Table Of ContentSpin polarization control in a 2-dimensional semiconductor
Ian Appelbaum and Pengke Li (李鹏科)
∗ †
Department of Physics and Center for Nanophysics and Advanced Materials, U. of Maryland, College Park, MD 20742
Longcarrierspinlifetimesareadouble-edgedswordfortheprospectofconstructing“spintronic”
logic devices: Preservation of the logic variable within the transport channel or interconnect is
essential to successful completion of the logic operation, but any spins remaining past this event
will pollute the environment for subsequent clock cycles. Electric fields can be used to manipulate
these spins on a fast timescale by careful interplay of spin-orbit effects, but efficient controlled
depolarization can only be completely achieved with amenable materials properties. Taking III-
6
VI monochalcogenide monolayers as an example 2D semiconductor, we use symmetry analysis,
1
perturbationtheory,andensemblecalculationtoshowhowthislongstandingproblemcanbesolved
0
by suitable manipulation of conduction electrons.
2
r
a
M I. INTRODUCTION (a) (b)
s
⊥
k E
9 Manipulation of electron spin orientation in polarized
2 ensembles provides a basis for new logic devices and cir-
l] cbuaistesdwdiethsigpnost.e1nIttiailsawdivdaenltyagbeeslieovveedr pthreastenwth-denayevcehrasrpgien- BD BBR k
l
a encodes logic state, semiconductor materials with the
h longest spin lifetime are the most suitable choice for
(c) (d)
s- transport channels between injection and detection con- Btotal s
e tacts. However, once a logic operation is completed, k
θ
m residual spins can – and will – interfere with those in- B B
D
. volved in future operations. Can we design a device
t
a with a controllable spin lifetime? In this scheme, oth-
m erwise robust spins would vanish from the channel by BBR BD
- an externally-induced, fast, and tunable depolarization
d mechanism upon completion of every logic operation. FIG. 1. Spin polarization control using pulsed spin-orbit
n
Thepresentpaperpresentsasolutiontothischallenge, fields. Panel (a) shows a charge carrier with quasimomen-
o
making use of two-dimensional semiconductor materials tumk andlong-livedspins ,perpendiculartotheplaneand
c parallel to the Dresselhaus⊥magnetic field B . In (b), elec-
[ having a strong uniaxial spin-orbit field anisotropy. In D
tric field E (cid:107) s creates a Bychkov-Rashba spin-orbit field
3 thisscheme,spinsareinitiallyalignedparallelorantipar- B ⊥E,k in t⊥he plane. Spins then precess about the total
alleltoalong-livedquantizationaxisatinjectionorgen- BR
v field B at angle θ , as shown in (c). When the electric
total B
7 eration. Afterspintransporttootherpartsofthedevice field vanishes, any residual in-plane spins s are quickly de-
2 and completion of a logic operation, a clocked voltage phased by the Dresselhaus field, as shown in(cid:107)(d).
5 pulse at an electrostatic gate generates an electric field
7 in the transport channel that induces a Bychkov-Rashba
0 effective magnetic field.2 This magnetic field, due to the
have identified several such two-dimensional materi-
.
1 structuralinversionsymmetry-breakingelectricfieldand
als with the requisite anisotropic spin-orbit properties.
0 spin-orbit interaction, is non-collinear to the spin axis
The most promising candidate material system we have
6 andthusrotatesthespinsviaprecessionontoanorthog-
found is the group-III metal–mono-chalcogenide mono-
1
onal axis. The physical logic environment is then reset
: layers (G3M-MCs). In this inversion-asymmetric two-
v for the next operation.
dimensional material system (such as GaSe and InS),
i
X One realization of such an anisotropic material is the the spin-orbit-induced k-dependent Dresselhaus effec-
zincblende [110] quantum well, whose spin relaxation tive magnetic field7 is oriented perpendicular to the
r
a properties have been thoroughly studied using optical monolayer plane and scales as a cubic function of the
orientation methods.3–6 However, fabrication of this sys- wavenumber k.8 Spin up and down are then the nat-
tem requires epitaxial growth and the active layer is ural eigenstates, immune to Dyakonov-Perel (DP) spin
buried deep within the bulk. An alternative approach relaxation which would otherwise cause precessional de-
tomeetourrequirementforanisotropywithoutsophisti- phasing upon momentum scattering for any other po-
catedcrystalgrowthortheconstraintofdeepencapsula- larizationaxis.9 Anelectrostatically-controlledBychkov-
tionisthroughuseofinversion-asymmetricvanderWaals Rashbafield–whichisalwaysperpendiculartoboththe
layered materials obtainable by exfoliation or vapor de- quasimomentumkandtheelectricfield z,thusoriented
position methods. in-plane – can be used to rotate spins tEoward the plane
Through detailed theoretical symmetry analysis, we and achieve depolarization.
2
II. MECHANISM AND MATERIALS ter Γ-point, while those of TMDCs are at the zone-edge
K(K ) points. As a result, the internal spin-orbit effec-
(cid:48)
tivemagneticfielddependenceoncrystalmomentumkis
The depolarization mechanism we describe is illus-
drastically different in each case: spin splitting scales as
trated in Fig. 1(a)-(d). In (a), spin-polarized electrons
k3 in G3M-MCs, vanishing at Γ due to Kramers’ degen-
oriented normal to the channel surface are injected elec-
eracythere,butinTMDCs,thespinsplittingclosetothe
trically from a ferromagnet with perpendicular magnetic
anisotropy(suchastheCoFeB/MgOsystem10andCo/Ni bandextremaisenormousinmagnitudeandindependent
orCo/Pdultrathinmultilayersystem11)orgeneratedvia of the wavevector k. The smallest spin-splitting can be
optical interband excitation with polarized light.8 The foundintheMoS2 conductionbandat4meV,equivalent
toamagneticfieldofmanytensoftesla;tocompetewith
out-of-plane spin-orbit Dresselhaus field stabilizes spins
it,thevoltage-inducedBychkov-Rashbafieldwillrequire
aligned (or anti-aligned) to it from extrinsic fluctuations
similarly enormous and impractically obtainable electric
(such as magnetic impurities, random strain gradient,
fields.
substrate potential fluctuation, etc), allowing them to
travel through the channel without appreciable depolar-
ization. In Fig. 1(b), we show that after a logic op-
III. PERPENDICULAR SPIN LIFETIME
eration is completed (such as spin-torque or exchange
frompolarizedelectronsmanipulatingthemagnetization
of a ferromagnetic contact12), a perpendicular electric We first justify our expectation of a long out-of-plane
field pulse provided by a transverse electrostatic gate in- spinlifetime,andmotivatethechoiceofconduction-band
ducesaBychkov-Rashbaeffectivemagneticfieldoriented electronmanipulationinn-typeG3M-MCmonolayers,as
in the plane. Its combination with the intrinsic Dressel- opposed to holes in p-type material.
haus field results in a total effective magnetic field mis-
aligned with the spins, at an angle θ . In Fig. 1(c),
B
A. Spin mixing
weshowspinprecessionaroundthetotalspin-orbitfield.
Withacarefullyengineeredgatevoltagepulseamplitude
and duration, spins precess into the plane, eliminating Although spins aligned to the Dresselhaus field are
the ensemble projection onto the original quantization immune to DP relaxation, they are still subject to the
axis when the electric field vanishes. As shown in Fig. Elliott-Yafet (EY) mechanism. EY spin relaxation is
1(d), the channel is then cleared of out-of-plane spin, driven by carrier scattering events that couple to minor-
and any remaining in-plane polarization is quickly de- ity spin components of the wavefunction. These impure
polarized by precessional dephasing. Residual spins are admixtures are introduced by the effect of spin-orbit in-
eliminated,preparingthechannelforthenextlogiccycle teraction and so EY is generally present in all materials
(whichmaybeaffectedbytheup/downorientationofthe regardless of inversion (a)symmetry.
injector ferromagnet from upstream circuit elements). Spin-orbitinteractioncanbetreatedasaperturbation
Severalquestionsmustbeansweredbeforethisscheme within k pˆ theory, where it generates two terms in the
can be considered viable: Which perpendicularly- envelopeH· amiltonian: ak-independentterm 4m(cid:126)20c2∇V×
polarized carriers (conduction electrons or valence holes, pˆ (cid:126)σ and a k-dependent term (cid:126)2 V kˆ (cid:126)σ. In two-
immune to DP) suffer the least relaxation by secondary · 4m20c2∇ × ·
dimensional systems when k z = 0, the latter takes on
spin-flip mechanisms? What is the magnitude of both ·
the form
theDresselhausandBychkov-Rashbacoefficientsforthis
band,andaretheycompatibletoachievecompletedepo- (cid:126)2 ∂V ∂V ∂V
(k σ k σ ) +k σ +k σ . (1)
larization in electric fields of reasonable strength? What 4m2c2 x y− y x ∂z y z ∂x x z ∂y
0 (cid:20) (cid:21)
istherelationshipbetweenoptimizedgatepulseduration
Sinceonlyσ andσ haveoff-diagonalelements,onlythe
and electric field, and is it consistent with the require- x y
ments imposed by an upper bound set by momentum firstterm(proportionalto ∂V)canperturbthewavefunc-
∂z
scatteringtime? Inthefollowingsections,weapplysym- tionwithoppositespinadmixtures. Thistermclearlyhas
metryanalysis,lowest-orderperturbationtheory,anden- thesamespatialsymmetrypropertiesasthepolarvector
semble integration to address these and other essential z; using the language and notation of group theory, it
questions. is a basis function for the irreducible representation (IR)
Beforeproceedingtothenextsection,wemustfirstad- Γ−2, as in Table I.
All components of the operator pˆ exist regardless of
dressanimportantissueregardingourchoicetofocuson
the dimensionality, so the k-independent spin-orbit in-
the lesser-known G3M-MC monochalcogenide materials
teraction is
(GaSe,InS,etc.),incontrasttothebetter-knownTMDC
di-chalcogenide system (WS2, MoSe2, etc.). In monolay- (cid:126) ∂V ∂V ∂V ∂V
σ p p +σ p p
ers of both materials, the internal Dresselhaus magnetic 4m2c2 x ∂y z− ∂z y y ∂z x− ∂x z
field (proportional to the spin-subband splitting) is al- 0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)
∂V ∂V
ways oriented out-of-plane.13 However, gap-edge states +σ p p . (2)
z y x
of G3M-MCs are located around the Brillouin zone cen- ∂x − ∂y
(cid:18) (cid:19)(cid:21)
3
TABLE I. Basis functions (BFs) of some irreducible repre- these states will have very small k and thus negligible
sentations (IRs) in Γ-point D3h group. The assignment of Γ+ spin admixtures. The valence band, however, has a
plus and minus superscripts to IRs follows the convention of 1
distorted‘caldera’shapeandsothermallyoccupiedholes
evenandoddparitywithrespecttotheoperationofin-plane
atthebandedge–onthecalderarim–haveasubstantial
mirror reflection σ .
h
k. Thisinturnleadstoastrongwavefunctionadmixture
IRs Γ+1 Γ+3 Γ−2 Γ−3 with components having Γ−2 character. As a result, we
BFs 1 {x, y} z {xz, yz} expect that the valence band states will be far more sus-
ceptible than conduction band states to spin relaxation
caused by the unavoidable presence of flexural phonons.
The σx,σy spin-mixing terms transform as the in-plane The k-independent perturbation Eq. (2) exacerbates
componentsofanaxial(pseudo-)vector xz,yz (theIR the problem for holes. This spin-orbit term leads to spin
{ }
Γ−3, see Table I). flipsinbothconductionandvalencebandsviascattering
With the assistance of the basis functions (Table I) with in-plane optical phonons sharing the same Γ−3 sym-
that capture the symmetries of spin-orbit perturbations metryofthetwospin-mixingtermsinEq.(2). Thecutoff
anddifferentbands,itisstraightforwardtoexaminehow energy of this type of phonon in G3M-MC monolayers is
spin-mixing is introduced. Here we focus on the gap- 25 meV17 and is therefore expected to seriously affect
edge states. The valence band spatial wavefunctions ≈spin relaxation at room temperature. The strength of
are invariant to all of the point-group symmetry oper- EY spin relaxation due to scattering with these optical
ations, and thus transform as a scalar, 1 (corresponding phonons is proportional to the minority-spin mixing am-
to IR Γ+1). The k-dependent spin-orbit perturbations plitude of the eigenstates, which is far larger in the Γ+1
thus cause first-order corrections of opposite spin from highestvalenceband(8%probabilityasopposedto0.1%
remote bands with Γ−2 (z-like) symmetry, since (cid:104)1|∂∂Vz|z(cid:105) in the conduction band)8 due to the close proximity of
isnonvanishing. Similarly, oppositespincomponentsare Γ−3 lower valence bands.
induced to the valence band by k-independent spin-orbit In light of these issues, we conclude that the spin life-
perturbationsfromremotebandswithΓ−3 ({xz,yz}-like) time for valence band holes is much shorter than elec-
symmetry. The same argument can be applied to the trons in the conduction band of G3M-MC monolayers.
conduction band, which is odd with respect to mir- We therefore restrict our subsequent analysis to the lat-
ror inversion about the plane, and so transforms like z ter carriers. For electrical injection of spin-polarized
(Γ−2). The conduction band wavefunction will thus ac- electrons into the conduction band, n-type conductiv-
quirespinadmixtureswithspatialsymmetriesofΓ+1 from ity is desirable, as is usually the case in GaS18 and
k-dependent and Γ+3 from k-independent perturbations. InSe19,whereasGaSeisusuallyp-type.18,20 Controllable
n-doping during synthesis is therefore desirable in this
case. On the other hand, spin injection via optical ori-
B. Phonon symmetry entation during photocarrier generation is insensitive to
the doping nature, while electron spin relaxation due
to exchange with holes (Bir-Aronov-Pikus mechanism21)
In-plane acoustic phonons in these materials have x,y
(Γ+) symmetry and therefore only play a role in spin- should play a minor role thanks to the relatively spin-
3 pure gap edge states (as compared with degenerate va-
preserving momentum scattering. These events couple
lenceedgestatesincubicsystems). Inbothdopingcases,
the spin-majority components of the wavefunctions and
back-gate bias tuning may be necessary to reduce the
affectthechargemobilitybutnotspinrelaxation. Inthe
background Bychkov-Rashba field induced from struc-
following we discuss the influence on spin relaxation due
tural inversion asymmetry introduced by the presence of
to carrier scattering with flexural phonons and optical
the substrate.
phonons, and justify that in both cases, the spin of elec-
trons in the conduction band is more robust than holes
in the valence band.
Because out-of-plane flexural phonons have no cut-off IV. CONDUCTION BAND DRESSELHAUS
and a quadratic dispersion relation to lowest order (and AND BYCHKOV-RASHBA COEFFICIENTS
hence a constant density of states, as opposed to the
vanishinglinearDOSforthein-planeacousticphonons), The proposed mechanism to exploit the spin-orbit
scatteringwiththemusuallydominatestheEYspinlife- anisotropyforchannelresetdependscruciallyonourabil-
time in two-dimensional materials.14–16 These phonons ity to generate an in-plane Bychkov-Rashba field that
have spatial symmetry of z (Γ−2) and so will drive spin rivals the out-of-plane Dresselhaus field in magnitude.
relaxation in both the valence and conduction bands by Onlythenwillasufficientcomponentofspinprecessinto
coupling majority spin to admixtures introduced by the the orthogonal in-plane orientation. Here, using third-
spin-flip terms of the k-dependent spin-orbit interaction orderperturbationtheorytocalculatethemagnitudesof
in Eq. (1). these two fields, we demonstrate the feasibility of this
The conduction band dispersion is quadratic around scheme.
the Brillouin zone center, so thermal electrons filling First of all, following the same scheme of evaluating
4
(a) (b) evaluated by
E
4
P5 E3 (cid:27)Γ−3c Γ−2c βc ≈|P1eE(cid:104)z(cid:105)αv|E1E+−23−, (4)
P g
5
Γ−2c z Eg where z is on the order of the monolayer thickness and
Q Q h i Γ+1v P1 ED1e+p−en3−d(cid:104)in(cid:105)isgtohnetehneeragvyerdaigffeerweanvceevebcettowreeonf tΓh+1evealencdtrΓon−3vs.,
P
P1 E21 Γ+1v α Γ−3v E1+−3− tc(hoarneBebvyeecnhtuknodevod-mRfiarnosahmnbtazeotrevoremru)pHttBhoeRa=wveβaalckuke(Dccorosemsφspσeaylhr−aabuslsienφtweσritxmh)
E1 Γ−3v γ k3sin3φσ .
c z
(cid:27)
Usingparametersappropriatefortheconductionband
FIG. 2. Matrix-element perturbation pathways allowing cal- of monolayer GaSe (α 0.3, E 3 eV, P h2/ma,
v g 1
≈ ≈ ≈
cmuolantoicohnalocofgsepniindesmploitntoinlagyeirns vthiae tΓw−2o dciosntidnuccttimonechbaannidsmosf: aBy≈ch3k.o7v5-˚AR,asEhb1+a−e3n−er≈gy0o.3n ethVe),orwdeerobotfa1inmaenVefxopreeclteecd-
(a) “bulk inversion asymmetry” Dresselhaus coefficient, and trons with k =√2m k T/(cid:126) 0.1π at T =300 K in an
(b)“structuralinversionasymmetry”Rashbacoefficient. Not ∗ B ≈ a
electricfield =1MV/cm,readilyobtainablewiththin-
to scale. E
film dielectric insulators and low gate voltages23, and of
thesameorderastheDresselhaussplittingatthesamek
along Γ K calculated from a tight-binding bandstruc-
−
the valence band Dresselhaus spin splitting,8 we can es- turecalculation(cubicpolynomialfittingtheDresselhaus
timatethemagnitudeγ oftheconductionbandDressel- spin-splitdispersiongivesγ 1.044eV˚A3).8,24–26 Byin-
c c ≈
haus term = γ k3sin3φσ using perturbation the- corporating on-site electrostatic energy into the calcula-
D c z
H
ory to third order in k pˆ. The dominant terms, rem- tion,wecanfitthelinearconductionbandsplittingalong
·
iniscent of those in the analogous calculation for III- Γ M (where Dresselhaus effect vanishes) to recover β :
c
V semiconductors22, correspond to perturbation paths Fo−r Ga(Se) atoms 1.2(2.3)˚A27 from the basal plane, our
through both the spin-orbit-split lower valence and up- numerical results yield β / 2.4 meV˚A/(MV/cm).
c
E ≈
perconductionbands. ReferringtoFig. 2(a),wherehor-
izontallinesrepresentthespin-dependentΓ-pointstates,
one obtains a magnitude V. ENSEMBLE SUMMATION
Ofcourse, notallelectronshavethesamekandhence
γ = (cid:126)3 (cid:104)Γ−2c|py|i(cid:105)(cid:104)i|py|j(cid:105)(cid:104)j|py|Γ−2c(cid:105) feel different Dresselhaus and Bychkov-Rashba fields.
c m30 EiEj Thus,theshortestpossibleelectrostaticgatepulse-width
i,j(cid:88)=Γ−3v,c optimizing precession-induced depolarization of the out-
1 1 of-plane component of these electrons is dependent on
= P QP , (3)
1 5
| |(cid:18)E2E3 − E1E4(cid:19) which states comprise the nondegenerate electron den-
sity n = D k T exp(E /k T), where D is the (con-
2d B F B 2d
stant) density of states, k T is the thermal energy, and
where P and P are proportional to off-diagonal ma- B
1 5 E < k T is the Fermi energy relative to the conduc-
trix elements of the momentum operator pˆx,y, i.e. tioFn ba−ndBminimum.
(cid:126)
P1(5) = m0(cid:104)Γ−2c|pˆx,y|Γ−3v(c)(cid:105), and Q is the matrix element An initially out-of-plane spin precesses around an ef-
(cid:126)
m0(cid:104)Γ−3v|pˆx,y|Γ−3c(cid:105). E1 and E2 (E3 and E4) are the ener- fective magnetic field at an angle θB with the surface
gies of the spin-split Γ−3v (Γ−3c) bands relative to Γ2c. normal, and therefore has an out-of-plane projection
Calculation of the electrostatic gate-induced Bychkov- S (ωt,θ )=cosωtsin2θ +cos2θ . (5)
z B B B
Rashbacoefficientcanbetreatedsimilarlywithinpertur-
bationtheory. AsshowninFig. 2(B),thedominantpath The in-plane spin components are
is via the closest Γ+1v and Γ−3v valence bands. Here, the
outofplaneelectricfield zdirectlycouplesthegapedge S (ωt,θ )=sinωtsinθ ,
Γpa−2criatyn.dTΓh+1vesktatepˆspbeerctauursEbeatthioenya[wreitohftohpepossaimteerepflaercatmioen- Sxy(ωt,θBB)=sin2 ωtsinB2θB,
· 2
ter P1 as in Eq. (3)] strongly couples the Γ−2c and Γ−3v
states that share the same in-plane planewave origin.8 where the y-direction lies in the plane formed by the ini-
Coupling between the two intermediate states is by the tial spin vector and the effective magnetic field.
k-independent spin-orbit term in Eq. 2, which is related Because the spin-orbit Hamiltonian terms (k,φ)
BR
to the strong spin-mixing coefficient α of the Γ+ va- and (k,φ)aretime-reversal-invariant,equilHibriumen-
v 1v HD
lenceband. TheBychkov-Rashbacoefficientcanthenbe semble averages over the in-plane components S and
x
(cid:104) (cid:105)
5
(a)
1
0.5
0.8
0.1 β =3meV˚A
c
k 0.6
i
)
a] φ S(tz 0.4 hcos2θBi
[π/ 0 0 Sz h0.2 6meV˚A
y
k
0
12meV˚A
-0.1 −0.20 5 10 15 20 25
time t [ps]
(b)
-0.5 1
-0.1 0 0.1 100
k [π/a] 0.8
x
)i0.6
FIG.3. Spinorientationink-spaceforathermalensembleof (t0
z 0.4
initially perpendicularly-polarized spins at T = 300 K, with S
β =6 meV˚A, at t = π(cid:126)/(cid:112)2β3/γ (cid:39) 3.22 ps. In-plane spin ps] 10 h0.2
c c c [
vcuecrtvoerssmaraerkplkott=ed(cid:112)foβr/s(eγversainl3stφa)t,eswhwehreereθSz==π0/.4.DaHsehreed, t0 t0 00 5 10 15 20 25
γ = 1.044 eV˚A3 andcm =c 0.655m , as apBpropriate for the βc [meV˚A]
c ∗ 0
conduction band of monolayer GaSe. 1 β0
c
0 5 10 15 20 25
β [meV˚A]
S for initially perpendicularly-polarized spins van- c
y
(cid:104)ish(cid:105)identically for all t. This can clearly be seen in FIG. 4. (a) Time evolution with β = 3,6, and 12 meV˚A.
c
Fig. 3, where we show the typical three-fold symme- Symbols indicate optimum times (pulsewidths) for minimum
try of in-plane spin components of thermally occupied out-of-plane projections. (b) Bychkov-Rashba parameter βc
dependenceofoptimizedpulsewidththatminimizesthisspin
states in k-space when the precession frequency ω is
projection. Inset: minimum spin projections over the same
set by the Bychkov-Rashba and Dresselhaus fields with
(cid:126)ω = (β k)2+(γ k3sin3φ)2, and the effective spin- βc range. Values of γc and m∗ are the same as in Fig. 3.
c c
orbitfieldorientationvariesastanθ =β /(γ k2sin3φ).
B c c
(cid:112)
The ensemble average over the out-of-plane compo-
nent S (t) does not vanish, except for precisely timed will it vanish completely (at an optimal time t(cid:48) < t0
z (cid:39)
(cid:104) (cid:105) 7 ps). This value can be approximated by the condition
gate pulses. Summing over all filled conduction electron
states in k-space (again assuming Boltzmann statistics), BBR =BD for thermal electrons, at βc0 (cid:39)γc2m∗(cid:126)k2BT and
we have t0 π(cid:126)2/β0√2m k T. If the in-plane Bychkov-Rashba
(cid:39) c ∗ B
fielddisappearsattheendofanelectricfieldpulseofthis
S (t) = 6C π/3 ∞S (ωt,θ )e Ck2kdkdφ, (6) duration, the ensemble will remain unpolarized and spin
(cid:104) z (cid:105) π z B − channel reset will be achieved.
(cid:90)0 (cid:90)0
Beyond t > t, the spin projections for β > β0 un-
(cid:48) c c
where C = (cid:126)2 , and we have exploited the sixfold dergo a damped oscillation, becoming negative before
symmetry of2tmh∗ekBDTresselhaus field magnitude in the an- passing through zero again and saturating at a positive
gular integration bound. value. The asymptotic values of these spin projections
Notice that Eq. (6) is independent of the Fermi en- as t correspond to the case where spins are fully
→ ∞
ergy E . The result of our calculation is therefore inde- dephased, and ensemble averages cosωt,sinωt = 0. In
F (cid:104) (cid:105)
pendent of the carrier density (which may change upon other words, only the incoherent part of the spin projec-
application of the Bychkov-Rashba field, due to capaci- tions [second term in Eq. (5)] remain.28
tivefield-effectfromthegatepotential),providedtheas- We can calculate the optimum time t for a range
(cid:48)
sumptionofnondegenerateBoltzmannstatisticsremains of Bychkov-Rashba parameter β values as shown in
c
valid. Fig.4(b). Forverysmallvaluesofβ <β0,whent >t0,
c c (cid:48)
Examples of this time evolution at T = 300 K are the ensemble out-of-plane spin component never reaches
shown in Fig. 4(a), for Bychkov-Rashba parameters zero. In this case, our calculated t corresponds to the
(cid:48)
β =3, 6, and 12 meV˚A (generated by electric fields minimum S . Using parameters appropriate for GaSe,
c z
1 5MV/cm),γ =1.044eV˚A3 andm =0.655m , thisconstr(cid:104)aint(cid:105)setsaminimumgate-inducedelectricfield
c ∗ 0
E ≈ −
as obtained from a tight-binding model for GaSe.8 The of 1 MV/cm, consistent with our previous calculation
≈
out of plane spin projection S (t) initially decreases, comparing the magnitudes of Dresselhaus and Bychkov-
z
but only for β larger than a c(cid:104)ritical(cid:105)value β0 6 meV˚A Rashba terms.
c c ≈
6
VI. DISCUSSION For this scheme to work, it is essential that there ex-
ist occupied regions in k-space where the magnitude of
Bychkov-Rashba field is greater than Dresselhaus field.
The short gate pulses of only several picoseconds sug- This statement does not necessarily imply that materi-
gestedheresetalowerboundforthespeedofdigitalspin- als with the smallest Dresselhaus coefficient should be
tronic devices making use of the proposed mechanism. sought: amoderatevaluestabilizesspinsagainstdephas-
However, this coherent precession scheme assumes that ingfromfluctuatingspin-orbitfieldsgeneratedbye.g. in-
carriersareinthecollisionlesslimitsetbythemomentum homogeneous strain.29 We thus suggest that, under the
scattering time upper bound. In practice, longer gate right conditions, other platforms with the right config-
pulses (and correspondingly lower perpendicular electric uration of spin-orbit coupling, such as zincblende [110]
fields) will likely be more practical; if this duration is QWs with a lowest-order Dresselhaus term kcosφσz,
∝
maintained far longer than the momentum scattering may also be effective in enabling spin-channel reset by
time, a DP-like dephasing and ensemble depolarization controlled depolarization.
will accomplish a similar result.
However long the gate pulse duration, its rising edge ACKNOWLEDGMENTS
must be abrupt to induce the coherent precession we
model. If the gate rise-time is substantially more than This work was supported by the Office of Naval Re-
theprecessionfrequency,theinitiallyperpendicularspins search under contract N000141410317, the National Sci-
will simply follow the instantaneous spin-orbit field via ence Foundation under contract ECCS-1231855, and
adiabatic passage; full depolarization of the out-of-plane the Defense Threat Reduction Agency under contract
spin will then be impossible. HDTRA1-13-1-0013.
[email protected] 16 W. Han, R. K. Kawakami, M. Gmitra, and J. Fabian,
∗
[email protected] Nature Nanotech. 9, 794 (2014).
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