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

1. 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

2. 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 kBT per cycle → extreme power dissipation at ultrahigh device densities. Tunneling & barrier fluctuations in nanoscale devices.

3. 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.

4. 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!

5. Alternative switching mechanism: Quantum interference • 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. David M. Cardamone, CAS & S. Mazumdar,Nano Letters 6, 2422(2006); CAS, D. M. Cardamone & S. Mazumdar, Nanotechnology18, 424014 (2007); U.S. Patent Application, Serial No. 60/784,503 (2007)

6. 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.

7. Nonequilibrium Green’s functions

8. Real-time Green’s functions

9. Molecular Junction Hamiltonian Coulomb interaction (localized orthonormal basis): Leads modeled as noninteracting Fermi gases: Lead-molecule coupling (electrostatic coupling included in Hmol(1)):

10. 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: Coulomb self-energy must be calculated approximately. G obeys the equation of motion: Once G is known, G< can be determined by analytic continuation on the Keldysh contour. Tunneling self-energy:

11. Electric and Thermal Currents Tunneling width matrix:

12. Elastic and inelastic contributions to the current

13. Elastic transport: linear response

14. 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 trans-stilbene up to 8-10eV: • Accurate to ~1% • U=8.9eV,t=2.64eV,ε=1.28 • Castleton C.W.M., Barford W., J. Chem. Phys. Vol 17 No. 8 (2002)

15. Enhanced thermoelectric effects near transmission nodes

16. Effect of a finite minimum transmission

17. 4. The Coulomb self-energy

18. Sequential-tunneling limit: ΣC(0) Nonequilibrium steady-state probabilities determined by detailed balance: Tunneling width matrix:

19. Correction to the Coulomb self-energy

20. 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.

21. Coulomb blockade in a diatomic molecule

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

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

24. Determining the lead-molecule coupling: thermopower • We can express the thermopower in terms of the transmission probability Find that mAu- m0 =-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 givesG=.63±.02eV • Experimentally the BDT junction’s Seebeck coefficient is found to be 7.0.2mV/K • Baheti et al, Nano Letters Vol 8 No 2 (2008)

25. 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)

26. Resonant tunneling through molecular excitons Justin P. Bergfield & CAS, Physical Review B 79, 245125 (2009)

27. 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…