Conductivity of Dirac fermions with phonon induced topological crossover
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Conductivity of Dirac fermions with phonon induced topological crossover
Zhou Li ∗ and J. P. Carbotte , Department of Physics, McMaster University, Hamilton, Ontario, Canada L8S 4M1 Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8 (Dated: January 27, 2018)We study the Hall conductivity in single layer gapped Dirac fermion materials including couplingto a phonon field, which not only modifies the quasi-particle dynamics through the usual self-energyterm but also renormalizes directly the gap. Consequently the Berry curvature is modified. As thetemperature is increased the sign of the renormalized gap can change and the material can crossover from a band insulator to a topological insulator at higher temperature (T). The effective Chernnumbers defined for valley and spin Hall conductivity show a rich phase diagram with increasingtemperature. While the spin and valley DC Hall conductivity is no longer quantized at elevatedtemperature a change in sign with increasing T is a clear indication of a topological crossover. Thechirality of the circularly polarized light which is dominantly absorbed by a particular valley canchange with temperature as a result of a topological crossover.
PACS numbers: 78.67.-n, 71.38.-k, 73.25.+i
I. INTRODUCTION
The isolation of graphene by exfoliation led to thediscovery of many of its exotic properties . It also ledto the fabrication and study of other two dimensionalsystems such as single layer group-VI dichalcogenides (e.g.
M oS ) and silicene , a buckled honeycomb lat-tice of silicon atoms. The Kane-Mele low energy Hamil-tonian can be used to describe these materials which haveDirac cones and valley degeneracy. The Dirac fermionscan acquire mass and spin-orbit coupling can spin polar-ize the bands. Manipulation of the size of the gaps in-volved can lead to a topological phase transition from aquantum spin Hall state to a band insulator. It is knownfrom consideration of the selection rules for the opti-cal transition involving circularly polarized light that thetwo valleys with indices τ = ± τ = +1( − .Here we go beyond a bare band model and include inthe Hamiltonian the coupling of the charge carriers toa phonon field . We find that the spin and valleydependent gap is renormalized by the phonons and thatthis renormalization depends strongly on temperature .Consequently the Berry curvature will be modified andthe topological quantum numbers are not well protected.For example, as we increase temperature, the gap canclose and reopen and as found by Garate the systemcan crossover from a band to a topological insulator.While reference [22] was concerned only with a proof inprinciple, here we show how this crossover can be mea-sured experimentally either using spin and/or valley DCHall conductivity or through the circular dichroism ofabsorbed light. The Kane-Mele model Hamiltonian is specified in section II along with electron-phonon cou-pling based on the Holstein model. The self energy dueto electron-phonon interaction directly modifies the gapwhich changes sign with increasing temperature (T). For-mulas for AC longitudinal and transverse optical conduc-tivity including self energy effects are presented in sec-tion III. The change in sign of the renormalized gap withincreasing T can be viewed as changing the underlyingChern number and we describe a pathway to determin-ing this topological crossover through DC valley and spinHall effects. Section IV deals with the temperature de-pendence of the dichroism of circularly polarized lightwith the handedness of the dominant absorption chang-ing sign with increasing temperature. Section V providesa brief conclusion. II. FORMALISM AND RENORMALIZED GAP
The Kane-Mele model Hamiltonian for gapped Diracfermions with spin-orbit coupling takes the form b H = ~ v ( τ k x ˆ σ x + k y ˆ σ y ) + ∆ z ˆ σ z − λ SO τ ˆ σ z ˆ S z , (1)where the valley index τ = ± λ SO is the spin orbitcoupling parameter, ∆ z a gap and v is the Fermi velocity.For definiteness in our calculations we will use λ SO =4 meV and v ≈ × m/s . The ˆ σ ′ s are the Paulimatrices for pseudospin and ˆ S z is the spin matrix for itsz-component ( s z ). Here we define the spin and valleydependent gap as ∆ τs z = ∆ z − λ SO τ s z .The Holstein electron-phonon interaction which hasbeen widely used, is written as H e − ph = − gω E X k , k ′ ,s c † k ,s c k ′ ,s ( b † k ′ − k + b k − k ′ )+ X q ω E b † q b q ,(2)where b † q creates a phonon of energy ω E and momentum q , g is a coupling between electrons and phonons and c † k ,s FIG. 1. (Color online) Temperature dependent renormal-ized gap ˜∆ τs z for spin up (top frame) and spin down (bot-tom frame) charge carriers, as a function of the bare massgap ∆ z . The vertical black dashed line indicate the line˜∆ τs z = ∆ τs z = Re Σ Z ( τ, s z ,
0) = 0. creates an electron of momentum k and spin s . The selfenergy is given by ˆΣ ( iω n ) = Σ I ( iω n ) ˆ I + Σ Z ( iω n )ˆ σ z (3)We are interested in the case when the chemical potential µ = 0, so that µ lies in the gap. In this case the selfenergy Re Σ I (0) = 0 so the chemical potential will notbe shifted by the electron-phonon interaction. HoweverΣ Z ( iω n → ω + i + ) will modify the gap ∆ τs z at ω = 0 andconsequently the Berry curvature. We give an analyticalexpression here for the ω = 0 limit Re Σ Z ( τ, s z ,
0) = g ω E ∆ τs z t π × [ln | ω E + | ∆ τs z | ω E + √ x cut | + ln | ( | ∆ τs z | − ω E ) / ( x cut − ω E ) | exp( ω E / ( k B T )) − √ x cut = p ~ v k + (∆ τs z ) and ~ vk max =3 . eV . The electron-phonon coupling g is chosen to -0.016 -0.008 0.000 0.008 0.0166080100120140160180200 C: TI (1,0)A: BI (0,1)
BI (0,1)TI (-1,0)BI (0,-1)TI (1,0)BI (0,-1) BI (0,1)TI (-1,0) BI (0,1)BI (0,-1) Z ( eV ) T ( K ) TI (-1,0)
A BC D D: TI (1,0)B: BI (0,-1)
FIG. 2. (Color online) The phase diagram in the ∆ z - T plane.The red and black line (phase boundaries) indicate ˜∆ τs z = 0for charge carriers in one valley with spin up and down re-spectively. The notation is BI (x,y) and TI (x,y) for bandand topological insulator respectively with spin Chern num-ber x and valley Chern number y. Increasing temperaturecan induce crossover from a band insulator to a topologicalinsulator or vice versa. make the effective mass correction to be about 0 . Cu x Bi Se found an order of magnitude larger ef-fective mass than what we have used here) of a typicalphonon coupling in semiconductors , ω E is chosen to be7 . meV and t = ~ v/a where a is the lattice constant,from this we know t ≈ . eV .For this set of parameters the first term in Eq. (4)which gives the zero temperature value of the renormal-ization Re Σ Z ( τ, s z ,
0) provides a 22% correction to ∆ τs z which is small. The second term in Eq. (4) however canbecome large as temperature increases and k B T >> ω E .The interacting Green’s function is given byˆ G ( k , iω n ) = 12 X s = ± (1 + s H k · σ ) G ( k , s, iω n ) (5)with H k = ( ~ vτ k x , ~ vk y , ∆ τs z + Σ Z ∗ ( iω n )) q ~ v k + | ∆ τs z + Σ Z ( iω n ) | (6)and G ( k , s, iω n ) = 1 iω n + µ − λτ s z / − E ( k , s, iω n ) (7) FIG. 3. (Color online) Color plot of the temperature depen-dent spin (top) and valley (bottom) Hall conductivity as afunction of temperature T and mass gap ∆ z . Superimposedas long dashed black lines are the phase boundaries of Fig. 2. where E ( k , s, iω n ) = s p | ∆ τs z + Σ z ( iω n ) | + ~ v k +Σ I ( iω n ). The self energy Σ I ( iω n ) directly renor-malizes the quasiparticle energies when there is no gap(∆ τs z = 0). However when ∆ τs z is non zero we see thatΣ Z ( iω n ) enters Eq. (6) and Eq. (7) to directly renormal-ize the gap (∆ τs z ). Thus the Berry curvature is renor-malized by the electron-phonon interaction through thenontrivial gap self energy Σ Z ( iω n ). At zero temperature,only the ω = 0 limit of Σ Z ( iω n → ω + i + ) enters theDC limit of the Hall conductivity which depends on therenormalized gap ˜∆ τs z = ∆ τs z + Re Σ Z ( τ, s z ,
0) and thisplays a critical role in this work. In Fig. 1 we show acolor plot of the magnitude of ˜∆ τs z as a function of thebare gap ∆ z and temperature T for fixed value of λ SO .The top frame is for one valley, τ = +1, and spin up,while the bottom frame is for spin down. The verticaldashed line at ∆ z = 4 and − meV respectively indicatesthe line ˜∆ τs z = ∆ τs z = Re Σ Z ( τ, s z ,
0) = 0. The renor-malized gap has other zeros as indicated by the whitecurves which separate positive and negative regions of˜∆ τs z . This change in the sign of the gap indicate possibletopological crossovers as we will discuss below. III. FORMALISM AND AC LONGITUDINALAND TRANSVERSE CONDUCTIVITY
The AC longitudinal conductivity σ xx ( ω ) and trans-verse Hall conductivity σ xy ( ω ) in the lowest order ap-proximation, following from the Kubo formula withoutvertex corrections are given by σ xx ( ω ) = e ~ Z , X k T r h σ x ˆ A ( k ,ω ) σ x ˆ A ( k ,ω ) i (8)where we are only interested in the absorptive part of theconductivity so no diamagnetic part is required , σ xy ( ω ) = e ~ Z , X k T r h τ z σ x ˆ A ( k ,ω ) σ y ˆ A ( k ,ω ) i (9)where ˆ A ( k , ω ) is the matrix spectral den-sity, R , = R ∞−∞ dω dω F ( ω ) and the function F ( ω ) = − ~ v π iω [ f ( ω ) − f ( ω )] ω − ω + ω + iδ . The optical conduc-tivity for circularly polarized light follows as Reσ ± ( ω ) = Reσ xx ( ω ) ∓ Imσ xy ( ω ) (10)Details on the impact of the self energies Σ I ( iω n ) andΣ Z ( iω n ) on the longitudinal and Hall optical conductiv-ity are found in the reference to which the reader isreferred.We begin our discussion with the zero temperature DClimit of the spin and valley Hall conductivity, which canbe associated to the spin and valley Chern numbers. Fi-nite temperature will obscure this relationship, but wewill still speak of an effective Chern number using thezero frequency renormalized gap calculated at finite T inthe zero temperature bare band expressions for the con-ductivities. At finite temperature the Hall conductivityis then given by Reσ xy = e π ~ X τ,s z τ Z kdk ~ v ˜∆ τs z [ f ( ε k , − ) − f ( ε k , + )][ | ˜∆ τs z | + ~ v k ] / (11)with f ( ω ) the Fermi-Dirac distribution function. For thespin and valley Hall conductivity τ in Eq. (11) should bereplaced by ~ τ s z / (2 e ) and 1 /e respectively. At T = 0Eq. (11) reduces to Reσ xy = e π ~ X τ,s z τ [ sgn ( ˜∆ τs z )] (12)thus the spin and valley Hall conductivity remains quan-tized at T=0 in our treatment of the electron-phononinteraction. Its only effect is to change the value of˜∆ τs z over its bare band value through the renormaliza-tion Re Σ Z ( τ, s z ,
0) of Eq. (4). The Chern number, spinChern number and valley Chern number are defined as C = P τ,s z τ [ sgn ( ˜∆ τs z )], C s = P τ,s z τ s z [ sgn ( ˜∆ τs z )]and C v = P τ,s z [ sgn ( ˜∆ τs z )]. A phase diagram for the ( eV ) I m xy () i n un i t o f ( e / ) s z =1 s z =1, with EPI s z =-1 s z =-1, with EPI T=250K z =-0.001eVT=160K T=10K
T=250K ( eV ) R e S xy () i n un i t o f ( e / ) No EPI With EPI
T=160K
T=10K
FIG. 4. (Color online) The real (right frames) and imaginary(left frames) of the transverse conductivity σ xy ( ω ) in unitsof e / ~ vs. photon energy in eV for three temperatures T =10 K , 160 K and 250 K from top to bottom. In all cases thegap ∆ z = 0 . eV . Black lines are for τ s z = 1 and red for τ s z = −
1. Solid curves are bare band results while dottedcurves include the electron-phonon renormalizations and arebased on Eq. (9). effective Chern numbers obtained in this way is given inFig. 2 with ∆ z on the horizontal axis and temperature T on the vertical axis. The solid black lines give the zerosof the renormalized gap ˜∆ τs z ≡ ∆ τs z + Re Σ Z ( τ, s z , , ± T = 0 calcu-lation for σ xy of Eq. (12) using the value of the renor-malized gap obtained at finite T . Of course for finite T one should really calculate the valley and spin Hallconductivity at the same T using the complete Eq. (9).Nevertheless Fig. 2 remains useful in defining the variousregions involved and we will see that the phase bound-aries defined in this figure remain imprinted in our finite T results although these boundaries are definitely fuzzedout by temperature. This is shown in Fig. 3 where wegive a color plot of the spin and valley Hall DC con-ductivity obtained from the finite temperature formulaEq. (11). This equation is a simplification of Eq. (9) forthe Hall conductivity. Eq. (9) includes the complete fre-quency dependent renormalization due to both Σ I ( iω n )and Σ Z ( iω n ). The first accounts for the quasiparticlerenormalizations which shifts the bare energies (real part) z =-0.015eV =+1 T = 30K + , spin up - , spin up + , spin down - , spin down R e + , - ( e / ) (eV) T = 300K
FIG. 5. (Color online) Temperature dependence of the dy-namic optical conductivity of the circularly polarized light.The top frame is for T = 30 K (low temperature) and thebottom frame for T = 300 K (high temperature). The valleyindex is τ = 1. Red lines are for right hand polarized lightand the black for left hand. The contribution from spin upand spin down bands are given separately. and provides damping (imaginary part). The second ac-counts for gap renormalization effects and also has fre-quency dependence and is complex. We have evaluatednumerically this complete equation for a number of pointsin Fig. 3 and have verified that these extra complicationsdo not change in an important way, the qualitative fea-tures shown in Fig. 3 which was drawn from the evalu-ation of the simpler Eq. (11). The numerical differencesbetween the two approaches agree to within a few per-cent at T ≈ K while at T ≈ K they differ at the20% level. But this difference is not important for theconclusions we will make. What is critical is the sign ofthe Hall conductivity rather than its exact magnitude.The boundary curves identified in Fig. 2 are repro-duced on Fig. 3 as long dashed black lines. Our finitetemperature data for the spin and valley Hall DC conduc-tivity clearly reflects some of the features of these sharpcrossovers but temperature does obscure their presence.Nevertheless our assignment of Chern numbers to vari-ous phases treating the gap as if it were effectively a zerotemperature value, provides considerable insight into theorigin of the complex variation of spin and valley Hallconductivity seen in Fig. 3. The left frames include theelectron-phonon renormalization while the right framesdo not and are for comparison. The top frames give thespin Hall and the lower frames give the valley Hall DCconductivity. We first note that in our units, the spinHall conductivity at T=0 is quantized and takes on thevalue zero for the larger values of | ∆ z | and − /π in theregion of smaller | ∆ z | . With a discontinuous crossoverat ˜∆ τs z = 0. Similarly the valley Hall conductivity jumpsfrom 1 /π to 0 and to − /π from right to left with bound-aries also at ˜∆ τs z = 0. As temperature is increasedslightly the left (with electron-phonon) and right (with-out electron-phonon) panels remain practically indistin-guishable. The crossover is no longer a jump but occursover a small region of variation in ∆ z . However, at highertemperatures they become radically different. Withoutelectron-phonon renormalization the sign of the spin Hallconductivity remains unchanged, while there can be signchanges when it is included. The region of small ∆ z isparticularly interesting as we go from a topological insu-lator with effective Chern spin Hall number -1 to +1 byincreasing T. The spin Hall conductivity goes from a neg-ative at T = 0 K to a positive value at T = 250 K and thischange in sign is taken as an indication of a topologicalcrossover. It occurs only when the zero frequency valueof the gap ˜∆ τs z changes sign with increasing T . Thiscan never occur in a bare band picture and is a resultof the electron-phonon renormalizations which becomelarger with increasing temperature and have the oppo-site sign to the bare gap ∆ τs z = ∆ z − λ SO τ s z . Note thatfor a given valley the renormalized gap must be positivefor one spin state and negative for the other so that bothcontributions add in the definition of the spin Hall con-ductivity to give a non zero value. Without a sign changethe two contributions would cancel leading to zero spinHall (trivial band insulator case).In Fig. 4 we present results for the transverse AC con-ductivity based on Eq. (9) with matrix spectral density ˆ A obtained from the fully interacting Green’s function spec-ified in Eq. (5) to (7). The left frame gives the imaginarypart of σ xy ( ω ) in units of e / ~ while the right gives thereal part of the spin Hall σ sxy ( ω ). Top to bottom framesgive three temperatures T = 10 K (top), T = 160 K (mid-dle) and T = 250 K (bottom). In all cases the red curvesare based on the bare bands for which Eq. (9) simpli-fies to Eq. (11). These are included for comparison withthe dotted curves which fully include the electron-phononrenormalizations through the self energies Σ I ( iω n ) andΣ Z ( iω n ) of Eq. (3). Considering first the imaginary part Imσ xy ( ω ) (left frames) we note that at low tempera-ture T = 10 K bare (solid) and renormalized (dotted)results are not qualitatively different from each other.This is not the case at high temperatures. In particularfor T = 160 K the sign associated with red ( τ s z = − τ s z = +1) curves is no longer the same. Thisarises because the sign of the renormalized results hasreversed while that of the bare band results is unaltered.This shows once more how phonons can introduce tem-perature variation in the conductivity not part of bareband calculations. For T = 250 K the situation is evenmore complex with renormalized conductivity crossingat ~ ω Reσ sxy ( ω )) shown onthe right in Fig. 4 we note again that renormalized (dot-ted) and bare band (solid) results agree qualitatively atlow T but that this changes as T increases. Particularlyimportant for this paper is the ω = 0 limit which givesthe DC value of the Hall conductivity used in Fig. 3. Atlow temperature bare and renormalized value agree whileat higher T they in fact carry opposite sign as has beenemphasized in constructing the color plots of Fig. 3. IV. CIRCULAR POLARIZATION ANDDICHROISM
Another consequence of the phonon renormalization ofthe effective gap is the change it can induce in the valleyHall optical selection rule for circularly polarized light.In Fig. 5 we show results for the frequency dependenceof
Reσ ± ( ω ) vs. ω in units of e ~ at T = 30 K (upperpanel) and at T = 300 K (lower panel). The solid redcurve is Reσ + ( ω ) for the spin up band and the red dot-ted is for spin down. Both show significant absorptionabove the main absorption edge determined by the effec-tive low temperature gap. By contrast the black curvesgive Reσ − ( ω ) and show that for this valley τ = 1, thereis comparatively little absorption of the left handed po-larized light. This selection rule can be reversed by in-creasing temperature through a change in the sign of theeffective gap brought about by the coupling to phononsas seen in the lower frame for T = 300 K . This tempera-ture is sufficiently large and the gap small enough that noabsorption threshold can now be seen but it is clear thatthe left handed light is now more strongly absorbed thanthe right handed light. Finally we note sharp phononstructures at energies (2 | ∆ τs z | + 2 ω E ) at low temperature.Of course once the sum over the two valleys τ = ± V. CONCLUSION
Coupling of Dirac fermions to a phonon field canchange the sign of their effective gap with increasingtemperature. This leads to a rich set of crossovers fromtopological to band insulators with recognizable imprintsin their DC spin Hall and valley Hall conductivity. Italso leads to a switch with increasing temperature in thepolarization from right to left handedness of the domi-nantly absorbed light by a given valley. Both effects pro-vide a pathway to the experimental verification of such aphonon induced topological crossover.
ACKNOWLEDGMENTS
This work was supported by the Natural Sciences andEngineering Research Council of Canada (NSERC), theCanadian Institute for Advanced Research (CIFAR).
REFERENCES ∗ [email protected] K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V.Khotkevich, S. V. Morozov, and A. K. Geim, Proc. Natl.Acad. Sci. USA , 10451(2005). K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y.Zhang, S. V. Dubonos, I. V. Grigorieva and A. A. Firsov,Science , 666 (2004). X. Zhang, Y.-W. Tan, H. L. Stormer and P. Kim, Nature , 201 (2005). A. K. Geim and K. S. Novoselov, Nat. Mater. , 183 (2007). T. Cheiwchanchamnangij and W. R. L. Lambrecht, Phys.Rev. B , 205302 (2012). Z. Y. Zhu, Y. C. Cheng and U. Schwingenschl¨ogl, Phys.Rev. B , 153402 (2011). W. Feng, Y. Yao, W. Zhu, J. Zhou, W. Yao and D. Xiao,Phys. Rev. B. , 165108 (2012). Zhou Li and J. P. Carbotte, Phys. Rev. B. , 205425(2012). B. Aufray, A. Kara, S. Vizzini, H. Oughaddou, C. L´eandri,B. Ealet and G. Lay, Appl. Phys. Lett. , 183102(2010). P. De Padova, C. Quaresima, C. Ottaviani, P. M.Sheverdyaeva, P. Moras, C. Carbone, D. Topwal, B.Olivieri, A. Kara, H. Oughaddou, B. Aufray, and G. Lay,Appl. Phys. Lett. , 261905 (2010). L. Stille, C. J. Tabert and E. J. Nicol, Phys. Rev. B ,195405 (2012). C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 226801(2005). D.Xiao, W. Yao and Q. Niu, Phys. Rev. Lett. , 236809(2007). D. Xiao, G. B. Liu, W. Feng, X. Xu and W. Yao, Phys.Rev. Lett. , 196802 (2012). H. Zeng, J. Dai, W. Yao, D. Xiao and X. Cui, Nature Nano. , 490 (2012). K. F. Mak, K. He, J. Shan and T. F. Heinz, Nature Nano. , 494 (2012). T. Cao, G. Wang, W. Han, H. Ye, C. Zhu, J. Shi, Q. Niu,P. Tan, E. Wang, B. Liu and J. Feng, Nature Communi-cations. , 887 (2012). Zhou Li and J. P. Carbotte, Physica B , 97 (2013),DOI: 10.1016/j.physb.2013.04.030. J. P. Carbotte, E. J. Nicol and S. G. Sharapov, Phys. Rev.B , 045419 (2010). T. Stauber and N. M. R. Peres, J. Phys.: Condens. Matter , 055002 (2008). A. Pound, J. P. Carbotte and E. J. Nicol, Phys. Rev. B , 125422 (2012). Ion Garate, Phys. Rev. Lett. , 046402 (2013). Takeshi Kondo, Y. Nakashima, Y. Ota, Y. Ishida, W.Malaeb, K. Okazaki, S. Shin, M. Kriener, Satoshi Sasaki,Kouji Segawa, and Yoichi Ando, Phys. Rev. Lett. ,217601 (2013). R. C. Hatch, D. L. Huber and H. H¨ochst, Phys. Rev. Lett. , 047601 (2010). E. Cappelluti and L. Benfatto, Phys. Rev. B , 035419(2009). L. Benfatto, S. G. Sharapov, N. Andrenacci and H. Beck,Phys. Rev. B71