Correlated electron current in a quantum point contact:
0.7 structure in conductance.
A.I. Milstein
G.I. Budker Institute of Nuclear Physics, Novosibirsk, Russia
Collaboration with C. Sloggett and O.P. Sushkov, UNSW, Sydney;
Eur. Phys. J. B 61, 427 (2008).
The conductance of a quantum
point contact (QPC) - a 1D constriction in a 2D electron gas is
quantized in units of G0 = 2e2/h ,
van
Wees
et
al,
Rev.Lett.60,848 (1988),
Phys.
Wharam et al, J. Phys.
L209 (1988).
C 21,
Thomas et al, Phys. Rev. Lett. 77, 135 (1996)
Typical parameters: n ∼ 3.6×1011 cm−2, F ∼ 10 meV, λF ∼ 40 nm,
le ∼ 9µm, L ∼ 1 µm, W ∼ 250 nm.
2
The observed conductance plateaus can be easily understood in the
single-electron picture (R. Landauer, Phys.Lett A 85, 91 (1981);
M. Büttiker, Phys. Rev. B 41, 7906 (1990). )
The non-equilibrium Fermi-Dirac distribution, n0k , under a small applied voltage V is
∂nf
eV 0
0
n0k = nf + s nf , nf = −
≈ δ(ε − µ) ,
2
∂ε
where nf = nf (E) is the equilibrium Fermi-Dirac distribution, the
sign s = k/|k| shows the direction of current flow, and µ is the chemical potential. Then we obtain the Landauer formula
Z
Ldk k
2e2
e
tεn0k =
tµV , G = G0tµ
J =2
L
2π m
h
where tµ is transmission probability.
3
U
x
Near the center of QPC
1
1
2
2
U = U0 − mω x + mωy2y 2 ,
2
2
Here m is the effective electron mass ( m ≈ 0.07me in GaAs), U0
depends on the gate voltage, typical values in experiment are ω ∼
1meV and ωy ∼ 4meV.
The transmission probability is
1
ε − ε0
tε =
, =
, ε0 = U0 + ωy (n + 1/2) .
−2π
ω
1+e
4
It is necessary to explain:
• ”0.7 structure” for n = 0 and absence of this structure for n 6= 0 ,
• Temperature dependence of conductance,
• Magnetic field dependence of conductance.
Thomas et al, Phys. Rev. Lett. 77, 135 (1996)
5
Different approaches
• Spontaneous magnetization;
Chuan-Kui Wang, Berggren, Phys. Rev. B 54, 14257 (1996); 57, 4552 (1998),
Calmes, Gold, Solid State Commun. 106, 139 (1998),
Zabala, Puska, Nieminen, Phys. Rev. Lett. 80, 3336 (1998),
Spivak,Zhou, Phys. Rev. B 61, 16730 (2000),
Bruus, Cheianov, Flensberg, Physica E 10, 97 (2001),
Starikov, Yakimenko, Berggren, Phys. Rev. B 67, 235319 (2003),
Reilly, Phys. Rev. B 72, 033309 (2005),
• Charge density wave;
Sushkov, Phys. Rev. B 64, 155319 (2001); 67, 195318 (2003);
• Kondo-type models;
Meir, Hirose, Wingreen, Phys. Rev. Lett. 89, 196802 (2002); Phys. Rev. Lett. 90, 026804
(2003),
Cornaglia, Balseiro, Europhys. Lett., 67, 634 (2004).
6
• Electron-electron interaction without assumptions of spin polarization or localized states;
Matveev, Phys. Rev. Lett. 92, 106801 (2004);
Kindermann, Brouwer, Phys. Rev. B 74, 125309 (2006),
Meidan, Phys. Rev. B 72, 121312(R) (2005).
Schmeltzer, Saxena, Bishop, Smith, Phys. Rev. B 71, 045429
(2005),
Syljuåsen, Phys. Rev. Lett. 98, 166401 (2007),
Rech, Matveev, Phys. Rev. Lett. 100, 066407 (2008),
Lunde, De Martino, Schulz, Egger, Flensberg, arXiv:0901.1183
7
Electron-electron interaction.
• For = 0 and n = 0, outside the barrier the electron-electron
Coulomb interaction is strongly screened. The interaction is unscreened only on the top of the barrier where the electron density
is low. For higher steps there are always lower channels penetrating the QPC. This leads to high electron density in the contact and
hence to screening of the effective interaction and the absence of
pronounced structures in higher conductance steps.
The e-e interaction in the n = 0 channel can be approximated as
e2
→ Hint = ωπ 2geδ(ξ1)δ(ξ2) ,
κ|x1 − x2|
√
where κ is the dielectric constant and ξ = mωx is the dimensionless distance. In GaAsFor, κp≈ 13 , ω ∼ 1meV, and the dimensionless
2
e
coupling constant ge ∼ π2κ m/ω ∼ 1 .
Interaction between electrons with opposite spins!
8
Single-particle wave function
Away from the barrier for k > 0,
1 ikx
1
−ikx
ψk (x) = √ e + Rk e
, x < 0; ψk (x) = √ Tk eikx, x > 0 .
L
L
Near the potential top
s
!1/4
√ −iπ/4
mvF2
ϕk (ξ)
eπ/2
√ , ϕk (ξ) =
ψk (x) ≈
Di−1/2( 2ξe
),
2ω
cosh(π)
L
where vF is the Fermi velocity far from the barrier, Dν is the
parabolic cylinder function. The probability density at the top of the
potential,
ρ() = |ϕk (0)|2 = √
π exp(π/2)
,
2
2 cosh(π) |Γ(3/4 − i/2)|
is peaked at ≈ 0.2. This results in enhancement of e-e interaction
and in the end leads to all effects considered.
9
4
ΡHΕL
ÈjHΞLÈ2 , Ε=0
5
3
2
1
-10
-5
0
Ξ
5
10
1.75
1.5
1.25
1
0.75
0.5
0.25
-2
-1
0
1
2
Ε
The peak in ρ() is not a resonance behavior such as that in the Wolff
model or the Kondo or Anderson models. These models assume a
single particle resonance or a virtual state that give a localized electron. The peak is due to semiclassical slowing and it is unrelated to a
resonance or virtual level.
10
3
Operator of electric current
ĵ is a single particle operator. Matrix elements hk|ĵ|k 0i are nonzero
only if k 0 = k or k 0 = −k,
k
k
2
hψk |ĵ|ψk i = e |Tk | = e t ,
m
m
k ∗
hψ−k |ĵ|ψk i = e Rk Tk .
m
We used the relation Rk∗ Tk + Rk Tk∗ = 0. Therefore ĵ can be represented as
eXk †
†
ĵ =
tak,σ ak,σ + Rk∗ Tk a−k,σ ak,σ ,
L
m
k,σ
The total electric current is given by
eXk
tnk,σ ,
J=
L
m
k,σ
†
nk,σ = hak,σ ak,σ i ,
11
†
ha−k,σ ak,σ i = 0 .
Account for e-e interaction
Two-leg diagrams, renormalization of potential. This effect does not
materially change the profile of the transmission coefficient.
Four-leg diagrams describing inelastic scattering gives a nontrivial
correction to the conductance.
12
Kinetic equation.
• Equilibration is due to collisions in leads;
• The equilibrium density matrix is diagonal in the basis of the scattering states that are stationary single particle states.
Kinetic equation which takes into account e-e interaction in the QPC:
∂nk
nk − n0k
=−
+ St(nk ) ,
∂t
Zτ
2
Ldk1 Ldk2 Ldk3 St(nk ) = 2π
Mkk1k2k3 2π 2π 2π
× nk2 nk3 (1 − nk )(1 − nk1 ) − nk nk1 (1 − nk2 )(1 − nk3 )
×δ(Ek + Ek1 − Ek2 − Ek3 ) .
Here τ is the relaxation time in the leads,
eV 0
n0k = nf + s nf ,
2
13
∂nf
0
nf = −
.
∂ε
Solution of kinetic equation at ge 1 and T ω.
X
2
T 2eV L3
.
[s
+
s
]
δ(E
−
µ)
−
s
−
s
M
St(nk ) = −
1
2
3
k
kk
k
k
1 2 3
12vF3
s s s
1 2 3
All legs in the matrix element are taken at the Fermi surface, so summations is performed over the directions s1, s2, s3.
T 2eV L3
2
2
St(nk ) = −
sδ(E
−
µ)
|M
|
+
|M
|
++−−
+++−
k
3
3vF
2
4
2
µ − ε0
π ge vF T
4
= −seV
ρ (µ̃)δ(Ek − µ) , µ̃ =
.
6 L ω
ω
The steady-state solution of the kinetic equation:
(
)
2
eV
π 4ge2 τ vF T
nk = nf +
sδ(Ek − µ) 1 −
ρ4(µ̃) .
2
3 L
ω
14
Conductance in units of G0 at ge 1 :
(
)
π 4ge2 τ vF T 2 4
tµ = tµ 1 −
ρ (µ̃) .
3 L
ω
1
tHΜL
0.8
0.6
0.4
0.2
-0.5
0
Μ
0.5
1
Conductance in units of 2e2/h versus µ̃ = (µ − ε0)/ω for different
vF
temperatures; ge 1, τ L
= 1. The uppermost curve corresponds to
geT = 0, while the lowest is geT ≈ 0.3K.
15
The constant is not small, ge ∼ 1, and virtual rescattering must be
taken into account.
a
b
c
d
Renormalization of the coupling constant due to rescattering:
ge → ge + δge ,
Z∞ Z∞
δge(µ) = 2ge2K(µ) ,
θ(1 − µ)θ(2 − µ)
1
ρ(1) ρ(2)
K(µ) =
4
2µ − 1 − 2
−∞−∞
θ(µ − 1)θ(µ − 2)
θ(µ − 1)θ(2 − µ)
+
−2
d1 d2 .
1 + 2 − 2µ
1 − 2
16
The second order correction alone is not sufficient. Since the kernel
K(µ) is independent of external momenta, the Brueckner approximation (summation of a geometrical progression) gives
ge
1
2
2
gR =
, or gR = ge R(µ) , R(µ) =
.
2
1 − 2geK(µ)
[1 − 2geK(µ)]
RHΜL
1.5
1
1
0.8
tHΜL
0.5
KHΜL
0
-0.5
0.6
0.4
0.2
-1
-0.5
-0.5 0
0.5
Μ
1
1.5
2
0
Μ
0.5
1
Plots of K(µ) and R(µ) for ge = 1 (solid line) and ge = 2 (long dashed
line). Plots of conductance tµ for different temperatures, using with
ge → gR(ge) for ge = 1.
17
Longitudinal magnetic field.
The interaction between electrons with parallel spins vanishes for the
contact Hamiltonian. Account for the longitudinal magnetic field is
equivalent to the replacement
1
tµ → (tµ + tµ0 ) , ρ4(µ) → ρ2(µ)ρ2(µ0) , µ0 = µ − 2gsµB B ,
2
with the corresponding substitutions in K(µ).
Since ρ(µ) is a peaked function, a magnetic field B ∼ ω/(2gsµB ) ∼
10T effectively switches off the interaction.
18
g = 1, T = 0.2K
g = 0.4, T = 0.5K
g = 0.2, T = 1K
ω = 6K; values of 2gsµB B/ω correspond to B from zero to 6T in
0.375T and also 12T. Dashed line: T = 0 and B = 0.
19
Conclusion.
• Within perturbation theory, we have considered transport of correlated electrons through a
quantum point contact. At zero temperature, the approach results in the usual Landauer formula
and the conductance does not show any structures.
• At nonzero temperature the electron-electron interaction gives rise to a current of correlated
electrons which scales as T 2 at very low temperatures. The corresponding correction to conduc2
2
tance is negative and strongly enhanced in the region 0.5 2eh ≤ G ≤ 1.0 2eh .
• We believe that the correction we found explains the ”0.7 structure” of conductance observed
experimentally.
• Our model is consistent with the experimental behavior of the 0.7 structure under a magnetic
field: a field smoothly “switches off” the effective interaction between electrons.
• The considered effects have a very simple physical origin: the electron wave function at the
barrier and hence the electron-electron interaction is strongly peaked when the transmission coefficient is slightly higher than 0.5.
20