Tuning phase transition between quantum spin Hall and ordinary insulating phases

Murakami, Shuichi; Iso, Satoshi; Avishai, Y.; Onoda, Masaru; Nagaosa, Naoto

doi:10.1103/physrevb.76.205304

Cited by 168 publications

(169 citation statements)

References 25 publications

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“…The position where the gap closing occurs in the BZ is dependent on the symmetry of the system. In an inversion-symmetric system, the gap closing occurs at TRIMs (i.e., at~k ¼G∕2 withG a reciprocal lattice vector) in the BZ, while the gap closes at points other than TRIMs in an inversion-asymmetric system (29,30). Because we are considering thin films with external electric fields, our system corresponds to the latter case.…”

Section: Resultsmentioning

confidence: 99%

“…Here, our intention is to change the band characters of the CBM states and VBM states to drive a topological phase transition. However, a small E ⊥ cannot affect the Z 2 invariant of the system, unless a singularity or a gap-closing point is encountered (29,30), because a topological invariant is robust under continuous deformation (it is a global property in the whole BZ). Fig.…”

Section: Resultsmentioning

confidence: 99%

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Topological quantum phase transitions driven by external electric fields in Sb ₂ Te ₃ thin films

Kim

et al. 2011

Proc. Natl. Acad. Sci. U.S.A.

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Using first-principles calculations, we show that topological quantum phase transitions are driven by external electric fields in thin films of Sb 2 Te 3 . The film, as the applied electric field normal to its surface increases, is transformed from a normal insulator to a topological insulator or vice versa depending on the film thickness. We identify the band topology by directly calculating the Z 2 invariant from electronic wave functions. The dispersion of edge states is also found to be consistent with the bulk band topology in view of the bulk-boundary correspondence. We present possible applications of the topological phase transition as an on/off switch of the topologically protected edge states in nano-scale devices.density functional theory | topological edge state T he concept of the topological order in condensed matter physics has provided a new perspective to the understanding of the origin of different phases and the exact quantization of Hall conductance in the quantum Hall effect (1, 2). Recently, nontrivial topological orders have been predicted theoretically and confirmed experimentally in both two-dimensional (2D) and threedimensional (3D) systems with the time-reversal invariance (3-11). These topologically nontrivial systems, called topological insulators (TIs), have intriguing properties that they develop robust conducting edge or surface states on the boundary with normal insulators (NIs) or vacuum following the bulk-boundary correspondence rule (12). These characteristic boundary states have a topological origin and are potentially useful for the design of nano-scale devices in spintronics or quantum computations.Manifestation of the nontrivial topology of occupied bands in a TI is attributed to the band inversion between occupied and unoccupied bands by large enough spin-orbit coupling (13). If the strength of the spin-orbit coupling should be reduced, the band topology would recover a trivial configuration via gap closing (14). Thus, modifying the spin-orbit strength can be a method to control the topology and induce a quantum phase transition between TI and NI phases. The band topology of a physical system may also be changed, for example, by adjusting lattice constants or internal atomic positions (15, 16). Based on this mechanism, a strain-induced topological phase transition can be driven if the original system is close to the phase boundary. In the case of 2D TIs (i.e., quantum spin Hall systems), still another factor affecting the band topology is an electrostatic scalar potential or an external electric field as an effective continuous model predicts that potential difference between upper and lower surfaces can transform topologically nontrivial thin films of Bi 2 Se 3 into topologically trivial ones (17,18). Actually, a model calculation shows that external electric fields can drive the quantum phase transition between TIs and NIs in HgTe quantum wells (19). It has been predicted that thin films of tetradymite semiconductors recently found to be 3D strong topological insulators...

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Topological quantum phase transitions driven by external electric fields in Sb ₂ Te ₃ thin films

Kim

et al. 2011

Proc. Natl. Acad. Sci. U.S.A.

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“…Therefore, under the combined operation of TRS and IS, E n,m (k) ¼ E n,k (k), hence, the energy band is doubly degenerate locally at each k. Under this condition, whenever an ABC occurs between the valence and conduction bands, a 3D DP with fourfold degeneracy can be generated. According to Murakami et al [4][5][6] , such an ABC can be achieved only under certain limited conditions because of the strong repulsion between degenerate bands. Namely, only when the valence and conduction bands have the opposite parities, an ABC can occur at a TRIM by tuning an external parameter m. In this case, a 3D DP appears at the quantum critical point (m ¼ m c ) between a normal insulator and a Z 2 topological insulator (see Fig.…”

Section: Resultsmentioning

confidence: 99%

“…For instance, as pointed out by Murakami 4 , at the timereversal invariant momentum k ¼ k TRIM , where k and À k are equivalent, all odd functions in H(k) vanish. In the case of (ii) with P ¼ cosyt z À sinyt x , a 1,2,3,4 (k TRIM ) ¼ 0.…”

Section: Methodsmentioning

confidence: 99%

“…However, the recent progress in the field of topological insulators has shown that stable 2D Dirac fermions exist ubiquitously on the surface of 3D topological insulators 1,2 . Moreover, through the careful studies on the topological phase transition between a 3D topological insulator and a normal insulator [4][5][6][7][8] , it is demonstrated that even the 3D Dirac fermions with the linear dispersion in all three momentum directions can be observed in the same material if we can reach the quantum critical point. As the 3D Dirac point (DP) with fourfold degeneracy does not carry a Chern number, the degeneracy at the gap-closing point can be easily lifted by small external perturbations, hence, the 3D Dirac fermions can be observed only at the single quantum critical point.…”

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Classification of stable three-dimensional Dirac semimetals with nontrivial topology

2014

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A three-dimensional (3D) Dirac semimetal (SM) is the 3D analogue of graphene having linear energy dispersion around Fermi points. Owing to the nontrivial topology of electronic wave functions, the 3D Dirac SM shows nontrivial physical properties and hosts various exotic quantum states such as Weyl SMs and topological insulators under proper external conditions. There are several kinds of Dirac SMs proposed theoretically and partly confirmed experimentally, but its unified picture is still missing. Here we propose a general framework to classify stable 3D Dirac SMs in systems having the time-reversal, inversion and uniaxial rotational symmetries. We show that there are two distinct classes of 3D Dirac SMs. In one class, the Dirac SM possesses a single Dirac point (DP) at a time-reversal invariant momentum on the rotation axis. Whereas the other class of Dirac SMs have a pair of DPs created by band inversion, and carry a quantized topological invariant.

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Hybridization of Topological Surface States and Emergent States

Murakami

2015

Topological Insulators

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IntroductionSince the time theoretical predictions were made of topological insulators (TIs) [1][2][3][4][5][6], various types of topological states have been studied theoretically and experimentally. Integer quantum Hall systems [7] are also among topological phases, which were discovered even before TIs. Nevertheless, an important feature of TIs is the formation of a topological state in absence of external magnetic fields. Another important and unique feature of TIs is that they can be realized in three dimensions (3D) [8][9][10]. This leads to the proposals of Weyl and Dirac semimetals (DSs). Both have 3D Dirac cones in their band structure.The surface states of TIs represent a manifestation of bulk topological order, and in this sense they are distinct from conventional surface states. In contrast to the conventional surface states, which crucially depend on the details of the surfaces, the surface states of TIs have some robust properties [4,5]. One typical form of the surface states of TI is a single Dirac cone in the surface two-dimensional (2D) Brillouin zone. Such a single Dirac cone in a 2D Brillouin zone cannot be realized in a purely 2D system, and is therefore unique to topological insulator surfaces. Because of such a topological origin and its unique band structure, it gives various interesting phenomena when one tries to modify such surface states.In this chapter, we review the band structures of such topological phases, in particular the TIs and Weyl semimetals (WSs). We also show emergent states when more than one surface Dirac cones hybridize each other. In Section 2.2 we review the TIs and Weyl semimetals. In addition, we show how the topological phase transitions between TIs and normal insulators (NIs) occur. In this discussion, we will see that in inversion-asymmetric systems, WS phases naturally emerge. In Section 2.3 we show various systems where more than one surface states of TIs hybridize with one another. Here, an important concept is the chirality of the surface Dirac cone. The analysis is then focused on the physics of thin films, where the surface states on the top surface and the bottom surface eventually hybridize for small thicknesses. Then we discuss the interface between two TIs, which offers an Topological Insulators: Fundamentals and Perspectives, First Edition. Edited

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Tuning phase transition between quantum spin Hall and ordinary insulating phases

Cited by 168 publications

References 25 publications

Topological quantum phase transitions driven by external electric fields in Sb ₂ Te ₃ thin films

Topological quantum phase transitions driven by external electric fields in Sb ₂ Te ₃ thin films

Classification of stable three-dimensional Dirac semimetals with nontrivial topology

Hybridization of Topological Surface States and Emergent States

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Tuning phase transition between quantum spin Hall and ordinary insulating phases

Cited by 168 publications

References 25 publications

Topological quantum phase transitions driven by external electric fields in Sb 2 Te 3 thin films

Topological quantum phase transitions driven by external electric fields in Sb 2 Te 3 thin films

Classification of stable three-dimensional Dirac semimetals with nontrivial topology

Hybridization of Topological Surface States and Emergent States

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Topological quantum phase transitions driven by external electric fields in Sb ₂ Te ₃ thin films

Topological quantum phase transitions driven by external electric fields in Sb ₂ Te ₃ thin films