Electron-electron interactions in a one-dimensional quantum wire spin filter
P. Devillard,1 A. Cr´pieux,2 K. I. Imura,3 and T. Martin2
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arXiv:cond-mat/0501145v3 [cond-mat.mes-hall] 6 Jun 2005

2

1
Centre de Physique Th´orique, Universit´ de Provence, Case 907, 13331 Marseille Cedex 03, France
e
e
Centre de Physique Th´orique, Universit´ de la M´diterran´e, Case 907, 13288 Marseille Cedex 9, France
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e
e
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3
Condensed Matter Theory Laboratory, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan

The combined presence of a Rashba and a Zeeman effect in a ballistic one-dimensional conductor
generates a spin pseudogap and the possibility to propagate a beam with well defined spin orientation. Without interactions transmission through a barrier gives a relatively well polarized beam.
Using renormalization group arguments, we examine how electron-electron interactions may affect
the transmission coefficient and the polarization of the outgoing beam.
PACS numbers: 71.70.Ej, 72.25.Dc, 72.25.Mk

Over the last decade, spintronics1 has emerged from
mesoscopic physics and nanoelectronics as a field with
implications in both quantum information theory2 and
for the storage of information. While the charge is routinely manipulated in nanoelectronics, the issue here is to
exploit the spin degree of freedom of electrons. In particular, spin filters are therefore needed to control the input
and the output of spintronic devices. A recent proposal3
explained the operation of a spin filter for a one dimensional wire under the combined operation of Rashba spin
orbit coupling and Zeeman splitting. Yet, electronic interactions in one dimensional wires are known to lead to
Luttinger liquid behavior and to the renormalization of
scattering coefficients. It therefore important to inquire
about the role of electronic interactions in the above mentioned spin filter. This is aim of the present paper.
A decade ago, a transistor based on the controlled precession of the electron spin due to spin orbit coupling was
proposed4 . Indeed, the Rashba effect5 in a semiconductor, can be modulated by a gate voltage, which controls
the asymmetry of the potential well which confines the
electrons. Since this proposal, many spintronic devices
based on the Rashba effect have been proposed, based
on a single electron picture6,7,8,9 . At the same time, onedimensional wires are now available experimentally. This
has motivated several efforts10 to study the interplay between the Rashba effect and electron interactions in infinite one-dimensional wires. However, little has been said
about Coulomb interactions in spintronic devices. Our
starting point stems from the fact that Rashba devices
are gated devices, in which the interaction are assumed
to be screened and weak. We will therefore use a perturbative renormalization group treatment of the Coulomb
interaction in order to address its consequence on the spin
dependent transmission. This approach will be justified
by estimating the strength of electron-electron interaction in single channel semiconductor devices.
Here, we consider a narrow ballistic wire which is submitted to spin orbit coupling with the Rashba term being
dominant. In order to obtain a spin-polarized beam of
electrons propagating in one direction, a small Zeeman
field is introduced3 . By confining in the y direction with
lateral gates, the problem becomes unidimensional and

the Hamiltonian for a one-dimensional wire reads3 :
H0 =

p2
α Ez
ǫZ
σ −
B.σ,
px σy +
∗ 0
2m
2B

(1)

where m∗ is the effective mass, Ez is the electric Rashba
field perpendicular to the layer and α depends on the
material used. B is the magnetic field, ǫZ is the Zeeman
energy. σ0 is identity matrix and σ = (σx , σy , σz ) the
usual Pauli matrices. The eigenvectors are thus products
of plane waves times a spinor. The eigenstates are:
2

E =

2m∗

2
2 2
kx ± 2 κ4 + kα kx ,
Z

∗

(2)

∗

where κ2 = m2 (ǫZ /2) and kα = m2 α Ez . The Rashba
Z
2
2
energy is defined through Eα = 2m∗ kα . The orientation
of eigenvectors is such that for E ≫ Eα , the spinor associated to the mode with larger wave-vector is directed
along |↑ y . In the interval [Eα − ǫZ /2 , Eα + ǫZ /2], there
is only one propagating mode (with two chiralities). The
other mode is evanescent. The dispersion relation for the
propagating mode is indicated in Fig. 1. The existence
of a pseudogap in a given energy interval has been used
in Ref. 3, to propose a spin-filtering device, where a potential step of height V1 corresponding to a gate voltage
permits to shoot in the middle of the pseudogap (Fig. 1).
Electron-electron interactions are taken into account,
with Vint (x), the Coulomb interaction potential. Following Ref. 11 which discusses the effect of weak electron interactions in a single mode 1D wire, we use a
Hartree-Fock approach followed by a poor man Anderson
renormalization in energy space. The Dyson equation in
Hartree-Fock approximation reads
ψk (x) = φk (x) +
+

dy

dy Gr (x, y) VH (y)ψk (y)
k
dz Gr (x, y)Vex (y, z)ψk (z), (3)
k

whith ψk (φk ) the one-electron wave function in the presence (absence) of interactions, and Gr (x, y) is the rek
tarded Green function. VH (x) = dy Vint (x − y)n(y), is

2
the Hartree potential, with:
2
n
| ψq (y) |2 ,

n(y) =
n=1 |q|<kF

(4)

(n)

the local density. Note that the sum is carried over the
n
two modes ψq (propagating and evanescent), for all wavevectors q with energies smaller than EF (at T = 0). kF (n)
are the wave-vectors of modes n at the Fermi energy. The
exchange potential (Fock) is given by the matrix
Vex (x, y) = −Vint (x − y)
4

×

∗

n=1 |q|<kF (n)

n
n
ψq,u (y)ψq,u (x)
0
∗
n
0
ψq,d (y)n ψq,d (x)

,
(5)

n
n
n
where ψq,u and ψq,d are the components of ψq (x) in the
basis {|↑ z , |↓ }z . Assuming a finite range d for the
Coulomb interaction, we will consider in our perturbative calculation only the terms in Eq. (3) which give the
leading logarithmic divergence for the scattering coefficients at the Fermi level, as in Ref. 11.

E+Eα

a)

EF
2.

1.

V1
0.
0.

k x /kα 0.3

b)
e

r 1,1 e

ik +x

−ik+x

t

t

1,1

1,2

v k+
v K+

v k+
v K−

e

iK + x

from the left of the step with an incident wave of wavevector k+ and group velocity vk+ , thus generating a reflected wave of amplitude r1,1 and two transmitted waves,
vk+ iK x
+
vK+ e

t1,1

and t1,2

vk+ iK x
−
,
vK− e

K+ being the larger

wave-vector. The two other diffusion states are obtained
by injecting electrons from the right of the step with
wave-vector Ki (i = + or −), thus generating reflected
vK
′
waves of the form ri,j vKi eiKj x and transmitted waves
j

t′
i,1

vKi
vk+

e−ik+ x , j = + or −. The square root factors

insure the unitarity of the S matrix12 .
In the geometry of Fig. 1, the S matrix thus takes the
form


′
r1,1 t′
1,1 t2,1
′
′
(6)
S =  t1,1 r1,1 r2,1  .
′
′
t1,2 r1,2 r2,2

We introduce the physical parameter η ≡ ǫZ /Eα which
is related to the ratio of the Zeeman energy to the Rashba
energy. The other relevant parameter is the ratio:
Γ=

EF
,
Eα

(7)

EF is the energy of the Fermi level measured as in Fig.
1a. In our case, EF is connected to the height V1 of the
step. In order to inject electrons from the left, in the
middle of the pseudogap, EF must be equal to Eα + V1 .
Next, in order to have four propagating modes at the
right of the step as in Fig. 1, V1 must be larger than
ǫZ /2. Thus, the minimum value of Γ is 1 + ǫz /2Eα .
If η → 0, Γ → 1. In the renormalization procedure,
following the usual procedure13 , one then rescales the
bandwidth from D0 , the real bandwidth, to D. Here D0
is assumed to be larger than the step height V1 . One then
obtains flow equations for the quantities t1,1 , t1,2 , t′ and
1,1
vk+
˜
t′ . It is more convenient to use ti,j = ti,j
, and
2,1
vKj

e

iK − x

V1

FIG. 1: a) The thin lines denote the energies E+Eα of the two
Rashba bands as a function of wave-vector ratio kx /kα both
on the left of the potential step and on the right, for η = 0.3
and Γ = 1.5. The potential step of height V1 is indicated
as a thick line. The dashed line is the Fermi energy EF . b)
One of the three scattering states. An incident wave eik+ x is
injected on the left hand side of the step and generates two
transmitted waves and one reflected wave.

In the case where the Fermi level lies in the middle
of the pseudogap (Fig. 1), there are three scattering
states. The first one corresponds to injecting electrons

˜i,1
t′

=

t′
i,1

vKi
vk+

. As an example, we quote the equation

˜
for t1,1 ,
˜
dt1,1
˜
˜′
= −(hv K )−1 t1,1 | r2,1 |2
d ln( D0 )
D
where v K =

1
2

vK+
′
˜ ′ ˜′ ∗
+t1,2 r1,1 r1,2 J1 ,
vK−
(8)

′
vK+ + vK− . The integral J1 is defined

′
′
′
as J1 = I1 − I0 where
′
I0 = −

vK− ˆ
V (KF + + KF − )
2v K

d
, (9)
2
vK ˆ (K+ − KF + ) + (K− − KF − )
= − −V
2v K
2
d
× ln (K+ − KF + ) + (K− − KF − )
, (10)
2
× ln (K+ + K− ) − (KF + +KF − )

′
I1

3

It has a determinant of −1 instead of +1 because the
number of propagating modes is not the same on each
side. The S matrix stays real symmetric under the renormalization group flow and thus, only two real equations
for t1,1 and t1,2 are needed.
When injecting electrons from the left of the step (exactly in the middle of the pseudogap as in Ref. 3), one
defines the polarization of the outgoing beam as:

1

2

2

t 1,2

0

0.5

FIG. 2: Solid line: location of the initial points. Upper dashed
line: renormalization group flow for Γ = 1. The starting point
is at t1,1 = 1. Lower dashed line: renormalization group
trajectory for Γ = 10. The dashed-dotted line is simply the
first diagonal, corresponding to totally unpolarized beams and
reaches t2 = 1 at t2 = 1.
1,1
1,2

We describe what happens first, if the height of the step
is small with respect to the Rashba energy, second, in the
reverse situation. In the first case, Γ is barely larger than
1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

The value p = 0 corresponds to a totally unpolarized
beam (experimentally not desirable). The value p = 1
corresponds to a perfectly polarized outgoing beam. According to Eq. 11, p is simply Γ−1 . It tends to zero as the
height of the step gets large with respect to the Rashba
energy. The propagating modes are either oriented along
the |↑ y or along the reverse direction.

Let l = ln D0 be the renormalization parameter and
D
′
A = J1 /(2π vK ). It is easier to express the renor¯
malization group equations in terms of the variables
x = t2 + t2 and y = t2 − t2 . They read:
1,1
1,2
1,1
1,2

polarization

(12)
total transmission

| t1,1 |2 − | t1,2 |2
.
p =
| t1,1 |2 + | t1,2 |2

starts is a curve in this plane, which is represented in Fig.
2 as a solid line. In addition, the behavior of some flow
trajectories is depicted in this figure.

t 1,1

ˆ
where V (k) denotes the Fourier transform of Vint . The
˜
˜1,1
˜2,1
renormalization equations for t1,2 , t′ and t′ have a
′
similar structure. J1 > 0 for repulsive interactions.
Any S matrix can be written as exp(−iu.λ) with u an
8-dimensional real vector and the components of λ are
˜
˜
˜1,1
˜2,1
real matrices (SU (3)). Since t1,1 , t1,2 , t′ and t′ are
generally complex numbers, our renormalization equations give 8 relations for real parameters and thus completely determine the flow of the S matrix. We now focus
on the limit η → 0 (small Zeeman coupling). The S matrix then simplifies and becomes a real and symmetric
matrix:
√
√ Γ−1 
 (Γ−1)2
2 Γ −2 Γ Γ+1
Γ+1




√
S = (Γ + 1)−1  2 Γ
0
Γ − 1  . (11)




√ Γ−1
Γ
−2 Γ Γ+1 Γ − 1
4 Γ+1

0
0

2

4

6

8

10

A Ln T in /T

(14)

FIG. 3: Total transmission and polarization as a function of
the logarithm of the temperature. The two curves marked
with left arrows represent the total transmission for Γ = 1.01
(solid line) and Γ = 10 (dashed line). The two other curves
represent the spin polarization for Γ = 1.01 (dash-dotted line)
and Γ = 10. (dashed line) respectively.

There are only 3 fixed points. The stable fixed point
x = 0, y = 0 corresponds to a perfectly reflecting step;
both t1,1 and t1,2 are zero. The unstable fixed point
x = y = 1 corresponds to t1,1 = 1 and t1,2 = 0, i.e.
perfect transmission with no mode conversion. The second unstable fixed point x = 1, y = −1 corresponds
to t1,2 = −1 and t1,1 = 0 (complete mode conversion).
When we have no electron-electron interactions, the location of the points where one starts the renormalization
procedure, in the plane (t2 , t2 ), depends on one pa1,2 1,1
rameter only, Γ. Thus, the ensemble of points where one

1 (1 ideally). We then start from an initial temperature
(bandwidth) Tin = D0 /kB , where t1,1 is close to 1 and
t1,2 close to zero, which means a perfect polarization of
the outgoing beam. The point moves first in a direction
such that the polarization diminishes significantly but
the total transmission remains approximately constant.
Then, the curve bends downwards and eventually moves
towards the origin (no transmission). The polarization
goes to a constant value.
In the second case (not desired in practice), Γ is large,
the initial polarization is already low (equal to 2/Γ) and

√
√
dx
= −A(x2 − y 2 ) 1 − x 1 − 1 − x ,
dl
dy
1
= − A(x2 − y 2 )y .
dl
2

(13)

4
still decreases towards a constant value. The transmission goes quickly to zero.
The renormalization procedure must be interrupted at
some stage given by D ≃ kB T . In the case where Γ is
close to 1 (small potential step), Eqs. (13) and (14) show
that the total transmission first decreases very slowly as
2
2
1 − rin − 8rin A ln2 T /Tin , where rin is the initial reflexion coefficient. rin goes to zero as Γ goes to 1. The
polarization decreases as ln(Tin /T ). For lower temperatures, the polarization goes to a constant Γ dependent
value, which is equal to 1/2 for Γ = 1. The total trans−1
mission goes to zero, asymptotically like ln(Tin /T ) 2 ,
but this regime is only attained for unrealistic values of
temperatures.
A characterization of such a spin filter requires the
comparison of both the total transmission coefficient, together with the evolution of the polarization under renormalization group flow. This information is illustrated in
Fig. 3 where both quantities are plotted as a function of
the logarithm of the inverse temperature. For Γ close to
1, the total transmission stays constant and close to unity
until the temperature is dropped by several orders of
magnitude, and then decreases in a monotonous fashion.
At the same time, the polarization drops faster than the
total transmission, signifying that the quality of the spin
filtering effect is polluted by electronic interactions before the total transmission is truly affected. For large Γ,
the total transmission first decreases monotonously (linear behavior in ln T ) starting from the initial bandwidth,
1
then giving place to a slower decrease ((ln T )− 2 ). The
polarization stays approximately constant in this case,
but at a deceptively low value of 0.2.
A useful way to quantify electron interactions is to
compute the Luttinger parameter g which is expected for
a gated heterostructure. To be specific we consider the
geometry of Ref. 14, where the Coulomb interaction in
the 1D channel is screened by the few transverse modes
in the wire and by the proximity of the 2DEG and gates:
V (r) ≃ (e2 /4πǫ0 ǫ)e−r/λs /r (λs ∼ 100nm, a fraction of
the width of the wire, W ∼ 20nm). Averaging over the

1

2

3
4
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7

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Rashba devices15 , but the typical parameters are comparable.
To summarize, we have looked at the effect of weak
electron electron interactions in one dimensional ballistic
quantum wires under the combined Rashba and Zeeman
effects. At the single electron level this device has the
advantage of working as a spin filter. We have characterized the influence of electron electron interactions in
this same device. We found that in the most relevant case
(EF −Eα ≪ 1) where the total transmission remains close
to unity for a range of bandwidths, the quality of spin
filtering properties decreases substantially. Although the
present work deals with a sharp step, the present approach can be extended to steps whose extension is much
larger than the Fermi wave length, using WKB-type approximations. Transmission through the step is likely to
be enhanced in this case, but the tendency for interactions to spoil filtering will remain valid. Also, this single
electron picture can be further complicated in practice:
for the case where two spin orientations are present (on
the right hand side of the step), it was shown16 that the
combined effect of spin orbit coupling and strong electron electron interactions provide some limitations to the
Luttinger liquid (metallic) picture, giving rise to spin or
charge density wave behavior instead. Nevertheless, the
present perturbative treatment of the Coulomb interaction is justified here because the one dimensional wire is
surrounded by nearby metallic gates in order to implement the Rashba effect.
We thank P. Stˇeda and F. Hekking for useful discusr
sions and comments. This work is supported by a CNRS
A.C. Nanosciences grant.

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