Table Of ContentTopological Imbert-Fedorov shift in Weyl semimetals
Qing-Dong Jiang,1,∗ Hua Jiang,2,∗ Haiwen Liu,1,3 Qing-Feng Sun,1,3,† and X. C. Xie1,3,‡
1International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, P.R. China
2College of Physics, Optoelectronics and Energy,
Soochow University, Suzhou 215006, P.R. China
3Collaborative Innovation Center of Quantum Matter, Beijing 100871, P.R. China
The Goos-Ha¨nchen (GH) shift and the Imbert-Fedorov (IF) shift are optical phenomena which
describe the longitudinal and transverse lateral shifts at the reflection interface, respectively. Here,
we report the GH and IF shifts in Weyl semimetals (WSMs)—a promising material harboring low
5 energy Weyl fermions, a massless fermionic cousin of photons. Our results show that GH shift in
1 WSMs is valley-independent which is analogous to that discovered in a 2D relativistic material—
0 graphene. However, the IF shift has never been explored in non-optical systems, and here we
2 show that it is valley-dependent. Furthermore, we find that the IF shift actually originates from
the topological effect of the system. Experimentally, the topological IF shift can be utilized to
n
characterize the Weyl semimetals, design valleytronic devices of high efficiency, and measure the
a
Berry curvature.
J
2
2
Introduction.—Modern quantum physics originates to characterizing WSMs and potential applications in
fromunderstandingthewave-particledualityofallparti- valleytronics28–31.
]
ll cles,amongwhichphotonisthefirstonebeingdiscovered In this paper, we report the GH effect and IF effect
a with such duality. Classically, the physics of a beam of for 3D Weyl fermions in WSMs. By using wave packet
h
light being reflected at an interface is governed by geo- method, we derive analytic results for the spatial shift
-
s metric optics law, where the photons are treated as clas- of the GH effect and IF effect. Our results show that
e sical particles. In contrast, when considering the wave the GH shift is valley-independent. By contrast, the
m
nature of photons, spatial shifts at the interface appear IF shift is valley-dependent[see Fig. 1], which give rise
t. as longitudinal shift in the incident plane1–3, or trans- to the valley-dependent anomalous velocities in the sys-
a verse shift normal to the incident plane4–9, which are tem. Due to the IF shift being perpendicular to the in-
m
knownastheGoos-H¨anchen(GH)effectandtheImbert- cident plane, it is a generic 3D effect and could never
- Fedorov(IF)effect,respectively. Duetoallparticlespos- appear in a 2D material, e.g. the graphene15. Further-
d
sessing the wave-particle duality, the spatial shifts are more, we demonstrate that the IF shift originates from
n
o alsoexpectedforotherparticles. Forexample, GHeffect the topological effect of the system, namely, the Berry
c has been shown to exist in the systems of electrons10–12, curvature of the system. Remarkably, the consequence
[ neutrons13, atoms14, etc. Particularly for the 2D mass- of the valley-dependent anomalous velocity is significant
1 lessDiracfermionsingraphenesystems,theGHshiftcan enoughtobedetectedexperimentally. Finally,wediscuss
v be manipulated from positive to negative by tuning an threeapplicationsofthevalley-dependentIFshift: (i)ef-
5 external electric field15. However, the IF effect has not fectivelycharacterizingtheWSMs;(ii)directlydetecting
3 been studied in non-optical systems. the Berry curvature; (iii) efficiently inducing valley cur-
5
Similar to photons, Weyl particles are also massless. rent with a high polarization rate.
6
But different from photons, Weyl particles are spin 1/2
0
. chiral fermions and described by the Weyl equation. Re- A
1 cently, Weyl semimetals (WSMs) have been proposed
A
0
as promising systems embedding Weyl fermions, gen- B
5
erating intensive interests16–22. In WSMs, the Weyl
1
: nodes always exist in pairs with opposite chiralities, and B
v each Weyl node corresponds to a valley index23,24. Sev-
i
X eral candidates are suggested to be WSMs including py-
rochloreirradiates16,topologicalinsulatorandnormalin-
r
a sulator heterostructures18, staggered flux states in cold
atom systems20, and photonic crystals based on double- FIG.1: IllustrationofGHeffectandIFeffectinWSMs. The
gyroid structures21. Despite of various theoretical pre- orange arrow represents the incident wave packet which in-
cludeWeylfermionsfromtwovalleys,whereasthegreen(red)
diction of WSMs, the experimental realization of WSMs
arrow represents the reflected wave packet of Weyl fermions
remain a challenge. This failure cannot be ascribed to
theimpedimentofmaterialgrowthtechnique25,26,butto onlyfromValleyA(B).∆GH denotestheGHshift,and∆AIF(B)
standsfortheIFshiftofvalleyA(B).Thisfigureonlyshows
the scarcity of the method of direct experimental iden-
the positive GH shift case.
tification of WSMs27. Due to the topological proper-
ties of the Weyl fermions, the WSMs may possess ex-
otic wave-packet dynamics, which indicates a new route Quantum GH and IF effects in WSMs.—The Hamil-
2
(cid:113)
tonian of the WSMs system is where κ = ((cid:126)v(cid:48)k )2+((cid:126)v(cid:48)k )2−(E−V)2/(cid:126)v(cid:48), β =
y y z z x
H=i=(cid:80)(cid:80)x,y,zvvii(cid:48)ppˆˆiiσσii+V(x) ((xx(cid:54)>00)) , (1) tvvaay(cid:48)n/ll−veyy1,(ik¯nγyW/=k¯SxM)v.z(cid:48)s/.Cvz≡, α¯sg=n[vtxavny−v1z(]vyisk¯tyh/evxckh¯xir)a,liatyndofθth=e
i=x,y,z
w(xhe>re0v)i; (pˆvii(cid:48))aries mveolomceitnytupmaraompeertaetrorfso,rVre(gxi)onisxpo(cid:54)ten0- (a) (b) 246 TRRoeetfagleliocntion TRRoeetfagleliocntion
tial, and σ stand for Pauli matrices. To guarantee spatialshift
i 0
the model Hamiltonian Hermitian, we let vx = vx(cid:48) in 2 GAH
this paper. This Hamiltonian indicates an interface lo- 4 IF
cated at x = 0 in the WSMs[see Fig. 2(a)]. We con- 6 IBF
90° 60° 30° 0 30° 60° 90°
sider a beam of Weyl fermions incident from the re- ValleyA(cid:144)B
gion x < 0 modeled by a Gaussian wave packet as (c) Θ=30° (d)2 ValleyB
ψgin(r)=(cid:82)−∞∞(cid:82)−∞∞dkyd√kzf(ky−k¯y)f(kz−k¯z)ψin(k,r), 68 GH shift ΘΘ==4650°° 1 IF shift ΘΘ==3405°°
Gtwhhaeeurmsesieaafnn(kdwsisa−tvreik¯bsvu)etciot=norf(u(k¯nx2c,tπik¯∆oyn,ksks¯)zo−)f1wweiitd−ht(hkss−∆=kkss)y2p/,2ez∆a.k2kseNdaoarteet 2024 (cid:1)(cid:1)(cid:1)(cid:1)642024690°(cid:1)60°(cid:1)30°030°60°90° 10 VallΘΘΘe===y346A050°°°
thatnoneofourresultsdependontheshapeofthewave 4 Θ=60°
2
packet. Here, ψin(k,r) is the incident wave function, 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
which is a solution of Weyl equation, i.e., Hψin = Eψin
for region x<0: FIG.2: (a)Schematicofp-njunctionwithaninterfaceatx=
0. v (v(cid:48))isthevelocityintheleft(right)sideoftheinterface.
ψin(k,r)= (cid:112)11+η2 (cid:18)ηe−eiiαα//22 (cid:19) eikxx+ikyy+ikzz, (2) Ejuniscittihoeni.F(ebrm)GieHnesrhgifyt,∆anGdHVanisdtIhFesphoiftte∆ntAIiaF(lBd)iofffevraelnlecyeoAf(tBhe)
versus incident angle θ. Total reflection happens at θ (cid:62) 30◦
where k = (cid:112)E2−((cid:126)v k )2−((cid:126)v k )2/(cid:126)v , α = (theyellowcolorshadowregion). λF =hvi/E standsforthe
x y y z z x
(cid:113) wave length of Weyl fermions. (c) Potential dependence of
tthane−r1e(flveyckteyd/vxwkavxe),paancdkeηt c=an bEEe−+w(cid:126)(cid:126)vvrzzitkktzze.n Aasnaψloreg(oru)sl=y, athnedG60H◦s(hdioftttfeodr).in(cdid)ePnotteanntgilaelθde=pe3n0d◦e(nscoeliodf),th4e5◦IF(dsahsihftedfo)r,
(cid:82)∞ (cid:82)∞ dk dk f(k − k¯ )f(k − k¯ )ψre(k,r)g, where incidentangleθ=30◦(solid),45◦(dashed),and60◦(dotted).
−∞ −∞ y z y y z z
ψre is the reflected wave function. ψre can be ob-
tained from the incident wave Eq.(2) by the substitution
We take the incident Fermi energy E =100 meV, po-
k (cid:55)→−k , α(cid:55)→π−αandmultiplicationwiththereflec-
x x tential V = 150 meV, and velocities v = v(cid:48) = 106 m/s,
tion amplitude r =|r|eiφr. The integrals of ψin and ψre i i
g g where i = x,y,z. In this case, β = γ = 1. Note that
givethecenterofthewavepackets, andthereforewecan
the two valleys (A/B) of a WSMs have opposite chiral-
obtain the spatial shifts in y, z directions32:
ities. Fig. 2(b) shows the spatial shifts versus the inci-
(cid:40)∆y =− ∂ φ (k¯ ,k¯ )∓ ∂ α(k¯ ,k¯ ) dent angle θ, where both the GH and IF shifts are odd
± ∂ky r y z ∂ky y z . (3) functionsofincidentangle,whichisconsistentwithsym-
∆z =− ∂ φ (k¯ ,k¯ )∓ ∂ α(k¯ ,k¯ )
± ∂kz r y z ∂kz y z metry analysis. Fig. 2(c) shows the valley independence
and potential dependence of GH shift ∆ , which can
GH
The spatial shifts for Weyl fermions are defined as the be tuned from positive to negative by external field V.
average shifts of the two spinor components: ∆y(z) = This feature is analogous to the GH shift in graphene15.
(∆y(z)+η2∆y(z))/(1+η2). InEq.(3),α(k¯ ,k¯ )represents Fig. 2(d) illustrates that the IF shift is independent of
+ − y z
the incident angle, and φ is the phase of the reflection potential, but depends on valley index. The IF shift can
r
coefficient, which can be obtained by matching the wave be utilized to manipulate the valley degree of freedom.
functionatx=0. Iftheincidentwavepacketisconfined TopologicaloriginoftheIFeffect.—Basedonthesemi-
in x-y plane, the in-plane shift ∆y and out-of-plane shift classical dynamics of wave packet, we show that the IF
∆z ofwavepacketcorrespondstotheGHshift∆GH and shift is closely related to the Berry curvature of the sys-
IF shift ∆IF: tem. Let us assume the velocity vi = vi(cid:48) (i = x, y,z) in
WSMs system. In this case, β = γ = 1, and Eq.(5)
∆GH = vvxy(cid:48)(cid:48) (1β+sβinsα¯inc2oα¯sα−¯κVE), (4) rWedeyulcensodtoe c∆anIFbe=reg−a(r(cid:126)dvevxdyvza)sEat1amnaθg.neCticommmononoplyo,lethine
(cid:126)v v 1 (cid:34)1− E(1−γ)(cid:35) k-space33, and thereby generates an effective magnetic
∆IF = −C| vx z|E tanθ 1− EV(1−β) , (5) field in k-space. The Berry curvature of Hamiltonian
y V Eq.(1) in region (x ≤ 0) is Ω± = ∓(cid:126)3vxvyvzk for con-
2E3
ductionbandandvalenceband1,respectively. Thesemi-
3
(a) TEM tip (b) (cid:92) (cid:91) semiclassical equation cannot be generally applied to all
cases of reflection process because of the breakdown of
(cid:93) A
B adiabatic approximation for some systems.
(cid:7247) (cid:720) (cid:721) WSMs Anomalous velocities induced by IF effect.—We
consider a well collimated beam of Weyl fermions
Ⅰ Ⅱ Ⅲ (cid:720)
propagating in the middle layer (Region II) of a sand-
V P M NR S wWicShMsst[sreuectFuirge.,3w(ah)ic].hTishecoanpsptlriuedcteeldecbtryicthproeteenltaiyaelrpsroo-f
Q
file is shown below the sandwich structure. The height
(c) (d)
of the structure is h and the width of the region II is d.
0.4 SNvalleyA There are both GH and IF shifts at the two interfaces
210 vvaAaB,,yy 00..02 QNvalleyB I(Fx s=hif±t,d2t)h.e mInirroorrdesrymtomoetbrsyeravbeotuhtexva=lle0y-pdleapneenndeeendt
1 vaA,z 0.2 to be broken32. Thus, we consider that the z direction
2 t o t avel laoncoitmyalous vaB,z 0.4 plane shift vveIIlo=citvie,sainndthveIItIh=reevre;gwiohnesraearesdthiffeerxenatn,dwyithdivrzIec=tiovnLs,
z z R
0 30° 60° 90° 120°150°180° 30° 60° 90° 120° 150° velocities are still identical vI = vII = vIII = v.
x/y x/y x/y
Without considering the spatial shifts at the interfaces,
FIG.3: (a)SchematicofthevalleysplitterforWeylfermions. the normal velocities in region II in y and z directions
Region I, II, and III are three WSM layers with different ve- are v = v sinθ cosφ and v = v sinθ sinφ, where
locities vI, vII, and vIII. (b) Illustration of wave packet n,y n,z
angles θ and φ characterize the incident direction of
trajectory (black arrow) of Weyl fermions in region II. The
the wave packet [see Fig. 3(b)]. We denote the GH
electrons are injected by TEM tip with incident polar angle
θ andazimuthalangleφ. Theparameterθ isfixed50◦ inour and IF shifts at the left (right) interface as ∆LG(HR)
calculation. In ordinary case, the trajectory of wave packet and ∆L(R), respectively. During multiple reflections
IF
staysinABMNplane. However,duetothevalley-dependent in region II, the GH and IF shifts are accumulated,
IF shift, the trajectory of Weyl fermions from valley A (B)
and induce average anomalous velocities: v =
a,y
wouldshifttoplaneABRS(ABPQ).(c)φdependenceofthe (cid:2)(∆L +∆R )cosφ+(∆L +∆R )sinφ(cid:3)/(2∆t) and
total anomalous velocities. (d) The propagating plane shift GH GH IF IF
v =[(∆L +∆R )sinφ+(∆L +∆R )cosφ]/(2∆t).
for two valleys. After considering IF shift, SN (QN) is the a,z GH GH IF IF
∆t = d/(v cosθ) represents the propagating time be-
final position shift for valley A (B).
tween two subsequent reflections. Therefore, the normal
and anomalous velocity result in the total velocity:
v =v +v .
classical equation of motion (EOM)1,35 of wave packet t,y(z) n,y(z) a,y(z)
is dr = ∂E(k) − dk ×Ω, where r and k are the center We set the parameters E = 100 meV, V = 150 meV,
dt (cid:126)∂k dt h = 10µm, d = 50 nm, vI = v = 1.2 × 106 m/s,
positions of the wave packet in phase space. We assume z L
vII = v = 106 m/s, vIII = v = 0.8×106 m/s, and
that the incident wave packet locates in the conduction z z R
velocities in other directions are all set as 106 m/s36.
band,andconsidertheincidentwavepacketinx-yplane
Fig. 3(c) shows φ dependence of the total anomalous
with k = 0. The variation of k by potential V and
z x velocities induced by the spatial shifts. The anomalous
nonzero Berry curvature Ω leads to the IF shift ∆ in
y IF velocities are valley-dependent, which implies that the
z-direction:
total velocities also depend on valley index. Eventually,
(cid:90) −kx (cid:126)v v 1 the different velocities of the two valleys lead to macro-
∆IF =− dkxΩy =−( vx z)E tanθ, (6) scopic separation in real space, which is experimentally
kx y detectable. Fig. 3(d) shows the opposite position shift
at the bottom of region II for valley A and valley B. GH
where tanθ = k /k . Remarkably, Eq.(6) and Eq.(5)
y x
shift cannot induce the position shift, because GH shift
completely coincide in the case of β = γ. This co-
always lie in the propagating plane. In contrast, IF shift
incidence does not depend on linear dispersion of the
system32. The consistency between Berry curvature cal- can induce the position shifts (µm order) for two valleys
(SN and QN)32.
culations and wave packet results strongly support that
the IF shift is mainly a topological effect. The IF shift Identification of WSMs.—The experimental verifica-
in WSMs is quite different from that in optical systems. tion of WSMs has not been justified mainly due to
Particularly, ∆ reaches the maximum in WSMs [see the scant of efficient detection method38–40. The di-
IF
Fig. 2(b)]comparingthezerovalueinopticalsystemsat rectARPESmeasurementsofenergydispersionsarecur-
θ =0◦6,8. Thisisbecausetheconservationofk ,k guar- rently scant due to the constraint of magnetic properties
y z
antee the Weyl Fermions stay in the same valley during of WSMs. Here, we suggest that the IF effect can be
thereflectionprocessesinWSMs. Incontrast, thepolar- used as an experimental identification of WSMs. It has
ization of the photons change during the reflection pro- been shown that the topological IF shift splits the inci-
cesses, which will severely influence ∆ . However,the dent wave packets into opposite directions respect to the
IF
4
(a) (c) contrast, after considering the anomalous velocities, Fig.
4(a),(b)and(c)showthedensitydistributionsforvalley
Valley B A,valleyB,andvalleyA&B,respectively. TheIFeffect
inducesoppositeshiftsiny-directionforvalleyAandval-
ley B. Since the Weyl fermions from valleys A and B are
Valley A Valley A well separated in space of micrometer order, pure valley
current can be generated in the green (red) regions[see
Fig. 4(c)]. This schematic set-up in Fig. 4(d) can be
utilized to generate pure valley current.
(b) (d)
1.0 DetectionofBerrycurvature.—TheBerrycurvature—
0.8 a gauge invariant quantity—should be detectable in
Valley B (cid:92) experiment1. However, up to date, there lacks an exper-
y 0.6 (cid:720)
sit (cid:93) imental feasible method to measure the Berry curvature
n
de 0.4 in real materials. The topological IF effect provides a
e
relativ 0.2 T1 T2 nweiwthwinavyertsoiomnesaysmurmeetthrye,BthererByecrurryvacuturvrea.tuFroerisaasnysetveemn
0 Valley B Valley A function of wave vector k, i.e., Ω(k) = Ω(−k)1, which
is usually satisfied in WSMs16–18. Considering a wave
FIG. 4: Valley density distribution shift caused by IF effect. packetpropagatinginx-y planewithkz =0,theIFshift
(a),(b)and(c)showtherelativedensitydistributionofvalley can be expressed as ∆ =2(cid:82)kxdk Ω (k), or
IF 0 x y
A,valleyB,andboth,respectively,afterconsideringGHand
IF effect. The color bar on the right represents the relative 1∂∆ (k)
density ρ/ρmax of Weyl fermions. Here, ρ (ρmax) represents Ωy(k)= 2 ∂IkF . (7)
theabsolute(maximum)densityofWeylfermionsatthebot- x
tom. The dashed lines indicate the location of the maximum To measure the Berry curvature Ω , one need to collect
y
intensity of density. Red (Green) box in Fig. (c) guides eyes
the IF shift in z direction ∆ (E,θ) as a function of
to the region where only valley B (A) exist (pure valley po- IF
energy E and incidence angle θ. The ∆ (E,θ) can be
larization). (d) Proposed setup for generating valley current. IF
transformedinto∆ (k ,k ),anditsderivativegivesout
ThedevicesT (red)andT (green)aretwoterminals,which IF x y
1 2
the Berry curvature Ω (k). In the same way, the IF
can extracted valley current out. y
shiftinotherdirectionscanbeusedtoobtainΩ (k)and
x
Ω (k).
z
valley index. Thus, the splitting of wave packets on the Summary.—We obtained analytical expressions of the
bottomofregionIIinFig. 3(a)canserveasahallmarkof GH shift and IF shift at the interface of WSMs. We
WSMs. Furthermore, even considering an incident wave demonstrate that the IF shift is valley-dependent, and
packet with finite angle range, this exotic splitting can can be attributed to the topological nature of the sys-
also exist. tem. This IF shift can lead to valley-dependent anoma-
Purevalleypolarization.—LetusconsideranTEMin- lousvelocity,whichisexperimentallydetectable. Finally,
jectorwithincidentanglerangeδ(θ ∈[θ −δ/2, θ +δ/2], we discuss three applications of the topological IF shift
c c
and φ ∈ [φ −δ/2,φ +δ/2]) at the top of Region II, including characterization of WSMs, fabrication of high
c c
and study the valley density distribution at the bottom efficientvalleytronicdevices,anddetectiontheBerrycur-
[see Fig. 4(d)]. We calculate the valley density distri- vature.
bution with parameters E = 100 meV, V = 150 meV, Acknowledgments.—This work was financially
h=10µm,d=80nm,δ =3◦,θ =32.5◦,andφ =90◦. supported by NBRP of China (2012CB921303,
c c
Without considering anomalous velocities, the density 2012CB821402, and2014CB920901)andNSF-Chinaun-
distributions for valley A and B are maximized at the derGrantsNos. 11274364, and91221302, and11374219.
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A. Vishwanath, Phys. Rev. X 4, 031035 (2014).
6
Appendix A: Supplementary Information for “Topological Imbert-Fedorov shift in Weyl semimetals”
Qing-Dong Jiang, Hua Jiang, Haiwen Liu, Qing-feng Sun, and X. C. Xie
I. Detail derivation of the GH and IF shifts.
To calculate the spatial shifts of the wave packet in y, z directions, one need to know the central positions of the
incident and the reflected wave packets at the interface (x=0). By expanding the phases α(k ,k ) and φ (k ,k ) to
y z r y z
the first order around (k¯y,k¯z), the integrals of ψgin and ψgre give that ψgin± ∝e−(y∓21∂∂kαy)2∆2ky/2 e−(z∓21∂∂kαz)2∆2kz/2 and
ψgr±e ∝ e−(y+∂∂φkyr±12∂∂kαy)2∆2ky/2 e−(z+∂∂φkzr±12∂∂kαz)2∆2kz/2, where the ± subscript corresponds to the first and the second
componentofthespinor,respectively. Thenonecanidentifythecenter(maximum)oftheincidentandreflectedwave
packets in real space. For the incident wave packet, the two spinor components are centered at (y¯in,z¯in), where
± ±
(cid:104) (cid:105)
yi±n =(cid:104)±12∂∂kyα(ky,kz)(cid:105)ky=k¯y,kz=k¯z . (A1)
zi±n = ±12∂∂kzα(ky,kz) ky=k¯y,kz=k¯z
For the reflected wave packet, the two spinor components are centered at (y¯re,z¯re), where
± ±
(cid:104) (cid:105)
yr±e =(cid:104)−∂∂kyφr(ky,kz)∓ 21∂∂kyα(ky,kz)(cid:105)ky=k¯y,kz=k¯z . (A2)
zr±e = −∂∂kzφr(ky,kz)∓ 12∂∂kzα(ky,kz) ky=k¯y,kz=k¯z
Considering above results, we can obtain the spatial shifts for two spinor components in y, z directions, which are
(cid:40)∆y =yre−yin =− ∂ φ (k¯ ,k¯ )∓ ∂ α(k¯ ,k¯ )
± ± ± ∂ky r y z ∂ky y z (A3)
∆z =zre−zin =− ∂ φ (k¯ ,k¯ )∓ ∂ α(k¯ ,k¯ )
± ± ± ∂kz r y z ∂kz y z
AsisshowninEq. (A3),inordertocalculatethespatialshiftattheinterface(x=0),weneedtoknowthereflection
phase φ (k¯ ,k¯ ) and α(k¯ ,k¯ ). α(k¯ ,k¯ ) represents the incident angle of the wave packet, and can be easily obtained
r y z y z y z
once we know the incident wave vector. Next, we elaborate on deriving the reflection phase φ . We consider the
r
total reflection case, i.e., the reflection probability |r|2 =1. Thus, the wave function must be evanescent in the region
x > 0[Fig. 2(a)]. We can calculate the reflection coefficient by matching the wave function ψin(k,r)+ψre(k,r) at
x = 0 to the evanescent wave. The continuity of wave function gives the reflection amplitude r = eiφr, where φr =
2 tan−1( ηcosα )−α−π/2. In this expression, ξ = (cid:126)(κ+ky) , where κ=(cid:113)((cid:126)v(cid:48)k )2+((cid:126)v(cid:48)k )2−(E−V)2/(cid:126)v(cid:48).
−ηsinα+ξ E−V+(cid:126)vz(cid:48)kz y y z z x
Substitute the expression of α and φ into Eq.(A3), and we can obtain the average shifts in y, z directions.
r
II. IF shift for nonlinear energy dispersion.
In the main text, we have shown that the Berry curvature is the origin of the IF shift for WSMs, which possess
linear energy band dispersions. Here, we show that the Berry curvature is still the origin of the IF shift for another
system without linear energy dispersions. To clarify this idea, we consider a system with a model Hamiltonian
(cid:18) Ak2 B(k −ik )(cid:19)
H(cid:48) = y x z , (A4)
B(k +ik ) −Ak2
x z y
(cid:113)
where A and B are two parameters. The eigenenergy of the Hamiltonian is E =± B2(k2+k2)+A2k4. Thus this
x z y
Hamiltonianobviouslydoesnothave3Dlinearizedenergydispersions. OnecanapplyagatevoltageV tothissystem
at the region x(cid:62)0; therefore, an interface appears at x=0.
We now consider that a wave packet incidents on the interface from the region x < 0. To calculate the IF
shift at the interface, we follow the same procedure in the main text. Analogously, we first construct the incident
wave packet in Gaussian profile with center momentum (k¯ ,k¯ ,k¯ ). Still we assume the wave packet as ψin(r) =
x y z g
(cid:82)∞ (cid:82)∞ dk dk f(k −k¯ )f(k −k¯ )ψin(k,r),wheref(k −k¯ )hasthesamedefinitionasinthemaintext. Similarly,
−∞ −∞ y z y y z z s s
the reflected wave packet can be constructed as ψre(r)=(cid:82)∞ (cid:82)∞ dk dk f(k −k¯ )f(k −k¯ )ψre(k,r). However,
g −∞ −∞ y z y y z z
7
since the Hamiltonian H(cid:48) here is different from that in the main text, one need to solve the wave functions ψin(k,r)
andψre(k,r)forH(cid:48). TheHamiltonianequationisH(cid:48)ψ =Eψ,whereψ =(ψ ,ψ )T isatwocomponentsspinor. Solve
1 2
this Hamiltonian equation and use continuity condition of the wave function at x=0, then we can get the solutions:
ψin(k,r)= (cid:112)11+ζ2 (cid:18)ζe−eiiαα((kkyy,,kkzz))//22 (cid:19) eikxx+ikyy+ikzz, (A5)
and
ψre(k,r)= (cid:112)1r+ζ2 (cid:18)i−ζiee−iiαα((kkyy,k,kzz))//22 (cid:19) e−ikxx+ikyy+ikzz. (A6)
(cid:114)
In the above expressions, kx = (cid:113)E2−(Aky2)2−(Bkz)2/B, α = tan−1(kz/kx), ζ = EE−+ααkky22, r = |r|eiφr is the
y
(cid:113)
reflection amplitude, and φ = 2tan−1( ζcosθ ) with ξ(cid:48) = B(κ+kz) and κ = (E−V)2−(Ak2)2−(Bk )2/B.
r −ζsinθ+ξ(cid:48) E−V+Ak2 y z
y
Substitute φ and α into Eq.(A3), and one can obtain the spatial shifts in y and z-directions. If the incident wave
r
packetisconfinedinx-y plane,thentheout-of-planespatialshiftinz-directioncorrespondstotheIFshiftandreads:
B2 k
∆ = x . (A7)
IF A Ek2
y
Alternatively, one can also obtain the IF shift ∆ from semiclassical equation for wave packet (Eq.(7) in the main
IF
text). Inordertoapplythesemiclassicalequation,oneneedcalculatetheBerrycurvatureΩ± fortheHamiltonianH(cid:48),
wherethesuperscript±correspondstotheconductancebandandvalenceband,respectively. TheBerrycurvatureof
thissystemisΩ± =(∓AB2k k , ∓AB2k2, ∓AB2k k )1. SubstitutetheBerrycurvatureexpressionintotheEq.(11)
2E3 x y 2E3 y 2E3 z y
in the main text, and one can obtain the analytic result of the IF shift ∆IF = BA2Ekkx2, showing the consistence with
y
Eq.(A7). The fact that the IF shift ∆ can be obtained from Eq.(7) in the main text provides another convincing
IF
argument that IF shift arises from the Berry curvature of the system.
III. Calculations of anomalous velocities and position shifts for two valleys.
(a) anomalous velocity induced by GH shift (b) anomalous velocity induced by IF shift
0.8
2
vA
1 0.4 a,y
vB
0 0 a,y
1 va,yHValleyA(cid:144)BL 0.4 vaA,z
2 v HValleyA(cid:144)BL vB
a,z 0.8 a,z
0 30° 60° 90° 120° 150° 180° 0 30° 60° 90° 120° 150° 180°
FIG.5: (a)TheanomalousvelocitiesinducedbytheGHshiftarevalley-independent. (b)Theanomalousvelocitiesinducedby
the IF shift are valley-independent.
In the main text, we have demonstrated that the GH shift does not depend on valley index, whereas the IF shift
depends on valley index. Although both of the GH shift and IF shift can bring anomalous velocities to the normal
velocities, only the anomalous velocity induced by IF shift is valley-dependent. There are both GH and IF shifts at
the left interface (x = −d) and the right interface (x = d). If the system has the mirror symmetry about the x = 0
2 2
plane,theGHshiftsareinthesamedirectionattheleftandrightinterfaces. ButIFshiftsareinoppositedirectionat
8
thesetwointerfaces,leadingtothecancellationoftheIFshiftsaftertwosubsequentreflections. Therefore,inorderto
observethevalley-dependentIFshift,themirrorsymmetryaboutx=0planeneedtobebroken. Soweconsiderthat
the z direction velocities in the three regions are unequal, with vI =v , vII =v, and vIII =v . For simplicity, the x
z L z z R
and y directions velocities are still with vI = vII = vIII = v. Figure S1 shows the anomalous velocities induced
x/y x/y x/y
by the GH shift and IF shift. The anomalous velocities are calculated with the same parameters as in the section
“Anomalous velocities induced by IF effect” in the main text. As one can see, the anomalous velocity induced by the
GH shift is valley-independent; by contrast, the anomalous velocity induced by the IF shift is valley-dependent. The
green curves and red curves correspond to valley A and valley B, respectively. The thick curves and dashed curves
correspond to the anomalous velocities in y-direction and z-direction, respectively.
Giventheincidentdirection(θ,φ)oftheincidentwavepacket,wecancalculatethefinalposition(x,y,z)atbottom
of the region II. The wave packet needs time T =h/v to reach to the bottom of the region II without considering
0 n,z
anomalous velocity. However, if we take the anomalous velocity into account, the wave packet needs time T =h/v
1 t,z
to reach to the bottom of the region II. Thus the anomalous velocities induced position shift is
∆L=v ×T −v ×T . (A8)
t,y 1 n,y 0
Put the total velocities of the valley A and valley B into Eq.(A8), we can get the final position shifts for the valley A
and valley B, respectively [see Fig. 3(d) in the main text].
IV. Calculations of valley density distribution induced by valley-dependent anomalous velocities.
(a) (b)
1.0
0.8
0.6
Valley A Valley B sity
en 0.4
d
e
v
ati 0.2
el
r
0
FIG. 6: Valley pattern before considering the IF shift. The figures (a) and (b) show the valley pattern of valley A and B,
respectively, before considering the anomalous velocities induced by the spatial shifts. The color bar on the right represents
the relative density of Weyl particles. Here, the relative density has the same meaning as that in the main text.
Sincetheincidentwavepacketsarelikelytobereflectedrepeatedlyattwointerfaces,oneonlyneedstoconsiderthe
totalreflectedcasessincethepartiallyreflectedwavepacketswouldeventuallydiminishafterbeingreflectedmultiple
times. A collimated incident wave packet usually has certain angle range, thus, we assume θ ∈ [θ −δ/2,θ +δ/2]
c c
and φ∈[φ −δ/2,φ +δ/2], where θ and φ are the angles of the center of the collimated incident wave packet and
c c c c
the δ is the angle spread range. In the numerical calculations, we take θ = 32.5◦ and φ = 90◦. In this case, the
c c
anomalous velocity v induced by the GH effect is negative, and the anomalous velocity induced by the IF shift is
a,y
relatively large. Hence, the total anomalous velocity has a remarkable valley-dependent effect.
For a wave packet of Weyl fermions propagating in the Region II, the total velocities in y, z directions are v and
t,y
v , respectively. Due to the valley-dependent anomalous velocities, the total velocities are also valley-dependent,
t,z
whichcanleadtomacroscopicshiftsofvalleys. Becauseofthereflectionatthetwointerfaces(Fig. 3(B)),thevelocity
in x direction vII in the Region II is a periodic function of ∆t, i.e., vII(t) = vII(t+2∆t). In the first period, the
x x x
velocity vII(t) reads
x
(cid:26) v t∈[0,∆t)∪[3∆t,2∆t]
vII(t)= x 2 2 . (A9)
x −v t∈[∆t,3∆t)
x 2 2
9
Theheightofthisstructureish, thusthewavepacketneedstimet =h/v toreachtothebottomoftheRegionII.
0 t,z
Hence, one can get the final position coordinate (x,y,z) of the wave packet, where x = (cid:82)t0dtvII(t), y = (cid:82)t0dtv ,
0 x 0 t,y
andz =−h. Asonecansee,asingleincidentdirection(θ,φ)ofthewavepacketcorrespondstoasinglepoint(x,y,z)
at the bottom of Region II. We further assume that the injector supplies an equal strength of incident beams in the
angle range θ ∈ [32.5◦ −δ/2,32.5◦ +δ/2] and φ ∈ [90◦ −δ/2,90◦ +δ/2]. Thus, there will be a pattern (density
distribution) at the bottom of Region II [see Fig. 4(d) in the main text]. The Figure S2 (a) and (b) show the density
distribution of valley A and valley B, respectively, before considering the IF shift. From the Figure S1(a) and (b) we
can find that the density distribution of valley A and valley B are located at the same place. Therefore, the Weyl
fermions from valley A and valley B are mixed together. However, if the anomalous velocity is taken into account,
the distribution of valley A and valley B will shift to opposite direction leading to the valley separation [as shown in
Fig. 4(a), (b) in the main text].
Supplementary References
∗ These authors contributed equally to this work.
† [email protected]
‡ [email protected]
1 D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 109 (2010).
