Quantum Transport at the Nanoscale

1
Quantum Transport at the Nanoscale
Kristian Sommer Thygesen Center for Atomic-scale
Materials Design (CAMD) Technical University of
Denmark
2
Moores Law
Gordon Moore (co-founder of Intel) 1965 The
density of transistors on a chip will double
every two years.
3
Moores Law
Si-based transistor 65 nm (2007).
ENIAC computer, 1947

• Future challenges for semiconductor device
  technology
• Leak currents through gate oxide (improvement by
  high k materials)
• Quantum confinement effects affect device
  functionality
• Lithographic top-down printing of ever smaller
  features
• Reliable doping of Si increasingly difficult

4
Molecular Electronics
Bottom-up design of electronic components using
single molecules. Advantages (Potentially)
cheap, truly nano-sized, flexibility in design
and functionality. Challenges Contacting
molecules by electrodes, large-scale integration
R. Stadler, M. Forshaw, and C. Joachim,
Nanotechnology 14, 138 (2003)
Example A molecular switch
E. Lortscher et al., Small 2, 973 (2006)
5
Programme
Part 1 Formalism and Methodology (40 min) Part
2 Coffee (10 min.) Part 3 Applications (40 min)
6
Part 1 Formalism and methodology

• Definition of the problem
• Two routes for solving the problem
• Wave function method (scattering theory)
• Greens functions
• Two limiting cases
• Adiabatic approximation
• Single resonant level model
• Combining transport and DFT

7
Definition of the problem 1
Calculate the IV curve!
I
8
Definition of the problem 2

• Non-interacting electrons
• Scattering free leads (perfect crystalline
  electrodes)
• Electrons incident from left/right are in
  thermal equilibrium with left/right reservoirs.
• Complete thermalization of electrons upon
  entering reservoir
• No back scattering at lead-reservoir interface

9
Ballistic transport single channel
Ballistic no scattering region. Single channel
one left/right moving state at each energy
Steady state current
A ballistic channel contributes to the
conductance by 1G0. Note This result is
independent of the band structure!
10
Ballistic transport many channels
Each transverse mode defines a transport channel.
M2
Steady state current
M is number of modes within the bias window
11
Scattering
In general, each incoming channel is scattered
into a linear combination of several forward and
backward moving channels.
Landauer formula
tnm(E)2 Probability that an electron with
energy E injected in mode n is transmitted into
mode m.
Elastic transmission function
Linear response conductance
12
Slowly varying potentials
Constricted quantum wire
Energy of transverse modes
Adiabatic ansatz
Adiabatic picture As the electron moves along
the x axis, energy is transferred from the
longitudinal motion to the transverse mode.
13
Conductance quantization
Numerical examples Transmission in 2d quantum
wires

• For slowly varying constrictions, the
  conductance is quantized in units of G0.
• Conductance quantization in quantum wires is a
  robust phenomenon.
• Conductance steps are observable for kT ltlt ?E,
  where ?E is the spacing of transverse modes.

14
Conductance quantization in gold chains
Chains of single metal atoms can be formed by
breaking a nanocontact using e.g. an STM or
mechanically controlled break junctions (MCBJ).
Yanson et al. Phys. Rev. Lett. 95 256806 (2005)
DFT molecular dynamics simulations by Sune Bahn.
15
Greens functions why?

• More complex than density, less complex than
  wavefunction
• Allow for systematic perturbation theory
• Easy to handle in a computer
• Straightforward to include interactions via
  self-energies, e.g. GW approximation
• Straightforward to describe open boundary
  conditions
• Seem rather abstract at first encounter
• Do not always provide much physical insight

16
Greens functions
Greens function (or resolvent) operator
Spectral representation
Spectral function (projected density of states)
(Use that )
17
Perturbation theory
Suppose the Hamiltonian can be split into two
parts
The unperturbed (or free) Greens function is
It is easy to show that G fullfills Dysons
equation
A perturbation series for G is obtained by
iteration
18
Perturbation theory Feynman diagrams
Consider the perturbation series
- and insert a complete set of states
More compactly (summation over internal
variables implied)
This can be represented graphically using Feynman
diagrams
Vnm
Gij
G0,ij
G0,in
G0,mj




19
Perturbation theory Feynman diagrams
Deriving the Dyson equation using Feynman
diagrams




(
)






1


1 -
( )-1 -
Q.E.D.
20
Embedding self-energy
Suppose the quantum system (Hilbert space) can be
divided into two regions
V
Take the coupling as the perturbation
The unperturbed Greens function reads
with
21
Embedding self-energy
The Dyson equation for the full Greens function
For the AA component

• The self-energy SB(z)
• Non-hermitian, energy-dependent potential
• Incorporates the coupling to region B (open
  b.c.s)

22
Embedding self-energy with Feynman diagrams
a
a




G0 only connects states within the same region V
only connects states in different regions
a
a
a
a
a
ß
a
ß




(
)






SB VABG0,BBVBA
23
Embedding self-energy an illustration
Projected DOS onto region A for a free electron
in 1D
x
Kinetic energy discretized on a grid.
With embedding self-energy
Without embedding self-energy

• Embedding broaden discrete levels into a
  continuum
• Embedding incorporates open boundary conditions

24
Current formula
No lead-lead coupling
Localized basis
The GF of the (finite) scattering region
GF of isolated left lead
Embedding of left lead
Transmission function
Equivalent to scattering theory result but less
transparent!
25
Single resonant level model
Greens function at the central site
Embedding self-energy
Imaginary part of embedding se, broaden the level.
Wideband approximation
GF, PDOS, transmission
26
Part 2 Applications

• DFT conductance calculations
• Conductance oscillations in atomic wires
• Conductance of a hydrogen molecule
• Inelastic scattering
• Electron-electron interactions and the DFT
  bandgap problem
• GW approach to quantum transport

27
Conductance from DFT
Plane wave density functional theory (DFT)
calculation for S and leads.
Extend potential to left and right of S by lead
potential.
Calculate Greens function of S.
Evaluate Kohn-Sham Hamiltonian in terms of a
localized basis set.
Evaluate the transmission function.
,
28
Setting up the Hamiltonian
VL
VL
VR
HL
HL
HC
HR
Generic shape of the Hamiltonian (Localized
basis set implied)
29
Atomic chains
Pt chain formed by breaking a nanocontact using a
break junction
Self-assembled Co chains at a stepped Pt surface.
Gambardella et al. Nature 416, 301 (2002)
Aloire et al. Phys. Rev. Lett. (2003)
30
Atomic chains
Molecular dynamics simulations with EMT
potentials, Sune Bahn 2001.
Chain rupture
Rubio-Bollinger et al. PRL 87 026101 (2001)
31
Conductance oscillations
Calculations
32
Conductance oscillations Resonant level model
Consider a chain of monovalent atoms
Hybridization with electrodes
EF
N elec-trons

• Every atom contributes 1 electron to the chain
  resonance system
• One resonance half-filled for odd N
• All resonances completely filled or empty for
  even N

The model explains (qualitatively) results for
simple metals Na,C, but fails for more complex
systems!
33
More complex chains Atomic Ag-O chains
Individual traces
Average over many chains
Thijssen et al. PRL 96, 026806 (2006)
length (A)

• What is the composition of the low-conducting
  Ag-Ox structure?
• Why does its conductance decrease with the wire
  length? Band gap effect??

34
Composition of Ag-O chains
Oxygen induced reconstruction of Ag(110) surface
DFT calculations of breaking force in infinite
wires
Bonini et al. PRB 69, 195401 (2004)
Alternating Ag-O chains are energetically and
thermodynamically stable.
35
Alternating Ag-O chains Calculations

• Long chain limit of 0.1G0 reproduced.
• Higher experimental conductance for short
  lengths due to pure Ag chains.
• Weak conductance oscillations.

?
M. Strange et al., Phys. Rev. Lett., accepted
(arXiv0805.3718)
36
Alternating Ag-O chains Calculations
Spin-polarized DFT bandstructure of infinite
alternating Ag-O chain
Spin-polarized DFT transmission function of
finite alternating Ag-O chains
Transport never on resonance ? low mean
conductance. Breakdown of resonant level model.
37
Looking into the litterature
G.B. Airy, Philosophical Magazine, 1832
38
Resonating chain model
k
L
39
Resonating chain model
0 Ballistic ? Fully reflecting ? max.
oscillation amplitude
Phase picked up in a roundtrip between the
contacts ? oscillation period
40
Calculating parameters for the Resonating-chain
model
Reflection probability and phase shift for
chain-bulk interface obtained from the scattering
state at Fermi energy
41
Results for Al chains

• Reflection phase shift close to p ? standing
  waves on chain ? resonant level model works well.
• Four-atom period since kFp/4a
• Large oscillations (i) max amplitude (ii)
  Large reflection

Al
Au
42
Results for Au chains
Al

• Vanishing reflection ? interference term almost
  zero.
• Two-atom period since kFp/2a

Au
43
Results for Ag-O chains

• Reflection phaseshift close to p/4 ?
  interference term finite but almost constant.
• Phaseshift of p/4 due to N1/2 Ag-O unit cells
  in chain.
• Two-atom period since kFp/2a

Ag-O
44
Extensions of the theory

• Inelastic scattering (electron-phonon
  interactions)
• Electronic correlations (electron-electron
  interactions)
• Finite bias effects
• Time dependent phenomena

45
Conductance of a hydrogen molecule
Break junction experiments at 4.2 K Presence of
hydrogen changes the conductance just before
breaking from 1.5G0 to 1G0 .
Pure Pt
With H2
Interpretation Single H2-molecular bridge.
First questions Why does H2 not dissociate? How
can H2 be conducting?
Smit et al. Nature 419, 906 (2002)
46
Electron-vibration coupling affects the dI/dV
Observation A drop in the conductance occurs
when bias exceeds the energy of a quantized
vibration of H2.
Explanation (For contacts with T?1 ) At low
temperature, electrons can only backscatter due
to Pauli blocking.
N. Agrait et al Chem. Phys. 281, 231 (2002)
D. Djukic et al PRB 71, 161402(R) (2005)
47
Simple view on inelastic scattering (T1)
h?
k, Egt
-k, E-h? gt
Interactions
eV
Fermis Golden rule for transitions
48
Vibrations of a molecular hydrogen bridge
Measured vibrations
Calculated vibrations
D. Djukic, K. Thygesen et al PRB 71, 161402(R)
(2005)
49
Theory of inelastic scattering
Hamiltonian
Free phonons
Electron-phonon interaction
50
Greens function
Lead coupling self-energies
Electron-phonon self-energy
Dyson equation with 1. Born approximation
Electron Greens function
Phonon Green function
51
Calculated dI/dV with electron-vibration
interactions
Longitudinal modes Conductance decreases at eVh?
Transverse modes No signal for Au electrodes
mode couples to d-band. For Pt Conductance
increases at eVh? in contrast to experiments!
52
DFT-transport success and limitations
Systems Junctions with small chemisorbed
molecules Metallic point contacts / atomic wires
Contacts with larger organic molecules linked
via S or N groups Calculated conductance up to
103 x experimental value!
Conductance 1G0
Thygesen Jacobsen PRL 94, 036807 (2005)
J. Reichert et. al PRL 88, 176804 (2002) J.
Heurich et al. PRL 88, 256803 (2002)
Physical effects Interference effects
(conductance fluctuations, even-odd effects)
Charge transfer, non-equilibrium charge
redistribution Correlation effects (Coulomb
blockade, Kondo effect) Inelastic effects (el-ph
coupling)
Lee et al. PRB 69, 125409 (2004)
53
The band gap problem in DFT

• DFT local xc-functionals underestimate
  HOMO-LUMO gaps
• Hartree-Fock is good for small molecules
  (SI-free), but overestimates the gap for extended
  systems
• GW includes screening in the exchange and this
  solves the gap problem.

(Carsten Rostgaard)
Screening correction
Bare exchange
54
GW in the central region
Assumption Electrodes and coupling may be
described by effective (DFT) Hamiltonian.
Correlation effects mainly important in
molecular region.
vxc
?GW
55
Self-consistent GW calculation
Electrode embedding self-energies
Dysons equation
Self-consistency ensures charge conservation and
yield a unique GF
GW equations
56
Two level model
57
Role of correlations in the two level model
Current
Molecular DOS

• The bias can affect the HOMO-LUMO gap
• The bias can affect the resonance width
• These effects must be taken into account to
  describe the IV

58
Impact of exchange-correlation on IV
Evolution of HOMO/LUMO levels in Hartree
(crosses), HF (triangles), and GW (circles).

• Hartree smears out features. Self-interaction
  effect.
• HF and GW agree well at low bias.
• Quasi-particle scattering reduces electronic
  lifetimes at finite bias.
• Dynamic screening reduce HOMO-LUMO gap at finite
  bias.

Thygesen, Phys. Rev. Lett. 100, 166804 (2008)
59
Summary

• Two methods for quantum transport
• Wave functions (scattering theory)
• Greens functions
• DFT-transport applies to strongly coupled
  systems (conductance 1G0)
• Conductance oscillations in atomic chains
• Electron-vibration coupling through self-energy
• GW scheme for quantum transport
• Reduced lifetime of molecular states
• Renormalization of molecular level positions
  (significant bias dependence)