Institut für Theoretische Festkörperphysik (Universität Karlsruhe) and Universidad Autonoma de Madrid (Spain)

Molecular Electronics: Experimental Techniques

1. Breakjunctions 2. Nanopores

3. Electrochemical methods 4. Electromigration

Molecular Electronic: Functional Structures1. Diode: Au-SAM-Ti-Au (Nanopore)4-thioacetatebiphenyl, M. Reed, APL (1997)

2. Swicht: Nanopore (60 K)M. Reed et al., Science (1999)

3. Reconfigurable Swicht: CatananeJ.R. Heath et al., Science (2000)

4. Single-electron transistor:Park et al., Nature (2003).

Molecular Electronics: Goal for the Theory. Understanding of the transport mechanisms at the molecular scale. Quantitative description of the transport properties.

Outline of this tutorial

1) Description of the elastic current: Landauer approach. i. Resonant tunneling ii. Temperature dependence of the conductance iii. Symmetry of the IVs: Rectification iv. Calculation of the transmission: Green´s functions v. Two level system: hydrogen molecule vi. Length dependence of the conductance vii. Ab initio calculations

2) Inelastic current: role of the vibration modes in the conduction i. Experimental motivation ii. Simple theoretical model: different transport regimes

3) Other transport mechanisms: correlation effects, hopping, transport, etc. (very brief)

4) Challenges and open problems

1. Landauer approach to electron transport in molecular contacts

Relation between electronic structure and electronic transport

real system

Sa(E) t(E)a(E)

r(E)a(E) Landauer formula

scattering problem

T(E)+R(E) = |t(E)|2+ |r(E)|2 =1

S

electron reservoirs

scattering region

EF

EF+eV

)(2 2

FETh

eG

spin degeneracy

Landauer approach to electron transport

1.1 Resonant tunneling (single level)

RL ffVEdETh

eVI ),(

2)(

220 )()(

4)(

RL

RL

EET

en ergy

Energy scheme of a molecular contact

Single-level model

Current formula

Resonant tunneling (single level)Off-resonant transport: moleculesas tunneling juncions

Single molecules as tunnel junctionsCui et al. (Lindsay), Science 294, 571 (2001)

1.2 Tunneling: temperature dependence

Voltage dependence:Again a tunnel junction!

Current independent of the temperature

Wang, Lee and Reed, PRB 68, 035416 (2003)

SHCHCH n 123 )(

Tunneling: temperature dependence

Conclusion

Off-resonant transport T independent On-resonant transport T dependent (as long as T ~ )

1.3 Symmetry of the IVs: Rectification “Molecular rectifiers”Arieh Aviram and Mark A. Ratner (Chem. Phys. Lett., 1974)

“The construction of a very simple electronic device, a rectifier,based on the used of a single organic molecule is discussed. Themolecular rectifier consists of a donor pi system and a acceptorpi system, separated by a sigma-bonded (methylene) tunnelingbridge. The response of such a molecule to an applied field iscalculated, and rectifier properties indeed appear.”

R. Metzger et al., JACS 1997

(… 23 years later)

Molecular conductivity takes shapeJ. Reichert et al., PRL 88, 176804 (2002)

(INT Karlsruhe)

asymmetric molecule

symmetric molecule

Symmetry of the IVs: Rectification

Molecular conductivity takes shapeJ. Reichert et al., PRL 88, 176804 (2002)

Single-level model: asymmetric coupling

Beware, Green‘s functions are coming!

1.4 Calculation of the transmission: Green‘s functions

jihHij

ijˆ

)0()(ˆ)0,(, jiji ctcTitG

Atoms Orbitals

Htci

dt

tdci

i ˆ),()(

..ˆˆˆˆˆˆ chVVHHHH RCLCCRL

1, ˆ)(ˆ HiEEG ar

Green´s function

Equation of motion?

Three subsystems: left (L), right (R) and center (C)

1.4 Calculation of the transmission: Green‘s functions

RCR

CRCCL

LCL

HV

VHV

VH

Hˆˆ0

ˆˆˆ0ˆˆ

ˆ 1,,, ˆ)(

arR

arLC

arC HiEEG

energies-self theare ),(ˆˆˆ)( where1, RLVHiEVE CC

ar

)(ˆ)()(ˆ)(Tr4)( EGEEGEET aCR

rCL

ttET ˆˆTr)( 2/12/1 ˆ 2ˆR

aCL Gt

rates scattering ---- Im a

1.5 Two-level conduction: Conductance of a hydrogen molecule Smit et al., Nature (2002); Leiden University

The hydrogen molecule forms a stablebridge between Pt electrodes.

The conductance is G ~ G0 and it is largelydominated by a single conduction channel.

Tunneling: two sites

Ht 0Bonding and antibonding states

2222

22

)()(

4)(

HtET

)Re(20

aH gtt

2t

Transmission:

Conductance of a hydrogen molecule

(i) Charge transfer between H2 and thePt leads and (ii) strong hybridizationbetween the molecule and the electrodes.

The Transport is dominated by the binding orbital.

DFT conductance calculation

JCC, J. Heurich, F. Pauly, W. Wenzel, G. Schön, Nanotechnology (2003)

1.6 Length dependence of conductanceThe conductance decays exponentially with the length of the moleculeWang, Lee and Reed, PRB 68,

035416 (2003)deGG 0

N-level bridge: n.n. interaction

0

{ r }{ l}

RL

1 . . . . N + 1

RL ffVEdETh

eVI ),(

2)(

( ) ( )20, 1 0 1( ) | ( ) | ( ) ( )L R

N NE G E E E T

01 12 , 10, 1

1 10

1 1 1 1ˆ ( ) ...B N NN

N N

G E V V VE E E E E EE E

0 0

1

1

2 LE E i

1 1,

1

1

2N N RE E i

G1N(E)

)(2 2

FETh

eG

1.7 Ab initio calculation (DFT)

VHHHH CRLˆˆˆˆˆ

J. Heurich, JCC, W. Wenzel, G. Schön, Phys. Rev. Lett. 88, 256803 (2002)

Three subsystems: central cluster and leads

Central cluster: Density functional calculation (DFT)

Leads: The reservoirs are modeled as two perfect semi-infinite crystals using a tight-binding parametrization (Papaconstantopoulos 1986).

Coupling: Hopping between the lead orbitals and the molecular orbitals of the central cluster.

i

iii ddH ˆ

i id

ijv

..ˆ chcdvV jiij

ij

Kohn-Sam energies Molecular orbitals

Theory: linear regime.

Total transmission and totaldensity of states of the two molecules

At the Fermi energy:

006.0

014.0

asym

sym

T

T

EF

From molecular orbitals to conduction channels

J. Heurich, JCC, W. Wenzel, G. Schön, Phys. Rev. Lett. 88, 256803 (2002)

Charge density of four molecular orbitals: (a) HOMO; (c) LUMO

The contribution to the transport of an individual molecular orbital depends onits character: extended or localized, weak coupling or strong coupling, etc.

I-V characteristics: symmetric molecule

I-V characteristics: symmetric molecule

2. Inelastic current

Role of the molecular vibrations in the electrical conduction

2.1 Experimental motivation: Inelastic tunneling spectroscopy

Ho and coworkers: ``Single molecule spectroscopy“ J. Chem. Phys. (2002)

2.1 Experimental motivation: hydrogen molecule

Smit et al., Nature (2002): measurement of the conductance of a hydrogen molecules between Pt leads

2.1 Experimental motivation: gold atomic chains

N. Agrait et al., PRL (2002): onset of energy dissipation in ballistic atomic wires

(See talk by NicolasAgrait this afternoon)

2.1 Experimental motivation: nanotubes

Satellite peaks: signature ofphonon-assisted tunneling

2.2 Inelastic current: toy model

)(ˆ bbccH phe

)1()()2(2

;)()(42 22

LLr

LinRLr

RLel ffEGdEih

eIffEGdE

h

eI

inelasticelastic III

electron-phonon coupling constant

Second order perturbation theory:

Simple model: resonant case

Simple model: resonant tunneling

Weak coupling regime: multiphonon peaksSee for instance S. Braig and K. Flensberg, PRB 68, 205324 (2003)

Rate equations:

Weak coupling regime: multiphonon processes

Theory: S. Braig and K. Flensberg PRB 68, 205324 (2003)

Microscopic modelsJanne Viljas and JCC (TB model, unpublished)

See talk by Thomas Frederiksen: T. Frederiksen et al. PRL 93, 256601 (2004)

3. Other mechanisms

3.1 Correlation effects

3.2 Incoherent hopping

........

0 = D

1 2 N

N + 1 = A

k 1 2

k 1 0 = k 0 1 e x p (-E 1 0 ) k N ,N + 1 = k N + 1 ,N e x p (-E 1 0 )

0 1,0 0 0,1 1

1 0,1 2,1 1 1,0 0 1,2 2

1, 1, , 1 1 , 1 1

1 , 1 1 1,

( )

( )N N N N N N N N N N N N

N N N N N N N

P k P k P

P k k P k P k P

P k k P k P k P

P k P k P

constant STEADY STATE SOLUTION

ET rate from steady state hopping

0,1 2,1 1 1,0 0 1,2 2

1, 1, ,

1 1,

1 1

0 ( )

0 ( )N N N N N N N

N N N N

N

k k P k P k P

k k P k P

P k P

/

1,0

1

1

B BE k T

D A N

N A D

kek k

k kN

k k

3.3 Conformation changes

3.4 Molecules as optoelectronic devices

4. Challenges and open problems Quantitative agreement with experimental results. Development of methods that describe the electronic transportthrough strongly correlated systems: interpolation between the weak coupling regime and strong coupling regime.

More extensive work on gating of molecular junctions. Characterizing transport junctions behavior in the presence of radiation. Effects of changing chemistry and doping on the bridge – can mechanisms be altered by chemical change, as in conducting polymers, and can we predict and control such behavior?

Elucidating the change in behavior from a single molecule conductance through junctions comprising a few molecules to molecular film conductors. Understanding noise Understanding heating , heat conduction and current induced chemical changes

Universität Karlsruhe Univ. Autónoma de MadridProf. Gerd Schön Prof. Alvaro MartínPriv. Doz. Dr. Wolfgang Wenzel (INT) Prof. Alfredo LevyDr. Jan E. Heurich Prof. Nicolas AgraitFabian Pauly Prof. Gabino RubioMichael Häfner Dr. Carlos UntiedtDr. Janne ViljasUniversität Konstanz Quantronics Group (Saclay) Leiden UniversityProf. Elke Scheer C. Urbina Prof. J. Van RuitenbeekProf. Peter Nielaba D. Esteve Dr. Bas LudophMarkus Dreher M.H. Devoret (now in Yale)