Many-body theory of electric and thermal transport in single-molecule junctions

INT Program “From Femtoscience to Nanoscience: Nuclei, Quantum Dots and Nanostructures,” July 31, 2009

Charles Stafford

1. Fundamental challenges of nanoelectronics(a physicist’s perspective)

Fabrication:

Lithography → self-assembly?

For ultrasmall devices, even single-atom variations from device to device (or in device packaging) could lead to unacceptable variationsin device characteristics → environmental sensitivity.

Contacts/interconnects to ultrasmall devices.

Switching mechanism:

Raising/lowering energy barrier necessitates dissipation of minimumenergy kBT per cycle → extreme power dissipation at ultrahigh device densities.

Tunneling & barrier fluctuations in nanoscale devices.

Molecular electronics

Fabrication: large numbers of identical “devices” can be readily synthesized with atomic precision. (Making the contacts is the hard part!)

But does not (necessarilly) solve fundamental problem of switching mechanism.

Single-molecule junction ≈ ultrasmall quantum dot

Similarities and differences:

Typically, π-orbitals of the carbon atoms are the itinerant degrees of freedom.

Charging energy of a single π-orbital: U ~ 9eV.

Charging energy of a benzene molecule: ‹U› ~ 5eV.

Nearest-neighbor π-π hopping integral: t ~ 2 – 3eV.

Lead-molecule coupling: Γ ~ 0.5eV (small parameter?).

Electronic structure unique for each molecule---not universal!

Alternative switching mechanism: Quantum interference

David M. Cardamone, CAS & S. Mazumdar, Nano Letters 6, 2422 (2006); CAS, D. M. Cardamone & S. Mazumdar, Nanotechnology 18, 424014 (2007); U.S. Patent Application, Serial No. 60/784,503 (2007)

(a) Phase difference of paths 1 and 2: kF 2d = π → destructive interference blocks flow of current from E to C.

All possible Feynman paths cancel exactly in pairs.

(b) Increasing coupling to third terminal introduces new paths that do not cancel, allowing current to flow from E to C.

2. The nonequilibrium many-body problem

•Mean-field calculations based on density-functional theory are the dominant paradigm in quantum chemistry, including molecular junction transport.

•They are unable to account for charge quantization effects (Coulomb blockade) in single-molecule junctions!

•HOMO-LUMO gap not accurately described; no distinction of transport vs. optical gap.

•Many-body effects beyond the mean-field level must be included for a quantitative theory of transport in molecular heterojunctions.

•To date, only a few special solutions in certain limiting cases (e.g., Anderson model; Kondo effect) have been obtained to the nonequilibrium many-body problem.

•There is a need for a general approach that includes the electronic structure of the molecule.

Nonequilibrium Green’s functions

Real-time Green’s functions

Molecular Junction Hamiltonian

Coulomb interaction (localized orthonormal basis):

Leads modeled as noninteracting Fermi gases:

Lead-molecule coupling (electrostatic coupling included in Hmol(1)):

Molecular Junction Green’s Functions

All (steady-state) physical observables of the molecular junctioncan be expressed in terms of G and G<.

Dyson equation:

Coulomb self-energy must be calculated approximately.

G obeys the equation of motion:

Once G is known, G< can be determined by analytic continuationon the Keldysh contour.

Tunneling self-energy:

Electric and Thermal Currents

Tunneling width matrix:

Elastic and inelastic contributions to the current

Elastic transport: linear response

3. Application to specific molecules:Effective π-electron molecular Hamiltonian

For the purpose of this talk we consider conjugated organic molecules.

• Transport due primarily to itinerantelectrons.• Sigma band is filled and doesn’t contribute appreciably to

transport.

Effective charge operator, including polarization charges induced by lead voltages:

† †ol

,m ,

,

ˆ ˆ ˆ ˆˆH1

2, .n n n n m nmn nm m

n nm nmntd d n d d U Q QH c

† 1nn n n

n

CQ d d

e

Parameters from fitting electronic spectra of benzene, biphenyl, and trans-stilbene up to 8-10eV:

Accurate to ~1% U=8.9eV,t=2.64eV,ε=1.28

2

11 ( /Ang)

nm nm nm

nm

UU U

R

Castleton C.W.M., Barford W., J. Chem. Phys. Vol 17 No. 8 (2002)

Enhanced thermoelectric effects near transmission nodes

Effect of a finite minimum transmission

4. The Coulomb self-energy

Sequential-tunneling limit:ΣC

(0)

Nonequilibrium steady-state probabilities determined by detailed balance:

Correction to the Coulomb self-energy

Self-consistent Hartree-Fock correction to theCoulomb self-energy of a diatomic molecule

•Narrowing of transmission resonances;•No shift of transmission peak or node positions;•No qualitative effect on transmission phase;•Correction small in (experimentally relevant) cotunneling regime.

Coulomb blockade in a diatomic molecule

Higher-order corrections to the Coulomb self-energy: RPA

5. Results for 1,4-benzenedithiol-Au junctions

Determining the lead-molecule coupling: thermopower

• Experimentally the BDT junction’s Seebeck coefficient is found to be 7.0.2V/K• Baheti et al, Nano Letters Vol 8 No 2 (2008)

Find that Au-0 =-3.22±.04eV, about 1.5eV above the HOMO level (hole dominated)• Experimentally the linear-conductance of BDT is reported to be 0.011G0

(2e2/h)•Xiaoyin Xiao, Bingqian Xu, and N.J Tao. Nano-letters Vol 4, No. 2 (2004)

• Comparison with calculated linear-response gives =.63±.02eV

• We can express the thermopower in terms of the transmission probability

1f

T E E dEE

SfeT T E dEE

Differential conductance spectrum of a benzene(1,4)dithiol-Au junction

•Junction charge quantized within ‘molecular diamonds.’•Transmission nodes due to quantum interference.•Resonant tunneling through molecular excited states at finite bias.

Justin P. Bergfield & CAS, Physical Review B 79, 245125 (2009)

Resonant tunneling through molecular excitons

Justin P. Bergfield & CAS, Physical Review B 79, 245125 (2009)

Conclusions•Electron transport in single-molecule junctions is a key example of a nanosystem far from equilibrium, and poses a challenging nonequilibrium quantum many-body problem.

•Transport through single molecules can be controlled by exploiting quantum interference due to molecular symmetry.

•Large enhancement of thermoelectric effects predicted at transmission nodes arising due to destructive quantum interference.

•Open questions:

Corrections to Coulomb self-energy beyond RPA

Fabrication, fabrication, fabrication…