Charles Stafford Manybody theory of electric and thermal

Charles Stafford 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

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 variations in device characteristics → environmental sensitivity. Contacts/interconnects to ultrasmall devices. Switching mechanism: Raising/lowering energy barrier necessitates dissipation of minimum energy k. BT 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 ~ 9 e. V. Charging energy of a benzene molecule: ‹U› ~ 5 e. V. Nearest-neighbor π-π hopping integral: t ~ 2 – 3 e. V. Lead-molecule coupling: Γ ~ 0. 5 e. V (small parameter? ). Electronic structure unique for each molecule---not universal!

Alternative switching mechanism: Quantum interference (a) Phase difference of paths 1 and 2: k. F 2 d = 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. 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)

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 junction can be expressed in terms of G and G<. Dyson equation: Tunneling self-energy: Coulomb self-energy must be calculated approximately.

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 itinerant p-electrons. • Sigma band is filled and doesn’t contribute appreciably to transport. Effective charge operator, including polarization charges induced by lead voltages: Parameters from fitting electronic spectra of benzene, biphenyl, and transstilbene up to 8 -10 e. V: Accurate to ~1% U=8. 9 e. V, t=2. 64 e. V, ε=1. 28 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 the Coulomb 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. 2 m. V/K • Baheti et al, Nano Letters Vol 8 No 2 (2008) • We can express thermopower in terms of the transmission probability Find that m. Au- m 0 =-3. 22±. 04 e. V, about 1. 5 e. V above the HOMO level (hole dominated) • Experimentally the linear-conductance of BDT is reported to be 0. 011 G 0 (2 e 2/h) • Xiaoyin Xiao, Bingqian Xu, and N. J Tao. Nano-letters Vol 4, No. 2 (2004) • Comparison with calculated linear-response gives G=. 63±. 02 e. V

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…