Spontaneous Emergence of Persistent Spin Helix from Homogeneous Spin Polarization

0.84 MB·English

by  V. A. Slipko

Spontaneous Emergence of Persistent Spin Helix from Homogeneous Spin Polarization
Valeriy A. Slipko,1,2 Ibrahim Savran,3 and Yuriy V. Pershin1,∗
1Department of Physics and Astronomy and USC Nanocenter,
University of South Carolina, Columbia, SC 29208, USA
2 Department of Physics and Technology, V. N. Karazin Kharkov National University, Kharkov 61077, Ukraine
3Department of Computer Science and Engineering,
University of South Carolina, Columbia, SC 29208, USA
We demonstrate that a homogeneous spin polarization in one-dimensional structures of finite
lengthinthepresenceofBychkov-Rashbaspin-orbitcouplingdecaysspontaneouslytowardapersis-
tent spin helix. The analysis of formation of spin helical state is presented within a novel approach
1
based on a mapping of spin drift-diffusion equations into a heat equation for a complex field. Such
1
a strikingly different and simple method allows generating robust spin structures whose properties
0
can be tuned by the strength of the spin orbit interaction and/or structure’s length. We generalize
2
ourresultsfortwo-dimensionalcasepredictingformationofpersistentspinhelixintwo-dimensional

n channels from homogeneous spin polarization.
a
J
PACSnumbers: 72.15.Lh,72.25.Dc,85.75.2d

1
3
The helical wave of rotating spin orientation is re- mation of homogeneous spin polarization into the per-

] ferred to as the spin helix. There is a significant interest sistent spin helix occurs when we introduce boundary
l
l to spin helix configurations in semiconductor materials conditions on electron space motion to describe finite-
a
h since the electron spin relaxation of such spin configura- length structures (in infinite systems the homogeneous
- tionscanbepartially[1–3]orevencompletelysuppressed spin polarization decays exponentially as predicted by
s
e [4, 5]. While a partial suppression of spin relaxation in D’yakonov-Perel’ theory [18]). Using a novel approach
m two-dimensional systems becomes possible in the pres- thatmapsspindrift-diffusionequationsintoaheattrans-
. ence of only Bychkov-Rashba [6] spin-orbit coupling (see fer equation for a complex field we find the exact time-
t
a Refs. [1, 2]), the complete suppression of spin relaxation dependenceofthespinpolarizationdynamics. Itisinter-
m
requires a specific combination of Bychkov-Rashba and esting that the amplitude of the resulting spin helix has
- Dresselhaus [7] interactions as it was demonstrated in anoscillatorydependenceonthesystem’slength. Below,
d
Ref. [4]. More generally, the relaxation of the spin he- we provide an intuitive explanation of this result based
n
lix is an example of situations [1–5, 8–17] when electron on properties of solution of heat equation. Moreover, it
o
c spin relaxation scenario deviates from the predictions of is necessary to emphasize that our theory is generalized
[ D’yakonov-Perel’ theory [18]. for the case of two-dimensional (2D) channels and can

1 Experimentally, the spin grating technique [19] is typ- be straightforwardly verified experimentally. In partic-
v ular, experimentally, the homogeneous spin polarization
ically used [3, 5] to create spin helical configurations in
9 canbeeasilycreatedusingtheopticalorientationbycir-
semiconductors. In this method, a sample is illuminated
3
cularly polarized light. Therefore, we believe that our
0 byapairofpumpbeamswithorthogonallinearpolariza-
approach would simplify tremendously the generation of
0 tions. Theinterferenceofsuchbeamsresultsinaspacial
long-living spin helical configurations in semiconductor
. modulation of light helicity. Correspondingly, through
2
structures and advance the field of spin storage in semi-
0 theopticalorientationeffect,amodulationofspinpolar-
conductors.
1 izationintheformofspinhelixisproduced. Moreover,a
1 spin injection from a ferromagnetic material into a semi-
:
v conductor can also be used to excite a spin helix [20].
t =0
i Inthisapproach, therotatingspinpolarizationiscaused
X
by coherent spin precession of electrons drifting in an
r
a applied electric field. However, the present authors are
notawareaboutanyexperimentalstudiesofspin helixes
excited by spin injection.
t →∞
In this Letter, we propose an alternative approach
to induce spin helical configurations. Specifically, we
demonstrate that in one-dimensional (1D) systems of fi-
FIG. 1: (Color online) Schematics of spontaneous transfor-
nitelengthwithBychkov-Rashbaspin-orbitcouplingthe
mationofhomogeneousspinpolarizationintopersistentspin
spin helical configurations emerge in the process of re- helixinafinitelengthsystemwithBychkov-Rashbaspin-orbit
laxation of homogeneous spin polarization (see Fig. 1). coupling.
Mathematically, such a strikingly unexpected transfor-
2
1.0 1.0
1.000 1.000
(a) (b)
0.7500 0.7500
0.8 0.8
0.5000 0.5000
) )
L L
of 0.6 0.2500 of 0.6 0.2500
nits 0 nits 0
u0.4 -0.2500 u0.4 -0.2500
n n
x (i x (i
0.2 S /S 0.2 S /S
z 0 x 0
0.0
0.02 0.04 0.06 0.08 0.10 0.02 0.04 0.06 0.08 0.10
Time (in units of L2/D) Time (in un its of L2/D)
FIG. 2: (Color online) Dynamics of formation of persistent spin helix from homogeneous spin polarization pointing in z
direction at t = 0. These plots were obtained using Eq. (13) at ηL = 15.45. This value of the parameter ηL corresponds to
the second local maximum of spin helix amplitude shown in Fig. 3.
Let us consider dynamics of electron spin polariza- of time the spin polarization is homogeneous and points
tion in a 1D system of a length L in x direction in the in z direction, that is
presence of Bychkov-Rashba spin-orbit coupling. In one-
S (x,t=0)=0, S (x,t=0)=S . (5)
dimensional limit, spin drift-diffusion equations [2] can x z 0
be written as Let us introduce a complex polarization S =S +iS .
x z
It is straightforward to show that Eqs. (1,3) and bound-
∂S ∂S
x = D∆S +C z −2γS , (1) ary conditions (4) can be rewritten in a more compact
∂t x ∂x x
form using S:
∂S
y = D∆S , (2)
∂t y ∂S ∂2S ∂S
=D −iC −2γS, (6)
∂∂Stz = D∆Sz−C∂∂Sxx −2γSz, (3) ∂t (cid:18)∂x2∂S ∂x(cid:19)
2D −iCS =0. (7)
∂x
where D = (cid:96)2/τ is the coefficient of diffusion, ∆ = Γ
∂2/∂x2, C = 2ηD is the constant describing spin rota- Defining a complex field u(x,t) by the relation
tions, γ =η2D/2 is the coefficient describing spin relax-
u(x,t)=e−iηxS(x,t), (8)
ation, η = 2αm(cid:126)−1 is the spin precession angle per unit
length, α isthespin-orbitcouplingconstant, mistheef- we find that Eq. (6) transforms into the heat equation
fectiveelectronmass,(cid:96)isthemeanfreepathandτ isthe
∂u ∂2u
momentum relaxation time. It follows from Eq. (2) that =D , (9)
y component of spin polarization, S , is not coupled to ∂t ∂x2
y
anyothercomponentofspinpolarization. Consequently, supplemented by Neumann (or second-type) boundary
selecting Sy(x,t=0)=0 we can safely take out Sy from conditions
our consideration. Eqs. (1,3) are complimented by stan- (cid:18) (cid:19)
∂u
dard boundary conditions [21] =0. (10)
∂x
Γ
Moreover, it is worth noticing that the initial conditions
(cid:18) (cid:19) (cid:18) (cid:19)
∂S ∂S
2D x +CS =0, 2D z −CS =0. (4) for u(x,t) are related to the initial conditions for S as
∂x z ∂x x
Γ Γ u(x,t=0)=e−iηxS(x,t=0). (11)
Here, Γ = [x = 0,x = L]. Mathematically, the bound-
Consequently, the initially homogeneous spin polariza-
aryconditions(4)areso-calledthird-typeboundarycon-
tion in z direction (Eq. (5)) corresponds to a spatially
ditions. This specific form of boundary conditions con-
modulated complex field
servesthespinpolarizationofelectronsthatscatterfrom
the sample edges. We assume that at the initial moment u(x,t=0)=S sin(ηx)+iS cos(ηx). (12)
0 0
3

The solution of Eq. (9) with the boundary condi- 1.0
tions (10) and initial condition (12) was obtained by the
methodofseparationofvariables. Itcanbepresentedin S
the form 0.8 x
de x S
S(x,t) =isin(ηL/2)eiη(x−L/2)+ itu x z
S0 ηL/2 pl 0.6
m
2ηLeiηx(cid:88)+∞ 1(η−L()−21−)n(eπ−ni)η2Le−π2Ln22Dt cos(cid:16)πLnx(cid:17). (13) ix A 0.4 x
n=1 el
H
This is our main analytical result describing dynamics n x
of spin polarization in 1D finite-length structures. Note pi 0.2
S
thatS andS componentsofspinpolarizationaregiven
x z
by real and imaginary parts of Eq. (13), respectively.
0.0
The first term in the right-hand side of Eq. (13) de- 0 5 10 15 20 25 30
η L
scribes the persistent profile of spin polarization (in the
form of spin helix) emerging at long times. Concerning
FIG.3: (Coloronline)Normalizedamplitudeofthepersistent
the second term in the right-hand side of Eq. (13), it
spin helix as a function of ηL. Insets show schematically
governs the dynamics of transformation of the initially
distributions of S and S at several specific values of ηL as
homogeneous spin polarization into the persistent spin x z
indicatedbyarrows. Positionsofminimaandmaximapoints
helix. Fig. 2 demonstrates dynamics of Sz and Sx com- of the amplitude are discussed in the text.
ponents of spin polarization given by Eq. (13). It is
clearlyseenthattheinitiallyhomogeneousspinpolariza-
tion in z directions transforms into the persistent spin
helix with an (infinitely) long lifetime. these times τ = L2/(π2D) provides the time scale of
1
Explicitly, in the long time limit, the spin polarization thetransformationprocess. Thedependenceofspinpro-
is given by cess on L is intuitively clear as electrons should "feel"
the system’s length before the transformation ends. It’s
sin(ηL/2)
Sx(x,t=+∞)=−S0 ηL/2 sin(η(x−L/2)), (14) alsointerestingthatsuchatimecanbelongerorshorter
then the relaxation time of homogeneous spin polariza-
Sz(x,t=+∞)=S0sinη(LηL/2/2)cos(η(x−L/2)). (15) tiniogntτhhat=τ1/<(Dτη2)w. hInenpaηrLtic<ulaπr,,tτh1e/τthim=es(ηaLr/eπt)h2emseaamne-
1 h
when ηL = π, and τ > τ when ηL > π (see also Fig.
In these equations the factor sin(ηL/2)/(ηL/2) defines 1 h
3).
reduction of the spin helix amplitude with the respect to
the initial amplitude of homogeneous spin polarization In order to obtain an additional insight on spin re-
S . We plot this function in Fig. 3. It is interesting laxation of the radial spin helix, we have performed ex-
0
thatthespinhelixamplitudeisanoscillatingfunctionof tensive Monte Carlo simulations employing an approach
the parameter ηL and takes zero values when ηL=2πn described in Refs. [8] and [22]. This Monte Carlo sim-
wherenisapositiveinteger. Thepositionsoflocalmax- ulation method uses a semiclassical description of elec-
ima can be found numerically. In particular, positions tron space motion and quantum-mechanical description
of four local maxima shown in Fig. 3 are 8.987, 15.450, of spin dynamics (the later is based on the Bychkov-
21.808, 28.132. Rashba coupling term). All specific details of the Monte
The heat equation is the best starting point to under- Carlosimulationsprogramcanbefoundinthereferences
stand the oscillatory dependence of spin helix amplitude cited above and will not be repeated here. A spin con-
on ηL depicted in Fig. 3. Accordingly to Eq. (11), the servationconditionwasusedforelectronsscatteringfrom
initially homogeneous initial condition (Eq. (5) for spin system boundaries. Generally, all obtained Monte Carlo
diffusion equations transforms into a modulated initial simulation results are in perfect quantitative agreement
condition for the heat equation. As the solution of heat withouranalyticalpredictionsthusconfirmingthemech-
equation in the given context represents simply the pro- anismofformationofpersistentspinhelixfromhomoge-
cess of temperature equilibration along the system, an neous spin polarization. A comparison of selected ana-
integer number of modulation periods results in zero av- lytical and numerical curves is given in Fig. 4.
erage "temperature" and, correspondingly in zero spin The results reported in this paper can be readily gen-
helix amplitude. Moreover, we would like to mention eralized for the persistent spin helix in two dimensions
that the spin helix formation process is described by a [4]. Indeed, it can be easily seen that in the case of
series of exponentially decaying terms whose time con- equalstrengthofBychkov-RashbaandDresselhausspin-
stants are given by τ = L2/(π2n2D). The longest of orbit interactions, α = β (where β is the Dresselhaus
n
4

trol of spin helix characteristics is achievable via appro-
0.3 Monte Carlo result priate choice of the above mentioned parameters. This
Analytical result suggested technique facilitates generation of spin helical
0.2 statesandcanbeusedinbothone-andtwo-dimensional
geometries.
0.1 I. S. acknowledges PhD scholarship from the Repub-
S0 lic of Turkey Ministry of National Education, Grant No:
S/z 0.0 [010] MEB1416.
-0.1 [100]
-0.2
∗ Electronic address: [email protected]
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z
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andstrengthofspinorbitinteraction. Therefore,thecon-