Gate-controlled valley transport and Goos-H\"{a}nchen effect in monolayer WS$_2$

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by  Hassan Ghadiri

Gate-controlled valleytransport andGoos-Ha¨ncheneffect inmonolayerWS
2
Hassan Ghadiri1 and Alireza Saffarzadeh2,3,∗
1Department ofPhysics, NorthTehranBranch, IslamicAzadUniversity, 16511-53311, Tehran, Iran
2Department of Physics, Payame Noor University, P.O.Box 19395-3697 Tehran, Iran
3Department ofPhysics, SimonFraserUniversity, Burnaby, BritishColumbia, Canada V5A1S6
(Dated:January25,2017)
Based on a Dirac-like Hamiltonian and coherent scattering formalism, we study spin-valley transport and
7
Goos-Ha¨nchenlike(GHL)effectoftransmittedandreflectedelectronsinagatedmonolayerWS2. Ourresults
1
showthatthelateralshiftofspin-polarizedelectronsisstronglydependentonthewidthofthegatedregionand
0
canbepositiveor negativeinbothKleintunnelingandclassical motionregimes. Theabsolutevalues ofthe
2
lateraldisplacementsatresonancepositionscanbeconsiderablyenhancedwhentheincidentangleofelectrons

n isclosetothecriticalangle. Incontrasttothetimereversalsymmetryforthetransmittedelectrons,theGHL
a shift of the reflected beams is not invariant under simultaneous interchange of spins and valleys, indicating
J the lack of spin-valley symmetry induced by the tunable potential barrier on WS2 monolayer. Our findings

4 provideevidenceforelectricalcontrolofvalleyfilteringandvalleybeamsplittingbytuningtheincidentangle
2 ofelectronsinnanoelectronicdevicesbasedonmonolayertransitionmetaldichalcogenides.

]
l
l I. INTRODUCTION n junctions [16], strained graphene [11, 17], graphene with
a
electric and magnetic barriers [18, 19], and valley [17] and
h
- Two-dimensional (2D) materials such as monolayers of spin[20]beamsplittersingraphene. Recently,theGHeffect
s ofelectronshasalsobeenstudiedinsilicene[21]andap-n-p
e transitionmetaldichalcogenides(TMDs)haveattractedcon-
junctionof MoS monolayer[22]. Itwas foundthatthe GH
m siderable attention due to their fascinating physical proper- 2
shiftofDiracfermions,whichhasamagnitudeoftheorderof
ties. Incontrasttothezero-gapgraphene,theyaredirectgap
.
t semiconductorswithsizablebandgapsinthevisiblespectrum Fermi wavelength [23, 24], can be amplified by multiple to-
a
talinternalreflections[22,24]. Moreover,thelateralshiftsof
m which make them promising materials in electronic and op-
Diracfermionsintransmissionthroughsemiconductorbarri-
toelectronic applications such as field-effect transistors and
-
ers[25],graphenesinglebarrier[23],doublebarrier[26],and
d light-emittingdiodes[1,2]. ThelayeredTMDssuchasMoS
2
n and WS monolayerspossess strong spin-orbitcouplingdue multiplebarrierstructures[27]havealsobeenreported.Such
2
o to the d orbitals of heavy metal atoms, causing a large spin displacements,calledGoos-Ha¨nchenlike(GHL)shifts,occur
c in the partial reflection regime and can be intensified by the
splitting ( 440 meV in WS monolayers) in the valence
[ ∼ 2 transmissionresonances(Fabry-Perotresonances). However,
bands [3, 4]. Moreover, the absence of inversion symmetry

1 inthecrystalstructureofthesematerialsallowsvalleypolar- itisstillachallengingproblemtoexperimentallyobservethe
v GH or GHL shifts of electrons due to the smallness of the
ization of carriers by optically exciting electrons with a cir-
8 shifts and also the difficulty in producing a well-collimated
cularly polarizedlight [5–10]. The valleysoccur at the non-
3 equivalentKandK′pointsinthehexagonalBrillouinzoneof electronbeam[24].Ontheotherhand,todesignabeamsplit-
9
ter effectively, the differencebetween lateral shifts of K and
6 TMDmonolayers[5]. Manipulatingboththespinandvalley
K′ valley electrons should be greater than the width of inci-
0 degree of freedomsin such monolayersmakes it possible to
. designnewgenerationofnanoelectronicdevicesforquantum dentelectronbeamwhichisabout100-1000λF,withλF the
1
Fermiwavelengthofelectrons[21,26].
computing[5,7,8,10].
0
7 It was demonstrated that the electrons and holes at the InthispaperwestudythequantumGHLeffectofspinpo-
1 band edges of MoS monolayers and other layered group- larized electrons in transmission through a gated monolayer
2
v: VI dichalcogenides can be well described by massive Dirac WS2, acting as a tunable potential barrier, as shown in Fig.
i fermions with strong spin-valley coupling [5]. On the other 1(a),intwocasesofKleintunnelingandclassicalmotion.We
X hand, due to the similarities between propagation of Dirac show that large positive and negative GHL shift values can
r fermions and propagation of electromagnetic waves in di- be obtained in the transmitted and reflected electron beams.
a electrics, many optical phenomena such as Brewster angles Moreover, the critical angle for total reflection of electrons
[11], collimation [12], Bragg reflection, electronic lenses attheinterfacebetweengatedandungated(normal)regions,
[13, 14] and Goos-Ha¨nchen (GH) shift [15, 16] have been is spin- and valley-dependent. Accordingly, it is possible
foundin single-layergraphene. Amongthem, the GH effect to choose incidentangles, so that spin-polarizedelectronsin
referstoalateraldisplacementofa lightbeamwhenitisto- onevalleyareallowedtopropagatethroughthegatedregion,
tally reflected from a dielectric interface. In the past several whereas the electrons in the other valley are blocked. We
years, moststudieson the electronicGH shifthave beende- also show that the GHL effect can generate transmitted and
voted to graphene-basednanostructuressuch as graphene p- reflectedvalley-polarizedbeamswithaseparationaslargeas
the width of the incident beam, when a spin-polarized elec-
tronwithaproperchoiceofenergyisincidentontheadjusted
electrostaticbarrier.
∗Electronicaddress:[email protected] Thepaperisorganizedas follows. In sectionII, we intro-
2
duceourmodelandformalismforcalculationofvalleytrans-
port and GHL shifts for both transmitted and reflected elec-
tronbeams. ItisshowninsectionIIIforsomelimitingcases
that our formalism can also reproduce the results of lateral
shiftsoftotallyreflectedbeaminMoS [22]andgraphenesin-
2
gleinterfaces[16],andalsopartiallyreflectedelectronsfrom
graphenebarrier[23]. Numericalresults and discussionsfor
spin-valleytransportandlateralshiftsinWS monolayer,by
2
tuning system parameters, are presented in Sec. IV. A brief
conclusionisgiveninSec. V.
II. MODELANDFORMALISM
To explore the effect of GHL shift in a TMD monolayer
FIG. 1: (color online) Schematic model for the gated monolayer
withatunablepotentialbarrier,weconsiderasingle-layerof
WS2 in which the gated region acts as a tunable potential barrier.
WS2inx yplaneandapplyatopgatevoltagetotheregion (b)Energyprofileofthemodelshownin(a)foranincidentelectron
II(0 < x−< d), while theregionsI (x < 0)andIII(x > d) withspinsz(s¯z)invalleyK(K′)andatypicalvalueVG < ∆. Ec
are kept at zero electrostatic potential, as shown in Fig. 1. andEvdenotetheenergyofconductionbandminimumandvalence
The top gate voltage induces a potential barrier with height bandmaximum,respectively. Theredandbluelinesshowthespin-
V (Fig. 1(b)), and turn the system into a n-p-n WS junc- dependent energy levels at the top of the valence band, due to the
G 2
tion.Thismeansthattheelectronictransportthroughthejunc- spin-orbit coupling. Notethatthespinsplittingat differentvalleys
areoppositeduetothetime-reversalsymmetry.
tioncanbestronglyaffectedbythegate. Weemphasizehere
thatthegatevoltageintheproposedmodelisonlyutilizedto
shifttheenergylevelsbutnottomodulatetheenergygap. In
dimension of monolayer along the y-axis is infinitely long.
thisnormal/gated/normalWS junction,thelow-energyelec-
tronsnearvalleysKandK′ca2nbedescribedbytheDirac-like Therefore, due to the translational invariance in the y direc-
tion,k isconserved,andtheelectronwavefunctionineach
Hamiltonian[5] y
regioncanbedescribedas
∆ σˆ 1
z
H = at(τkxσˆx+kyσˆy)+ 2σˆz −λτ 2− sˆz ψ = 1 A1 ei(kxx+kyy)
1 B τatkeiτϕ
+VGΘ(x)Θ(d x), (1) 1 (cid:18) (cid:19)
where a is the lattice consta−nt, t is the effective hoppingin- +rBsτz τatAk1e−iτϕ ei(−kxx+kyy) , (3)
tegral, and τ = 1( 1) is the valley index corresponding to 1 (cid:18)− (cid:19)
K (K′) point. k −denote the components of electron wave
x,y
vspeacntonrinmgetahseucreodndfruocmtioKnaannddvKa′l,eσncxe,ys,ztaateresitnhethPeatuwliomvaatlrleicyess, ψ2 = Bα τatkA′e2iτϕ′ ei(kx′x+kyy)
2 (cid:18) (cid:19)
∆isthebandgapbetweenvalenceandconductionbands,and
2λ is the spin-splitting at the valenceband maximumdue to +Bβ τatkA′2e−iτϕ′ ei(−kx′x+kyy), (4)
thespin-orbitinteraction. sˆ isthez componentofPaulima- 2 (cid:18)− (cid:19)
z
trixforelectronspinandΘ(x)istheHeavisidestepfunction.
Accordingto Eq. (1), the dispersionrelationin thegatedre- tτ A
ψ = sz 3 ei(kxx+kyy), (5)
gionIIisgivenby 3 B τatkeiτϕ
3 (cid:18) (cid:19)
(2E−2VG−τszλ)2−(∆−τszλ)2 =(2atk′)2, (2) where ϕ = tan−1(ky/kx) is the incident angle of elec-
trons on the barrier, rτ is the reflection amplitude, ϕ′ =
wherek′ = kx′2+ky2andsz =±1whichstandsforspinup tan−1(ky/kx′)istheansgzleofrefraction,andtτsz isthetrans-
( )and spinqdown( ). Inthe regionsI and III,however,the mission amplitude of electrons. The coefficients Ai and Bi
re↑lationcan be obta↓inedby setting VG = 0 and replacingk′ (i = 1,2,3) in Eq. (3-5) are expressed as A1 = A3 =
withk = kx2+ky2. E−τszλ+∆/2,A2 =A1−VG,B1 =B3 = A21+(atk)2,
andB = A2+(atk′)2.
Now wqe consider the propagation of spin-polarized elec- 2 2 p
The total internal reflection of electrons occurs if the in-
tronsin the WS2 junction which can be scattered by the po- p
cident electron waves strike the potential barrier at a critical
tentialbarrier,inducedbythegatevoltageV . Itisassumed
G
angleϕ whichisgivenby
that the spin-polarized electrons are injected by a magnetic c
contact,suchasaferromagneticsemiconductor[10,28],into
theleftnormalregion.Forsimplicity,thesourceofspininjec- ϕ =arcsin (2E−2VG−τszλ)2−(∆−τszλ)2 . (6)
tionisnotincludedinthemodel. Wefurtherassumethatthe c s (2E τszλ)2 (∆ τszλ)2
− − −
3
When the angle of incidence ϕ is greater than the critical Therefore, the two components of ψ represent two Gaus-
in
angle, the electron wave vector in the electrostatic barrier, sians of the same width, centered at the two different mean
k′ = k′2 k2, becomes imaginary which indicates an coordinatesy¯+inandy¯−inalongthey-axis.
x − y Usingasimilaranalysis,thetransmittedelectronbeamcan
evanescqentwave in the gatedregion,andhence, the electron
beexpressedas
beam will experiencea total reflection from the electrostatic
pbaarrtriiaelrl.yInretflheeccteadseanodfϕpa<rtiaϕllcy,htroawnesmveirt,tethde. eTlhecetrcoonefwficavieenitss ψtr(x,y) = √2Bπ∆ky ei[kx(ky0)x+ky0y]|tτsz(ky0)|
rτ ,α,β,andtτ canbeobtainedbyapplyingthecontinuity 3
osfzthewavefunscztionsatthetwointerfaces,x = 0andx = d A3 e−21(y−y¯+tr)2∆2ky (12),
showninFig. 1. Thetransmissiontτsz andreflectionrsτz coef- × τatke−12(y−y¯−tr)2∆2kyei[τϕ(ky0)+Φtτsz(ky0)] !
ficientsforelectronwavesareexpressedas
wherey¯tr = k˙ (k )x Φ˙ (k )andy¯tr = k˙ (k )x
2ss′ F1 cosϕcosϕ′e−ikxd Φ˙ (k +) −τϕ˙(xk y)0. N−otettτshzatΦy0 in E−q. (1−2)xandyi0n th−e
tτ = F2 , (7) tτsz y0 − y0 tτsz
sz 2ss′ F1qcosϕcosϕ′cos(k′d) iD mean coordinatesy¯±tr, denotesthe phase of transmission co-
F2 x − efficient tτ . Accordingly, the GHL shifts of the upper and
q lowercomspzonentsofthetransmittedelectronsaregivenby
[ 2ss′ F1 eiτϕsin(τϕ′)+i(1 F1e2iτϕ)]sin(k′d) σ± = y¯±tr x=d y¯±in x=0
rsτz = − q2sFs2′ F1 cosϕcosϕ′co−s(kF′2d) iD x , = −Φ˙|tτsz(k−y0)−|k˙x(ky0)d. (13)
F2 x −
q (8) Sincethetwocomponentshaveequalshiftvalues,theaverage
whereD =[(1+F1) 2ss′ F1 sinϕsinϕ′]sin(k′d),F = GHLshiftofthetransmittedbeamcanbeexpressedas
F2 − F2 x 1
aAn2d/As′1=, Fs2gn=(A(E).−VG −∆q/2)/(E−∆/2),s = sgn(A1), σtτr,sz =−Φ˙tτsz(ky0)−k˙x(ky0)d. (14)
2
Applyingsimilar considerationsto the lateral shiftof the re-
TostudytheGHLeffect,we consideranincidentelectron
einrgtyheEf,ormofaGaussianwavepacketofwidth∆ky anden- flσe+ct+ed2τbϕe˙a(mky,0)w.eBfiyncdonthsiadterσin+gt=he−prΦo˙brsτazb(ikliyty0)ofanudppσe−ran=d
lowercomponentsofthebeams,theaverageshiftofreflected
+∞ 1 electronswillbe
ψ (x,y) = dk f(k k )
in Z−∞ yA y− y0 B1 σrτe,sz = B12[A12σ++(atk)2σ−]
1 ei[kx(ky)x+kyy] , (9) 1
× τatkeiτϕ(ky) 2τa2t2k2
(cid:18) (cid:19) = Φ˙ (k )+ ϕ˙(k ). (15)
− rsτz y0 B 2 y0
wheretheGaussianf(ky −ky0) = e−(ky−ky0)2/2∆2ky shows Inthecaseoftransmittedbeam,wen1eedtocalculateΦτ
the angular distribution of electron beam around the central tsz
from Eq. (7), and substitute into Eq. (14). Thus, the lateral
incidence angle ϕ = arcsin(k /k). Therefore, the trans-
0 y0 GHL shift of transmitted electrons throughthe tunable elec-
mittedelectronbeamcanbegivenas
trostaticpotentialcanbeobtainedby
+∞ tτ (k )
ψtr(x,y) = Z−∞ dkyAf(3ky −ky0e)i[ksxzB(k3yy)x+kyy]. (10) σtτr,sz = [(8+24kkcx202os+2(2kx′kkx′d022))+sin2(kk2x2kx′kk04dx′x′2d)s−in22(kkkx′02x′2d]d)tanϕ , (16)
× τatkeiτϕ(ky)
(cid:18) (cid:19) wherek2 = ss′γkk′+2k2andγ = F /F (1+F /F ).
0 − y 2 1 1 2
Inthecaseof∆ky ≪ k,wecanexpandϕ(ky)andkx(ky) On the other hand, by computing Φprsτz from Eq. (8) and
tofirstorderaroundk , substitutein Eqs. (9)and(10),and substitutingintoEq.(15),theGHLshiftforthereflectedwave
y0
evaluatetheintegrals,toobtainthespatialformoftheincident canbewrittenas
and transmitted electron beams. In this regard, the incident
2τ C +C C
beamisgivenby στ =στ + 1 2− 3 , (17)
re,sz tr,sz k B2(1+C )
x (cid:20) 1 0 (cid:21)
ψ (x,y) = √2π∆ky A1 e−(y−y¯+in)2∆2ky/2 where
in B1 τatke−(y−y¯−in)2∆2ky/2eiτϕ(ky0) ! F k2 (k2 k2) F
×ei[kx(ky0)x+ky0y] , (11) C0 = "F21 +4k′y2 −2 xk−2 y #F12
wherey¯+in = −k˙x(ky0)x, andy¯−in = −k˙x(ky0)x−τϕ˙(ky0). 4ss′ ky2 (1+ F1) F1 , (18)
Here, the dot indicates the derivative with respect to ky. − kk′ F2 rF2
4
A. singleinterfaceonTMDmonolayers
Asalreadymentioned,forincidentanglesgreaterthanthe
criticalangle,ϕ>ϕ ,thewavevectork′ becomesimaginary
c x
andhence,thetotalreflectionoccurs.Insuchacase,Eq. (17)
whichisgivenforϕ < ϕ isalsovalidforϕ > ϕ ,provided
c c
thatwemakethesubstitutionk′ iκ.Afterthissubstitution
x →
andtakingthelimitd ,thesecondterminEq.(17)does
→∞
notchange,butthefirsttermbecomes
[8k2κ2+2k2(κ2 k2)]tanϕ
x 0 − x . (22)
κ(4k2κ2+k4)
x 0
Therefore, the GH shift of the totally reflected beam from
a step potential (single interface) applied on a monolayer of
TMD, can be easily calculated by replacing the first term of
Eq. (17)by(22).
In Ref. (22), the probabilitiesof upper and lower compo-
nentsofthereflectedelectronshavenotbeenincludedincal-
FIG.2: (color online) Dependence of transmission probabilityand culationsandthe averageGH shiftwas simply computedby
1Gtwh9Hhe.e5Lgr◦eas,thVeaidnGfdtr=oeϕ3gfcietohV=ne.itT5nr0ahn(.e1as,0ombt◦)hitieKtnerldKpeiae-nrvleaatcmultlnreeonyten,erslϕsinwcagri=teah:n2s(d0az.,(b5=c)1,d◦E1)iocn=lnaKst1′hs-.iev7ca5walllieedmVyt,ho,(tϕdcio,do=n)f, (aσ1nrτ7de),sfibzne=aclolym21(etσhse+kτGx+[H1σ−s−h)iFF.ft2122Oo−nftt2hhsiesst′boqatasFFilsl12y,(t1rhee−flseecFFct21eo)dnkkdb′]et/ea(rm1m+cianCnE0bq)e,.
E =5.0eV,ϕ=20.5◦,andϕc =20.88◦inK-valley,ϕc =21.73◦ simplifiedas
inK′-valley.Notethatdisnormalizedwithkx′.
τ [(τκ+k )2 k′2F1]+ 2ss′k′(κ+τk ) F1 cosϕ
στ = kx y − F2 τκ y F2 .
C =a2t2k2 A2F12+(a2t2k2 A2) 2ky2 + ky2−kx2 F1 , retotal,sz (τκ+ky)2+k′2FF21 −2ss′k′(τκ+ky) qFF12 sinϕ
1 − 1F22 − 1 "k′2 k2 #F2 q(23)
ThisequationisexactlythesameasEq. (8)giveninRef. 22
(19)
forMoS monolayers.
2
1 F3
C =ss′ k2(a2t2k2+A2) k2(a2t2k2 3A2) 1 ,
2 x 1 − y − 1 kk′sF23 B. graphenebarrier
(cid:2) (cid:3) (20)
By setting λ = 0 and ∆ = 0 in Eqs. (1), (16), and (17),
C =ss′ k2(a2t2k2+A2)+k2(3a2t2k2 A2) 1 F1 . theHamiltonianofgaplessgraphenewithatunablepotential
3 x 1 y − 1 kk′ F barrierandthecorrespondingGHLshiftsfortransmittedand
r 2
(cid:2) (cid:3) (21) reflectedbeamscanbeobtained,respectively[23]. Inthisre-
Itis importantto notethatthe GHL shiftfor the transmit- gard,thelateralshiftsofthereflectedandtransmittedelectron
ted wave is dependenton the productof sz and τ, as can be beamsingraphenearethesameandexpressedas
seenfromtheexpressions,F ,F ,s,s′,andEq. (2). Inother
words, the GHL shift for th1e tra2nsmitted wave has a s τ [(2+ k0′2 + k0′2)sin(2kx′d) k0′2]dtanϕ
(spin-valley)symmetry [29], whereas such a symmetryzca·n- σre =σtr = cokx2s2(k′kdx′2)+2kkx′0′4d si−n2(kkx′2′d) ,
notbeseenintheGHLshiftofthereflectedwave,duetothe x k2k′2 x
x x
presenceofthesecondterminEq. (17). Infact,thereflected (24)
beam is not only dependenton the product of τ and sz, but wherek0′2 =−ss′kk′+ky2. Notethatinthecaseofλ=0and
alsoonvalleyindexτ,separately,indicatingthatthereflected ∆=0(gappedgraphene),weobtainσre =σtr.
electron does not remain invariantunder simultaneousinter- E6 q. (24) at ss′ = 1 is the same as6Eqs. (11) and (19)
changeofspinsandvalleysinthissystem. inRef. 23, respectivel±y. Moreover,inthecaseofss′ = 1,
substitutingk′ iκforϕ>ϕ intoEq. (24),andtaking−the
x → c
limitd ,onecanobtainaftersomecalculation
→∞
III. LIMITINGCASESINGHLSHIFT k2 kk′
σ = y− . (25)
κk k
x y
Now we consider some limiting cases to show how the
Finally,usingEq. (2),weobtain
present formalism is also able to reproduce the analytic re-
sults of previousstudies in MoS and graphenesingle inter- sin2ϕ+1 V /E
2 σ = − G . (26)
faces[16,22],andalsographenebarrier[23]. κsinϕcosϕ
5
FIG.4: (coloronline)Dependenceofcriticalanglesonincidenten-
ergyforelectronswithsz =1invalleysKandK′atVG=3eV.
shiftsaregivenby
FIG.3:(coloronline)DependenceofGHLshiftinthereflectedbeam
onthewidthdofthegatedregionin(a,b)Kleintunnelingand(c,d)
coltahsesricpaalrammoetitoerns,awrhe:er(ea)foEra=llp1l.o7t5seϕV=an1d0ϕ◦can=d5V0G.1=0◦,3(ebV).ETh=e σtτr,sz|kx′d=nπ = k02d2ktax′2nϕ . (27)
1.75eVandϕc = 20.51◦,(c)E = 4.45eVandϕc = 14.75◦,(d)
E =4.45eVandϕc =15.55◦.Notethatdisnormalizedwithkx′. NotethatonlythefirstthreemaximaareshowninFig. 2(b)
and(d). IntheKleintunnelingregime(Fig. 2(a)and(b)),the
transmission probabilities and the lateral shifts are strongly
dependent on the valleys K and K′, whereas such a valley-
Thisequationdescribesthe GH shiftof electronsin totalre-
dependenttransportisweakerintheclassicalmotionasshown
flection from a step potential (single interface) on graphene.
inFig. 2(c)and(d).Althoughtheangleofincidenceiscloser
Itisinterestingtonotethatthisexpressionisexactlythesame
to the criticalangles in the classical regime comparedto the
asEq. (11)givenin Ref. 16. Therefore,ourpresentformal-
Kleintunnelingregime,theGHLeffectintheKleintunneling
isminlimitingcasesisabletoreproducethepreviousresults can more significantly separate the K and K′ valley electron
ofGHshiftsforsingleinterfacesongraphene[30]andTMD
beams compared to the classical motion. For instance, the
monolayers[22],andalsoGHLshiftongraphenebarrier[23].
differencebetween the two lateral displacementsat the third
maximumabsolutevalueis 1065nmintheKleintunneling,
∼
whileitis 227nmintheclassicalregime.
∼
FromEq. (16),itisclearthatthelateralshiftisdependent
on the angle of incidence. Moreover, k′ is zero at critical
IV. RESULTSANDDISCUSSION x
angle ϕ . Therefore, at incident angles ϕ close to ϕ , the
c c
maximumabsolutevaluesoftheshiftsgivenbyEq. (27),can
WepresentournumericalresultforgatedmonolayerWS2 increaserapidlysothatthespatialseparationbetweenKand
asafamilymemberoflayeredTMDmaterials,usingthepa- K′ electron waves in both regimes can exceed the width of
rameterst=1.37eV,∆=1.79eV,2λ=0.43eV,anda=3.197A˚ incidentbeam.Itshouldbementionedthatduetothepresence
[5]. Fromthedispersionrelation,Eq. (2),theenergyregions ofs τ symmetryinEq. (16),theresultsgiveninFig.2with
z
for electrons in propagating mode are given by ∆ < E < s =·+1andτ = 1(τ = +1)are thesame asthose with
2 z −
V ∆+τs λandE >V +∆associatedwiththeKleintun- s = 1andτ =+1(τ = 1)(notshown).
neGli−ng2effectz[23,31–33](Gss′ =2 1)andtheclassicalmotion zWe−now study the effect−of gated region on the reflected
(ss′ = 1), respectively. In both−regimes, transmission prob- electron beam with some parameters different with those in
ability, T = tτ 2, and lateral shift στ of electrons with Fig. 2. The GHL shifts as a function of normalized bar-
energyvalueE| sazn|dagivenV ,arestrotnr,gslzydependentonthe rier width k′d in the both regimes are shown in Fig. 3. In
G x
widthdofthegatedregion.Therefore,tocomparethesequan- the Klein tunneling effect (Fig. 3(a) and (b)), the reflected
titiesinthetwoenergyregions,weshowinFig. 2thetrans- K valley electrons exhibit positive or negative values of the
missionprobabilitiesandtheGHLshiftsofatransmittedelec- lateral shifts, depending on the d values, whereas the GHL
tronbeamwithspinup(s =1)throughthegatedmonolayer, shifts of electrons belonging to the K′ valley are only neg-
z
asthebarrierwidthd(normalizedbyk′)isincreased. Atthe ative. In the classical motion regime (Figs. 3(c) and (d)),
x
resonanceconditionsk′d=nπ(n =0, 1, 2, ),thepo- however,thereflectedK′ valleyelectronscantakepositiveor
x ± ± ···
tentialbarrierinbothenergyregionsbecomesfullytranspar- negative shift values, whereas the values are solely positive
ent, T = 1, and the maximumabsolute values of the lateral for the reflected beam belongingto the K-valley. The peaks
6
longing to the K-valley are transmitted, due to the different
trajectoriesimposedbythegatevoltageforeachvalley,indi-
catingavalleyfilterwithwidetunability.
The energy dependence of GHL shift for the transmitted
K and K′ valley electrons with s = 1 through the gated
z
region with d = 4 nm is shown in Fig. 5. The lateral
shifts of electrons at various angles of incidence (Figs. 5(a)
and (b)) demonstrate successive peaks with different abso-
lute values, corresponding to the transmission resonances,
in both Klein tunneling and classical motion regimes with
positive and negative lateral shifts, respectively. The almost
zero lateral shift region between Klein tunneling and clas-
sical motion is the transmission gap [34] and is given by
V + 1τs λ 1 4a2t2k2+(∆ τs λ)2 < E < V +
G 2 z − 2 y − z G
1τs λ+1 4a2t2qk2+(∆ τs λ)2,whichincludestheen-
2 z 2 y − z
ergygapasqwell.
Clearly the transmission gap can be wider and the abso-
FIG. 5: (color online) The GHL shift of the transmitted K and K′ lutevaluesofthelateralshifts,aswellastheirdifferencebe-
longing to different valleys, can be enhanced by increasing
electronsasafunctionofincidentenergyat(a,b)differentincident
theangleofincidence. Moreover,bycomparingtheenergies
anglesand(c,d)differentgatevoltages. In(a,b)VG = 3.0eV,and
(c,d)ϕ=20◦.Thewidthdofthegatedregionisfixedat4nm. ofmaximumabsolutevaluesofGHLshiftsforKandK′elec-
trons,wecanseethatthevalleyseparationinthecaseofKlein
tunneling is stronger than that in the classical motion [10].
inστ correspondtotheperiodicaloccurrenceoftransmis- Figs. 5(c)and(d)illustratetheeffectofvariousgatevoltages
sionrree,sszonances,as can beseen in the transmittedbeam (see onGHLshiftofthetransmittedKandK′electrons,whenthe
Fig. 2). Nevertheless,thelateralshiftvaluesattheresonance incidentangleis20◦. Here,onecanseethatthetransmission
positionsforthereflectedelectronbeamsarephysicallymean- gapalsoincreaseswithincreasingtheexternalgate. Inaddi-
inglesswhichisduetothefactthatthereflectionprobability, tion,withincreasingthegatevoltage,thewholespectrumof
R = 1 T, at these positionsis zero. In the vicinity of the thelateralshiftsmovestowardshigherenergiesandthemaxi-
resonan−ces, the lateral shift of the reflected beam (similar to mumabsolutevaluesoftheshiftsaswellastheirdifferences,
that of the transmitted beam) can increase rapidly when the increase.
incidentangleapproachesthecriticalangles. Note that due to the time reversal symmetry, the lateral
doAessnaolrteeaxdiystminenσtτioned.,Tthheersezfo·τres,ythmemveatlruyesseoefnliantetrhaelσshtτri,fstzs cshhiafntsgeodf,inifcidweentinetleerccthroannsgewitthhesvpainllesyzs=in −th1e, trreamnsaminitutend-
forreflectedK-vallreey,sezlectronswithspins aredifferentfrom beam, while the lateral shifts will be somewhat different by
z
those for K′-valley electrons with spin s¯ . As a result, the interchangingthevalleysinthereflectedelectronbeams(not
z
reflected electron beams from such electrostatic barriers are shown) due to the absence of sz τ symmetry, as discussed
·
fullyspin-andvalley-polarized.However,ascanbeseenfrom previously.
Fig. 3, the lateral shift difference for the reflected electrons Inordertoexploretheinfluenceofacontinuesrangeofgate
withthesamevalueofs τ (inbothregimes)isnotenough voltageson the GHL shift, in Fig. 6 we showthe calculated
z
to be detectedexperimenta·lly,whileit is detectableforelec- results for transmitted K and K′ electrons with spin s = 1
z
tronswithdifferentvaluesofs τ,whentheincidentangleis andenergyE =2.5eV,whentheexternalgateisvariedfrom
z
sufficientlyclosetoϕ (notshow·n). 0.5 to 4.5 eV. The lateral shifts of both K and K′ electrons
c
FromEq.(6)itisclearthatthecriticalangleϕ forincident arepositiveatlowgatevalues,duetotheclassicalmotionof
c
spin-polarized electrons with energy E is valley-dependent. electrons which satisfy the condition E > VG + ∆2 for any
Fig. 4showsthathowthisangleforanincidentelectronwith incidentangleϕ < ϕ . However,athighergatevoltages,the
c
s = +1andτ = 1changeswithenergy. We canseethat incidentelectrons experiencethe Klein tunnelingeffect, and
z
±
the values of ϕ within a specific energy window are quite hence, the lateral shifts become negative. Again, there is a
c
differentfor the two valleys. This suggests that the incident gap-likefeatureinthelateralshiftvaluesbetweentheclassi-
angle ϕ can be tuned so that only electrons from a specific calmotionandKleintunnelingregimes,correspondingtothe
valleytraversethroughtheelectrostaticbarrier,whiletheelec- transmissiongap. As the incidentangleincreases, the maxi-
trons from the other valley are fully reflected and blocked. mumabsolutevaluesofthelateralshiftsaswellastheirdif-
For instance, for electrons with s = +1 and incident en- ferencesincreaseandthegap-likeregionbecomesbroader.
z
ergy E = 1.75 eV, the critical angles in two valleys are Comparing the gate voltages at which the absolute values
ϕ (τ =1)=50◦andϕ (τ = 1)=20◦(seeFig.4).There- of the lateral shifts in the two valleys are maximum, we can
c c
fore,iftheincidentangleϕra−ngesbetween20◦ and50◦,the see that at some gate values the shift of K′ valley electrons
K′ valleyelectronsarebackscattered,whiletheelectronsbe- in the Klein tunneling is almost zero, while the shift of K
7
valley electrons is maximum. As already mentioned, these
maximum values correspond to the transmission resonances
anddemonstratethevalleysplittingofchargecarriersinWS
2
monolayers, which can be generated and controlled by gate
voltage[8,10].
V. CONCLUSION
In conclusion, the spin-valley transport and the GHL ef-
fectofthetransmittedandreflectedelectronsinaWS mono-
2
layer with a tunable electrostatic barrier have been theoret-
ically studied in both Klein tunneling and classical motion
regimes. Interestingly,itwasfoundthattheGHLshiftofthe
reflected electrons does not remain invariant under simulta-
neousinterchangeof spins and valleys. Therefore, the spin-
valleysymmetryinthelateralshiftsofthereflectedelectrons
fromthebarrierisdifferentcomparedtothatinthetransmit-
tedelectrons. Thelateraldisplacementofelectronbeamscan
be positive or negative dependent on the width of the gated
regionand/orincidentenergy.Ourfindingsshowthatavalley
filter and valley beam splitter can be achieved by tuning the
incidentangle of electronsandexternalgate voltage in WS
2
monolayers. Thesefeaturesindicatethepossibilityofmanip-
ulating valley index in the TMD monolayers which can be
utilizedforquantuminformationapplications[10].
FIG. 6: (color online) Gate voltage dependence of GHL shift for
transmittedKandK′electronswithsz =1atthreedifferentincident
Acknowledgement
angles. TheincidentenergyE andthewidthdofthegatedregion
arefixedat2.5eVand4nm,respectively.
ThisworkispartiallysupportedbyIranScienceElitesFed-
eration.
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