Direct observation and temperature control of the surface Dirac gap in a topological crystalline insulator

Wojek, Bastian M.; Berntsen, Magnus H.; Jonsson, Viktor; Szczerbakow, A.; Dziawa, P.; Kowalski, B.J.; Story, T.; Tjernberg, Oscar

doi:10.1038/ncomms9463

Cited by 58 publications

(46 citation statements)

References 29 publications

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“…Among all, in the Z 2 topological insulator (TI) class, the band inversion occurs at an odd number of time-reversal symmetric points 13,14 in the first Brillouin zone (BZ), whereas in topological crystalline insulators (TCI) it occurs at an even number of crystalline mirror symmetric points. 5,[15][16][17][18] Both the TI 8,[19][20][21][22] and TCI [15][16][17][23][24][25][26][27] states have been thoroughly studied experimentally. Moreover, the topological phase transition (Fig.…”

Section: Introductionmentioning

confidence: 99%

Magnetooptical determination of a topological index

et al. 2017

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When a Dirac fermion system acquires an energy-gap, it is said to have either trivial (positive energy-gap) or non-trivial (negative energy-gap) topology, depending on the parity ordering of its conduction and valence bands. The non-trivial regime is identified by the presence of topological surface or edge-states dispersing in the energy gap of the bulk and is attributed a non-zero topological index. In this work, we show that such topological indices can be determined experimentally via an accurate measurement of the effective velocity of bulk massive Dirac fermions. We demonstrate this approach analytically starting from the Bernevig-HughesZhang Hamiltonian to show how the topological index depends on this velocity. We then experimentally extract the topological index in Pb 1-x Sn x Se and Pb 1-x Sn x Te using infrared magnetooptical Landau level spectroscopy. This approach is argued to be universal to all material classes that can be described by a Bernevig-Hughes-Zhang-like model and that host a topological phase transition.npj Quantum Materials (2017) 2:26 ; doi:10.1038/s41535-017-0028-5 INTRODUCTIONThe concept of band topology in condensed matter systems has revolutionized our understanding of quantum phases of matter.

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Section: Introductionmentioning

confidence: 99%

Magnetooptical determination of a topological index

et al. 2017

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“…This raises the open question about the classification of interacting TCIs protected by spatial symmetries. On the experimental side, a growing body of interaction-driven phenomena has been found in existing TCI materials, including spontaneous surface structural transition and gap generation [6][7][8] and anomalous bulk band inversion [26]. Moreover, new TCI materials have been predicted in transition-metal oxides [27,28] and heavy fermion compounds [29,30], where strong electron interactions are expected.…”

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Theory of interacting topological crystalline insulators

2015

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We study the effect of electron interactions in topological crystalline insulators (TCIs) protected by mirror symmetry, which are realized in the SnTe material class and host multivalley Dirac fermion surface states. We find that interactions reduce the integer classification of noninteracting TCIs in three dimensions, indexed by the mirror Chern number, to a finite group Z 8 . In particular, we explicitly construct a microscopic interaction Hamiltonian to gap eight flavors of Dirac fermions on the TCI surface, while preserving the mirror symmetry. Our construction builds on interacting edge states of U (1) × Z 2 symmetry-protected topological phases of fermions in two dimensions, which we classify. Our work reveals a deep connection between three-dimensional topological phases protected by spatial symmetries and two-dimensional topological phases protected by internal symmetries. The prediction and observation of topological crystalline insulators (TCIs) in the SnTe material class has expanded the scope of topological matter and gained wide interest [1][2][3][4][5]. These TCIs possess topological surface states that are protected by mirror symmetry of the rocksalt crystal and become gapped under symmetry-breaking structural distortions [6][7][8][9]. These surface states are predicted to exhibit a plethora of novel phenomena ranging from large quantum anomalous Hall conductance [1,10,11] to strain-induced pseudo-Landau levels and superconductivity [12], which are currently under intensive study [13][14][15].According to band theory, TCIs protected by mirror symmetry are classified by an integer topological invariant, the mirror Chern number [16]. However, recent theoretical breakthroughs [17][18][19][20][21][22][23][24] have found that the classifications of interacting systems are markedly different from noninteracting systems in various classes of topological insulators and superconductors protected by internal symmetries [25]. This raises the open question about the classification of interacting TCIs protected by spatial symmetries. On the experimental side, a growing body of interaction-driven phenomena has been found in existing TCI materials, including spontaneous surface structural transition and gap generation [6][7][8] and anomalous bulk band inversion [26]. Moreover, new TCI materials have been predicted in transition-metal oxides [27,28] and heavy fermion compounds [29,30], where strong electron interactions are expected.Motivated by these theoretical and experimental developments, in this work we study the effect of electron interactions in mirror-symmetric TCIs. Our main result is that interactions reduce the classification of three-dimensional (3D) TCIs from Z in the noninteracting case to Z 8 . We obtain this result by introducing a "domain wall" construction of interacting surface states of 3D TCIs, which exploits the nonlocal nature of mirror symmetry. This construction builds on interacting edge states of two-dimensional (2D) TCIs or U (1) × Z 2 symmetryprotected topological (SPT) phases, which...

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“…Pairs of Dirac cones with spin-momentum locking are located near theX points of the surface Brillouin zone, forming a Kramers pair. At low temperatures the surface undergoes a structural transition into a ferroelectric state and one of the mirror symmetries is spontaneously broken [14,15], while the other remains intact. As a result, two of the surface Dirac cones become massive, while the other two remain massless [16] (see inset in Fig.…”

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Quantum Nonlinear Hall Effect Induced by Berry Curvature Dipole in Time-Reversal Invariant Materials

2015

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It is well known that a nonvanishing Hall conductivity requires broken time-reversal symmetry. However, in this work, we demonstrate that Hall-like currents can occur in second-order response to external electric fields in a wide class of time-reversal invariant and inversion breaking materials, at both zero and twice the driving frequency. This nonlinear Hall effect has a quantum origin arising from the dipole moment of the Berry curvature in momentum space, which generates a net anomalous velocity when the system is in a current-carrying state. The nonlinear Hall coefficient is a rank-two pseudotensor, whose form is determined by point group symmetry. We discus optimal conditions to observe this effect and propose candidate two-and three-dimensional materials, including topological crystalline insulators, transition metal dichalcogenides, and Weyl semimetals. DOI: 10.1103/PhysRevLett.115.216806 PACS numbers: 73.43.-f, 03.65.Vf, 72.15.-v, 72.20.My Introduction.-The Hall conductivity of an electron system whose Hamiltonian is invariant under time-reversal symmetry is forced to vanish. Crystals with sufficiently low symmetry can have resistivity tensors which are anisotropic, but Onsager's reciprocity relations [1] force the conductivity to be a symmetric tensor in the presence of time-reversal symmetry. Hence, when the electric field is along its principal axes the current and the electric field are collinear, at least to the first order in electric fields. However, this constraint is only about the linear response and does not necessarily enforce the full current to flow collinearly with the local electric field.In this Letter we study a special type of such nonlinear Hall-like currents. We will demonstrate that metals without inversion symmetry can have a nonlinear Hall-like current arising from the Berry curvature in momentum space. The conventional Hall conductivity can be viewed as the zeroorder moment of the Berry curvature over occupied states, namely, as an integral of the Berry curvature within the metal's Fermi surface. The effect we discuss here is determined by a pseudotensorial quantity that measures a first-order moment of the Berry curvature over the occupied states, and hence we call it the Berry curvature dipole. This nonlinear Hall effect has a quantum origin arising from the anomalous velocity of Bloch electrons generated by the Berry curvature [2], but it is not expected to be quantized.In a time-reversal invariant system, the Berry curvature is odd in momentum space, Ω a ðkÞ ¼ −Ω a ð−kÞ, and hence its integral weighed by the equilibrium Fermi distribution is forced to vanish, because Kramers pair states at k and −k are equally occupied. However, the second-order response is determined by the integral of the Berry curvature evaluated in the nonequilibrium distribution of electrons computed to first order in the electric field. Since the

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Direct observation and temperature control of the surface Dirac gap in a topological crystalline insulator

Cited by 58 publications

References 29 publications

Magnetooptical determination of a topological index

Magnetooptical determination of a topological index

Theory of interacting topological crystalline insulators

Quantum Nonlinear Hall Effect Induced by Berry Curvature Dipole in Time-Reversal Invariant Materials

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