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Transcript of Institut für Theoretische Festkörperphysik (Universität Karlsruhe) and Universidad Autonoma de...
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)