Download Scattering of electrons from an interacting region

Scattering of electrons from an interacting
region
Abhishek Dhar
Raman Research Institute,
Bangalore, India.
Collaborators:
Diptiman Sen (IISc, Bangalore)
Dibyendu Roy (Weizmann Institute, Israel).
Phys. Rev. Lett. 101, 066805 (2008)
Scattering of electrons from an interacting region – p.1/23
The problem of transport
L
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A
A
I
V
∆V
I=
= G∆V
R
1
G=
= conductance of system
R
For macroscopic systems it is usual to define the
resistivity and conductivity of the material
RA
ρ=
l
1
σ= .
ρ
Scattering of electrons from an interacting region – p.2/23
Calculating conductivity
Microscopic theories for σ:
◮
Kinetic theory (Boltzmann transport theory). Think
of diffusing electrons with mean collision time τc .
ne2 τc
σ=
m
.
◮
Green-Kubo formula:
σ = () lim lim
τ →∞ L→∞
Z
τ
dthJ(0)J(t)i
0
Infinite system size limit necessary.
Scattering of electrons from an interacting region – p.3/23
Small systems
What about transport in mesoscopic systems and
nanosystems ?
Scattering of electrons from an interacting region – p.4/23
Small systems
Scattering of electrons from an interacting region – p.5/23
Calculating conductance
Above theories are not directly applicable. It is not
meaningful to talk of conductivity. Rather one is
interested in the conductance:
I
.
G=
∆V
Main question: How do we calculate this?
Conductivity: Intrinsic property of system.
Conductance: Properties of reservoirs (leads) and
contacts important and should be incorporated into
calculation.
Scattering of electrons from an interacting region – p.6/23
Non-interacting electrons:
Landauer formalism
This is the most popular approach in mesoscopic
physics. Views conduction as a quantum mechanical
transmission problem. Simplest version:
Scatterer
µL T
µR TR
L
1D leads containing
non−interacting electrons
(initially in thermal equilibrium)
Electronic scattering states given by:
ψk (x) = eikx + rk e−ikx
left lead
= tk eikx right lead
Transmission: T (ǫk ) = |tk |2 .
Scattering of electrons from an interacting region – p.7/23
Landauer formula
µL
µR
e
I=
2π~
Z
dǫT (ǫ)[f (µL , TL , ǫ) − f (µR , TR , ǫ)]
For TL = TR = 0:
e
e2
I =
T (ǫF )∆µ =
T (ǫF )∆V
2π~
2π~
I
e2
G =
= T (ǫF ) .
∆V
h
The Landauer formula. →Keldysh formalism, quantum
Langevin equations also give this. Scattering of electrons from an interacting region – p.8/23
Interacting electrons
What happens when electrons DO NOT interact while
in the leads, but DO interact in the sample region. This
is a harder problem.
HR
HL
HS=H0
+ VI
S
V
LS
µL
TL
VRS
µR TR
HS0 + HL + HR is non-interacting (quadratic
Hamiltonian).
Coupling VC = VLS + VRS is also quadratic.
VI is non-quadratic and represents interactions in sample.
Scattering of electrons from an interacting region – p.9/23
General approach
Finding density matrix of nonequilibrium steady state
(NESS): Solution in two stages.
◮
Start with VC = VI = 0 and
eq
(µ
,
T
)
⊗
ρ
⊗
ρ
ρ(t = 0) = ρeq
L
L
S
R (µR , TR )
L
◮
Let H0 = HS0 + HL + HR + VC . Evolve for infinite time
ESS
.
using H0 and find ρN
0
◮
ESS
. Evolve again for an infinite time
Start with ρN
0
ESS
using H0 + VI and find ρN
.
I
ˆ N ESS ].
Calculate: J = T r[Jρ
I
Our contribution: Solving this problem at zero
temperature .
Scattering of electrons from an interacting region – p.10/23
Zero temperature case
ESS
→ |φi which is a many-particle state
In this case ρN
0
satisfying:
H0 |φi = E|φi .
The state |φi is known exactly. It is formed of single
particle states, |φi = |k1 , k2 , .....kN i, and consists of right
moving states (k > 0) filled up to µL and left moving
states (k < 0) filled up to µR .
With this as the “incident” state we try to find the
“scattering state” |ψi satisfying the equation:
(H0 + VI )|ψi = E|ψi .
ESS
and we calculate the
|ψi corresponds to ρN
I
ˆ
current using hψ|J|ψi.
Scattering of electrons from an interacting region – p.11/23
Lippman-Schwinger theory
(H0 + V )|ψi = E|ψi
For “incident” state |φi the solution is given by:
1
|ψi = |φi +
VI |ψi
E + iη − H0
= |φi + G0 VI |ψi
= |φi + G0 V |φi + G0 VI G0 VI |φi + ... ,
1
where G = E+iη−H
is the non-interacting Green’s
0
function.
Can find scattering state (and thus the nonequilibrium
steady state) perturbatively.
Scattering of electrons from an interacting region – p.12/23
Earlier work
◮
Mehta and Andrei, PRL (2006)
For particular model with δ-function interaction
find exact many-particle scattering state by
Bethe-Ansatz. Find a solution corresponding to the
correct incident state. Use this to find exact
steady state current.
◮
Goorden and Buttiker, PRL (2007)
Find two-particle scattering state in a two channel
problem with interactions in a local region.
◮
Nonequilibrium Kondo problem: Results from
nonequilibrium Green’s function, Numerical RG,
Density Matrix RG.
Scattering of electrons from an interacting region – p.13/23
Model of 1D spinles Fermions
Dot
−4
−3 −2 −1
Left Lead
◮
0
1
U
2
3
4
5
Right Lead
Hamiltonian of the model,
HL = −
∞
X
(c†x cx+1 + c†x+1 cx ),
x=−∞
VI = U n 0 n 1 ,
◮
Single particle state: φk (x) = eikx
Energy ǫk = −2 cos k and −π < k ≤ π.
Scattering of electrons from an interacting region – p.14/23
Two particles
◮
Two particle incoming state specified by
k = (k1 , k2 ) given by:
φk (x) = ei(k1 x1 +k2 x2 ) − ei(k2 x1 +k1 x2 )
Ek = ǫk1 + ǫk2 .
◮
Two-particle scattering state can be found
exactly. Let 0 = (1, 0).
ψk (x) = φk (x) + U KEk (x)ψk (0)
where KEk (x) = hx|G+
0 (Ek )|0i
¯
φk (0)
ψk (0) =
¯ (0)]
[1
−
U
K
Ek
¯
¯
Scattering of electrons from an interacting region – p.15/23
Two particle S-matrix
◮
Two electrons from the noninteracting leads with
initial momenta (k1 , k2 ) emerge, after scattering,
with momenta (k1′ , k2′ ). At x = (x1 , x2 ) one has
x1
x2
=
.
′
′
sin(k1 )
sin(k2 )
◮
Energy is conserved, i.e., Ek = Ek′ ; but momentum
is not conserved because the interaction term
U n0 n1 breaks translation invariance.
◮
Probably not solvable by Bethe-Ansatz.
◮
Bound states.
Scattering of electrons from an interacting region – p.16/23
Wavepacket dynamics
◮
We numerically study time evolution of a
two-particle wave-packet which passes throught
the interacting region.
◮
We form the wave-packet with the complete set
of the exact two-particle scattering eigenstates
and determine their time-evolution through:
Z π
Z q1
1
−iEq t
Ψ(x, t) =
dq
dq
a(q)ψ
(x)
e
,
1
2
q
2
(2π) −π
−π
X
where a(q) =
Ψ(x, t = 0) ψq∗ (x).
x1 >x2
Scattering of electrons from an interacting region – p.17/23
Wavepacket dynamics
(a) incident wave-packet, (b) after passing through
the origin with U = 0, (c) after passing through the origin with U = 2.
Scattering of electrons from an interacting region – p.18/23
Two-particle current.
◮
Current is given by the expectation value of the
operator jx = −i(c†x cx+1 − h.c.) in the scattering
state |ψk i = |φk i + |Sk i.
◮
Current in the incident state is given by
hφk |jx |φk i = 2[sin(k1 ) + sin(k2 )]N ,
N = total number of sites in the entire system.
◮
Change in current due to scattering,
δj(k1 , k2 ) = hSk |jx |Sk i + hSk |jx |φk i + hφk |jx |Sk i
2|φk (0)|2 Im[KEk (0)]
=
[sgn(k1 ) + sgn(k2 )]
¯
¯
2
|1/U − KEk (0)|
¯
where sgn(k) ≡ |k|/k. δj ∼ U 2 .
Scattering of electrons from an interacting region – p.19/23
N-particle generalisation
◮
For incident state |φkN i = |k1 , k2 , ...kN i we cannot
find |ψkN i exactly for N ≥ 3.
◮
Scattered wave is given by |ψkN i = |φkN i + |SkN i.
Do second order perturbation theory
|SkN i = G0 VI |kN i + G0 VI G0 VI |kN i + ....
◮
Three processes at O(U 2 ).
◮
Consider only two-particle scattering.
Scattering of electrons from an interacting region – p.20/23
Change in Landauer current
◮
hkN |ĵ|kN i gives the Landauer current.
◮
Change in current value is given by:
δjN = hψkN |ĵ|ψkN i − hkN |ĵ|kN i
Z
Z
1
=
dk1 k2 δj(k1 , k2 ) .
2
2(2π)
We find that the Landauer current e2 /h is reduced by
a term of order U 2 . In the presence of impurities
reduction is O(U ).
Scattering of electrons from an interacting region – p.21/23
Other applications
◮
More general dot with applied gate voltage.
◮
Study of resonance behaviour in systems like
parallel and series double dots.
Dot
−4
−3 −2 −1
0
1
2
U
Left Lead
3
4
5
Right Lead
II
U
−4
−3
−2
−1
0
0
1
2
3
4
I
Scattering of electrons from an interacting region – p.22/23
Other applications
◮
Parallel conductors in proximity to each other and
interacting in some localised region.
−4
−3 −2 −1
0
1
2
3
4
−4 −3 −2 −1
0
1
2
3
4
◮
Electrons with spin, interactions on more sites.
◮
Entanglement by interactions.
Lippman-Schwinger scattering theory provides a nice
framework to study zero temperature nonequilibrium
steady states of electrons driven across an interacting
region by a finite chemical potential bias.
Scattering of electrons from an interacting region – p.23/23