Magnetic-Field Induced Semimetal in Topological Crystalline Insulator Thin Films
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Magnetic-Field Induced Semimetal in Topological Crystalline Insulator Thin Films
Motohiko Ezawa
We investigate electromagnetic properties of a topological crystalline insulator (TCI) thin film under externalelectromagnetic fields. The TCI thin film is a topological insulator indexed by the mirror-Chern number. Itis demonstrated that the gap closes together with the emergence of a pair of gapless cones carrying oppositechirarities by applying in-plane magnetic field. A pair of gapless points have opposite vortex numbers. Thisis a reminiscence of a pair of Weyl cones in 3D Weyl semimetal. We thus present an a magnetic-field inducedsemimetal-semiconductor transition in 2D material. This is a giant-magnetoresistance, where resistivity is con-trolled by magnetic field. Perpendicular electric field is found to shift the gapless points and also renormalizethe Fermi velocity in the direction of the in-plane magnetic field.
Introduction:
Topologically stable states such as topolog-ical insulator are among the most exciting topics in mod-ern condensed matter physics. Topological crystalline insu-lator (TCI) is a topological insulator protected by the crys-tal symmetry . Its experimental realizations in Pb x Sn − x Teexcite studies of TCI . A thin film made of TCI providesus with a new platform of 2D electron system . Whenthe film is thin enough, the gap opens due to hybridizationbetween the front and back surfaces, and it turns the systeminto a topological insulator. The TCI thin film is a topologi-cal insulator indexed by the mirror-Chern number . Veryrecently, a TCI thin film made of SnTe was experimentallymanufactured .Weyl semimetal has recently been found to be topo-logically robust in three dimensions (3D). The emergence ofa Weyl semimetal is always accompanied by a pair of Weylcones with opposite chiralities subject to the fermion doublingtheorem . Each Weyl cone has a gapless point in momen-tum space, carrying the opposite monopole number. A pairof monopoles cannot annihilate each other dynamically, sincethey are parts of the ground-state texture of the Berry curva-ture. The semimetal is topologically stable provided two Weylcones are separated. Indeed, when two Weyl cones meet head-on by controlling system parameters, they disappear and thegap opens in the system. These are the basic features of the3D Weyl semimetal.We investigate the band structure and the topological prop-erty of TCI thin film by applying the in-plane magnetic fieldand the perpendicular electric field. As these external fieldsare increased, the gap reduces and closes, forming a gaplessDirac cone at certain critical fields. Then the gapless Diraccone splits into a pair of gapless cones with opposite chirali-ties. They are akin to a pair of Weyl cones in 3D. Indeed, theycarry the opposite vortex numbers. We may call the gaplesscone (point) "Weyl" cone (point) in 2D based on the similarity.We find a flat band emerges to connect them in a nanoribbon,as is a reminiscence of the Fermi arc connecting the twoWeyl points in the surface of a Weyl semimetal. It is to be em-phasized that the emergence of the Weyl points is solely due tothe in-plane magnetic field, while the electric field only shiftsthe position of the Weyl points and renormalizes the Fermivelocity. When we change the direction of in-plane mag-netic field, the gap remains open and the positions of the Weylpoints rotate in parallel to the magnetic field direction. Thepair of Weyl points never annihilate each other provided they are separated by the in-plane magnetic field. Thus we havepresented a magnetic-field induced semimetal-semiconductortransition in 2D material. Topological Crystalline Insulator Thin Film:
Gapless Diraccones emerge on the surface of a topological insulator. Weconsider the [0,0,1] TCI surface, where there are gapless Diraccones at the X and Y points. When the thickness is very thin,the gap opens due to hybridization between the front and backsurfaces. We explicitly investigate the low-energy physicsnear the Fermi energy around the X point, but the same analy-sis can be carried out also around the Y point. It is well knownthat the low-energy physics in the vicinity of the Dirac pointis described precisely by the Dirac theory. Hence we are ableto present analytic formulas. Nevertheless, we also carry outnumerical studies based on the tight-binding model to confirmanalytical results.The effective low-energy Hamiltonian of the TCI thin filmhas been derived in the vicinity of the X point . It is expressedin terms of × matrices, H = [ v x k x σ x − v y k y σ y ] τ y + mτ x , (1)where σ = ( σ x , σ y , σ z ) and τ = ( τ x , τ y , τ z ) represent thespin and surface degrees of freedom; v i and k i are the Fermivelocity and the momentum into the i -direction; mτ x repre-sents the tunnelling term between the two surfaces. We haveset ~ = 1 for simplicity. The Hamiltonian H has the mir-ror symmetry, M H ( k ) M − = H ( k ) , with the generator M = − i σ z τ x .Without the external fields, the Hilbert space is divided bythe eigenvalues ( M = ± i ) of the mirror operator . Themirror-Chern number C M is defined by the difference of theChern numbers in these two sectors. The total Chern num-ber is zero ( C = 0 ) and the mirror-Chern number is given by C M = sign ( m ) for each cone. It is a mirror-Chern insula-tor. When we analyze a nanoribbon based on the tight-bindingmodel , gapless edge modes appear, as signals the topologi-cal nature of the bulk [Fig.1(a)].We introduce the external field terms H ext = E z τ z + B x σ x + B y σ y . (2)The first term is induced by applying electric field perpen-dicular to the TCI thin film. The in-plane Zeeman terms areinduced by applying in-plane magnetic field.We explore the system with H = H + H ext . The bandstructure changes as a function of the external fields. The band FIG. 1: Band structure of bulk and nanoribbon of the TCI thin filmbased on the tight-binding model. (a) Without external field ( B x =0 , B y = 0 ). (b) At the phase transition point ( B x = m, B y = 0 ),where the band gap closes. (c) In the semimetallic phase ( B x =2 m, B y = 0 ). (d) In the semimetallic phase ( B x = 2 m, B y = m ).We have set m = 0 . . gap is located at k x = k y = 0 , where the energy spectrumreads E = ± (cid:12)(cid:12)(cid:12)q B x + B y ± p m + E z (cid:12)(cid:12)(cid:12) . (3)The gap closes when B x + B y = m + E z , where a topologicalphase transition occurs [Fig.1(b)].Before the gap closes the system is an insulator. However, itis no longer a mirror-Chern insulator since the mirror symme-try is broken. When we examine a nanoribbon based on thetight-binding model , edge modes are gapped. The mirror-Chern number C M becomes a continuous function of E z , B x and B y and is no longer quantized. For instance, we find C M = m p m + E z , (4)when we apply only E z . It is reduced to the quantized value, C M = sign ( m ) , in the limit E z = 0 . Magnetic field induced Semimetal:
We investigate the TCIthin film after the phase transition point, where the gap closes[Fig.1(c)]. We may choose the direction of the in-plane mag-netic field as the x -axis without loss of generality. To study thephenomenon analytically we examine the energy spectrum ofthe Dirac theory, E = ± r v y k y + (cid:16)p v x k x + m + E z ± B x (cid:17) . (5)We show the band structure in Fig.2. The phase transitionpoint is B x = m + E z . Beyond the point, a gapless Diraccone is decomposed into two Weyl cones located at ( k x , k y ) = FIG. 2: Band structure near the X point based on the 4-band theory.(a) Without external field ( B x = 0 , B y = 0 ). (b) At the phase tran-sition point ( B x = m, B y = 0 ), where the band gap closes. (c) Inthe semimetallic phase ( B x = 2 m, B y = 0 ). (d) In the semimetallicphase ( B x = 2 m, B y = m ). We have set m = 0 . . ( ± k Xx , as in Fig.2(c) with k Xx = v − x p B x − m − E z .We also refer to these Weyl points as the X ± points. It is to beemphasized that the decomposition is made possible by the in-plane magnetic field. We also find that flat edge modes appearconnecting the two Weyl points in a nanoribbon [Fig.1(c)],which would correspond to the Fermi arc connecting the twoWeyl points in the surface of a 3D Weyl semimetal.We employ the effective × Hamiltonian by taking onlytwo bands nearest to the Fermi energy. We may derive it in thesecond order perturbation theory around the X ± point as fol-lows. Since the quantization axis is σ x and τ x at the Γ point, itis convenient to make the cyclic rotation of the Pauli matricesand diagonalize them. In the new basis the Hamiltonian reads H = [ v x k x σ z − v y k y σ x ] τ x + mτ z + E z τ x + B x σ z , (6)which is explicitly written as H = (cid:18) H TT † H (cid:19) , (7)with H = ( m − B x ) σ z − v y k y σ x , (8) H = ( − m − B x ) σ z − v y k y σ x , (9) T = − v x k y σ z − iE z σ y . (10)The dominant term is H . In the second-order perturbationtheory, we obtain the effective Hamiltonian H eff = H − T † H − T = ( B x − m − ( m + B x )( v x k x + E z )( m + B x ) + v y k y ) σ z − v y k y (1 + v x k x + E z ( m + B x ) + v y k y ) σ x . (11)By neglecting higher order terms in k y and changing the Paulimatrices inversely, we obtain the effective Hamiltonian H eff = (cid:18) B x − m − v x k x + E z m + B x (cid:19) σ x − v y k y σ y , (12) FIG. 3: Topological phase diagram. In the B x - E z plane, with thetopological semimetal (TS) and the insulator ( C = 0 ). with the energy spectrum being E = ± s(cid:18) v x k x + m + E z − B x m + B x (cid:19) + v y k y . (13)It agrees with (5) up to the order of k x .At the transition point B x = m + E z [Fig.2(b)], the energyspectrum is highly anisotropic. The dispersion is Schrödinger-like in the k x direction, and Dirac-like in the k y direction.Beyond the transition point B x > m + E z [Fig.2(c)], werewrite the Hamiltonian (12) as H eff = − v x (cid:0) k x − k Xx (cid:1) (cid:0) k x + k Xx (cid:1) m + B x σ x − v y k y σ y . (14)We may approximate k x ± k Xx = ± k Xx around the X ± point, H X ± eff = ∓ ˜ v x (cid:0) k x ∓ k Xx (cid:1) σ x − v y k y σ y , (15)with the renormalized velocity ˜ v x = 2 v x k Xx / ( m + B x ) . Theenergy spectrum in the vicinity of the X ± points is E ± = ± q ˜ v x ˜ k x + v y k y . (16)The role of E z is to shift the position of the Weyl points andrenormalize the Fermi velocity.It is easy to consider the general in-plane field B x = 0 and B y = 0 when E z = 0 . We find that the gap closes at ( k Xx , k Xy ) and ( − k Xx , − k Xy ) , where ( k Xx , k Xy ) = ( v − x | B x | , v − y | B y | ) / q − m /B k , (17)with B k = B x + B y . The positions of Weyl cones are parallelto the direction of the magnetic field. We have also confirmedthe results by calculating the band structure of the bulk and ananoribbon based on the tight-binding model. Namely the gapremains closed for an arbitral direction of the in-plane field, aswe show an example in Fig.1(e).We discuss the topological stability of the X ± points. TheHamiltonian is of the form H eff = R ( n x σ x + n y σ y ) , (18)where n x and n y are normalized fields subject to n x + n y = 1 .It has two eigen-spinors with the eigen-energy E ± = ±| R | . FIG. 4: Spin direction based on the 2-band theory. (a) In the topolog-ical insulator ( M x = 0 ). (b) At the transition point ( M x = m ). (c)In the topological semimetal ( M x = 2 m ), where a hedgehog struc-ture is found at the X + point, and an anti-hedgehog structure at the X − point. The vortex number is nontrivial at these points. The state corresponding to the filled band is given by thespinor | S i = 1 √ (cid:18) e − iφ/ − e + iφ/ (cid:19) , (19)when we parametrize ( n x , n y ) = (cos φ, sin φ ) . The spin ofthe state is given by s i = h S | σ i | S i , which we show in Fig.4.We note that s i = − n i . The spin direction forms a hedgehogstructure in the vicinity of the X + point, while it forms ananti-hedgehog structure in the vicinity of the X − point. Thereare a source and a sink of the spin flow at these points. Theyare described by a topological charge.We may define the vortex number for the spin texture by Q = 12 π I dk α [ s x ( k ) ∂ k α s y ( k ) − s y ( k ) ∂ k α s x ( k )] , (20)where the integration is carried out along the boundary of theBrillouin zone. It is trivial to see Q = 12 π I dk µ ∂ k µ φ. (21)In general, it yields Q = 0 , ± , ± , · · · , because φ is definedonly modulo π . For the specific field configuration given in(15), we find Q = ± for the Weyl cones at X ± . These val-ues are topologically stable because any perturbation cannotchange the quantized value of the topological charge Q . Wemay also evaluate the topological number for the spin config-uration at the phase transition point [Fig.4(a)] and also beforethe phase transition point (i.e. in the insulator phase) to findthat Q = 0 . We may interpret that a pair creation of Weylcones with Q = ± occurs from the topological trivial Diraccone with Q = 0 at the phase transition point. It is interestingto note that a flat edge mode appear to connect the two X ± points in a nanoribbon [Fig.1(c)]. Note that the total topo-logical charge (20) is zero ( Q = 0 ) both before and after thephase transition. Nevertheless, the semimetallic phase is sta-ble topologically due to the presence of a pair of two Weylcones with Q = ± generated by the in-plane field, preciselyas in the 3D Weyl semimetal.The same analysis is carried out for the Dirac cone at the Y point, from which a pair of Weyl cones emerge located at the Y ± point under in-plane magnetic field. Discussions:
We have demonstrated that a semimetallicphase emerges in a TCI thin film by applying in-plane mag-netic field. The semimetallic phase is characterized by theexistence of a pair of gapless Weyl cones as in the case ofthe 3D Weyl semimetal. In conclusion, we have proposed amagnetic-field induced semimetal-semiconductor transition in2D material. This is a giant-magnetoresistance, where resis-tivity is controlled by magnetic field. The transition betweenthe insulator phase and the semimetallic phase will be exper- imentally detectable by electric transport measurement. TheTCI thin film has already been manufactured . Our findingwill open a way to magneto-nanoelectronics based on the TCIthin film.I am very much grateful to N. Nagaosa, Y. Ando and Y.Tanaka for helpful discussions on the subject. This work wassupported in part by Grants-in-Aid for Scientific Researchfrom the Ministry of Education, Science, Sports and CultureNo. 25400317. L. Fu, Phys. Rev. Lett. 106, 106802 (2011). T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil and L. Fu, Nat.Comm. 3, 982 (2012). J. Liu, et.al. Nat. Mat. 13 178 (2014). C. Fang, M. J. Gilbert, B. A. Bernevig, Phys. Rev. Lett. 112,046801 (2014). J. Liu, W. Duan and L. Fu.Phys. Rev. B 88 241303(R) (2013). Y. Okada, M. Serbyn, H. Lin, D. Walkup, W. Zhou, C. Dhital, M.Neupane, S. Xu, Y. J. Wang, R. Sankar, F. Chou, A. Bansil, M.Z. Hasan, S. D. Wilson, L. Fu, V. Madhavan, Science 341 1496(2013). M. Ezawa, Phys. Rev. B 89, 195413 (2014). Y. Tanaka, Z. Ren, T. Sato, K. Nakayama, S. Souma, T. Takahashi,K. Segawa and Y. Ando, Nat. Phys. 8, 800 (2012). Su-Yang Xu, Chang Li, N. Alidoust, M. Neupane, D. Qian, I.Belopolski, J.D. Denlinger, Y.J. Wang, H. Lin, L.A.Wray, G. Lan-dolt, B. Slomski, J.H. Dil, A. Marcinkova, E. Morosan, Q. Gibson,R. Sankar, F.C. Chou, R. J. Cava, A. Bansil and M.Z. Hasan, Nat.Com. 3, 1192 (2012). P. Dziawa, B. J. Kowalski, K. Dybko, R. Buczko, A. Szczerbakow,M. Szot, E. Lusakowska, T. Balasubramanian, B. M. Wojek, M.H. Berntsen, O. Tjernberg and T. Story, Nat. Mat. 11, 1023 (2012). M. Ezawa, New J. Phys. 16, 065015 (2014). H. Ozawa, A. Yamakage, M. Sato, Y. Tanaka, Phys. Rev. B 90,045309 (2014). Jeffrey C.Y. Teo, Liang Fu, C.L. Kane, Phys. Rev. B 78, 045426(2008). R. Takahashi and S. Murakami, Phys. Rev. Lett. 107, 166805(2011). A. A. Taskin, F. Yang, S. Sasaki, K. Segawa, and Y. Ando, Phys.Rev. B 89, 121302(R) (2014). S. Murakami, New J. Phys. 9, 356 (2007). P. Hosur, X. L. Qi, Comptes Rendus Physique 14, 857 (2013). T. O. Wehlinga, A. M. Black-Schaffer, and A. V. Balatsky, Adv.Phys. 76, 1 (2014). O. Vafek, A. Vishwanath, Ann. Rev. Cond. Mat. Phys. 5, 83(2014). A.A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205 (2011). P. Hosur, S. A. Parameswaran, and A. Vishwanath, Phys. Rev.Lett. 108, 046602 (2012). Gabor B. Halasz and L. Balents, Phys. Rev. B 85, 035103 (2012). X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys.Rev. B 83, 205101 (2011). A. A. Zyuzin and A. A. Burkov, Phys. Rev. B 86, 115133 (2012). A. Sekine, K. Nomura, J. Phys. Soc. Jpn. 82, 033702 (2013). H. B. Nielsen and M. Ninomiya (1981) Phys. Lett. B 105, 219. M. Ezawa, Phys. Lett. A 378, 1180 (2014).28