はじめにtomi/kougi_note/IntroTIOhtsuki.pdfCdTe has a band ordering similar to GaAs with a s-type (! 6) conduction band, and p-type valence bands (! 8, ! 7) which are separated

はじめに• 講義資料：大槻東巳のホームページ，講義資料からダウ

ンロードする

• 今日の授業と資料を基に1月29日までにA4用紙1枚でレポートを作成。課題はトポロジカル絶縁体とは何か？

• 提出先：4-389A

トポロジカル絶縁体入門物理学序論

上智大学物理領域大槻東巳

2016年のノーベル物理学賞

サウレスハルデインコスタリッツ½ ¼ ¼

”for theoretical discoveries of topological phase transitions and topological phases of matter”

トポロジー

https://www.nobelprize.org/nobel_prizes/physics/laureates/2016/fig_fy_en_16_topology.pdf

アウトライン• トポロジカル絶縁体(topological insulator, TI)とは何か？

• 昔から知られていたトポロジカル絶縁体 : 量子ホール効果(QHE)における量子ホール絶縁体quantum Hall insulator (QHI)

• 量子ホール絶縁体以外のトポロジカル絶縁体の予言と発見 à2次元量子スピンホール系 (quantum spin Hall systems (QSHE)(HgTe))と3次元 TI (Bi2Se3)

• CdTe/HgTe/CdTe量子井戸における量子スピンホール効果

• GaN/InN/GaN量子井戸の可能性

References :1) 東北大学金属材料研野村健太郎准教授による講義ノートhttp://www-lab.imr.tohoku.ac.jp/~nomura/note.html2) Review: M. Hasan, C. Kane: Rev. Mod. Phys. 82 (2010) 30453) Review: X.-L. Qi, S.-C. Zhang: Rev. Mod. Phys. 83 (2011) 10574) Miao et al., Topological Insulator Transition in a GaN/InN/GaN Quantum Well,

PRL 109, 186803 (2012) 5) Photonic topological insulator 1: Haldane, Raghu: PRL 100, 013904 (2008)6) Photonic topological insulator 2: Khanikaev et al.: Nature materials, 12

(2013)233

1. トポロジカル絶縁体とは ?

• バンドギャップ絶縁体でギャップ内に端/表面状態をもつもの。

• ただの表面状態でなく，トポロジカルな要因

で保護されているため，ランダムネス，電子

間相互作用，電子格子相互作用の影響を受け

ない。

• 電流やスピンの方向が特徴的

2. 昔から知られていたトポロジカル絶縁体，QHI in QHE@>)#$>=�A)**�%--%($

4*"$'"#+B�C,12) )#2�D%??%1�EFGHI

2

1 1,2,3,jhR j

j e� � �

@>)#$>=�A)**�%--%($

4*"$'"#+B�C,12) )#2�D%??%1�EFGHI

2

1 1,2,3,jhR j

j e� � �

σ yx =e2

hj , σ xx = 0

絶縁体

異なるタイプの絶縁相

量子ホール効果をトポロジカル数で解釈する

• 久保公式

周期ポテンシャルの場合，ブロッホ関数が固有関数。

ストークスの定理

TOPOLOGY OF HALL CONDUCTANCE 349

tity remains invariant under this transformation. From Eq. (3.8) the corresponding transformation of ji(k,, k2) is given by

A’@, , kd = &k,, kz) + zV,f(k,, k,). (3.11)

It is easy to see from Eqs. (3.9) and (3.11) that og is invariant under the transformation (3.10).

The non-trivial topology arises when the phase of the wavefunction cannot be determined uniquely and smoothly in the entire magnetic Brillouin zone. The transformation (3.10) implies that the overall phase factor for each state vector 1 uklk2) can be chosen arbitrary. This phase can be determined, for example, by demanding that a component of the state vector u~,,Jx(~), y”‘) = (x(O), ~“‘1 uklk2) is real. However, this convention is not enough to fix the phase on the entire magnetic Brillouin zone, since z.Q&x’~), y(O)) vanishes for some (k,, k2). The existence of zeros of z+,,Jx, y) has been shown in Section II. For the sake of simplicity, consider the case where u,,,,(x(~), y(O)) vanishes only at one point (k\“, kh”) in the magnetic Brillouin zone. See Fig. 1. Divide T2 into two pieces HI and H,, such that H, con- tains (k!O), k$O)). We adopt a different convention in H, so that another component of the state vector z.++(x’~), y”)) = (x’~‘, y”‘] u~,~,) is real, where (x(l), y(l)) and H, are chosen such that LQ,,~~(x(‘), y’“) does not vanish in H,. Thus the overall phase is uniquely determined on the entrie T*. In Fig. 1, a phase of one component of the state vector u~,~~(x”‘, y(O)) = (.~~O’y(‘)l uklk2) is schematically drawn.

IT rtt T 0

kl 27T - qa

FIG. 1. Schematic diagram of a phase of a wavefunction in the magnetic Brillouin zone. The Brillouin zone is actually a torus, so the edges (k,, kZ) = (0, k2) and (2z/qa, k,); and also the edges (k, . 0) and (k, , 2x/b) must be identified.

Thouless et al. PRL 1982

v=dH(k)/dk に注意

量子ホール効果から得られた教訓

• バルクの波動関数が非自明な位相構造を持つ à この位相構造をトポロジカル数で定義à 量子ホールコンダクタンスはe2/h x トポロジカル数となり厳密に量子化

• トポロジカル数は整数のみを取るので，ある程度の摂動を受けてもコンダクタンスは変化しない à 10-9の精度で量子化

• この議論の弱点：トポロジカル数はブロッホ関数で定義されているが，量子ホール効果は乱れた2次元電子系で観測されている

バルク vs. エッジ描像• バルクのトポロジカル数が nの場合， n本のエッジ状態

がサンプルの端に現れる。

• àバルクの波動関数のトポロジーをトポロジカル数(Chern number)で定義する代わりに，実験的には試料のエッジ状態を調べればよい

http://physics.aps.org/articles/v2/15

エッジ描像の利点

• エッジ電流は電流測定に直接きいてくる

• ランダムネスがあっても定義できる

• エッジ状態がランダムネス・相互作用に対して安定かどうかはある程度直感的に分かる

3. 量子ホール絶縁体以外のトポロジカル絶縁体に向けて

• 量子ホール効果の発見1980年，分数量子ホール効果が1982年，それぞれにノーベル賞が授与され済み

• 2000年代前半 à スピントロニクスの研究の発展o 電流ではなく，スピンを流したい。しかも磁石や磁場を使わず

• 2次元系でスピンを流す：時間反転対称性のある量子ホール効果 à 量子スピンホール効果（QSHE）

• 2010年前後：3次元のトポロジカル絶縁体

• いずれもスピン軌道相互作用がキー

実際の物質o 2D HgTe QSHE 後で詳しく述べる

o 3D • Bi2Se3 (2009, Yu-Qi Xia, Zahid Hasan), • Bi0.9Sb0.1 (2008, David Hsieh, Zahid Hasan)• Bi2Te3• TlBiSe2

H Zhang et al., Nature Physics, 2009

With randomness

ARPES

QSEに向けて：HgTeと CdTeの比較

1D gapless edge states that lie inside the bulk insulating gap.The edge states have a distinct helical property: Two stateswith opposite spin polarization counterpropagate at a givenedge (Kane and Mele, 2005a; Wu et al., 2006; Xu andMoore, 2006). For this reason they are also called helicaledge states, i.e., the spin is correlated with the direction ofmotion (Wu et al., 2006). The edge states come in Kramersdoublets, and TR symmetry ensures the crossing of theirenergy levels at special points in the Brillouin zone (BZ).Because of this level crossing, the spectrum of a QSH insu-lator cannot be adiabatically deformed into that of a topo-logically trivial insulator without helical edge states.Therefore, in this sense, the QSH insulator represents a newtopologically distinct state of matter. In the special case thatSOC preserves a Uð1Þs subgroup of the full SU(2) spinrotation group, the topological properties of the QSH statecan be characterized by the spin Chern number (Sheng et al.,2006). More generally, the topological properties of the QSHstate are mathematically characterized by a Z2 topologicalinvariant (Kane and Mele, 2005b). States with an even num-ber of Kramers pairs of edge states at a given edge aretopologically trivial, while those with an odd number aretopologically nontrivial. The Z2 topological quantum numbercan also be defined for generally interacting systems andexperimentally measured in terms of the fractional chargeand quantized current on the edge (Qi, Hughes, and Zhang,2008a), and spin-charge separation in the bulk (Qi and Zhang,2008; Ran et al., 2008).

In this section, we focus on the basic theory of the QSHstate in the HgTe/CdTe system because of its simplicity andexperimental relevance and provide an explicit and pedagog-ical discussion of the helical edge states and their transportproperties. There are several other theoretical proposals forthe QSH state, including bilayer bismuth (Murakami, 2006),and the ‘‘broken-gap’’ type-II AlSb/InAs/GaSb quantumwells (Liu, Hughes et al., 2008). Initial experiments in theAlSb/InAs/GaSb system already show encouraging signa-tures (Knez et al., 2010). The QSH system has also beenproposed for the transition metal oxide Na2IrO3 (Shitadeet al., 2009). The concept of the fractional QSH state wasproposed at the same time as the QSH state (Bernevig andZhang, 2006) and has been recently investigated theoreticallyin more detail (Young et al., 2008; Levin and Stern, 2009).

A. Effective model of the two-dimensionaltime-reversal-invariant topological insulatorin HgTe/CdTe quantum wells

In this section we review the basic electronic structure ofbulk HgTe and CdTe and presented a simple model firstintroduced by Bernevig, Hughes, and Zhang (2006) (BHZ)to describe the physics of those subbands of HgTe/CdTequantum wells which are relevant for the QSH effect. HgTeand CdTe crystallize in the zinc blende lattice structure. Thisstructure has the same geometry as the diamond lattice, i.e.,two interpenetrating face-centered-cubic lattices shiftedalong the body diagonal, but with a different atom on eachsublattice. The presence of two different atoms per lattice sitebreaks inversion symmetry and thus reduces the point groupsymmetry from O h (cubic) to Td (tetrahedral). However, even

though inversion symmetry is explicitly broken, this has onlya small effect on the physics of the QSH effect. To simplifythe discussion, we first ignore this bulk inversion asymmetry(BIA).

For both HgTe and CdTe, the important bands near theFermi level are close to the ! point in the Brillouin zone[Fig. 2(a)]. They are a s-type band (!6), and a p-type bandsplit by SOC into a J ¼ 3=2 band (!8) and a J ¼ 1=2 band(!7). CdTe has a band ordering similar to GaAs with a s-type(!6) conduction band, and p-type valence bands (!8, !7)which are separated from the conduction band by a largeenergy gap ($ 1:6 eV). Because of the large SOC present inthe heavy element Hg, the usual band ordering is inverted :The negative energy gap of %300 meV indicates that the !8

band, which usually forms the valence band, is above the !6

band. The light-hole !8 band becomes the conduction band,the heavy-hole band becomes the first valence band, and thes-type band (!6) is pushed below the Fermi level to liebetween the heavy-hole band and the spin-orbit split-offband (!7) [Fig. 2(a)]. Because of the degeneracy betweenheavy-hole and light-hole bands at the ! point, HgTe is azero-gap semiconductor.

When HgTe-based quantum well structures are grown, thepeculiar properties of the well material can be utilized totune the electronic structure. For wide QW layers, quantumconfinement is weak and the band structure remains inverted.However, the confinement energy increases when the wellwidth is reduced. Thus, the energy levels will be shifted and,eventually, the energy bands will be aligned in a ‘‘normal’’way, if the QW thickness dQW falls below a critical thicknessdc. We can understand this heuristically as follows: for thinQWs the heterostructure should behave similarly to CdTeand have a normal band ordering, i.e., the bands withprimarily !6 symmetry are the conduction subbands and

FIG. 2 (color). (a) Bulk band structure of HgTe and CdTe;(b) schematic picture of quantum well geometry and lowest sub-bands for two different thicknesses. From Bernevig et al., 2006.

Xiao-Liang Qi and Shou-Cheng Zhang: Topological insulators and superconductors 1061

Rev. Mod. Phys., Vol. 83, No. 4, October–December 2011

s軌道

heavy hole

light holes軌道

Hamiltonian s-orbital: Kramers doublet |s↑> and |s ↓>p-orbital: |px+ i py ↑>, |-(px- i py) ↓>Near the Γ point: |s+>, |px+ i py ↑>, |s->, |-(px- i py) ↓>

E(k) = ε(k)± (M − Bk2 )2 + A2k2

（Science ‘06）

H(k), 2x2行列

M = Es-Ep @ Γ点

sとpの間のエネルギーMを変えると

M>0 M=0 M<0

s

ps

p

TIと他の表面状態の違い

エッジ状態はなぜ安定か？

(ランダムネス，相互作用などに対して)

偶数個の表面バンドà不安定，表面バンドの数は0か1àZ2型

バルクのトポロジカル数がゼロで

ないと表面・端状態が現れるわけ

• 真空ではトポロジカル数が0なので界面においてトポロジカル数が不連続になってしまう

• トポロジカル数が0（真空）と有限の領域（TI）を

つなぐため，ギャップレスの表面状態が現れる必要がある

a ba b

symmetry as an applied magnetic field would), in simplified models introduced in around 2003 it can lead to a quantum spin Hall effect, in which electrons with opposite spin angular momentum (commonly called spin up and spin down) move in opposite directions around the edge of the droplet in the absence of an external magnetic field2 (Fig. 2b). These simplified models were the first steps towards understanding topological insulators. But it was unclear how realistic the models were: in real materials, there is mixing of spin-up and spin-down electrons, and there is no conserved spin current. It was also unclear whether the edge state of the droplet in Fig. 2b would survive the addition of even a few impurities.

In 2005, a key theoretical advance was made by Kane and Mele3. They used more realistic models, without a conserved spin current, and showed how some of the physics of the quantum spin Hall effect can survive. They found a new type of topological invariant that could be computed for any 2D material and would allow the prediction of whether the material had a stable edge state. This allowed them to show that, despite the edge not being stable in many previous models, there are realistic 2D materials that would have a stable edge state in the absence of a magnetic field; the resultant 2D state was the first topological insulator to be understood. This non-magnetic insulator has edges that act like perfectly conducting one-dimensional electronic wires at low tempera-tures, similar to those in the quantum Hall effect.

Subsequently, Bernevig, Hughes and Zhang made a theoretical prediction that a 2D topological insulator with quantized charge con-ductance along the edges would be realized in (Hg,Cd)Te quantum wells4. The quantized charge conductance was indeed observed in this system, as a quantum-Hall-like plateau in zero magnetic field, in 2007 (ref. 5). These experiments are similar to those on the quantum Hall effect in that they require, at least so far, low temperature and artificial 2D materials (quantum wells), but they differ in that no magnetic field is needed.

Going 3DThe next important theoretical development, in 2006, was the realization6–8 that even though the quantum Hall effect does not general-ize to a genuinely 3D state, the topological insulator does, in a subtle way. Although a 3D ‘weak’ topological insulator can be formed by layering 2D versions, similar to layered quantum Hall states, the resultant state is not stable to disorder, and its physics is generally similar to that of the 2D state. In weak topological insulators, a dislocation (a line-like defect

in the crystal) will always contain a quantum wire like that at the edge of the quantum spin Hall effect (discussed earlier), which may allow 2D topological insulator physics to be observed in a 3D material9.

There is also, however, a ‘strong’ topological insulator, which has a more subtle relationship to the 2D case; the relationship is that in two dimensions it is possible to connect ordinary insulators and topologi-cal insulators smoothly by breaking time-reversal symmetry7. Such a continuous interpolation can be used to build a 3D band structure that respects time-reversal symmetry, is not layered and is topologically non-trivial. It is this strong topological insulator that has protected metallic surfaces and has been the focus of experimental activity.

Spin–orbit coupling is again required and must mix all components of the spin. In other words, there is no way to obtain the 3D strong topologi-cal insulator from separate spin-up and spin-down electrons, unlike in the 2D case. Although this makes it difficult to picture the bulk physics of the 3D topological insulator (only the strong topological insulator will be discussed from this point), it is simple to picture its metallic surface6.

The unusual planar metal that forms at the surface of topological insulators ‘inherits’ topological properties from the bulk insulator. The simplest manifestation of this bulk–surface connection occurs at a smooth surface, where momentum along the surface remains well defined: each momentum along the surface has only a single spin state at the Fermi level, and the spin direction rotates as the momentum moves around the Fermi surface (Fig. 3). When disorder or impurities are added at the surface, there will be scattering between these surface states but, crucially, the topological properties of the bulk insulator do not allow the metallic surface state to vanish — it cannot become local-ized or gapped. These two theoretical predictions, about the electronic structure of the surface state and the robustness to disorder of its metallic behaviour, have led to a flood of experimental work on 3D topological insulators in the past two years.

Experimental realizationsThe first topological insulator to be discovered was the alloy BixSb1−x, the unusual surface bands of which were mapped in an angle-resolved photoemission spectroscopy (ARPES) experiment10,11. In ARPES exper-iments, a high-energy photon is used to eject an electron from a crystal, and then the surface or bulk electronic structure is determined from an analysis of the momentum of the emitted electron. Although the surface structure of this alloy was found to be complex, this work launched a search for other topological insulators.

Figure 1 | Metallic states are born when a surface unties ‘knotted’ electron wavefunctions. a, An illustration of topological change and the resultant surface state. The trefoil knot (left) and the simple loop (right) represent different insulating materials: the knot is a topological insulator, and the loop is an ordinary insulator. Because there is no continuous deformation by which one can be converted into the other, there must be a surface where the string is cut, shown as a string with open ends (centre), to pass between the two knots; more formally, the topological invariants cannot remain

defined. If the topological invariants are always defined for an insulator, then the surface must be metallic. b, The simplest example of a knotted 3D electronic band structure (with two bands)35, known to mathematicians as the Hopf map. The full topological structure would also have linked fibres on each ring, in addition to the linking of rings shown here. The knotting in real topological insulators is more complex as these require a minimum of four electronic bands, but the surface structure that appears is relatively simple (Fig. 3).

195

NATURE|Vol 464|11 March 2010 PERSPECTIVE INSIGHT

194-198 Insight Moore NS.indd 195194-198 Insight Moore NS.indd 195 3/3/10 11:30:563/3/10 11:30:56

© 20 Macmillan Pu blishers Limited. All rights reserved10

a ba b












195




トポロジカル絶縁体

通常の絶縁体，もしくは真空a ba b












195




表面状態

右側通行と左側通行の例（物材機構胡博士のスライドより）

4. 2D realization in HgTe

Comparison of HgTe with Cd Te

1D gapless edge states that lie inside the bulk insulating gap.The edge states have a distinct helical property: Two stateswith opposite spin polarization counterpropagate at a givenedge (Kane and Mele, 2005a; Wu et al., 2006; Xu andMoore, 2006). For this reason they are also called helicaledge states, i.e., the spin is correlated with the direction ofmotion (Wu et al., 2006). The edge states come in Kramersdoublets, and TR symmetry ensures the crossing of theirenergy levels at special points in the Brillouin zone (BZ).Because of this level crossing, the spectrum of a QSH insu-lator cannot be adiabatically deformed into that of a topo-logically trivial insulator without helical edge states.Therefore, in this sense, the QSH insulator represents a newtopologically distinct state of matter. In the special case thatSOC preserves a Uð1Þs subgroup of the full SU(2) spinrotation group, the topological properties of the QSH statecan be characterized by the spin Chern number (Sheng et al.,2006). More generally, the topological properties of the QSHstate are mathematically characterized by a Z2 topologicalinvariant (Kane and Mele, 2005b). States with an even num-ber of Kramers pairs of edge states at a given edge aretopologically trivial, while those with an odd number aretopologically nontrivial. The Z2 topological quantum numbercan also be defined for generally interacting systems andexperimentally measured in terms of the fractional chargeand quantized current on the edge (Qi, Hughes, and Zhang,2008a), and spin-charge separation in the bulk (Qi and Zhang,2008; Ran et al., 2008).

In this section, we focus on the basic theory of the QSHstate in the HgTe/CdTe system because of its simplicity andexperimental relevance and provide an explicit and pedagog-ical discussion of the helical edge states and their transportproperties. There are several other theoretical proposals forthe QSH state, including bilayer bismuth (Murakami, 2006),and the ‘‘broken-gap’’ type-II AlSb/InAs/GaSb quantumwells (Liu, Hughes et al., 2008). Initial experiments in theAlSb/InAs/GaSb system already show encouraging signa-tures (Knez et al., 2010). The QSH system has also beenproposed for the transition metal oxide Na2IrO3 (Shitadeet al., 2009). The concept of the fractional QSH state wasproposed at the same time as the QSH state (Bernevig andZhang, 2006) and has been recently investigated theoreticallyin more detail (Young et al., 2008; Levin and Stern, 2009).

A. Effective model of the two-dimensionaltime-reversal-invariant topological insulatorin HgTe/CdTe quantum wells

In this section we review the basic electronic structure ofbulk HgTe and CdTe and presented a simple model firstintroduced by Bernevig, Hughes, and Zhang (2006) (BHZ)to describe the physics of those subbands of HgTe/CdTequantum wells which are relevant for the QSH effect. HgTeand CdTe crystallize in the zinc blende lattice structure. Thisstructure has the same geometry as the diamond lattice, i.e.,two interpenetrating face-centered-cubic lattices shiftedalong the body diagonal, but with a different atom on eachsublattice. The presence of two different atoms per lattice sitebreaks inversion symmetry and thus reduces the point groupsymmetry from O h (cubic) to Td (tetrahedral). However, even

though inversion symmetry is explicitly broken, this has onlya small effect on the physics of the QSH effect. To simplifythe discussion, we first ignore this bulk inversion asymmetry(BIA).

For both HgTe and CdTe, the important bands near theFermi level are close to the ! point in the Brillouin zone[Fig. 2(a)]. They are a s-type band (!6), and a p-type bandsplit by SOC into a J ¼ 3=2 band (!8) and a J ¼ 1=2 band(!7). CdTe has a band ordering similar to GaAs with a s-type(!6) conduction band, and p-type valence bands (!8, !7)which are separated from the conduction band by a largeenergy gap ($ 1:6 eV). Because of the large SOC present inthe heavy element Hg, the usual band ordering is inverted :The negative energy gap of %300 meV indicates that the !8

band, which usually forms the valence band, is above the !6

band. The light-hole !8 band becomes the conduction band,the heavy-hole band becomes the first valence band, and thes-type band (!6) is pushed below the Fermi level to liebetween the heavy-hole band and the spin-orbit split-offband (!7) [Fig. 2(a)]. Because of the degeneracy betweenheavy-hole and light-hole bands at the ! point, HgTe is azero-gap semiconductor.

When HgTe-based quantum well structures are grown, thepeculiar properties of the well material can be utilized totune the electronic structure. For wide QW layers, quantumconfinement is weak and the band structure remains inverted.However, the confinement energy increases when the wellwidth is reduced. Thus, the energy levels will be shifted and,eventually, the energy bands will be aligned in a ‘‘normal’’way, if the QW thickness dQW falls below a critical thicknessdc. We can understand this heuristically as follows: for thinQWs the heterostructure should behave similarly to CdTeand have a normal band ordering, i.e., the bands withprimarily !6 symmetry are the conduction subbands and

FIG. 2 (color). (a) Bulk band structure of HgTe and CdTe;(b) schematic picture of quantum well geometry and lowest sub-bands for two different thicknesses. From Bernevig et al., 2006.

Xiao-Liang Qi and Shou-Cheng Zhang: Topological insulators and superconductors 1061

Rev. Mod. Phys., Vol. 83, No. 4, October–December 2011

HgTeの厚さを変えるとバンド反転

ではどうやってスピン電流を確認したか？

Quantized 4 Termianl Conductance (Konig et al., Science 2007)

Ref. 7: Miao et al., Topological Insulator

Transition in a GaN/InN/GaN Quantum Well

PRL 109, 186803 (2012)

ultrathin InN layers embedded into GaN

inverted-band transition, a small SOI is sufficient to drivethe TI transition. Furthermore, we found that the largepolarization field induces a considerable Rashba SOI.The Rashba SOI in QWs has attracted attention becauseit can be adjusted by gate voltage [23,24] and band engi-neering [25,26]. It thus provides a controllable approach toenhancing spin-polarized transport in nonmagnetic semi-conductors. From the eight-band Kane model, we estimatethe strength of this Rashba SOI to be on the order of 1 to2 meV [Fig. 3(d)]. This is comparable to the Rashba SOIinduced by an external electric field in InAs and HgTequantum wells [11,12]. A Rashba SOI of this magnitudeusually occurs only in systems containing heavier atoms.The unusually large Rashba SOI inGaN=InN=GaNQWs isdue to the strength of the polarization field, which easilyexceeds 10 times the strength of an applied electric fieldresulting from a gate voltage. Such a large Rashba effectwas noted in a previous study on a GaN=In0:5Ga0:5N=GaNQW [27].

The proposed inverted band semiconductor structurebased on thin InN QWs and the presence of a TI state offerconsiderable advantages over other systems includinggraphene [3], the Bi chalcogenides [7], and the Heusler

compounds [8]. GaN=InN=GaN QW structures can beintegrated in nitride-based transistors, which are alreadyextensively used in high-frequency and high-power de-vices [28]. Because charge carriers can screen the polar-ization fields, the polarization potential can be controlledby adjusting the carriers densities in the QW, and thereforethe TI transition can be controlled by doping or applying abias voltage.High quality InN layers in GaN matrix with a thickness

of 1–2 ML and atomically sharp interfaces have alreadybeen fabricated [29,30]. Both photoluminescence [29] andelectroluminescence [31] originating from electron-holerecombination have been observed. We have estimatedthat the critical thickness of InN layers pseudomorphicallygrown on a GaN substrate is as large as 17 A, whichcorresponds to more than 5 InNMLs [19]. This is sufficientto allow the topological insulator transition to occur.In summary, based on first-principles calculations and

an effective low-energy k ! p model, we have demon-strated that ultrathin GaN=InN=GaN QWs can undergoan inverted band transition and become a topologicalinsulator. This quantum phase transition is driven by thestrong polarization fields originating from the wurtzitesymmetry and lattice mismatch, and by the Rashbaspin-orbit interaction resulting from the field. This is thefirst demonstration of the formation of a TI phase causedby intrinsic polarization in commonly used semiconduc-tors with weak intrinsic SOI. Since polarization fieldsoccur in many materials, a similar mechanism may applyto other systems as well. Our approach may pave the waytoward integrating controllable TIs with conventionalsemiconductor devices.We thank Professor X.-L. Qi for valuable discussions.

M. S.M. was supported as part of the Center for EnergyEfficient Materials, an Energy Frontier Research Centerfunded by the U.S. DOE-BES (Grant No. DE-SC0001009).M. S.M. also thanks the ConvEne-IGERT Program (NSF-DGE0801627) and the MRSEC program (NSF-DMR1121053). Q.Y. was supported by the UCSB SolidState Lighting and Energy Center. C.V. dW. was supportedby NSF under Grant No. DMR-0906805. W.K. L, L. L. L.and K. C. were supported by the National Basic ResearchProgram of China (973 Program) under GrantNo. 2011CB922204 and the NSF of China under GrantNo. 10934007. The electronic structure calculations madeuse of NSF-funded TeraGrid resources under GrantNo. DMR07-0072N.

*[email protected]†[email protected]

[1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82 , 3045(2010).

[2] X. L. Qi and S. C. Zhang, Rev. Mod. Phys. 83 , 1057(2011).

FIG. 3 (color online). (a) Band structure of a 16 AGaN=InN=GaN QW obtained from the six-band effectiveHamiltonian. (b) Schematic of an infinite long spin Hall barwith a width (along y) of 1000 A. The thickness along the [0001]growth direction (labeled as z) comprises the InN QW plus two200-A-thick GaN barrier layers on either side. The yellow linesshow the helical edge states, and the green arrows show the spinorientation. The short black arrows indicate the polarization-induced electric field. (c) Band structure of the Hall bar obtainedby solving the effective six-band model. The insets show thedensity distributions of one Kramers pair of edge states: on theleft the spin-up state at kk ¼ # 0:1 !A# 1, on the right the spin-down state at kk ¼ 0:1 !A# 1. (d) Rashba spin splitting (RSS) ofelectron (green), HH (blue) and LH (red) subbands.

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polarization can invert the bands of a GaN=InN=GaN QWwhen the InN thickness exceeds three monolayers (ML)–aML being defined as a double layer of In and N (with athickness of 3.12 A in strained InN) along the [0001] axis.We then use an eight-band k ! p model that includes bothSOI and the strong polarization field to prove that such asystem can become a TI and possess edge states in theenergy gap of a Hall bar structure.

The proposed TI has many unique advantages, including(1) it can be realized based on commonly used semicon-ductors and be integrated into various devices, (2) it isdriven by large intrinsic polarization fields, (3) the TI statecan be manipulated by applying external fields or injectingcharge carriers and can be adjusted by standard semicon-ductor techniques, including doping, alloying and varyingtheQW thickness, and (4) the polarization field can induce alarge Rashba SOI in this system containing only lightelements, which provides a new approach to manipulatingspin freedom in such systems.

Our proposed QW consists of InN layers sandwichedbetween GaN along the [0001] direction [Fig. 1(b)].Group-III nitrides (III-N), including InN, GaN, and AlN,have been intensively studied because of their applicationsin light emitters, high-frequency transistors, andmany otherareas. An important feature of III-N compounds is thatalthough their band gap varies from 0.7 eV to 6.2 eV, allthree compounds and their alloys are stable in the samewurtzite structure with relatively modest variations in alattice constant. This feature, combined with advancedgrowth techniques, allows the fabrication of high-qualityalloys and heterojunctions. InN has a fundamental gap ofonly " 0:7 eV [13], resulting in strong coupling between theelectron and hole states and leading to a large nonparabo-licity of the bands around the! point [Fig. 1(c)] aswell as anexceedingly small electron effective mass of 0:067m0 [14].

Our first-principles calculations are based on densityfunctional theory (DFT) with a plane-wave basis set andprojector augmented waves [15], as implemented in theVASP program [16]. Because the accuracy of the band gapis of key importance for this Letter, we employ a hybridfunctional [17]. Recent calculations have demonstrated thereliability of this approach for producing structural pa-rameters as well as band gaps in agreement with experi-ment. Using a standard mixing parameter of 0.25, we foundthe band gaps of GaN and InN to be 3.25 and 0.62 eV(Ref. [18]). The lattice parameters a ¼ b and c are found tobe 3.182 and 5.175 A for GaN and 3.548 and 5.751 A forInN. The GaN=InN=GaN QW is modeled by a supercellconsisting of 1 to 5 atomic layers of InN and 23 or 24atomic layers of GaN (periodicity requires that the totalnumber of atomic layers be even). The SOI is not includedin the first-principles calculations, but its effect is includedin the k ! p model.

Figures 2(a) and 2(b) show band structures around the !point calculated with DFT using the hybrid functional of

Heyd, Scuseria and Ernzerhof (HSE) [17], and Fig. 2(c)shows the band gap as a function of the number of InNlayers. At 2ML, the system has a gap of 0.82 eV (larger thanthe fundamental gap due to strain and quantum confine-ment), but at 3 ML this is reduced to 0.06 eV [Fig. 2(c)]. Inboth these cases, the band structures are still normal in thesense that the heavy hole (HH) and light hole (LH) states aredegenerate at the ! point (!6) and are lower in energy thanthe electron state (E) (!1). At 4 ML, however, the systemexhibits an inverted band structure, in which the !6 statesare 0.10 eV higher than the !1 state. In Figs. 2(a) and 2(b),the states around the! point are denoted by their symmetry.Such an inverted band structure is a signature of the tran-sition to a TI state.We now discuss the details of this transition to an

inverted band structure. For an ultra-thin QW, the gapbetween the valence and conduction states is determinedby the interplay of three factors, namely quantum confine-ment, polarization field, and strain. The quantum-confinement effect is large for these thin QWs, explaining

FIG. 2 (color online). (a) and (b) Band structure of aGaN=InN=GaN QW around the ! point for 2 and 4 ML ofInN, based on first-principles DFT-HSE calculations. The greenlines represent electron states, red lines light-hole states, andblue lines heavy-hole states. (c) Calculated energy gaps asa function of the thickness of InN layers for polar ([0001])(blue squares) and nonpolar (½10"10%) (orange diamonds)GaN=InN=GaN QWs. The thicknesses were scaled to the thick-ness of 1 ML of InN in a polar QW and therefore correspond tothe number of InN layers in the polar case. The inset shows a½10"10% QW with two InN layers. (d) Polarization field as afunction of number of inserted InN layers calculated by DFT-HSE (blue squares) and based on theoretical polarization con-stants (orange circles). The blue and red block arrows in the insetshow the polarization direction of GaN and InN regions [19].

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T. Nakaoka’s idea

5 layers of InN(1.6nm thickness) Tunneling à 3D TI

Weyl semimetal• Weyl semimetal：H=±v (p±p0)・σ, E=±v |p±p0|• surface states appear in certain direction.

Huang et al., Nat. Commun. 6, 7373 (2015).Xu et al., Nature Phys. 11, 748–754 (2015)

Xu et al., Nature Phys. 11, 748–754 (2015) Weyl cones in NbAs

Weyl semimetal• Weyl semimetal：H=±v (p±p0)・σ, E=±v |p±p0|• surface states appear in certain direction.

Huang et al., Nat. Commun. 6, 7373 (2015).Xu et al., Nature Phys. 11, 748–754 (2015)

Xu et al., Nature Phys. 11, 748–754 (2015) Weyl cones in NbAs

band insulator to WSM

p

s

randomness

band Insulator

パラメータを変える

k

ε

Weyl semimetalE=vk

WSMの表面状態

β=0.45W=0.8

β=0.45W=1.7

β=0.45W=2.2

まとめ• トポロジカル絶縁体 à 電流やスピンの向きが偏った

端・表面状態をもつ

• これらの状態は摂動によっても壊されない

• トポロジカル絶縁体は2種類に分かれる: 量子ホール型(Z型）と量子スピンホール型(Z2型)

Documents

はじめにtomi/kougi_note/IntroTIOhtsuki.pdfCdTe has a band ordering similar to GaAs with a s-type (! 6) conduction band, and p-type valence bands (! 8, ! 7) which are separated