Optical control of transport through
molecular junctions
Ulrich Kleinekathöfer
School of Engineering and Science, Jacobs University Bremen
Warwick, August 2009
W.H. Zurek, Los Alamos Science 27, 2 (2002). quant-ph/0306072
i~
d
|Ψ(x, t)i = H(x, t)|Ψ(x, t)i
dt
mi
~ i (t)
d 2R
~ i = −∇V
~ i (t))
~ (R
=F
dt 2
Outline
1
Density matrices and damped harmonic oscillators
2
Light-harvesting in purple bacteria
3
Transport through molecular wires
4
Hole Transfer in DNA Driven by Solvent Fluctuations
Outline
1
Density matrices and damped harmonic oscillators
2
Light-harvesting in purple bacteria
3
Transport through molecular wires
4
Hole Transfer in DNA Driven by Solvent Fluctuations
Different systems: one theory
harmonic oscillator or coupled two-level-systems
coupled to bosonic thermal bath
Thermal
bath
System
tight-binding model coupled to two fermionic reservoirs
Laser
Reservoir
System
Reservoir
Reduced density matrix formalism
Goal: description of ultra-fast (fs) processes in dissipative
systems / molecular wires
full quantum dynamics including dephasing, energy dissipation
but also coherences and accurate laser-matter interaction
splitting in relevant system and bosonic / fermionic reservoirs
Laser
Reservoir
System
σ - density matrix of the full system
(relevant system + bath)
i~
dσ(t)
= [H(t), σ(t)]
dt
Reservoir
Reduced density matrix formalism
Goal: description of ultra-fast (fs) processes in dissipative
systems / molecular wires
full quantum dynamics including dephasing, energy dissipation
but also coherences and accurate laser-matter interaction
splitting in relevant system and bosonic / fermionic reservoirs
Laser
Reservoir
System
Reservoir
reduced density-matrix:
ρ = trB (σ) - density matrix of the relevant system
i~
dρ(t)
= [HS (t), ρ(t)]+D(t)ρ(t)
dt
System-bath coupling
Hamiltonian
H = HS + HB + HSB
every environmental degree of freedom only slightly distorted
⇒ modeled by harmonic oscillators
how strongly does the environment absorb energy?
⇒ spectral density J(ω)
perturbation theory in the system-bath coupling HSB
either time-nonlocal theory (time-convolution)
Z t
dρ(t)
= LS ρ(t) +
dt 0 K (t 0 )ρ(t 0 )
dt
0
or time-local theory (time-convolutionless)
Z t
dρ(t)
= LS ρ(t) +
dt 0 K (t 0 )ρ(t)
dt
0
Hierachical scheme: Damped harmonic oscillator
population dynamics of third excited state
1
1
population
0.8
0.6
0.4
0.2
0
0
N=2
N=4
N=6
N=8
0.8
population
N=2
N=4
N=6
N=8
0.6
0.4
0.2
2
4
t/t0
6
time-nonlocal
8
10
0
0
2
4
t/t0
6
8
10
time-local
M. Schröder, M. Schreiber, U. Kleinekathöfer, J. Chem. Phys. 126, 114102 (2007)
Outline
1
Density matrices and damped harmonic oscillators
2
Light-harvesting in purple bacteria
3
Transport through molecular wires
4
Hole Transfer in DNA Driven by Solvent Fluctuations
Molecular dynamics simulation of LH-II
LH-II complex of
Rhodospirillum molischianum
about 110 000 atoms
using parallel MD code NAMD2
500-3000 snapshots every 2 fs
Energy gaps of single BChls
Fast quantum chemical calculation for each snapshot
configuration: ZINDO for each separate BChl incl. point charges
using ORCA
ground states
excited states
Energy gaps of single BChls
Fast quantum chemical calculation for each snapshot
configuration: ZINDO for each separate BChl incl. point charges
using ORCA
ground states
excited states
Spectral density of the protein environment
Autocorrelation function of the energy gap ∆Ej
#
"
16
N−i
1 X
1 X
∆Ej (ti + tk )∆Ej (tk )
C(ti ) =
16
N −i
j=1
k =1
Spectral density
2
J(ω) = tanh
π
ω
2kb T
Z∞
dt C(t) cos ωt
0
30
Spectral Density [a.u.]
25
20
Simulation length
t=1ps (B800)
t=3ps (B800)
t=6ps (B800)
t=6ps (B850)
15
10
5
0
0
0.05
0.1
0.15
ω [eV]
0.2
0.25
Quantum mechanical model for the B850 ring
16 coupled two-level systems
coupled to a thermal bath,
characterized by its spectral density J(ω)
only transfer between neighboring sites
determination of spectra in perturbation theory
Absorption spectra for B850 ring
B850
B800
1.0
I(ω)
0.8
0.6
0.4
0.2
0.0
1.3
1.4
1.5
ω [eV]
red: experiment
green: direct from MD simulation
blue: quantum mechanical model
1.6
1.7
M. Schröder, U. Kleinekathöfer, M. Schreiber, J. Chem. Phys. 124, 903 (2006)
A. Damjanović, I. Kosztin, U. Kleinekathöfer, K. Schulten, Phys. Rev. E 65, 919 (2002)
Absorption spectra for B850 ring
B850
B800
1.0
I(ω)
0.8
0.6
0.4
0.2
0.0
1.3
1.4
1.5
ω [eV]
red: experiment
green: direct from MD simulation
blue: quantum mechanical model
1.6
1.7
Quantum Biology
M. Schröder, U. Kleinekathöfer, M. Schreiber, J. Chem. Phys. 124, 903 (2006)
A. Damjanović, I. Kosztin, U. Kleinekathöfer, K. Schulten, Phys. Rev. E 65, 919 (2002)
Outline
1
Density matrices and damped harmonic oscillators
2
Light-harvesting in purple bacteria
3
Transport through molecular wires
4
Hole Transfer in DNA Driven by Solvent Fluctuations
Molecular wires
first reproducible experiments on molecular wires
break junctions, STM setups, DNA wires, . . .
influence of laser light on molecular wires gives an
opto-electronic coupling
with femtosecond laser pulses: high spatial as well as temporal
resolution
The model
E(t)
E
E
JL(E)
EF,l n1
JR(E)
∆
n2
E
F,r
The model
H(t) = HS (t) + HL + HSL
X
†
HS (t) =
(En + Un (t))cn† cn − ∆(cn† cn−1 + cn−1
cn )
n
HL =
X
ωq cq† cq
q
HLS
X
=
(Vq c1† cq + Vq∗ cq† c1 )
q
E(t)
E
E
JL(E)
EF,l n1
JR(E)
∆
n2
E
F,r
Population dynamics
E(t)
E
E
JL(E)
EF,l n1
JR(E)
∆
n2
E
F,r
Un (t) = A(t)δ1n − A(t)δ2n
E1 = E2 = EF ,r + 1∆ = EF ,l − 1∆
Coherent destruction of tunneling (CDT)
E(t)
E
E
JL(E)
EF,l n1
Un (t) = A(t)δ1n − A(t)δ2n
JR(E)
∆
n2
A(t) = A0 sin(ωt)
E
F,r
3
2
1
0
-1
-2
E
∆ = 0.1 eV
I [e]
0.01
0
I [e]
1 [e] ≈ 2.4 × 10−4 A
0.02
0.01
0
0
50
100
150
time [fs]
200
250
J. Lehmann, S. Camalet, S. Kohler, P. Hänggi, Chem. Phys. Lett. 368, 282 (2003)
Coherent destruction of tunneling (CDT)
E(t)
E
E
JL(E)
EF,l n1
JR(E)
∆
n2
E
F,r
Un (t) = A(t)δ1n − A(t)δ2n
A(t) = A0 sin(ωt)
J. Lehmann, S. Camalet, S. Kohler, P. Hänggi, Chem. Phys. Lett. 368, 282 (2003)
Coherent destruction of tunneling (CDT)
E(t)
E
E
JL(E)
EF,l n1
JR(E)
∆
n2
Un (t) = A(t)δ1n − A(t)δ2n
A(t) = A0 sin(ωt)
E
F,r
J. Lehmann, S. Kohler, V. May, P. Hänggi, J. Chem. Phys. 121, 2278 (2003)
CDT: Short laser pulse
E(t)
E
E
Un (t) = A(t)δ1n − A(t)δ2n
−(t − T )2
A(t) = A0 exp
σ2
JL(E)
n1
JR(E)
∆
n2
3
2
1
0
-1
-2
E
F,r
E
EF,l I [e]
0.02
0.01
I [e]
0
0.01
0
0
100
200
300
time [fs]
400
500
U. Kleinekathöfer, G.-Q. Li, S. Welack, M. Schreiber, Europhys. Lett. 75, 139 (06).
Predefined current pattern
goal: determine laser field which creates predefined current
(target is non-local in time)
Z
J0 (E) = dt {P(t) − tr{I(t)ρS (t)}}2
Z
λ tf (E(t) − Ẽ(t))2
.
J(t, E) = J0 (E) +
dt
2 t0
s(t)
P(t) - predefined current pattern
Ẽ(t) - laser field of the previous iteration
functional derivative of J with respect to E(t)
E(t) = Ẽ(t)
Z
s(t)
δρ (τ )
−
dτ {2tr{I(τ )ρS (τ )} − 2P(τ )} tr{I(τ ) S
}
λ
δE(τ )
Z
s(t)
δI(τ )
−
dτ {2tr{I(τ )ρS (τ )} − 2P(τ )} tr{
ρ (τ )}
λ
δE(τ ) S
Optimal control with Gaussian target
E(t)
E
E
JL(E)
n1
JR(E)
∆
n2
E
F,r
2
I [nA]
1.5
1
0.5
0
0.3
E [eV]
EF,l 0.2
0.1
0
0
100
Time [fs]
200
G.-Q. Li, S. Welack, M. Schreiber, U. Kleinekathöfer, Phys. Rev. B 77, 075321 (08).
Optimal control with step function target
E(t)
E
E
JL(E)
n1
JR(E)
∆
n2
E
F,r
2
I [nA]
1.5
1
0.5
0
0.4
E [eV]
EF,l 0.2
0
-0.2
0
100
Time [fs]
200
G.-Q. Li, S. Welack, M. Schreiber, U. Kleinekathöfer, Phys. Rev. B 77, 075321 (08).
Optimal control with oscillatory target
E(t)
E
E
JL(E)
JR(E)
∆
n2
E
F,r
I [nA]
n1
2
1.5
1
0.5
0
2
1
0
-1
0.8
0.6
0.4
0.2
0
0
I [nA]
E [eV]
EF,l 100
200
300
Time [fs]
G.-Q. Li, S. Welack, M. Schreiber, U. Kleinekathöfer, Phys. Rev. B 77, 075321 (08).
Spin current through quantum dot: CDT
E(t)
E
E
JL(E)
EF,l
JR(E)
ε
EF,r
E
ε
3
2
1
0
-1
-2
-3
0.02
spin-up current
I [nA]
0.015
spin-down current
total current
0.01
0.005
0
0.02
spin-up current
spin-down current
total current
_
I [nA]
0.015
0.01
0.005
0
0
200
400
Time [fs]
600
800
A. F. Amin, G.-Q. Li, A. H. Phillips, U. Kleinekathöfer, Eur. Phys. J. B 68, 103 (09).
Spin current through quantum dot: CDT
E(t)
E
E
JL(E)
EF,l
JR(E)
ε
EF,r
E
ε
3
2
1
0
-1
-2
-3
0.015
spin-up current
spin-down current
total current
I [nA]
0.01
0.005
0
spin-up current
spin-down current
total current
I [nA]
_
-0.005
0.005
0
0
200
400
Time [fs]
600
800
A. F. Amin, G.-Q. Li, A. H. Phillips, U. Kleinekathöfer, Eur. Phys. J. B 68, 103 (09).
Spin current through quantum dot: Optimal control
E(t)
E
E
JL(E)
EF,l
JR(E)
ε
EF,r
0.01
I [nA]
ε
0.005
E [eV]
0
0.2
0.1
0
200
400
600
800
Time [fs]
A. F. Amin, G.-Q. Li, A. H. Phillips, U. Kleinekathöfer, Eur. Phys. J. B 68, 103 (09).
Outline
1
Density matrices and damped harmonic oscillators
2
Light-harvesting in purple bacteria
3
Transport through molecular wires
4
Hole Transfer in DNA Driven by Solvent Fluctuations
Computational Methodology
double-stranded DNA species of sequences GTn GGG with n =
1, 2, 3, 4, 5, 7, 10 and 14
first, a classical MD simulation of the DNA
CT parameters: TB Hamiltonian consisting of site energies and
electronic couplings based on the SCC-DFTB method
time-dependent Schrödinger equation
i~
∂
Ψ = HΨ
∂t
initial state on one end and sink on the other end
large fluctuations of site energies in the order of 0.4 eV,
dramatically reduced barrier heights
solvent fluctuations introduce a significant correlation between
neighboring sites
Survival of hole P(t) vs. time in GTGGG
a) 100 simulations, 20 ps each.
b) survival with the static model (50 × longer time scale)
Occupation of bridge A by the hole in GTGGG
a) averaged time dependence from dynamical simulations and
the result with the completely static model
b) occupation of A-bridge in all GTn GGG sequences; the
averaged time dependence from dynamical simulations
Rate constant of hole transfer in GTn GGG
data from full MD-based calculations as well as those based on
constant site energies and on constant electronic couplings
T. Kubař, U. Kleinekathöfer, M. Elstner, J. Phys. Chem. B (in press) (09).
Rate constant of hole transfer in GTn GGG
parameters calculated with the inclusion of environment
(QM/MM) and without that (‘in vacuo’)
T. Kubař, U. Kleinekathöfer, M. Elstner, J. Phys. Chem. B (in press) (09).
Comparison of dynamical and statistical models
T. Kubař, U. Kleinekathöfer, M. Elstner, J. Phys. Chem. B (in press) (09).
Survival of hole in GT14 GGG
parameter set generated in dynamical simulation (a) and the
statistical model (b)
Acknowledgments: The Group
R. Schulz, G.-Q. Li, J. Liebers, S. Pezeshki, C. Olbrich, A. Amin,
(L. Moevius, S. Welack, M. Schröder)
K. Schulten and his team (Urbana-Champaign)
T. Kubar and M. Elstner (TU Braunschweig)