トポロジカル絶縁体入門 - 上智大学大学院理工学研究科物理学領域

トポロジカル絶縁体入門上智大学物理領域
大槻東巳
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 à 2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
h.p://www-‐‑lab.imr.tohoku.ac.jp/~nomura/note.html
2) Review: M. Hasan, C. Kane: Rev. Mod. Phys. 82 (2010) 3045
3)  Review: X.-‐‑L. Qi, S.-‐‑C. Zhang: Rev. Mod. Phys. 83 (2011) 1057
4)  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
@>)#$>= A)** %--%($
@>)#$>=
A)** %--%($
縁体，QHI in QHE
1 h
Rj
j 1, 2,3,
2
j e1 h
Rj
j 1, 2,3,
2
je
e2
σ yx = j , σ xx = 0
h
絶縁体
4*"$'"#+B C,12) )#2 D%??%1 EFGHI
4*"$'"#+B C,12) )#2 D%??%1 EFGHI
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
It is easy to see from Eqs. (3.9) and (3.11) that og is invariant
mation (3.10).
The non-trivial topology arises when the phase of the wave
determined uniquely and smoothly in the entire magnetic B
transformation (3.10) implies that the overall phase factor fo
1uklk2) can be chosen arbitrary. This phase can be determined
demanding that a component of the state vector u~,,Jx(~), y”‘) =
real. However, this convention is not enough to fix the phase on
Brillouin zone, since z.Q&x’~), y(O)) vanishes for some (k,, k2
zeros of z+,,Jx,
y) has been shown in Section II. For the sake of
Thouless et al. PRL 1982
the case where u,,,,(x(~), y(O)) vanishes only at one point (k\“, k
Brillouin zone. See Fig. 1. Divide T2 into two pieces HI and H,
tains (k!O), k$O)).We adopt a different convention in H, so that
of the state vector z.++(x’~), y”)) = (x’~‘, y”‘] u~,~,) is real, wher
are chosen such that LQ,,~~(x(‘),y’“) does not vanish in H,. Thu
is uniquely determined on the entrie T*. In Fig. 1, a phase of on
state vector u~,~~(x”‘, y(O)) = (.~~O’y(‘)luklk2) is schematically dra
周期ポテンシャルの場合，ブロッホ関数が固有関数。
v=dH(k)/dk　に注意
ストークスの定理
IT rtt
0
kl
T
-27T
qa
FIG. 1. Schematic
diagram
of a phase of a wavefunction
in the magnetic
Brillouin
zone is actually
a torus, so the edges (k,, kZ) = (0, k2) and (2z/qa,
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
h.p://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
ARPES
With randomness
H Zhang et al., Nature Physics, 2009
c theory of the QSH
of its simplicity and
explicit and pedagoges and their transport
oretical proposals for
h (Murakami, 2006),
InAs/GaSb quantum
al experiments in the
encouraging signaystem has also been
de Na2 IrO3 (Shitade
ional QSH state was
state (Bernevig and
stigated theoretically
vin and Stern, 2009).
QSEに向けて： HgTe と CdTeの比較
l
tor
lectronic structure of
a simple model first
Zhang (2006) (BHZ)
bands of HgTe/CdTe
he QSH effect. HgTe
lattice structure. This
diamond lattice, i.e.,
bic lattices shifted
fferent atom on each
way, if the QW thickness dQW falls below a critical thickness
dc . We can understand this heuristically as follows: for thin
QWs the heterostructure should behave similarly to CdTe
and have a normal band ordering, i.e., the bands with
primarily À6 symmetry are the conduction subbands and
s軌道
heavy hole
s軌道
light hole
Hamiltonian （Science ‘06）
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) ↓>
H(k), 2x2行列
M = Es-‐‑Ep @ Γ点
E(k) = ε (k) ± (M − Bk 2 )2 + A 2 k 2
sとpの間のエネルギーM
を変えると
p
s
s
p
M>0
M=0
M<0
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1àZ2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
NATURE|Vol 464|11 March 2010
a
NATURE|Vol 464|11 March 2010
NATURE|Vol
464|11 March 2010
PERSPECTIVE INSIGHT
表面状態
通常の絶縁体，
a
b
トポロジカル絶縁体
a
もしくは真空
b
Figure 1 | Metallic states are born when a surface unties ‘knotted’ electron
PERSPECTIVE INSIGHT
defined. If the topological invariants are always defined for an insulator,
b
4. 2D realization in HgTe
c theory of the QSH
of its simplicity and
explicit and pedagoges and their transport
oretical proposals for
h (Murakami, 2006),
InAs/GaSb quantum
al experiments in the
encouraging signaystem has also been
de Na2 IrO3 (Shitade
ional QSH state was
state (Bernevig and
stigated theoretically
vin and Stern, 2009).
Comparison of HgTe with Cd Te
l
tor
lectronic structure of
a simple model first
Zhang (2006) (BHZ)
bands of HgTe/CdTe
he QSH effect. HgTe
lattice structure. This
diamond lattice, i.e.,
bic lattices shifted
fferent atom on each
way, if the QW thickness dQW falls below a critical thickness
dc . We can understand this heuristically as follows: for thin
QWs the heterostructure should behave similarly to CdTe
and have a normal band ordering, i.e., the bands with
primarily À6 symmetry are the conduction subbands and
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)
are 0.10 eV higher than the À1 state. In Figs. 2(a) and 2(b),
the states around the À point are denoted by their symmetry.
Such an inverted band structure is a signature of the transition to a TI state.
We now discuss the details of this transition to an
inverted band structure. For an ultra-thin QW, the gap
between the valence and conduction states is determined
by the interplay of three factors, namely quantum confinement, polarization field, and strain. The quantumconfinement effect is large for these thin QWs, explaining
Ref. 7: Miao et al., Topological Insulator tors and be integrated into various devices, (2) it is
en by large intrinsic polarization fields, (3) the TI state
be manipulated by applying external fields or injecting
rge carriers and can be adjusted by standard semicontor techniques, including doping, alloying and varying
QW thickness, and (4) the polarization field can induce a
e Rashba SOI in this system containing only light
ments, which provides a new approach to manipulating
freedom in such systems.
Our proposed QW consists of InN layers sandwiched
ween GaN along the [0001] direction [Fig. 1(b)].
up-III nitrides (III-N), including InN, GaN, and AlN,
e been intensively studied because of their applications
ght emitters, high-frequency transistors, and many other
s. An important feature of III-N compounds is that
ough their band gap varies from 0.7 eV to 6.2 eV, all
e compounds and their alloys are stable in the same
tzite structure with relatively modest variations in a
ce constant. This feature, combined with advanced
wth techniques, allows the fabrication of high-quality
ysultrathin InN layers embedded into GaN
and heterojunctions. InN has a fundamental gap of
y $0:7 eV [13], resulting in strong coupling between the
tron and hole states and leading to a large nonparaboy of5 layers of InN
the bands around the À point [Fig. 1(c)] as well as an
eedingly
small electron effective mass of 0:067m0 [14].
(1.6nm thickness)
Our first-principles calculations are based on density
ctional theory (DFT) with a plane-wave basis set and
ector augmented waves [15], as implemented in the
T. Nakaoka’s idea
P program [16]. Because the accuracy of the band gap
f key importance for this Letter, we employ a hybrid
ctional [17]. Recent calculations have demonstrated the
PHYSICAL REVIEW LETTERS
Transition in a GaN/InN/GaN Quantum Well compounds [8]. GaN=InN=Ga
PRL 109, 186803 (2012) integrated in nitride-based tran
extensively used in high-frequ
vices [28]. Because charge carr
ization fields, the polarization p
by adjusting the carriers densitie
the TI transition can be controll
bias voltage.
High quality InN layers in Ga
of 1–2 ML and atomically sha
been fabricated
[29,30]. Both ph
Tunneling à 3D TI
electroluminescence [31] origi
recombination have been obse
that the critical thickness of InN
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)
•  GaN/InNで実現できるかも知れないàどうやって確認す
る？

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