Table Of ContentSpin transport of electrons through quantum wires with
spatially-modulated strength of the Rashba spin-orbit interaction
4
0 X. F. Wang∗
0
2
Concordia University Department of Physics
n
a 1455 de Maisonneuve Ouest
J
9 Montr´eal, Qu´ebec, H3G 1M8, Canada
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(Dated: February 2, 2008)
]
l
l
a Abstract
h
-
s We study ballistic transport of spin-polarized electrons through quantum wires in which the
e
m
strength of the Rashba spin-orbit interaction (SOI) is spatially modulated. Subband mixing, due
.
t
a toSOI,between thetwolowest subbandsis taken intoaccount. Simplifiedapproximate expressions
m
- for the transmission are obtained for electron energies close to the bottom of the first subband
d
n and near the value for which anticrossing of the two lowest subbands occurs. In structures with
o
c
periodically varied SOI strength, square-wave modulation on the spin transmission is found when
[
1 only one subband is occupied and its possible application to the spin transistor is discussed. When
v
4 two subbands are occupied the transmission is strongly affected by the existence of SOI interfaces
4
3
as well as by the subband mixing.
1
0
4
PACS numbers: 72.25.Mk,73.23.Ad, 72.25.Dc
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t
a
m
-
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c
:
v
i
X
r
a
∗
Electronic address: [email protected]
1
I. INTRODUCTION
With the development of nanotechnology, manipulation and measurement of spin-orbit
interaction (SOI) in semiconductor nanostructures [1, 2, 3, 4, 5, 6, 7] have been realized as
well as injection and detection of spin-polarized current. [8, 9, 10, 11, 12, 13, 14, 15, 16,
17] As a result, in the past years increasing attention has been drawn to the spin-related
behavior of quasi-one-dimensional (Q1D) electron systems in the presence of SOI, especially
the Rashba SOI term, of strength α, which results from asymmetric electric confinement in
nanostructures. [18]Thishasbeenalsogreatlystimulatedbytheproposal[19]ofestablishing
a spin transistor, among other novel spintronic devices, and its potential application to the
promising quantum computing. The SOItheory developed earlier for bulk materials [18]and
two-dimensional electron systems [20] has been applied to the electronic band structure and
spectral properties ofrealistic quantum wires. [21, 22, 23, 24]Intriguing transportproperties
through quantum wires have been predicted as a result of peculiar features in their band
structures introduced by the SOI, such as additional subband extrema and anticrossings. It
has been found that the spatial distribution of the spin orientation in quantum wires can be
greatly influenced by the subband mixing and the existence of interfaces between different
SOI strengths. [23] Furthermore, the Rashba SOI has interesting effects on the shot noise
for spin-polarized and entangled electrons, [25] and on the spectral properties of interacting
quantum wires. [24] The former may lead to another way of measuring the SOI strength in
quantum wires.
To run a spin transistor based on the SOI in a Q1D system, a spin filter is required to
provide the initial spin-polarized current. One of the realistic options is to inject a spin
current from ferromagnetic semiconductors or metals. [8, 9, 10, 11, 12, 13, 14, 15, 16, 17]
The spin polarizationofthe injected current is independent of theexistence of the SOIin the
Q1D system [26] and this makes it possible to separate the study of the spin transistor from
that of the spin filter. Recently, several efforts have been made to describe in more detail
the behavior of spin polarized electrons in Q1D systems in the presence of the Rashba SOI.
A numerical tight-binding simulation has been carried out to study ballistic transport [27]
throughaquantumwireinwhichoneSOIsegmentisadiabaticallyconnectedtotwosegments
without SOI. The results illustrate that a uniform spin precession along the wire should
be observed provided the Rashba SOI strength is weak and subband mixing is negligible.
2
For strong SOI, however, spin modulation becomes energy-dependent and can be strongly
suppressed at finite temperatures. Lately, a square-wave modulation of spin polarization
and a good spin transistor behavior have been predicted in transport through periodically
stubbed waveguides for weak SOI and subband mixing due to it occuring only in the stubs.
[28] Although in general subband mixing results in disagreeable effects on spin precession in
quantum wires, it can offer further control of spin polarization at low temperatures in some
situations. [27, 28, 29, 30]
Inthispaper we study ballistic transportof Q1Dspin-polarizedelectron gases inthepres-
ence of a spatially-modulated Rashba SOI strength α and take into account the subband
mixing between the first and second subbands. This modulation can be achieved exper-
imentally by external gates [3] and may result in further modulation of spin currents, as
pointed out in Refs. 27 and 23, and reported in Ref. 31 when subband mixing is neglected.
Different from the periodically stubbed waveguides, with the same strength α everywhere,
studied previously, [28, 29] here we consider waveguides without stubs but with spatially-
modulated α, which, from an experimental point of view, are easier to realize and control.
We assume that the electric confinement, that gives rise to the SOI, is strong enough that
excited states due to this confinement, as observed in Ref. 7, are not occupied.
The paper is organized in the following way. In Sec. II we propose a model Hamiltonian
with the Rashba SOI term to obtain the band structure and wave function. In Sec. III
we formulate the transfer-matrix description of the transmission process and in Sec. IV we
present and discuss the results of spin transport. Conclusions follow in Sec. V.
II. MODEL
We consider a Q1D electron system, an InGaAs/InAlAs quantum wire fabricated by
confining a two-dimensional (2D) electron system in the x-y plane by an infinitely high,
square-well potential V(x). The wire has width w along the x direction, as shown in Fig.
1(a). In the presence of the Rashba SOI the Q1D one-electron Hamiltonian reads
ˆ 2
H = λ iα(σ ) +V(x), (1)
z
− ∇ − ×∇
whereλ = ~2/2m∗. = (∂/∂x,∂/∂y,0)istheLaplaceoperator,αisthestrengthoftheSOI,
∇
and σ = (σ ,σ ,σ ) denotes the spin Pauli matrices. In the σ representation and with the
x y z z
3
l l
1 2 (a)
y
x w a 1 a 2 a a
i i+1
A B C D i i+1
k (b)
15 c
k+ k-
V) 10 2 2 k+1 k-1 Ec
e
m
(
E 5
0
-2 0 2
k (108 m-1)
y
FIG. 1: (a) A quantum wire along the y direction composed of a series of segments i = 1,2,3,
···
of SOI strength α and length l . (b) Energy dispersion of the lowest two subbands in a InGaAs
i i
quantum wire 600˚A wide. The thick (thin) solid, dashed, dotted, dash-dotted curves present
branches ε− (E−), ε+ (E+), ε− (E−), and ε+ (E+) respectively.
1 1 1 1 2 2 2 2
use of the eigenfunction of the Q1D Hamiltonian h(x) = λ 2 +V(x) without the Rashba
− ∇x
term, the eigenfunction is φ (x) = 2/wsin(mπx/w) for 0 x w and m = 1,2,3, .
m
≤ ≤ ···
p
Then the wave function of Eq. (1) can be expressed as Ψ(k,r) = eiky φ (x)Cσ σ with
mσ m m| i
P
σ = (1,0)T for spin up and (0,1)T for spin down, with T denoting the transpose of the
| i X
column matrix .
X
For the sake of simplicity, while retaining the subband mixing effects, we assume that
only the lowest two subbands ( m = 1 and 2) are involved in the transport. This can be
considered as the actual case when the temperature and the electron density are not too
4
high. Then the secular equation HˆΨ = EΨ takes the form
+
E E αk 0 δ C
1 1
− −
αk E E δ 0 C−
1 1
− = 0, (2)
0 δ E E αk C+
2 2
−
δ 0 αk E E C−
− 2 − 2
where E = E0 + λk2 and E0 is the mth subband bottom in the absence of SOI; δ =
m m m
α dxφ (x)φ′(x) = 8α/3w is the mixing term due to SOI between the first and the second
2 1
R
subband. The resulting eigenvalues εσ(k) and eigenvectors Ψσ(k) are
n n
φ 2δφ /B
ε±(k) = (E +E ∆E )/2, Ψ±(k) = 1 1 ∓ 2 ±k ,
1 1 2 − ∓ 1 F±k φ1 +2δφ2/B±k
± (3)
φ 2δφ /B
ε±(k) = (E +E +∆E )/2, Ψ±(k) = 1 2 ∓ 1 ∓k .
2 1 2 ± 2 F∓k φ2 +2δφ1/B∓k
±
Here ∆E = [(∆E 2αk)2+4δ2]1/2, ∆E = E0 E0, B = (∆E 2αk)+∆E , and
± 12 12 2 1 ±k 12 −
± − ∓
F = [2+4δ2/B2 ]1/2. Setting δ = 0 in Eq. (2) gives the eigenfunctions without subband
±k ±k
mixing n, = φ ( 1,1)T/√2 with corresponding energy E± = E αk. The resulting
| ±i n ± n n ±
wavevector difference between the two spin branches of each subband for the same energy
E = E+(k+) = E−(k−), which we denote as the intrasubband SOI splitting, is constant for
n n
any energy and has the value
k = k− k+ = 2m∗α/~2. (4)
α
−
In Fig. 1(b) we plot the energy spectrum of the lowest two subbands with (thick curves)
and without (thin curves) the mixing term taken into account. This energy spectrum is
essentially the same as that shown in Fig. 2 of Refs. 29 and 30. The intersections of the
Fermi energy, here taken equal to the anticrossing energy E as shown by the horizontal bar,
c
with the energy spectrum define the wave vectors of the propagating modes. E is given by
c
E = E+(k ) = E−(k ) = 5E /2+9~2E2/(8m∗α2) where the branches without mixing E+
c 1 c 2 c 1 1 1
and E− (E− and E+) anticross each other at the wave vectors k ( k ). Without mixing,
2 1 2 c c
−
the Fermi wave vector difference between the two spin branches of each subband remains
constant. Withmixing, however, thedifference between ε− andε+ branches shows anenergy
n n
dependence. Furthermore, in the branches E± electrons have the same spin orientation, but
n
5
in the branches ε±, which are not pure spin states, the average electronic spins rotate
n
continuously from their low-energy orientation to their opposite high-energy orientation, as
also discussed recently in Ref. 23. At the anticrossing energy, the maximum mixing effect
happens between forward propagating electrons in the ε+ and ε− branches (corresponding
1 2
to B = 2δ in Eq. (3)) and between backward ones in the ε− and ε+ branches (B = 2δ).
k 1 2 −k
We denote the wave vector difference between the ε± and ε∓ branches at this energy as the
1 2
intersubband SOI splitting 2k related to k . For α < 2~2/m∗w, we have
δ c
m∗δ 8k2
k = αw. (5)
δ ≈ ~2k 9π2
c
III. TRANSFER MATRIX
We consider a quantum wire with a variable strength of SOI. It may be composed of a
series of SOI segments separated by SOI interfaces. In each SOI segment, the SOI strength
is approximately uniform and the Hamiltonian described in the previous section applies. To
describetheelectronicbehaviorpropagatingthroughthequantumwire,webeginconsidering
the transmission process of an electron with energy E through one SOI interface. The
electron is incident from the left to the interface joining two segments (labelled i and i+1)
with different SOI strength α and α as shown in Fig. 1(a). Taking into account only the
i i+1
lowesttwosubbands, wewritethewavefunctioninsegmenti,intermsofoftheeigenfunction
Ψ± given by Eq. (3) as
ni
ϕi(x,y) = Xhc±1iΨ±1i(k1±i)eik1±iy +c¯±1iΨ±1i(−k1∓i)e−ik1∓iy +c±2iΨ±2i(k2±i)eik2±iy +c¯±2iΨ±2i(−k2∓i)e−ik2∓iyi.
±
(6)
To obtain proceed we follow the approach of Ref. 26: we match the wave function and its
flux at the interfaces between the i and i+1 segments. The velocity operator is given by
∂H i ~ ∂ α
vˆ = = − m∗ ∂y ~ . (7)
y
∂py α i ~ ∂
~ − m∗ ∂y
The continuity of the wave function at the interface y = y , between the i and
i,i+1
i + 1 segments, gives ϕ (x,y ) = ϕ (x,y ) and that of the flux vˆ ϕ (x,y) =
i i,i+1 i+1 i,i+1 y i |yi,i+1
vˆ ϕ (x,y) . Multiplying the two equations by Ψ∗±(k) and Ψ∗±(k) respectively, then
y i+1 |yi,i+1 1i 2i
integrating over x we obtain eight linear equations for the eight coefficients of the wave func-
tions. Here we drop the subscript and superscript pertaining to k since they are the same as
6
those for Ψ as shown in Eq. (6). Because electrons in branches Ψ+ and Ψ− are decoupled
1i 2i
from electrons in branches Ψ− and Ψ+, a result of the symmetric property of the wave func-
1i 2i
tion,theseequationsaregroupedintotwosimilarbutindependent equationgroupswitheach
composed of four equations. The group corresponding to modes Ψ+ and Ψ− connects the
1i 2i
column matrix Rˆ = (c+,c¯+,c−,c¯−)T and column matrix Lˆ = (c+ ,c¯+ ,c− ,c¯− )T
i 1i 1i 2i 2i i+1 1i+1 1i+1 2i+1 2i+1
and reads:
SˆRˆ = Qˆ Lˆ . (8)
i i i+1 i+1
Denoting the scalar product Ψ Ψ′ in all matrix products by the integral
h | i
Ψ∗T(x,y)Ψ′(x,y)dx and the direct product of the column matrix with the row ma-
X
R
trix by , the upper (2 4) part of the 4 4 matrix Sˆ is given by and
i i i
Y X ⊗ Y × × A ⊗ B
ˆ
the lower part by while the upper (2 4) part of the 4 4 matrix Q is given
i i i+1
A ⊗ C × ×
by and the lower part by . Here = Ψ+(k) , Ψ−(k) T, =
Ai ⊗ Bi+1 Ai ⊗ Ci+1 Ai h 1i | h 2i | Bi
(cid:0) (cid:1)
Ψ+(k) , Ψ+( k) , Ψ−(k) , Ψ−( k) and = ξ+(k) , ξ+( k) , ξ−(k) , ξ−( k)
| 1i i | 1i − i | 2i i | 2i − i Ci | 1i i | 1i − i | 2i i | 2i − i
(cid:0) (cid:1) (cid:0) (cid:1)
with ξ = vˆ Ψ .
y
| i | i
If the SOI exists only in segment i+1 (α = 0 and α = α), we discard all the subscripts
i i+1
i+1 in the matrices Sˆ and Qˆ and explicitly express them as
i i+1
1 1 0 0
0 0 1 1
ˆ
Si = (9)
k0/m∗ k0/m∗ 0 0
1 1
−
0 0 k0/m∗ k0/m∗
2 − 2
and
1 1 2δ 2δ
Θ1 Θ¯1 Θ2Λ2 Θ¯2Λ¯2
2δ 2δ 1 1
Qˆi+1 = √2 1 (−αΘ+1Λ1k1+) 1 (−αΘ¯1Λ¯1k1−) 2δ (αΘ1+ k2−) 2δ (αΘ¯2 k2+) . (10)
Θ1 ~2 m∗ Θ¯1 ~2 − m∗ Θ2Λ2 ~2 m∗ Θ¯2Λ¯2 ~2 − m∗
2δ (α k1+) 2δ (α + k1−) 1 (α k2−) 1 (α + k2+)
Θ1Λ1 ~2 − m∗ Θ¯1Λ¯1 ~2 m∗ Θ2 ~2 − m∗ Θ¯2 ~2 m∗
Here k0 = [2m∗(E E0)]1/2/~ (k0 = [2m∗(E E0)]1/2/~) is the wave vector of the electrons
1 1 2 2
− −
in the first (second) subband in the segment without SOI and Λ = B , Λ¯ = B ,
1 k+ 1 −k+
1 1
Λ = B , Λ¯ = B , Θ = F , Θ¯ = F , Θ = F , and Θ¯ = F are defined in the
2 k− 2 −k− 1 k+ 1 −k+ 2 k− 2 −k−
2 2 1 1 2 2
segment with SOI of strength α.
The above complex matrix can be simplified approximately in the low-energy limit (E &
E0)andtheanticrossing energylimit(E ≈ E ). Iftheelectrondensity issufficiently lowthat
1 c
7
the Fermi energy is close to the bottom of the first subband E0 in segment i, the correction
1
to the wave function caused by subband mixing is negligible or we have Θ = Θ¯ = √2 and
δ = 0 in Eqs. (9) and (10). We find all the spin modes in the quantum wire are decoupled
from each other and the transfer equations for all modes have similar forms. The transfer
+
equation for the mode Ψ has the form
1
+ +
1 1 c 1 1 c
1i = 1i+1 , (11)
k0 k0 c¯+ K K c¯+
1 − 1 1i 2 − 2 1i+1
with K = (1/~)[2m∗(E E0 + ε + V ) + (m∗α/~)2]1/2 and ε δ2/∆E the energy
2 1 0 0 0 12
− ≈
correction to the first subband as a result of the SOI subband mixing. V denotes the
0
conduction band offset in the segment i+1 reckoned from the conduction band bottom in
segment i, which may be introduced by material mismatch at the interface or by an external
gate bias.
When the SOI strength is in the range ~2/m∗w < α < 2~2/m∗w, the anticrossing energy
is higher then the second subband bottom but lower than the third subband bottom and the
intersubbandSOIsplittingismuchsmallerthantheintrasubbandSOIsplitting,i.e. k k .
δ α
≪
Eqs. (9) and (10) can be greatly simplified if the electron energy is near to the anticrossing
energy, i.e. E E . In this case, we use the approximation V = 0, Λ 2δ k−/k+,
c 0 1 2 1
≃ ≈
p
Λ 2δ k+/k−, Λ¯ 8αk0, Λ¯ 8αk0, Θ 2 k+/k0, Θ 2 k+/k0, Θ¯ Θ¯ √2,
2 1 2 1 1 2 2 1 1 1 2 2 2 1 2
≈ ≈ ≈ ≈ ≈ ≈ ≈
p p p
k+ k +k , k− k k , k− k +k , k+ k k , k0 k +k /2, k0 k k /2,
1 c δ 2 c δ 1 c α 2 c α 1 c α 2 c α
≈ ≈ − ≈ ≈ − ≈ ≈ −
δ/Λ¯ 0. Eq. (8) reduces to
≈
c+ k+/2k0 0 k−/2k0 0 c+
1i 1 1 2 1 1i+1
p p
+ +
c¯ 0 1 0 0 c¯
1i 1i+1
= . (12)
c− k+/2k0 0 k−/2k0 0 c−
2i − 1 2 2 2 2i+1
p p
c¯− 0 0 0 1 c¯−
2i 2i+1
We see that, in the anticrossing energy limit, the mixing happens mainly between the
two modes involved in the anticrossing (c+ and c−) and the modes corresponding to the
1 2
coefficients c¯+ and c¯− remain almost intact when transmitting from segment i to segment
1 2
i+1 though there is a wave vector mismatch between the two segments.
OncethematricesSˆ−1 andQˆ inEq. (8)areknown, thetransfermatrixfortheinterface
i i+1
joining segment i and i+1 is obtained simply: M = Sˆ−1Qˆ . For a quantum wire with
i,i+1 i i+1
8
n segments, the total transfer matrix then reads
Mˆ = Pˆ Mˆ Pˆ , (13)
1 i,i+1 i+1
Y
i=1,n−1
where the transfer matrix for the i-th segment of length l is expressed as
e−ik1+il 0 0 0
0 eik1−il 0 0
Pˆi = . (14)
0 0 eik2−il 0
0 0 0 e−ik2+il
Atransfer matrixsimilar toMˆ given by Eq. (13)isobtainedfor themodesΨ− andΨ+ by
1i 2i
applyingthesameprocessasaboveandwillnotbeshownhere. Finallya8 8transfermatrix
×
Tˆ is obtained connecting the wave-function coefficients of the electron with energy E at the
left inlet end, Iˆ= (c±,c¯±,c±,c¯±)T, and at the right outlet end, Oˆ = (c± ,c¯± ,c± ,c¯± )T in
1I 1I 2I 2I 1O 1O 2O 2O
ˆ ˆˆ
the form I = TO.
IV. RESULTS AND DISCUSSION
In the following we present results for ballistic transport of electrons, incident with spin
up (along the z direction), through a quantum wire in which segments with and without SOI
alternate periodically. If not otherwise specified, zero temperature and parameters w = 600
˚A, l = 2500 ˚A, α = 3.45 10−11 eVm, m∗ = 0.05, and V = 0 will be assumed. The strength
0
×
of SOI can be adjusted experimentally [3, 4, 5, 6] and here we use the same value of α as
used in Ref. 30 for the sake of comparison. At zero temperature, only electrons of the Fermi
energy contribute to the transport. In the inlet and outlet segments of the quantum wire,
we assume there is no SOI and the group velocity of the electrons is proportionalto the wave
vector (k0 = [2m∗(E E0)]1/2/~, n = 1,2) and the density of states to its inverse. When
n − n
using the normalized coefficients for the input wave function (c± = 1/ 2k0,c± = 1/ 2k0),
1I 1 2I 2
p p
we express the zero-temperature spin-up (spin-down) partial conductance G+(G−) and the
n n
partial transmission T+(T−) at the outlet end via the subband n and the reflection R+ (R−)
n n n n
from the inlet end as:
G± = e2T± = e2kn c+ c− 2; R± = kn c¯+ c¯− 2. (15)
n h n h 2 nO ± nO n 2 nI ± nI
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
9
The finite-temperature conductance G±(T) is obtained by integrating over energy the above
n
zero-temperature conductance multiplied by the Fermi distribution function f :
n
df (E,T)
G±(T) = dEG±(E,T = 0) n . (16)
n −Z n dE
A. The low-energy limit
If only the first subband is occupied and E and E0 are much above the Fermi energy,
c 2
Eq. (11) can be used to estimate the transmission through the quantum wire. In a quantum
wire with only one SOI segment, of strength α = α and length l = l, sandwiched between
2 2
two segments without SOI as the part A-B-C-D shown in Fig. 1(a), the total transmission
takes the form
t
T = (17)
tcos2(K l)+sin2(K l)
2 2
wheret = 4(k0)2K2/[(k0)2+K2]2. ThisisasinusoidaldependencewithamaximumT = 1
1 2 1 2 max
for sin(K l) = 0 and a minimum T = t. For a quantum wire of fixed width, the stronger
2 min
is the SOI strength and the less is the energy of the incident electrons, the more efficient is
the modulation of the transmission. Another feature of this interface-induced transmission
modulation is that it does not affect the spin polarization. The output percentage of spin-
up and spin-down electrons remains the same as predicted neglecting the interface effect,
T+ = T cos2(θ/2) and T− = T sin2(θ/2) with θ = k l. These features offer the possibility
α
of independent control of the total transmission and of the spin polarization and will be
advantageous when designing a spin transistor employing this system.
In Fig. 2, we plot T+ (thick solid curves), T− (thin solid), and R+ (dotted) as a function
of the main parameters of the quantum wire. Since only spin-up electrons are incident and
there is no spin-flip mechanism, spin-down reflection is not observed. The spin-up and spin-
down electron output flux show, in Fig. 2(a), a modulated sinusoidal dependence on the
length l of the SOI segment instead of the simple one as the interface effect is neglected. [19]
For electrons near the bottom of the first subband, a strongly energy-dependent reflection
happens(Fig. 1(b))duetothewavevectormismatchbetweenelectronsinsegmentswithand
without SOI. A similar energy dependence of the conductance (transmission) has also been
found in Ref. 27. In some cases, e.g., when an extra gate bias is applied, an offset V exists
0
between the conduction bands of the material in the segment with SOI and the one without
10
