1. Self-consistent GW calculations for single-molecule transport: Bridging the theory-experiment gap in single-molecule transport Kristian S. Thygesen Center for Atomic-scale Materials Design (CAMD) Department of Physics Technical University of Denmark

2. A single molecule HOMO LUMO Energy

3. EFEF EFEF HOMO LUMO Energy Closed shell Weak coupling to electrodes

4. Chemical bond with electrodes Energy EFEF EFEF

5. Finite bias voltage Energy EFEF EFEF

6. EFEF EFEF How does electron-electron interactions and the nonequilibrium conditions affect the electronic structure and transport properties of molecular junctions? Finite bias voltage

7. Outline  DFT Conductance of BDT/BDA junctions  GW-transport scheme  Free molecules  Molecule-solid interface  Finite bias

8. Example: Engineering molecules for electronics What happens to the conductance of parent molecules (BDT/BDA) when different functional groups are attached?

9. DFT-based conductance calculations Qualitative effect as expected, but… Effect of side groups very weak, and … Calculated conductance 10-100 times larger than experimental values! D. J. Mowbray, G. Jones, and K. S. Thygesen, JCP 128, 111103 (2008)

10. Effect of side group on HOMO levels Correction for SI errors and image charge interactions. Image charge effect partially saves the day for DFT. D. J. Mowbray, G. Jones, and K. S. Thygesen, JCP 128, 111103 (2008) Correction for SI errors

11. Comparison to experiment (BDA@Au) D. J. Mowbray, G. Jones, and K. S. Thygesen, JCP 128, 111103 (2008)

12. M. Strange et al., PRL 101, 096804 (2008) W. H. Thijssen et al. PRL 96, 026806 (2006) M. Strange, et al. JCP 128, 114714 (2008) Some success stories Why does DFT work well in some cases while it fails in other cases?

13. Beyond the single-particle approximation Time-dependent DFT Stefanucci and Almbladh, Euro. Phys. Lett. 67, 14 (2004) Di Ventra and Todorov, J. Phys.:Cond.Mat. 16, 8025 (2004) Linear response Kubo formula Bokes, Jung and Godby, Phys. Rev. B 76, 125433 (2007) Rate equations + exact diagonalization Hettler, Wenzel, Wegewijs, Schoeller, Phys. Rev. Lett. 90, 076805 (2003) Many-body perturbation theory Darancet, Ferretti, Mayou and Olevano Phys. Rev. B 75, 075102 (2007) Thygesen and Rubio, J. Chem. Phys. 126, 091101 (2007) Ferretti, Calzolari, Di Felice, and Manghi, Phys. Rev. B 72, 125114 (2005)

14. The band gap problem of 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. Hartree-Fock exchange Screening correction Schilfgaarde, Kotani, and Faleev, PRL 96, 226402 (2006)

15. Many-body approach to quantum transport: GW in the central region Thygesen and Rubio, J. Chem. Phys 126, 091101 (2007) ; Phys. Rev. B 77, 115333 (2008) v xc  GW Two problems: Conventional GW gives quasi-particle excitations of the groundstate, but transport is a nonequilibrium phenomenon How to deal with interactions in infinite, non-periodic systems? Two (possible) solutions: Formulate GW on the Keldysh contour Assume that leads can be described at the mean-field (DFT) level, and include correlations only in a central region

16. Hamiltonian Non-interacting part: Kohn-Sham Hamiltonian: Interactions: MB Hamiltonian:

17. Non-equilibrium GFs and current The retarded GF of the central region: Embedding self-energies Interaction self-energy Correlation functions: Current from lead α into central region: Symmetrized current:

18. Non-equilibrium GW equations Interaction Full GF with coupling to leads Self-consistent solution of Dyson equation

19. Program- flow

20. Keldysh nonequilibrium formalism Dyson equation solved fully self-consistent with GW self-energy (charge conservation, no G 0 dependence) Full dynamical dependence of all quantities (no plasmon-pole approximation, no linearized quasi-particle equation) All quantities calculated in real time/frequency (no analytic continuation) Valence-core exchange included (known in PAW) Non-conventional features: Overview of GW-transport method Localized atomic orbital basis. Dynamical dependence sampled uniformly in real frequency/time. Ions described by PAW + frozen core approximation Product basis technique to reduce size of four-index quantities (W and P). Parallelization over orbitals and time/frequency grid.

21. PBE underestimates position of occupied states due to self- interaction errors. PBE0 improves PBE slightly. HF yields too deep-lying levels because of neglect of orbital relaxations Self-consistent GW corrects the HF energies by including dynamical screening (orbital relaxation). Small molecules (2-8 atoms) containing : H, C, N, Cl, O, F, S, P, Na, Li, Si Calculated HOMO level for 35 gas molecules C. Rostgaard, K. W. Jacobsen, and K. S. Thygesen, submitted

22. Calculated HOMO level for 35 gas molecules GW shifts occupied levels up in energy compared to HF GW systematically improves HF energies

23. Limitations: Strong vs. weak correlations Spectral functions:  Local interactions (Hubbard) - > strong correlations  Long-range interactions (1/R) - > weak correlations Entropy of reduced density matrix: Degree of correlation: Θ = S/S max, 0< Θ <1 Θ =0.52 Θ =0.11

24. Molecule@surface: Dynamic polarization effects S. Kubatkin et al. Nature, 425, 698 (2003) Experiments on OPV5 molecule transistors. J. Repp et al. PRL, 94, 026803 (2005) STS of pentacene adsorbed at NaCl/Cu thin films.

25. Energy cost of adding an electron to the LUMO is given by spectral function: Molecule@surface: Dynamic polarization effects

26. Microscopic model of metal-molecule interface

27. Dependence on metal-molecule interaction Gap reduction due to screening in metal (image charge formation). Open squares: Exact difference in total groundstate energy with an extra electron (hole) on the molecule. All many-body eigenstates are single Slater determinants: weakly correlated system Vanishing t hyb (weak physisorption) Thygesen and Rubio, Phys. Rev. Lett. 102, 046802 (2009)

28. Dependence on metal band width Vanishing t hyb (weak physisorption) Small t  Large metal DOS at E F  Large density response  Efficient screening GW quasiparticle is not just total energy difference, i.e. the QP has overlap with excited N+1 particle states of the metal.

29. Dependence on metal-molecule hopping The density response of the molecule increase with the coupling. Intra-molecular screening occurs via charge-transfer to the metal. Suggests a direct correlation between chemisorption bond strength and HOMO-LUMO gap reduction. Thygesen and Rubio, Phys. Rev. Lett. 102, 046802 (2009)

30. First-principles GW calculations: Physisorbed benzene z=4.5 Å G 0 W 0 calculations performed with the Yambo code(*). Yambo: G 0 W 0  LDA, Plane wave basis, norm-conserving pseusopotentials, plasmon pole approximation. (*) A. Marini, C. Hogan, M. Grüning, D. Varsano, arXiv:0810.3118 (2009) See also: J. B. Neaton et al. Phys. Rev. Lett. 97, 216405 (2006)

31. GW and LDA benzene HOMO-LUMO gaps on different surfaces LDA gaps are independent of substrate GW gaps show large variation across different surfaces GW gap sensitive to atomistic details, e.g. surface plane (BaO) J.M.Garcia, A. Rubio and KST, submitted 4.5 Å

32. Classical image charge model Best-fit values for  and z 0 : Electrostatic energy of point charge above a polarizable medium: Classical model describes the physics of the gap reduction qualitatively.

33. Variation of HOMO and LUMO levels GW: Symmetric effect on HOMO and LUMO. Exceptions Li and BaO(111) LDA: HOMO level agrees better with GW than does LUMO Very good agreement between LDA and GW for HOMO at metallic surfaces (error cancellation in LDA between self-interaction and image charge)

34. General trends in level shifts Semiconductors: Gap reduction increases with decreasing substrate band gap. Metals: Gap reduction increases with increasing substrate DOS at E F Li and BaO(111) deviate from general trend!

35. GW self-energy to second order in V Renormalization of single electronic level, , by non-local interactions with substrate electrons (time-ordered quantities): Substrate joint density of states weighted by particle-hole transitions

36. Quasiparticle self-consistent equation: Graphical solution to QP equation (“generic” ∆ corresponding to constant V kk’ ): Retarded self-energy: Effective interaction strength: Microscopic origin of general trends Substrate joint density of states weighted by particle-hole transitions Trends for both metals and semiconductors can be explained by assuming constant and system independent V kk’

37. Pt-H 2 -Pt: Density of states GW/HF SZ/DZP basis set. Image charge effect

38. Pt-H 2 -Pt: Transmission  More Pt atoms in GW region  Basis set  General trend: G DFT > G GW > G HF Dynamical screening (image charge effects)

39. Applying a bias voltage 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 Molecular DOS Current:

40. Impact of exchange-correlation on IV Peaks in dI/dV:  Position and width influenced by the bias!

41. Evolution of HOMO/LUMO levels in Hartree (crosses), HF (triangles), and GW (circles). HF and GW agree at low bias Quasi-particle scattering reduces electronic lifetimes at finite bias Enhanced dynamic screening reduces GW gap Impact of exchange-correlation on IV Thygesen, Phys. Rev. Lett. 100, 166804 (2008)

42. Acknowledgements Carsten Rostgaard Karsten W. Jacobsen Juan Maria Garcia Lastra Angel Rubio Collaborators: CAMD, Technical University of Denmark University of the Basque Country, Spain Funding: The Lundbeck Foundation Danish Center for Scientific Computing (DCSC)

43. Conclusions Bridging the ”experiment – theory gap” in single-molecule transport rely on proper incorporation of correlation effects beyond the mean-field (DFT) approximation. Band gap problem of DFT: Why does DFT-transport work for some systems? HF works well for isolated molecules Dynamical screening important at molecule-surface interfaces -> DFT levels may not be that wrong (error cancellations) Importance of correlation effects increase out of equilibrium

44. Anderson model