relating the electron densityto the property in question, the prediction of materials’ properties from first principles should bepossible.

One important complicating factor that should be mentioned, is that the exact functional forfinding the system energy from the electron density is unknown. The system energy can beexpressed as [3]:

E0[ρ0(r)] = Tni[ρ0(r)] + Vne[ρ0(r)] + Vee[ρ0(r)] + ∆T [ρ0(r)] + ∆Vee[ρ0(r)] (4.1)

Here, the first term is a functional for the kinetic energy of a fictitious system of non-interactingelectrons, while the second and third terms are the classical nuclear–electron and electron–electron (Coulombic) interaction. The fourth term is a correction to the kinetic energy due to theelectrons interacting, while the final term contains all non-classical corrections to the interaction

4.1. Density functional theory 37

energy such as contributions arising from Fermi-statistics. In short, the problem lies with the finaltwo terms in the above equation. These are often treated jointly as an exchange–correlation en-ergy functional, Exc[ρ0(r)], the exact form of which is unknown. Since the exchange–correlationfunctional is unknown, various approximations have been developed in an attempt to accountfor these effects. Even though DFT is in principle variational, thereby allowing us to determinethe ground state density through energy minimization, the use of these exchange–correlation ap-proximations introduces an uncertainty. There is no longer any way to be certain that the electrondensity with the lowest energy is the exact ground state density.

However, comparisons of the calculated electron density with that observed through diffractionexperiments show that the correspondence is good for simple materials such as Si and Ge, thoughthere is a larger discrepancy for more complex materials [4, 5, 6]. In practical terms, the mostnotable failure of modern DFT calculations is a tendency to underestimate the band gap in semi-conductors, often by as much as 10–30%. For example, several transition metal oxides that inreality are semiconductors or insulators, are predicted to be metals. These failures are indeedusually ascribed to the the various exchange–correlation approximations that are used.

Most DFT calculations explicitly calculate ground state properties of the system. Therefore,comparison of experimental data with such calculations is often not formally justified as mostexperiments investigate the system far from the ground state. This is perhaps most obviouswith experimental techniques that deliberately bring the system into an excited state, such as thevarious spectroscopies where energy is absorbed by an electron. The comparison of the groundstate density of states with such experiments can be subject to large errors [7, 8].

Despite these challenges, DFT has seen great application and success in many areas of solidstate physics and molecular chemistry. In general the geometries and energetics predicted formolecules are in good agreement with observations [3], and the crystal structure, phase stabili-ties, bulk moduli, heats of formation, and electronic structure of solids are successfully predictedfor many solids [9, 10]. In practical terms, calculations based on density functional theory havebecome an indispensable aid in materials science, and solid state physics and chemistry for in-terpreting experimental results and predicting materials’ properties.


4.2 X-ray photoelectron and Auger electron spectroscopy

X-ray photoelectron spectroscopy (XPS) is a technique for studying the composition, chemicalstate, and electronic structure of a material by irradiating a sample with X-rays and measuringthe kinetic energy of the emitted photoelectrons. Only electrons with binding energy Eb equal toor less than the photon energy hν are emitted from the sample, and their binding energy can thenbe found through the relation Eb = hν − Ek − φ. Here φ is the instrumental work function andEk is the measured kinetic energy.

The plot of photoelectron intensity versus binding energy is closely related to the occupied den-sity of states (DOS) of the material. However, since the excitation of a photoelectron generatesa core hole, the observed binding energy is not equal to the orbital energy ε. When the core holeis present, the surrounding electrons are rearranged in response to the changed potential at theatom. The binding energy is then given by Eb = ε − R, where R is a relaxation energy takinginto account this rearrangement and the effects of the changed potential [11].

In most commercially available XPS instruments, the kinetic energy is analyzed in an elec-trostatic hemispherical energy analyzer, often reaching a relative energy resolution ∆E/E =

5.3 · 10−3 [12]. Figure 4.1 shows an XPS survey spectrum from a sample consisting of CoP3

single crystals embedded in a tin-matrix. The spectrum shows sharp peaks from photoelectronsemitted from the energy levels of the various elements in the sample. Also present are peaksfrom the emitted Auger electrons, these features are discussed further below.

To a first approximation, the binding energy of an electron in a particular atomic orbital is an ele-mental characteristic, and can be used to identify the elements present in a sample. Furthermore,the peak intensities from the various elements are related to the composition of the sample. Thus,elemental quantification is possible, often with an accuracy of 0.1 at % [13].

XPS is often referred to as a tool for surface characterization. Even though the incident X-raysmay penetrate many micrometers into the sample, the majority of the observed photoelectronsoriginate only a few nanometres below the surface. Still, this may be in the order of 10 unit cells,and is often considered to be sufficient for the signal to be representative for the bulk electronicstructure without any effects of the low-dimensionality of the surface. A more pressing problemis the presence of surface contamination such as oxides and hydrocarbons. In order to avoid suchproblems, the sample is studied in high-vacuum, often after sputtering the surface with ions of

4.2. X-ray photoelectron and Auger electron spectroscopy 39

Figure 4.1: XPS survey scan from a sample consisting of CoP3 single crystals embedded in a tinmatrix.

argon or xenon.

Although the binding energy is an elemental characteristic, it is affected by the chemical envi-ronment of the element. The binding energy Ei of the electron emitted from the energy level i

can be described through the equation [13]:

Ei = Ei,0 + kqi + eVi (4.2)

Here, Ei,0 is the neutral free atom binding energy, qi is the charge on the atom in the sample, whileVi is a contribution to the energy due to the charge on the surrounding atoms Vi =

∑j 6=i

qj

rij. If

the same element is present in two different chemical environments, a chemical shift may beobserved:

∆Ei = k∆qi + e∆Vi (4.3)


This chemical shift will reflect ground state charge transfer to or from the atom in question,and is an important tool in investigating the electronic structure and bonding in the material.For example, if charge is transferred away from the element when bonds are formed, the intra-atomic screening is reduced giving a higher binding energy. At the same time, the extra-atomiccharge may increase. Therefore the two terms in equation (4.3) may have opposite sign, oftencausing the observed chemical shift to be very small. Thus, accurate measurements of chargetransfer through binding energy studies are prone to the uncertainties introduced by a lack ofaccurate energy referencing due to e.g. charging of the sample. This problem may be somewhatremedied by calibrating the energy scale at known features such as the carbon 1s edge fromsurface contamination.

In order to reduce the problems associated with energy referencing the concept of the Auger-parameter was introduced by Wagner [14]. As the photoelectron leaves the sample, the atomis left in an excited state with a hole in the orbital that the electron previously occupied. De-excitation occurs with a transition of an electron from a higher energy orbital to the orbital witha hole, and a rearrangement of the valence electrons in response to the core hole (screening).Energy conservation then dictates that the difference in binding energy must be expelled fromthe atom, either in the form of a photon or — more relevant to our discussion — in the form ofyet another excited electron. These are called Auger electrons, and an example of such a processis sketched in figure 4.2. Here, a K shell photoelectron is excited by an incident X-ray photon,leaving a core hole. The atom then is de-excited through a transition of a L1 electron into the K

shell, and the excess energy is transferred to an L3 electron.

These processes are denoted by the electron shells involved. Under the assumption that theattractive potential of the core holes and extra-atomic environment can be ignored, the kineticenergy of the Auger electron is given by [15]:

Ek(KL1L3) ≈ Eb(K)− Eb(L1)− Eb(L3) (4.4)

This assumption is not strictly valid, but is acceptable in order to gain a first estimate of the kineticenergy. The kinetic energy of the Auger electron is characteristic for the atomic species in muchthe same way as the binding energy of the photoelectrons. Furthermore it is independent of theenergy of the incident photon, the only requirement being that the incident photon is energeticenough to create the initial core hole.

4.2. X-ray photoelectron and Auger electron spectroscopy 41

1s (K)

2s (L1)

2p1/2

(L2)

2p3/2

(L3)

Figure 4.2: Production of a KL1L3 Auger electron. A K shell core hole is generated through absorp-tion of an X-ray photon (left). The atom is then de-excited though the transition of an L1 electron,filling the core hole and exciting the an L3 electron (right).

Wagner realized that the difference between two observed kinetic energies is accurately mea-surable since any problems with energy referencing due to e.g. sample charging will be can-celled. This led to the definition of the Auger parameter, a modified version of which is givenby [16, 17, 11, 18]:

α′ = Ek(C1C2C3)− Ek(C) + hν = Ek(C1C2C3) + Eb(C) (4.5)

Here Ek(C1C2C3) is the observed kinetic energy of an Auger electron originating from the pro-cess involving the C1C2C3 core levels of an element, and Ek(C) and Eb(C) are the kinetic andbinding energies of a photoelectron originating from the C orbital of the same atom.

If one assumes that the chemical shift due to ground state charge transfer is the same for allorbitals of an atom, equations (4.5) and (4.4) show that:

∆α′ = ∆Ek(C1C2C3) + ∆Eb(C)

= ∆Eb(C1)−∆Eb(C2)−∆Eb(C3) + ∆Eb(C) = 0 (4.6)

Thus, at a first glance, the Auger parameter is insensitive to changes in the core binding energiesdue to charge transfer. However, the Auger parameter measures the response of the system when


an atom is ionized by removing a core electron. These relaxation effects are more pronounced inthe Auger process as there are two core holes that contribute. Therefore often |∆Ek(C1C2C3)| >|∆Eb(C)|, and the Auger parameter shift is nonzero and depends on the response to the coreholes.

If Auger parameters for an element are measured in different bonding environments, shifts in theAuger parameter are related to how the bonding affects the response to the core hole. It can beshown that the shift in Auger parameter between two different bonding environments is givenby [17]:

∆α′ = ∆

[q

dk

dN+

(k − 2

dk

dN

)(dq

dN

)+

dU

dN

](4.7)

Here q is the ground state charge on the atom in question and k is the change in potential whena valence electron is removed. Furthermore, N is core orbital occupation number, and U isthe potential from the surrounding atoms. The first term describes the response of the valenceelectrons to the core hole, while the second and third terms take into account transfer of chargein response to the core hole and the polarization of the surrounding atoms. Thus, equation (4.7)states that changes in the Auger parameter between two chemical environments is caused bydifferences in response to the core hole. Under certain assumptions, e.g. for metals, the secondand third terms in the above equation can be ignored, and equation (4.7) becomes

∆α′ = ∆qdk

dN(4.8)

and the Auger parameter shifts are then directly related to the ground state charge transfer be-tween the two different chemical environments [17].

As an example, figure 4.3 shows the phosphorus 2p-KLL Auger parameter for several phospho-rus containing compounds. Significant shifts are observed indicating different degrees of groundstate charge transfer in the various compounds.

4.3. Electron Energy-Loss Spectroscopy 43

1986,4

1986,6

1986,8

1987,0

1987,2

1987,4

1987,6

1987,8

1988,0

1988,2

0 10 20 30 40 50 60 70 80 90 100

at% P

P 2

p-KLL A

ug

er

Pa

ram

ete

r (e

V)

Ni3P Ni

12P

5

Ni5P

4

InP

CoP3

P4S

10

Zn3P

2

GaP

ZnP2

Pure P

NiP3

Figure 4.3: The P 2p-KLL Auger parameter for several compounds.

4.3 Electron Energy-Loss Spectroscopy

Electrons entering a material may undergo energy losses that contain information about the com-position and electronic structure of the sample. Using a nearly monochromatic incident electronbeam, these energy losses can be measured with an electron energy-loss spectrometer (EELS). Ifthe experiment is performed in a transmission electron microscope (TEM), the sample is usuallyonly some tens of nanometres thick, and the incident electrons are transmitted through the spec-imen. After the electrons leave the sample, their energy is measured in a spectrometer, therebyshowing how much energy was transferred from the electrons to the specimen.

Figure 4.4 shows a sketch of a so-called post column magnetic prism spectrometer. In practicalterms, these spectrometers are attached to the bottom of the TEM, below the viewing screen. Theelectrons traverse the TEM column from the electron gun, through the sample, and into the thespectrometer through an entrance aperture. Once the electron beam has entered the spectrometer,it is subjected to a uniform magnetic field which acts on the electrons through the Lorentz forceF = e(v ×B), thereby spatially separating the electrons according to their velocity, i.e. kineticenergy. After passing through a number of electron lenses (not shown in the figure) the electronsarrive at a detector which records the beam intensity as a function of position [19]. Thus, theenergy distribution of the electrons may be retrieved.


B

Incre

asin

g e

ne

rgy lo

ss

Electron beam cross-over

Magnetic prism

Detector

Figure 4.4: Sketch of a magnetic prism spectrometer. After passing through the sample, the fastelectrons enter the magnetic prism, where they are subjected to a uniform magnetic field. This fieldcauses the electrons to be dispersed according to their kinetic energy, and their spatial distribution isrecorded by a detector.

4.3.1 Core excitations

There are several processes by which the incident electrons may transfer energy to the specimen.One of the most important is through excitation of the constituent atoms, whereby the incidentelectrons impart some energy E and momentum q to a core electron. The core electron thusmakes a transition from an occupied initial state |Ψi〉 to an unoccupied final state |Ψf〉 above theFermi–level, see figure 4.5.

In order for this transition to occur, the core electron must according to the Pauli exclusionprinciple receive an amount of energy E ≥ EF − EC , where EF is the Fermi-energy and EC isthe binding energy of the core level. In the energy loss spectrum, sharp features may be observedat the threshold energy signaling the onset of these transitions. The electron energy-loss spectrummeasures the probability that such excitations occur, and reflects a convolution of the occupiedpart of the DOS with the unoccupied part.

The probability that an electron in an initial state |Ψi〉 will be scattered into one of the final states|Ψf〉 is given by the double differential cross-section [20]:


2p

3s

3p

3d

4s

EF

e-

Conduction bandE

Figure 4.5: Sketch of an EELS excitation process. The incident electron transfers energy to a 2p coreelectron, which is then excited into an empty 3d state above the Fermi-level.

d2σ(E,q)

dE dq∝ 4γ2

a20q

4

∑

f

|〈Ψf |eiq·r|Ψi〉|2

=4γ2

a20q

4ρ(Ei + E)|〈Ψf |eiq·r|Ψi〉|2 (4.9)

Here a0 is the the Bohr radius and γ the relativistic correction factor. The final and initial statesare chosen so that the difference in their energies Ef − Ei equals the energy loss E of theincident electron. There can be many possible final states for the electron; under the assumptionthat the matrix element term |〈Ψf |eiq·r|Ψi〉|2 is a constant for the scattering process in question,the multiple final states can be accounted for by the sum in the first line of equation (4.9), givingthe DOS term ρ(Ei + E) in the final expression.

The exponential appearing in the matrix element term |〈Ψf |eiq·r|Ψi〉|2 can be expanded in aTaylor series:

eiq·r = 1 + iq · r +(iq · r)2

2!+ · · · (4.10)

In most experiments the majority of electrons transfer only a small amount of momentum tothe sample, and any contribution to the spectrum from electrons undergoing a large momentum


transfer may further be limited using an angle limiting collection aperture in front of the spec-trometer. In this case, the product q · r is small and the Taylor expansion can be truncated afterthe second term. Inserting the Taylor expansion into equation (4.9) and remembering that theinitial and final states are orthogonal to each other, we then get:

d2σ(E,q)

dE dq∝ 4γ2

a20q

4ρ(Ei + E)|〈Ψf |iq · r|Ψi〉|2 (4.11)

The limit of small q is called the dipole approximation. If valid, the electronic transitions arelimited to those with a change in orbital angular momentum quantum number ∆l = ±1, similarto the transitions that take place upon absorption of a photon. In this case, the DOS term in equa-tion (4.11) should be interpreted as an angular momentum (symmetry) selected DOS, while thematrix term |〈Ψf |iq·r|Ψi〉|2 takes into account the radial overlap of the initial and final states. Formoderate to high energy loss transitions, the initial states are essentially atomic and are stronglylocalized near the atomic core. Non-zero values for the overlap are therefore only achieved forfinal states localized on the same atom. Thus, the intensity and shape of a peak in the energy lossspectrum represents a site and symmetry selected density of unoccupied states. Furthermore, theenergy loss intensities for high energy transitions is reduced as the spatial overlap of the initialand final states is small for large differences in binding energy.

This interpretation is a useful first approximation in understanding the features observed in theenergy-loss spectrum. However, it should be noted that, as with photoelectron spectroscopy,a core hole is generated when the excitation occurs. As a consequence the potential felt by thesurrounding electrons is changed, and the final states |Ψf〉 are not those found in the ground state.These effects are important when comparing observed energy-loss spectra with calculations.

While the initial states are located close to the core and are mainly atomic in nature, the finalstates are further from the core and located above the Fermi-level. The energy-loss spectrumwill therefore reflect the valence electron configuration of the atom. An example of this can beseen in figure 4.6(a), which shows the energy-loss features (called ’white lines’) due to the L2,3

excitations in the first row transition metals. These features appear when transition metal (TM)2p electrons are excited into the empty 3d and 4s states. The radial overlap of the 2p and 4sstates is small compared to that of the 2p and 3d states [21], and the transition probability istherefore dominated by the 2p→3d transitions. When moving from Ti, with a valence electronconfiguration [Ar]3d24s2, towards Cu, [Ar]3d104s1, the 3d states are gradually filled by additional


Energy Loss (eV)

Inte

nsity

Cu L2,3

Cu

Cu2O

CuO

910 970930 950

(b)In

ten

sity

Energy Loss (1 div = 50 eV)

Cu

Ni

Co

Fe

Cr

V

Ti

(a)

Figure 4.6: The L2,3 edges of (a) seven pure transition metal films, adapted from Pearson [23] and(b) Cu in pure copper and its oxides, adapted from Keast [24].

valence electrons. As the number of available states above the Fermi-level decreases, the sharp,intense peaks observed for Ti are reduced, until virtually no such peaks are observed in Cu [22,23]. Indeed, it has been shown that with appropriate normalization and correction for the matrixelement of equation (4.11), the total white line intensity is linearly correlated to the number of3d holes [23].

The final states are heavily influenced by the interaction with the neighbouring atoms, and aretherefore strongly affected by the bonding and coordination of the excited atom. A most dramaticexample of this is the case of copper and its oxides. Figure 4.6(b) shows the L2,3 edges of Cu,Cu2O, and CuO. As already mentioned, the white lines are almost completely absent in purecopper, as the 3d band is completely filled. However, upon oxidation, the sharp peaks associatedwith the 2p→3d transitions are readily observed. This is interpreted as an emptying of the Cu3d states upon bonding with oxygen, consistent with an ionic model where charge is transferredfrom copper to oxygen. This approach has proven useful in studying the alloying behaviour ofcompounds containing transition metals [25, 26].

4.3.2 Collective excitations and the joint density of states

As mentioned previously, the energy-loss spectrum is related to a convolution of initial states withfinal states. However, the above treatment explicitly refrains from treating the energy distribution


of initial states. This is adequate in the case of energy losses associated with the excitation ofcore electrons. The initial states of these transitions are usually sharply localized in energy, andthe spectrum can be interpreted as a convolution of the final states with delta-function-like initialstates.

However, for transitions where both the initial and final states are located close to the Fermi-level,this treatment is no longer sufficient. The initial states are then highly distributed in energy, andfor any given energy loss there are several possible transitions. In this case the convolution mustbe treated explicitly, and the observed energy-loss intensity can be described by [20]:

I(E) ∝∫ EF

EF−E

|〈Ψf |eiq·r|Ψi〉|2ρ(Ei + E)ρ(Ei)dEi (4.12)

where the integral is over all initial state with energy between EF and EF−E. Thus, this energy-loss spectrum should be considered as reflecting a joint density of states of the material, and notjust the DOS above the Fermi-level. As both the initial and final states are located close to theFermi-level, they are heavily influenced by the bonding arrangements in the material, and thelow loss EELS spectrum may contain much useful information.

The low loss spectrum is, however, not dominated by these single electron transitions. Rather, themost intense features visible are usually the so-called plasmons. As the incident electron entersthe material, the electric field surrounding it repulses the electrons in the material, forcing theelectron distribution out of equilibrium with regard to the atomic potential. After the electron hasleft the material, the restoring force between the electrons and nuclei cause a collective oscillationof the valence electrons about the equilibrium position.

A good understanding of this phenomenon can be achieved by studying the macroscopic equa-tions of electrodynamics. If a material is subjected to an oscillating external electric field E(ω,k),we know from classical electrodynamics that it is polarized according to [27]:

P(ω,k) = ε0 [ε(ω,k)− 1]E(ω,k) (4.13)

where ε(ω,k) = ε1 + iε2 is the complex dielectric function of the material, with real and imagi-nary parts ε1 and ε2. The electric displacement of the medium is given by


D(ω,k) = ε0E(ω,k) + P(ω,k)

= ε0E(ω,k) + ε0 [ε(ω,k)− 1]E(ω,k) = ε(ω,k)ε0E (4.14)

We notice that the polarization exactly cancels the external field when ε(ω) = 0. In the freeelectron model, the electrons act as an oscillating plasma, and it can be shown [28] that thisresonance frequency, called the plasma frequency, is given by:

ωp =

√Ne2

V m0ε0

(4.15)

where N is the number of valence electrons in the unit cell, V is the unit cell volume, ε0 is thepermittivity of free space, e the electron charge, and m0 the free electron mass.

Simple metals and semi-conductors such as Be, B, Al, Ge, GaAs, and Si display sharp and promi-nent plasmon peaks, and peaks at multiples of the plasmon energy due to the incident electronexciting several plasmons before leaving the sample. As an example, the low loss spectrum fromSi is shown in figure 4.7 where the energy loss peaks from three plasmon excitations are readilyobserved. In the case of such materials, the free electron model is quite successful in predictingthe plasmon energy Ep = ~ωp [29, 19].

On the other hand, wide band gap ionic materials and elemental materials of transition metalstend to exhibit broad plasmon peaks, with only the first plasmon visible, as exemplified by thelow loss spectrum of Co in figure 4.7. In these materials it is not adequate to describe the valenceelectrons as a free electron plasma with a single, common, resonance frequency. Additionally,there is often no obvious choice for the number of contributing valence electrons, and the simpleDrude model may fail dramatically to give adequate estimates of the plasmon energy.

In the dielectric formulation, the double differential cross section is given by [19, 20]:

d2σ

dΩdE=

1

π2a0m0v2na

1

Θ2 + Θ2E

Im

( −1

ε(E,q)

)(4.16)


Inte

nsity (

arb

. units)

Energy Loss (eV)

-50 0 50 100 150

Si

Co

Co M2,3

Zero Loss Peak

1st plasmon

2nd plasmon

3rd plasmon

Figure 4.7: The low loss region of Si and Co. In the case of simple metals and semi-conductors,multiple, sharp plasmon peaks are usually observed in the low loss region, as is the case for Si seenin the figure. For more complex metals such as Co a single broad peak is observed. Also seen is theCo M2,3 edge at approximately 60 eV.

Here a0 is the Bohr-radius, m0 the electron mass, v the speed of the incident electron, and na thenumber of atoms per unit volume. In addition, Θ and ΘE are the scattering angle of the electron,and characteristic angle for an energy loss E [19]. The final term is referred to as the loss functionand can be interpreted as a renormalization of imaginary part of the dielectric function:

Im

( −1

ε(E,q)

)=

ε2

ε21 + ε2

2

(4.17)

Through equation (4.16) the close connection between the observed energy loss spectrum andthe imaginary part of the dielectric function is established, and the full dielectric function canthe be retrieved through a Kramers–Kronig analysis [27]. As an example, figure 4.8 showsa comparison of the dielectric function obtained from EELS experiments, with the dielectricfunction from DFT calculations.

There are several advantages to using this approach to determine the dielectric function as com-pared to traditional optical methods. EELS experiments are be expected to be less sensitive

4.3. References 51

Re(ε

) (a

rb. units)

Energy Loss (eV)

0 5 10 15 20

(a)

Im(ε

) (a

rb. units)

Energy Loss (eV)

0 5 10 15 20

(b)

Figure 4.8: (a) The real and (b) imaginary parts of the dielectric function of CoAs3 as obtained fromEELS and DFT. The experimentally obtained function is shown with a fully drawn line, while thetheoretical function is a dashed line. From Prytz et al. [30].

to surface contamination than e.g. optical reflectance experiments, and may yield the dielectricfunction over a wider energy (frequency) range, although with poorer energy resolution.

To summarize, in the initial study of new materials, only small amounts may be readily available.Thus, measurements of e.g. thermoelectric properties may at first be difficult and unrealiable.However, after having determined the crystal structure and thermal parameters by diffractiontechniques, the electronic structure can be studied combining DFT calculations and experimen-tal studies using advanced analytical TEM. This instrument is highly versatile with the opportu-nity of performing EELS, precise convergent beam electron diffraction and electron holography.These techniques can be used on small volumes of materials to determine their electronic struc-ture and bonding, from which their thermoelectric properties can be inferred. Thus, these TEMbased techniques may be of great value when searching for new materials.

References

[1] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density func-tionals. Review of Modern Physics 71, 1253 (1999).

[2] Hohenberg, P. & Kohn, W. Inhomogeneous electron gas. Physical Review 126, B864(1964).


[3] Cramer, C. J. Essentials of Computational Chemistry – Theories and Models, chap. 8,233–273 (John Wiley and Sons Ltd., Chichester, 2002).

[4] Zuo, J. M., Kim, M., O’Keeffe, M. & Spence, J. C. H. Direct observation of d-orbital holesand Cu–Cu bonding in Cu2O. Nature 401, 49 (1999).

[5] Zuo, J. M. Measurements of electron densities in solids: a real-space view of electronicstructure and bonding in inorganic crystals. Reports on Progress in Physics 67, 2053 (2004).

[6] Wu, L. et al. Valence-electron distribution in MgB2 by accurate diffraction measurements.Physical Review B 69, 064501 (2004).

[7] Lie, K., Brydson, R. & Davock, H. Band structure of TiB2: Orientation-dependent EELSnear-edge fine structure and the effect of the core hole at the B K edge. Physical Review B

59, 5361 (1999).

[8] Lie, K., Høier, R. & Brydson, R. Theoretical site- and symmetry-resolved density of statesand experimental EELS near-edge spectra of AlB2 and TiB2. Physical Review B 61, 1786(2000).

[9] Sutton, A. P. Electronic structure of materials, chap. 11, 204–214 (Oxford University Press,Oxford, 1993).

[10] Hafner, J. Atomic-scale computational materials science. Acta Materialia 48, 71 (2000).

[11] Thompson, M., Baker, M. D., Christie, A. & Tyson, J. F. Auger Electron Spectroscopy,vol. 74 of Chemical Analysis (John Wiley and Sons, New York, 1985).

[12] Drummond, I. W. XPS: Instrumentation and performance. In Briggs, D. & Grant, J. T.(eds.) Surface analysis by Auger and X-ray photoelectron spectroscopy, chap. 5, 117–144(IM Publications and SurfaceSpectra Limited, Charlton and Manchester, 2003).

[13] Grant, J. T. XPS: Basic principles, spectral features and qualitative analysis. In Briggs,D. & Grant, J. T. (eds.) Surface analysis by Auger and X-ray photoelectron spectroscopy,chap. 2, 31–56 (IM Publications and SurfaceSpectra Limited, Charlton and Manchester,2003).

[14] Wagner, C. D. Chemical shifts of auger lines, and the auger parameter. Faraday Discussions

of the Chemical Society 60, 291 (1975).

4.3. References 53

[15] Grant, J. T. AES: Basic principles, spectral features and qualitative analysis. In Briggs,D. & Grant, J. T. (eds.) Surface analysis by Auger and X-ray photoelectron spectroscopy,chap. 3, 57–88 (IM Publications and SurfaceSpectra Limited, Charlton and Manchester,2003).

[16] Moretti, G. The auger parameter. In Briggs, D. & Grant, J. T. (eds.) Surface analysis

by Auger and X-ray photoelectron spectroscopy, chap. 18, 501–530 (IM Publications andSurfaceSpectra Limited, Charlton and Manchester, 2003).

[17] Thomas, T. D. & Weightman, P. Valence electron structure of AuZn and AuMg alloysderived from a new way of analyzing Auger-parameter shifts. Physical Review B 33, 5406(1986).

[18] Gaarenstroom, S. W. & Winograd, N. Initial and final state effects in the ESCA spectra ofcadmium and silver oxides. The Journal of Chemical Physics 67, 3500 (1977).

[19] Egerton, R. F. Electron Energy Loss Spectroscopy in the Electron Microscope (PlenumPress, New York, 1986), 1 edn.

[20] Rez, P. Energy loss fine structure. In Ahn, C. C. (ed.) Transmission Electron Energy

Loss Spectrometry in Materials Science and the EELS Atlas, chap. 4, 97–126 (Wiley-VCHVerlag GmbH & Co. KGaA, Weinheim, 2004).

[21] Muller, D. A., Singh, D. J. & Silcox, J. Connections between the electron-energy lossspectra, the local electronic structure, and the physical properties of a material: A study ofnickel aluminum alloys. Physical Review B 57, 8181 (1998).

[22] Leapman, R. D., Grunes, L. A. & Fejes, P. L. Study of the L2,3 edges in the 3d transitionmetals and their oxides by electron-energy loss spectroscopy with comparison to theory.Physical Review B 26, 614 (1982).

[23] Pearson, D. H., Ahn, C. C. & Fultz, B. White lines and d-band occupancies for the 3d and4d transition metals. Physical Review B 47, 8471 (1993).

[24] Keast, V. J., Scott, A. J., Brydson, R., Williams, D. B. & Bruley, J. Electron energy-lossnear edge structure – a tool for the investigastion of electronic structure on the nanometrescale. Journal of Microscopy 203, 135–175 (2001).

[25] Ouyang, H. & Jen-Tai, K. Occupancy of the 3d electron shell of Co and Cr in nanosizedCoCrPt magnetic thin films. Journal of Applied Physics 92, 7510 (2002).


[26] Pearson, D. H., Ahn, C. C. & Fultz, B. Measurements of 3d occupancy from L2,3 electronenergy-loss-spectroscopy spectra of rapidly quenched CuZr, CuTi, CuPd, CuPt and CuAu.Physical Review B 50, 12969 (1994).

[27] Yu, P. Y. & Cardona, M. Fundamentals of Semiconductors (Springer-Verlag, Berlin, 2003),3 edn.

[28] Kittel, C. Introduction to Solid State Physics (John Wiley & Sons, Inc., 1996), 7th edn.

[29] Raether, H. Excitation of Plasmons and Interband Transitions by Electrons, vol. 88 ofSpringer Tracts in Modern Physics (Springer-Verlag, Berlin, 1980).

[30] Prytz, Ø., Løvvik, O. M. & Taftø, J. Comparison of theoretical and experimental dielectricfunctions: Electron energy-loss spectroscopy and density-functional calculations on skut-terudites. Physical Review B 74, 245109 (2006).

Chapter 5

Overview of papers

The scientific results of this Ph.D. study are presented in the four papers included in this thesis.The main methods used are density-functional theory, electron energy-loss spectroscopy, andx-ray photoelectron spectroscopy. A brief summary of the articles is given below.

Paper I

As discussed in a previous chapter, filling the voids in the skutterudite structure can dramaticallydecrease the thermal conductivity of the material. Based on the rattling box picture of this effect,it can be expected that the magnitude depends on the the size of the filling atom relative to the sizeof the void: a relatively small atom will decrease the thermal conductivity more than a relativelylarge atom [1].

This paper investigates the crystal structure, thermodynamic stability, and electronic structureof La-, Y-, and Sc-filled CoP3 using density functional theory (DFT). While these elements arechemically similar, they have different atomic radii, with Sc the smallest and La the largest. Itcan therefore be expected that filling the structure with Sc would have a greater effect on thethermal conductivity than filling with La.

The solubility of La, Y, and Sc in CoP3 is calculated to be around 5, 3-6 %, and below 1% at 0K, respectively. Although these are rather small values, this is expected to increase considerably

55

56 Chapter 5. Overview of papers

if Fe is substituted for Co. Calculations of the density of states and band structure indicate thatthe main effect of filling is to push the Fermi-level into the conduction band, while other featuresremain rather unchanged.

Paper II

In this article, the possibility of combining density functional theory (DFT) and electron energyloss spectroscopy (EELS) to determine the dielectric function of materials is investigated. Thebinary skutterudites CoP3, CoAs3, and CoSb3 serve as model systems, and the theoretically andexperimentally obtained low energy-loss spectra and plasmon energies are compared.

The skutterudites display multiple sharp plasmon peaks, similar to those observed in simplemetals and semiconductors. When compared with the plasmon energies calculated using the freeelectron model, the experimental plasmons are found at systematically higher energy.

The DFT calculated plasmon energy also deviates significantly from the observed values, butin a non-systematic way. While the plasmon energy of CoP3 and CoAs3 is underestimated, thecalculated value for CoSb3 is overestimated by more that 6 %. This is contrary to what the caseof some crystals with less complicated electronic structure, where good agreement is found [2].When the theoretical and experimental low loss region below the plasmon peaks are compard, aqualitative agreement is obtained for the skutterudites.

In the case of CoAs3, a Kramers–Kronig analysis of the EELS spectra gives a dielectric functionin good agreement with the theoretic calculations. Some of the remaining discrepancies may becaused by the challenge of refining the experimental spectra before Kramers–Kronig analysis.

Paper III

While a predominantly covalent bonding is expected in the binary skutterudites, a partial ioniccharacter cannot be discounted. In this paper, the transition-metal 3d occupancy of a series ofthermoelectric skutterudites is investigated using electron energy-loss spectroscopy.

The intensity of the transition-metal L2,3 edges of CoP3, CoAs3, CoSb3, and NiP3 are investi-

5.0. References 57

gated and compared those of the pure metals. In the case of the cobalt based skutterudites, asignificant increase of the edge intensity is observed, the largest change is observed in the case ofCoP3. Previous studies have shown that the intensity of these edges correlates with the 3d occu-pancy [3]. Based on this it concluded that there is an emptying of the 3d states in these materialscompared to the situation in pure metals. The reduction in 3d occupancy is 0.77 electrons/atomfor CoP3, and about 0.4 electrons/atom for the arsenic and antimony based skutterudites. Onlysmall changes in occupancy are observed for NiP3.

In comparison, the intensity of the L2,3 edges of iron in LaFe4P12 is significantly decreased,signalling a filling of the 3d states. This is consistent with the idea that each interstitial La atom(rattler) donates three electrons to compensate for missing valence electron of iron as comparedto cobalt, and indicates that the compensation mainly takes place on the iron site.

Paper IV

In this article, the Auger parameter is used to study the valence electron distribution in CoP3.The electron transfer between Co and P is estimated using the model of Thomas and Weight-man [4] which relates changes in Auger parameter values to charge transfer. In the EELS studypresented in Paper III, an emptying of the cobalt 3d states was found. Here it is found that eachphosphorus atom gains approximately 0.24 electrons. Based on stoichiometric considereations,this is equivalent to a donation from cobalt of about 0.72 electrons/atom, which is in excellentagreement with the EELS studies of Paper III.

References


69, 139 (2001).

[2] Keast, V. J. Ab initio calculations of plasmons and interband transitions in the low-losselectron energy-loss spectrum. Journal of Electron Spectroscopy and related phenomena

143, 97 (2005).

58 Chapter 5. Overview of papers

[3] Pearson, D. H., Ahn, C. C. & Fultz, B. White lines and d-band occupancies for the 3d and4d transition metals. Physical Review B 47, 8471 (1993).

[4] Thomas, T. D. & Weightman, P. Valence electron structure of AuZn and AuMg alloys derivedfrom a new way of analyzing Auger-parameter shifts. Physical Review B 33, 5406 (1986).

Paper I

O. M. Løvvik and Ø. Prytz

Density-functional band-structure calculations for La-, Y-, and Sc-filled

CoP3-based skutterudite structures

Physical Review B 70, 195119 (2004)

59

Density-functional band-structure calculations for La-, Y-, and Sc-filled CoP3-basedskutterudite structures

O. M. Løvvik and Ø. PrytzUniversity of Oslo, Center for Materials Science and Nanotechnology, P.O.Box 1126 Blindern, N-0318 Oslo, Norway

(Received 6 April 2004; revised manuscript received 30 August 2004; published 23 November 2004)

The crystal structure, thermodynamic stability, and electronic structure of La-, Y-, and Sc-filled CoP3 arepredicted from density-functional band-structure calculations. The size of the cubic voids in the skutteruditestructure is changed much less than the difference in size between the different filling atoms, and we expectthat the larger rattling amplitude of the smaller Sc and Y atoms may decrease the lattice thermal conductivityof Sc- and Y-filled structures significantly compared to La-filled structures. The solubility of La, Y, and Sc inCoP3 is calculated to be around 5, 3-6 %, and below 1% at 0 K, respectively. Based on similar systems, this isexpected to increase considerably if Fe is substituted for Co. Fe substitution is also expected to compensate theincreased charge carrier concentration of the filled structures that is seen in the calculated electron density ofstates. In conclusion, Sc- or Y-filled FeCoP3 skutterudite structures are promising materials for thermoelectricapplications.

DOI: 10.1103/PhysRevB.70.195119 PACS number(s): 61.18.j, 71.20.b

I. INTRODUCTION

There has recently been considerable interest in materialswith the skutterudite structure because of their newly discov-ered potential for thermoelectric applications.1,2 These mate-rials have the general formula TX3, with T being a transitionmetal (typically Co, Rh, Ir, etc.) and X a pnictogen (P, As,Sb, with possible substitution by Sn, etc.). They crystallize in

the cubic Im3 space group (No. 204), with a unit cell con-sisting of eight smaller cubes with the transition metal on thecube corners, six of which are filled with mutually perpen-dicular “rings” of four pnictogens. This leaves a rather largevoid in each of the two unoccupied cubes, corresponding tothe 2a sites in Wyckoff notation.

Thermoelectric materials are usually rated based on thethermoelectric figure of merit Z=S2 /, where S and arethe Seebeck coefficient and electrical conductivity, and isthe thermal conductivity of the material. A high value for Zindicates that a given material is suitable for thermoelectricapplications. The skutterudite materials usually display ahigh value for S and , but unfortunately they also exhibit ahigh thermal conductivity, thus reducing their suitability forthermoelectric applications.

In 1994, Slack and Tsoukala predicted that filling thevoids in the skutterudite materials could reduce their latticethermal conductivity through scattering of the thermalphonons by the rattling of the filling atoms in an oversized“box” in the structure.3 This effect has been thoroughly dem-onstrated for several compositions, with the filling elementstypically being a rare-earth metal, e.g., La, Ce, Nd, Sm, Eu,Th, etc. Depending on the rattling amplitude, this fillingatom lowers the lattice thermal conductivity by as much asan order of magnitude, with a strong correlation between therattling amplitude and the thermal conductivity.4–8 When allthe empty boxes in the structure have been filled (fillingfraction 100%), the general formula is MT4X12, where M isthe rare-earth filling element. The possibility of “tailoring”the thermal conductivity in this way has caused great interest

in the skutterudite materials for thermoelectric applications.The filled skutterudite crystal structure is shown in Fig. 1.

Based on the “rattling box” picture of this effect, it can beexpected that a large ion in a relatively small box has asmaller effect on the thermal conductivity than a small ion ina relatively large box. Indeed, this has been confirmed in thecase of Ce filling of the CoFeP3 and CoFeSb3 systems.9

The antimony based skutterudite structures have much largerlattice constants a=903−913 pm than the phosphorusbased system a=770−779 pm, leaving more free space forthe Ce ion in which to “rattle.” This results in a 90% reduc-tion of the thermal conductivity of the antimony-based skut-terudite structure compared to only a 20% reduction for the

FIG. 1. The filled skutterudite crystal structure. The compositionis MT4X12, where M is the rare-earth filling atom (large grey balls),T is the transition metal (black balls), and X is the pnictogen (whiteballs). The conventional cubic unit cell containing two formulaunits is shown, only with the origin translated by (0.25,0.25,0.25).

PHYSICAL REVIEW B 70, 195119 (2004)

1098-0121/2004/70(19)/195119(6)/$22.50 ©2004 The American Physical Society70 195119-1

phosphorus-based structure when filling with Ce.9 This trendis general, and even for several different materials the ther-mal conductivity decreases by an order of magnitude almostmonotonically when the rattling amplitude increases from 15pm (in CeFe4P12) to 29 pm (in CeFe4Sb12).2

It is evident that the smaller unit cell of the phosphorus-based skutterudite structures requires a smaller filling ele-ment to achieve a comparable reduction in thermal conduc-tivity to that seen in the larger antimony based materials. Inthis article we explore the feasibility of filling the CoP3structure with La, Y, and Sc. The rattling amplitude is de-fined as the difference between the void filler covalent radiusand the radius of the cage.10 Since the covalent radius de-creases from 169 to 144 pm when going from La to Sc, wemay expect that the rattling amplitude decreases in the orderof 25 pm, even though the cell parameters may be slightlydifferent in those materials. Following the general trend forfilled skutterudites,2 we may thus expect that Sc-filled skut-terudites have a lattice thermal conductivity of at least anorder of magnitude smaller than corresponding La-filledstructures. The covalent radius of Y is 162 pm, not too dif-ferent from La; the expected reduction in lattice thermal con-duction is from this around a factor of 2. To explore thefeasibility of filling with Y or Sc, we calculated the thermo-dynamical stability of such filled skutterudite structures bycomparing the calculated free energy of various compounds.At the same time we calculate their optimized cell param-eters by performing full relaxation of unit cell size andshape, as well as optimization of ionic positions.

The detailed electronic structure of these hypothetical ma-terials is also investigated in this work to clarify how theelectronic conductivity may be influenced by the filling at-oms. Both the density of states (DOS), local DOS, and bandstructure of the materials are presented, and for unfilled CoP3the results are compared to previous calculations.11,12

A number of theoretical studies have previously investi-gated various properties of skutterudite-type materials. Thesemiempirical extended Hückel tight-binding method hasbeen used to calculate the electronic structure of filled andunfilled skutterudite structures.13–15 CoP3 and NiP3 havebeen investigated by using the linear muffin-tin orbitalmethod in the atomic sphere approximation.11 Singh and co-workers have used the linearized augmented plane wavemethod within the local-density approximation to calculatevarious properties of IrSb3 , CoSb3, and CoAs3;16 CoP3 andLaFe4P12;

12 and La,CeFe4Sb12.17,18

II. COMPUTATIONAL METHOD

Periodic density-functional calculations within thegeneralized-gradient approximation as implemented in theVienna ab initio simulation package (VASP)19,20 was used inall the calculations, except the DOS mentioned below. Theprojector augmented wave (PAW) method was used to spanout the electron density,21 and the gradient correction wasPW91.22 The PAW method is a generalization of the linear-ized augmented plane-wave method, and gives access to thefull wave function;23 meanwhile, it has comparable effi-

ciency to the pseudopotential approach. The criterion forself-consistency in the electronic structure determination wasthat two consecutive total energies differed by less than 0.01meV. A cutoff energy of 500 eV was used throughout, andthe density of k points in the irreducible wedge of the Bril-louin zone was sufficient to ensure that the overall error dueto the mentioned numerical sources was of the order of 1meV per unit cell. The number of irreducible k points in theBrillouin zone varied between 24 for the smallest unit cells,consisting of eight CoP3 units, to 7 for the largest unit cells,consisting of 32 CoP3 units. Some of the smaller unit cellsfor calculating the thermodynamical stability had even morek points, up to the CoP structure, with 216 irreducible kpoints in the Brillouin zone. This latter structure has fourformula units in the conventional unit cell. For the DOScalculations the number was increased further, to a total of76 irreducible k points in the Brillouin zone for the Co8P24model. The smearing of partial occupancies of the wavefunctions are done using the tetrahedron method with Blöchlcorrections.

Relaxations have been performed by using the RM-DIISimplementation of the quasi-Newton method. The ionic co-ordinates and the unit cell size and shape were optimizedsimultaneously to eliminate structures with internal stress.The structure was considered relaxed when all the forceswere less than 0.05 eV/Å. A single calculation using highaccuracy was performed after the completion of the relax-ation, to determine the accurate total free energy.

The electronic density of states (DOS) was calculatedboth using VASP and the ADF-BAND24,25 packages. The latteris based on linear combinations of numeric and Slater-typeatomic orbitals, and is, in principle, exact (at the GGA level)when all numerical issues such as basis sets, integration ac-curacy, etc., have been taken care of. This thus serves as aconvenient check of the calculated DOS by using VASP.

III. RESULTS AND DISCUSSION

A. Crystal structure

When creating periodic models with various filling frac-tions, some symmetry is necessarily broken. Table I lists thesymmetry of the unit cells used to represent the filling frac-tions of this study. The model with 25% filling is made from

TABLE I. The models being used to create the compoundsMxCo4P12, where M is La, Y, and Sc and x is the filling fraction ofM. The number of atoms in the unit cell Natoms and the space group(SG) have been listed in addition to the filling fraction x.

x Natoms SG SG No.

0 32 Im3 204

12.5 133 Im3 204

25 65 Pmmm 47

50 33 Pm3 200

100 34 Im3 204

O. M. LØVVIK AND Ø. PRYTZ PHYSICAL REVIEW B 70, 195119 (2004)

195119-2

a 112 conventional (cubic) cell, and is thus the onlymodel that has intrinsically broken the cubic symmetry andthe isotropic distribution of the filling atoms. The model withx=12.5% has been constructed using a 222 primitive(rhombohedral) cell in order to preserve cubic symmetry.The other models use the conventional cubic cell.

The relaxation procedure allowed the unit cell to changesize and shape and the ionic positions to relax. None of themodels changed space group during the relaxation, but allthe models with 25% filling lost the cubic shape of their unitcells. The resulting lattice constants are shown for all themodels in Table II. We can see that the models with isotropicdistribution of the filling atoms (i.e., all except the 25%filled), retained the cubic shape of their unit cells. The 25%La-filled model is also not far from cubic; the differencebetween the smallest and largest lattice constants is only 0.6pm. This difference has grown to 1.5 pm and 1.1 pm for Yand Sc, respectively. These structures are still cubic withinthe uncertainty of our relaxation method, however: Whenstarting from different input structures, the lattice constantsof the same resulting structure may differ by up to 3 pm withour current accuracy.26 Since there is no reason to expect thatsymmetry in the real material is broken in the same way as inthe 25% models, we conclude that the real structure of filledCoP3 will most probably be cubic.

The lattice constants increase with increasing filling frac-tion, and most for the larger filling atoms. This has beendepicted in Fig. 2, where the cubic cell volume has beenplotted as a function of filling fraction for the three fillingelements La, Y, and Sc. The volume expansion is very closeto linear, and the small deviations from linearity are withinthe error limits of our calculations. The expansion is morethan twice as large for filling with La than for Sc, with Y inthe middle, reflecting the variation in their size. But the dif-ference in size of the cubic voids (around 6 pm for 100%filling, and not more than 1 pm for more realistic filling

fractions) is not nearly as large as the size difference of 25pm between La and Sc.28 Our founding hypothesis has thusbeen confirmed, and based on the general trend of fully filledskutterudite structures,2 we may expect an order of magni-tude decrease of the lattice thermal conductivity of Sc-filledskutterudite structures compared to that of La-filled struc-tures. Similarly, the lattice thermal conductivity of Y-filledskutterudite structures should be approximately halved com-pared to the La-filled ones. This is, of course, speculative anddepends, among several other things, on the thermodynamicsolubility of the filling atoms.

B. Phase stability

The thermodynamic stability of the current models hasbeen calculated by checking different decomposition routesfrom the filled structure. The total free energy of all the com-pounds participating in these hypothetical reactions has beencalculated in the same manner as the skutterudite structures;by a full relaxation of both the unit cell and ionic positions,followed by a final high precision calculation. The moststable end point was chosen in each case to evaluate thestability of the filled structure. The investigated decomposi-tion routes have been presented for the 50% filled models inTable III. The possible routes are similar for the other fillingfractions, except for the 100% Y-filled CoP3, where YP5 can-not form together with Co-P compounds. The most favorableroute is similar for all the filling atoms; for M-filled CoP3 itis to MP, CoP3, and CoP2.

This has been used to calculate the decomposition en-thalpy Hdecomp defined as

HdecompMT8X24 = EMT8X24 − 8ETX3 − EM 1

for the first decomposition route, etc. Here ETX3 andEMT8X24 are the uncorrected total free energy of the un-filled and filled skutterudite-type structures as provided byVASP and EM is that of the filling metal in its standardstate. This has been used to create a plot of the decomposi-tion enthalpy as a function of filling fraction for the threefilling elements in Fig. 3. None of the models in this studyare stable compared to the end products, implying that thesolubility of La, Y, and Sc is less than 12.5% in CoP3 at 0 Kassuming ordered distribution of the filling element. Apart

TABLE II. The resulting lattice constants for the different mod-els in study in the form MxCo4P12. The resulting cell angles were all90°, and the resulting space groups were as in Table I. The experi-mental lattice constant of CoP3 (no filling) is 771.1 pm (Ref. 27).

Filling atom M Filling fraction alatt blatt clatt

(%) (pm) (pm) (pm)

0 773.5 773.5 773.5

La 12.5 774.8 774.8 774.8

La 25 776.5 776.4 775.9

La 50 779.7 779.7 779.7

La 100 784.6 784.6 784.6

Y 12.5 774.3 774.3 774.3

Y 25 776.1 775.7 774.6

Y 50 777.8 777.8 777.8

Y 100 781.4 781.4 781.4

Sc 12.5 773.7 773.7 773.7

Sc 25 775.2 774.7 774.1

Sc 50 775.8 775.8 775.8

Sc 100 778.2 778.2 778.2

FIG. 2. The relaxed volume in nm3 of the models MxCo4P12 asa function of the filling fraction x, with M =La, Y, and Sc. Theexperimental volume of CoP3 is 0.4585 nm3 (Ref. 27).

DENSITY-FUNCTIONAL BAND-STRUCTURE … PHYSICAL REVIEW B 70, 195119 (2004)

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from the behavior of the Y-filled models, which crosses thelinear curve of Sc filled CoP3 between 25 and 50 % Y, all thecurves closely follow a linear behavior, and we thus feelrelatively safe to linearly extrapolate the curves to Hdecomp=0 which is indicative of a stable filled structure. This hasbeen done in Fig. 3 by extrapolating linearly from the 25 and12.5 % data points. The point from the 25% Y-filled model ishigher than should be expected from the linear behavior, anda linear extrapolation using the 50% point instead of the 25%point would cross the abscissa at a higher value. We do nothave a reasonable physical explanation for this deviationfrom linearity; most probably it is reflecting the numericaluncertainty of our procedure. The predicted solubility inCoP3 at 0 K may hence be read out of Fig. 3: 5% La, 3-6 %Y, and less than 1% Sc.

Our results could be slightly altered if the zero-point mo-tion of the ions were included. This has not been done fortwo reasons. We expect that this effect would more or less becanceled when comparing different systems. In addition,

such calculations would be prohibitively expensive, particu-larly for the large systems in study.

Furthermore, we expect both temperature effects and en-tropy to increase the solubility of the filling atoms whenmoving to room temperature. Since no experimental investi-gations of these compounds have yet been published, wemust compare to another system: The solubility of Ce inCoSb3 is around 7%,2 increasing to 100% when Co is fullysubstituted by Fe. Similarly, it is known that the solubility ofLa and Y is 100% in FeP3.29,30 From this we expect that it ispossible to fill mixed FeCoP3 skutterudite structures withboth La, Y, and Sc.

The substitution of Fe for Co in skutterudite-type struc-tures is usually motivated by charge compensation for thefilling. We shall see how this can be understood from theelectronic structure in what follows.

C. Electronic structure

The electronic density of states (DOS) and the band struc-ture are important to judge whether a material is promisingfor thermoelectric applications—the optimal charge carrierconcentration for a thermoelectric material is around1019 cm−3, which means that it should be a semiconductor ora semimetal.2 Calculated electronic structures have alreadybeen published for CoP3 by two previous studies using dif-ferent techniques and achieving very different results.11,12

Llunell and co-workers used the atomic sphere approxima-tion within the linearized muffin tin orbital method (LMTO-ASA), and found that the band structure exhibited apseudogap separating the valence and conduction bands.11

Fornari and Singh, using the linearized augmented planewave (LAPW) formalism, found that this gap is crossed by aphosphorus p band above the Fermi level, so that CoP3 isexpected to be metallic.12 The two studies also presentedDOS plots that were relatively similar, but they also differedin some important details, particularly around the Fermilevel.

We have calculated the total DOS of CoP3 using two otherapproaches: the planar augmented wave method of VASP

(Refs. 19,20) [shown in Fig. 4(a)] and the linear combinationof numeric and Slater-type atomic orbitals as implemented inADF-BAND [shown in Fig. 4(b)]. These two plots do not differby more than details, even though the methods used are quite

TABLE III. Possible decomposition routes for 50% filled CoP3.The decomposition enthalpy Hdecomp is given for each route in eVper formula unit. A negative decomposition enthalpy means that thefilled CoP3 is unstable compared to the end products.

Decomposition route Hdecomp

(eV)

LaCo8P24→La+8CoP3 1.579

LaCo8P24→LaP+7CoP3+CoP2 −1.243

LaCo8P24→LaP+7.5CoP3+0.5CoP −1.210

YCo8P24→Y+8CoP3 0.107

YCo8P24→YP+7CoP3+CoP2 −2.825

YCo8P24→YP+7.5CoP3+0.5CoP −2.792

YCo8P24→YP5+3CoP3+5CoP2 −1.282

YCo8P24→YP5+5.5CoP3+2.5CoP −1.119

ScCo8P24→Sc+8CoP3 0.445

ScCo8P24→ScP+7CoP3+CoP2 −2.198

ScCo8P24→ScP+7.5CoP3+0.5CoP −2.166

FIG. 3. The decomposition enthalpy of filled CoP3 as a functionof filling fraction x. The enthalpy has been normalized to the con-ventional cell, that is, Co8P24. The dotted lines are linear extrapo-lations of the data points at 25 and 12.5 %.

FIG. 4. The total DOS of CoP3 calculated by ADF-BAND (a) andVASP (b). The energy is in eV relative to the Fermi level.


195119-4

different. Both methods should in principle be able to pro-duce results close to the DFT limit, without any further im-portant approximations than the description of the exchange-correlation potential. It is important to note that, while VASPcalculates the electron density within the generalized gradi-ent approximation, the electron density has been found self-consistently within the local-density approximation in theADF-BAND. The VASP curve was produced using spin-polarized calculations, while the ADF-BAND curve did not;this is actually the reason for the difference in the valenceband far from the Fermi level. We have performed VASP testcalculations with spin polarization and without, and the onlynon-negligible difference in the DOS was found at the highend of the valence band. Our results are quite similar to theDOS curves for CoP3 presented by the previous LMTO-ASAstudy11 and LAPW study,12 apart from a range of details.

When turning to the band structure, however, the differ-ences are much clearer. We have plotted the band structurefor CoP3 in Fig. 5. There is no indirect gap at the Fermilevel, as was found in Ref. 11, and we find the same bandcrossing the Fermi level close to the point as was found inRef. 12. In most of the overall features, our results corre-spond nicely with those of the latter study. There are a num-ber of details that differ, though, particularly around the Hpoint. This does not change the main picture, however; thatof a metallic CoP3 with relatively low DOS directly belowthe Fermi level.

We now turn to the effect on the electronic structure fromadding the rattling atom. The effect on the DOS of adding Lato CoP3 is shown in Fig. 6. First of all, it is noted that theoverall shape of the DOS is changed only to a small extentwhen filling with La. The main difference when increasingthe filling fraction is that the Fermi level is moved upwardsinto what used to be the conduction band. The same effect isseen for Y- and Sc-filled CoP3 (not shown) with a quantita-tively similar impact on the Fermi level. This indicates thatthe same electronic changes are caused by filling with eitherof the three elements. In all cases the DOS plots show thatfilled CoP3 has a larger carrier concentration than the unfilledstructure, giving a higher electrical conductivity and thuscontributing to a higher figure of merit. However, it is likelythat the change in DOS also will have a detrimental effect onthe Seebeck coefficient, thereby reducing the figure of merit.

This is even more visible in the band structure, shown for50% La-filled CoP3 in Fig. 7. Here we see that the higher

Fermi level pushes the band that previously crossed theFermi level down into the valence band. Meanwhile, severalbands that previously belonged to the conduction band arenow crossing the Fermi level from above. Most of the otherqualitative properties of the band structure are unchanged, sothe main effect of adding La has been to increase the carrierconcentration. We know from La-filled FeP3 that this situa-tion may improve when substituting Fe for Co. By carefullychoosing the optimal Fe content, it should be possible tomove the Fermi level back down to the area with scarcelypopulated electron states, just below the point where theDOS increases dramatically, that is, around −0.4 eV. If thismay be done without destroying the other benign propertiesof the band structure, we would expect that Y- or Sc-filled

FIG. 5. The band structure of CoP3. The energy is in eV relativeto the Fermi level.

FIG. 6. The total DOS of 12.5, 25, 50, and 100 % La-filledCoP3. The energy is in eV relative to the Fermi level.

FIG. 7. The band structure of LaCo8P24. The energy is in eVrelative to the Fermi level.

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FeCoP3 should have a thermoelectric figure of merit supe-rior to that of La-filled structures. This is the subject of aforthcoming study.

IV. CONCLUSIONS

Accurate density-functional band-structure calculationshave been performed to predict the crystal structure and solu-bility of La, Y, and Sc in CoP3. A number of models wereused to vary the content of the filling atoms in CoP3 from12.5 to 100 %. Full structural relaxations showed that cubicsymmetry was, within the accuracy of the method, main-tained in all cases. The lattice expansion due to increasingfilling fraction was linear, with the La-filled models expand-ing more than twice as much as the Sc-filled models, andY-filled CoP3 in between. The difference in expansion is neg-ligible compared to the difference in size between the fillingatoms, and we expect that Sc- and Y-filled structures mayhave significantly reduced thermal conductivities because ofthe larger rattling amplitude of the filling atoms.

The thermodynamic solubility of the filling atoms in CoP3was found by extrapolating curves of the decomposition en-thalpy of the filled models to stable filling. The present cal-culations were performed at 0 K, and at this temperature wepredict a solubility of La, Y, and Sc in CoP3 of 5%, 3-6 %,

and less than 1%, respectively. Similarly to Ce in CoSb3, weexpect this to increase if substituting Fe for Co.

The calculated DOS and band structures show that theelectronic effect of adding La, Y, or Sc is quite similar. Itcauses the Fermi level to move into the conduction band,effectively increasing the charge carrier concentration andthus the electrical conductivity. It may also, however, de-crease the Seebeck coefficient, hence reducing the figure ofmerit. Based on this, we do not expect the hypothetical ma-terials of this study to be too promising for thermoelectricapplications. This may easily change, however, if the in-creased charge concentration is compensated by replacing Fefor Co—we then expect that the Sc- or Y-filled structuresmay have thermoelectric performances superior to that ofLa-filled structure. We therefore propose that synthesis of Sc-or Y-filled FeCoP3 skutterudite-type structures is at-tempted, and that their thermoelectric properties be investi-gated.

ACKNOWLEDGMENTS

The authors gratefully acknowledge stimulating discus-sions with Johan Taftø, Terje Finstad, and Helmer Fjellvåg.Computing time and excellent support from the NOTURconsortium are also appreciated.

1 C. Uher, Semicond. Semimetals 69, 139 (2001).2 G. Chen, M. S. Dresselhaus, G. Dresselhaus, J. P. Fleurial, and T.

Caillat, Int. Mater. Rev. 48, 45 (2003).3 G. A. Slack and V. G. Tsoukala, J. Appl. Phys. 76, 1665 (1994).4 B. C. Sales, D. Mandrus, and R. K. Williams, Science 272, 1325

(1996).5 G. S. Nolas, G. A. Slack, D. T. Morelli, T. M. Tritt, and A. C.

Ehrlich, J. Appl. Phys. 79, 1 (1996).6 B. C. Sales, D. Mandrus, B. C. Chakoumakos, V. Keppens, and J.

R. Thompson, Phys. Rev. B 56, 15 081 (1997).7 G. S. Nolas, J. L. Cohn, and G. A. Slack, Phys. Rev. B 58, 164

(1998).8 A. Grytsiv, P. Rogl, S. Berger, C. Paul, E. Bauer, C. Godart, B.

Ni, M. M. Abd-Elmeguid, A. Saccone, R. Ferro, and D. Kaczo-rowski, Phys. Rev. B 66, 094411 (2002).

9 A. Watcharapasorn, R. C. DeMattei, R. S. Feigelson, T. Caillat,A. Borshchevsky, G. J. Snyder, and J. P. Fleurial, J. Appl. Phys.86, 6213 (1999).

10 D. J. Braun and W. Jeitschko, J. Less-Common Met. 72, 147(1980).

11 M. Llunell, P. Alemany, S. Alvarez, V. P. Zhukov, and A. Vernes,Phys. Rev. B 53, 10 605 (1996).

12 M. Fornari and D. J. Singh, Phys. Rev. B 59, 9722 (1999).13 D. W. Jung, M. H. Whangbo, and S. Alvarez, Inorg. Chem. 29,

2252 (1990).14 M. Partik and H. D. Lutz, Phys. Chem. Miner. 27, 41 (1999).

15 D. H. Galvan, Int. J. Mod. Phys. B 17, 4749 (2003).16 D. J. Singh and W. E. Pickett, Phys. Rev. B 50, 11 235 (1994).17 J. L. Feldman, D. J. Singh, I. I. Mazin, D. Mandrus, and B. C.

Sales, Phys. Rev. B 61, R9209 (2000).18 J. L. Feldman, D. J. Singh, C. Kendziora, D. Mandrus, and B. C.

Sales, Phys. Rev. B 68, 094301 (2003).19 G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).20 G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11 169 (1996).21 G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).22 J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R.

Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671(1992).

23 P. E. Blöchl, Phys. Rev. B 50, 17 953 (1994).24 G. te Velde and E. J. Baerends, Phys. Rev. B 44, 7888 (1991).25 G. te Velde and E. J. Baerends, J. Comput. Phys. 99, 84 (1992).26 O. M. Løvvik, S. M. Opalka, H. W. Brinks, and B. C. Hauback,

Phys. Rev. B 69, 134117 (2004).27 W. Jeitschko and D. Braun, Acta Crystallogr., Sect. B: Struct.

Crystallogr. Cryst. Chem. 33, 3401 (1977).28 URL http://www.webelements.com29 W. Jeitschko, A. J. Foecker, D. Paschke, M. V. Dewalsky, C. B.

H. Evers, B. Kunnen, A. Lang, G. Kotzyba, U. C. Rodewald,and M. H. Moller, Z. Anorg. Allg. Chem. 626, 1112 (2000).

30 I. Shirotani, Y. Shimaya, K. Kihou, C. Sekine, and T. Yagi, J.Solid State Chem. 174, 32 (2003).


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Paper II

Ø. Prytz, O. M. Løvvik, and J. Taftø

Comparison of theoretical and experimental dielectric functions: electron

energy-loss spectroscopy and density-functional calculations on skutterudites.


67

Comparison of theoretical and experimental dielectric functions: Electron energy-lossspectroscopy and density-functional calculations on skutterudites

Ø. Prytz,1 O. M. Løvvik,1 and J. Taftø2

1University of Oslo, Centre for Materials Science and Nanotechnology, P.O.Box 1126 Blindern, N-0318 Oslo, Norway2University of Oslo, Department of Physics, P.O.Box 1048 Blindern, N-0316 Oslo, Norway

Received 13 July 2006; revised manuscript received 2 October 2006; published 13 December 2006

We explore the possibility of combining density functional theory DFT and electron energy loss spectros-copy EELS to determine the dielectric function of materials. As model systems we use the skutteruditesCoP3, CoAs3, and CoSb3 which are prototypes for thermoelectric materials. We achieve qualitative agreementbetween the theoretically and experimentally obtained low energy-loss spectra and dielectric function. Some ofthe remaining discrepancies may be caused by the challenge of refining the experimental spectra beforeKramers-Kronig analysis. However, contrary to what is the case for some crystals with less complicatedelectronic structure, the DFT calculated plasmon energies deviate significantly from the experimental values.The great accuracy with which the plasmon energy can be determined by EELS, suggests that this techniquemay provide valuable inputs in further efforts to improve DFT calculations. The use of EELS as the experi-mental technique may become particularly powerful in studies of small volumes of materials.

DOI: 10.1103/PhysRevB.74.245109 PACS numbers: 71.45.Gm, 79.20.Uv, 71.20.b

I. INTRODUCTION

Many electron energy-loss spectroscopy EELS studieshave focused on the energy loss edges corresponding to theexcitation of core electrons, usually occurring above 100 eV.These excitations normally involve dipole allowed transi-tions from an initial state which is highly localized in energy,to a final state above the Fermi level. The relative intensity ofthese features can be used for quantitative studies of compo-sition, while the fine structure will reflect the local density ofempty states. Numerous studies have used these core-lossedges to investigate the electronic structure and bonding ofmaterials.1–7

In comparison, the energy-loss features appearing atlower energies below 50 eV are less studied, even thoughthey can be orders of magnitude more intense than the corelosses discussed above. These low energy-loss spectra con-tain a wealth of information, e.g., the dielectric function ofthe material can be extracted through a Kramers-Kroniganalysis of high quality spectra. The energy loss processestaking place in this energy region can be divided into twomain categories.

One category is the collective excitations of the valenceelectrons, which can be considered as a plasma. These exci-tations occur when the incident electron perturbs the outervalence electrons of the material, causing collective oscilla-tions called plasmons with eigenfrequency p. The idealfree electron plasma frequency is fully determined by thevalence electron density. These plasmons are usually thesingle most intense feature available in the energy-loss spec-trum. However, studies of the plasmon energies reveal thatthe electrons in real materials do not behave exactly as a freeelectron plasma. This is because the electron gas oscillates ina solid with a potential set up by the atom cores. Further-more, single electron excitations may contribute to raising orlowering the plasmon energy. In principle, careful investiga-tion of the plasmon energy of materials could reveal impor-tant information on their electronic structure.8 Unfortunately,

an interpretation of the observed spectra along these lines isextremely demanding, and in practice the plasmon energieshave mainly been used to study changes in valence electrondensity, e.g., resulting from strain or hydration of metals.9,10

In addition to the plasmons, the low energy-loss regioncontains low energy interband transitions of single electrons.These interband transitions cannot be interpreted with refer-ence to the unoccupied density of states only. The initialstates of these transitions are located close to the Fermi level,and are usually not localized in energy. The energy lossesmust therefore be discussed in light of a convolution of ini-tial states below the Fermi level with final states above it,reflecting a dipole selected joint density of states. Due to thedependence on both initial and final states we can expectthese low energy-loss interband transitions to be very sensi-tive to bonding effects in the material, potentially revealinginformation on the electronic structure. As with the plasmonenergies, the interpretation of these spectra is very challeng-ing and has not been widely applied to study the electronicstructure of materials. For this region of the energy-lossspectrum to be more extensively used to study the electronicstructure, there is a need for strong interaction betweentheory and experiment.

However, the calculation of electronic excitations fromfirst principles is a highly non-trivial problem, which so farhas no satisfactory solution.11,12 Some methods are in prin-ciple exact, but their scaling behavior with system size pro-hibits their use except for the smallest model systems—theconfiguration-interaction method13 and Green’s functionbased quasi-particle approaches14,15 are examples of this.Time-dependent DFT, on the other hand, is also in principleexact and has much better scaling behavior, but the correctform of the exchange-correlation function and its variationkernel are not known, and no good general approximationshave been found so far.16–18 This is a very active field ofresearch, and a number of possible solutions areemerging.19–23 However, these are still usually accessibleonly for very small systems.

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Thus, for calculating these properties for larger unit cells,one has to turn to simpler DFT response calculations, inwhich the Kohn-Sham eigenvalues are taken as quasiparti-cles from which optical excitation energies may be calcu-lated. This is not formally justified, but direct comparisons ofcalculated core loss spectra with experimental results haveshown good correspondence for some systems.24–27 How-ever, the correspondence of theoretical and experimental lowenergy-loss spectra is less well investigated.

Recently, Keast28 compared the theoretical plasmon en-ergy and low energy-loss spectra for a number of elementalsolids and some binary alloys with experiments with goodresults. Also, Moreau and Cheynet29 compared band struc-ture calculations with EELS experiments on BN filaments.Furthermore, a satisfactory agreement between experimentaland calculated dielectric function has already been achievedfor simple crystals such as Si.30

We intend to approach more complicated materials thanthe tetrahedrally coordinated semiconductors: crystals with alarger unit cell, without a well-defined number of valenceelectrons and more than one type of nearest neighbor bonds.At the same time we look for a material system withoutstoichiometry problems and crystals of high quality. Interest-ing candidates are materials with the skutterudite structure.These are line compounds, and virtually defect free asjudged by transmission electron microscopy. In addition, ma-terials with this structure are highly interesting thermoelec-tric materials which strengthen their candidacy as excellentmodel systems.

We study the binary skutterudites CoP3, CoAs3, andCoSb3. The crystal structure of these materials is cubic. Cohas six P nearest neighbors d2sp3 hybrid bonds, while thepnicogen is tetrahedrally coordinated with two Co and two Pnearest neighbors sp3 hybrid bonds.31 Several band struc-ture calculations have been performed on these and similarmaterials using DFT.32–34 However, apart from measure-ments of the electronic conductivity and Seebeck coefficient,there are few direct comparisons between calculations andexperimental results for these materials, particularly when itcomes to detailed information about their electronicstructure.35–38

The aim of the present work is to compare DFT calculatedand EELS extracted dielectric functions for crystals withmore complex bonding than has previously been approached.We believe that this combination of experimental and theo-retical methods to study the electronic structure may becomea powerful synergistic approach towards understanding theelectronic properties of materials. A major motivation forusing EELS is that this technique may become particularlypowerful for nanomaterials, and when only small amounts ofmaterials are available, since EELS spectra can be acquiredfrom nanometer sized regions.

II. EXPERIMENTAL PROCEDURES, DATA ANALYSIS,AND THEORETICAL METHODS

A. Synthesis and sample preparation

The skutterudite samples were prepared by direct reactionof the constituent elements during heating in evacuated and

sealed silica tubes. The homogeneity of the samples weresubsequently checked by x-ray powder diffraction in trans-mission with a Siemens D5000 diffractometer using mono-chromated Cu-K1 radiation. The samples were found to bealmost completely single phase, with only small amounts ofimpurity phases present. The lattice constant a and positionparameters y and z were refined using the General StructureAnalysis System GSAS.39 The experimental values of thisstudy show good agreement with those reported earlier.40–42

Samples for transmission electron microscopy TEM andelectron energy loss spectroscopy EELS were prepared bygrinding in ethanol in an agate mortar and deposited on acarbon film suspended on a copper mesh. The composition ofthe samples was then checked using energy dispersive x-rayanalysis EDS in a JEOL 2000FX TEM operated at 200 kVfitted with a Tracor Northern EDS detector and a Noran Sys-tem Six analysis system. The EDS analyses showed that thesamples were of the correct composition within the experi-mental accuracy. Furthermore, electron diffraction and imag-ing in TEM showed that the crystals were of excellent qual-ity.

B. Electron energy loss spectroscopy

The EELS studies were performed using a post columnGatan imaging filter fitted to a field emission JEOL 2010FTEM operated at 197 kV. The TEM was operated in imagemode with a collection semiangle =6 mrad, and a slightlyconvergent electron beam with incidence semiangle ap-proximately 3 mrad.

The spectra were obtained with an energy resolution of1.2 eV, as determined by the full width at half maximum ofthe zero-loss peak, and a spectrometer dispersion of0.2 eV/channel. The energy scale was further calibrated us-ing the plasmon peak of pure Si as a standard. The spectrawere corrected for channel-to-channel gain variations in thedetector, and the dark current was subtracted. The spectrawere obtained from sample areas of thickness between 1 and2 times the mean free path of the fast electrons. This ensuredthat the relative contribution of the surface plasmons wasnegligible.

Figure 1 shows an example of the spectra obtained in thisfashion, the multiple plasmon peaks appear because an inci-dent electron may lose energy to plasmon excitation morethan once. The effects of multiple scattering were removedthrough a Fourier-log deconvolution as implemented in theGatan EELS Analysis Software see Egerton43 for details.Thus the single scattering distribution SE was obtained.

The subtraction of the broad tails of the zero loss peak isof special importance when studying the low energy-lossarea of the spectrum. A zero loss profile obtained without thesample was used in the deconvolution of each spectrum toidentify the elastic contribution. This premeasured profilewas fitted to the spectra in the region 1.5–2.5 eV. As a resultof this fitting, spectral information in this area may be lost.Figure 2 shows an example of how this fitting was done.

The energy loss suffered by an incident electron is deter-mined by the electronic response of the material, which isdescribed by the complex dielectric function E. The di-

PRYTZ, LØVVIK, AND TAFTØ PHYSICAL REVIEW B 74, 245109 2006

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electric function is related to the experimental single scatter-ing distribution through

SE Im−1

E =

2

12 + 2

2 . 1

The full dielectric function can be retrieved from the ex-perimental spectrum through a Kramers-Kronig analysis.This was also done using the Gatan EELS Analysis software,see Egerton43 for details.

C. Density functional theory

The calculations were performed using density functionaltheory at the generalized gradient approximation level.Structural relaxations and densities of states DOS were cal-culated using the Vienna ab initio Simulation PackageVASP.44,45 For VASP, the energy cutoff for the plane waveexpansion was 650 eV, the k-point distances were less than0.20 Å−1, the criterion for self-consistency was energy con-vergence within 10−5 eV, and spin polarization was allowed.Ionic relaxation was said to be achieved when all forces wereless than 0.05 Å−1.

The dielectric function of the skutterudites was obtainedfrom the Optic package of the WIEN2K code.46 In these calcu-lations, the dipole approximation is assumed valid, and onlytransitions satisfying the dipole selection rule l= ±1 areconsidered. Convergence of the WIEN2K results were achievedwith the product RMTKmax=7, the largest vector in the chargedensity Fourier expansion Gmax=14, and the number of kpoints chosen to be 1000.

The calculated dielectric function can be compared to theexperimental dielectric function obtained as described above.Alternatively, a theoretical spectrum can be calculated ac-cording to Eq. 1 and compared directly to the experimentalresults. The theoretical plasmon energy of the three com-pounds was obtained from the maximum of this theoreticalspectrum.

III. RESULTS AND DISCUSSION

The cell parameters and positional parameters y and zobtained from the fully relaxed cell of the DFT calculations

are listed in Table I along with previously reported XRDvalues and those obtained in this study. The overall agree-ment between theory and experiments is very good, althoughthe DFT calculations seem to slightly overestimate the cellsize less than 1%.

The total density of states DOS and the symmetry pro-jected local density of states for the three skutterudites areshown in Fig. 3. In general, the DOS for the three com-pounds show great similarity. The Co d states are split intostates of t2g and eg symmetry due to the octahedral environ-ment of the Co atoms, with the majority of the Co d statesstrongly peaked in an area approximately 2–2.5 eV belowthe Fermi level. The pnicogen p states are rather uniformlydistributed above and below the Fermi level, hybridizingwith the Co d states and contributing heavily to the totalDOS in the −7 to +5 eV region. The Co s and p states con-tribute only to a small degree to the total DOS, while thepnicogen s states dominate in the region 9–15 eV below theFermi level. In CoP3, the Fermi level is located at the verylower edge of the conduction band, while in CoAs3 andCoSb3 it is shifted slightly downward towards the valenceband, into the pseudogap where the density of states is verysmall. It is also evident that this pseudogap is largest forCoP3, and decreases in width in the skutterudites with theheavier pnicogens and larger unit cells. These results are ingood agreement with previous work32,34,35 and will later beused in the interpretation of experimental and theoreticalspectra obtained in this study.

A. Plasmon excitations

The experimental energy-loss spectra from the skutteru-dites show rather sharp plasmon peaks, and pronounced ad-ditional peaks associated with multiple plasmon losses arevisible as exemplified in Fig. 1. This behavior is very similarto that displayed by semiconductors and simple metals, whiletransition metals and their oxides usually display only asingle broad plasmon. The Fourier-log deconvolution suc-cessfully removed the contributions from multiple scattering,giving the single scattering distributions shown in Fig. 4a.

FIG. 2. Example of the zero-loss peak removal and Fourier-logdeconvolution. The unprocessed raw data is shown with a fullydrawn line, while the fitted zero-loss peak is the dotted line. Theresulting spectrum after subtraction of the zero loss peak and de-convolution is shown with a dashed line.

FIG. 1. In the raw data obtained from the EELS experimentbefore deconvolution, multiple plasmons are observed. Here isshown an unprocessed experimental spectrum from CoAs3 wherewe observe a plasmon energy of slightly more than 20 eV.

COMPARISON OF THEORETICAL AND EXPERIMENTAL… PHYSICAL REVIEW B 74, 245109 2006

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We immediately recognize the intense plasmon peaks asso-ciated with collective oscillations of the valence electrons.Also visible are energy loss features associated with the M4,5and N4,5 excitations of outer core electrons of As and Sb,respectively.

The theoretically calculated spectra for the compoundsare shown in Fig. 4b. Similar to the experimental results,these show a single sharp plasmon peak, and loss features onthe high energy side of the plasmon peak associated with theexcitation of outer core electrons. The experimental plasmonpeaks are significantly broader than those of the theoreticalspectra, a feature that can be explained by broadening causedby the limited energy resolution of the experiments. The ex-perimentally obtained plasmon energies are given in Table II,alongside theoretical values obtained from the DFT calcula-tions.

The plasmon energy of materials displaying narrow, welldefined plasmon peaks is often successfully predicted by the

free electron Drude model. In this model, the plasmon en-ergy is given by Ep

D= pD, where p

D is the frequency of theplasma oscillation given by8

pD = ne2

0m0. 2

TABLE I. The lattice constant a, and position parameters y and z for the three skutterudites obtained byXRD and DFT. Previously reported experimental values are listed for comparison. The DFT cell parameterfor CoP3 is from a previous study Ref. 47.

a Å y z

CoP3

XRD this work 7.71631 0.348123 0.1428529Previously reported XRD Ref. 40 7.71128 0.348956 0.145136Previously reported DFT Ref. 47 7.735 0.348 0.146

CoAs3

XRD this work 8.20562 0.33843 0.148125Previously reported XRD Ref. 41 8.1953 0.34311 0.15031DFT this work 8.277 0.341 0.152

CoSb3

XRD this work 9.04636 0.32996 0.15606Previously reported XRD Ref. 42 9.03853 0.335374 0.157884DFT this work 9.117 0.334 0.160

FIG. 3. Color online Total and local projected density of statesfor CoP3, CoAs3, and CoSb3. The total density of states for CoP3

was published in a previous study Ref. 47.

FIG. 4. Color online Experimental and theoretical low energy-loss spectra for the three skutterudites. a Experimental spectrashowing a single plasmon peak. b Theoretical spectra. The energyloss peaks associated with the As M4,5 transitions and the Sb N4,5

transitions are visible in both the experimental and theoreticalspectra.


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Here, n is the density of valence electrons, e and 0 arethe elementary electric charge and the permittivity of freespace, while m0 is the free electron mass. Table II also liststhe plasmon energy found by using Eq. 2 and assumingthree valence electrons for Co and five electrons for P, As,and Sb, giving a total of 144 valence electron per unit cell forall the skutterudites

Using this valence electron count, the Drude model pre-dicts a reduction of Ep as the unit cell size increases and thevalence electron density decreases. We see that this is in factthe case, with CoSb3 having the largest unit cell and thelowest plasmon energy, while CoP3 has the smallest cell sizeand thereby the largest plasmon energy. Other than this, wenote that the Drude model significantly, and consistently, un-derestimates the plasmon energy of the skutterudites. Thedeviation between experimental and predicted plasmon ener-gies ranges from 4 to nearly 7%.

The basic premise of the Drude model is that the valenceelectrons can be considered as simple harmonic oscillators.The plasmon frequency p

D is then identified as the fre-quency of resonant oscillation. However, when transitionmetals are involved, this simple picture is complicated by thefact that these elements have an outer shell consisting of alarge number of d electrons that are bound to a varying de-gree. Thus, the transition metals and their oxides usually dis-play a single, broad plasmon peak, and the Drude modeloften fails to predict their plasmon energies. In comparison,simple metals such as Be, B, Na, and Al, and semiconductorssuch as Si, Ge, and GaAs have sharp plasmon peaks near thepredicted value, and peaks associated with the multiplelosses are readily observed.8,43,48 The skutterudites representa different class of materials where the plasmons display thecharacteristics of simple metals or semiconductors, but theDrude model fails to predict their energy.

When turning to the DFT calculations, we note that thecorrespondence with the experimental plasmon energies isnot much better than for the Drude model; the plasmon en-ergy of CoSb3 is significantly overestimated 6% whilethose of CoP3 and CoAs3 are underestimated by 2–4%.

Despite this, the results presented here are only slightlypoorer than that achieved for simpler systems,28 and for Siusing more sophisticated methods.30 We have also performedspin-polarized calculations not presented, but these give aneven worse result both in the position of the plasmon peakand the line shape relative to the experimental ones. Onepossible reason may be a fortuitous cancellation of errors inthe spin-restricted calculations. To improve these numbers

further, more sophisticated methods than standard DFT mustbe used, such as time-dependent DFT or Green’s functionbased approaches such as Hedin’s GW method49 includingelectron-hole interactions—this is beyond the scope of thecurrent work.

Thus, the main advantage of the DFT calculations, com-pared to the free electron model, is that also the line shapesof the absorption spectra, as well as the dielectric function,are relatively easily achieved. This is due to the former beingbased on the calculated band structure of the compounds.

B. Single electron transitions in the low energy-loss region andthe dielectric function

The intensity of the EEL spectrum associated with singleelectron transitions is given by50

I feik·ri2

= f1 + ik · r +ik · r2

2!+ ¯ i2

fik · ri2. 3

Here i and f designate the occupied initial and unoc-cupied final states, k is the momentum transfer from theincident electron to the excited electron, while r is the posi-tion of the electron with the atom core as the origin. If theproduct k ·r is sufficiently small, the higher powers of theTaylor series in Eq. 3 vanish, and the dipole approximationis to be considered valid. All observed transitions then obeythe dipole selection rule l= ±1. This situation is usuallyachieved by using a small entrance aperture for the spec-trometer, which limits the scattering vector k.

In core-loss spectroscopy, the initial states are atomic innature and are usually assumed to be highly localized inenergy. The spectral features are therefore mainly due tovariations in the density of empty states above the Fermilevel. However, we wish to examine the fine structure of thelow energy-loss region below the plasmons. In these experi-ments, both the initial and final states are located close to theFermi level and distributed in energy. The observed energyloss features are then related to a joint density of states. Weexpect this fine structure to be very sensitive to the bondingin the skutterudites. In principle, it is also possible to mea-sure the band-gap of materials using EELS. However, this ispractical only for wide band-gap semiconductors and insula-tors using systems with sub-eV energy resolution.51

Figure 5 shows the low energy-loss region of the threeskutterudites after deconvolution and with the zero loss peakremoved, together with the calculated spectra. Here, theoret-ical spectra calculated both with and without a 0.3 eVLorentzian broadening are shown. At the high energy side ofthese figures, we see the rising intensity of energy loss due tothe plasmon excitations. The plasmon peaks extend outsidethe figures.

On the low energy side of the plasmon peaks, there aretrailing shoulders in both the theoretical and experimentalspectra. These shoulders are associated with the low energysingle electron transitions close to the Fermi level. We notethat for all three materials this shoulder is rather featureless,

TABLE II. Experimental plasmon energy Epexp for the three skut-

terudites. We also list the plasmon energy calculated from theDrude model Ep

D and the plasmon energy found from the ELF basedon the dielectric function Ep

DFT. The relative deviation of thecalculated vs the experimental energy is also listed.

Epexp/eV Ep

D / eV EpDFT/eV

CoP3 21.67±0.08 20.78 −4.1% 20.9 −3.55%

CoAs3 20.34±0.11 18.95 −6.8% 19.9 −2.16%

CoSb3 16.88±0.09 16.37 −3.0% 17.9 +6.04%


245109-5

reflecting the uniform unoccupied density observed in Fig. 3.Regardless of this, the qualitative shape of this low energy-loss region varies somewhat in the three materials, and thisvariation is reasonably reproduced in the calculated spectra.Furthermore, for CoAs3 it is possible to identify individualfeatures in both the theoretical and experimental spectra. Inparticular, features in the energy-loss spectra just below10 eV and around 13 eV could represent dipole allowedtransition from the very sharp As s feature located approxi-mately 8–10 eV below the Fermi level to available stateswith p symmetry, see Fig. 3. Even though the details avail-able in the experimental spectra are rather limited in com-parison to the theoretical spectra, the overall correspondence

is good, indicating that the calculated electronic structurefairly accurately reflects the real bonding in the skutterudites.

At these low energies, the possible breakdown of the di-pole approximation needs to be considered. As explainedearlier, the DFT calculations only take into account electrontransitions satisfying the dipole selection rule l= ±1. How-ever, since we are considering transition of the outer shellelectrons, the product k ·r may approach or exceed unity asr is very large. This may lead to additional energy loss in-tensity in our spectra, and cause the dipole transitions to bepartially hidden by a “background” of nondipole transitions.However, we observe no obvious effects indicating a break-down of the dipole selection rule, except possibly a generalbroadening of the features.

Based on the good agreement between the experimentaland theoretical spectra, we wish to compare the theoreticaldielectric function with the dielectric function derived di-rectly from the experiments. The complex dielectric functionof a material describes the electrical response of the materialto external fields, and is usually obtained from optical ex-periments, but can also be retrieved from the low energy-lossspectra through deconvolution and subsequent analysis usingthe Kramers-Kronig relation.43 This approach has several ad-vantages over the conventional optical methods: the experi-ments can be performed on minute amounts of nontranspar-ent material and may yield the dielectric function over alarger range of frequencies.

FIG. 6. a The real and b imaginary parts of the dielectricfunction of CoAs3. The experimentally obtained function is shownwith a fully drawn line, while the theoretical function is a dashedline.

FIG. 5. The experimental fully drawn line, and theoreticalenergy-loss spectra of a CoP3, b CoAs3, and c CoSb3. Thetheoretical spectra have been calculated with dashed and withoutdotted a 0.3 eV Lorentzian broadening. The relative scaling of thetheoretical data with respect to the experimental data is arbitrary, nodirect comparison of the absolute intensity is possible.


245109-6

The Kramers-Kronig analysis successfully extracted thereal and imaginary parts of the dielectric function from ourexperimental data from CoAs3, but was less successful forthe other two materials. The experimental dielectric functionof the As-based skutterudite is shown in Fig. 6 together withthe theoretical dielectric function. Both the real and imagi-nary parts of the experimental dielectric function displaypeaks in the lower parts of the spectrum. These occur atapproximately 1.8 and 2.4 eV, respectively, which is in ex-cellent agreement with the DFT calculations that indicatemaxima at 1.4 and 2.5 eV. The real part of the experimentaldielectric function crosses zero at about 6.1 eV, which issignificantly higher than the DFT value of 2.9 eV. There isalso a significant difference in the slope of both the real andimaginary parts in the 4–10 eV region. Despite these differ-ences the overall agreement between experiment and theoryis reasonable, suggesting that using EELS to investigate thedielectric function and comparing to theoretical calculationsmay be an important tool in studying the electronic structureof novel materials.

IV. CONCLUSION

We achieve a reasonable agreement between the DFT cal-culated and EELS extracted dielectric functions in our studyof binary skutterudites. However, there are also significantdifferences. The cause of these differences is not likely to bethat our experiments are performed materials that deviatefrom the nominal compositions, nor defects in the materialthat disturbs the measurements. When it comes to details ofthe dielectric function there may be inaccuracies on the ex-perimental side in the procedures used to correct the spectrafor multiple scattering. However, EELS represents a highlyrobust and accurate method to measure the plasmon energy.Thus, the significant discrepancy between EELS measuredand DFT calculated plasmon energies may inspire to furtherimprovements of first-principle calculations. An interestingcharacteristic of EELS is that the technique can be used onnanometer sized volumes. Thus, plasmon energy measure-ments by EELS may become a valuable technique to testDFT calculations on nanomaterials, a parallel to use calori-metric data to test DFT calculations on bulk materials.52

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39 A. C. Larson and R. B. von Drele unpublished.40 W. Jeitschko, A. J. Foecker, D. Paschke, M. V. Dewalsky, C. B.

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WIEN2k, an Augmented Plane Wave+Local Orbitals Program forCalculating Crystal Properties Karlheinz Schwarz, TU Wien,Austria, 2001.

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Paper III

Ø. Prytz, J. Taftø, C. C. Ahn, and B. Fultz

Transition metal d-band occupancy in skutterudites studied by Electron

Energy-Loss Spectroscopy.


77

Transition metal d-band occupancy in skutterudites studied by electron energy-loss spectroscopy

Ø. PrytzCentre for Materials Science and Nanotechnology, University of Oslo, P.O. Box 1126 Blindern, N-0318 Oslo, Norway

J. TaftøDepartment of Physics, University of Oslo, P.O. Box 1048 Blindern, N-0316 Oslo, Norway

C. C. Ahn and B. FultzDivision of Engineering and Applied Science, California Institute of Technology, M/C 138-78, Pasadena, California 91125, USA

Received 18 August 2006; revised manuscript received 28 November 2006; published 13 March 2007

The transition-metal 3d occupancy of a series of thermoelectric skutterudites is investigated using electronenergy-loss spectroscopy. We find that bonding causes an emptying of the 3d states in the binary skutteruditesCoP3, CoAs3, CoSb3, and NiP3, while compared to the pure Fe the 3d occupancy in LaFe4P12 is significantlyincreased, consistent with the idea that each interstitial La atom rattler donates three electrons to compensatefor missing valence electron of Fe as compared to Co. These experimental results are in agreement withprevious models suggesting a predominantly covalent bonding between transition metal and pnictogen atomsin skutterudites, and provide evidence of charge transfer from La to the Fe-P complex in LaFe4P12.

DOI: 10.1103/PhysRevB.75.125109 PACS numbers: 79.20.Uv, 71.20.Be, 72.20.Pa

I. INTRODUCTION

In the search for new and more effective thermoelectricmaterials, the class of materials with the so-called skutteru-

dite crystal structure space group Im3 has received muchattention. These materials have the formula unit TMPn3,where TM is transition metal, usually one of the column 9elements Co, Rh, or Ir, while Pn is one of the pnictogens P,As, or Sb. The cubic unit cell of the skutterudites is made upof eight smaller primitive cubes of the transition metal, six ofwhich are filled by rectangular four-member rings of thepnictogen.1 This leaves two rather large voids in the center ofthe two remaining metal cubes.

The skutterudites are especially interesting due to theirtolerance for modification using other atomic species whileretaining their crystal structure. For example, filling of thevoids with heavy atomic species called “rattlers” has beenobserved to lead to dramatic reductions in the thermal con-ductivity of the material, which is beneficial for the thermo-electric properties. Furthermore, substitution on the transi-tion metal or pnictogen sites can control the electronicproperties. A virtual continuum of modified skutteruditesmay be formed, with compositions such asLnyTM4−xFexPn12. Here, Ln is one of the lanthanides, oftenLa or Ce, TM is one of the column 9 transition metals asabove, Pn is one of the pnictogens, and a column 8 transitionmetal such as Fe is introduced to maintain charge balance.2

Tailoring their electronic and thermal properties is thereforepossible, and modified skutterudite materials are indeedamong the most promising of a new generation of thermo-electric materials.2–4 Several theoretical5–8 andexperimental9–13 studies of the electronic structure of Co-based skutterudites have been performed. A particular inter-esting signature of the bonding and electronic structure is theoccupancy of the 3d states of Co. Anno et al.12 used x-rayphotoelectron spectroscopy XPS to study the occupiedstates of CoAs3 and CoSb3. They found evidence of a small

charge transfer from metal to pnictogen atoms and suggest ahybridization between the metal d and pnictogen p states. Inthis study, we use the complementary technique of electronenergy-loss spectroscopy EELS to investigate the effects ofbonding by probing the density of empty transition metal 3dstates above the Fermi level of CoP3, CoAs3, and CoSb3, aswell as NiP3 and the filled skutterudite LaFe4P12. The EELSanalyses are done in a probe forming transmission electronmicroscope TEM with 200 keV incident electrons. Theability to form a small probe with a TEM means that EELScan be used on small volumes of materials.

Electron energy-loss spectra are obtained by analyzing theenergy distribution of electrons transmitted through thesample. Some of these electrons will have lost energythrough inelastic scattering in the sample, for example, byexciting core electrons from their ground state. These exci-tations involve transitions of electrons from the occupiedcore levels into empty states above the Fermi level. Theprobability of a transition occurring is dependent on the spa-tial overlap of the initial and final states, effectively restrict-ing the transitions to final states centered on the same atomas the initial states. Since the initial core levels are highlylocalized in energy, these transitions are mainly sensitive tofinal-state effects. The occupancy of these final states maychange due to bonding through charge transfer or hybridiza-tion, and this will be reflected in the EEL spectrum. Thus,using EELS, we probe the local density of empty states ofthe material, potentially revealing a wealth of information onboth bonding and crystal structure.14–18

The spectra obtained in this study include sharp energy-loss features associated with dipole transitions of the typeTM 2p1/2→3d3/2 L2 and 2p3/2→3d5/2,3/2 L3. In the firstrow transition metals, these excitations take place below1000 eV, an energy range well suited for EELS studies. In asingle electron approximation, the intensity of these featuresis then given by

I u3d3deik·r2p2. 1

PHYSICAL REVIEW B 75, 125109 2007

1098-0121/2007/7512/1251096 ©2007 The American Physical Society125109-1

Here, u3d is the unoccupied density of states with tran-sition metal 3d symmetry, k is the momentum transferfrom the fast electron to the core electron, and r is the posi-tion of the electron with the atom core as the origin.

Previous studies of transition metals19,20 and theiroxides21 have revealed a linear relationship between the in-tensity of the energy-loss features associated with these tran-sitions called “white lines” and the 3d-band occupancy.This can be used to quantitatively determine the occupancyof the d states. The outer 3d electrons may participate inbonding through charge transfer and hybridization with wavefunctions from neighboring atoms. Electron energy-loss ex-periments probe the part of the molecular orbital which isprojected onto the atom with the original 2p state. Althoughone may not be able to distinguish between the effects ofcovalent or ionic bonding,22 important information on bond-ing, charge compensation, and hybridization upon compoundformation can still be revealed.16,23,24

The cobalt-based skutterudites are among those whichhave received most attention in studies of skutterudites asthermoelectric materials. Using EELS, we explore the localelectronic structure of cobalt in CoP3, CoAs3, and CoSb3,thereby investigating the bonding of cobalt in differentatomic environments. Furthermore, we study NiP3, a mate-rial which is unique in that it forms the skutterudite structurewith a noncolumn 9 transition metal.

Finally, we investigate the d-band occupancy of Fe in thefilled skutterudite LaFe4P12. Binary skutterudites do not formwith column 8 transition metals such as iron, as these lackthe outer 3d electrons needed to form the bonds that stabilizethe structure. However, when the so-called rattlers such as Laor Ce are introduced in crystal, Fe can be fully substitutedonto the Co sites. Thus, we wish to probe the local electronicstructure of iron in this material in order to reveal the mecha-nisms by which this material forms.

The spectra obtained from these materials are comparedto those of the pure metals, and the results are discussed interms of the bonding mechanisms of the skutterudites.

II. EXPERIMENTAL PROCEDURES AND DATA ANALYSIS

Samples of CoP3 and LaFe4P12 were synthesized using atin-flux technique,25 while the CoAs3, CoSb3, and NiP3 weresynthesized by direct reaction of the constituent elements insealed and evacuated silica tubes. Thin areas for TEM stud-ies were obtained by crushing the samples in an agate mortarand then deposited on a carbon film suspended on a coppergrid.

Electron transparent thin films of pure Fe and Ni wereprepared by thermal evaporation onto single-crystal rocksaltsubstrates. These films were then floated off the substrates inwater and picked up with copper TEM grids. The cobaltsample was prepared by mechanical grinding and subsequentelectrochemical polishing using a mixture of 95% methanoland 5% perchloric acid as an electrolyte.

The samples were studied in diffraction coupling mode ina Philips EM 430 TEM operated at 200 kV, fitted with aGatan 666 parallel-detection electron energy-loss spectrom-eter. The spectrometer collects data only in 1022 channels

simultaneously, thus limiting the energy range of each spec-trum at high dispersion settings. Low-loss and core-lossspectra were therefore obtained separately. Between five andten pairs of spectra were obtained at several different loca-tions on each sample.

The composition of all the samples was verified usingenergy dispersive x-ray analysis, while the crystal structurewas checked using electron diffraction. Furthermore, thesamples were investigated for oxidation using the oxygen Kedge in the EEL spectra. No significant traces of oxygenwere found.

The spectra were corrected for the spectrometer darkcount, and each spectrum was divided by the spectrometerresponse function, obtained by uniformly illuminating thedetector with electrons. The spectrometer was set to a 0.5 eVdispersion, and each spectrum was obtained as shifted 1.5 eVfrom the previous spectrum to average out gain variations inthe photodiode array. The energy resolution of these experi-ments was approximately 1.5–2.0 eV, as determined by thefull width at half maximum of the zero-loss peak. The back-ground below the core loss edge was removed using apower-law model of the form Ae−BE fitted to a 30 eV win-dow of the pre-edge intensity,26 as seen in Fig. 1. Further-more, the spectra were deconvolved using the Fourier-ratiomethod26 with the obtained low-loss spectra as input. Thespectra were then aligned and summed. No attempt wasmade to measure any chemical shift in the onset of the tran-sition metal white lines.

The white-line intensity was obtained by integrating thenumber of counts over a region which for the cobalt contain-ing samples extended typically 26.5 eV beyond the edge on-set. For the samples containing Fe and Ni, the white-lineintegration regions were approximately 22 and 28 eV, re-spectively. The contribution of the 2p→continuum transi-tions was estimated by a linear function extending from theedge onset to the start of the postedge continuum. This con-tribution was then subtracted from the edge integral. Theresulting intensity was then normalized to a 50 eV window50 eV beyond the edge onset see Graetz et al. for details.21

In the dipole approximation, only transitions resulting in achange of the angular momentum quantum number of l

FIG. 1. Example of how a fitted power-law model was used toremove the background contribution at the Fe L2,3 edge.

PRYTZ et al. PHYSICAL REVIEW B 75, 125109 2007

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= ±1 are allowed. For 2p electrons, the dipole allowed tran-sitions are then 2p→4s and 2p→3d. Although nondipoletransitions occur, it has been shown that the contribution ofthese transitions becomes significant only several hundredeV above the L2,3 edge onset.27 Furthermore, dipole allowedtransitions to the 4s states are not expected to contributesignificantly,28 and we consider only transitions of the type2p→3d.

Under these assumptions, the measured and normalizedintensity is then given by20

I u3d

E0

E1

3dr2p2d

E0+50

E0+100

dr2p2d

= u3dMwhite line

Mcontinuum.

2

Here, u3d is the density of unoccupied TM 3d statesand d is a continuum state with energy and angular mo-mentum quantum number l=2. The numerator 3dr2p2 isthe transition probability from the 2p to 3d states, while thedenominator is the same for transitions to the continuum.The ratio of the matrix elements Mwhite line /Mcontinuum wascalculated by Pearson et al. in an earlier work.20 The totalTM 3d occupancy n3d can now be obtained through the re-lation

n3d = 10 − u3d = 10 − IMcontinuum

Mwhite line. 3

The statistical error in our experiments is small. We reportstandard deviations in the measured 3d occupancy less than0.16 electrons/atom, and in some measurements as low as0.01 electrons/atom. However, a higher uncertainty is asso-ciated with the absolute value of the d occupancy. This un-certainty is primarily introduced by the different possibleboundary choices for the white-line integration and themethod of continuum subtraction. Graetz et al.21 estimatethat this error is approximately ±0.4 electrons/atom. How-ever, the relative errors are small, and given the observedstandard deviations, we expect the relative changes in d oc-cupancy we obtain to accurately reflect changes in the elec-tronic structure upon compound formation.

III. RESULTS

Normalized spectra from pure Co and the three Co-basedskutterudites are shown in Fig. 2. The background subtrac-tion in the case of CoSb3 was complicated by the delayededge of the Sb M4,5 transitions, which extends several hun-dred eV beyond their onset at 528 eV. Furthermore, the SbM2,3 edges at 766 and 812 eV add to the complications ofsubtracting the background. This causes an apparently pro-tracted onset of the Co L2,3 edges for CoSb3, while the otherthree compounds display a rather sharp onset.

The spectra display a large increase in the white-line in-tensity of CoP3 and CoAs3 compared to that observed in pureCo. The L3 peak of CoSb3 is slightly less intense than the L3

peak of pure cobalt, while the L2 peak is more intense. Weattribute this difference in ratio to the mentioned problemswith background subtraction. The L2,3 splitting is approxi-mately 14.5 eV for all the Co-based compounds.

Using the observed white-line intensities and Eq. 3, withthe matrix element correction factor given by Pearson etal.,20 we calculate the 3d occupancy of the studied com-pounds. These are given in Table I. We see that the 3d occu-pancy of pure Co is 8.3 electrons, suggesting that the el-emental cobalt is close to a 3d84s1 valence state, consistentwith the assumptions made by Pearson et al. In comparison,the Co 3d orbitals of the skutterudites are significantly de-pleted, displaying occupancies of 7.53, 7.90, and 8.02 elec-trons for CoP3, CoAs3, and CoSb3, respectively.

As mentioned above, Sb M transitions extend beyond theCo L2,3 onset. This causes an imperfect fit of the power lawto the background below the L2,3 peaks. To estimate the ef-fect of this imperfect fit, we apply a power-law backgroundmodel to a spectrum from pure Sb data from the EELSAtlas30. When fitted to the same 30 eV window used in theanalysis of CoSb3, we are left with a nonuniform residual.

FIG. 2. Normalized EEL spectra of the Co L2,3 edge of purecobalt and the three Co-based skutterudites.

TABLE I. Average observed transition metal 3d occupancyn3d for the pure metals and the skutterudites. We also list thechange of occupancy in the skutterudite vs the appropriate puretransition metal n3d. The final column is the relevant TM-pnictogen electronegativity difference in Pauling units Ref.29. A negative value signifies the higher electronegativity of thepnictogen relative to the TM.

Compound TM n3d n3d

Co 8.30±0.07

CoP3 7.53±0.05 −0.77 −0.31

CoAs3 7.90±0.12 −0.40 −0.30

CoSb3 8.02±0.15 −0.28 −0.17

CoSb3 with correction 7.89±0.16 −0.41 −0.17

Ni 8.71±0.01

NiP3 8.59±0.09 −0.13 −0.28

Fe 6.87±0.06

LaFe4P12 7.66±0.11 +0.79 −0.36, +0.73 Fe-La

TRANSITION METAL d-BAND OCCUPANCY IN… PHYSICAL REVIEW B 75, 125109 2007

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The effect of this residual is to underestimate the white-lineintensity relative to the continuum, thereby overestimatingthe 3d occupancy. To correct this error, we use the data fromthe EELS Atlas to estimate the relative error in the back-ground fit. This correction is then applied to the CoSb3 data,reducing the 3d occupancy by approximately 0.1 electrons/atom. Although this correction may be reasonable, it relieson the assumption that the Sb M edges are unaltered inCoSb3 as compared with the pure Sb. These correctionsshould therefore be treated with some care. However, weexpect the Sb 3d→continuum transitions to dominate, andthe edge will therefore be unaltered in the compound com-pared to the pure element.

The normalized spectra for Ni and NiP3 are shown in Fig.3. We note that the white-line intensity of these compoundsare significantly reduced compared to those of the Co-basedcompounds, corresponding to a filling of the 3d states due tothe extra Ni electron. The 3d occupancy of nickel obtainedfrom our experiments is 8.71 electron/atom, which is fairlyclose to the expected value with Ni in a 3d94s1 valence state.Although the spectral features of the NiP3 and pure metalseem quite different, the obtained 3d occupancy of the skut-terudite is only slightly reduced compared to the pure metal.The 3d occupancy of NiP3 is found to be 8.59 electrons/atom, a depletion of only 0.13 electrons relative to the puremetal see Table I.

The most dramatic effect is observed for the white-lineintensity of iron in the filled skutterudite LaFe4P12. Figure 4shows the normalized spectrum for this compound togetherwith that of pure iron. We immediately note that the white-line intensity of the pure metal is significantly larger thanthat observed in the Co- and Ni-based compounds. The3d-occupancy obtained for pure iron is 6.9 electrons/atom, inexcellent agreement with the expected 3d74s1 valence con-figuration. In comparison, the Fe L2,3 lines of LaFe4P12 aresignificantly decreased, signaling a filling of the 3d statescorresponding to +0.77 electrons/atom relative to the puremetal. This filling gives a total 3d occupancy of iron in theskutterudite of 7.66 electrons/atom, close to that observed forcobalt in the binary skutterudites.

IV. DISCUSSION

The results presented in the previous section suggest sig-nificant changes in the local electronic structure of the tran-

sition metals upon formation of the binary skutterudites. Inthe cobalt-based skutterudites, the depletion of the d bandranges from −0.77 to −0.28 electrons/atom. However, theseresults do not give an indication whether the cause of thisdepletion is charge transfer from cobalt to the pnictogens orhybridization of outer valence states increasing the amountof d character above the Fermi level. We observe a generalcorrelation between the difference in electronegativity and the changes in d-band occupancy see Table I. Thissuggests that there may be an ionic contribution to the bond-ing of the skutterudites. Furthermore, Anno et al.12 foundsmall chemical shifts in the TM 2p binding energy of CoAs3and CoSb3 which may further indicate charge transfer fromCo to the pnictogen, and thus an ionic component to thebonding. Our results are consistent with these findings.

However, the details of our experimental results show aless straightforward relationship. For example, is verydifferent for CoAs3 and CoSb3, but the observed depletion isvirtually the same. Furthermore, only a small depletion isobserved in NiP3, even though is only slightly less thanin CoP3. A purely ionic picture based on considerations ofelectronegativity is therefore not sufficient for a systematicdescription of bonding in the skutterudites. Indeed, there isample evidence of strong covalent bonding between the tran-sition metals and pnictogens. First, the difference in elec-tronegativity between the Pn and TM atoms is 0.3 or less. Incomparison, other compounds usually considered to exhibitionic bonding e.g., CoO display a difference of 1.5 ormore. We, therefore, do not expect any significant ionic char-acter in the bonding based on the electronegativity values.Second, the observed TM-Pn bond length is close to the sumof the elemental covalent radii.2 We, therefore, expect theTM-Pn bonding in the skutterudites to be mainly covalent innature, with only a small degree of charge transfer betweenthe elements.

The most frequently used model for covalent bonding inskutterudites is due to Dudkin.31 In skutterudites, the transi-tion metal is octahedrally coordinated by pnictogen atoms,giving rise to octahedral d2sp3 hybrid orbitals. This allowsthe transition metal to form strong bonds with the six sur-rounding pnictogen atoms. The Dudkin model is generally

FIG. 3. Normalized spectra from pure nickel and NiP3. FIG. 4. Normalized spectra from the pure iron and LaFe4P12

samples. The sharp absorption peaks at high energy side are fromthe La M4,5 transitions.


125109-4

confirmed by site- and symmetry-projected density of statesfrom DFT calculations. Typically, a large overlap betweenthe TM 3d and Pn 3p states is observed,7,9,32 indicatingbonding through hybridization.

The Dudkin model successfully predicts that the binaryskutterudites with column 9 transition metals Co, Rh, andIr should be diamagnetic semiconductors. The skutteruditestructure does not generally form if the these transition met-als are substituted by those of column 8 or 10. An exceptionto this is the binary NiP3, which in the Dudkin model has an“excess” of one electron, thereby giving metallic conduction.This is confirmed by experimental33 and theoretical6 investi-gations.

Our experiments suggest that this hybridization increasesthe amount of available d-character above the Fermi level forthe Co-based skutterudites. This effect is largest in CoP3where the change from pure cobalt is −0.77 electrons/atom,giving a total 3d occupancy of 7.53 electrons/atom. In com-parison, the d-band occupancy of NiP3 is only slightly re-duced compared to the pure metal, giving a total occupancyof 8.59 electrons. This depletion is far less than what weobserve in CoP3, but it is interesting to note that the totaloccupancy in the nickel-based skutterudite is raised almostexactly one electron/atom above that of the cobalt-basedcompound. This suggests that the excess electron introducedto the structure is quite accurately described in a rigid bandapproach.

An experimental comparison with FeP3 is not possible asthis compound does not form. In the Dudkin model, it isconsidered electron deficient, as iron lacks one electron toform the hybrid d2sp3 orbitals required to stabilize the tran-sition metal in an octahedron of pnictogens. However, fillingthe voids in the skutterudite structure with electropositiveelements such as La allows the formation of filled skutteru-dites with iron substituted for cobalt. This is understood interms of La in a 3+ valence state after donating the necessaryelectrons to saturate the Fe-P bond sufficiently to stabilizethe structure. This would entail a transfer of charge awayfrom La into the bonding region between Fe and P, occupy-ing the hybrid bonding orbitals.

In LaFe4P12, we observe a filling of states with Fe 3dcharacter relative to pure iron. The filling is rather large,corresponding to +0.79 electron/atom giving the greatly re-duced Fe white lines observed in Fig. 4. This filling raisesthe d-band occupancy of iron up to the level observed forcobalt in CoP3, and the two compounds are in a sense locallyisoelectronic. Furthermore, if we assume that each La atomdonates three electrons to the four nearest Fe atoms, this

would correspond to a change of +0.75 electrons/iron atom.This is very close to the observed value of +0.79 electrons/atom and indicates that the charge compensation takes placesolely on the Fe sites.

Although the d-band occupancies we observe in the skut-terudites are consistent with the Dudkin model, further ex-perimental studies and theoretical investigations using band-structure calculations are needed to understand the bondingmechanisms in the skutterudites. In particular, energy-lossedges related to the pnictogens should be investigated, andchanges in the calculated local density of states DOS of Niin NiP3 and Fe in LaFe4P12 should be compared to the ex-perimental results of this work. Also, further experimentalstudies using EELS with higher-energy resolution would beuseful to further investigate hybridization effects in thesematerials.

V. CONCLUSION

We have used electron energy-loss spectroscopy to studythe unoccupied density of states of a series of binary skut-terudites. These studies reveal a significant increase in the CoL2,3 white lines as compared to pure Co metal, indicating anemptying of the Co 3d states upon formation of the skutteru-dite compounds. This is consistent with the conclusions ofAnno et al.,12 who investigated the occupied states below theFermi-level, while we have probed the unoccupied statesabove the Fermi level. Since the Co-Pn bonding in thesematerials is predominantly covalent as suggested byDudkin,31 we attribute this effect mainly to d2sp3 hybridiza-tion causing an increase in the number of empty states with dcharacter. In comparison, only a small emptying of d states isobserved for NiP3.

Furthermore, our experimental results on LaFe4P12 indi-cate a filling of the Fe 3d states, allowing iron to form thed2sp3 hybrid states needed to stabilize the skutterudite struc-ture. This supports previous models of bonding where chargeis transferred from the La atoms to the Fe-P complex andindicates that the charge compensation mainly takes place onthe Fe sites.

ACKNOWLEDGMENTS

The authors would like to acknowledge support from theUniversity of Oslo through the FUNMAT@UiO programand the U.S. DOE through Grant No. DE-FC36-05GO15065.One of the authors Ø.P. would also like to thank Shu Miaofor help with the TEM and spectrometer during his stay atCaltech.

1 A. Kjekshus and G. Pedersen, Acta Crystallogr. 14, 1065 1961.2 C. Uher, Semicond. Semimetals 69, 139 2001.3 G. S. Nolas, D. T. Morelli, and T. M. Tritt, Annu. Rev. Mater. Sci.

29, 89 1999.4 G. Chen, M. S. Dresselhaus, G. Dresselhaus, J. P. Fleurial, and T.

Caillat, Int. Mater. Rev. 48, 45 2003.

5 D. J. Singh and W. E. Pickett, Phys. Rev. B 50, 11235 1994.6 M. Llunell, P. Alemany, S. Alvarez, V. P. Zhukov, and A. Vernes,

Phys. Rev. B 53, 10605 1996.7 M. Fornari and D. J. Singh, Phys. Rev. B 59, 9722 1999.8 O. M. Løvvik and Ø. Prytz, Phys. Rev. B 70, 195119 2004.9 Ø. Prytz, O. M. Løvvik, and J. Taftø, Phys. Rev. B 74, 245109

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2006.10 G. J. Long, B. Mahieu, B. C. Sales, R. P. Hermann, and F. Grand-

jean, J. Appl. Phys. 92, 7236 2002.11 I. Devos, M. Womes, M. Heilmann, J. Olivier-Fourcade, J. C.

Jumas, and J. L. Tirado, J. Mater. Chem. 14, 1759 2004.12 H. Anno, K. Matsubara, T. Caillat, and J. P. Fleurial, Phys. Rev. B

62, 10737 2000.13 I. Lefebvre-Devos, M. Lassalle, X. Wallart, J. Olivier-Fourcade,

L. Monconduit, and J. C. Jumas, Phys. Rev. B 63, 1251102001.

14 R. D. Leapman, L. A. Grunes, and P. L. Fejes, Phys. Rev. B 26,614 1982.

15 J. Tafto and O. L. Krivanek, Phys. Rev. Lett. 48, 560 1982.16 T. I. Morrison, M. B. Brodsky, N. J. Zaluzec, and L. R. Sill, Phys.

Rev. B 32, 3107 1985.17 R. F. Klie, Y. Zhu, G. Schneider, and J. Tafto, Appl. Phys. Lett.

82, 4316 2003.18 L. A. J. Garvie and P. R. Buseck, Am. Mineral. 89, 485 2004.19 D. H. Pearson, B. Fultz, and C. C. Ahn, Appl. Phys. Lett. 53,

1405 1988.20 D. H. Pearson, C. C. Ahn, and B. Fultz, Phys. Rev. B 47, 8471

1993.21 J. Graetz, C. C. Ahn, H. Ouyang, P. Rez, and B. Fultz, Phys. Rev.

B 69, 235103 2004.

22 V. J. Keast, A. J. Scott, R. Brydson, D. B. Williams, and J. Bru-ley, J. Microsc. 203, 135 2001.

23 D. H. Pearson, C. C. Ahn, and B. Fultz, Phys. Rev. B 50, 129691994.

24 H. Ouyang and K. Jen-Tai, J. Appl. Phys. 92, 7510 2002.25 A. Watcharapasorn, R. C. DeMattei, R. S. Feigelson, T. Caillat,

A. Borshchevsky, G. J. Snyder, and J. P. Fleurial, J. Appl. Phys.86, 6213 1999.

26 R. F. Egerton, Electron Energy Loss Spectroscopy in the ElectronMicroscope, 1st ed. Plenum, New York, 1986.

27 R. D. Leapman, P. Rez, and D. F. Mayer, J. Chem. Phys. 72,1232 1980.

28 D. A. Muller, D. J. Singh, and J. Silcox, Phys. Rev. B 57, 81811998.

29 P. W. Atkins, Quanta: A Handbook of Concepts, 2nd ed. OxfordUniversity Press, Oxford, 1991.

30 C. C. Ahn and O. L. Krivanek, EELS Atlas Gatan, Warrendale,1983.

31 L. D. Dudkin, Sov. Phys. Tech. Phys. 3, 216 1958.32 K. Koga, K. Akai, K. Oshiro, and M. Matsuura, Phys. Rev. B 71,

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125109-6

Paper IV

S. Diplas, Ø. Prytz, O. B. Karlsen, J. F. Watts, and J. Taftø

A quantitative study of valence electron transfer in the skutterudite compound

CoP3 by combining x-ray induced Auger and photoelectron spectroscopy.

Journal of Physics: Condensed Matter 19, 246216 (2007)

85

IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 19 (2007) 246216 (13pp) doi:10.1088/0953-8984/19/24/246216

A quantitative study of valence electron transfer in theskutterudite compound CoP3 by combining x-rayinduced Auger and photoelectron spectroscopy

S Diplas1,2,4, Ø Prytz1, O B Karlsen2, J F Watts3 and J Taftø2

1 Centre for Materials Science and Nanotechnology, University of Oslo, PO Box 1126, Blindern,NO-0318 Oslo, Norway2 Department of Physics, University of Oslo, PO Box 1048, Blindern, NO-0316 Oslo, Norway3 The Surface Analysis Laboratory, School of Engineering, University of Surrey, Guildford,Surrey GU2 7XH, UK

E-mail: [email protected]

Received 15 March 2007, in final form 4 May 2007Published 24 May 2007Online at stacks.iop.org/JPhysCM/19/246216

AbstractWe use the sum of the ionization and Auger energy, the so-called Augerparameter, measured from the x-ray photoelectron spectrum, to study thevalence electron distribution in the skutterudite CoP3. The electron transferbetween Co and P was estimated using models relating changes in Augerparameter values to charge transfer. It was found that each P atom gains0.24 e−, and considering the unit formula CoP3 this is equivalent to a donationof 0.72 e− per Co atom. This is in agreement with a recent electron energy-lossspectroscopy study, which indicates a charge transfer of 0.77 e−/atom from Coto P.

1. Introduction

Compounds with the skutterudite-type structure have attracted much attention over the lastdecade, owing to their possible use as thermoelectric materials. These compounds have thegeneral formula TX3, with TM being a transition metal (typically Co, Rh, Ir) and X one of thepnicogens P, As or Sb. The skutterudites belong to the cubic space group Im3 (Kjekshus andPedersen 1961, Kjekshus and Rakke 1974); the metal atoms are octahedrally coordinated bythe pnicogen atoms, while the pnicogens have two metal and two pnicogen nearest neighboursin a tetrahedral environment.

In an effort to understand the electronic properties and atomic bonding in binaryskutterudites, several band structure calculations have been performed (Llunell et al 1996,Singh and Pickett 1994, Fornari and Singh 1999, Løvvik and Prytz 2004). X-ray photoelectron

4 Author to whom any correspondence should be addressed.

0953-8984/07/246216+13$30.00 © 2007 IOP Publishing Ltd Printed in the UK 1

http://dx.doi.org/10.1088/0953-8984/19/24/246216

mailto:[email protected]

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J. Phys.: Condens. Matter 19 (2007) 246216 S Diplas et al

spectroscopy (XPS) studies of transition metal monophosphides, LaFe4Sb12, CeFe4Sb12,CoAs3, CoSb3 and RhSb3 (Anno et al 2000, Lefebvre-Devos et al 2001, Nemoshalenko et al1983, Grosvenor et al 2005, 2006) employed photoelectron peak shifts and shapes, energy lossfeatures and valence band spectra to study charge transfer and bonding phenomena. Smallchemical shifts in the core binding energies of both the transition metals and pnicogens wereobserved for the antimonides and arsenides, indicating limited charge transfer between theconstituent elements. This is consistent with the small difference in the electronegativity ofthese elements. A predominantly covalent bonding scheme is therefore assumed, in whichthe transition metal is bonded to its octahedrally coordinated pnicogens through d2sp3 hybridorbitals, while the bonds between the pnicogens have sp3 hybrid character (Uher 2001, Dudkin1958, Kjekshus and Pedersen 1961). This bonding scheme is in qualitative agreement withband structure calculations (Koga et al 2005, Fornari and Singh 1999), while somewhat largershifts for the phosphides indicate bonding of more ionic nature (Grosvenor et al 2006).

Recently, studies using electron energy-loss spectroscopy (EELS) indicated an emptyingof the Co 3d states in CoP3, CoAs3 and CoSb3 relative to that of the pure metal (Prytz et al2007). The effect was largest for CoP3, showing a reduction of approximately 0.77 e−/atom,while smaller changes were observed for CoAs3 and CoSb3 (∼0.4 e−/atom). However, EELSprobes a dipole-selected local density of empty states, and is therefore sensitive to both chargetransfer away from the Co atoms, and hybridization effects causing a change in the degree ofd-character of the valence electrons (Keast et al 2001). In an effort to deconvolute these effects,we compare these EELS results with results from XPS and x-ray induced Auger electronspectroscopy (XAES) which probe occupied energy states. XPS and XAES are based on theexcitation of core levels, and although core electrons are not directly involved in bonding theyrespond to changes in the atomic environment (and to charge transfer/redistribution phenomena(a)) via energy shifts of their spectral peaks. We characterize the CoP3 compound produced ina Sn flux with SEM and XRD, and we supplement XPS with XAES to probe the bondingin the CoP3 crystals. In addition to the common practice of using core-level energy shifts,we monitor changes in the P 2p–Co 2p energy separation and employ the Auger parameter inthe Thomas and Weightman model (1986) in order to probe atomic bonding and charge transferphenomena. The results are discussed and compared to the literature data of pure Co, P (white),Co–P compounds and other transition metal phosphides.

2. Methods and materials

2.1. Synthesis and characterization

Single crystals of CoP3 were grown using a Sn-flux technique (Watcharapasorn et al 1999).Cobalt filings (Goodfellow 99.9%), pieces of red phosphorus (Koch-Light 99.999%), andtin granules (Fluka 99.999%) were loaded into a silica glass ampoule in the atomic ratioCo:P:Sn = 1:3:25, and the ampoule was evacuated and sealed. The ampoule was kept at780 C for one week before being slowly cooled to room temperature.

The resulting ingot was cut into two vertical sections. Light microscopy and scanningelectron microscopy (SEM) combined with energy-dispersive x-ray analysis (EDX), showedthat the ingot contained single crystals of CoP3 dispersed in a Sn matrix (see figure 1(a)). Noother phases were detected. SEM showed that the crystals (up to 300 μm in diameter) all hada composition consistent with CoP3 (see figure 1(b)). Part of the unused ingot was immersedin dilute HCl (water:HCl = 1:1), which dissolved the Sn matrix, leaving the CoP3 crystalsun-attacked (see figure 1(b)). X-ray diffraction (XRD) was performed with a Siemens D-5000 diffractometer in Bragg–Brentano geometry using Cu Kα1 radiation. The XRD data were

2


Figure 1. SEM images from (a) CoP3 single crystals embedded in the Sn flux, and (b) example ofa single crystal after etching away the Sn flux as described in the text.

Table 1. The cell parameter a of CoP3 obtained in this study, compared with that of previousworkers.

Cell parameter, a (pm)

This work 770.8Watcharapasorn et al (1999) 770.73Jeitschko et al (2000) 770.5

refined using the general structure analysis system (GSAS) (Larson and Von Dreele 2004) withthe interface EXPGUI (Toby 2001). The XRD analysis showed that the synthesized crystals,shown in figure 1(b), consisted of a single phase, with a diffraction pattern corresponding tothe cubic CoP3 skutterudite. The cell parameter obtained, a, together with that reported byprevious workers, is shown in table 1.

XPS and XAES were performed on CoP3 crystals embedded in the Sn matrix usinga VG Scientific ESCALAB Mk II fitted with a Thermo Electron Corporation Alpha 110hemispherical analyser and Mg Kα radiation (hν = 1253.6 eV). Survey and high-resolutionspectra were acquired at pass energies of 100 and 20 eV respectively. Several areas (crystals)were analysed to ensure reproducibility of results. The use of the continuous Bremsstrahlungradiation generated by the 12 keV electrons entering the target enabled excitation of P 1selectrons with accompanying emission of P KLL Auger electrons (Castle and West 1979,1980). The surface of the sample was analysed after Ar ion etching (4 keV) for 6 min untilthe C 1s and O 1s signals were minimized. By monitoring the Co and P 2p as well as theC and O 1s peak intensity during etching we were confident that etching did not affect thesample stoichiometry, but it only diminished the surface contamination. Data processing wasperformed using the CasaXPS software (www.casaxps.com).

2.2. Auger parameter and charge transfer calculations

A common practice to reduce energy referencing effects utilizes binding energy shifts betweentwo different chemical environments. This shift (E) can be expressed as

E(i) = U − R. (1)

3

http://www.casaxps.com
















The first term expresses initial-state contributions arising from the dependence of the potentialto the changes in valence charge as well as from the Coulomb interaction between thephotoelectron and the surrounding charged atoms. R is the final-state contribution expressedas the relaxation energy change arising from the response of the atomic and extra-atomicenvironment to the screening of the core hole. The higher the electronic polarizability ofthe surrounding electron cloud is, the larger the relaxation energy becomes. However, thebest practice to eliminate energy referencing involves measurement of the energy separationbetween two different spectral features on the same spectrum and subsequent comparison of thisseparation between different chemical environments. This was the initial concept of the Augerparameter (α) as introduced by Wagner (1975) and modified by Gaarenstroom and Winograd(1977), who showed that energy referencing problems are completely removed by using thesum of the ionization energy (I ) and the kinetic energy of the Auger electron (K ) involving thesame primary excitation. They defined therefore the modified Auger parameter α′ as follows:

α′ = I + K . (2)

The difference in the Auger parameter between two different states 1 and 2 is given by

α′ = α′1 − α′

2 = I + K . (3)

The above definition of α′ separates final-state from initial-state effects, and it is knownas the final-state Auger parameter. By assuming all core level shifts between two differentenvironments to be the same, it can be shown that the changes in the final-state Auger parameterα′ are equal to twice the change in relaxation energy (R):

α′ = 2R = 2(Rval + Rea) (4)

where Rval is the change in R due to differences in the number of final-state valenceelectrons and Rea is the contribution due to changes in extra-atomic relaxation (Moretti1998). In general, large α′ values are observed in metals and semiconductors due to superiorelectron screening, while smaller ones are observed for insulators. Equation (4) is not reliablefor transition metals (Kleiman and Landers 1998), where d–s interband charge transfer has tobe taken into account.

Thomas and Weighman (1986) have developed a model relating Auger parameter changesto initial-state atomic charge transfer, electron screening and polarization of the surroundings:

α′ = [q( dk/ dN) + (k − 2 dk/ dN)( dq/ dN) + ( dU/ dN)] (5)

where k is the change in core potential when a valence atom is removed, q is the valencecharge, N is the occupancy of core orbitals and U represents the contribution from thechemical environment. The above model assumes that k and q depend linearly on N . Furtherdevelopment of the model has taken into account the dependence of k on the valence charge(Cole et al 1994, Cole and Weightman 1994).

3. Results and discussion

3.1. P and Co 2p core level and Co–P bond ionicity

Figure 2 shows the Mg Kα excited P 2p and 2s peaks together with their plasmon peaks. The Sn4s peak is also shown, since the CoP3 crystals were embedded in the Sn matrix (figure 1(a)) andthe irradiated area is large enough (a few mm2) to pick up signal from the Sn matrix surroundingthe CoP3 crystals. Small-area XPS was not performed because the use of a monochromatorwould not allow for the Bremsstrahlung induced excitation of the P KLL. The peak–plasmonseparation was found to be 22.2±0.5 and 22.5±0.5 eV for the P 2p and P 2s peaks respectively.

4


Figure 2. P 2p, P 2s high-resolution XPS spectra of CoP3 and the associated plasmons. The Sn 4sof the Sn matrix is also shown.

These values are in good agreement with the plasmon energy of 21.7 eV obtained by electronenergy-loss spectroscopy (Prytz et al 2006).

The Co and P 2p binding energy values are listed in table 2 together with the literaturevalues (Wagner et al—NISTdatabase). We observe a shift towards lower binding energiesfor both Co and P 2p peaks, compared to the binding energies in the elemental solids. Theshift for P 2p was found to be −0.8 eV compared with pure P whilst the shift for Co 2p wassmaller (−0.4 eV). In comparison, previous studies of CoP (Grosvenor et al 2005) and Co2P(Nemoshalenko et al 1983) report virtually zero or small (−0.2 eV) shift in the Co 2p bindingenergy of CoP and Co2P respectively, while the binding energy of P 2p was lowered by 0.5 eVin CoP and 0.6 eV in Co2P. For CoP3, Grosvenor et al monitored a positive shift (+0.4 eV) forCo 2p in CoP3 compared to pure Co, accompanied with a significant reduction (−0.7 eV) of theP 2p considering white P in elemental form (Grosvenor et al 2006). We attribute the oppositeshifts of Co 2p in CoP3 between the present study and that of Grosvenor et al (2005) to energyreferencing issues. In measuring the chemical shift we used the literature value for pure Co2p and this could well be the reason for the discrepancy. Absolute energy values depend onthe sample/spectrometer work function, and this can be important when, for example, differentspectrometers are used. For this reason we focus our analysis on the use of the Auger parameterwhich, as explained in section 2.2, is completely free of energy referencing inconsistencies. ForCoSb3 no shift in Co 2p and a very small one (−0.1 eV) for Sb were observed. Grosvenor et al

5


Table 2. Peak positions of Co and P 2p obtained from CoP3 and literature values. Photoelectronpeaks are given in binding energy.

Compound Co 2p3/2 (eV) P 2p (eV) Eba (eV)

CoP3 777.9 129.2 648.7Pure Co (lit) 778.3Pure P white (lit) 130.0CoP (lit) 778.4 129.5 648.9Co2P (lit) 778.2 129.4 648.8Phosphidesb 129.1Co oxides, sulfidesc 780.5Co and P ionicd 651.4Co and P covalente 648.3

a Eb: difference in Co 2p and P 2p binding energy.b Average of P 2p literature values of various phosphides (http://srdata.nist.gov/xps/).c Average of Co 2p literature values of various oxides and sulfides (http://srdata.nist.gov/xps/).d Subtraction of P 2p values in b from Co 2p values in c.e Subtraction of P 2p value of pure P from the Co 2p value of pure Co.

interpreted that as evidence for a greater covalent character of the antimonides compared tophosphides. In their study the ionic character of the Co–P bond in CoP3 was further supportedby monitoring the intensity of the Co 2p energy loss peak (Grosvenor et al 2006). The observedreduction of the intensity of the loss feature in CoP3 compared to pure Co was interpreted as areduction of the Co valence electrons. Their interpretation differs from that of Hillebrecht et alfor the Ni 2p shake up satellite (Hillebrecht et al 1983), who attributed its presence to a highdensity of unoccupied states just above EF and the sharp decrease in satellite intensity to thefilling of the Ni 3d band. The use of unmonochromated Mg Kα x-ray radiation did not allowus to measure accurately the intensity ratio between the satellite and the Co 2p main peak, dueto the overlap of the x-ray satellite (Mg Kα4) with the shake up satellite. The x-ray satellite ofCo 2p1/2 appears at ∼5–6 eV higher than the Co 2p3/2 peak. This is approximately the energyregime of the occurrence of the Co 2p3/2 shake up (or plasmon) satellite. An attempt to subtractthe x-ray satellite gave an Isat/ICo 2p value of 0.06, which is close to the 0.1 reported for CoP3

by Grosvenor et al (2006).The difference in electronegativity between Co and P is not large: cobalt has an

electronegativity of 1.88 in Pauling units, while for phosphorus the value is 2.19 (Atkins 1991).These values would suggest charge transfer from Co towards P, which should manifest itself inthe XPS spectrum as a shift of the Co 2p peak to higher binding energies and that of P 2p tolower binding energies. Thus, the Co 2p–P 2p binding energy separation in ionic compoundsshould be larger than in covalent compounds. As shown in figure 3 and table 2, the Co 2p–P2p energy separation of the CoP3 compounds is higher than that obtained considering pure Coand P (covalent bonding), and comparable to previously reported values of CoP and Co2P. Theenergy separation for CoP3 and other Co–P compounds in figure 3 is significantly smaller thanwhat is observed in very ionic environments such as Co in for example oxides and sulfides andP in other phosphides. Therefore the Co–P bonding should not be considered as strongly ionic,but rather as covalent with a partial ionic character. Previous studies on CrP, MnP, FeP andCoP showed a decreased ionicity of the metal–phosphorus bond on progressing from CrP toCoP. In addition, experimental valence band spectra coupled with theoretical studies indicatedthat the metal t2g states (one component of the 3d crystal-field splitting) increases throughthe series CrP to CoP (Grosvenor et al 2005). It was suggested that these states are due tometal–metal bonding and their increase was in agreement with the shortening of the average

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Figure 3. Comparison of Co 2p–P 2p energy separation between Co phosphides (from the literatureand the current study) and the boundary conditions of Co and P in both ionic and covalent (pureelemental) environments.

Table 3. Co and P Auger parameter values obtained for CoP3 and literature values for pure Co andP in several allotropes and compounds. Photoelectron and Auger peaks are given in binding andkinetic energy respectively.

Co 2p3/2

(eV)P 2p(eV)

Co LMM(eV)

P KLL(eV)

Co 2p-LMMAP (α′) (eV)

P 2p-KLLAP (α′) (eV)

CoP3 777.9 129.2 597.6 1858.7 1375.5 1987.9Pure Co (lit) 778.3 598 1376.3Pure P white (lit) 130.0 1857.2 1987.2ZnP2 (lit) 129.6 1857.3 1986.9Zn3P2 (lit) 138.65 1858.2 1986.85P4S10 (lit) 133.05 1853.45 1986.5GaP (lit) 129.2 1857.5 1986.7InP 128.65 1858.65 1987.3

metal–metal bond lengths. Similarly, the population of the P 3p states involved in the P–P andTM–P bonding decreases in agreement with the increase in the length of the P–P bond. Thevalence band spectrum in our study was dominated by the Sn 3d band and therefore was of noanalytical use. However, our results regarding Co–P ionicity are in general agreement with thetrend shown in the monophosphides study (Grosvenor et al 2005).

3.2. The Auger parameter and charge transfer

Table 3 shows Co 2p-LMM and P 2p-KLL α′ values for CoP3 from this study as well as Pα′ values from the literature for P-containing compounds (XPS database). The Co LM23M23

peaks in figure 4 were used together with the Co 2p3/2 (not shown) for the measurement ofα′ for Co whilst the P 2p and P KLL shown in figures 2 and 5 respectively were used for theP Auger parameter (AP) values. The use of α′ allows us to probe the response of the core

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Figure 4. High-resolution Co LMM after subtraction of a Shirley-type background.

Figure 5. High-resolution Bremsstrahlung induced P KLL of CoP3. The energy of the P KLLplasmon as determined by the separation between the main peak and the first plasmon peak, as wellas between the first and second plasmon peak, is 22.8 ± 0.5 eV.

potential to changes in the atomic environment when localized core levels and core–core–core(CCC) Auger transitions are used. For this reason we used the Co L23M23M23 peak (CCCtransition) instead of the most intense L2M23M45 which corresponds to a core–core–valence

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Figure 6. Comparison of P 2p KLL Auger parameter values between CoP3 in this study and theliterature values for pure P, metal phosphides and a phosphorus pentasulfide.

(CCV) transition. As shown in both figure 6 and table 3, α′ for P in CoP3 (and in the literatureNi–P compounds) is larger than the literature value for pure P. On the other hand the α′ valuefor Co in CoP3 is reduced compared to that for pure metal. Furthermore, as shown in figure 6,the value of α′ for P in InP, ZnP2, Zn3P2, GaP and P4S10 are close to or smaller than the valuesfor pure P.

In the case of metals, α′ in (5) can be interpreted as charge transfer (q) since thesecond and third terms in equation (5) become zero due to the assumption of perfect screening(dq/ dN = 1) and highly polarizable surroundings (U = 0). In this context the increaseof the P AP would correspond to a negative q , thus indicating electron donation from Co toP (dk/ dN in equation (5) is a negative quantity). However, in covalent semiconductors suchas Si and P the radial maximum of sp3 hybrid orbitals is greater than the inter-nuclei distanceof adjacent sites and thus it is expected that these compounds are characterized by delocalizedscreening. In this context the last two terms in equation (5) are important.

The higher α′ of P in CoP3 compared to pure P shows an increased core hole screeningefficiency and therefore demonstrates a ‘metallic’ character of the CoP3 in the same contextas Ni–P compounds in previous studies (Franke et al 1991, Franke 1987). In contrast, α′of Co shows the opposite behaviour as we move from pure Co to CoP3. Assuming a metalliccharacter of the CoP3 compound, both Co and P AP changes can be interpreted in terms ofelectron transfer. Using equation (5) for the metal case we calculate the charge transfer for Pfor s- and p-type to be −0.34 and −0.24 e−/atom respectively (see table 4) with the negativesign denoting electron gain. In the periodic table, phosphorus (3s23p3) is followed by sulfur(3s23p4), and according to the equivalent cores approximation it is more likely for a core holein the photoionized P atom to be screened by a p valence electron than by an s electron. Thisis particularly important in the context of p–d coupling and/or s–p–d hybridization. In bothcases the contribution of P 3p and Co 3d charge is a central issue. EELS suggested (Prytzet al 2007) that Co donates 0.77 e−/Co atom. From stoichiometric considerations one Co atomcorresponds to three P atoms in CoP3; therefore, P gains 0.77/3 = 0.26 e−/atom. We see thatthe above value lies closer to p-type charge transfer for P.

We now consider the Co contribution to charge transfer. Assuming that the differences ind screening following ionization are equal to differences in d charge (Thomas and Weighman

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Table 4. Rea values, charge transfer values and potential parameters used for the charge transfercalculations of P and Co in CoP3. A positive sign in the charge transfer values denotes electrondonation and a negative sign electron gain.

ElementRea

(eV)dk/dNP

a

(eV)dk/dNCo

b

(eV)dk ′/dNCo

b

(eV)

s-chargetransfer(e−/atom)

p-chargetransfer(e−/atom)

d-chargetransfer(e−/atom)

P s charge −2.04 −0.34P p charge −2.09 −0.24Co s charge −2.27 −4.09 0.08Co d charge −4.57 −4.60 +0.27 to +0.93P +0.462Co −0.4

a Values taken from Jackson et al (1995).b Values taken from Gregory et al (1993).

1986), equation (5) can be written as

α = qd( dkd/ dN − dks/ dN) + qs( dks/ dN). (6)

The terms dks/dN and dkd/dN are the potential parameters (see table 4) for the s shelland d shell respectively (Gregory et al 1993), whilst qd and qs are the charge transfercontribution of the Co d band and s band respectively. Assuming overall charge neutrality, theCo s valence band contribution is given by

xq(Co)s = −(1 − x)q(P) (7)

where x is the P molar fraction (0.75) and q (P) is the P charge transfer. Considering onlyp-type P charge we calculate the s- and d-type Co charge transfer. Using equations (6) and (7)and the two sets of potential parameters in table 4, we calculate +0.27 e−/atom qCo

d +0.93 e−/atom (table 4). The higher value in the above range corresponds to potentialparameters for free atoms and the lower for potential parameters normalized in order to takeinto account the compression of the valence band in the solid (Thomas and Weighman 1986,Gregory et al 1993). We recall that the electron transfer away from Co deduced from EELSinvestigations (Prytz et al 2007) was 0.77 e−/atom. This lies within the 0.27–0.93 e−/atomrange as estimated with XPS in this study. The accuracy of the estimated charge transferwould depend, to a certain extent, upon the choice of the potential parameters incorporatedin the calculations. For the P contribution to charge transfer we used potential parameter values(Jackson et al 1995) which take into account the nonlinear dependence of k and q on N. Forthe Co contribution the only literature data we found (Gregory et al 1993) referred to potentialparameters which were calculated assuming a linear dependence of k and q on N .

In the above analysis P was treated as having ‘metallic’ behaviour in CoP3 due to theincreased screening compared to pure P and other P-containing compounds with knownsemiconducting or insulating behaviour (see figure 6 and table 5). However, it is possible that,due to its semiconducting nature, α′ for P could reflect differences in screening efficiency asrepresented in Weightman’s models by the dielectric constant (Weightman 1998) rather thancharge transfer. Although electrical and optical measurements on CoP3 have suggested thepresence of a small band gap, the semiconducting nature of the compound is in doubt due todisagreements amongst calculations (Grosvenor et al (2006) and references 10, 13, 15 and 16therein). Therefore, the metallic character of the compound is not an unreasonable assumptionand the phosphorus AP dependence on initial-state charge transfer gains ground.

The use of the Bremsstrahlung induced P KLL peak has an additional advantage. Thematrix element of an Auger process usually involves the wavefunction of a core orbital and

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Table 5. Band gap values for various TM phosphides shown in figure 5 and difference in s andd orbital radius between Co and various TMs. Values refer to free atoms. A positive differencemeans that the Co radii are larger.

Compoundelements Compound

Difference in sorbital radiusa

between TMs(%)

Difference in dorbital radiusa

between TMs (%) Band gap (eV)

Co–Ni +3.4 +5.9Co–In +6.9 +27.1Co–Zn +9.9 +17.8Co–Ga +24.7 +25.6

InP 1.42b

Zn3P2 1.5c

α-ZnP2 2.05d

CaP 2.32a

a Radii values taken from www.webelements.com.b Kittel (1986).c Suda and Kakishita (1996).d Karajamaki et al (1980).

the Auger transition will only sample the local electronic structure (Matthew and Komninos1975). Thus, the localized character of the final states of the Auger transition contrasts thedelocalized nature of screening in semiconductors such as P (see also section 3.3); thereforeit is difficult to distinguish contributions of the local and non-local density of states (DOS) inAuger profiles of semiconductors involving the valence band (Weightman 1998). By acquiringthe Bremsstrahlung induced P KLL in the present study we probe only core levels and weovercome the above problem probing thus only the P DOS.

3.3. The Auger parameter and changes in polarizability/screening efficiency

It has been shown that the shift of P 1s (P1s) is different from that of P 2p (P2p), and moreprecisely m = P1s/P2p ≈ 1.14 (Moretti 1998 and references 88, 89, 90 therein). Therefore,equation (4) changes to

α = 2R + (m − 1)E2p = 2R + 0.14E2p. (8)

E2p between pure P and P in CoP3 is a negative quantity (see table 2). Equation (8)shows that, without correcting for unequal P 1s and P 2p shifts, the increase in relaxationenergy accompanying photoelectron and Auger electron emission (and/or P core hole electronscreening) in CoP3 as compared to pure P is underestimated. Using equation (8) and tables 2and 3 we measure the relaxation energy (R) associated with the emission of 2p photoelectronsand the associated KLL and LMM Auger electrons in the P and Co atoms of CoP3 (see table 4).R between P in CoP3 and in pure P is +0.46 eV. Accordingly the R value for Co is −0.4 eV.The negative sign indicates a reduction in the extra-atomic relaxation (and/or screening) energyin the case of Co in CoP3.

Although to a good approximation α is only related to final-state effects in the form ofthe relaxation energy (R) transferred from the surroundings to the photoionized atom, theelectronic polarizability of the surroundings also depends on the nature of chemical bonding.Therefore apart from initial-state effects (Madelung potential, charge transfer), the final-stateeffects also depend on the ground-state properties of the system (Moretti 1990 and references1–4 therein). The position of CoP3 in figure 6 can be interpreted in terms of the polarizabilityof the surrounding electrons. As we progress from Co to Ni, Zn, Ga, In, the valence band of

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the metal becomes more localized and therefore less polarizable (see table 5). The P atoms inNiP3, InP, Zn3P2, ZnP2, and GaP are surrounded by less polarizable electrons compared to Patoms in CoP3. As mentioned in section 1, P in CoP3 has two Co and two P nearest neighbours.The reduced polarizability of the valence electrons in the Ni–P compounds compared to CoP3

is counterbalanced by the increased number of the average P–Ni ligands in Ni5P4, Ni12P5

and Ni3P. In these compounds the Ni-rich environment of the P atoms provides polarizableelectrons which screen the P core hole more effectively than the P-rich environment of P atomsin NiP3.

An alternative explanation of the relative positions of the compounds in figure 6 can begiven by considering the band gaps of the semiconductors (see table 5). The increase inband gap would lead to a reduction in screening efficiency (as shown by the lower α′ value)since the conduction electrons would ‘strangle’ more to screen a core hole. However, such anexplanation should have as a prerequisite that the relative position of the Fermi level does notvary significantly from compound to compound and that we assume more or less ‘intrinsic’semiconductors.

4. Conclusions

The Co–P 2p binding energy difference indicates that the Co–P bonding in CoP3 is covalentwith partial ionic character. We used the Bremsstrahlung radiation to acquire the P KLL Augerpeak and to measure the P 2p KLL Auger parameter. The reduced P 2p KLL Auger parameter ofpure P compared to CoP3 showed that the P core holes are screened better in the compound thanin the pure element, while the Co 2p LMM Auger parameter showed that electron screening ofthe Co core holes is stronger in the elemental environment. Considering a metallic environmentfor the P atoms in CoP3 compared to pure P, the charger transfer between Co and P atoms wasestimated using Auger parameter data and the Thomas and Weightman model. It is suggestedthat upon bonding to form the CoP3 compound, Co donates 0.27–0.93 e−/atom and P gains0.24 e−/atom. These values correspond well with Co losing 0.77 e−/atom as deduced fromprevious EELS studies and thus P gaining 0.77/3 = 0.26 electrons/atom (Co:P = 1:3).Additional explanations of the increased P Auger parameter in CoP3 compared to other P-containing compounds include a reduced band gap and an increased polarizability of the Patomic environment in this compound.

Acknowledgment

Financial support from FUNMAT@UiO to one of us (ØP) is gratefully acknowledged.

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