Fermi arc plasmons in Weyl semimetals

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by  Justin C. W. Song

Fermi arc plasmons in Weyl semimetals
Justin C. W. Song1,2 and Mark S. Rudner3
1Division of Physics and Applied Physics, Nanyang Technological University, Singapore 637371
2Institute of High Performance Computing, Agency for Science,
Technology, and Research, Singapore 138632 and
3 Center for Quantum Devices and Niels Bohr International Academy,
Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
IntherecentlydiscoveredWeylsemimetals,theFermisurfacemayfeaturedisjoint,opensegments
– the so-called Fermi arcs – associated with topological states bound to exposed crystal surfaces.
7 Here we show that the collective dynamics of electrons near such surfaces sharply departs from
1 that of a conventional three-dimensional metal. In magnetic systems with broken time reversal
0 symmetry, the resulting Fermi arc plasmons (FAPs) are chiral, with dispersion relations featuring
2 open, hyperbolic constant frequency contours. As a result, a large range of surface plasmon wave
vectors can be supported at a given frequency, with corresponding group velocity vectors directed
n
along a few specific collimated directions. Fermi arc plasmons can be probed using near field
a
J photonics techniques, which may be used to launch highly directional, focused surface plasmon
beams. The unusual characteristics of FAPs arise from the interplay of bulk and surface Fermi arc
1
carrier dynamics, and give a window into the unusual Fermiology of Weyl semimetals.
3

]
l Three dimensional Weyl semimetals (WSMs) are pro-
l
a totypical topological metals, featuring one or more pairs
h
of protected band degeneracy points in the Brillouin
-
s zone[1–7]. These“Weylpoints”actasquantizedsources
e and sinks of Bloch band Berry flux [8]. The nontrivial
m
bulk band topology is manifested in the appearance of
t. peculiar branches of gapless states bound to certain ex-
a
posed surfaces, featuring open segment Fermi surfaces,
m
i.e., “Fermi arcs”(see Fig. 1a) [1–6, 9]. At the single par-
-
ticle level, these unusual features are responsible for a
d
n variety of intriguing transport phenomena [10–17].
o FIG.1: a. DisjointFermisurfaceofatype-IWeylsemimetal:
Here we show that the coupled, collective dynamics
c in addition to the bulk closed Fermi surfaces (red and blue
[ of electrons on both the open-segment and closed WSM spheres), the WSM features open Fermi arcs (orange curve)

Fermi surfaces sharply departs from that expected in on certain exposed crystal surfaces. b. The velocity den-
1
conventional metals. This unusual situation gives rise sity can be separated into bulk (v ) and surface (v ) compo-
v b s
2 to “Fermi arc plasmons” (FAPs), which are confined to nents, with vs carried by the chiral Fermi arc states. c. The
Fermi arc plasmon dispersion has three chiral branches, fea-
2 certain exposed surfaces and are characterized by highly
turinghyperbolicfrequencycontoursoverawidebandwidth.
0 anisotropicdispersions(Figs.1b,c). Inconventionalmet-
0 Thesemodesarisefromthehybridizationofconventionalsur-
als, surface plasmons are nearly dispersionless and fea-
0 faceplasmons,supportedbybulkfreecarriers,andcollective
. ture closed, elliptical constant frequency contours. In modes of carriers in the topological surface states. The or-
2 contrast, we find that FAPs in WSMs with broken time dinary surface plasmon frequency is denoted by ±ωsurf, and
0 pl
reversal symmetry are hyperbolic over a wide range of topologicalsurfacestatesarecharacterizedbythevelocityv0.
7 s
frequencies, featuring open iso-frequency contours that
1
: do not close on themselves (Fig. 2). Such contours sup-
v
i portawiderangeofwavevectorsateachfrequency,with semimetal’s bulk and surface carriers, including the bulk
X a wavenumber-independent group velocity direction for anomalous Hall conductivity [18], and the chiral Fermi
r large wave vectors. Together, these features allow for arcsurfacestatevelocity[1–3]. Consequently,FAPsmay
a
tight focusing of collimated, non-reciprocal surface plas- alsoprovidemeanstodynamicallyprobethepeculiarcar-
mon waves, with frequency-dependent directionality. rier kinematics of Weyl semimetals at zero-field [18, 19].
FAPs arise from the hybridization of chiral collective We expect that FAPs can be realized in WSMs that
modesassociatedwithcarriersintopological“Fermiarc” break time reversal symmetry (TRS), such as candidate
surfacestates(slantedplane,Fig.1c),withordinarysur- magnetic WSMs [1–3, 20–24]. We note parenthetically
face plasmon modes supported by bulk carriers close to thathyperbolicplasmonscanbeartificiallyengineeredin
the surface (horizontal gray planes, Fig. 1c). This hy- metamaterials,wherehyperboliccharacteristicsareman-
bridizationandtheresultingdispersionrelationsaresen- ifested for plasmon wavelengths larger than the feature
sitive to the unusual constitutive relations of the Weyl sizes of resonator components [25]. In contrast, WSMs
2
that host FAPs are intrinsic hyperbolic plasmonic mate- where σ(r,ω) ∝ Θ(−z) is the bulk conductivity tensor,
rials,andmayexhibithyperbolicbehavioratshortwave- φ is the electric potential, and e = −|e| is the elec-
lengths, not limited by fabrication or patterning tech- tron charge. The conductivity σ(ω) includes an intrinsic
niques. Indeed, FAPs may arise for plasmon wave vec- anomalous Hall component σ = σ = −σ , due to
H zx xz
tors up to the characteristic Weyl node separation, k , the Berry flux between the Weyl nodes [18], and a longi-
∗
which is typically an O(1) fraction of the inverse lattice tudinalpartσ =σ =σ . Notethatweworkatzero
xx yy zz
constant: k ∼ 1nm−1. This scale corresponds to wave- applied magnetic field, and do not access the finite mag-
∗
lengthsmuchsmallerthantypicalpatternedfeaturesizes, netic field WSM collective dynamics [10, 16, 17, 26–28].
which may be on the order of >∼100nm [25]. At the surface z = 0, the continuity relation for the
Previous studies have revealed a number of interest- surface density picks up a source term arising from bulk
ing plasmonic phenomena, including non-reciprocity [26] currents impinging on the surface:
and signatures of the chiral anomaly [10, 16, 26–28] as-
(cid:12)
sociated with the collective behavior of carriers on the iωns(rs,ω)+∇·vs(rs,ω)=zˆ·vb(cid:12)0−. (3)
(closed) bulk Fermi surfaces of WSMs [29–31]. Here we
aim to expose the new collective phenomena and cou- To arrive at Eq. (3), we have used Eq. (1) and the full
pling mechanisms that arise due to the chiral Fermi arc continuity equation, along with ∂ Θ(−z)=−δ(z). Plas-
z
modes on the WSM surface. mons emerge from Eqs. (2) and (3) as self-sustained col-
lective oscillations, with the electric potential generated
Collective carrier dynamics—Tobegin,weexam-
internallybydensityfluctuationsδn(r,ω)=n(r,ω)−n .
ine collective charge carrier dynamics in a 3D WSM in a 0
Here n is the equilibrium electron density.
semi-infinite slab geometry, occupying the region z < 0, 0
As in a conventional metal, bulk carriers (denoted M)
see Fig. 1b. For concreteness, we consider a model dis-
may pile up at a surface when the system is pushed
persion with two bulk Weyl nodes at zero energy, situ-
out of equilibrium, giving a surface density contribution
ated at points (0,±k ,0) in the Brillouin zone (i.e., sep-
∗
n =nM [42]. In equilibrium, nM ≡0. Importantly, the
arated along the ky direction) [32]. For each fixed value s s s
associatedsurfacevelocitydensity,vM,vanishesatlinear
of ky with |ky| < k∗, the two-dimensional dispersion s
orderinthedeviationfromequilibrium[32]. Inthelinear
ε (k ,k ;k ) is gapped, with upper and lower bands
2D x z y
regime, Eq. (3) together with the non-retarded Coulomb
featuringChernnumbers+1and-1, respectively. Atthe
interactionyieldsthefamiliarnon-dispersivesurfaceplas-
surface z = 0, this nontrivial topology yields a branch
mon mode of a 3D metal [34].
of gapless chiral surface states for each |k |<k . When
y ∗
For the WSM, in addition to the conventional surface
the Fermi energy is close to zero, the filling of the chiral
density nM, we must also account for the dynamics of
surface states is characterized by an open segment Fermi s
carriers in the topological surface states (denoted F).
surface – the “Fermi arc” (see Fig. 1a, and Refs. [1–3]).
We thus express the total surface density as n (r ,t) =
Asweshow, theseFermiarccarriersbringdramaticnew s s
nM(r ,t)+nF(r ,t)[43],wherenF(r ,t)istheFermiarc
features to WSM surface plasmons. s s s s s s
surface state density distribution (see Figs. 1a,b). Here
To describe FAPs in the slab geometry, it is useful
nF includes allelectrons inthe topologicalsurface states
to partition the particle density field n and the velocity s
up to the Fermi arc, and is thus sensitive to details of
density field v into bulk and surface contributions. For
thedispersionfarbelowtheFermisurface[44]. However,
each Fourier mode with angular frequency ω, we write:
the electrodynamic response only depends on deviations
n(r,ω) = n (r,ω)Θ(−z)+n (r ,ω)δ(z), from equilibrium, i.e., deformations of the Fermi arc.
b s s
The topological surface states carry a finite (in-plane)
v(r,ω) = v (r,ω)Θ(−z)+v (r ,ω)δ(z). (1)
b s s
surface velocity density, analogous to the persistent cur-
Herethesandbsubscriptsofn andv denotesurface rents that flow in quantum Hall edge states. For
s,b s,b
and bulk contributions, respectively, and rs = (x,y) is a straight Fermi arc with ky-independent dispersion,
the two-dimensional coordinate on the surface z = 0. εs(kx,ky)=vs0kx, the surface velocity density is propor-
The surface densities n and v account both for surface tionaltonF andorientedalongxˆ. Herev0 isthevelocity
s s s s
charges that may build up from the motion of bulk free of the chiral surface states on the surface z = 0; it has
carriers, as in a usual metal [33], and for the behavior of the same sign as σH [32]. For a more general topological
carriers in the unconventional topological surface states. surfacestatedispersion,non-equilibriumdeformationsof
We consider the case of low but finite doping, so that k the Fermi arc may also lead to currents parallel to yˆ,
∗
is large as compared to the typical Fermi wave vectors similar to a conventional 2D system with a closed Fermi
in the bulk. Note that v is always oriented within the surface. WeaccountforDrude-likedynamicsalongyˆ us-
s
plane of the two-dimensional surface, so that vz =0. ing a phenomenological Drude weight D [32] and write:
s
The bulk carrier density obeys the continuity relation
(cid:20) (cid:21)
iωnb+∇·vb =0, evb(r,ω)=σ(r,ω)(cid:2)−∇φ(r,ω)(cid:3), (2) vs(rs,t)=(cid:2)vs0nFs(rs,t)(cid:3)xˆ+ iDeω(cid:2)−∂yφ(rs,t)(cid:3) yˆ. (4)
3
Note that D = 0 for a system with k -independent sur-
y
face state dispersion (as described above).
Wenowanalyzethesurfacedensitydynamics,Eq.(3).
We focus on the collisionless limit, ω (cid:29) 1/τ , where
surf
τ is a relaxation time characterizing the scattering
surf
between F and M species on the surface [45]. In this
limit, nF and nM obey separate continuity equations:
s s
(cid:12)
ieωnMs +σxx∇zφ(cid:12)0− =0
ieωnF +σ ∇ φ+e∇·v =0. (5)
s zx x s
FIG. 2: a. Contour plots of the FAP dispersion for branch
The two fluids M and F are coupled by the Coulomb
1 in Fig. 1c, obtained from Eq. (9). Wave vectors are scaled
interaction, encoded in φ(r,ω). In the linear regime, vs by q0 = ωpl/vs0. Fine dashed contours are hyperbolic, and
is associated solely with the F carriers (see above). do not close on themselves at large wave vectors. In a small
Notice that the surface density dynamics is explic- frequency interval near |ω| <∼ ωpsulrf, elliptical contours are
itly coupled to bulk currents in Eq. (5). In particular, found (bold yellow dashed lines, see discussion in text). The
σxx∇zφ(cid:12)(cid:12)0− captures the current density of bulk free car- FquAePncgyrocuopntvoeulorcsit(ybliascokriaenrrtoewdst)r.anAsvleornsgetaogtihveencohnystpaenrtbofrleic-
riers impinging on the metal surface from z < 0. Simi-
frequency contour, the group velocity points along a single
larly, σ ∇ φaccountsforthefactthatanin-planeelec-
zx x direction for a large range of q values, allowing for focused
y
tric field induces changes in the topological surface state
propagationofplasmonwaves. b. LinecutofFAPdispersion
density, via anomalous Hall currents associated with un- for q = 0. Parameters used: dimensionless Drude weight
y
dergap carriers at momenta far from the Weyl nodes. D˜ = 6.0, see Eq. (10), and dimensionless Hall conductivity
σ˜ =4πσ /[(κ+1)ωsurf]=2.0 [41].
H H pl
FermiarcplasmonsinaWeylsemimetal—FAPs
arise as surface-bound plane wave solutions to the equa-
tions of motion (1)-(5), of the form δn,φ ∼ eiωt−iq·rs. Eq. (5) with Eq. (4) for a plane wave, we relate the sur-
Here q = (qx,qy). Below we algebraically eliminate the face density fluctuations to the electric potential φ˜<:
q
particle and velocity density fields from the equations of
mmoottiioonn,oofbstuarifnaicnegaandcobmuplkacctardreiescrsripintitoenrmofstohfeφc(or,uωp)le.d δn˜χ =Gχφ˜<, GM =−qσxx, GF = qxσzx+ Dωqy2. (8)
s,q q ieω e(ω−v0q )
Away from the surface at z = 0, the Fourier modes s x
φ˜q(z) inside (<) and outside (>) the WSM obey the Note that the chirality of the Fermi arc carriers is exhib-
Poisson equation (in the non-retarded limit): ited in Eq. (8) through the pole in GF, which arises for a
single value of q for a given ω.
4πe x
(q2−∂2)φ˜<(z)= n˜ (z), (q2−∂2)φ˜>(z)=0, (6) Using Eq. (8) in Eq. (7), and the explicit forms for
z q κ b,q z q φ˜>(z)andφ˜<(z),weseekthevaluesofqandωsuchthat
q q
where q = |q|, and κ is the dielectric constant in the the boundary conditions on φ˜ and ∂ φ˜ are satisfied.
q z q
WSM (the region denoted > is vacuum); the ω depen- This yields the FAP dispersion relation:
dencies of φ˜ and n˜ (z) are implicit. We eliminate n˜
q b b,q
in Eq. (6) using Eq. (2), iωnb =(1/e)∇·(σ(ω)∇φ) [46], (ω−v0q )(cid:104)κ+1 − ωp2l(cid:105)− 4πqxσ − 4πDqy2 =0. (9)
and obtain solutions φ˜<(z)=α eqz, φ˜>(z)=β e−qz. s x κ ω2 κ q H κωq
q q q q
To fully determine the plasmon potential φ˜ (z), we
q Here we have used 4πσ /κ = ω2/iω, valid in the colli-
mustspecifyappropriateboundaryconditionsatthesur- xx pl
sionless limit; ω is the bulk plasmon frequency.
face z = 0. First, φ˜ (z) must be continuous at z = 0. pl
q
Second, note that FAPs involve a dynamical modulation Characteristics of Fermi arc plasmons — The
ofthesurfacechargedensity,en . Asaresult,theelectric roots of Eq. (9) yield the frequencies of Fermi arc plas-
s
displacement field across the surface exhibits a jump: mons. We find three distinct branches of solutions (la-
beled 1, 2, and 3), as shown in Fig. 1c and Fig. 2. These
∂zφ˜q(cid:12)(cid:12)0+ −κ∂zφ˜q(cid:12)(cid:12)0− =−4πeδn˜s,q. (7) branchesarisefromthehybridizationofconventionaldis-
persionless surface plasmons with the collective excita-
Here we use δn =δnM+δnF, with nχ(r ,ω)=nχ,(0)+ tionsinthetopologicalsurfacestates. Thehybridization
s s s s s s
δnχ(r ,ω), where χ = {M,F}. Superscript (0) denotes iscontrolledbyσ andD,whichcouplethebulkandsur-
s s H
the equilibrium carrier densities. Note that there is no facedensities[seeEqs.(4)and(5)]. Wenotethatbranch
jump in the displacement field in equilibrium. 3 is related to branch 1 via the transformation ω → −ω
To obtain a closed set of equations for the potential andq→−q. However,ateachqtherearethreedistinct
φ˜ (z), we must eliminate δn˜ from Eq. (7). Using solutions,associatedwithmodeswithdifferentvelocities.
q s,q
4
In Fig. 2a we show a contour plot of the dispersion note that retardation effects may alter the jump discon-
relation of the FAP branch labeled 1 in Fig. 1c. Inter- tinuity at very small q via hybridization with the light-
x
estingly, most of these contours are open, and resemble cone[26]. Thesplittingω(1)−ω(1) isproportionaltoσ ,
+ − H
hyperbolae: at large |q|, the open contours asymptoti- andthusprovidesthebasisforadynamicalmeasurement
cally approach the line θ =θ∞(ω˜), with of the anomalous Hall conductivity in WSMs. Interest-
ingly,forsmallbulkdensities,theroleoftheseanomalous
sin2θ∞ =−ω˜(cid:16)1− 1 (cid:17), D˜ = 4πD , (10) Hallcurrentsbecomesdominant,andtheFAPfrequency
cosθ∞ D˜ ω˜2 (κ+1)vs0ωpsulrf becomes insensitive to ωpl: ω+(1) →4πσH/(κ+1).
Inadditiontothehyperbolicitydiscussedabove,FAPs
where q = q(cosθ,sinθ), ω˜ = ω/ωpsulrf, and ωpsulrf = with elliptic constant frequency contours also occur in
(cid:112)
ω κ/(κ+1). Eq. (10) was obtained by taking the small frequency windows near ±ωsurf, see yellow dashed
pl pl
large |q| limit of Eq. (9). Since sin2θ /cosθ is even lines in Fig. 2a. Focusing on the case ω > 0 (branch 1),
∞ ∞
under θ →−θ , there are two asymptotes, see Fig. 2. closed contours exist at frequency ω if there is a finite,
∞ ∞
The FAP velocity on a given branch j, and at a positive value of q that solves Eq. (9) for q =0,q <0.
y x
given wave vector q, dω(j)/dq, is directed perpendic- This condition is realized for ω(1) < ω < ωsurf. Similar
− pl
ular to the corresponding constant frequency contour considerations hold for branch 3.
at q (Fig. 2a, black arrows). Due to their hyperbolic
Discussion — In this work, our aim has been to ex-
character, large wave vector FAPs at a given frequency
posetheuniquephenomenologyofsurfaceplasmonsaris-
in Eq. (10) all propagate along the specific direction
ing from the coupled dynamics of bulk and topological
vˆ = |sinθ |xˆ − sgn[sinθ ]cosθ yˆ, independent of
pl ∞ ∞ ∞
|q| (for |q| <∼ k∗ where our treatment is valid). Conse- surface state carriers in Weyl semimetals. The hyper-
boliccharacterofFAPscanbeprobed,e.g.,viascanning
quently,theseFAPspropagateascollimatedbeams. Fur-
near-field optical microscopy (SNOM) [37, 38]. On the
ther, as allowed by broken TRS, hyperbolic FAP modes
surface of a WSM, the near-field tip may launch a pair
are non-reciprocal: as shown in Fig. 2, the FAP velocity
of collimated beams along the two directions of vˆ con-
is predominantly directed towards +xˆ. pl
sistent with the applied excitation frequency; the propa-
The asymptotic pitch θ of each hyperbolic constant
∞
gation is biased along the direction of the chiral velocity,
frequency contour, and hence the direction vˆ of colli-
pl
v0xˆ. In contrast, in hyperbolic materials with TRS, col-
matedFAPpropagation,iscontrolledbythefrequencyω, s
limated plasmon beams are launched in fours [39, 40],
see Eq. (10). Approaching the conventional surface plas-
mon frequency ωsurf from above, the RHS of Eq. (10) with reflection symmetry in x and y.
pl
approaches zero from below and the linear asymptotes We note that, even without topological surface states,
bend towards −xˆ: θ → ±π. Consequently, the FAP the presence of a bulk anomalous Hall conductivity in
∞
velocity vˆ cants towards yˆ, the direction perpendicu- a magnetic system can itself yield chiral surface or edge
pl
lar to the chiral surface state velocity v0xˆ, see Eq. (4). plasmons [26, 35, 36]. Additionally, FAPs exhibit hyper-
s
Thisbehaviorforω >∼ωpsulrf resultsfromtheresonanthy- bolicity out to large frequencies (eventually limited by
bridization of intra-Fermi-arc oscillations with the con- thetopologicalsurfacestatebandwidth),farbeyondthat
ventional metallic surface plasmon mode. of ωpsulrf (see, e.g., branch 1 in Fig. 1c). Thus, directed
plasmonbeamswithalargebandwidth, characteristicof
A further striking signature of the coupled bulk and
hyperbolic FAPs, provide an optical signature of WSM
surface carrier dynamics of the WSM can be found in
Fermi arc surface states [1].
the FAP dispersion for q = 0 (see Fig. 2b), where the
y
Drude weight D drops out of Eq. (9). Analyzing the Before closing, we briefly outline the material char-
limits q →0± in Eq. (9), we find a discontinuity in the acteristics that favor FAP observation. For example,
x
FAP dispersion (within branch 1) between q →0+ and plasmon excitations with energies in the bulk interband
x
q →0−: particle-holecontinuumaredampedbyelectron-holepair
x
creation. Hence it is desirable to have a small surface
ω(1) =(cid:113)(cid:2)2πσ /(κ+1)(cid:3)2+[ωsurf]2± 2πσH , (11) plasmon frequency ωpsulrf (around which the FAP char-
± H pl (κ+1) acteristics are most pronounced), which is below the in-
terband threshold. Additionally, since ωsurf is directly
pl
where ω(1) = ω(1)(q → 0±,q = 0). A similar splitting determined by total carrier density, whereas the inter-
± x y
also arises for branch 3. However, there is no splitting band pair creation threshold is determined by the Fermi
for branch 2 since ω(2)(q =0,q =0)=0. energy relative to the Weyl nodes, WSMs with low car-
x y
The discontinuity in Eq. (11) arises from bulk anoma- rier densities, but with moderately high Fermi energy,
lous Hall currents directed into or out-of the surface. are favored. In particular, these considerations seem to
Thesecurrentsaredrivenbythein-planeelectricfieldas- favor WSMs with only a few pairs of Weyl nodes.
sociated with FAP waves eiωt−iqxx that propagate along Several candidate TRS breaking WSMs have been re-
+xˆ (q >0)or−xˆ (q <0),respectively[26,35,36]. We cently synthesized [20, 21]; numerous other candidates
x x
5
have been proposed [1–3, 22, 23]. Most recently, mag- [14] X. Huang, et al., Observation of the chiral-anomaly-
neticHeusleralloyshavebeenpredictedtohostonlytwo induced negative magnetoresistance in 3D Weyl
Weyl nodes at the Fermi energy [24]. semimetal TaAs, Phys. Rev. X 5, 031023 (2015).
[15] P. J. W. Moll, et al., Transport evidence for Fermi-
ThetwinhyperbolicandchiralcharacterofFAPsmay arc-mediated chirality transfer in the Dirac semimetal
yield new photonic tools such as an intrinsically non- Cd3As2, Nature (2016).
reciprocal Purcell enhancement of spontaneous emission [16] A.V.Andreev,B.Spivak,Magnetotransportphenomena
for emitters placed close to a WSM surface. Along with related to the chiral anomaly in Weyl semimetals, Phys.
Rev. B 93, 085107 (2016).
the prospects of using FAPs as dynamical probes of the
[17] Y.Baum,etal,.Currentatadistanceandresonanttrans-
peculiarfeaturesofWSMs,theseopportunitiesmakethe
parency in Weyl semimetals, Phys. Rev. X 5, 041046
search for optimal materials for realizing FAPs, and de-
(2015).
tailed material-specific modeling of FAP characteristics, [18] A. A. Burkov, Anomalous Hall effect in Weyl metals,
important directions for future exploration. Phys. Rev. Lett. 113, 187202 (2014).
[19] D. T. Son, N. Yamamoto, Berry curvature, triangle
Acknowledgements — We thank Y. D. Chong, K. anomalies, and the chiral magnetic effect in Fermi liq-
Flensberg, and M. S. Schecter for helpful comments on uids, Phys. Rev. Lett. 109, 181602 (2012).
the manuscript. This work was supported by the Singa- [20] A. B. Sushkov et al., Optical evidence for a Weyl
semimetal state in pyrochlore Eu Ir O , Phys. Rev. B
pore National Research Foundation (NRF) under NRF 2 2 7
92, 241108 (2015).
fellowshipawardNRF-NRFF2016-05(J.C.W.S.),andby
[21] S. Borisenko et al., Time-reversal symmetry breaking
the Villum Foundation (M.S.R).
type II Weyl state in YbMnBi , arXiv: 1507.04847
2
(2015).
[22] T. Guan et al., Evidence for half-metallicity in n-type
HgCr Se , Phys. Rev. Lett. 115, 087002 (2015).
2 4
[23] D. Bulmash, C.-X. Liu, X.-L. Qi, Prediction of a Weyl
[1] X. Wan, A. M. Turner, A. Vishwanath, S. Y. Savrasov, semimetal in Hg1−x−yCdxMnyTe, Phys. Rev. B 89
TopologicalsemimetalandFermiarcsurfacestatesinthe 081106 (2014).
electronic structure of pyrochlore iridates, Phys. Rev. B [24] Z. Wang et al., Time-Reversal-Breaking Weyl Fermions
83, 205101 (2011). inMagneticHeuslerAlloys,Phys.Rev.Lett.117,236401
[2] A.ABurkov,andL.Balents,Weylsemimetalinatopo- (2016).
logicalinsulatormultilayer,Phys.Rev.Lett.107127205 [25] A. Poddubny et al., Hyperbolic metamaterials, Nature
(2011). Photonics 7, 948-957 (2013).
[3] K.-Y. Yang, Y.-M. Lu, Y. Ran, Quantum Hall effects [26] J. Hofmann, S. Das Sarma, Surface plasmon polaritons
in a Weyl semimetal: Possible application in pyrochlore intopologicalWeylsemimetals,Phys.Rev.B93,241402
iridates, Phys. Rev. B 84, 075129 (2011). (2016).
[4] X. Gang et al., Chern semimetal and the quantized [27] Panfilov,I.,A.A.Burkov,D.A.Pesin,Densityresponse
anomalousHalleffectinHgCr Se ,Phys.RevLett.107, in Weyl metals, Phys. Rev. B 89, 245103 (2014).
2 4
186806 (2011). [28] J. Zhou, H.-R. Chang, D. Xiao, Plasmon mode as a de-
[5] B. Q. Lv et al., Experimental discovery of Weyl tection of the chiral anomaly in Weyl semimetals, Phys.
semimetal TaAs, Phys. Rev. X 5, 031013 (2015). Rev. B 91 035114 (2015).
[6] S.-Y. Xu et al., Discovery of a Weyl fermion semimetal [29] B. Rosenstein, H. C. Kao, and M. Lewkowicz, Nonlocal
andtopologicalFermiarcs,Science349,613-617(2015). electrodynamics in Weyl semimetals, arXiv:1508.01604
[7] A. Turner and A. Vishwanath, Beyond Band Insula- (2015).
tors: Topology of Semi-metals and Interacting Phases, [30] Y. Ferreiros and A. Cortijo, Unconventional electromag-
arXiv:1301.0330 (2013). neticmodeinneutralWeylsemimetals,Phys.Rev.B93,
[8] F.D.M.Haldane,BerryCurvatureontheFermiSurface: 195154 (2016).
Anomalous Hall Effect as a Topological Fermi-Liquid [31] O. V. Kotov and Yu. E. Lozovi, Dielectric response and
Property, Phys. Rev. Lett. 93, 206602 (2004). novel electromagnetic modes in three-dimensional Dirac
[9] F. D. M. Haldane, Attachment of Surface “Fermi Arcs” semimetal films, Phys. Rev. B 93, 235417 (2016).
to the Bulk Fermi Surface: “Fermi-Level Plumbing” in [32] See Supplementary Information for a discussion of
Topological Metals, arXiv:1401.0529 (2014). WeylsemimetalswithmultiplepairsofWeylnodes,van-
[10] D.T.Son,B.Spivak,Chiralanomalyandclassicalnega- ishingbulkmetalliccontributiontosurfacevelocityvM,
s
tivemagnetoresistanceofWeylmetals,Phys.Rev.B88, and surface velocity density in the topological surface
104412 (2013). states.
[11] S. A. Parameswaran, T. Grover, D. A. Abanin, D. A. [33] A. Fetter, Edge magnetoplasmons in a two-dimensional
Pesin, and A. Vishwanath, Probing the Chiral Anomaly electron fluid confined to a half-plane. Phys. Rev. B 33,
withNonlocalTransportinThree-DimensionalTopolog- 3717 (1986).
ical Semimetals, Phys. Rev. X 4, 031035 (2014). [34] R. H. Ritchie, Plasma losses by fast electrons in thin
[12] A.C.Potter,I.Kimchi,A.Vishwanath.Quantumoscilla- films, Phys. Rev. 106, 874 (1957).
tionsfromsurfaceFermiarcsinWeylandDiracsemimet- [35] A.A.Zyuzin,V.A.Zyuzin,Chiralelectromagneticwaves
als, Nature Comm. 5, (2014). in Weyl semimetals, Phys. Rev. B 92, 115310 (2015).
[13] J. Xiong, et al., Evidence for the chiral anomaly in the [36] J. C. W. Song, M. S. Rudner, Chiral plasmons with-
Dirac semimetal Na3Bi, Science 350, 413-416 (2015). out magnetic field, Proc. Natl. Ac. Sci. 113, 4658-4663
6
(2016). carriers.
[37] J. Chen et al., Optical nano-imaging of gate-tunable
graphene plasmons, Nature 487, 77-81 (2012).
[38] Z. Fei et al., Gate-tuning of graphene plasmons revealed Velocity density of topological surface states
by infrared nano-imaging, Nature 487, 82-85 (2012).
[39] J. S. Gomez-Diaz, M. Tymchenko, A. Alu, Hyperbolic
A. Chiral velocity sign, and bulk-edge correspondence
plasmonsandtopologicaltransitionsoveruniaxialmeta-
surfaces, Phys Rev. Lett. 114, 233901 (2015).
[40] A. Nemilentsau, T. Low, G. Hanson, Anisotropic 2D Here we resolve the sign of the chiral velocity v0,
s
materials for tunable hyperbolic plasmonics, Phys. Rev. Eq. (4) of the main text, associated with the topolog-
Lett. 116, 066804 (2016). ical surface states. Consider a potential V that con-
[41] In order to illustrate the properties of Fermi arc plas-
fines WSM carriers to the region z < 0. This confining
mons, we have used parameters σ˜ =2.0, and D˜ =6.0.
H potential produces a force in the −zˆ direction, so that
Inarrivingattheseballparkvalues,weusedκ∼10[20],
e(−∂ V)∝−zˆ. Next we note that besides confining the
and σ = e2(2k )/h with 2k ∼ 0.02 − 0.1A−1. We z
H ∗ ∗
also adopted a simple model for D ∼ n¯e2/m, where n¯ carriers, V may also give rise to an anomalous Hall cur-
is a typical surface density between the Weyl nodes; we rent σxz(−∂zV). Due to the bulk-edge correspondence,
used m ∼ E /v¯2, with E the Fermi energy. Taking theundergapanomalousHallcurrentsandedgecurrents
F F
ωpsulrf ∼ 20 − 60meV, and a typical y-velocity on the from topological surface states move in the same direc-
surface v¯ ∼ 108cms−1, we arrive at order of magnitude tion. As a result, the surface velocity density satisfies
estimates of σ˜ ∼2−15, and D˜ ∼5−50.
H
[42] Here we neglect the role of incidental, trivial surface sgn(v0)=sgn(−σ /e2)=sgn(σ ). (12)
s xz H
states that may be present for certain surface termina-
tions.
Whyisthereaminussigninthemiddleexpression? The
[43] While topologically trivial surface states may also exist,
confining electric field points towards −(1/e)zˆ, giving an
theydonotgiverisetothepeculiarFAPphenomenology
electric current in the direction sgn(−σ /e)xˆ. Further,
that we unveil. xz
[44] A natural cut-off for the topological surface state band- the velocity and electrical current are related by another
width, and hence for n0, can be estimated from the en- factorofe,givingthefactorof1/e2. Hencethesignofv0
s s
ergy of the Lifshitz point where the Weyl cones touch. is independent of the sign of the carrier charge, e. In the
[45] At low temperatures, 1/τsurf → 0, while the oscillation last expression we used σH =σzx =−σxz. Note that on
frequency ω remains finite.
the opposite surface the confining force would point in
[46] Note that only the symmetric part of σ contributes to
the +zˆ direction, yielding an opposite sign for the chiral
the divergence.
velocity on that surface: sgn(v0,opp)=sgn(−σ ).
s H
SUPPLEMENTARY INFORMATION B. Drude model for Fermi arc yˆ-direction velocity density
Vanishing bulk metallic contribution to surface The topological surface states of the WSM are char-
velocity acterized by a chiral velocity, oriented in the xˆ direction
(i.e., the direction perpendicular to the k-space line con-
nectingthebulkWeylnodes). Additionally,forageneric
Inanordinarymetalwithnotopologicalsurfacestates,
an excess surface density nM(ω) may accumulate when topological surface state dispersion, a non-equilibrium
s
Fermi arc carrier distribution may carry a velocity den-
the metal is pushed out of equilibrium. This surface
charge layer may host a surface current density evM = sity in the yˆ-direction, parallel to the line connecting the
s
σM(−∇φ), where σM is the surface conductivity associ- bulkWeylnodes. Theprecisefeaturesofthesey-currents
s s
dependonthedetailsofthetopologicalsurfacestatedis-
ated with the bulk metallic carriers accumulated on the
persion, and the scattering mechanisms on the surface.
surface. Taking a simple Drude model for the longitu-
dinal surface conductivity, we write σM =nMe2/(miω). However, we note that the carriers may move in both
s s
the +yˆ and −yˆ directions, and at equilibrium (for zero
Here m is the effective mass of the carriers. Writing
nM = nM,(0) + δnM and noting that, in equilibrium, electricfield),thevelocitydensityintheyˆ-directionvan-
nsM,(0) =s 0, we finsd σM ∝ δnM. Similarly, because ishes, vs(0,y) = 0. Hence, we use a simple phenomenolog-
φ(sr,ω)=e(cid:82) U(r,r(cid:48))δn(sr(cid:48),ω)dr(cid:48)issgeneratedbytheplas- ical Drude model to capture the y-currents induced by
in-plane electric fields:
mon’s density inhomogeneity, it also depends directly on
δn. (Here U(r,r(cid:48)) is the Coulomb kernel.) As a result, D
the surface velocity vsM supported by bulk carriers accu- evys =σysyEy = iω+γEy, (13)
mulated at the surface goes as (δn)2. Hence, to linear
order in δn, the surface current v , Eq. (4) of the main where E =−∂ φ is the electric field in the yˆ direction,
s y y
text, is supported solely by the Fermi arc surface state and γ is the transport scattering rate along the surface.
7
Inthecollisionlesslimit,ω (cid:29)γ,theconductivityreduces that the longitudinal conductivity σ˜ is the same for all
xx
to σs →D/iω. Weyl nodes. Here σ˜ is the contribution to the total
yy xx
We note that, unlike the response to y-fields described longitudinalconductivitycomingfromasingleWeylnode
above, the chiral surface states do not possess a Drude pair, i. Finally, the associated Fermi arcs for each pair
type response to x-directed electric fields. While the ap- ofWeylnodespossessasurfacevelocitydensitygivenby
plication of an electric field may impart momentum to
tehnesucraersritehrasta,ltoongleaxˆd,itnhgeolrindeearr(,acthfiirxaelddidsepnesristiyo)n,athloenregixˆs vs,i(rs)=(cid:104)vs(0,i)nFs,i(rs)(cid:105)d(cid:98)⊥i −(cid:104)Diω0(d(cid:98)(cid:107)i·∇)φ(rs)(cid:105)d(cid:98)(cid:107)i. (14)
no change to the x-component of velocity density. How-
ever, as described in the main text, an x-directed elec- Here d(cid:98)(cid:107) and d(cid:98)⊥ are unit vectors (in the x-y plane) that
i i
tric field causes the topological surface state density to describe the directions parallel and perpendicular to the
change due to impinging currents brought about via the vector connecting the i-th pair of Weyl nodes, respec-
bulk Hall conductivity. Thus the velocity density vs on tively, with d(cid:98)⊥i =d(cid:98)(cid:107)i ×zˆ. D0 describes the Drude weight
the surface responds to electric fields in the xˆ and yˆ di- for a single Fermi-arc, i.
rections very differently, as captured by Eq. (4) of the Similar to Eq. (8) of the main text, the total accu-
main text. mulated surface density can be written in terms of the
electric potential, δn˜ =(cid:80) Gχφ˜<, with
s i,χ i
FAPs in WSMs with multiple Weyl node pairs
GM =−qσ˜xx, GF = σH,i(d(cid:98)⊥i ·q)+ Dω0(d(cid:98)(cid:107)i ·q)2.
FermiarcsalsoexistinWSMswithmultiplebulkWeyl i ieω i e(ω−v(0)d(cid:98)⊥·q)
s,i i
node pairs, labeled by an index i = 1,2,.... Extending (15)
the two-fluid model used in the main text, we can asso- The collective modes (surface plasmons) are found by
ciate bulk and surface densities n and nχ with each seekingthecombinationsofωandqsuchthatthebound-
b,i s,i
(whereχ={M,F}labelsthebulkfreecarrierandtopo- ary conditions of continuous φ˜ and of the jump in the
q
logical surface state contributions to the surface density, electric displacement field can be satisfied, as described
respectively). In the same limit ωτ (cid:29)1 considered in above Eq. (9) in the main text. Thus we must solve the
surf
themaintext, thefieldsnF andnM obeydistinctequa- generalized secular equation
s,i s,i
tions of motion, similar to Eq. (5). Surface plasmons
(cid:88)
arising from the collective dynamics of these carrier den- −2q+4πe Gχ =0. (16)
i
sities can be obtained in the same fashion as detailed in
i,χ
the main text, with all components coupled through the
common electrical potential φ. For simplicity, through- To derive the dispersion relation, analogous to Eq. (9),
out this section we take κ=1. we first write 4πσ˜ /κ = ω2/(2iω), where ω is the
xx pl pl
To illustrate this approach, we confine ourselves to a bulk plasmon frequency of the WSM. Here, the factor
system with two pairs of Weyl nodes, i = 1,2. We take of 2 in the denominator on the right hand side reflects
all four Weyl nodes to be situated in the k -k plane, at the fact that, in this example, the two Weyl node pairs
x y
±d(cid:107)(i.e.,d(cid:107)·zˆ=0). EachpairofWeylnodescontributes eachcontributehalfthetotaldensityofbulkfreecarriers.
abuilkHalliconductivityσ withsign(andorientation) Collecting terms with factors (ω−v(0)d(cid:98)⊥ ·q) and (2−
H,i s,i i
consistentwiththatdescribedinEq.(12);wewillassume ω2/ω2), we obtain:
pl
(cid:104) (ωsurf)2(cid:105)
2(ω−vs(0,1)d(cid:98)⊥1 ·q)(ω−vs(0,2)d(cid:98)⊥2 ·q) 1− pωl2 +M1+M2 =0, (17)
√
where ωsurf =ω / 2 (taking κ=1), and the hybridization amplitudes are given by
pl pl
(cid:104) 4πD |q| (cid:105)
M1 = (ω−vs(0,2)d(cid:98)⊥2 ·q) −4πσH,1(qˆ·d(cid:98)⊥1)− ω0 (qˆ·d(cid:98)(cid:107)1)2 ,
(cid:104) 4πD |q| (cid:105)
M2 = (ω−vs(0,1)d(cid:98)⊥1 ·q) −4πσH,2(qˆ·d(cid:98)⊥2)− ω0 (qˆ·d(cid:98)(cid:107)2)2 . (18)
We note that for non-vanishing M1,2, the collective oscillations in the chiral branches (ω − vs(0,i)d(cid:98)⊥i · q) = 0 and
the conventional surface plasmon mode described by (1−(ωsurf)2/ω2) = 0 will hybridize, giving rise to FAPs. The
pl
8
structure of the resulting plasmon bands will depend on the magnitudes of the hybridization amplitudes, as well as
the positions and orientations of the pairs of Weyl nodes (as encoded in the {d }).
i
Seeking hyperbolicity, we note that, at large q, the terms going as O(q2) dominate Eq. (17). Using this large q
limit, we determine the asymptotic contours via:
2(v(0)d(cid:98)⊥·q)(v(0)d(cid:98)⊥·q)(cid:104)1− (ωpsulrf)2(cid:105)=−4πD0|q|(cid:2)v(0)d(cid:98)⊥·q(qˆ·d(cid:98)(cid:107))2+v(0)d(cid:98)⊥·q(qˆ·d(cid:98)(cid:107))2(cid:3). (19)
s,1 1 s,2 2 ω2 ω s,2 2 1 s,1 1 2
As a sanity check, we note that for a WSM with two contours that asymptotically approach the lines θ =
pairs of Weyl nodes that point in the same direction, θ (ω˜) at large q, with
∞
d(cid:107) =d(cid:107) (and v(0) =v(0) =v ), Eq. (19) yields the same
1 2 s,1 s,2 0
hyperbolae as described in the main text (setting κ = 1
cos3θ −sin3θ ω˜ (cid:104) 1 (cid:105)
and recalling D0 =D/2). ∞ ∞ = 1− , (21)
We now illustrate the contours of FAPs in a situation cosθ∞sinθ∞ D˜0 ω˜2
with multiple (2) pairs of Weyl nodes, with d(cid:107) and d(cid:107)
1 2
pointing in different directions. For demonstration, we
take a simple model with d(cid:107) along xˆ and d(cid:107) along yˆ: where ω˜ = ω/ωpsulrf. Here D˜0 = 2πD0/(vωpsulrf), as in the
1 2 main text (with κ = 1). A complex pattern of asymp-
d(cid:107) =xˆ, d(cid:107) =yˆ, totic contour lines θ = θ∞(ω˜) arises from the hybridiza-
1 2 tion of ordinary surface plasmons modes with collective
d⊥ =d(cid:107)×zˆ=−yˆ, d⊥ =d(cid:107)×zˆ=xˆ, (20) oscillations in the Fermi arc surface states, and from hy-
1 1 2 2
bridization between the collective modes of each of the
and v(0) = v(0) = v. Substituting the relations in separate branches of Fermi arc surface states (associated
s,1 s,2
Eq. (20) into Eq. (19), we find open constant frequency with the Weyl node pairs i=1,2).