Electrical tuning of Rashba spin-orbit interaction in multigated InAs nanowires

1.2 MB·

by  Zoltán Scherübl

Electrical tuning of Rashba spin-orbit interaction in multigated InAs nanowires
Zolt´an Scheru¨bl,1 Gerg˝o Fu¨l¨op,1 Morten H. Madsen,2 Jesper Nyg˚ard,2 and Szabolcs Csonka1
1Department of Physics, Budapest University of Technology and Economics and Condensed Matter
Research Group of the Hungarian Academy of Sciences, 1111 Budapest, Budafoki u´t 8., Hungary
2Center for Quantum Devices, Niels Bohr Institute,
University of Copenhagen, 2100 Copenhagen, Denmark
(Dated: January 11, 2016)
Indiumarsenide(InAs)nanowires(NWs)areapromisingplatformtofabricatequantumelectronic
devices, among others they have strong spin-orbit interaction (SOI). The controlled tuning of the
SOIisdesiredinspinbasedquantumdevices. Inthisstudyweinvestigatethepossibilityoftuning
6
the SOI by electrostatic field, which is generated by a back gate and two side gates placed on the
1
opposite sides of the NW. The strength of the SOI is analyzed by weak anti-localization effect.
0
We demonstrate that the strength of SOI can be strongly tuned by a factor of 2 with the electric
2
field across the NW, while the average electron density is kept constant. Furthermore a simple

n electrostaticmodelisintroducedtocalculatetheexpectedchangeofSOI.Goodagreementisfound
a betweentheexperimentalresultsandtheestimatedRashbatypeSOIgeneratedbythegate-induced
J electric field.

8
PACSnumbers: 71.70.Ej,73.63.-b,72.25.Rb,85.35.-p,73.20.Fz

]
l
l
a I. INTRODUCTION time-reversed electron path pairs interfere destructively
h in zero magnetic field leading to enhanced conductance.
-
s Recently III-V semiconductor nanowires (NWs) have Breaking the time-reversal symmetry of the system, i.e.
e by applying a magnetic field, suppresses the interfer-
attracted increasing attention in the field of quantum
m
encetermoftheconductance, resultinginnegativemag-
electronics. The strong spin-orbit interaction (SOI) is
. netoconductance (MC). WAL was used to extract the
t an attractive property of InAs and InSb NWs: it results
a strength of SOI in InAs and InSb NWs.28–35 In most of
large g-factor, which leads to large Zeeman-splitting,1–4
m the studies the WAL signal was investigated in field ef-
it allows to effectively address spins by electric fields
- (EDSR),5,6 furthermore to define new types of quan- fect transistor (FET) configuration, where a global back
d
gate (BG) electrode is placed under a silicon oxide layer,
n tum bits, so called spin-orbit qubits,7–9 where the quan-
which changes the electron density and the electrostatic
o tum information is stored by coupled spin and orbital
field in the NW at the same time.28–33 Other studies in-
c degree of freedom. Very recently theoretical proposals
1 [ siduegaglesptleadt1fo0–r1m6 tthoaitndthuecseetoonpeoldoigmiceanlsisounpaelrcNoWndsucatrievitayn, tNrWodubcyesaasotloidpeglaetcet,roblyestied,ePtMheMBAG34, soerpaarHatfOed2flraoymer.t3h5e
Here we report results on InAs NWs using a new ge-
v and they may host pairs of so-called Majorana bound
4 states, which are key building blocks for topological ometry where two SGs are added beside the BG. In this
5 way similar to FET structure the BG serves to change
quantum computation. The strong SOI is also a key in-
8 the electron density, while the SGs could be used to in-
gredient of these Majorana wires. The models assume a
1 duce a transverse electric field in the wire by applying
Rashba-type SOI, which in combination with magnetic
0
opposite voltages without changing the overall electron
. field leads to the essential locking of spin and momen-
1 density.
tum in the 1D band structure. Several groups have re-
0 ported the signature of these elusive particles17–21 as a Here we show that the total SOI strength can be en-
6
presence of a zero bias anomaly. Since the experimental hancedwithafactorof2,onlybyapplyingelectricfieldin
1
: conditionsneededtoreachthistopologicalsuperconduct- the NW. Furthermore a simple electrostatic model is in-
v ing state are challenging, it would be highly desirable to troduced which shows that this tuning is consistent with
Xi have further experimental control parameters to analyze the external field-induced Rashba-type SOI.
r andtunetheMajoranawires,e.g. atunableRashbaSOI
a strength could give a better insight.
In this paper we demonstrate the tunability of the II. DEVICE AND METHODS
Rashba SOI strength by means of local gating. We have
fabricatedInAsNWdeviceswithpairsoflocalsidegates InAs NWs were grown by molecular beam epitaxy.36
(SGs) (see Fig. 1a&b). By applying voltages with op- After growth, they were deposited onto a thermally
posite sign on these two SGs an electrostatic field can oxidized, degenerately doped n-Si substrate from a 2-
be generated. The influence of the electrostatic field on propanol dispersion. The thickness of the SiO layer is
2
the strength of SOI is studied via weak anti-localization 400 nm and the underlying doped Si layer serves as the
(WAL) measurements. global BG electrode. The source (S), drain (D) and SG
WAL arises from interference effects,22–27 where the (SG1&2) electrodes were defined in a two step e-beam
2
lithography process and were deposited by UHV e-beam a) b)
S
evaporation (Ti/Au 10/90 nm), after Ar ion beam etch- NW VSG1 VSG2
ing to remove the native oxide from the surface of the SG1 SG1 B SG2
NW
NW. A false color SEM picture of the sample is shown
SG2 SiO
on Fig. 1a. The SG separation is 220 nm, the separation 2
1 μm n-Si (BG)
of the S and D electrodes is L=1 µm and the diameter
D VBG
oftheNWisW =77nm. Fig.1bshowsthecrosssection
of the device perpendicular to the NW at the center of c) VBG (V) d)
the SG electrodes on Fig. 1a. -40 -20 0 20 40 60 ,
2.5
In contrast to previously used top gated geometries, SG1, VBG = VSG2 = 0 V
2.0 SG2, V = V = 0 V
an important advantage of the side gated sample is that BG SG1
1.5 BG, V = V = 0 V
the gate electrode is not in direct contact with the NW. )G0 SG1 SG2
This way formation of charge traps at the interface of G (1.0
0.5 T = 50 K
theNWandthetopgateelectrodeisprevented. Further
0.0
advantageisthatelectricfieldcanbeinducedintheNW
-10 -5 0 5 10 15
without significant change of the conductance by apply- e) VSG1/2 (V) f) VS G 1 (V) = -α VSG2
ingLoowppotesmitepevroalttuagreestoranntshpeorStGms.easurements were per- 00..0036VVSSGG12 == {--V1S1G,1-/100.7, ... -1} V 0.04 VVSSGG11 == <<--1116....--16>> VV
formed in a liquid helium cryostat with variable tem- )G00.00VBG = 15 V G))000.02 VfitSsG1 = <-7..3> V
perature insert system which is equipped with a super- G ( G (0.00
conducting magnet. Prior to cool down the sample was Δ-0.03 Δ
-0.02
pumped overnight at room temperature to remove the
-0.06
-0.1 0.0 0.1 0.2 0.3 0.4 -0.1 0.0 0.1 0.2 0.3 0.4
water contamination from the surface of the NW.
B (T) B (T)
The conductance, G follows a typical n-type FET
characteristic of the NW, as it is shown on Fig. 1c at FIG.1. a)FalsecolorSEMpictureofarepresentativedevice.
T = 50 K. Both SGs and the BG can individually in- b)Crosssectionofthedevice,perpendiculartothenanowire
crease the conductance up to 1.5 G (G = 2e2/h is the (NW) axis along the center of side gate electrodes. Two side
0 0
conductance quantum), with a saturation like tendency, gates (SG1&2) and a back gate (BG) electrode can be used
andcancompletelydepletetheNW,i.e. quenchthecon- to apply electric field on the NW. Beside that magnetic field
perpendicular to the substrate and NW was used. c) The
ductance,whiletheothergatesarekeptatzeropotential.
conductance of the NW as a function of different gate volt-
The transconductance curves measured as a function of
agesatT=50K.Notethatthex axisforBGandSGcurves
V and V (black and red curves on Fig. 1c) are
SG1 SG2 are different. d) Variation of magnetoconductance (MC) i.e.
practically overlapping, which indicates that the capaci-
∆G(B)=G(B)−G(B =0) as a function of the asymmetric
tivecouplingtotheSGsarealmostequal. TheSGshave
SG voltage. For the value of α see Eq. 2. e) Typical MC
a 4 times stronger effect than the BG due to closer spac- curves, vertical cuts from panel d). All curves show a peak
ing to the NW (70 nm, 400 nm respectively). Note that around zero magnetic field due to the weak anti-localization
differenthorizontalaxesareusedfortheBGandtheSGs. (WAL) and oscillations at higher B due to universal conduc-
To determine the strength of spin relaxation the mag- tance fluctuation (UCF). f) Averaged MC curves in a 10 V-
netoconductance of the device was measured in various wide gate voltage window are shown. Here < . > indicates
the gate range of averaging. The effect of SG-induced elec-
gate settings at T = 4.2 K. The magnetic field was per-
tric field can be seen. As stronger electric field is applied,
pendicular to the substrate (see Fig. 1b). In the used
theminimumoftheWALcurveshiftstohigherBfieldvalues
gate geometry with 70 nm spacing between the NW and
anddecreases,whichindicatesshorterl ,strongerSOI.Red
SO
SGs, SG voltages (V ) up to 20 V and difference of
SGi curves are fits with theory of WAL (see Eq. 1).
30 V between V and V could be applied without
SG1 SG2
electric breakdown.
Changes of the MC in low magnetic field contains the
spinrelaxation-relatedWALsignal,thereforewefocuson B or the gate voltage is changed.
the variation of the conductance from the zero magnetic The WAL signal allows us to extract important trans-
field value, i.e. ∆G = G(B)−G(B = 0 T). Fig. 1d&e port parameters, like the phase coherence length and
show a typical set of measurements of ∆G as a function the spin relaxation length.22,24 To relieve the WAL sig-
ofB andgatevoltages. TheMCcurvesshowtwogeneral nal from UCF the MC curves were averaged in moving
features, first every curve has a local maximum around gate voltage window31 with a width of 10 V. Three aver-
zeromagneticfield,whichisasignatureofWAL.Second aged MC curves are shown in Fig. 1f, the corresponding
athighermagneticfields(B >0.1T),furtheroscillations gate voltage windows are indicated with colored bars in
appear in ∆G , which depend on the gate voltages in a Fig. 1d, above the upper horizontal axis. Due to the av-
randomfashion. This randomfluctuation (calleduniver- eraging the strongly varying features at high magnetic
salconductancefluctuation,UCF)isduetothevariation fields disappear and only a characteristic peak around
oftheinterferenceconditionofcoherentelectronpathsas B = 0 T remains with a dip and a monotonic increase
3
towardshigherB field. Thislineshapeisacharacteristic We present results measured on one particular device,
WAL signal. butsimilareffectswereobservedonseveralotherdevices.
TheaveragedMCcurvesarefittedwiththetheoretical
formula of WAL in one dimensional systems,22,24,25,33
III. RESULTS
 (cid:32) (cid:33)−1/2
2e2 3 1 4 1
∆G(B)=−  + +
hL 2 l2 3l2 l2
φ SO B a)240 b)0.95
1(cid:32)1 1 (cid:33)−1/2 200 VBG = 15 V llϕSO 0.90 m) lϕ lSO G
− 2 lφ2 + lB2 , (1) m) 160 )G00.85 l (n
l (n 120 E G ( 0.80 G (G0)
where lφ is the phase coherence length, lSO is the spin B
relaxation length, L = 1 µm is the separation of S and 80 0.75
-12 -8 -4 0 4 8 12 -12 -8 -4 0 4 8 12
D electrodes, h is the Planck constant, e is the electron-
V (V) = -α V V (V) = -αV
charge and SG1 SG2 SG1 SG2
c) 240 d) 1.2
lϕ lSO G
8(cid:126)2 200
l2 = √ , 1.0
where B is the magBneticefi2Beld2Wan2d 3W is the diameter of l (nm) 112600 B E )G (G00.8
theNW.Notethatthegeometricalfactor(i.e. theareaof
80 0.6
thecrosssection),W2inRef.22and24hasbeenreplaced
√ 0 20 40 60 0 20 40 60
by 3 3W2 to adapt the formula to the hexagonal cross
8 V (V) V (V)
section of the NW, following the idea of Ref. 33. BG BG
This formula is valid in the dirty limit, i.e. the elas-
FIG. 2. a) The fitted l and l values as a function of
SO φ
tic scattering length, l is much smaller than the wire
e the asymmetric side gate (SG) voltage. For the value of α
d(cid:112)ia(cid:126)m/eeBter>, a>ndWin. tOhuerloswammpalegniesticclofiseeldtolifmulifit,lli.teh.elmfirs=t sinedeicEaqt.es2.thTehteunpienagk-olifkSeOreIdwuictthiona foafcltSoOr oofn2t,hwehriecdhcisurdvuees
condition, since the diameter of the NW is W = 77 nm, to the electric field across the nanowire (NW). (inset) The
and the elastic scattering length is l ≈ 10−20 nm, de- geometryofthemeasurementwiththedirectionoftheelectric
e
terminedfromtransconductancemeasurements.33,37 The andmagneticfields. b)TheconductanceoftheNWatB =0
second inequality implies a |B|<0.1 T condition for the changes less than 20% while the asymmetric SG voltage is
tuned. (inset)Combineddataofa)andb),l andl values
fitting. The MC curves were fitted in the 0 < B < 0.3 SO φ
asafunctionoftheNWconductance. Thelackofcorrelation
T interval, since it is needed to contain the minimum of
of the conductance and the fitting parameters indicates that
theWALcurveforareliablefitting. Theformulahastwo
thetuningisnottheresultofthechangeofconductance. c),d)
fitting parameters, the phase coherence length (l ) and
φ Same as a)&b) for the back gate (BG) tuned measurement.
the spin relaxation length (l ). Assuming that the SOI
SO At high positive BG voltage the electric field results in the
dominates the spin relaxation, lSO is used to measure reduction of lSO by a factor of 2, while the conductance and
the strength of SOI.23,38 According to Eq. 1 a signature so the electron density is strongly increased.
of enhanced SOI (i.e. reduced l ) is the shift of the
SO
minimum of the WAL curve to higher B values.
The three experimental curves presented in Fig. 1f
In our geometry (see Fig. 1a&b) there are 3 gate elec-
show a clear shift of the minimum of the WAL signal
trodes which can induce an electric field inside the NW.
astheasymmetricSGvoltageisincreased. Itisinagree-
Inthefollowingtheinfluenceoftwodifferentelectricfield
ment with the expected enhancement of SOI as higher
profiles on l is studied. In the first case asymmetric
electric field is induced by larger gate voltage. In order SO
potential is applied on the SGs, while the BG voltage is
to extract l and l these curves were fitted by Eq. 1.
φ SO fixed. In the second case finite voltage is applied only
ThefittedMCcurvesareshownwithredlinesinFig.1f.
on BG, while the SGs are grounded. In the following we
There is a reasonable agreement between the measured
showthattheinducedexternalelectricfieldcanstrongly
and fitted curves, even though the magnetic field win-
enhance the SOI in both cases.
dow used for the fit is somewhat wider than the validity
ofthemodel. Theextractedl parametersforthethree
SO
plotted measurements are 175, 144 and 92 nm (from top
to bottom), showing an enhancement of SOI in increas- A. Tuning with in-plane electric field
ing electrostatic fields. In the following, using the fitting
proceduredescribedabove,l andl areextractedfrom In the standard FET geometry the BG voltage can be
φ SO
averaged MC traces measured at different gate settings. used to induce electric field in the NW, however V
BG
4
changes the strength of the electric field and the elec- plane electric field. In order to have a comparison, in
tron density at the same time (see Fig. 1c). The main a second set of measurements the BG was used as the
advantage of the 3-gate-configuration is that it allows to source of the electric field, while the SG electrodes were
generateanelectrostaticfield, without changingthecon- grounded. This setting is similar to the standard FET-
ductanceandtheaverageelectrondensityintheNW,by like gating.28–33 The main difference compared to case
applying opposite voltages on the two SGs (see Fig. 2a of asymmetric SG voltages is that in addition to the
inset). This asymmetric gating induces an electric field strength of the electrostatic field, the electron density
paralleltothesubstrateplane, perpendiculartotheNW alsochangeswithV ,resultinginthechangeofthecon-
BG
axis. ductance,asitisshowninFig.2d. Notethattheconduc-
In order to keep the conductance of the NW constant tancehereisextractedfromtheaveragedMCdata. Also
while the electrostatic field was enhanced, the asym- notethatthetransconductancecurveisshiftedtohigher
metric SG voltages were swept in the following way: V values compared to the blue curve on Fig. 1c, since
BG
V =−αV , where the high BG voltages (up to 60 V), required to achieve
SG1 SG2
(cid:40) highelectrostaticfieldsacrosstheNW,causedthefilling
0.7 if V >0V
α= SG2 , (2) of charge traps in the SiO2 in the close proximity of the
1/0.7 if VSG2 <0V NW, which screens the effect of the BG. The rearrange-
ment of these static charges results a in difference be-
i.e. on the negatively charged SG electrode a voltage
tween the extracted physical quantities at the same gate
smallerbyafactorof0.7wasappliedbecauseofthenon-
configuration for the two different set of measurements.
linear gate dependance of the conductance (see Fig. 1c).
The corresponding fitted l and l parameters are
In the measurement, shown on Fig. 2a&b V was φ SO
SG1 plotted in Fig. 2c. l shows a clear tendency as a func-
swept between -16 and 15 V, but on the voltage axis the SO
tion of V . At small gate voltage values l is around
data is plotted in the [-11 V, 10 V] interval, since it was BG SO
150nm, whileforlargerV andtherebyenhancedelec-
averagedin a window of 10 V. The BG voltage was fixed BG
trostaticfieldl monotonouslydecreases. ApplyingBG
at15Vtomaintaintheconductanceabove0.5G ,which SO
0 voltageofmorethan60V,l isreducedevento70nm.
ensures the visibility of the WAL signal. Fig. 2b shows SO
In some of the previous studies investigating the SOI
the B = 0 T conductance extracted from averaged MC
via WAL in standard FET geometry in InAs and InSb
measurements as a function of asymmetric SG voltages
NWs no tuning was observed,31 while in others a tuning
applied according to Eq. 2. The conductance was kept
effect up to 30% was reported.28,29 All of these studies
close to constant within 0.2 G .
0 used smaller gate voltage range than us. A strong tun-
On the main panel of Fig. 2a the l and l , deter-
φ SO ing, comparable with our results was only observed in
mined by fitting the averaged MC curves with Eq. 1,
devicesusingatopgate35 orelectrolytegate.34 Ourmea-
are shown. At zero SG voltages the spin relaxation
sured l values at low gate voltage values are in good
lengthisabout170nm,whichiscomparablewithearlier SO
agreement with these studies.
results.28,29,31,34 By applying the asymmetric voltage on
Thechangeofthephasecoherencelengthisnotsignif-
theSGs,andthusgeneratinganelectricfieldperpendicu-
icant, in the used range of BG voltage l increases less
lartotheNW,l canbesignificantlyreducedto 90nm. φ
SO
than 20% from 180 nm to 220 nm. Interestingly its ten-
The MC data shown in Fig. 1d,e&f also corresponds to
dencyisjustoppositetothecaseofasymmetricSGs(see
this measurement.
Fig. 2a). The gate voltage-induced enhancement can be
The conductance also varied slightly with the applied
explainedwiththereductionofelectron-electroninterac-
SG voltages. However plotting l and l as a function
φ SO
tion at higher carrier densities (higher conductance).
of the conductance (Fig. 2b inset) reveals that there is
no correlation between the conductance and the fitting
parameters. It indicates that the reduction of l is not
SO
the result of the change of the conductance. IV. SIMULATION
IncreasingtheelectricfieldacrosstheNW,theelectron
density is dominantly pushed to the surface of the NW In the previous section it was shown that by applying
(seeSec.IV),whichcanexplainthereductionofthephase an electric field on the NW with gate electrodes the SOI
coherence length with increasing electric field (see the canbeenhancedbyafactorof2. Inthissectionasimple
black curve in Fig. 2a), due to the increased scattering electrostatic model is introduced, which is used to calcu-
on the surface-roughness or residual contamination. late the electrostatic field (E(r)) profile inside the NW,
Investigatingotherdevicesthehighestreductionfactor and to estimate the electric field-induced Rashba SOI.
of l was 3 for V =18 V. A good agreement was found between the numerical re-
SO SG1
sults for l and the measured values presented in the
SO
previous section.
B. Tuning with perpendicular electric field Forsimplificationthemeasureddeviceismodeledwith
its2dimensionalcrosssection, showninFig.1banddis-
In the previous section it was shown that l can cussedinSec.II.TheNWismodeledbyahexagon,with
SO
be strongly tuned by an asymmetric SG induced in- 40nmlengthofedge(whichcorrespondstoW =80nm).
5
Inelectrostaticviewpointasingleconductionbandisas- the following formula:41
sumed with parabolic dispersion, and hard-wall confine-
(cid:126)2
ment potential. Fermi level pinning at the surface of the
l = , (5)
NW gives rise to band bending,39,40 which for simplic- SO,R m∗eα (cid:104)E(cid:105)
0
ity is neglected in our model. Furthermore we fixed the
Fermienergyattheedgeoftheconductionbandforzero where α0 = 1.17 nm2 for InAs. (cid:104)E(cid:105) is the spatially av-
gate voltages. eraged electric field within the NW, weighted with the
In the simulation the electric potential, V(r) was cal- DOS at Fermi energy (see Eq. A4).
culated by solving the Poisson-equation, In the measurements the lSO was finite at zero gate
voltages, which indicates that other sources of spin re-
ρ(r) laxation are present. To take them into account, we in-
∆V(r)= , (3)
(cid:15)0(cid:15)ir troduce the built-in spin relaxation length, lSO,Bi. The
built-in spin relaxation processes may originate from
where, (cid:15)i is the relative dielectric constant of medium i breaking the inversion symmetry of the system by the
r
((cid:15)SriO2 = 3.9, (cid:15)IrnAs = 15.15), (cid:15)0 is the vacuum permit- crystal structure, confinement potential, or scattering
tivity, and ρ(r)=−en(r) is the electron charge density, centers. For the sake of simplicity we assume that these
whichisonlynon-zerowithintheNW.Theelectronden- processes are independent from the external gate volt-
sityoftheNW,n(r)isgivenbythebulkdensityofstates ages. For the different spin relaxation contributions the
(DOS) of the InAs at zero temperature, following sum rule was used:
l−2 =l−2 +l−2 (6)
3 (cid:18)2m∗(cid:19)3/2 SO SO,Bi SO,R
n(r)= [eV(r)]3/2, (4)
4π2 (cid:126)2 The value of l was chosen such that the measured
SO,Bi
andthecalculatedl valuescoincidewitheachotherat
SO
where e is the electron charge, (cid:126) = h/2π is the Planck zeroasymmetricSG/BGvoltageinthetwodemonstrated
constant, m∗ =0.023me is the effective mass of the elec- measurements. Thesimulationcontainsnofurtherfitting
trons in the conduction band, and me is the free elec- parameters.
tron mass. In the simulation Eq. 3&4 are solved self- The measured and simulated l curves are shown in
SO
consistently in an iterative manner. The details of the Fig. 3c&f for the two different electrostatic field profiles.
simulation can be found in the Appendix. Themeasuredl curvesarethesameasinFig.2a&c. In
SO
The calculated electric potential and the DOS at the both cases the simulated curves reproduce well the main
Fermi energy for the two type of measurements are findings of the measurement, that the SOI is enhanced
shown in Fig. 3a-b&d-e. The black lines represent byincreasingelectricfield. Furthermore,despitethesim-
the NW/SiO2/vacuum interfaces, the scale is shown in plicityofthemodelandthelownumberofmodelparam-
Fig. 3a. The SG&BG electrodes are not shown on the eters fairly good quantitative agreement was found.
graphs, the SG1 (SG2) is on the left (right) 70 nm away In both cases l tends to saturate at higher voltages,
SO
from the edge of the NW, the BG is 400 nm below the whichistheresultofelectrostaticscreeningbyelectrons.
NW/SiO2 interface. As it can be seen from the DOS graphs in Fig. 3b&e,
Fig. 3a&b correspond to the asymmetric SG mea- the electron density can reach high values in the NW
surement with gate voltage values of VSG1 = 11 V, close to the positively charged gate electrode. At higher
VSG2 = −7.7 V and VBG = 15 V. The electron den- electrondensitiesthescreeningismoreefficient,inagree-
sity(whichcanbedirectlycalculatedfromtheDOSwith mentwiththestaticscreeningtheory, whichleadstothe
Eq. A1&A2) is concentrated to the left side of the NW, decreasing enhancement of SOI as the gate voltages are
asaresultoftheattractiveforceofthepositivelycharged furtherincreased. Toquantifythescreeningeffectwees-
SG1 electrode. For this gate configuration the numerical timatethescreeninglengthintheframeworkofThomas-
calculation yield an average electric field of 20 mV/nm Fermimodel.42Inthismodelthescreeninglengthisgiven
inside the NW. by
Fig. 3d&e correspond to the BG measurement with
grounded SG electrodes with VSG1 = VSG2 = 0 V and r =(cid:18)e2D(εF)(cid:19)−1/2.
VBG =58 V gate voltage values. The electron density is TF (cid:15) (cid:15)
0 r
concentratedtothebottompartoftheNW.Theaverage
electric field within the NW is about 25 mV/nm for this With a typical value of DOS at Fermi level, D(ε ) =
F
gate configuration. 1·1031 eV−1cm−3, for InAs ((cid:15)InAs = 15.15) the screen-
r
The small areas of low DOS values at the surface of ing length is r ≈ 10 nm, which is consistent with the
TF
the NW in Fig. 3b&e are artifacts arising from the grid simulated electric potential profile (see Fig. 3a&d).
of the simulation (5 nm). Wehaveshownthattheintroducedelectrostaticmodel
The spin relaxation length associated to the external can explain our experimental results with an external
field-induced Rashba SOI can be calculated from magni- field-induced Rashba SOI, assuming another, constant
tude of the electric field supplied by the simulation with source of spin relaxation.
6
Following the standard framework of WAL28 we as- a) d)
sumethattheSOIisthemainmechanismofthespinre-
laxation and the measured spin relaxation length gives a
goodmeasureofthestrengthoftheSOI.UsingEq.5and
themeasuredlSO values,onecanestimatethestrengthof 40 nm
the SOI, α =α e(cid:104)E(cid:105)=(cid:126)2/m∗l , which is reasonable
R 0 SO
at high external fields, when the induced Rashba SOI
dominates the built-in contribution. Tuning l from
SO
150 to 70 nm corresponds to αR ≈ 2·10−11 eVm and b) 2x 1031e)
4.3·10−11 eVm, respectively.
In a very recent work of van Weperen et al. (see
Ref. 35) a more realistic model of WAL theory is pre-
sented, which takes the 3D structure of the NW into ac-
count more precisely and allows us to extract spin-orbit
induced spin relaxation length l and the Rashba
SO,R
spin-orbitcouplingparameterα moreaccurately. How-
R
ever, the parameter range of the model does not cover c) f)160
Measurement Measurement
the condition of our measurements since for our sam- 200 Simulation Simulation
ple W/l is around 4 − 8. We believe that additional 120
model caelculation would be desired in order to give an nm) 160 nm)
accurate estimation of the SOI strength αR, which goes l (SO120 V = 15 V l (SO 80V = V = 0 V
beyondsomesimplifications: Asthesourceofspinrelax- BG SG1 SG2
l = 280 nm l = 160 nm
ation mechanism most models consider a simple Rashba 80 -8 SO,Bi 0 8 0SO,Bi 20 40 60
type SOI induced by a homogenous electrostatic field in V (V) V (V)
SG1 BG
the entire NW. In our device the electrostatic field is
not homogeneous and in addition other SOI term, like FIG. 3. Results of the numerical simulation. a),b) The
D’yakonov-Perelhastobetakenintoaccountduetolack electric potential and density of states (DOS) at Fermi en-
ergy, respectively, in the vicinity of the nanowire (NW), for
of inversion symmetry of wurtzite NW structure. Never-
V = 11 V, V = −7.7 V, V = 15 V. The black lines
theless it does not influence our finding that l can be SG1 SG2 BG
SO mark the NW/SiO /vacuum boundaries. The low DOS val-
tuned by gate voltages even if the conductance and the x
ues at the surface of the NW is the result of the relatively
average density is kept constant.
coarseresolutionofthegrid(5nm). c)Spinrelaxationlength
in the asymmetric side gate configuration, the measurement
data is same as in Fig. 2a (black squares), and the result of
V. CONCLUSION the numerical calculation (red curve). d),e) Same as a) and
b), for BG-tuning measurement with V = V = 0 V,
SG1 SG2
V =58 V. f) Same as c), for tuning with BG.
The SOI of InAs NWs was studied by WAL measure- BG
mentsinasidegatedgeometry. Wehaveshownthatthe
strength of the SOI can be enhanced by up to a factor
of 2, applying electric field across the NW. The tuning
bytheinternalinversionasymmetryofthewurtzitecrys-
wasdemonstratedatanearlyconstantconductancewith
tal structure and the quantized conductance channels of
asymmetric voltages applied on opposite SGs. Further-
the NWs. Such a more realistic theory could shine light
more, a similarly large tuning was observed applying a
on the limits of the validity of the simple 1D Rashba
high BG voltage, which strongly changed the electron
SOI Hamiltonian for the description of spin physics in
density as well.
semiconductor NWs.
An electrostatic model was introduced to calculate
strength of Rashba SOI induced by the external electric
field. Onthecomparisonoftheexperimentalresultsand
the numerical calculations good agreement was found,
ACKNOWLEDGMENTS
whichsupportsthattheRashbaSOIistheoriginofgate-
induced changes of the WAL signal. The possibility to
tune the Rashba SOI strength in-situ without changing We acknowledge useful discussions with Andras Palyi,
the electron density of the device is highly promising for BalazsDora,AttilaGeresdi,PeterMakk,ChristianScho-
various quantum electronic devices. nenberger, Stefan Oberholzer, Samuel d’Hollosy, Balint
InordertogainabetterinsightintotheSOIinNWs,a FulopandThomasSandJespersen. Weacknowledgesup-
more realistic model calculation would be desired which port from EU ERC CooPairEnt 258789, FP7 SE2ND
takes into account the inhomogeneous electrostatic field 271554, Hungarian Grant No. OTKA K112918. S.C.
in the NW, the presence of NW confinement potential, was supported by the Bolyai Scholarship and G. F. was
the band bending at the interface, the SOI term caused a SCIEX fellow (project NoCoNano)
7
Appendix A: Simulation electron-filling,
(cid:90) ∞
Inthissectionthedetailsoftheelectrostaticsimulation n(r)= D(ε)f(ε)dε (A2)
are discussed. 0
ThethicknessoftheSGelectrodesis100nm,thesepa-
wascalculatediteratively,wheref(ε),theFermi-function
rationofthemis220nm. Thethicknessoftheoxidelayer
wasapproximatedwithzerotemperatureform,theHeav-
is 400 nm. The boundary of the geometry is defined by
iside step-function.
a 1 µm × 1 µm square, which lower edge corresponds to
Toassuretheconvergenceofthealgorithmforboththe
the BG. The potential of the upper edge is fixed at zero
electron density and the potential the weighted average
potential. Thepotentialoftherightandleftedgesbelow
of the last two iteration step was used as new value, i.e.
(above) the SG electrode are defined by linear interpola-
tionbetweenthepotentialofBGandSG(SGandupper n(i) →(1−η )n(i−1)+η n(i)
n n
edge). Square lattice with 5 nm resolution is used in the V(i) →(1−η )V(i−1)+η V(i). (A3)
V V
simulations.
The NW is modeled with the properties of bulk InAs In the simulations η =η =0.2 was used.
n V
with 3 dimensional DOS TothecalculationofSOItheelectricfieldiscalculated
from the potential within the NW. Since only electrons

 1 (cid:16)2m∗(cid:17)3/2√ε if ε≥0 at the Fermi-surface contribute to the conductance, the
D(ε)= 2π2 (cid:126)2 , (A1) electric field is averaged with respect of the DOS at the
0 if ε<0 Fermi-level,
with hard-wall confinement potential, neglecting the (cid:82) d2r|E(r)|D(eV(r))
(cid:104)E(cid:105)= (cid:82) (A4)
band bending. d2rD(eV(r))
DuringthesimulationthePoisson-equation(Eq.3)and
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