Table Of ContentSpin polarization effects in a two dimensional mesoscopic electronic structure with
Rashba spin-orbit and lateral confinement
Stefano Chesi and Gabriele F. Giuliani
Department of Physics, Purdue University, West Lafayette, IN 47907, USA
(Dated: February 5, 2008)
7
0 Because of the peculiar coupling of spatial and spin degrees of freedom effected by the Rashba
0 spin-orbit interaction, lateral confinement of a two dimensional electronic system leads to a finite
2 transverse spin polarization near the longitudinal edges of a current carrying quantum wire. The
n sign of this component of the polarization is opposite at the two edges and can be reversed upon
a inversion of the current. Interestingly for small spin orbit coupling this is the largest contribution
J to the total polarization, its magnitude being of second order in the coupling constant. As a
7 consequence this phenomenon cannot be revealed in lowest order perturbative approaches. An in
1 plane spin polarization component is also present that is perpendicular to the current. Within the
samemodelthiscomponentwouldbealsopresentinthebulk. Ontheotherhandwhileinthelatter
] case its magnitude is linear in the coupling constant, we find that it only represents a third order
ll effect in the wire geometry. Our results are consistent with a general rigorous argument on the
a parityofthecomponentsofthespinpolarization withrespecttothesign ofthespinorbitcoupling
h
constant.
-
s
e PACSnumbers: 72.25.Dc,85.75.-d,71.70Ej,73.63.Nm
m
.
t We consider in this paper the effects of the lateral have moreover chosen to describe the latter by means
a
confinementonthe properties ofatwo-dimensional(2D) of infinite potential barriers, an assumption that allows
m
electron liquid in the presence of linear Rashba spin or- us to present a transparent and explicit discussion while
- bit. The basic idea underlying our work stems from the at the same time affording us a full description of the
d
n simple observation that in order to construct a localized physical phenomenon. A great advantage of our model
o wavefunctiononemustsuperimposeplanewaveswithop- is that an analytic perturbative expansion in the spin-
c positecomponentsofthewavevectoralongthedirection orbit coupling constant can be developed, a procedure
[
of localization. Because spin-orbit forces the latter to that shows how the effect would not be revealed by low-
1 haveoppositespincomponentsalongtheunrestricteddi- estorderperturbationtheories. Ageneralizationtomore
v rection,itisclearthatinthedirectionofconfinementthe generalsmoothgatingpotentialsisreadilyimplemented.
5 electronicstatesmustdisplayacorrespondinglyinhomo- That a steady state current can lead to a spin polar-
1 geneous spin polarization whose direction also changes ization in a 2D electron liquid subject to Rashba spin
4
in space. In the particular case in which a 2D electronic orbit coupling is hardly unexpected. It is in fact well
1
0 paramagnetic system is laterally confined to produce a known that a current forced along the x direction of a
7 quantum wire, the superposition involves spins that are bulk 2D electronic system will be accompanied by an in
0 parallel to the plane of motion and therefore combine to plane polarization along y.2,3,4,5,6 For small spin orbit
/ give an out of plane polarization component. Since the this polarization is linear in the coupling constant.
t
a sign of this component of the magnetization is opposite
As we shall see however, while this component of the
m for states with opposite momenta along the wire, it is
polarization is still present, confinement leads to a spin
- clearthatanetpolarizationcanbe establishedwhenthe polarization that, for small spin orbit, is mostly out of
d
occupation of such states is unbalanced as in the case in
plane. Interestingly, as already stated, this polarization
n
whichanetsteadystatecurrentisforcedthroughthesys-
o can be flipped simply by inverting the current.
tem. As we will show this net perpendicular static spin
c The magnitude of the effect strongly depends on the
: polarization is most pronounced near the lateral bound-
v strength of the spin orbit coupling. A quick order of
aries and displays an oscillatory behavior. Inverting the
i magnitude estimate can be obtained by inspecting the
X direction of the currentsimply flips the polarization. No
caption of our Figure 7 while making reference to our
r spin current is involved in the problem.
model hamiltonian (Eq. (1)).
a
In order to demonstrate and exemplify this idea, we Whiletheproblemcanbecompletelysolvedbynumer-
present here a complete analysis of the simplest possi- ical means we will show that a careful study of the per-
ble model hamiltonian describing a system of non inter- turbation expansion in the spin orbit coupling constant
acting electrons subject to a homogeneous neutralizing is crucial in understanding the interplay of the various
background, a linear Rashba spin orbit coupling and a components of the polarization.
lateral confining potential.1 The effect of impurities is Thepaperisorganizedasfollows: SectionIdefinesour
neglected although impurity scattering in the channel model and details our formulation; Section II describes
will not change our main conclusions. Although more the numerical and perturbative solutions to the single
realistic model potentials could be readily employed we particle problem providing a discussion of various prop-
2
ertiesoftheeigenfunctionsandeigenenergies. SectionIII k k
y y
is devoted to the study of the total spin polarization in
thepresenceofasteadystatecurrent. Manyofourmost
relevant results are presented in this Section. Finally
Section IV provides a discussion of the most interesting
physics including that of the crossover of the in plane
component of the polarization and our conclusions. kx kx
I. FORMULATION OF THE PROBLEM AND
BOUNDARY CONDITIONS
FIG. 1: Possible free space states that can be superimposed
The model Hamiltonian we consider is the following:
to obtain confined eigenstates. On right we show a situation
pˆ2 forwhichthetwostatesoftheupperbranchhaveRe[ky]=0.
Hˆ = +α(σˆ pˆ σˆ pˆ )+V(yˆ) , (1)
x y y x
2m −
where the motion is restricted to the x,y plane and the
Atagivenenergyǫ,thereareonlyfourfreespaceeigen-
lateral confining potential is given by:
states that can be combined to form solutions satisfying
the zeroboundaryconditionoftheinhomogeneousprob-
0 for y <W/2
V(y)= | | (2) lem. This is depicted in Figure 1: for each spin subband
for y W/2 .
(cid:26) ∞ | |≥ wehavetwostateswithoppositevalueofk . Thefour
y
±possible values of k are k±, with
Since the system is translationally invariant along the y ± y
x direction, the eigenfunctions of the problem can be
constructedbylinearsuperpositionoffreespace(V(y)= k± = K2 k2 , (6)
y ±− x
0) eigenstates with definite values of kx. Furthermore q
the reflection symmetry x x allows one to generally where K± are given in Eq. (5) above.
assume k >0. →− Since we are interested in spatially undamped solu-
x
Forreference,werecallherethatthefreespaceelectron tions, we always take kx < K+. However, it will nec-
eigenfunctions and eigenenergies are given by essary to include the case K− < kx < K+, where
k− =ik− ispurelyimaginary. Wealsodefinetheangles
y | y |
ϕ(k0,)±(x,y) = ei(k√xx2+Lk2yy) (cid:18)ie±iφ1k (cid:19) , (3) φ± by means of: k±
y
φ =arctan . (7)
±
k
x
¯h2(k2+k2)
ǫk± = 2xm y ∓α¯h kx2+ky2 , (4) If ky− = i|ky−| we have that also φ− is purely imagi|nk−a|ry
whereListhelinearsizeofthesysteqmandφkistheangle anIdtfirsocmontvheeniaebnotvteofcoornmsiudlearwtheegfeotllφow−i=ngitawroctcaonuhplekyxso.f
between the direction of the wave vector and the x-axis. (unnormalized) states, which are also eigenstates of the
Interestingly the Rashba spin-orbit coupling forces each reflection symmetry y y:
plane wave state to have its own distinct orientation of →−
thespinquantizationaxis. Thisdirectionliesinthex y cos(k±y φ±)
ptwlaonsepalintdbmanadkseswahnicahngalreeocfhπa2rwacittehrikz.edTbhyesoepsptaotseitsefoc−rhmi- ψ±±(ǫ,kx,y)= ±i cos(kyy±y+− φ22±)!eikxx , (8)
rality in the sense that states with the same wave vector sin(k±y φ±)
hovaevre,obpepcaousisteeosfptinimdeirreecvteiorsnasl,ininththeetwproesbeanncdeso.7f,8RMasohrbea- ψ±∓(ǫ,kx,y)= ±i sin(kyy±y+− φ22±) !eikxx . (9)
spin orbit, states within the same band with opposite The first subscript refers to the chirality of the spin
±
wavevectorshaveoppositespins. Thecorrespondingsur- subbandwhilethesecond subscriptreferstothereflec-
±
faces (lines) at constant energy ǫ are concentric circles tion operation y y. The eigenstates in the confined
→ −
with radii K . These are given as the positive solutions system are then expressed in the following way:
±
of the equation
ϕ+(k ,y)=c++ψ +c−+ψ , (10)
n x n ++ n −+
¯h2K2 ϕ−(k ,y)=c+−ψ +c−−ψ , (11)
ǫ= ± ¯hαK . (5) n x n +− n −−
±
2m ∓
wherethesuperscriptinϕ± referstothereflectionparity,
n
ConstantenergylinesareschematicallydisplayedinFig- and not to the chirality. As one would expect, the solu-
ure 1. tions of the confined system are an admixture of states
3
7 7
6 6
5 5
LV4 LV4
e e
m m
H H
3 3
Ε Ε
2 2
1 1
-2·108 -1·108 0 1·108 2·108 -2·108 -1·108 0 1·108 2·108
kx Hm-1L kx Hm-1L
FIG. 2: Energy spectrum of a W = 0.1 µm wide quantum FIG. 3: Same as in Figure 2 but with a larger value of the
wire. Forclaritytheexactenergybandsareonlyshown(dots) spin orbit coupling: ¯hα= 5 10−9 meVm. Strong deviations
for discrete values of kx, while the perturbative results from from perturbation theory are dueto anti-crossing of bands.
Eq. (23) are also plotted (solid lines) for comparison. The
parameters values are as follows: ¯hα = 2 10−9meVm and
m = 0.3m0. The horizontal dashed line marks a Fermi level
of 6 meV and is significant in relation to Figures 5, 6, and 7. II. SOLUTION OF THE PROBLEM
The corresponding electron density is 0.67 1012 cm−2.
A. Exact solution at kx =0
A case in which the solutions canbe expressedin sim-
with different chirality. A spin subband index will be
pleclosedformisthatofk =0. Forthisparticularvalue
introduced as an approximate quantum number in the x
ofk itis immediate to seethat Eq.(1)admits solutions
case of small spin orbit in a following Section where we x
whichareeigenstatesofσˆ . Assumingthefollowingform
present a perturbative treatment of the problem. x
for the wavefunctions
Imposing the boundary condition at y = W/2 for the
ϕ±n leads to the following conditions: e∓imh¯αyΦ(y) x , (15)
|±i
cos(ky+W−φ+) sin(ky−W−φ−) c++ Φ(y) is found to satisfy:
2 2 n =0 , (12)
iscions((kky+y+WW2−+φφ++)) −ciossin((ky−ky−WW−2+φφ−−))!(cid:18)cc−n++−(cid:19) −2¯hm2 ∂∂y22 +V(y)− 12mα2 Φ(y)=ǫΦ(y) . (16)
2 2 n =0 . (13) (cid:20) (cid:21)
isin(ky+W2+φ+) −icos(ky−W2+φ−)!(cid:18)c−n−(cid:19) This result is quite generalholding for anarbitraryform
oftheconfiningpotentialV(y). Forthecaseofaninfinite
At fixed kx, the the determinants of (12) and (13) are well we obtain:
functions of ǫ, through the implicit dependence of k±
y
and φ . The zeros of these determinants provide two ¯h2 πn 2 1
± ǫ = mα2 , (17)
sets of discrete energies ǫ±n(kx), corresponding to states n 2m W − 2
of opposite parity. (cid:16) (cid:17)
where n is a positive integer. For every eigenvalue there
If one is not interested in separating states with dif-
is an additional twofold spin degeneracy.
ferent reflection parity, one can combine the conditions
ensuing from (12) and (13) into the compact equation:
B. Numerical results
(1 cosk+W cosk−W)sinφ sinφ (14)
− y y + −
+(1−cosφ+cosφ−)sinky+W sinky−W =0 , We reproduce in Figures 2-6 numerical results, calcu-
latedwiththemethodsofthepreviousSection. InFigure
whichgivesthewholespectrumforagivenkx. Theeigen- 2 and3 we presenttwo examplesof the energyspectrum
functions,droppingthereflectionparityindex,arehence- as function of k for different values of the spin-orbit
x
forth quite generally denoted as ϕν(kx,y). coupling α.
We finally notice that for values of ǫ and k such that The propertiesof the eigenfunctions are perhaps more
x
k− ispurelyimaginary,thestructureoftheaboveformu- interesting and can be best exemplified by the corre-
y
lasstillholds,andoneneedonlysubstitutehyperbolicfor spondingnumberandspinpolarizationdensities. Figures
trigonometric functions. 4, 5 and6 refer to these propertiesof the eigenfunctions,
4
andhavebeenobtainedwiththesameparametersofFig- Higher order terms are also readily obtained.
ure2andanoccupationcharacterizedbyaFermienergy The necessary condition for the validity of the per-
ǫ =6 meV. In particular, we show there for each state turbative expansion is that the matrix elements (22) be
F
the relative number density (k ,y) and spin polariza- smaller than the unperturbed energy differences. The
ν x
tion density ~ (k ,y). TheNse quantities are defined as latter are of order h¯2 . This gives:
Pν x mW2
follows:
¯h ¯h
α and α . (24)
ν(kx,y) = ϕν(kx,y′)δ(y yˆ′)ϕν(kx,y′) , (18) ≪ mkxW2 ≪ mW
N h | − | i
Clearly the perturbative expansion fails for large values
of k , or for a channel with large W.
~ (k ,y) = ϕ (k ,y′)~σˆδ(y yˆ′)ϕ (k ,y′) . (19) x
ν x ν x ν x The perturbation theory allows us to explicitly calcu-
P h | − | i
late a number of interesting quantities. In particular the
From simple symmetry considerations, it follows that
z-component of the spin polarization density of the con-
thepolarizationisvanishinginthexdirection,alongthe
fined states (see Eq. (19)) is given by:
wire,andstateswiththesameenergyandoppositevalues
of kx have the same density but opposite spin polariza- Pnz±(kx,y)=hϕn±(kx,y′)|σˆzδ(y−yˆ′)|ϕn±(kx,y′)i (25)
tion. As it will be discussed in a following Section, this 2
αm αmW
fact allows in principle to produce a net spin polariza- = F (y/W) k G (y/W)+O(α3) ,
n x n
∓ ¯h − ¯h
tion along the y and z direction, by driving an electrical (cid:18) (cid:19)
current along the wire. where the functions F and G are given by:
n n
64 ′ nm πny πmy
F (y/W)= sin sin ,
n π2 (n2 m2)2 W W
C. Perturbation theory m −
X
256 ′ nm πny πmy
G (y/W)= sin sin , (26)
Although the numerical analysis provides the com- n π4 (n2 m2)3 W W
m −
pleteanswertotheproblem,usefulinsightcanbegained X
from an examination of the perturbative approach. We where the sum extends over positive even (odd) integers
therefore consider next the case of small spin orbit cou- m for odd (even) n.
pling and will assume that the Rashba term can in- In the same way,while Pnx±(kx,y) vanishes, the corre-
deed be treated as a perturbation. In order to sim- sponding perturbative resultfor the polarizationalongy
plifythe notation,inthis Sectionweconsiderapotential of the n states can be expressed as:
±
that confines the electrons in 0 < y < W rather than y (k ,y)= ϕ (k ,y′)σˆ δ(y yˆ′)ϕ (k ,y′) (27)
W/2 < y < W/2. For simplicity we will label the per- Pn± x h n± x | y − | n± x i
− 2 πny αm 2
turbed eigenstates via the (approximate)quantum num- = sin2 WH (y/W)+O(α3) ,
n
±W W ∓ ¯h
bers of the unperturbed solutions. Accordingly we write
(cid:16) (cid:17)
where the function H is defined as follows:
n
ϕ (k ,y)=ϕ0 (k ,y)+O(α) , (20)
n± x n± x 128 ′ m2n2 πny
H (y/W)= sin2
where n π4 (n2 m2)4 W
"
m −
X
2 nπy ′ m2nn′ πny πn′y
ϕ0 (k ,y)= eikxxsin , (21) +2 sin sin
n± x LW W |±iy (n2 m2)2(n2 n′2)(m2 n′2) W W
r m,n′ − − −
X
with n a positive integer. We also find convenient to ′ mm′n2 πmy πm′y
+ sin sin , (28)
choose the spinors along y, in such a way that the states (n2 m2)2(n2 m′2)2 W W
areeigenstatesoftheyreflectionsymmetry. Therelevant mX,m′ − − #
matrix elements of the spin orbit interaction are: where the sums extend over the positive integers. In
particular while n′ is of the same parity as n, both m
hϕ0n′±|Rˆ0|ϕ0n±i=∓¯hαkxδnn′δkxkx′ , (22) and m′ are of opposite parity. Moreovern′ 6=n.
2h¯αnn′(1 ( 1)n+n′) Notice that as expected, in all these formulas the sub-
hϕ0n′∓|Rˆ0|ϕ0n±i=∓ W (n−2 −n′2) δkxkx′ . stitution kx → −kx and ± → ∓ gives a state with the
− same energy and opposite spin polarization.
We can easily obtain the form of the perturbation ex-
pansion, up to second order:
III. EDGE SPIN POLARIZATION
¯h2 πn 2 ¯h2k2
ǫ (k )= + x
n± x 2m W2 2m We considerhere the spin accumulationatthe bound-
(cid:16) (cid:17)1 aries of the wire in the presence of an electric current in
¯hαkx mα2+O(α3) . (23) the x direction.
∓ − 2
5
k Hm-1L
x
-2·108 -1·108 1·108 2·108
FIG. 4: NumberdensityNν(y)(in m−1 units) of individualeigenstates at theFermilevel of aquantumwire. Theparameters
are the same as in Figure 2. The wave vector values are determined by the crossings of the bands with the horizontal dashed
line at 6 meV in Figure 2 and are given at the top (dots on the kx line). The corresponding Nν(y) functions are shown in
the panels, where the numbers label the first twelve wave vectors, counted from left to right on the top kx line. States with
opposite wavevectors havethesame density.
k Hm-1L
x
-2·108 -1·108 1·108 2·108
FIG. 5: Spin polarization density Pz(y) along the z direction (in m−1 units) of the same eigenstates of Figure 4. States with
ν
opposite wavevectorshavepolarization of opposite sign. Solid linescorrespond tothefirst twelve kx values,counted from left
to right on the top kx line, while dashed lines refer to −kx.
Although the response of a system of electrons sub- tential drop occurring at the contacts at x= .
±∞
ject to Rashba spin orbit and an applied electric field Withinthisframework,thetotalspinpolarizationden-
is quite complicated and requires a careful analysis,2,3 sity due to the currentflux canbe obtainedin the linear
for simplicity sake we will model here a current carrying regime from the relation:
steady state by assuminga modified electronoccupation
gµ ˜ δµ
distribution. In particular we will assume that a finite ~(y) = B ~ (k ,y) , (29)
ν Fν
chemicalpotentialdifferenceδµ=µ+−µ− isestablished M ± 2 Xν 2π¯hvFνP
between the right moving and left moving states. This
where the sum is restricted to the occupied one-
is depicted in Figure 2 by the two short horizontal solid
dimensional subbands with k and v being the (pos-
lines (whose separationfrom the equilibrium Fermi level Fν Fν
itive) Fermi wave vector and Fermi velocity relative to
- dashed - is not in scale). The present model is meant
the occupied band as labeled by the index ν. In this for-
to describe ballistic transport with all the chemical po-
mulathepositivesignreferstothecaseofholetransport.
6
k Hm-1L
x
-2·108 -1·108 1·108 2·108
FIG. 6: Spin polarization density Py(y) along the y direction (in m−1 units) of the same eigenstates of Figure 4. States with
ν
opposite wavevectorshavepolarization of opposite sign. Solid linescorrespond tothefirst twelve kx values,counted from left
to right on the top kx line, while dashed lines refer to −kx.
Asaconsequencethedirectionofthemagnetizationonly of the one dimensional subband ν, a length scale that
depends on the current direction and not on the type of we assume much larger than the width W of the wire.
carrier involved. By taking for simplicity the same scattering time for all
The perturbative result for the z component is subbands τ = τ the ensuing magnetization can be ob-
z ν
M
readily obtained from Eqs. (26): tained. The results for (y) are plotted in Figure 9.
z
M
We notice that the oscillations of the magnetization are
mgµ mαW 2 ˜ somewhatdamped as comparedto the ballistic case. On
B
(y) δµ G (y/W) , (30)
Mz ≃ ∓ 2π¯h2 ¯h n theotherhand,theoverallsignoftheeffectisunchanged,
(cid:18) (cid:19) Xn the behavior being in qualitative agreementwith the ex-
perimental results of Ref. 9. We haste to state however
where the sum is limited to the positive integers corre-
that our model is rather different from the situation of a
sponding to the occupied bands. The results of the (ex-
dirty three-dimensional sample dealt with in the experi-
act)numericalcalculationsof (y)areshowninFigure
z
M ment.
7.
Interestingly the expression given in (27) is not suf-
ficient to obtain the leading corresponding perturbative
IV. DISCUSSION
expression for (y). The reason is simply that in the
y
M
confined geometry the first finite contribution to this
quantity is clearly of order α3 (see also below). This As we have noted, while for small spin orbit the out
term can be of course readily calculated. The results for of plane (z-axis) polarization z(y) induced by an x-
M
the exact calculation of (y) are plotted in Figure 8 axis current is quadratic in the coupling constant,the in
y
alongside the perturbativMe expression. plane (y-axis) polarization y(y) is much smaller, first
M
It is remarkable that the z component of the magneti- appearing in third order. This result is only in apparent
zationislargerforsmallvaluesofthespin-orbitcoupling contradictionwiththewellknowncorrespondingbulkre-
constant while the two components become comparable sult in two dimensions which entails a linear dependence
afonrdladrgiseprlavaylsueasn. Fausyrtmhmeremtroirce,bwehhailveiMor za(cyr)ocshsatnhgeeswsiirgen, mofoMdeyl(uys)eodnitnheSeccotuiponlinIgIIcosnuscthanat.bWehaitvhiionrtchaensibmeprleifiaedd-
(y) remains of the same sign and is instead symmet- ily derived by assuming the distorted momentum space
y
Mric. electrondistribution function of Figure 10. The final ex-
Asexpected,inallcases,thereisgoodagreementwith pression is given by
the approximate analyticalformula (30) for small values 3
1 αm2 δµ
of α. σˆ = , (31)
L2h yi π2¯h2√2ǫ +mα2
Ourtheorycanbeextendedtothecaseofnonballistic F
transport. Asimpletreatmentofsuchasituationcanbe where δµ = 2h¯v δk. As expected for small α (y) is
F y
obtained by setting δµ eEv τ in Eq. (29). Here E linear in the coupling constant.10 M
Fν ν
→
is the magnitude of the driving electric field and v τ The difference lies with a quenching of the y-axis po-
Fν ν
is the elastic mean free path appropriate for the states larization due to the confining potential. As it can be
7
0.6
0.2
L 0.4 L
2m 0.2 2m 0.15
Μ Μ
(cid:144) (cid:144)
ΜB 0 ΜB 0.1
H H
-0.2
Mz My0.05
-0.4
-0.6 0
-40 -20 0 20 40 -40 -20 0 20 40
y HnmL y HnmL
3 2.5
L 2 L
2m 2m 2
Μ 1 Μ
(cid:144) (cid:144) 1.5
B 0 B
Μ Μ
H H 1
-1
z y
M -2 M 0.5
-3 0
-40 -20 0 20 40 -40 -20 0 20 40
y HnmL y HnmL
FIG.7: Spinpolarization densityinthez directionofaW = FIG. 8: Spin polarization in the y direction corresponding
0.1 µm wide quantum wire, for carriers with effective mass to the same parameters of the previous Figure 7. In the
m = 0.3m0 and g = 2. The Fermi energy is ǫF = 6 meV, top panel ¯hα = 1.5 10−9 meVm while in the bottom one
which corresponds to an electron density of 0.67 1012 cm−2. ¯hα = 3.5 10−9 meVm. The leading perturbative contribu-
We used δµ = 0.1 meV which gives a current in the wire of tion (cubicin α) is also plotted (dashed).
23 nA. In the top panel h¯α = 1.5 10−9 meVm while in the
bottomoneh¯α=3.510−9 meVm. Theperturbativeresultis
also plotted (dashed). As a comparison of the magnitude of
theeffect, the Pauli susceptibility is χP =72.5 µB/(µm2T). 0.6
L 0.4
2
m
Μ 0.2
(cid:144)
surmisedbyinspectingEq.(27),thiseffectcanbereadily B 0
Μ
seentooriginatefromthelowestordercancellationofthe H-0.2
contributionsto (y)stemmingfromstateswithinthe z
My M -0.4
same spin-split one dimensional band. The cancellation
-0.6
only occurs when the spin-splitting is smaller than the
energyquantizationassociatedwith the one-dimensional -40 -20 0 20 40
y HnmL
confining potential. This is of course not operational in
the bulk since in such a situation the energy spectrum is
FIG. 9: Magnetization density of a W =0.1 µm wide quan-
continuum.
tumwire,forcarrierswithsameparametersasthefirstpanel
The crossover between these two regimes approxi-
of Figure 7. We used E = 1 kV/m and a constant value for
mately occurs when the spin-splitting equals the energy τ = 7.3 10−13 s, which gives the same current in the wire of
spacing between the eigenvalues of the confining poten-
Figure 7. The perturbative result is also plotted (dashed).
tial along the y direction, i.e. for mαW 1. In order to
h¯ ≃ The magnetization is somewhat different from the previous
exemplify this phenomenonwe haveplotted inFigure11 case, where the oscillations of the magnetization are more
the spatially averagedmagnetization density pronounced. The overall sign of theeffect is thesame.
1 W/2
= (y)dy , (32)
y y
M W M
Z−W/2 Exactly the opposite mechanism is responsible for the
as a function of the value h¯α of the spin orbit coupling caseof (y), which,aswehaveseen,is insteadvanish-
z
M
constant. With the parameters chosen in Figure 11 the inginthebulk. Interestingly,asonecaninferfromFigure
crossoveroccursapproximatelyforh¯α=1.710−9meVm. 11, the expression of Eq. (31), formally valid in the bulk
The dashed curve superimposed to the ǫ = 2.5 meV for finite values of α when in principle the perturbative
F
dottedline ispurely cubic andprovidesauseful guide to condition is inapplicable, still gives good results for the
the eye for that case. case of a rather narrow channel with W =0.15 µm.
8
k
y cussed are compatible with a quite general and rigorous
argumentregardingtheparityof (y)and (y)with
z y
K +δk respect to the sign of the spin oMrbit couplinMg constant.
K −δk +
+ Considerthe effect onthe hamiltonian(including contri-
butions from the electron electron interaction and non
magnetic impurities) of the application of the operator
kx σˆz. Clearly σˆzHˆ(α)σˆz = Hˆ( α). Then it immediately
−
K −δk followsthatgivenaneigenstateoftheproblemforagiven
− K +δk value of α, the correspondingsolution for α canbe ob-
−
−
tained simply by application of σˆ . This in turn implies
z
that, quite generally, while (y) is even with respect
z
M
to α, (y) is odd.
y
M
FIG.10: Distorted momentumspaceoccupation distribution We have shown that a net spin polarization is estab-
usedtocalculatethebulkvalueofthemagnetization. K± are lishedwhenasteadystatecurrentisrunthroughthewire
defined in Eq. (5).
obtained by laterally confining a two dimensional elec-
tron gas in the presence of Rashba spin orbit. Reversing
the current inverts the spin polarization. The effect is
4
phenomenologically analogous to, yet distinct from the
3.5
spin-Hall effect that has recently been the subject of in-
L
2 3 tense discussion (for a useful review, see Ref. 11) and
m
Μ 2.5 was demonstrated experimentally.9,12,13,14 The mecha-
(cid:144)
B 2 nismrelevanttothepresentdiscussionsimplystemsfrom
Μ
H 1.5 the structure of the electronic wavefunctions in a con-
(cid:143)(cid:143)(cid:143)(cid:143)(cid:143)My 1 fiapnpedeagletoomseptriny.cuArcrceonrtdsinogrlyim, tphuerrietyisscnaottneeriendg.toMmoarkee-
0.5
over its geometric nature should leave the phenomenon
0 2 Ñ4Α Hme6V´nm8L 10 12 mostly unchangedin the presence ofa moderate amount
of impurities in the wire. A similar effect should also be
relevant in the case of electronic states localized by an
FIG.11: Averagespin polarization densityin they direction
impurity potential as well as in the transition to a low
as a function of α. The parameters used are: W =0.15 µm,
density Wigner crystal in the presence of Rashba spin
ǫF =2.5,10,40 meV, δµ=0.05 meV, and m=0.3m0 (mass
orbit.
as in previous Figures). Here the dots represent the (exact)
numerical result, while the dotted curves are obtained from
Eq. (31) in the text. The dashed line is a guide to the eye
representing a purecubic behavior.
Acknowledgments
We discussnexthowthe differentbehaviorsofthetwo TheauthorswouldliketothankLeonidRokhinsonand
components of the spin polarization we have just dis- Yuli Lyanda-Geller for vigorous,useful discussions.
1 That the effects of the electronic interactions can be ne- Awschalom, Science 306, 1910 (2004).
glected is usually a rather dangerous assumption. A sys- 10 It must be noted for completeness that within the more
tematic study of the interplay of the Coulomb interaction rigorousmicroscopicapproachinwhichtheeffectofimpu-
andRashbaspin-orbitinatwodimensionalelectronliquid rities is accounted for (see for instance Ref. 2 and 3) the
can befound in S.Chesi and G. F. Giuliani, unpublished. coefficient ofthelinearcontributiontoMy(y)dependson
2 A. G. Aronov and Y. B. Lyanda-Geller, JETP Lett. 50, thespecific electron impurity potential assumed.
431 (1989). 11 H.-A. Engel, E. I. Rashba, and B. I. Halperin, cond-
3 A.G.Aronov,Y.B.Lyanda-Geller,andG.E.Pikus,Sov. mat/0603306 (2006).
Phys. JETP 73, 537 (1991). 12 J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth,
4 V. M. Edelstein, Solid StateCommun. 73, 233 (1990). Phys. Rev.Lett. 94, 047204 (2005).
5 Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. 13 V.Sih,R.C.Myers,Y.K.Kato,W.H.Lau,A.C.Gossard,
Awschalom, Phys. Rev.Lett. 93, 176601 (2004). and D. D. Awschalom, NaturePhys.1, 31 (2005).
6 A. Y. Silov, P. A. Blajnov, J. H. Wolter, R. Hey, K. H. 14 V.Sih,W.H.Lau,R.C.Myers,V.R.Horowitz,A.C.Gos-
Ploog, and N. S. Averkiev,Appl.Phys. Lett. 85 (2004). sard, and D. D. Awschalom, Phys. Rev. Lett. 97, 096605
7 Y.A.Bychkovand E. Rashba, JETP Lett. 39, 78 (1984). (2006).
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