Time-dependent phenomena in quantum transport web: ag-gross Max-Planck Institute for Microstructure...

Time-dependent phenomena in quantum transport

web: http://users.physik.fu-berlin.de/~ag-gross

Max-Planck Institute forMicrostructure Physics

Halle

E.K.U. Gross

Electronic transport: Generic situation

left lead Lcentral

region Cright 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.


left lead Lcentral




Goal 1: Calculate current-voltage characteristics I(V)


left lead Lcentral




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


left lead Lcentral




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 uniqueGoal 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 Lcentral

region C right lead R

EE V,ET dE h

e )V(I 21 ff EE V,ET dE h

e )V(I 21 ff

Transmission function T(E,V) calculatedfrom static (ground-state) DFT

2

eV E F1,2

2

eV E F1,2

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!)

R

C

L

R

C

L

stat

RR

stat

RC

stat

CR

stat

CC

stat

CL

stat

LC

stat

LL

E

HH0

HHH

0HH

Define Green’s functions of the static leads

1 EGHE statL

statLL

1 EGHE statR

statRR

Substitute L and R in equation for central region

(HCLGLHLC + HCC + HCRGRHRC) C = E C

Effective KS equation for the central region

L := HCL GL HLC R := HCR GR HRC

g = ( E - HCC - L - R )-1

L L L -i - L L L -i - R R R -i - R R R -i -

gg R L racet T gg R L racet T

ChrysazineChrysazine

Chrysazine (a)

Chrysazine (b)

Chrysazine (c)

O

O

OH OH

0.0 eV 3.35 eV

0.54 eV 3.41 eV

1.19 eV 3.77 eV

Relative Total Energies and HOMO-LUMO Gaps

-7.21

-6.28

-4.49

-3.39-1.99

-4.10

-6.47

-6.75

-7.36-7.38

-7.59

En

erg

y (e

V)

-8

-7

-6

-5

-4

-3

-2

-5 0 5 10 15 20Density of States (1/eV)

0.00001 0.0001 0.001 0.01 0.1 1

Transmission Function (1/eV)

0

0.02

0.04

0.06

0.08

0.1

0 2 4 6 8

Cu

rren

t (

A)

Voltage (V)

(a) (b) (c)

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


Steady state is treated with ground-state DFT In particular, transmission function has peaks at the excitation energies of the gs KS potential.


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).


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)


• TD many-body perturbation theory (Kadanoff-Baym equation) --theory straightforward --numerically difficult

• TD denstity-functional theory (TD Kohn-Sham equation)




• TD denstity-functional theory (TD Kohn-Sham equation) --theory complicated (xc functional) --numerically simple




• 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 Lcentral


TDKS equation (E. Runge, EKUG, PRL 52, 997 (1984))

'rr

t'r'rdrtvrt't'rv 3

KSvxc[ (r’t’)](r t)

rt rtvm2

rtt

i jKS

22

j

t

t

t

tHtHtH

tHtHtH

tHtHtH

t

t

t

ti

R

C

L

RRRCRL

CRCCCL

LRLCLL

R

C

L

Electronic transport with TDDFT

left lead Lcentral


TDKS equation

Propagate TDKS equation on spatial grid

,t,r,t,rt 21A vector

nHamiltonia TDKS of blocks-grid ingcorrespondtHAB

is purely kinetic, because KS potential is local

BAtH for AB

RCCRLCCL HHHH , , , are time-independent

0HH RLLR

with grid points r1, r2, … in region A (A = L, C, R)

tHttHtHtt

i RCRCCCLCLC

tHttHt

i CRCRRR

tHttHt

i CLCLLL

Hence:

t

t

t

tHtHtH

tHtHtH

tHtHtH

t

t

t

ti

R

C

L

RRRCRL

CRCCCL

LRLCLL

R

C

L

L

C

R

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

L R

'tt't,tGtHt

i LLL

'tt't,tGtHt

i RRR

Define Green’s Functions of left and right leads:

0tHt

iG

0tHt

iG

RRRR

LLLL

r.h.s. of L solution of hom. SE

r.h.s. of R solution of hom. SE

explicity:

00,tiG'tH't,tG'dtt LL

t

0

CLCLL

00,tiG'tH't,tG'dtt RR

t

0

CRCRR

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, 035308 (2005)

ttHtt

i CCCC

source term: L → C and R → C charge injection

memory term: C → L → C and C → R → C hopping

00,tGiH00,tGiH RRCRLLCL

t

0

CRCRCRLCLCL 'tH't,tGHH't,tGH'dt

Note: So far, no approximation has been made.

Necessary input to start time propagation:

• lead Green’s functions GL, GR

• initial orbitals C(0) in central region as initial condition

for time propagation

Calculation of lead Green’s functions:

'tt't,tGtHt

i LLL

leadleft

extHxcnuclLL t,rVt,rVrVTtH

tUHtUrVrVTtH LstatLL

leadleft.s.g

statHxcnuclLL

tUHtH RstatRRRR tUtU, RL 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

0R

0C

0L

0R

0C

0L

statRR

statRC

statCR

statCC

statCL

statLC

statLL

E

HH0

HHH

0HH

Gives effective static KS equation for central region

0C

0C

statRC

statR

statCR

statLC

statL

statCL

statCC EHEGHHEGHH

In the traditional Landauer + static DFT approach, this equation is used to calculate the transmission function. Here we use it onlyto calculate the initial states in the C-region.

UV(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

2tforU

2t0fortsinUtU

0

20

L

same steady state!

U0 V(x)

left lead right leadcentral 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, 256802 (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) = Uosin(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 effectsAnalytical: G. Stefanucci, Phys. Rev. B, 195115 (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:

)t(II)t(I )D()S(

(D)bb' b b'

b,b '

I (t) sin[( )t]

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-stateoscillations 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

QD T biasL,R

ˆ ˆ ˆ ˆ ˆH t H H H H t

†T link 1 , 0

, i 1

ˆ ˆ ˆH V c c h.c.

i 1 , i ,i 1

ˆ ˆ ˆH t Vc c h.c.

bias i ,, i 1

ˆ ˆH t U t n

QD ext 0 0 0ˆ ˆ ˆ ˆH v n Un n

KS QD,KS T biasL,R

ˆ ˆ ˆ ˆ ˆH t H t H H H t

QD,KS KS 0 0ˆ ˆH t v n t n

0 0n t n t

KS 0 ext 0 xc 0

1v n t v Un t v n t

2

Solve TDKS equations (instead of fully interacting problem):

LDA functional for vxc 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, 146402 (2003)

1 1LDAxc xc xcv n 1 n v n n 1 v 2 n

1xc link

1 n nv n Un 2V cos cos

2 2

We use this functional as Adiabatic LDA (ALDA) in the TD simulations.

LDAxcv n has a discontinuity at n = 1 Note:

S. Kurth, G. Stefanucci, E. Khosravi, C. Verdozzi, E.K.U.G., Phys. Rev. Lett. 104, 236801 (2010)

Is this Coulomb blockade?

Is this Coulomb blockade?

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

Steady-state density as function of applied bias for KS potential with smoothened discontinuity

UL/V

Fingerprint of Coulomb blockade

S. Kurth, G. Stefanucci, E. Khosravi, C. Verdozzi, E.K.U.G., Phys. Rev. Lett. 104, 236801 (2010)

SFB 450SFB 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 450SFB 658SPP 1145

Research&Training Network

Define Green’s functions of the static leads

1 EGHE statL

statLL

1 EGHE statR

statRR

Effective static KS equation for central region

0C

0C

statRC

statR

statCR

statLC

statL

statCL

statCC EHEGHHEGHH

For bound states, this equation is solved iteratively.

For continuum states, the imaginary part, Im Gcc(E), of the

corresponding Green’s function is diagonalized.

Position dependence of current

TD current averaged over one period of traveling wave

Time-dependent phenomena in quantum transport web: ag-gross Max-Planck Institute for Microstructure...

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