Thermal robustness of the quantum spin Hall phase in monolayer WTe 2 Antimo Marrazzo1 1Dipartimento di Fisica Universit a di Trieste Strada Costiera 11 I-34151 Trieste Italy

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Thermal robustness of the quantum spin Hall phase in monolayer WTe2

Antimo Marrazzo1, ∗

1Dipartimento di Fisica, Universit`a di Trieste, Strada Costiera 11, I-34151 Trieste, Italy

(Dated: February 13, 2023)

Monolayer 1T’-WTe2has been the ﬁrst two-dimensional crystal where a quantum spin Hall phase

was experimentally observed. In addition, recent experiments and theoretical modeling reported

the presence of a robust excitonic insulating phase. While ﬁrst-principles calculations with hybrid

functionals and several measurements at low temperatures suggest the presence of a band gap of

the order of 50 meV, experiments could conﬁrm the presence of the helical edge states only up

to 100 K. Here, we study with ﬁrst-principle simulations the temperature eﬀects on the electronic

structure of monolayer 1T’-WTe2and consider the contributions of both thermal expansion and

electron-phonon coupling. First, we show that thermal expansion is weak but tends to increase

the indirect band gap. Then, we calculate the eﬀect of electron-phonon coupling on the band

structure with non-perturbative methods and observe a small reduction of the band inversion with

increasing temperature. Notably, the topological phase and the presence of a ﬁnite gap are found

to be particularly robust to thermal eﬀects up to and above room temperature.

In 2014 [1], Qian et al. predicted through ﬁrst-

principles simulations that two-dimensional (2D) tran-

sition metal dichalcogenides (TMDs) in the 1T’ phase

would exhibit a sharp quantum spin Hall eﬀect (QSHE),

characterized by strong band inversions and relatively

large band gaps. These TMDs are deﬁned by the chemi-

cal formula MX2, where M is the transition metal (W or

Mo) and X is the chalcogenide (Te, Se or S), and they

exhibit a variety of polytypic structures such as 1H, 1T,

and 1T’. In particular, the 1T structure in MX2TMDs is

typically unstable in freestanding conditions and under-

goes a spontaneous lattice distortion in the xdirection

to form a period-doubling 2 ×1 distorted superstructure,

the 1T’ structure, made of 1D zigzag chains along the

ydirection [2]. In all the TMDs, but WTe2[3], the 1T’

structure is dynamically stable but it does not correspond

to the lowest energy polymorph [1] and it is a metastable

phase, although it can be stabilized under appropriate

conditions [2] thanks to the large energy barrier between

the 1H and the 1T’ phases [1]. In WTe2instead, the 1T’

structure corresponds to the most stable phase [2,3].

Conductance experiments have conﬁrmed the presence

of a quantum spin Hall insulating (QSHI) state in 1T’-

WTe2until 100 K [4,5].

As QSHI material, 1T’-WTe2excels in many aspects.

First, it exhibits a strong band inversion [1] of about

1 eV and a band gap above kBTR[6], where TRis

room temperature. In addition, WTe2is a simple binary

compound, which simpliﬁes the experimental synthesis–

especially with bottom-up approaches–compared to the

numerous ternary QSHI compounds [7]. Finally, WTe2

has a layered crystal structure and the binding energy be-

tween the layers is relatively low [8] (around 30 meV/˚

A2),

such that the material is exfoliable into monolayers [9].

It is then compelling that monolayers of 1T’-WTe2have

become among the most promising 2D materials to re-

∗antimo.marrazzo@units.it

alize the QSHE and the target of several experimental

investigations [4,5,10–13].

Although predictions report an extremely robust band

inversion and indirect band gap above room temperature,

experiments could ﬁnd signatures of the QSHI phase un-

til 100 K only, suggesting a possible role of temperature

on the band structure. Here we show that the topologi-

cal phase and the presence of a ﬁnite gap are particularly

robust to thermal eﬀects up to high temperatures, sug-

gesting that extrinsic eﬀects or thermally activated bulk

conductance [14] might be responsible for the measured

transition temperature.

The presence of a ﬁnite indirect band gap in mono-

layer WTe2has been debated. Density-functional theory

(DFT) simulations [1,10] with the semilocal PBE func-

tional predict a metallic state, with bands overlapping

at the Fermi level and a direct gap throughout the Bril-

louin zone (BZ). On the contrary, calculations [6,15,16]

with the HSE hybrid functional predict an indirect gap

of about 50 meV. The predicted band gap is also strongly

sensitive to the lattice constant [6,11,17], where even a

small amount of strain can open a gap as predicted at the

DFT level with diﬀerent functionals such as LDA, PBE

and PBEsol [17] and conﬁrmed experimentally [11]. In

this work, we calculate the PBE and HSE band structure

both with spin-orbit coupling (SOC) on the equilibrium

structure relaxed at the DFT-PBE level, as reported in

Fig. 1. The Wannier-interpolated HSE band structure

has a ﬁnite indirect gap of 62 meV, while at the PBE

level the system is metallic.

Angle-resolved photoemission spectroscopy [6]

(ARPES) and scanning-tunneling microscopy and

spectroscopy [10] (STM/STS) experiments indicate the

presence of a ﬁnite gap of around 50 meV, in agreement

with the hybrid-functional predictions. Some recent

STM/STS experiments instead suggested the presence of

a metallic band structure with a Coulomb gap induced

by disorder [18]. Finally, more recent transport [13] and

STM/STS [12] experiments, backed-up by many-body

calculations [13,19,20], provided strong evidence of the

arXiv:2210.11258v2 [cond-mat.mes-hall] 10 Feb 2023

Y°X

°0.50

°0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

Energy [eV]

HSE with SOC

Y Γ X

−1.0

−0.5

0.0

0.5

1.0

1.5

Energy (eV)

PBE with SOC

HSE with SOC

FIG. 1. Left panel (a): top and lateral views of monolayer 1T’-WTe2, tungsten (tellurium) atoms are marked in light blue

(orange) and the primitive call is drawn. Right panel (b): band structure for monolayer 1T’-WTe2with spin-orbit coupling

(SOC) calculated at the DFT-PBE (black) and HSE (red) level. The energy zero is at top of the valence band, circles denote

direct calculations while solid lines represent Wannier-interpolated states.

presence of an excitonic insulating state. In addition,

Ref. [12] explicitly ruled out the scenario of a band

insulator, even in the presence of strain.

The band structure of topological insulators is char-

acterized by two distinct quantities, the indirect band

gap and the band inversion [21,22]. The presence

of a ﬁnite gap guarantees the insulating nature of the

bulk and ensures that, in an undoped ﬁnite crystal-

lite, electronic transport can occur only through the

topologically-protected metallic edge states. In all QSHI,

the band gap is entirely driven by SOC and it is typically

very weak, on the order of tens to hundreds of meV [7].

However, the strength of the topological phase is more

connected to the size of the band inversion, which also de-

termines the localization of the edge states [23]. Follow-

ing Refs. [10,24], we deﬁne the band inversion [21,22] for

WTe2as the energy diﬀerence at Γ (the high-symmetry

point where the band inversion occurs) between the low-

est unoccupied band d+

yz , formed by a combination of

tungsten dyz orbitals with positive overall parity, and

the occupied d−

z2state (with negative parity), the latter

sits at about 0.5 eV below the valence band maximum

(VBM). The d−dband inversion in WTe2is driven by

the 1T’ distortion and it is present also in absence of

SOC [10,24].

Temperature is known to aﬀect the band structure

of semiconductors [25], through the combined eﬀects of

thermal expansion (TE) and electron-phonon coupling

(EPC). Both eﬀects can occur also at zero temperature

owing to the presence of zero-point motion, that can in-

ﬂuence both the equilibrium lattice constant and renor-

malize the band gap. The inclusion of these eﬀects can

lead to rather diﬀerent predictions with respect to cal-

culations performed on the average atomic conﬁguration

at zero temperature; for instance zero-point renormaliza-

tion (ZPR) alone has been estimated to modify the band

gap of diamond by 0.6 eV [26]. It is then compelling to

assess how TE and EPC aﬀect topological phases, and

in particular the persistence of the small indirect gap.

In fact, typical predictions assume negligible tempera-

ture eﬀects and deduce critical temperatures based on

the zero-temperature structure with averaged atomic po-

sitions. However, temperature-dependent studies of the

electronic structure of topological materials can be com-

putationally rather challenging, especially because they

involve the calculation of EPC by considering both SOC

and, as in the case of WTe2, beyond-DFT methods like

hybrid functionals or many-body perturbation theory at

the GW level. Hence, fully ﬁrst-principles studies of tem-

perature eﬀects in topological insulators are scarce and

limited to 3D structures [27–33]. Overall, the results of

these recent studies [27–33] provide a general picture of

temperature eﬀects in 3D topological insulators, where

there seems to be no general trend for the sign of the

correction to the band gap: temperature can promote or

suppress the topological phase, depending on the very

details of the system.

As mentioned above, in WTe2the electronic bands

around the Fermi level and, in particular, the presence

of an indirect gap, are very sensitive to the lattice con-

stant [6,11,17], suggesting a possible important eﬀect

of TE on the band structure at ﬁnite temperature. In

absence of an applied external pressure, we can calculate

the equilibrium structure at any temperature T by min-

imizing the Helmholtz free energy F({ai}, T ) = U−T S

with respect to all the independent geometrical degrees of

freedom {ai}. Monolayer WTe2is orthorhombic with a

rectangular unit cell, hence we set {ai}= (a, b), where we

name athe shortest lattice parameter and bthe longest

one, the latter corresponds to the period-doubling di-

rection y. We adopt the quasi-harmonic approximation

(QHA), where the vibrational free energy F is written

as in the case of a perfectly harmonic crystal but where

anharmonic eﬀects are included through the dependence

of the phonon frequencies on the lattice parameters. The

QHA free energy can be then calculated as

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ThermalrobustnessofthequantumspinHallphaseinmonolayerWTe2AntimoMarrazzo1,1DipartimentodiFisica,UniversitadiTrieste,StradaCostiera11,I-34151Trieste,Italy(Dated:February13,2023)Monolayer1T'-WTe2hasbeenthersttwo-dimensionalcrystalwhereaquantumspinHallphasewasexperimentallyobserved.Inaddition,recente...

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Thermal robustness of the quantum spin Hall phase in monolayer WTe 2 Antimo Marrazzo1 1Dipartimento di Fisica Universit a di Trieste Strada Costiera 11 I-34151 Trieste Italy

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