Time-dependent phenomena in quantum transport web: Max-Planck Institute for Microstructure Physics Halle E.K.U. Gross

Electronic transport: Generic situation left lead L central region C right lead R Bias between L and R is turned on: U(t)V for large t A steady current, I, may develop as a result.

Electronic transport: Generic situation left lead L central region C right lead R Bias between L and R is turned on: U(t)V for large t A steady current, I, may develop as a result. Goal 1: Calculate current-voltage characteristics I(V)

Electronic transport: Generic situation left lead L central region C right lead R Bias between L and R is turned on: U(t)V for large t A steady current, I, may develop as a result. Goal 1: Calculate current-voltage characteristics I(V) Goal 2: Analyze how steady state is reached, determine if there is steady state at all and if it is unique

Electronic transport: Generic situation left lead L central region C right lead R Bias between L and R is turned on: U(t)V for large t A steady current, I, may develop as a result. Goal 1: Calculate current-voltage characteristics I(V) Goal 2: Analyze how steady state is reached, determine if there is steady state at all and if it is unique Goal 3: Control path of current through molecule by laser

left lead right lead Control the path of the current with laser

left lead right lead Control the path of the current with laser Necessary: Algorithm to calculate shape of optimal laser pulse with quantum optimal control theory

Outline Standard Landauer approach (using static DFT ) Why time-dependent transport? Computational issues (open, nonperiodic system) Recovering Landauer steady state within TD framework Transients and the time-scale of dephasing Electron pumping Undamped oscillations associated with bound states Oscillations associated with Coulomb blockade

Standard approach: Landauer formalism plus static DFT left lead L central region C right lead R Transmission function T(E,V) calculated from static (ground-state) DFT Comparison with experiment: Qualitative agreement, BUT conductance often 1-3 orders of magnitude too high.

eigenstates of static KS Hamiltonian of the complete system (no periodicity!) Define Green’s functions of the static leads

Substitute  L and  R in equation for central region (H CL G L H LC + H CC + H CR G R H RC )  C = E  C Effective KS equation for the central region  L := H CL G L H LC  R := H CR G R H RC g = ( E - H CC -  L -  R ) -1

Chrysazine Chrysazine (a) Chrysazine (b) Chrysazine (c) 0.0 eV3.35 eV 0.54 eV3.41 eV 1.19 eV3.77 eV Relative Total Energies and HOMO-LUMO Gaps

Energy (eV)

Possible use: Optical switch A.G. Zacarias, E.K.U.G., Theor. Chem. Accounts 125, 535 (2010)

Use ground-state DFT within Landauer formalism Fix left and right chemical potentials Solve self-consistently for KS Green’s function Transmission function has resonances at KS levels No empirical parameters, suggests confidence level of ground-state DFT calculations Summary of standard approach

Two conceptual issues:  Assumption that upon switching-on the bias a steady state evolves

Two conceptual issues:  Assumption that upon switching-on the bias a steady state evolves  Steady state is treated with ground-state DFT In particular, transmission function has peaks at the excitation energies of the gs KS potential.

Two conceptual issues:  Assumption that upon switching-on the bias a steady state evolves  Steady state is treated with ground-state DFT In particular, transmission function has peaks at the excitation energies of the gs KS potential. Hence, resonant tunneling occurs at wrong energies (even with the exact xc functional of gs DFT).

Two conceptual issues:  Assumption that upon switching-on the bias a steady state evolves  Steady state is treated with ground-state DFT In particular, transmission function has peaks at the excitation energies of the gs KS potential. Hence, resonant tunneling occurs at wrong energies (even with the exact xc functional of gs DFT). One practical aspect:  TD external fields, AC bias, laser control, etc, cannot be treated within the static approach

Three different ab-initio approaches TD many-body perturbation theory (Kadanoff-Baym equation) TD denstity-functional theory (TD Kohn-Sham equation) TD wave-function approaches (TD many-body Schrödinger equation)

Three different ab-initio approaches TD many-body perturbation theory (Kadanoff-Baym equation) --theory straightforward --numerically difficult TD denstity-functional theory (TD Kohn-Sham equation) TD wave-function approaches (TD many-body Schrödinger equation)

Three different ab-initio approaches TD many-body perturbation theory (Kadanoff-Baym equation) --theory straightforward --numerically difficult TD denstity-functional theory (TD Kohn-Sham equation) --theory complicated (xc functional) --numerically simple TD wave-function approaches (TD many-body Schrödinger equation)

Three different ab-initio approaches TD many-body perturbation theory (Kadanoff-Baym equation) --theory straightforward --numerically difficult TD denstity-functional theory (TD Kohn-Sham equation) --theory complicated (xc functional) --numerically simple TD wave-function approaches (TD many-body Schrödinger equation) --theory known --numerically very difficult --most accurate of all (when feasible)

Three different ab-initio approaches TD many-body perturbation theory (Kadanoff-Baym equation) --theory straightforward --numerically difficult TD denstity-functional theory (TD Kohn-Sham equation) --theory complicated (xc functional) --numerically simple TD wave-function approaches (TD many-body Schrödinger equation) --theory known --numerically very difficult --most accurate of all (when feasible)

Electronic transport with TDDFT left lead L central region C right lead R TDKS equation(E. Runge, EKUG, PRL 52, 997 (1984)) v xc [  (r’t’)](r t)

Electronic transport with TDDFT left lead L central region C right lead R TDKS equation

Propagate TDKS equation on spatial grid is purely kinetic, because KS potential is local are time-independent with grid points r 1, r 2, … in region A (A = L, C, R)

Hence: L CR Next step: Solve inhomogeneous Schrödinger equations, for  L,  R using Green’s functions of L, R, leads L R

Define Green’s Functions of left and right leads: r.h.s. of L solution of hom. SE r.h.s. of R solution of hom. SE explicity: insert this in equation C

Effective TDKS Equation for the central (molecular) region S. Kurth, G. Stefanucci, C.O. Almbladh, A. Rubio, E.K.U.G., Phys. Rev. B 72, (2005) source term: L → C and R → C charge injection memory term: C → L → C and C → R → C hopping Note: So far, no approximation has been made.

Necessary input to start time propagation: lead Green’s functions G L, G R initial orbitals  C (0) in central region as initial condition for time propagation

Calculation of lead Green’s functions: likewise total potential drop across central region Simplest situation: Bias acts as spatially uniform potential in leads (instantaneous metallic screening)

initial orbitals in C region eigenstates of static KS Hamiltonian of the complete system Gives effective static KS equation for central region In the traditional Landauer + static DFT approach, this equation is used to calculate the transmission function. Here we use it only to calculate the initial states in the C-region.

U V(x) left lead right leadcentral region Numerical examples for non-interacting electrons Recovering the Landauer steady state Time evolution of current in response to bias switched on at time t = 0, Fermi energy  F = 0.3 a.u. Steady state coincides with Landauer formula and is reached after a few femtoseconds U

Transients Solid lines: Broken lines: Sudden switching same steady state! U0U0 V(x) left leadright lead central region barrier height: 0.5 a.u.  F = 0.3 a.u.

ELECTRON PUMP Device which generates a net current between two electrodes (with no static bias) by applying a time- dependent potential in the device region Recent experimental realization : Pumping through carbon nanotube by surface acoustic waves on piezoelectric surface ( Leek et al, PRL 95, (2005) )

Archimedes’ screw: patent 200 b.c. Pumping through a square barrier (of height 0.5 a.u.) using a travelling wave in device region U(x,t) = U o sin(kx-ωt) (k = 1.6 a.u., ω = 0.2 a.u. Fermi energy = 0.3 a.u.)

Experimental result: Current flows in direction opposite to sound wave

Current goes in direction opposite to the external field !!

Bound state oscillations and memory effects Analytical: G. Stefanucci, Phys. Rev. B, (2007)) Numerical : E. Khosravi, S. Kurth, G. Stefanucci, E.K.U.G., Appl. Phys. A93, 355 (2008), and Phys. Chem. Chem. Phys. 11, 4535 (2009) If Hamiltonian of a (non-interacting) biased system in the long-time limit supports two or more bound states then current has steady, I (S), and dynamical, I (D), parts: Note: -  bb’ depends on history of TD Hamiltonian (memory!) Questions: -- How large is I (D) vs I (S) ? -- How pronounced is history dependence? Sum over bound states of biased Hamiltonian

1-D model: start with flat potential, switch on constant bias, wait until transients die out, switch on gate potential with different switching times to create two bound states note: amplitude of bound-state oscillations may not be small compared to steady-state current History dependence of undamped oscillations

amplitude of current oscillations as function of switching time of gate question: what is the physical reason behind the maximum of oscillation amplitude ?

So far: systems without e-e interaction Next step: TDKS, i.e. inclusion of e-e- interaction via approximate xc potential time-dependent picture of Coulomb blockade

Model system U

Solve TDKS equations (instead of fully interacting problem): LDA functional for v xc is available from exact Bethe-ansatz solution of the 1D Hubbard model. N.A. Lima, M.F. Silva, L.N. Oliveira, K. Capelle, PRL 90, (2003)

We use this functional as Adiabatic LDA (ALDA) in the TD simulations. has a discontinuity at n = 1Note:

S. Kurth, G. Stefanucci, E. Khosravi, C. Verdozzi, E.K.U.G., Phys. Rev. Lett. 104, (2010)

Is this Coulomb blockade?

Steady-state equation has no solution in this parameter regime (if v KS has sharp discontinuity) !

Steady-state density as function of applied bias for KS potential with smoothened discontinuity U L /V Fingerprint of Coulomb blockade S. Kurth, G. Stefanucci, E. Khosravi, C. Verdozzi, E.K.U.G., Phys. Rev. Lett. 104, (2010)

SFB 450 SFB 658 Research&Training Network

PRIZE QUESTION No 4 Consider the “standard approach” to the calculation of I(V), i.e. the combination the Landauer formula with ground-state DFT. Suppose you have the exact (ground-state) KS potential of the junction. Which features of the I(V) characteristics do you expect to come out correctly, and which do you expect to come out wrong (even though the calculation is done with the exact functional). What will go wrong, in addition, if the xc functional is approximated by LDA.

SUMMARY Standard static DFT + Landauer approach: Chrysazine as optical switch TDDFT approach to transport properties -- Electron pumping -- Persistent current oscillations from transitions between bound states -- Memory effect: amplitude of oscillations depends on history -- TD picture of Coulomb blockade -- Discontiuity of xc potential of crucial importance Optimal laser control of -- Chirality of current in quantum rings -- Path of wave packet in real space

SFB 450 SFB 658 SPP 1145 Research&Training Network

Define Green’s functions of the static leads Effective static KS equation for central region For bound states, this equation is solved iteratively. For continuum states, the imaginary part, Im G cc (E), of the corresponding Green’s function is diagonalized.

Position dependence of current TD current averaged over one period of traveling wave