Measuring Rashba spin orbit interaction strength zero magnetic field

0.11 MB·English

by  T. P. Pareek

Measuring Rashba spin orbit interaction strength without magnetic field
T. P. Pareek
Harisch Chandra Research Institute, Chaatnag Road, Jhunsi, Allahabad - 211019, India
We propose a multi-terminal(Non-magnetic) mesoscopic electrical transport measurement for
measuring Rashba spin orbit strength. The method proposed is different from the usual mag-
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netotransport measurements, which requiremagnetic fields for e.g. Shubnikov-deHass oscillations
0
or weak anti-localization measurements. Our methods uses three terminal mesoscopic device, in
0
which one of the terminal acts as a voltage probe. We show that the Voltage measured at the
2
non-magneticvoltage probe dependson the strength of Rashbaspin-orbit interaction.

n
a PACSnumbers: 72.25-b,72.25.Dc,72.25.Mk
J

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A central theme of spintronics research is to realize standourproposalwerecapitulatebrieflythegeneralized

] Datta-Das spin field effect transistor(Spin-FET) or its Landauer-Bu¨ttikertheory for multi-terminalspin-charge
l
l equivalent which would allow manipulation of spin at transportdevelopedbyus recently [12]. For spinquanti-
a
h mesoscopic level[1]. Datta-Das Spin-FET relies on the zation axis to be along uˆ, pointing along (θ, φ) where θ
- Rashbaspin-orbitinteractionwhichcanbe controlledby andφareusualsphericalangles,thenetspinandcharge
s
applying a gate voltage [2]. Accordingly it is know as currents flowing through terminal m are given by;
e
m gate controlled spin-orbit interaction. The key idea is
application of gate voltage on the top of device causes
. 2
mat cchonantrgoelss iRnaasshybma-mspeitnr-yorobfitco(nRfiSnOin)gipnotetreancttiaiolnanwdhhicehncies Imq =Imσ +Im−σ ≡ eh X {(TnαmσVm−TmσαnVn} (1)
n6=m,σ,α
dependant on the interface electric field. This gate con-
-
d trol of RSO interaction have been experimentally veri-
n fied in many two dimensional narrow gap III-IV semi- Is = Iσ −I−σ
o conductor heterostructures for e. g. in InAs/AlSb [3], in m m m
c e2
[ IGna0S.5b2A-Inl0A.4s8-AGsaShbetqeruoasnttruumctuwreelslsb[y4]NaintdtainanIndxGDaa1s−extAasl/. ≡ h n6=Xm,α{(Tnαmσ −Tnαm−σ)Vm+(Tm−nσα−Tmσnα)V(n2})
2 [5]. Experimentally,Shubnikov-deHass(SdH)oscillation
v are measuredas a function ofmagnetic field andapplied whereIq ischargecurrentandIs isspincurrentflowing
5 m m
gate voltage. Zero field spin splitting manifest itself as a in terminal m. Tα,σ are spin resolved transmission coef-
1 nm
beating pattern in SdH oscillation due to two close fre- ficientsfromterminalm tonandV isappliedpotential
1 m
2 quencycomponentswithsimilaramplitudes arisingfrom at terminal m.
1 spin-split levels. The spin splitting at zero field is ex- Now let us consider the special case where the volt-
4 tracted from finite field data by linear extrapolation [7]. ages at terminal 1 and 2 are respectively V1=0 and V2
0 Howeverdue to finite magnetic field, Zeemansplitting is and terminal third is a voltage probe, i.e, Iq = 0 (see
3
t/ always present which makes it ambiguous to extract the Fig.1, here we would like to stress that we consider a
a
spin orbit parameter. Further the top gate voltage also Y shaped three terminal conductor since it is symmetric
m
changes the concentration of charge carriers in spin sub with 120 deg rotation hence the effect due to geometri-
- bands. This affects the beating pattern in SdH oscilla- calasymmetrywouldbeminimizedwhichwillnotbethe
d
n tion which crucially depends on the difference of carrier case if one considersother three terminal geometries for,
o concentration between tow spin sub-bands. In-fact SdH e.g.,aTshapedconductor). Withthisconditiononecan
c oscillationsvanishesforhighervaluesoftopgatevoltage, determinethevoltage,V3,atthirdterminalusingtheset
v: since the carrier concentration in the channel saturate of equation (1) and is given by [13],
i without filling the second sub-band [8]. The spin-orbit
X coupling parameter thus can not be extracted beyond a V3 T32
= (3)
ar certain gate voltage. Hence the measurement of Rashba V2 T13+T23
spin-orbit parameter with SdH oscillation is not free of
ambiguities as was pointed out recently [9],[10] [11]. where Tnm are total transmission coefficient (summed
over spin indices) from terminal m to n.
In light of above discussion, it is clear that it would It is know from studies in mesoscopic physics that the
be interesting and highly desirable if one can design presence of third lead can change the conductance, since
an experiment which would allow to measure the spin- itactsasascatterer[13]. Henceinsimilarwayifthescat-
orbit parameter with out the application of magnetic tering is changed within the sample that should also re-
field. Weproposeandshowinthisletterthatmesoscopic flect in three terminal measurement. This is seen clearly
multi-terminal charge transport measurement provides from(3)thatthevoltageV3 atthirdnon-magnetictermi-
such a way to measure sin-orbit interaction. To under- nal depends on the ratio of transmission coefficients T32
2
and T13 + T23. Presence of spin-orbit interaction does
lead to spin conserved and spin flip scattering which in
V
turn changes spin resolvedtransmission coefficient Tα,σ, 3 3
nm
and , hence the total transmission coefficient. Accord- NM
ingly the voltage V3 at third terminal will show changes
as one varies spin-orbit interaction strength through top
gate voltage. In other words V3 will show depen- V
dence on the gate voltage. So by calibrating these 1 d/2
1 d
changes one can extract the Rashba spin-orbit parame-
ter. Wewouldliketostressthatwedonotneedtoapply NM
d/2
magneticfieldwhichisthecaseforSdHoscillationaswell
V
for weak localization measurement. Hence this method y 2
willnothavethe difficulties whichwasthe case forother 2
methods as discusses in introduction.
z x NM
Here we would like to point out that In Ref.[12] we
hadstudiedthe problemofhowtomeasurethespincur-
rents when terminal third is Ferromagnetic. Third ter-
FIG. 1: Y-shaped three terminal junction with applied volt-
minal being Ferromagnetic selectively allowed electrons
whose spins were parallel to the magnetization of lead, ages V1, V2 and V3 as depicted. Terminal third (labeled 3)
is a voltage probe and draws no charge current . All three
and, hence it was sensitive to the spin currents flowing.
terminals are non-magnetic.
Furtherinref. [12]spincurrentsandcorrespondingvolt-
age were studied as a function of quantization axis for
a fixed spin-orbit interaction strength. Since the origin
found in Ref.[14]. The numerical result presented takes
of spin-orbitinteraction was due to presence of impurity
the quantum effect and multiple scattering into account
whichisfixedforagivensampleandcannotbechanged.
andis exactatthe singleparticle level. Forthe modelof
WhileRashbainteractioncanbechangedthroughanex-
disorderwe takeAndersonmodel, where on-siteenergies
ternalgatevoltage,whichisthetopicofstudyinpresent
are distributed randomly within [-U/2, U/2], where U is
manuscript. Further in this article all leads are non-
the width of distribution.
magnetic and there is no magnetic filed either. Since
weareinterestedinthestrengthofspin-orbitinteraction To present quantitative results we measure energy in
which depends on the ratio of total transmission coeffi- units ofhoppingparametert fortightbinding model, we
cient in different leads.Hence in present study topic of set it equal to 1 above the band bottom. Dimensionless
investigation is charge transport when all the leads are Rashbacouplingisdefinedasλrso =α/2∗t∗λf,whereλf
non-magneticandhowcanoneuseittomeasurestrength isFermienergy. Experimentallyobservedvaluesofthese
ofspin-orbitinteractionwhichiseasiertomeasureexper- parameters are are Ef ≈ 80−120meV, corresponding
imentally as well is different from spin current study as λf ≈70−200Ao andtheRSOcouplingαvariesbetween
wasdone inRef. [12]. Sinceleadsarenon-magneticthey 0.4×10−11eV −m to 1.2×10−11eV −m [3, 4, 5, 6].
are insensitive to different componentof spin currents in Accordingly our dimensionless Rashba coupling parame-
otherwordseventhoughthe spincurrentsflowingacross ter λrso varies between 0.005 to 0.04. In our numerical
x,y andz axiswillbe differentbutthe voltagemeasured simulation we have varied λrso upto 0.05.
at third lead will be same and measures the strength of
spin-orbit interaction.
To obtain quantitative results we perform numerical
In Fig.2 we show charge current(I1 =−I2 flowing be-
simulation on a Y-shaped conductor shown in Fig. 1(we q q
tweenterminalone andtwoandvoltageatthe third ter-
have discussed above that Y-shaped conductor being
minal (which is voltage probe, i.e.,I3 = 0 ) for ballistic
mostsymmetric amongthe three terminaldevices which q
system. We see that varies as spin orbit coupling is var-
wouldminimizedanyspuriousresultsduetogeometrical
ied. Since the system is ballistic , hence the scattering
asymmetry). We model the conductor on a square tight
occursonlyduetospin-orbitinteractionandaccordingly
binding lattice[14] with lattice spacing a and we use the
correspondingtightbindingmodelincludingRashbaspin the V3/V2 and Iq1 shows variation. We notice that both
chargecurrent and voltagedepends quadratically on the
orbit interaction given by
spin-orbit coupling strength λ . This is expected since
rso
H =α(σ k −σ k ) (4) only non zero scattering matrix element due to RSO are
Rashba x y y x
ofsecondorderinperturbationtheory. Henceweseethat
where σ’s are Pauli matrices and k and k are mo- inathree terminalgeometryonecanextractRashbapa-
x y
mentum vector along x and y axis respectively and α is rameterjustbymeasuringchargecurrentorequivalently
Rashbaspin-orbitinteractionparameter. Forthecalcula- voltage. We stress that in two terminal though the con-
tion ofspin resolvedtransmissioncoefficient, we use the ductance will showvariationas a function of Rashbapa-
recursive green function method. Details of this can be rameter , but these variation will be overshadowed due
3
3.93 0.097 2.7 0.26
3.92 0.096 2.65 0.255
0.095
units) 3.91 0.094 V2 units) 2.6 0.25 V2
arb. 3.9 0.093 V/3 arb. 2.55 0.245 V/3
1I(q3.89 1I(q 2.5 0.24
0.092
3.88 0.091 2.45 0.235
3.87 0.09 2.4 0.23
0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05
λ λ λ λ
rso rso rso rso
FIG.2: Chargecurrentandvoltageversusspin-orbitinterac- FIG.4: Chargecurrentandvoltageversusspin-orbitinterac-
tion strength λrso for Ef=1.0t and U=2.0t. tion strength λrso for Ef=1.0t and U=1.0t.
3.5 0.18 2.1 0.38
3.48 0.17 2.05 0.37
1I(arb. units)q33..4446 00..1165 V/V32 1I(arb. units)q11.9.295 000...333465 V/V32
3.42 0.14
1.85 0.33
3.40 0.01 0.02λ 0.03 0.04 0.050.130 0.01 0.02λ 0.03 0.04 0.05 1.80 0.01 0.02λ 0.03 0.04 0.050.320 0.01 0.02λ 0.03 0.04 0.05
rso rso rso rso
FIG.3: Chargecurrentandvoltageversusspin-orbitinterac- FIG.5: Chargecurrentandvoltageversusspin-orbitinterac-
tion strength λrso for Ef=1.0t and U=0.5t. tion strength λrso for Ef=1.0t and U=2.0t.
to beating pattern arising out of band mismatch at the
forming disorder averages and hence the RSO strength
contact between reservoirs and the system, which give
can be calibrated for diffusive case as well.
risetopotentialwellatthe contacts. Duetomultiple re-
flections in this potential well the conductance will show
beating pattern arising out of interference between two
Rashba spilt spin sub-bands [17]. Hence a two terminal
measurement will suffer from the same difficulties and
ambiguitiesaswasthe caseforSdHoscillation. Therefor
Finallywediscussabouttheobservebilityofthiseffect.
a three terminal measurement is necessary and useful.
We see that for experimentally accessible regime of λ
rso
the currentandvoltagechangesrespectivelyby60%and
10%. These changes are large enough to be observable
InFig.3,Fig.4andFig.5wepresentresultsfordiffu- in experiments. Further since changes in current I1 is
q
sive regimecorrespondingto Andersondisorderstrength an order of magnitude larger compared to the changes
U=0.5t,1tand2t,respectively. Foragivensamplesince in voltage, hence through charge measurement it will be
disorder strength is fixed hence variations in charge cur- easiertoobservethiseffectandcorrespondinglymeasure
rentandvoltagesarejustduetochangeinRSOstrength Rashba spin orbit interaction strength.
asisseenintheFigures. Howeverfoelargevaluesofλ InconclusionwehaveproposedandshownthatRashba
rso
there are qualitative changes between voltages and cur- spinorbitinteractioncanbemeasuredinabsenceofmag-
rents. Thisissosincewehavenotperformeddisorderav- netic field just by measuring charge current or voltages
erage,hence for a given distribution asymmetry is intro- inathreeterminalmesoscopicdevice. Wehavediscussed
ducedbetweendifferentbranchesofY-shapedconductor, advantages of this method over the standard SdH oscil-
which reflects in the qualitative difference between volt- lation measurement for which applied magnetic field is
ages and charge current. The asymmetry introduced for essential. Further the effect for charge current is of the
a givendistributionof impurities can be reduced by per- order of 60% and for voltage it is of the order of 10%.
4
Hencebyjustmeasuringchargecurrentinathreetermi- interactionstrength. Wehopethatthiswillopenupnew
nalmesoscopicsystemonecanextractRashbaspinorbit opportunities in the field of spintronics.
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