All-electrical detection of the relative strength of Rashba and Dresselhaus spin-orbit interaction in quantum wires

0.17 MB·English

by  Matthias Scheid

All-electrical detection of the relative strength of Rashba and Dresselhaus
spin-orbit interaction in quantum wires
Matthias Scheid,1,2,∗ Makoto Kohda,1 Yoji Kunihashi,1 Klaus Richter,2 and Junsaku Nitta1
1Graduate School of Engineering, Tohoku University,
6-6-02 Aramaki-Aza Aoba, Aoba-ku, Sendai 980-8579, Japan
2Institut fu¨r Theoretische Physik, Universit¨at Regensburg, 93040 Regensburg, Germany
We propose a method to determine the relative strength of Rashba and Dresselhaus spin-orbit
9 interaction from transport measurements without the need of fitting parameters. To this end, we
0 make use of theconductance anisotropy in narrow quantumwires with respect to the directions of
0 an in-planemagnetic field,thequantumwire and thecrystal orientation. Wesupportourproposal
2
by numerical calculations of the conductance of quantum wires based on the Landauer formalism

n which show theapplicability of themethod toa wide range of parameters.
a
J PACSnumbers: 71.70.Ej,73.20.Fz,73.63.Nm

1
2
With conventional electronics expected to reach crit- ments; therefore methods are highly desirable that allow

] ical boundaries for its performance soon, a new field of one to determine the ratio α/β from transport measure-
l
l research utilizing the spin of the electron has evolved in ments. In principle, this can be achieved by fitting weak
a
h recentyears. Withinthisfieldcalledspintronicsmuchat- antilocalization (WAL) data from magneto-conductance
- tentionhasbeenfocussedonspin-orbitinteraction(SOI) (MC) measurements to analytical predictions [11, 12].
s
because it provides a way of controlling the spin degree However, the results usually bear a certain ambiguity,
e
m offreedomelectricallyin(non-magnetic)semiconductor- since onehasto fit the data withseveralparametersand
based systems without the need of external magnetic the possible error margins are thus quite large.
.
t
a fields. However, SOI in two-dimensional electron gases Hence, in this Letter we propose an alternative, all-
m (2DEG)isadouble-edgedsword,sincespinrelaxationin electricalmethodtodeterminetherelativestrength,α/β,
- disordered 2DEGs, which is typically dominated by the ofRashbaand DresselhausSOI frommeasuringthe con-
d D’yakonov-Perel’ (DP) mechanism [1], is enhanced for ductance of narrow quantum wires defined in a 2DEG
n
strong SOI. Since many promising semiconductor spin- subject to an in-plane magnetic field. The method is
o
c tronics device proposals, e.g. the Datta-Das spin field based on the fact, that only for a field parallel to the
[ effect transistor (SFET) [2], rely on coherent spin trans- effective magnetic field due to SOI the weak localization

port, it is desirable to efficiently suppress the spin re- (WL) correction to the conductance survives, while it is
1
v laxation. In 2DEGs formed in III-V semiconductor het- suppressedforallotherdirections. No fitparametersare
2 erostructures, there are typically two main SOI contri- required,andα/βisstraightforwardlyrelatedtothisspe-
8 butions, namely, Rashba SOI due to structural inversion cific field direction, where the conductance is minimal.
2 asymmetry[3]andDresselhausSOIduetobulkinversion WenumericallycalculatetheconductanceGofadisor-
3
asymmetryofthesemiconductorcrystal[4]. Aninterest- dered quantum wire realized in a 2DEG with SOI linear
.
1 ing situation occurs when the k-linear Rashba and Dres- in momentum. The single-particle Hamiltonian of the
0
selhaustermsareofequalstrength,i.e.α=β. Then,spin quantum wire in x-direction reads [13]
9
isagoodquantumnumberandDPspinrelaxationisab-
0
v: sent[5]. Latelytherehasbeenmucheffortintothisdirec- H= πx2+πy2 +U(x,y)+ µBg∗ B~ +B~ (~π) ·~σ, (1)
i tion both theoretically with new device proposals [5, 6], 2m∗ 2 || so
X and experimentally with the aim to achieve α = β [7]. (cid:0) (cid:1)
r Naturally, a precise control of the ratio α/β is essential with the effective spin-orbit field
a
for spin manipulation and the operability of many spin-
tronics devices. Since the strength β of the Dresselhaus B~ (~π) = 2 eˆ απ +β(π cos2φ−π sin2φ) (2)
so µBg∗~ x y x y
SOI is fixed in a given quantum well the most promis- h (cid:0) (cid:1)
ing tool to modify α/β is the control of the Rashba SOI + eˆy −απx−β(πxsin2φ+πycos2φ)
strength α via gate voltages [8]. (cid:0) (cid:1)i
To operate spintronics setups relying on the value of and the external in-plane magnetic field
α/β requires the ability to measure this ratio with high
accuracy. Although it is possible to determine α/β by B~ =B (cos(θ−φ)eˆ +sin(θ−φ)eˆ ). (3)
|| || x y
using optical techniques [7, 9, 10], this is not always
an option. If, e.g., the semiconductor heterostructure The vector potential components A in π = (p +eA )
i i i i
is covered by a top gate used to tune the Rashba SOI arise due to the perpendicular magnetic field B whose
z
strength,it is verydifficult to carryoutopticalmeasure- contributiontotheZeemaneffectweneglect. InEq.(2)α
2
andβistheRashbaandDresselhausSOIstrengthrespec- a) b) c)
W0 α = 0.03
tivelyandφ/θistheanglebetweenthequantumwire/in-
0.1
planemagneticfieldandthe[100]directionofthecrystal 2.8
]
for a zinc-blende heterostructure grown in the [001] di- [G0 ]G0
creocntfiionnin.gTphoeteenlteicatlrfoosrtatthiecqpuoatnentutimalwUir(ex,ayn)ditnhceluddiseosrtdheer ∆ G 0 [G ββ == 00..001 λλ == 01/3
potentialfromstaticnon-magneticimpuritiesinaregion 2.8W0 2.6 ββ == 00..0023 λλ == 110/3
of length L. For the calculations we use a discretized -0.1
versionofthe Hamiltonian(1) that allowsus to evaluate 0 1Φ/Φ2 0 1Φ/Φ2 0 1Φ/Φ2
s 0 s 0 s 0
thetransportpropertiesofthewirebycomputinglattice
Green functions. For details see, e.g., Ref. [14]. FIG. 1: (Color online) Magnetoconductance of a quantum
Thedimensionlessnumericalparametersusedinthislet- wire plotted against the magnetic flux Φs = W02Bz in units
ter (denoted by a bar) are related to real physical quan- of Φ0 = h/e. a) ∆G = G(Φs) −G(0) for α¯ = 0.03 (i.e.
tities as follows (for square lattice spacing a): Energy LαSO ≈5.2W0), β¯= 0.0 and widths W = W0, 1.3W0, 1.8W0,
Eβ¯¯ == ((m2m∗a∗/a~2/2)~β2.)EA, sSOaItystpriecnagltlhesngα¯th=sca(lme∗fao/r~t2h)eαsaimnd- 2φ.3=Wπ0/,22.f8oWrs0evfreoramlvtaolpuetsoobfoβ¯tt.ocm).α¯b=) αβ¯¯==00..0033,,WW ==22..33WW00,,
φ=π/2 and θ=π for several values of λ.
ulations we introduce W = 20a. In the calculations,
0
the disorder potential is modelled by Anderson disor-
der with strength U¯ . The mean free path is given by
0
l=2.4W E¯ /U¯2,whereE¯ isthescaledFermienergy. L, where L in the numerical simulation takes the role of
0 F 0 F
The condupctance of the wire is obtained by averaging the phase coherence length in the experiment.
over N disorder configurations and unless stated other- Wenowinvestigatetheinfluenceofanadditionalin-plane
d
wisethefollowingparametersarefixed: E¯ =0.5(corre- magnetic field on the conductance of a quantum wire
F
spondingto4propagatingmodesforawireofwidthW ), where WAL is suppressed. For convenience, we intro-
0
L=7.5W0, U¯0 =1.4 (i.e. l ≈0.87W0) and Nd =10000. duce the ratio λ = B||/|B~so(kx)| which is the relative
strength of the in-plane magnetic field and the effective
Tounderstandthemechanismforthedetectionofα/β, magnetic field due to SOI for a k-vector along the quan-
which requires finite B~ , we first study the conductance tum wire, see Eqs. (2),(3). In Fig. 1c we show the MC
||
of quantum wires at B =0. Specifically we present the for the case α = β for several values of λ: The con-
||
MCfortwocases,whereWAL issuppressed: (a)Rashba ductance at Φ = 0 is enhanced by a finite B . The
s ||
and Dresselhaus spin precession lengths larger than the formoftheMCcurvesinFig.1ccanbeunderstoodfrom
width of the wire W, i.e., Lα = (π~2/m∗α) ≫ W, the expression for the WL/WAL conductance correction
SO
Lβ =(π~2/m∗β)≫W and (b) α=β. from diagrammatic perturbation theory [23]. It is of the
SO
1
In Fig. 1a, we plot G(Φ ) − G(0) for wires with fixed form ∆G ∝ (C − C ), where the first (sin-
s 00 m=−1 1m
α 6= 0,β = 0 and different widths W, showing that for glet) term C00 contriPbutes positively to the conductance
smaller W WAL is suppressed, which is in line with ear- and is responsible for the typical WAL peak in systems
lierexperimentalresults[15]andconfirmsanalytical[16] withSOI.ItisunaffectedbyDPspinrelaxationbutsup-
and numerical treatments [15]. Since spin relaxation is pressed by an in-plane magnetic field [24]. The second
essential for WAL, the mechanism for the suppressionof (triplet) term gives a negative conductance contribution
WAL can be attributed to an enhancement of the spin- and is suppressed for short spin relaxation times [23].
scatteringlength innarrowwires[17, 18], and moregen- For the parametersusedin Fig.1c, C is suppressedfor
00
erally, in confined geometries [19, 20]. λ ≥ 0.15, thus in the respective curves shown in Fig. 1c
In the case (b), α = β, B~ points uniformly into the only the triplet term is present in ∆G resulting in pos-
so
[¯110]-directionforallk-vectorsandaso-calledpersistent itive MC (∂G/∂Φs) > 0. While for λ = 0 we observe
spin helix forms [21]. There the spin state of an electron WL due to α=β, increasing λ gives rise to a transition
is determined only by its initial and final position inde- to ∂G/∂Φs ≈ 0 at λ ≈ 1 and back to WL for λ ≫ 1.
pendent of the exactpath in-between. Therefore,charge This can be understood by the change of the spin relax-
carriers do not acquire an additional phase due to SOI ation in the system: For finite B~|| in a direction differ-
upon return to their initial positions, resulting in con- ent from [¯110] (θ = 3π/4), the resulting magnetic field
structive interference of the wavefunctions connected by B~ (~π)=B~ +B~ (~π) will not be uniformly in the [¯110]
tot || so
time reversal,hence WL [22]. This behavioris shownfor direction anymore, but cause spin relaxation, which is
fixedW andαbutvariableβ inFig.1bwhereweobserve strongest for comparable strengths of B~ and B~ and
|| so
that WAL is suppressed for α=β. yields a reduction of the triplet term (green diamonds
In both cases shown in Figs. 1a,b the absence of WAL in Fig. 1c). For in-plane magnetic fields which distinctly
is caused by the suppression of spin relaxation with the exceed the effective magnetic field (λ≫1), on the other
spin relaxation length exceeding the length of the wire hand, WL is restored to some degree (blue triangles in
3
π
a) b) c)
0.9
min3π/4
θ
]
G0
G [ 0.85 π/2-1 0 1
λ = 0 (α−β)/(α+β)
λ = 1/2
λ = 1
0.8 λ = 3 FIG.3: (Coloronline)θmin determinednumericallyforasys-
0 π/2 0 π/2 0 π/2 π tem with W0, α¯+β¯= 0.04, U¯0 = 1.2, (µBg∗m∗a2/~2)B|| =
θ θ θ
0.01 and Nd = 20000. Black squares: φ = π/2; red circles:
φ = π/4; blue diamonds: φ = 0. The solid lines represent
FIG. 2: (Color online) Conductance of a wire of width W0 Eq.(4) for therespective angles φ.
at Φs = 0, φ = π/2 and fixed (µBg∗m∗a2/~2)|B~so(kx)| =
p2(α¯2+β¯2) = 0.02 with respect to θ for different values of
λ. a) α¯=β¯b) α¯=0 c) α¯=3β¯. since spin is still a good quantum number. In Fig. 2
we observe that the minimum of G appears at the angle
which corresponds to the respective effective magnetic
Fig. 1c) since the resulting B~ (~π) is strongly aligned in field direction for a k-vector along the wire direction.
tot
the directionof B~ and spin relaxationis reduced again. Inview ofthe resultsofFig.2,weconjecture thatalso
||
The enhancement of G(Φ = 0) in an in-plane magnetic for a quasi-one-dimensional quantum wire with W ≪
s
field is anisotropic with respect to the direction of B~ . Lα/β theangleatwhichtheminimumintheconductance
|| SO
For θ = (3/4)π, spin remains a good quantum number appearsisgivenbythedirectionoftheeffectivemagnetic
due to B~|| kB~so. Thus DP spin relaxation is absent, re- field B~so(kx) for a k-vector along the wire direction xˆ:
sulting in WL.This behaviorcanbe observedinFig.2a,
αcosφ+βsinφ
whereG(θ) atΦs =0isshownforaslightly differentge- θmin =arctan − . (4)
(cid:18) βcosφ+αsinφ(cid:19)
ometry. Contrarytothecaseconsideredhere,insystems
showing WAL for B = 0, the transition from WAL to
|| In Fig. 3, we plot Eq. (4) for three different wire orien-
WL is observed with increasing B [25, 26] due to the
|| tations φ (solid lines), whose validity is nicely confirmed
reduction of the singlet term caused by B~ .
|| by extracting θmin from the numerical G(θ) dependence
We now investigate the conductance subject to an in-
(suchasFig.2)fordifferentratiosofα/β (symbols)with
planemagneticfieldinquantumwireswhereWALissup-
fixed α+β. In order to use this feature for the deter-
pressed due to a much smaller width with respect to the
mination of the ratio α/β we suggest to measure G(θ)
spinprecessionlengths. InFig.2weplotthedependence
for quantum wires oriented either along the [100] or the
oftheconductanceontheangleθforthreedifferentratios
[010]-direction. Then the angle of the minimum conduc-
α/β. In order to understand the increase of G at λ > 0
tance directly provides the unambiguous value for the
for all but one angle θ, we consider the case of a strictly
relative strength and signs of α and β. Choosing, e.g.
one-dimensional quantum wire (1DQW) with SOI. We
φ = π/2 this ratio is given by α/β = −cot(θ ), which
min
follow this approach,since for the systeminvestigatedin
is representative for the whole sample, since the influ-
Fig. 2 the width of the wire is much smaller than the
ence of the lateral confinement on the strength of the
phase coherence length, a situation where it is sufficient
SOI is negligible [27]. Considering quantum wires real-
to take into account only the transversal zero-mode for
izedinanInAlAs/InGaAsheterostructure(typicalvalues
the calculationofthe quantumcorrectionto the conduc- m∗ =0.05m ,g∗ =3)andfixingthewidthW =350nm,
0 0
tance[16]. Adisordered1DQWexhibits WL evenifSOI
we see that the parameters used in Fig. 3 (l ≈ 412nm,
of the Rashba and/or Dresselhaus type is present, since B ≈0.17Tandα+β ≈3.5·10−12eVm)arewellinreach
||
the spin is a conserved quantity in this limit. The effec-
of present day experiments [25, 28].
tivemagneticfieldexperiencedbytheelectronsisexactly
We have neglected effects due to the cubic Dresselhaus
opposite for electrons travelling in +xˆ or −xˆ-direction,
SOIterm,whichbecomesincreasinglyimportantforwide
and thus no additional phase in the wavefunction is ac-
quantumwells. Ingeneral,itinducesadditionalrandom-
quired by electrons returning to their original position.
ization of the spin state, which for the case of a very
However, a finite in-plane magnetic field can suppress
strong cubic Dresselhaus contribution can result in the
the WL and induce an increase in the conductance. If
absence of the suppression of WAL [22]. Nevertheless,
B~ ∦ B~ (k ), the direction of the total magnetic field,
|| so x sincecubic Dresselhauscouplingis smallestfork-vectors
B~||+B~so(kx),isdifferentforelectronstravellingin+xˆor along[100]or[010]directions,wehaveneglecteditforthe
−xˆ-direction,resultinginspinrelaxationandanincrease determinationofα/β,sinceinourproposalthequantum
of G (reduction of WL). A minimum in G(θ) exists for wireis assumedto be orientedinone ofthose directions.
B~ k B~ (k ), where no DP spin relaxation takes place However,incontrastto a1DQW,itmighthaveaneffect
|| so x
4
α + β ity may help to facilitate the design of semiconductor-
0.01 0.02 0.03 0.04 based building blocks for spintronics.
0.91π Acknowledgements We acknowledge valuable discus-
min 0.9π sions with M. Wimmer, I˙. Adagideli and D. Bercioux.
θ 0.89π JN and MK acknowledgefinancial support fromMEXT,
MS from JSPS and the Studienstiftung des Deutschen
0.88π
0.5 1 1.5 2 2.5 3 Volkes, and KR from DFG through SFB 689.
l / W
0
FIG. 4: (Color online) Numerically determined θmin for W0,
φEi=theπr/t2h,e(µmBega∗nmfr∗eae2p/~a2th)Bl|f|o=rfi0x.0ed1,α¯N+dβ¯==8000.002a(nbdluαe/cβirc=les3). ∗ Electronic address: [email protected]
or α¯+β¯ for fixed l ≈ 0.87W0 (red squares) was varied. The [1] M. I. D’yakonov, and V. I. Perel’, Fiz. Tverd. Tela 13,
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(2004).
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SO arXiv:0803.3705v1 (2008), to appear in the Ency-
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SO
be applied at sufficiently small B (λ ≪ 1). As can be Lett. 94, 026809 (2005).
||
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