Mesoscopic Stern-Gerlach spin filter by nonuniform spin-orbit interaction

Abstract

A novel spin filtering in two-dimensional electron system with nonuniform spin-orbit interactions (SOI) is theoretically studied. The strength of SOI is modulated perpendicular to the charge current. A spatial gradient of effective magnetic field due to the nonuniform SOI causes the Stern-Gerlach type spin separation. The direction of the polarization is perpendicular to the current and parallel to the spatial gradient. Almost 100 % spin polarization can be realized even without applying any external magnetic fields and without attaching ferromagnetic contacts. The spin polarization persists even in the presence of randomness. Comment: 6 pages, 5 figures (2 color figures), to appear in Phys. Rev. B, Rapid Commun

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arXiv:cond-mat/0409161v2 [cond-mat.mes-hall] 14 Jun 2005
Mesoscopic Stern-Gerlach spin filter by nonuniform spin-orbit interaction
Jun-ichiro Ohe1, Masayuki Yamamoto2, Tomi Ohtsuki1and Junsaku Nitta3
1Department of Physics, Sophia University, Kioi-cho 7-1, Chiyoda-ku, Tokyo 102-8554, Japan
2I. Institut f¨ ur Theoretische Physik, Universtit¨ at Hamburg, Jungiusstrasse 9, 20355 Hamburg, Germany
3NTT Basic Research Laboratories, NTT Corporation,
Atsugi-shi, Kanagawa 243-0198, Japan, and CREST-JST,
4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan
(Dated: February 2, 2008)
A novel spin filtering in two-dimensional electron system with nonuniform spin-orbit interactions
(SOI) is theoretically studied. The strength of SOI is modulated perpendicular to the charge current.
A spatial gradient of effective magnetic field due to the nonuniform SOI causes the Stern-Gerlach
type spin separation. The direction of the polarization is perpendicular to the current and parallel
to the spatial gradient. Almost 100 % spin polarization can be realized even without applying
any external magnetic fields and without attaching ferromagnetic contacts. The spin polarization
persists even in the presence of randomness.
Considerable attention has been devoted to the manipulation of the electron spin in semiconductor systems.1,2To
control the spin of electrons in semiconductors, the spin-orbit interaction (SOI) due to the lack of the inversion symme-
try in two dimensional electron system (2DES) is quite useful since its strength can be controlled by an additional gate
voltage.3,4The spin field-effect transistor proposed by Datta and Das is a hybrid structure of ferromagnetic electrodes
(FM) and a semiconductor (SM) 2DES channel.5This hybrid device, however, requires a complex and careful fabri-
cation process. Furthermore, spin injection efficiency from FM into SM is poor because of a conductance mismatch
between FM and SM.6Therefore, it is desired that spin polarized carriers in semiconductor channels can be generated
and manipulated without attaching any ferromagnetic contacts and without applying any external magnetic fields.
Several devices based on the SOI are proposed to have spin polarized carriers in semiconductor channel.7,8,9,10,11,12,13
From experimental point of view, the spin polarization mechanism should be robust against disorder.
One of the historical experiments of the spin separation is the Stern-Gerlach experiment.14They have considered
a particle with spin propagating through the nonuniform magnetic field. Because the derivative of the magnetic field
plays a role of the spin-dependent potential, particles with up and down spins are accelerated in opposite directions.
However, for electrons, this effect is hard to observe because of the effect of the Lorentz force acting on a electron
beam in transverse directions.15
The importance of the modulated SOI on the transport properties has recently been stressed.16,17,18,19,20Since the
effect of SOI on the propagating electrons is, in some respect, similar to the effective magnetic field, the modulated
SOI is expected to cause the Stern-Gerlach like spin separation. In this paper, we theoretically demonstrate that
a mesoscopic Stern-Gerlach spin filter is feasible by using a nonuniform spin-orbit interaction as shown in Fig. 1.
The strength of the SOI is modulated along the direction perpendicular to the charge current. We demonstrate that
nearly 100% spin polarization (perpendicular both to the current and the direction normal to 2DES) can be achieved
without a ferromagnetic contact and an external magnetic field. The above results are obtained both from the wave
packet dynamics and from the transfer matrix calculation of the transmission coefficients. We have found that the
large polarization is obtained when the electron propagates via the lowest channel where the transverse mode of the
wave function contains only single antinode. We also investigate the effect of randomness. The results show that the
polarization of the current survives as long as the randomness is not so strong.
We consider a 2DES in x-y plane and the current is assumed to flow in the x-direction, while fixed boundary
condition is imposed in the y-direction. A square lattice is considered for modeling 2DES and only the nearest
neighbor hopping is taken into account. The tight binding lattice spacing a and the hopping energy V0=
m∗is the effective electron mass are taken as the unit length and the unit energy, respectively. The region in which
the SOI exists is Lx× Ly= 60 × 30, where Lxand Lyare the length and the width of the system. We attach ideal
leads to both sides of this region. To include the SOI, we use the Ando model21described by the Hamiltonian,
¯ h2
2m∗a2 where
H =
?
i,σ
Wic†
iσciσ−
?
<i,j>,σ,σ′
Viσ,jσ′c†
iσcjσ′
(1)
with
Vi,i+ˆ x=
?
cosθ(y)
−sinθ(y) cosθ(y)
sinθ(y)
?
,(2)

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and
Vi,i+ˆ y=
?
cosθ(y)
−isinθ(y)
−isinθ(y)
cosθ(y)
?
, (3)
where c†
along x-(y-)direction. Wiis the random potential distributed uniformly in the range [−W/2,W/2]. Unless explicitly
stated, we consider the impurity free case. The strength of SOI is characterized by θ(y)
iσ(ciσ) is a creation (annihilation) operator of an electron on the site i with spin σ and ˆ x(ˆ y) the unit vector
θ(y) =2θmax
Ly
y − θmax. (4)
The Hamiltonian of the Rashba SOI22is generally described as
HR=1
¯ h{α(y)pxσy−(α(y)pyσx+ pyα(y)σx)
2
}, (5)
where α(y) is the strength of the SOI. The relation between α(y) and θ(y) is given by
α(y) ≃ 2θ(y)V0a( for θ(y) ≪ 1).(6)
The spin separation mechanism we propose in this paper is as follows. Equation (5) means that the effective
Zeeman field in the y-direction appears when an electron propagates in the x-direction. If the effective Zeeman field
has gradient in the y-direction, up and down spin electrons are accelerated in opposite directions. Unlike the Stern-
Gerlach experiment, this effect is expected to be easily observed even if the particles have charge. Since the spins are
expected to align in the y-direction, hereafter we concentrate on the polarization in the y-direction.
It is possible to make a spatial gradient of the SOI in the y-direction e.g. by using two gate electrodes which partially
cover the channel, and the change in the SOI strength between the two electrodes ∆α has been experimentally obtained
to be 0.4 − 0.8 × 10−11[eVm].3,4Let us assume a = 30 [nm] and m∗= 0.05m0where m0is the free electron mass.
Then the system area is 1.8 × 0.9 [µm2], V0≃ 0.85 [meV] and ∆α ≃ 0.64 × 10−11[eVm] by setting ∆θ(y) = 0.04π.
We therefore choose θmaxto be 0.02π.
To investigate the electron transport in the nonuniform SOI system, we calculate the time evolution of the
wave packet by the equation-of-motion method based on the exponential product formula.23The charge density
?
assumed to be
σ(|?↑ |ψσ?|2+ |?↓ |ψσ?|2) of the initial wave packet is shown in Fig. 2. The initial wave packet with spin σ is
ψσ(t = 0) = Asin
?
πy
Ly+ 1
?
exp
?
ikxx −δk2
xx2
4
?
χσ, (7)
with
χ↑=
1
√2
?1
i
?
, χ↓=
1
√2
?
1
−i
?
. (8)
We set kx= 0.5 and δkx= 0.2.
The wave packet after time evolution is shown in Fig. 3. The charge density splits into the upper and the lower parts
with its spin polarizations opposite. The maximum value of?
|?↓ |ψσ?|2)| are almost the same, which suggests that nearly 100 % spin filtering has been achieved.
To reinforce the above finding, we have also calculated the polarization of the current by the Landauer formula.24
Figure 4 shows the polarization of the current as a function of the electron Fermi energy E in units of the hopping
energy. Inset of this figure is the schematic view of the system. Three-terminal geometry has been employed to
distinguish the current through the upper and lower part of the system. The system area including the nonuniform
SOI is again 60 × 30 and the width of the upper and lower leads in the right hand side is 5. We assume that the
chemical potential of the reservoir attached to the left hand side lead is E + eV with V the voltage, while both
of the upper and lower leads in the right hand side are attached to the reservoirs with chemical potential E. The
transmission coefficients are calculated via the transfer matrix method25extended to include spins.26In order to
realize the multi-terminal system, the static potential is assumed in the middle of the right leads. We confirmed that
the result of the transfer matrix method agrees with the result of the Green function method27up to 4 digits.
The polarization of the current is defined as
σ(|?↑ |ψσ?|2+|?↓ |ψσ?|2) and that of |?
σ(|?↑ |ψσ?|2−
Py=T↑− T↓
T↑+ T↓,(9)

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where Tσ=e2
with spin σ. We focus on the component from the lowest channel of the left lead which corresponds to the result of
the time evolution of the wave packet. As shown in Fig. 4 spin filtering effect is clearly obtained in the nonuniform
SOI system. It should be noted that the polarization along z-direction can be obtained in the uniform SOI system
by considering multi-terminal geometry.8However, the polarization has the strong energy dependence and the energy
region where the large polarization occurs is very narrow.
In this calculation, we consider the current in the lowest channel of the left lead where there are no nodes in the
wave function along the transverse direction (see Eq. (7)). In order to obtain the high polarization of the current, we
found that the lowest channel is expected only in the left lead while the channel mixing is allowed in the sample. Inset
of Fig. 4 shows the polarization which takes into account the whole channel. The spin separation effect of the higher
channel becomes weaker due to the fact that the transverse wave function has several antinodes along the y-direction.
Each antinode splits but the trajectory is quite different, which causes the cancellation of the polarization. One can
avoid this difficulty by attaching a narrow lead or fabricate a point contact in the left hand side so that only the
first channel opens. The complex trajectory is mainly due to the scattering by the hard wall located to separate the
channel to the upper and lower leads. Using the soft wall potential may improve the situation.
From the experimental point of view, we have to consider the effect of randomness. Figure 5 shows that the average
of the polarization in the presence of impurities. The energy of the electron is fixed to E/V0= 3.0 and average over
10000 samples has been performed. From this figure, we see that the finite polarization of the current persists even in
the presence of impurities. Further increase of the randomness significantly destroys the polarization. The strength
of the disorder is related to the mean free path in 2DES without spin-flip scattering as
h
?
σ′|tσ,σ′|2. Here tσ,σ′ is the transmission coefficient from the left lead with spin σ′to the right leads
W =
?
6λ3
F
π3a2Lm
?1/2
· E(10)
where λF denotes the Fermi wavelength and Lm the mean free path.28Inset of Fig. 5 shows the replot of the
polarization as a function of Lm, which indicates that finite polarization remains as long as Lmexceeds the system
length. The electron mean free path in the InGaAs 2DES channel exceeds 1µm in the wide range of the gate voltage,4
and the spin filtering effect by nonuniform SOI strength is expected to survive in actually fabricated devices. We have
also changed the distribution of the impurities to the binary distribution, where the site energy Wi takes only two
values, 0 and W > 0. In this case the variance of the potential fluctuation becomes p(1−p)W2with p the probability
that Wi takes W. We set p ≈ 0.092 so that p(1 − p) = 1/12 corresponding to the uniform distribution considered
above. No significant difference is observed by this change of distribution. We have also confirmed that distributing
attractive impurities, i.e., W < 0 does not influence the results either.
In conclusion, we have investigated the novel spin filtering due to the nonuniform spin-orbit interaction (SOI)
system. The strength of SOI is modulated perpendicular to the charge current, which yields the gradient of the
effective Zeeman field, and the spin separation occurs as in the Stern-Gerlach experiment. Both the time evolution of
the wave packet and the transmission coefficient indicate the large polarization which survives even in the presence
of impurities. In this mechanism, the direction of the spin polarization can be easily switched by the gate voltages
between the two gate electrodes. When the fully polarized current (which can be produced by the device proposed
here) is injected to this system, one expects that the current flows along one side of the system and the currents in
the upper and lower leads become totally different, i.e., the information of the spin polarization is transformed into
conductance. Therefore this system can also be used as a detector of the polarized current. It should be emphasized
that this effect is expected to survive even at relatively high temperature. This is because the mechanism proposed
here does not rely on the quantum interference which is easily destroyed by dephasing. Also the weak dependence of
Pyon energy in the lowest channel suggests that smearing of the Fermi distribution function is not important.
The authors are grateful to B. Kramer, K. Dittmer and K. Slevin for valuable discussions. This work was supported
in part by the Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology, No. 14540363.
One of the authors (J.O.) was supported by Research Fellowships of the Japan Society for the Promotion of Science
for Young Scientists.
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FIG. 1:
interaction. Stern-Gerlach type spin separation occurs when un-polarized electrons go through the nonuniform SOI region
between the two gate electrodes.
Top view of Stern-Gerlach spin filter. Vg1 and Vg2 are gate voltages to produce a spatial gradient of spin-orbit
FIG. 2: Initial wave packet (t = 0) propagating to the right. Yellow region indicates the area where SOI is present.

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FIG. 3: Wave packet after time evolution (t = 70 · ¯ hV−1
the charge density?
0
). The strength of the SOI is modulated in the y-direction. Upper:
σ(|?↑ |ψσ?|2+ |?↓ |ψσ?|2), and lower: the corresponding polarization,?
σ(|?↑ |ψσ?|2− |?↓ |ψσ?|2).
?
?
??
FIG. 4: Polarization Py (Eq. (9)) of the current as a function of the Fermi energy. Filled (Open) circle shows the polarization
of the upper (lower) lead in the right hand side. Main figure shows the component from the lowest channel of the left lead.
Inset shows the result in which the whole channel is taken into account. The schematic view of the three-terminal geometry is
also shown in the inset.