Time-dependent phenomena in quantum transport web: ag-gross Max-Planck Institute for Microstructure Physics Halle E.K.U.
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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.
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.
Goal 1: Calculate current-voltage characteristics I(V)
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.
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 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.
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
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 Lcentral
region Cright lead R
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
region Cright lead R
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
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