Table Of ContentPhonon-induced topological transitions and crossovers in Dirac materials
Ion Garate
Department of Physics, Yale University, New Haven, CT 06520, USA
(Dated: September24, 2012)
Weshowthatelectron-phononinteractionscanalterthetopologicalpropertiesofDiracinsulators
and semimetals, both at zero and nonzero temperature. Contrary to the common belief that in-
creasingtemperaturealwaysdestabilizestopologicalphases,ourresultshighlightinstancesinwhich
phononsmay lead totheappearance of topological surface states above acrossover temperature in
a material that has a topologically trivial ground state.
2
1 PACSnumbers:
0
2
Introduction.— Recent years have witnessed the un- crease the bandgap of ordinary semiconductors mono-
p
expected discovery of novel topological phases in cer- tonically as a function of temperature [16]. However, it
e
S tain electrically insulating [1] and semimetallic [2] solids has not been recognized that, at least in Dirac materi-
whose low-energy bulk excitations are Dirac fermions. als, phonon-induced renormalizationof the bandgap can
1
Topological phases in these “Dirac materials” are char- culminate in a band inversion both at zero and nonzero
2
acterized by nonzero integers (topological invariants), temperature. In insulators with non-topological ground
] whichmanifestthemselvesthroughpeculiarandrobustly states, a band inversion arising at nonzero temperature
l
e gaplessstateslocalizedattheboundariesofthematerial. implies the emergence of peculiar gapless surface states
-
Partlyenticedbyavisionoftransistorsthatwouldoper- above that temperature. This may be regarded as a
r
t atebyswitchingtopologicalinvariantsonandoff,thereis “topologicalthermalcrossover”,whichhasnotbeenpre-
s
. keeninterestinfindingwaystoinducetopologicalphases viously discussed.
t
a in intrinsically non-topologicalmaterials. Electron-phonon self-energy.– Consider electrons in a
m
In centrosymmetric crystals, quantum phase transi- perfectperiodiclattice withcrystalHamiltonianh(0)(k),
d- tions between topological and non-topological insulators whose energy eigenvalues Ekα and eigenstates |Ψkαi ≡
n occur when the sign of the Dirac fermions’ mass gets in- exp(ik·r)|kαicharacterizetheelectronicstructure. Here
o verted at an odd number of time-reversal-invariant mo- r is the positionvector, k is the crystalmomentum, α=
c menta (TRIM) in the bulk Brillouin zone. In equilib- 1,...,N is the band index and |kαi is a N-component
[
rium, such “band inversions” can result from applying eigenspinor. The change in the electronic structure due
1 pressure [3] or dc electric fields [4], from changing the to lattice vibrations is approximately captured by the
v compound stoichiometry [5], or from strong alloying [6]. electron-phonon self-energy [17, 18]
5
8 Aeffweactyivferloympreoqduuilciebraiubma,ndacinevleercstiroonm[a7g],naetnidc efinegldinsemeraedy Σαα′(iωn,k)= gq2hkα|k−qβihk−qβ|kα′i
6 Xq,β
4 coupling to dissipative baths presumably induces topo-
. logical phases in driven cold atomic systems [8]. The 1+nq−fk−qβ nq+fk−qβ
9 × + , (1)
0 latter two cases illustrate how the topological invariant (cid:20)iωn−ξk−qβ +ωq iωn−ξk−qβ −ωq(cid:21)
2 foranelectronsystemmaybealteredaftercouplingitto where ω ≡ (2n+1)πT (n ∈ Z) is the Matsubara fre-
n
1 a non-electronicenvironment,andmotivate the question
quency, T is the temperature, ξ = E − ǫ , ǫ is
: kα kα F F
v of whether analogous environment-induced effects could the Fermi energy, ω is the phonon dispersion, n =
q q
Xi occur in equilibrium. [exp(ωq/T) − 1]−1 is the phonon occupation number,
Incrystalsthatareinthermodynamicequilibrium,lat- f =[exp(ξ /T)+1]−1isthefermionoccupationnum-
r kα kα
a tice vibrations form the most ubiquitous bath that cou- ber, and g is the electron-phonon coupling. For con-
q
plestoelectrons. Althoughelectron-phononcouplingcan creteness and simplicity we concentrate on the deforma-
hardlybe engineeredinagivenmaterial,its influence on tion coupling to longitudinal acoustic phonons [17], so
electronic properties is strongly temperature-dependent. that g2 = ~C2q/(2ρa3c ) and ω = c q, where C is the
q s q s
Even as a recent formal study suggests that topological deformation potential, a is the lattice constant, ρ is the
invariants are not robust under dephasing [9], a growing atomic mass density and c is the sound velocity.
s
body of work[10–15] treats phonons as mere actors that Inpresenceoflatticevibrations,theelectronicGreen’s
do not alter the topology of the underlying electronic function G obeys G−αα1′(iωn,k) = (iωn−ξkα)δαα′ −
stage. The objective of our paper is to demonstrate that Σαα′(iωn,k), where δ is Kronecker’s delta. The off-
(i) phonons canmodify topologicalproperties of anelec- diagonal matrix elements of the self-energy are gener-
tronic structure,and(ii) temperaturecanbe usedasthe ally nonzero and lead to avoided band crossings ex-
“agent” that drives those changes. cept at certain (e.g. topologically protected) degener-
Electron-phonon interactions are known to often de- acy points. Upon diagonalization, the Green’s function
2
obeys G−αα1′(iωn,k) = (iωn+ǫF −χα(iωn,k))δαα′, and Adopting the spherical approximation of Eq. (3) near
the solutions of k=0andintroducingaUVmomentumcutoffk =π/a,
c
Eq.(4)canbeevaluatedanalyticallyinthelimitsT ≪ω
∗ ∗ c
Ekα+iΓkα =χα(Ekα−ǫF +iΓkα,k) (2) andT ≫ωc,whereωc ≡cskcistheDebyefrequency. We
obtain
∗
yield the energy dispersion E and broadening |Γ | of
kα kα
dressed quasiparticles at temperature T. Σ =U2f (T), (5)
0,z 0,z
Phonon-induced topological transitions in insulators.–
A minimal lattice model that captures the generic low- where U ≡ gq/(aq)1/2 is a constant in units of energy,
energy physics of 3D Dirac insulators reads [19] and f0,z(T) are functions of electronic structure param-
eters and of temperature, with
h(0)(k)=ǫ (k)1 +d(k)·στx+M(k)1 τz, (3)
0 4 2
f (T ≪ω )≃−y 1−x2ln(1+1/x2) /4+O(T4)
z c
where 1n is an n × n identity matrix, σi and τi are f (T ≫ω )≃−(T(cid:2)/ω )y 1−xtan−1((cid:3)1/x) , (6)
z c c
Pauli matrices in “spin” and “orbital” space (respec-
(cid:2) (cid:3)
tively), ǫ0 = 2γ(3− icos(kia)), di = −2λsin(kia), x ≡ (2/π)λ/(t2 − γ2)1/2 and y ≡ t/(t2 − γ2). Simi-
M(k) = m+2t(3− Picos(kia)), i ∈ {x,y,z}, (γ,λ,t) larly, f0(T)= −(γ/t)fz(T). In the derivation of Eq. (6)
arebandparametersaPndmistheDiracmass. Anticipat- we have neglected ω from the denominators of Eq. (1),
q
ing that |m| . |t| in typical Dirac materials, this model whichisagoodapproximationasconfirmednumerically.
describes an insulator with an energy gap Eg = 2m at This approximation leads to the cancellation of Fermi
k = 0: it is topological if mt < 0 and non-topological factors in Eq. (1) and thus highlights the fact that ω
c
if mt > 0. It contains two degenerate conduction governs the T-dependence of m∗.
bands (Ek1 = Ek2) and two degenerate valence bands From Eq. (5), a topological transition or crossoveroc-
(Ek3 = Ek4 = −Ek1). Since |Ekα| ≫ |m| for k 6= 0 curs at U =Uc(T), where
TRIM,phonon-inducedchangesintopologicalproperties
are realized via band inversions at k =0. Thus we focus U (T)= −m/f (T). (7)
c z
onΣαα′(iωn,0),whichcanberecastedinmatrixformas p
Whether phonons favor topological or non-topological
Σˆ(iω ,0)=Σ (iω )1 +Σ (iω )1 τz, (4) phases depends on the band parameters of the perfect
n 0 n 4 z n 2
crystal. If t2 >γ2, sgn(f )=−sgn(t) and phonons tend
z
with Σ (iω ) ≡ (Σ (iω ,0) + Σ (iω ,0))/2 and to drive the system into the topological insulator phase.
0 n 11 n 33 n
Σ (iω )≡(Σ (iω ,0)−Σ (iω ,0))/2. If t2 <γ2, phonons instead stabilize the non-topological
z n 11 n 33 n
At T = 0, a topological quantum phase transition oc- insulating phase. Since |f (T)| is a monotonically de-
z
∗
curswhentherenormalizedDiracmass,m ≡m+Σ (0), creasing function of x, with f → 0 as x → ∞, phonons
z z
crosseszeroandchangessign[20]. Duetothefrequency- are more likely to affect the band topology in materials
dependence of the self-energy, 2m∗ is in general not with flatter bands (smaller λ).
equal to the renormalized energy gap E∗ ≡ E∗ − E∗, Some reasonable parameter values are [10, 22] t ≃
g c v
∗ ∗ ∗
where E ≃ m + Σ (E − ǫ ) + Σ (E − ǫ ) and 0.25eV, γ ≃ 0.15eV, λ ≃ 0.18eV, C ≃ 35eV, ρ ≃
c 0 c F z c F
E∗ ≃ −m+Σ (E∗ −ǫ )−Σ (E∗ −ǫ ). However, in 7800kg/m3, c ≃ 1.7km/s and a ≃ 1nm. Then,
v 0 v F z v F s
the toy model treated here, Σαα′(ω,k)≃Σαα′(0,k) and U ≃ 70meV and Uc(0) ≃ 34 m[meV]. Accordingly,
∗ ∗ ∗
thus E ≃ 2m . In other words, the sign change of m phonons may drive a topologicapl quantum phase transi-
g
occurs nearly simultaneously with the band inversion of tion in small-gap semiconductors (m . 10meV). More-
the dressed bulk quasiparticle spectrum. over, because U (T ≫ ω ) ≪ U (0), a material that is
c c c
One interesting aspect of phonons is that their influ- topologically trivial at T = 0 may develop topological
ence on the electronic structure is strongly temperature- surface states at higher temperature.
dependent below the melting point of the solid, because Aside from the Dirac mass, electron-phonon inter-
∗
the Debye temperature is of the order of room tempera- actions renormalize the Fermi energy, ǫ (T) ≃ ǫ −
F F
∗
ture [21]. Accordingly, m is T-dependent and may in Σ (0) [23]. Though not explicitly, U in Eq. (7) depends
0 c
∗
principle change sign as a function of temperature at on ǫ implicitly because screening (ignored above) leads
F
some T > 0. Provided that quasiparticle broadening to g → g /κ(q), where κ(q) ≃ 1 + q2 /q2 and q
c q q TF TF
∗
is small compared to |m (T)|, the topological proper- is the Thomas-Fermi wavevector. Therefore, the obser-
ties of G are identical to those of h(0) with m ≡ m∗(T). vation of phonon effects on topological properties may
∗
Therefore, a sign change of m (T) at T =T implies the be contingenton havinglow bulk doping (q ≪k ). In
c TF c
∗ ∗
emergence(ifm (0)t>0)orevanescence(ifm (0)t<0) viewofthis,electricallygatedthinfilmsarethepreferred
of topological surface states at T & T . Angle-resolved venue to tune the chemical potential inside the bulk gap
c
photoemission may be the most direct probe to access and to observetransport and thermodynamic signatures
this “thermal topological crossover”. of phonon-induced surface states.
3
m=25 meV m=20 meV
60 14
III 10 III
10
40 ωU/c 6 II 6 II
V) 2 I 2 I
e 20
m 1 2 3 1 2 3
E* ( 0 10 m=10 meV 10 m=5 meV
III III
8 8
-20 m*>0 m*=0 m*<0 II 6 II
T=0 T=Tc T>Tc 6 I 4
4 I
-0.1 0 0.1 -0.1 0 0.1 -0.1 0 0.1 2
k a 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5
z T/ω
c
FIG. 1: Dressed bulk quasiparticle dispersion along [001],
FIG. 2: Phonon-induced topological quantum phase transi-
obtained from Eqs. (1), (2) and (3). Parameter values are
tion (at T = 0) and topological crossover (at T > 0) for
U ≃ 57meV, γ = 0.15eV, λ = 0.18eV, t = 0.25eV,
different values of the bare energy gap. The parameters
ωc = 8meV, ωq = ωc/(1.24π)aq and m = 20meV (where (U,γ,λ,t,ωc)arethesameasinFig.1. I:non-topological in-
1.24π/aistheradiusoftheDebyesphere). Becauseelectron- ∗ ∗ ∗
sulator (m >0); II: “crossover” regime (m <0 and |m |<
phonon interactions preserve lattice and time-reversal sym- ∗ ∗
4T);III:topological“insulator”(m <0and|m |>4T). Be-
metries, dressed bands are two-fold degenerate. Left: T = 0
cause ωc is fixed, the dashed lines in all the graphs converge
(topologically trivial insulator). Middle: T ≃ 0.7ωc (topo- to thesame high-T asymptote (c.f. Eq. (6)).
logical crossover temperature). Right: T = 1.2ωc (inverted
regime). The bandbroadening |Γkα|(not shown) grows with
T,but remains .1meV throughout thethree panels.
insulatorsexistsuchthat |E∗|&8T deep inthe inverted
g
regime. In the “crossover” region, where surface states
are emerging, m∗ < 0 but |E∗| . 8T. In order to have
The thermal topological crossover introduced above g
a trivial insulator at T = 0 turn into a topological “in-
is meaningful only if the quasiparticle band broaden-
sulator” at high T, it helps if m/ω is larger because (i)
ing in the vicinity of TRIM remains small compared to c
|m∗|. The reduced phase space for scattering near k =0 at T = 0, Uc increases for larger m (irrespective of ωc)
∗
and (ii) at high T, |m | grows faster with T for smaller
helps keep the broadening small. For example, it fol-
ω (irrespective of m).
lows from inspection that there exists a finite window, c
Phonon-inducedtopologicaltransitionsinsemimetals.–
1 . U2f (T)/m . 2/(1+γ/t), for which quasiparticle
z A simple lattice model for a 3D Dirac semimetal is
bands are inverted and yet sharp (Γ =0) at the same
0α
time. Here we have neglected the broadening due to e.g. h(k)=h(0)(k)+∆σz1 , (8)
2
impuritiesandbulk-surfacescattering. Thelatterwillbe
∗
suppressed at k ≃0 insofar as |m |>(ω ,T).
c where ∆ >0 is an exchange field due to magnetic order
Along similar lines, the topological surface states de- that may originate e.g. from bulk dopants in a Dirac in-
∗ ∗
rivedfromGarehardlydetectableunless|E |≃2|m |≫ sulator. When m ∈ (−∆,∆), the energy spectrum of h
g
T. Whether this condition is met or not depends on contains a pair of Weyl nodes of opposite chirality sep-
∗
material parameters, because |m | ∝ T at high tem- arated in momentum space (along [001]): the system is
perature (c.f. Eq. (6)). In our toy model, using the a Weyl semimetal (WS) and has Fermi arcs [2] at the
∗
parameter values enlisted above, |E | ≫ T translates sample boundaries. For |m| ≥ ∆, the two Weyl nodes
g
into ωc ≪ 25meV, which can only be satisfied in soft merge and the crystal turns into an insulator. Here we
Dirac materials with low Debye temperatures. This up- showthatlatticevibrations,whichpreservetranslational
perboundonωcbecomeslarger(andthuseasiertofulfill) invariance only on average, can destroy or create Weyl
when λ is smaller and/or when γ is closer to t. nodes both at T =0 and T >0.
∗
Next,weproceedtosolvethefulllatticemodelnumer- We evaluate E by combining Eqs. (1), (2) and (8).
kα
ically. On one hand, Fig. 1 displays the dressed quasi- The resulting phase diagram (Fig. 3) evidences that, in
particlespectrum,illustratingabandinversionatT >0. presenceofphonons,(i)∆remainsnearlyunchanged,(ii)
∗
On the other hand, Fig. 2 shows a topological crossover theWSphaseisstableform (T)∈(−∆,∆),and(iii)the
∗
as a function of temperature. The topological “insula- Weylnodesoccurawayfromzeroenergyduetoǫ →ǫ .
F F
∗
tor” at T > 0 is defined as a phase where m < 0 and These generic observations are likely to hold for more
|E∗| & 8T, the latter attribute being rather arbitrary. elaboratemodelsthanEq.(8). Inaddition,Fig.3proves
g
∗
For many semiconductors, ∂E /∂T ∼ 1 − 10 at high that a material which is insulating at T = 0 may turn
g
T [10,16,24];henceitisconceivablethatsomerealDirac intoWSatnonzerotemperature. Inordertoobservethis
4
1.5
V) 100
1 E* (me-1000 E*-55000 [1] SLeeett.e.9g8. 1L0.6F80u3, (C2.0L0.7)K;aRn.eR, oayn,dPhEy.Js..RMeevl.eB, P7h9y,s1.95R3e2v2.
-0.2-0.10 0.10.2 -0.2-0.1 0 0.1 (2009); J. Moore, Nature 464, 194 (2010); M.Z. Hasan
ωT/c Insulatorkz a Weyl Semimetkaz la aQnidanCd.LS..-KCa.nZeh,aRnegv,.RMevod..MPohdy.sP.h8y2s,.38034,51(0250710()2;01X1.)-.L.
0.5 [2] S. Murakami, New J. Phys.9, 356 (2007); G.E. Volovik,
Insulator JETP Lett. 91, 55 (2010); X. Wan, A.M. Turner, A.
m</∆ m>/∆ V(2i0s1h1w)a;nAat.Ah.aBndurSk.oYv.,SMav.Dra.sHovo,oPkhayns.dRLe.vB.Bale8n3t,s,20P5h1y0s1.
Rev. B 84, 235126 (2011).
-1 0 1 2 3
m/∆ [3] M. Bahramy, B.-J. Yang, R. Arita and N. Nagaosa, Na-
ture Communications 3, 679 (2012).
FIG. 3: Phase diagram for Eq. (8) in presence of electron- [4] M. Kim, C.H. Kim, H.-S. Kim and J. Ihm, Proc. Nat.
phonon interactions. The parameters (U,t,γ,λ,ωc) take the Acad. Sciences 109, 671 (2012).
same values as in Fig. 1, with ∆ = 20meV. For simplic- [5] M.Z. Hasan, D. Hsieh, Y. Xia, L.A. Wray, S.-Y.Xu and
ity we have neglected any T-dependence of ∆ by assuming C.L. Kane, arXiv:1105.0396 (2011).
that themagnetic order producingit is robust up toT ∼ωc. [6] J. Liu, R.-L. Chu, J.K. Jain and S.-Q. Shen, Phys. Rev.
When m > m>, increasing T may induce a WS phase; for Lett.102,136806(2009);C.W.Groth,M.Wimmer,A.R.
m ∈ (m<,m>), increasing T destroys the WS phase. When Akhmerov, J. Tworzydlo and C.W.J. Beenakker, Phys.
m < m<, the system is insulating for all T. The thick hori- Rev.Lett.103,196805(2009);H.-M.Guo,G.Rosenberg,
zontallinesegmentisthewindowfortheWSphaseinabsence G. Refael and M. Franz, Phys. Rev. Lett. 105, 216601
of electron-phonon interactions and at T = 0. Insets: repre- (2010).
sentative quasiparticle energy spectra for the insulating and [7] N.H.Lindner,G.RefaelandV.Galitski,NaturePhysics
WS phases, along [001]. Arrows indicate nodes with nonzero 7, 490 (2011).
topological invariant. [8] S. Diehl, E. Rico, M.A. Baranov and P. Zoller, Nature
Physics 7, 971 (2011).
[9] J.E. Avron, M. Fraas, G.M. Graf and O. Kenneth, New
J. Phys.13, 053042 (2011).
thermally-induced WS phase, the temperature at which [10] B.-L. Huang and M. Kaviany, Phys. Rev. B 77, 125209
themagneticordergivingriseto∆dissolvesmustnotbe (2008).
[11] W. Cheng and S.-F. Ren, Phys. Rev. B 83, 094301
low compared to the Debye temperature.
(2011).
Conclusions– We have given a proof-of-principle for
[12] X. Zhu, L. Santos, R. Sankar, S. Chikara, C. Howard,
the impact of electron-phonon interactions on the band
F.C.Chou,C.ChamonandM.El-Batanouny,Phys.Rev.
topology of insulators and semimetals, with an empha- Lett. 107, 186102 (2011).
sis on thermally driven topological crossovers. Although [13] R.C. Hatch, M. Bianchi, D. Guan, S. Bao, J. Mi, B.B.
we have focused on phonons, other bosonic baths may Iversen, L. Nilsson, L. Horneker and P. Hofmann, Phys.
likewise play a role in inducing or destroying topological Rev. B 83, 241303 (2011).
[14] S.GiraudandR.Egger,Phys.Rev.B83,245322(2011);
phases in thermodynamic equilibrium.
S. Giraud, A. Kundu and R. Egger, Phys. Rev. B 85,
A necessary future task will consist of searching for
035411 (2012).
real Dirac materials with appropriate band parameters
[15] Z.-H. Pan, A.V. Fedorov, D. Gardner, Y.S. Lee, S. Chu
to realize the qualitativeeffects discussedherein. Gener- and T. Valla, Phys.Rev.Lett. 108, 187001 (2012).
ally, the more promising candidates will be soft crystals [16] P.B.AllenandV.Heine,J.Phys.C9,2305(1976); P.B.
with strong electron-phonon coupling and a small zero- Allen and M. Cardona, Phys.Rev.B 23, 1495 (1981).
temperature bandgapthatis nonetheless largerthan the [17] G.D. Mahan, Many Particle Physics (3rd ed., Kluwer
Academic/Plenum Publishers, New York,2000).
Debye temperature. Specifically, BiTl(S1−δ Seδ)2 seems [18] For simplicity we keep only the “rainbow” diagram and
interesting because both the magnitude and sign of its
ignore the“Debye-Waller” diagram of Ref. [16].
bandgap are tunable [5]. However, it is not yet known [19] G. Rosenberg and M. Franz, Phys. Rev. B 82, 035105
howstrongthephonon-inducedbandgaprenormalization (2010).Althoughweconcentrateona3Dmodel,thecon-
is in this material. Should BiTl(S1−δ Seδ)2 become fer- cepts and ideas we discuss can be applied to 2D Dirac
romagnetic upon magnetic doping, there would also be materials.
anopportunitytoalternatebetweeninsulatingandWeyl [20] Z.WangandS.-C.Zhang,Phys.Rev.X2,031008(2012);
Z. Wang and B. Yan,arXiv:1207.7341 (2012).
semimetallic phases by tuning δ and the temperature.
[21] In contrast, the self-energy due to static disorder (c.f.
IacknowledgevaluableinteractionswithV.Albert,M.
Ref. [6]) is T-independent, and the self-energy due to
Franz,L.Glazman,J.Moore,J.D.SauandJ.Vayrynen. purely electronic Coulomb interactions is weakly T-
This project has been funded by Yale University, and dependent because the typical bandwidth of the elec-
hasalsobenefitedfromthekindhospitalityofthe Aspen tronic dispersion is ∼ 100 times larger than the Debye
Center for Physics through NSF Grant No. 1066293. temperature.
5
[22] H.Zhang,C.-X.Liu,X.-L.Qi,X.Dai,Z.FangandS.-C. [24] P.B. Allen and M. Cardona, Phys. Rev. B 27, 4760
Zhang, NaturePhysics 5, 438 (2009). (1981); S. Logotheditis, J. Petalas, H. Polatoglou and
[23] Wefindthephonon-inducedrenormalizations oft,λand D. Fuchs,Phys. Rev.B 46 4483 (1992).
γ to be veryweak.
