Gate-tunable Lifshitz transition of Fermi arcs and its nonlocal transport signatures Yue Zheng1Wei Chen1Xiangang Wan1and D. Y. Xing1 1National Laboratory of Solid State Microstructures School of Physics

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Gate-tunable Lifshitz transition of Fermi arcs and its nonlocal transport signatures

Yue Zheng,1Wei Chen,1, ∗Xiangang Wan,1and D. Y. Xing1

1National Laboratory of Solid State Microstructures, School of Physics,

and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

One hallmark of the Weyl semimetal is the emergence of Fermi arcs (FAs) in the surface Brillouin

zone that connect the projected Weyl nodes of opposite chirality. The unclosed FAs can give rise

to various exotic eﬀects that have attracted tremendous research interest. The conﬁgurations of the

FAs are usually thought to be determined fully by the band topology of the bulk states, which seems

impossible to manipulate. Here, we show that the FAs can be simply modiﬁed by a surface gate

voltage. Because the penetration length of the surface states depends on the in-plane momentum, a

surface gate voltage induces an eﬀective energy dispersion. As a result, a continuous deformation of

the surface band can be implemented by tuning the surface gate voltage. In particular, as the saddle

point of the surface band meets the Fermi energy, the topological Lifshitz transition takes place for

the FAs, during which the Weyl nodes switch their partners connected by the FAs. Accordingly, the

magnetic Weyl orbits composed of the FAs on opposite surfaces and chiral Landau bands inside the

bulk change its conﬁgurations. We show that such an eﬀect can be probed by the nonlocal transport

measurements in a magnetic ﬁeld, in which the switch on and oﬀ of the nonlocal conductance by the

surface gate voltage signals the Lifshitz transition. Our work opens a new route for manipulating

the FAs by surface gates and exploring novel transport phenomena associated with the topological

Lifshitz transition.

I. INTRODUCTION

In the last two decades, the research on novel topo-

logical materials has seen rapid progress, involving the

discoveries of the topological insulators1–3 and topolog-

ical semimetals4–10. The latter possess gapless energy

spectra but nontrivial band topology, which can give rise

to interesting eﬀects stemming from both the bulk and

surface states. According to the features of the band

degeneracies, topological semimetals can be further clas-

siﬁed into several types, such as Weyl semimetal (WSM),

Dirac semimetal4–6 and nodal-line semimetal10. As the

counterparts of the massless Weyl and Dirac fermions in

condensed matter physics, the quasiparticles with linear

dispersion in the WSM and Dirac semimetals provide an

interesting platform for exploring novel eﬀects predicted

by high-energy physics11–26. These eﬀects are manifested

as anomalous transport and optical properties which can

be probed using a standard approach of condensed mat-

ter physics27–37.

The nontrivial band topology of the WSMs is embod-

ied in the monopole charge (or Chern number of the

Berry curvature ﬁeld) carried by the Weyl nodes. Ac-

cording to the no-go theorem38,39, the Weyl nodes of op-

posite chirality must appear in pairs. The manifestation

of the nontrivial band topology of the WSM is the un-

closed Fermi arcs (FAs) spanning between Weyl nodes

of opposite chirality projected into the surface Brillouin

zone. The emergence of the FAs is a unique property

of the WSMs, without any counterpart in high-energy

physics, which can not only serve as the hallmark of the

WSMs11–21,23,24, but also lead to several novel phenom-

ena4,5,40–46. Given that the FAs are the Fermi surface of

the topological surface states, one might think that all

its properties, especially how they connect pairs of Weyl

nodes are completely determined by the band topology

of the bulk states through the bulk-boundary correspon-

dence. Therefore, it seems that the only way to modify

the conﬁgurations of the FAs is to change the bulk prop-

erties of the WSMs.

Interestingly, recent research progress shows that the

conﬁgurations of the FAs are quite sensitive to the details

of the sample boundary47–49, which opens the possibility

for manipulating the FAs through surface modiﬁcations.

In particular, the topological Lifshitz transition50 of the

FAs can be induced by surface decoration48 or chemi-

cal potential modiﬁcation49, which changes the sizes and

shapes of the FAs, and especially, the way they connect

pairs of Weyl nodes. The existing experiments show that

FAs with diﬀerent conﬁgurations can be realized in dif-

ferent samples47–49,51–54, but whether it is possible to

continuously modify the FAs in a given sample remains

an open question. It is of great interest to explore the

possibility of manipulating the FAs by external ﬁelds, in

which both continuous deformation and abrupt Lifshitz

transition of the FAs can be achieved.

In this work, we show that the FAs of the WSMs can

be continuously tuned by a surface gate voltage. Because

the penetration length of the surface state depends on

the in-plane momentum, the surface gate voltage acts un-

equally on these surface modes and induces a momentum-

dependent potential energy, or eﬀectively, an additional

dispersion of the surface band52. As a result, a contin-

uous deformation of the surface band and so the FAs

can be achieved by simply tuning the gate voltage. It is

shown that the existence of the saddle point in the surface

band is responsible for the topological Lifshitz transition

of the FAs. Speciﬁcally, the transition takes place when

the saddle point coincides with the Fermi energy. At the

same time, the Weyl nodes switch their partners that are

connected by the FAs. A direct physical result is that

the magnetic Weyl orbits composed of the FAs on oppo-

arXiv:2210.12956v1 [cond-mat.mes-hall] 24 Oct 2022

( 0)Uy

−

| ( )|



/y nm

(a) (b)

WSM

FIG. 1. (a) Schematic of the electrostatic potential imposed

on the top surface of WSM. (b) The dispersion of fkkalong y-

direction, the color of curves is corresponding to solid squares

in (a) for diﬀerent kzchannels, here kx=k1.

site surfaces and the chiral Landau bands inside the bulk

change their conﬁgurations. We show that such an eﬀect

can be probed by the nonlocal transport measurements

in a magnetic ﬁeld, in which the nonlocal current can be

switched on and oﬀ by the gate voltage, thus providing a

clear signature of the gate-voltage induced Lifshitz tran-

sition. In addition to the study on the eﬀective model,

we also calculate the surface electrostatic potential in-

duced by the gate voltage in a speciﬁc WSM ZrTe using

the ﬁrst-principles calculations. It shows that the gate

voltage can induce large potential energy which is suﬃ-

cient for driving the Lifshitz transition of FAs. Our work

not only uncovers the scenario of the Lifshitz transition

of the FAs but also paves the way for its continuous ma-

nipulation by a surface gate voltage.

The rest of this paper is organized as follows: in Sec. II,

we study the Lifshitz transition of the FAs based on both

the continuous and lattice models. In Sec. III, we study

the transport signatures of the surface gate induced Lif-

shitz transition. In Sec. IV, we discuss the experimental

implementation of our proposal. Finally, we give a brief

summary in Sec. V.

II. LIFSHITZ TRANSITION OF FERMI-ARC

INDUCED BY SURFACE GATE

We start with an eﬀective model of WSM with four

Weyl nodes52

H(k) = M1(k2

1−k2

x)σx+vykyσy+M2(k2

0−k2

y−k2

z)σz,

(1)

here vyis the velocity in the ydirection, k0,1and M1,2are

parameters, σx,y,z are the Pauli matrices in the pseudo-

spin space. The two bands are degenerate at four Weyl

nodes kW= (±k1,0,±k0) with two FAs spanning be-

tween them respectively [cf. Fig. 2(a)]. Expanding

H(k) around the Weyl points yields four Weyl equations

h(k) = ∓2M1k1kxσx+vykyσy∓2M2k0kzσz.

We are interested in the FA surface states, which can

be solved under the open boundary condition in the ydi-

rection. Consider a semi-inﬁnite WSM that occupies the

space of y > 0 as shown in Fig. 1(a) and make the sub-

stitution ky→ −i∂yin Eq. (1). For a given kz, Eq. (1)

reduces to a 2D system in the x-yplane. It can be ver-

iﬁed that for |kz|<|k0|, such an eﬀective 2D system

possesses a nonzero Chern number with the chiral edge

state that appears at its boundary. The edge states of

all kzslices comprise the FA surface states. By solving

the eigenequation [H(kx,−i∂y, kz)−ε0]ψ(kx, y, kz) = 0

under the open boundary condition ψ(y= 0) = 0, we ob-

tain the dispersion and wave function of the surface state

ε0=M1(k2

x−k2

1),

ψ(kk) = fkz(y)eikxx+ikzza

b,(2)

with M2>0, fkz(y) = η(eλ1y−eλ2y) the spatial dis-

tribution function, η=2M2(k2

0−k2

1−4M2

2(k2

0−k2

z)the normalization

coeﬃcient, and λ1,2=−1

2M2±q1

4M22+ (k2

z−k2

0).

Here, the key point is that the wave function ψ(kk)

of the surface state relies on the in-plane momentum

kk= (kx, kz). Speciﬁcally, wave function for diﬀerent

kkpossesses unequal spatial spreading in the ydirection

as shown in Fig. 1(b). This property opens a new route

for manipulating FA surface states by a surface potential,

which can be induced by a surface gate voltage. Consider

an electric potential imposed on the surface of the WSM,

which can be captured by Us(y) = Uδ(y−0+). It in-

duces an energy shift of the surface states, which can be

evaluated by the overlap integral

δε =

∞

Uδ(y−0+)|fkk(y)|2dy. (3)

The energy shift has a kzdependence and can be ex-

panded as δε '(−Ak2

z+B)Uto the second order of kz

with two constants A, B > 0 for M1, M2>0. Physically,

such potential energy induces an eﬀective surface disper-

sion and so the full energy of the surface states becomes

˜ε(kk) = ε0+δε 'M1(k2

x−k2

1)+(−Ak2

z+B)U. (4)

It means that the band shape of the surface states and

then the FAs can be continuously tuned by the surface

electric potential.

The interesting case occurs for U > 0, such that

the coeﬃcients before k2

xand k2

zhave opposite signs.

This means that the surface band bends towards oppo-

site directions for kxand kzand becomes a hyperbolic

paraboloid; see Figs. 2(e,f). The hyperbolic paraboloid

surface band contains a saddle point, which is the key to

understand the Lifshitz transition of the FAs. Setting the

Fermi energy to zero, the FAs deﬁned by the intersection

curves between the surface band and Fermi surface un-

dergo continuous modiﬁcations as the surface potential

Uincreases. The critical point for the Lifshitz transition

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Gate-tunableLifshitztransitionofFermiarcsanditsnonlocaltransportsignaturesYueZheng,1WeiChen,1,XiangangWan,1andD.Y.Xing11NationalLaboratoryofSolidStateMicrostructures,SchoolofPhysics,andCollaborativeInnovationCenterofAdvancedMicrostructures,NanjingUniversity,Nanjing210093,ChinaOnehallmarkoftheWeylse...

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Gate-tunable Lifshitz transition of Fermi arcs and its nonlocal transport signatures Yue Zheng1Wei Chen1Xiangang Wan1and D. Y. Xing1 1National Laboratory of Solid State Microstructures School of Physics.pdf

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Gate-tunable Lifshitz transition of Fermi arcs and its nonlocal transport signatures Yue Zheng1Wei Chen1Xiangang Wan1and D. Y. Xing1 1National Laboratory of Solid State Microstructures School of Physics

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