atsuo shitade- quantum spin hall effect in transition metal oxides

Master thesis

Quantum spin Hall effect in transitionmetal oxides

Supervisor: Professor Naoto Nagaosa

February, 2009

Department of Applied Physics

School of Engineering

University of Tokyo

76505 Atsuo Shitade

Contents

1 Introduction 3

2 Some backgrounds 52.1 QH effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 QH effect and the first Chern number . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Charge polarization and the first Chern number . . . . . . . . . . . . . . . . 62.1.3 Haldane model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 SH effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 Theories of the SH effect in doped semiconductors . . . . . . . . . . . . . . . 92.2.2 Experiments in doped semiconductors . . . . . . . . . . . . . . . . . . . . . . 102.2.3 Experiments in metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.4 Theories in metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 QSH effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.1 Kane-Mele model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.2 TR polarization and the Z2 number . . . . . . . . . . . . . . . . . . . . . . . 152.3.3 QSH effect and the TR polarization . . . . . . . . . . . . . . . . . . . . . . . 192.3.4 Experimental realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Relativistic Mott insulator Sr2IrO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 QSH effect in Na2IrO3 293.1 Model construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Transfer integrals to the NN sites . . . . . . . . . . . . . . . . . . . . . . . . 303.1.2 Transfer integrals to the NNN sites . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Paramagnetic phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Edge AFM phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3.1 Mean field approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3.2 Fractional charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4 Experimental proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 First principles approach to topological insulators 434.1 Na2IrO3 with the honeycomb lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1.1 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

1

CONTENTS 2

4.1.2 Discrepancy between calculations and experiments . . . . . . . . . . . . . . . 444.1.3 Extension of the tight binding model . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Lu2Ir2O7 with the pyrochlore lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.1 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Summary and future problems 53

Chapter 1

Introduction

The non-trivial topology of wave functions has attracted intensive interests in condensed matterphysics. The Hall conductivity in two-dimensional electron systems was reported to be quan-tized as long as the Fermi energy lies in localized states [1]. This quantum Hall (QH) effectis experimentally surprising because disorder, the electron correlation, the finite size effect, andthree-dimensionality etc. are expected to give some corrections to the Hall conductivity. Thouless,Kohmoto, Nightingale, and den Nijs (TKNN) found that the QH effect can be described by thefirst Chern number associated with the Berry curvature [2, 3, 4, 5]. Haldane studied a model onthe honeycomb lattice showing the QH effect without Landau levels [6]. In this model, total flux isequal to zero and only breaking of time-reversal (TR) symmetry is essential. Topological numbersgive a chance to characterize quantum liquids without the Ginzburg-Landau scheme of symmetrybreaking.

A recent breakthrough is theoretical predictions and experiments on the quantum spin Hall(QSH) effect in TR invariant systems [7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Intuitively it can beregarded as two copies of QH systems with up and down spins, but is driven by the relativisticspin-orbit coupling (SOC). Opposite spins have opposite signs of the Hall conductivity and thespin Hall (SH) conductivity is quantized. The Kane-Mele model on the honeycomb lattice [7, 8] isthe case. However, the SH conductivity is not quantized generally because spin is not conserved[10]. Then what characterizes a QSH insulator? It is known to be robust against spin non-conserving interactions such as the Rashba coupling, disorder, and the weak electron correlation[7, 8, 17, 18, 19]. In fact, the Z2 number distinguishes a QSH insulator from an ordinary bandinsulator [7, 20, 21, 22, 23, 13]. It is closely related to Kramers doublets protected by TR symmetryand corresponds to the number of gapless helical edge modes. A HgTe/CdTe quantum well waspredicted to show the QSH effect [11] and later experimentally observed [12]. A three-dimensionalBi1−xSbx alloy was found to have the odd number of pairs of helical edge modes by using angle-resolved photoemission (ARPES) [15], and now the existence of this novel state of matter hasbeen firmly established. So far, however, QSH insulators are limited only in semiconductorsand at low temperature. This is bacause the large SOC and fine tuning of band structures areneeded. Therefore one important development is search for more robust QSH insulators at highertemperature with a wider gap.

Another development is to study interplay between the non-trivial topology and the electron

3

CHAPTER 1. INTRODUCTION 4

correlation. Needless to say, the electron correlation and related interplay between spin, orbital,and charge degrees of freedom bring a variety of unusual properties such as high temperaturesuperconductivity (HTSC) in cuprates, the colossal magnetoresistance (CMR) effect in manganites,and so on [24, 25, 26]. Its research targets have been mainly 3d transition metal oxides in whichthe SOC is negligible or at most perturbative. One example is the single-band (x2 − y2) Hubbardmodel based on cuprates, in which an orbital angular momentum is quenched. Another exampleis the Dzyaloshinsky-Moriya (DM) antisymmetric interaction in spin systems without inversionsymmetry [27, 28]. It can be derived from the second order perturbation of the SOC plus theexchange coupling and can bring the weak FM or the helical magnetism.

On the other hand, when we focus on 5d elements going down from 3d elements in the periodictable, the electron correlation becomes smaller because orbitals get more extended, while the SOCbecomes larger with increasing atomic number. Both the electron correlation and the SOC playimportant roles with the same order of magnitudes. A layered perovskite oxide Sr2IrO4 providesa suggestive example, which is a Mott insulator [29, 30, 31] in sharp contrast to the fact that its4d version Sr2RhO4 can be described as a Fermi liquid [32, 33]. Recently Kim et al. found that anarrow band with an effective total angular momentum jeff = 1/2 is formed and even the relativelyweak electron correlation leads to a Mott insulator [34].

Thus 5d elements provide realistic stages to fuse the two separate mainstreams in condensedmatter physics, i.e. spintronics in semiconductors and the electron correlation in 3d elements.The final goal of this thesis is the very term “fusion” through the study of 5d transition metaloxides from the topological point of view. Although the umklapp scattering term in the strongelectron correlation opens up a gap in a helical liquid associated with spontaneous breaking ofTR symmetry [18, 19], it is an open problem what about non-magnetic insulators in geometricallyfrustrated lattices. The Kondo problem in a helical liquid is rather different from that in a spinfulTomonaga-Luttinger (TL) liquid [18, 35]. Interplay between the non-trivial topology and theelectron correlation in topological transition metal oxides contains such challenging problems.And precisely, the first problem is whether topological insulators can be realized in transitionmetal oxides or not.

Organization of this thesis is following. In Chapter 2, some backgrounds are explained. Thebasic notions of topological insulators, both with and without TR symmetry, are reviewed. Sincethe QSH effect may not be familiar, (not quantum) SH effect is also introduced. At the lastSection, the electronic structure of Sr2IrO4 is discussed, which provided the above big motivationof this thesis.

Chapter 3 is devoted to the QSH effect and the related magnetic transition in Na2IrO3. Theeffective model is found to be a QSH insulator and a guiding principle to design QSH transitionmetal oxides is derived. A new mechanism of the magnetic transition is proposed based on itstopological nature. This work is summarized in a paper [36].

In Chapter 4, first principles band calculations are performed to examine the validity of theeffective model on Na2IrO3 and to search for topological transition metal oxides.

Chapter 2

Some backgrounds

In this Chapter, some basic notions of topological insulators are summarized. The QH effect is aphenomenon in TR breaking topological insulators in two dimensions, which is characterized bythe first Chern number as TKNN pointed out [2, 3, 4, 5]. Before explanations of the main subjectof this thesis, the QSH effect, the SH effect is briefly introduced. The QSH effect is a phenomenonin TR invariant topological insulators in two and three dimensions, which is characterized by theZ2 number [7, 21, 22, 23]. The physical meanings of these topological numbers are also explained.Finally, we focus on an Ir oxide Sr2IrO4, in which the SOC and the electron correlation drasticallychange its electronic structure.

2.1 QH effect

In a strong magnetic field and at low temperature, the Hall conductivity σH is quantized and thereexist gapless edge modes, which is called the QH effect. Although plateaus of σH as a function ofa magnetic field are due to localization, we can see the Kubo formula of the Hall conducitvity isassociated with the Berry phase of wave functions, leading to the first Chern number at T = 0[2, 3, 4, 5]. The charge polarization in one-dimensional insulators provides the physical meaningof the first Chern number [37, 38]. Haldane invented the model showing the QH effect withoutLandau levels [6].

2.1.1 QH effect and the first Chern number

For simplicity, consider the non-interacting Hamiltonian. According to the Kubo formula, the Hallconductivity is given by

σH =1

iL2

∑

n6=m

∑

~k

f(εn(~k))− f(εm(~k))

(εn(~k)− εm(~k))2〈n;~k|Jx(~k)|m;~k〉〈m;~k|Jy(~k)|n;~k〉

=1

iL2

∑n

∑

~k

f(εn(~k))∑

m6=n

〈n;~k|Jx(~k)|m;~k〉〈m;~k|Jy(~k)|n;~k〉 − (x ↔ y)

(εn(~k)− εm(~k))2,

(2.1)

5

CHAPTER 2. SOME BACKGROUNDS 6

where |n;~k〉 is a Bloch wave function with the eigenvalue εn(~k), and f(ε) is the Fermi distribution.

The matrix elements of ~J(~k) = ~∇kH(~k) can be rewritten as

〈m;~k| ~J(~k)|n;~k〉 = (εn(~k)− εm(~k)〈m;~k|~∇k|n;~k〉 (2.2)

for m 6= n, and then (2.1) is transformed into

σH =1

L2

∑n

∑

~k

f(εn(~k))i

[∂

∂kx

〈n;~k| · ∂

∂ky

|n;~k〉 − (x ↔ y)

]

=1

L2

∑n

∑

~k

f(εn(~k))[~∇k × ~An(~k)]z,

(2.3)

where ~An(~k) ≡ 〈n;~k|i~∇k|n;~k〉 is the Berry connection. At T = 0, n runs over occupied bands,leading to

σH =1

2π

occ∑n

∫d2k

2π[~∇k × ~An(~k)]z ≡ 1

2π

occ∑n

Cn. (2.4)

Cn is called the first Chern number and proved to be an integer as follows.First, we can see that the first Chern number is invariant under U(1) gauge transformation

|n;~k〉 → eiχn(~k)|n;~k〉, which leads to ~An(~k) → ~An(~k) − ~∇kχn(~k). This is based on a conjecture

that phases of wave functions χn(~k) can be taken smoothly in the entire Brillouin zone, which is

not correct generally. For example, we try to fix a gauge so that 〈a|n;~k〉 is real. However, the

Brillouin zone may contain such ~k0 as 〈a|n;~k0〉 = 0, where this gauge can not be adopted. Then

we adopt another gauge so that 〈b|n;~k〉 is real in the finite region D containing ~k0. Thus a phase

mismatch |n;~k〉D = eiχn(~k)|n;~k〉D occurs at ∂D, leading to the non-zero first Chern number

Cn =

∫

D

d2k

2π[~∇k × ~AD

n (~k)]z +

∫

D

d2k

2π[~∇k × ~AD

n (~k)]z

=1

2π

∫

∂D

d~k · [ ~ADn (~k)− ~AD

n (~k)] =1

2π

∫

∂D

dχn(~k).

(2.5)

In this way, the first Chern number reflects the non-trivial phase structure of wave functions.

2.1.2 Charge polarization and the first Chern number

The first Chern number characterizes change of the charge polarization for a periodic cycle in onedimension. In fact, a QH insulator on a cylinder can be regarded as a one-dimensional insulator.Here we begin with the charge polarization in terms of the Berry connection.

Consider a one-dimensional insulator under periodic boundary conditions. The eigenstates canbe written as Bloch wave functions 〈y|ψn(k)〉 = eiky〈y|n; k〉. We can define Wannier wave functionslocalized at the site Y as

〈y|n; Y 〉 =

∫ π

−π

dk

2πeik(y−Y )〈y|n; k〉. (2.6)


The charge polarization is the shift of the center of Wannier wave functions and can be written as

Pρ =occ∑n

〈n; Y |(y − Y )|n; Y 〉 =

∫ π

−π

dk

2πA(k), (2.7)

where again A(k) is the U(1) Berry connection (but summed over occupied bands here). Note thatthe charge polarization is well-defined only mod 1. In fact, by performing gauge transformation|1k〉 → eiχ1(k)|1k〉, Pρ is transformed into Pρ− [χ1(π)−χ1(−π)]/2π. This additional term must bean integer due to periodicity of the Brillouin zone.

However, change of Pρ induced by adiabatic change of the Hamiltonian H(t) is well-defined.The eigenstates |n; t, k〉 and the Berry connection can be defined at each time t. Then change ofPρ is given by

Pρ(t2)− Pρ(t1) =

∫ π

−π

dk

2π[Ak(t2, k)− Ak(t1, k)]

=

∫ π

−π

dk

2π[Ak(t2, k)− Ak(t1, k)] +

∫ t2

t1

dt[At(t,−π)− At(t, π)]

=

∫ π

−π

dk

2π

∫ t2

t1

dt[~∇(t,k) × ~A(t, k)]z.

(2.8)

In the second line, the vanishing term At(t,−π) − At(t, π) = 0 is added and the line integral isperformed around a loop in the (t, k)-space, which makes the Stokes theorem applicable in thethird line. For a periodic cycle H(t + T ) = H(t), Pρ(T/2) − Pρ(−T/2) is given by the integralin (2.8) over the entire torus. This is the first Chern number itself. Thus the first Chern numberchracterizes change of the charge polarization in a periodic cycle.

Next we relate two-dimensional QH insulators to one-dimensional insulators. Consider a two-dimensional insulator on a cylinder threaded by flux Φ in Figure 2.1 (a). Flux Φ corresponds avector potential Ax = −Φ/L, leading to an electric field Ex = −Ax. From a general argument onelectrons in a magnetic field, a momentum kx is substituted by kx−Ax = kx +2π/Lx ·Φ/Φ0, whereΦ0 = 2π is a flux quanta. Thus a two-dimensional insulator can be regarded as a parametrized one-dimensional insulator, and kx can be identified with Φ. End states of the cylinder as a function ofΦ reflects edge modes as a function of kx. The first Chern number, which is defined as the numberof charge moving from one end to another for a cycle, guarantees the presence of gapless chiraledge modes as shown in Figure 2.1 (b).

2.1.3 Haldane model

At the end of this Section, the Haldane model on the honeycomb lattice [6] is introduced. Thismodel shows that the QH effect results from breaking of TR symmetry, and in principle evenwithout net flux. It is also important as a forerunner of the Kane-Mele model showing the QSHeffect [7, 8].

The Haldane model is a spinless fermion model including (1) transfer integrals to the nearest-neighboring (NN) sites t1, (2) transfer integrals to the next-nearest-neighboring (NNN) sites t2e

iφ


(a) (b)

Figure 2.1: (a) A cylinder threaded by flux Φ. (b) Schematic edge modes as a function of kx forC = 1. A shaded area shows bulk states, and red solid and dashed lines indicate evolution oflocalized states at y = 0 and L. A black dashed line is the Fermi energy.

(a) (b)

Figure 2.2: (a) The Haldane model defined on the honeycomb lattice. A green arrow indicatestransfer integrals t2e

iφ. (b) Phase diagram of the first Chern number ν in the (φ,M/t2)-space.The phase boundaries follow |M/t2| = 3

√3| sin φ| [6].

which break TR symmetry, and (3) staggered on-site potentials ±M on 1/2 sublattices whichbreak inversion symmetry. It is explicitly written as

H = t1

NN∑ij

c†icj + t2

NNN∑ij

eiνijφc†icj + M∑

i

ξic†ici. (2.9)

Here νij ≡ 2/√

3[d1× d2]z = ±1, where d1 and d2 are unit vectors along the two bonds connectingthe site i and j. ξi = ±1 gives staggered on-site potentials ±M . |t2/t1| < 1/3 is assumed toguarantee that the lower band is fully occupied. As seen in Figure 2.2 (b), the TR breaking phaseφ 6= 0, π makes this model non-trivial without a uniform magnetic field.

2.2 SH effect

In doped semiconductors and metals, transverse charge currents with up and down spins flow inthe opposite direction due to the SOC, leading to a pure spin current. This effect is called theSH effect, which has been the active field in spintronics. It was first proposed by D’yakonov andPerel, which originates from the extrinsic skew scattering [39]. On the other hand, the intrinsic


SH effect was theoretically predicted in p-type semiconductors [40] and two-dimensional n-typesemiconductors [41]. In fact, both the intrinsic and extrinsic contributions are important. Inoueet al. showed that for the Rashba Hamiltonian the vertex correction exactly cancels the bare SHconductivity in the clean limit [42]. Experimentally, Kato et al. observed the SH effect in n-typesemiconductors [43], and Wunderlich et al. observed in p-type semiconductors [44]. In this Section,these theoretical and experimental studies are briefly introduced. The SH effect in metals, whichis more promising candidates for application to spintronics, is also explained.

2.2.1 Theories of the SH effect in doped semiconductors

In p-type semiconductors, six-fold degeneracy of p-orbitals is partially lifted by the SOC, leadingto four-fold degenerate states with a total angular momentum j = 3/2 and two-fold degeneratestates with j = 1/2. The latter is well-separated from the former and can be neglected. Thus thevalence band can be described by j = 3/2-states, which are split into two doubly degenerate bandscalled the heavy-hole (HH) and light-hole (LH) bands. The k · p theory results in the effectiveHamiltonian near the Γ-point,

H(k) =1

2m

[(γ1 + 5γ2/2)k2 − 2γ2

∑a

k2aj

a2 − 2γ3

∑

a 6=b

kakb(jajb + jbja)

], (2.10)

where the sign of energy is inverted. γ1, γ2, and γ3 are the Luttinger parameters, which are oftenapproximated as γ2 = γ3. Within this approximation, the Hamiltonian is written as

H(k) =1

2m

[(γ1 + 5γ2/2)k2 − 2γ2(k · j)2

]. (2.11)

Murakami et al. calculated the SH conductivity based on the semi-classical equation of motion,

σSH =1

12π2(3kHH

F − kLHF ), (2.12)

where kHHF and kLH

F are the Fermi wave numbers for the HH and LH, respectively [40].In n-type semiconductors, the SOC is small because the conduction band mainly consists of

an s-orbital. However, in two-dimensional electron gas (2DEG) in heterostructures, structuralinversion-asymmetry (SIA) induces the Rashba SOC, which reads

H(~k) =k2

2m+ α(σxky − σykx). (2.13)

The coupling constant α can be tuned by the gate voltage. Sinova et al. obtained the universalSH conductivity

σSH =1

8π(2.14)

from the Kubo formula [41].The above results are assumed the absence of impurities. There are three impurity effects; (1)

the self-energy correction (Figure 2.3 (b)), (2) the side-jump (Figure 2.3 (c)), and (3) the skew


(a) (b) (c) (d)

+ c.c.

+ c.c.

Figure 2.3: Diagrammatic expressions for (a) the bare SH conductivity, (b) the self-energy cor-rections, (c) the side-jump terms, (d) the skew scattering terms. The lower panels of (c) and (d)show schematic pictures of the scattering processes. Red and blue arrows indicate motion of upand down spins.

scatering (Figure 2.3 (d)). The self-energy correction (1) can be included by the finite τ−1 and isregarded as correction of the intrinsic SH effect. The side-jump (2) and the skew scattering (3)give the extrinsic SH effect. Inoue et al. evaluated the side-jump effect for the Rashba Hamiltonian(2.13) within the ladder approximation, which exactly cancels the intrinsic term (2.14) in the cleanlimit [42]. They assumed isotropic short-range potentials, and in addition, many groups found thatthe SH conductivity vanishes for long-range potentials. On the other hand, for the Luttinger model(2.11) the side-jump effect vanishes within the ladder approximation [45].

2.2.2 Experiments in doped semiconductors

After these theoretical predictions, two seminal experiments were performed. Both observed thespin accumulation at the edges; one is in n-type semiconductors by the Kerr rotation [43], and theother in p-type semiconductors by a light-emitting diode (LED) [44].

Kato et al. prepared thin films of n-GaAs and n-In0.07Ga0.93As with n = 3 × 1016cm−3 [43].When an electric field is applied in the x-direction, an spin current polarized in the z-direction flowsin the y-deirection. As a result, spins polarized in the ±z-direction are accumulated at the edgesof the samples. Such distribution of spins can be detected by the Kerr rotation. In this method, alinearly polarized beam is incident normal to the sample. The polarization of the reflected beamis rotated by an angle proportional to the z-component of spins. Moreover, when a transversemagnetic field is applied, spins precess around it. Thus the rotation angle decrease monotonicallyas A0/[(ωLτs)

2 +1], where ωL is the Larmor frequency and τs is the spin lifetime. This is known asthe Hanle effect. Figure 2.4 shows the experimental data on n-GaAs [43], which strongly supportsthe SH effect. They also roughly estimated the SH conductivity as σSH = 5 × 10−3Ω−1cm−1 forn-In0.07Ga0.93As [43].

Distribution of spins can be detected by a circularly polarized light. Wunderlich et al. fabricated


Figure 2.4: The SH effect in GaAs at T = 30K. (A) Schematic picture of the experimental setup.(B) Kerr rotation as a function of an applied magnetic field B for x = −35µm (red) and x = +35µm(blue) for E = 10mVµm−1. (C) Kerr rotation as a function of x and B for E = 10mVµm−1. (D,E, F) Spatial dependence of the peak Kerr rotation A0, the spin lifetime τs, and reflectivity R,respectively. (G) Kerr rotation as a function of E and B at x = −35µm. (H, I) E dependence ofA0 and τs, respectively [43].


(a)

(b)

(c)

(d)

Figure 2.5: (a) Schematic cross section of the LED. (b) Scanning electron microscope (SEM) imageof the LED. (c) Circular polarization along the z-axis for LED 1 for two opposite IP. (d) Circularpolarization along the z-axis for fixed IP for LED 1 and 2 [44].

LEDs in (Al,Ga)As/GaAs heterostructures in Figure 2.5 (a) and (b) [44]. In usual LEDs, whena forward bias is applied, electrons in the conduction band and holes in the valence band arerecombined accompanied with light emission. Now that spins of holes are polarized due to theSH effect, the emitted light is circularly polarized to conserve an angular momentum. Namely,inversion of circular polarization of light in Figure 2.5 (c) and (d) directly shows inversion of spinpolarization, which supports the SH effect in p-type semiconductors.

2.2.3 Experiments in metals

The spin Hall effect in metals is more promising than that in semiconductors because (1) the SHconductivity is much larger because of the larger number of carriers, (2) a contact with metallicferromagnets does not suffer from impedance mismatch, and (3) they are more robust againstdisorder and thermal excitations due to the larger Fermi energy. Recently, 4d and 5d metals withthe larger SOC have been focused on [46, 47]. On the other hand, the optical detections used insemiconductors can not be applied to metals because the spin diffusion length is much shorter.Therefore in metals, the inverse SH effect, in which a spin current induces a charge current, iswidely used. Here we see one of the methods to inject a spin current, the non-local spin injectionfrom metallic ferromagnets [48].

Valenzuela et al. prepared a CoFe/Al device as shown in Figure 2.6 (a), in which a ferromagneticelectrode (FM1) was used to inject a spin-polarized current, and Al was oxidized to make atunneling barrier [48]. A current I flows from FM1 into the left side of the Al film in Figure 2.6(b), and a spin current is injected due to spin diffusion. Therefore we can observe the inverse


(a)

(b) (c)

(d)

Figure 2.6: (a) Atomic force microscope (AFM) image of the device. (b) SH measurement. Acurrent I flows from FM1 into the left side of the Al film, and a SH voltage VSH is measuredbetween the two Hall probes at a distance LSH from FM1. (c) Spin transistor. I flows from FM1into the right side of the Al film, and a voltage V is measured between FM2, which is located ata distance LFM from FM1, and the left side of the Al film [48].

SH effect without suffering from the anisotropic magnetoresistance, the ordinary and anomalousHall effects in FM1. FM1 and another ferromagnetic electrode (FM2) construct a spin transistorin Figure 2.6 (c), which was used to determine the spin polarization P , the spin diffusion lengthλsf , and the angle θ of the magnetization M . With the configuration in Figure 2.6 (b), the SHresistance at LSH is written as

RSH =∆RSH

2sin θ (2.15)

with

∆RSH =PσSH

tAlσ2e−LSH/λsf , (2.16)

where tAl is the thickness of the Al film. Exponential decay in (2.16) was clearly seen in Figure 2.6(d), and they estimated σSH = (3.4± 0.6)× 101Ω−1cm−1 for tAl = 12nm and σSH = (2.7± 0.6)×101Ω−1cm−1 for tAl = 25nm, leading to the SH angle of αSH ≡ σSH/σ = (1− 3)× 10−4 at heliumtemperature.

Recently, Kimura et al. observed the SH effect in Pt with the SH conductivity of σSH =2.4× 102Ω−1cm−1 and the SH angle of αSH = 3.7× 10−3 at room temperature [46]. More recently,the SH angle of αSH = 0.113 was reported in a FePt/Au device [47].

2.2.4 Theories in metals

As well as in semiconductors, both the intrinsic and the extrinsic contributions are important inmetals. Guo et al. performed relativistic first principles band calculations for Pt and calculated theintrinsic SH conductivity based on the Kubo formula (Figure 2.7 (a)) [49]. They obtained σSH =2200Ω−1cm−1 at T = 0 and 240Ω−1cm−1 at T = 300K, which is very close to the experimentalvalue [46]. This giant SH conductivity originates from the enhanced Berry curvature Ωz(k) at X-


(a) (b)

(c)

Figure 2.7: (a) Relativistic band structures (left) and the SH conductivity (right) for Pt. Reddashed and blue solid curves indicate scalar relativistic and full relativistic calculations. (b) Berrycurvature Ωz(k). (c) n-decomposed Berry curvature Ωn

z (k). The inset shows the temperaturedependence of the SH conductivity [49].

and L-points due to the SOC-induced anti-crossings as shown in Figure 2.7 (b) and (c). In Pt, theintrinsic contribution is dominant.

On the other hand, 4d and 5d metals including Pt were investigated based on the nine-orbital(s, p, and d) tight binding model [50, 51]. Kontani et al. found that the vertex correction dueto local impurities is less than 5% within the Born approximation [50]. This is because a p-level,which gives the non-vanishing vertex correction, is 20eV higher than the Fermi energy. Tightbinding models are convenient to capture an intuitive origin; the effective Aharonov-Bohm phasedue to the SOC involving s- and d-orbitals.

Recently, motivated by the giant SH effect in a FePt/Au device [47], the role of the electroncorrelation was pointed out [52]. Since the Fermi surface of Au is mainly composed of an s-orbital,the intrinsic contribution is expected to be small. They considered three possibilities; (1) Auvacancies, (2) Pt impurities, and (3) Fe impurities. The cases (1) and (2) do not change theelectronic structure near the Fermi energy, while LDA+U in the case (3) does change. t2g-orbitalsshow the resonant peak at the Fermi energy and play the major role in the SH effect. On theother hand, eg-orbitals are 2eV higher than the Fermi energy, leading to the Kondo effect atlow temperature. It should be emphasized that the energy splitting between eg- and t2g-orbitalsresults not from a crystalline field but from the electron correlation. The SH angle due to the skewscattering is expressed as

αSH =

∫dΩI(θ)S(θ) sin θ∫dΩI(θ)(1− cos θ)

=3

5

=(e−2iδ1 − 1)(e2iδ+2 − e2iδ2−)

3 sin2 δ+2 + 2 sin2 δ−2

, (2.17)

where δ±2 = δj=2±1/2 is the d-wave phase shift, and δ1 is the p-wave one. This quantity was found


to be the order of 0.1 when the resonant peak is well-split due to the SOC as shown in the insetof Figure 2.8 (c). In this sense, it can be referred to as the resonant skew scattering.

2.3 QSH effect

The QSH effect was first theoretically proposed by Kane and Mele in graphene [7, 8]. Intuitively itcan be regarded as two copies of QH insulators, but the SH conductivity is not generally quantizedbecause spin is not conserved. On the other hand, a topological number called the Z2 numberand helical edge modes persist even in the presence of disorder and the weak electron correlation[18, 19]. In this Section, the TR polarization is defined, which provides the physical meaning ofthe Z2 number, and then we relate the Z2 number to the QSH effect [20, 13]. This procedureis completely analogous to the charge polarization and the QH effect. Generalization to threedimensions is easy, in which a set of four Z2 numbers distinguishes a strong topological insulator(STI) from a weak topological insulator (WTI) [21, 22, 23]. Experimental realizations are alsoreviewed.

2.3.1 Kane-Mele model

To begin with, the Kane-Mele model on the honeycomb lattice [7, 8] is explained. This model isjust a generalization of the Haldane model showing the QH effect [6] and is explicitly written as

H = tNN∑ij

c†icj + iλSO

NNN∑ij

νijc†iszcj + iλR

NN∑ij

c†i (s× dij)zcj + λv

∑i

ξic†ici. (2.18)

The second term is the SOC which involves the spin-dependent and complex transfer integrals tothe NNN sites, corresponding to φ = σπ/2 in the Haldane model. The third term is the Rashbainteraction, which breaks mirror symmetry and spin conservation. Note that all these terms donot break TR symmetry, which plays a key role in the QSH effect.

If λR = 0, spin is conserved and the spin Chern number can be defined as the difference of thefirst Chern number between up and down spins, i.e. Cs ≡ C↑−C↓, which leads to the quantized SHconductivity. Also, edge modes crossing at k = 0 or π are spin-filtered in the sense that electronswith opposite spin propagate in opposite directions. This is the very intuitive but special case.

As shown in Figure 2.9 (a), the region is finite where a pair of gapless helical edge modes exists,which is characterization of the QSH effect. In this region, the charge conductance is predicted to bequantized as G = 2e2/h for the two-terminal geometry in Figure 2.9 (b). Another characterizationby the Z2 number and robustness of helical edge modes against disorder are discussed later.

2.3.2 TR polarization and the Z2 number

In this Subsection, the TR polarization for one-dimensional TR invariant insulators is introduced,and the Z2 number is defined as its change for half a cycle [20]. And then in the next Subsection,


Figure 2.8: Partial DOSs for Au with (a) 3.1% Au vacancies, (b) 3.1% Pt impurities, (c) 3.1% Feimpurities in non-magnetic state, and (d) in ferromagnetic state [52].


(a)

(b)

Figure 2.9: (a) The band dispersions for the zigzag edge geometry in (left) the QSH phase withλv/t = 0.1 and (right) the insulating phase with λv/t = 0.4. In both cases λSO/t = 0.06 andλR/t = 0.05. The inset shows the phase diagram in the (λv, λR)-space [7]. (b) Schematic picture ofthe two-terminal measurement. The charge conductance is quantized as G = 2e2/h. Right picuresindicate the population of edge modes [8].

relation between the Z2 number in one dimension and the QSH effect in two dimensions is explainedin the same manner of the first Chern number and the QH effect.

In the TR invariant cases, the Kramers theorem guarantees that every Bloch wave functionwith k is degenerate with its TR partner with −k. Therefore 2N eigenstates can be divide into Npairs which satisfy

T |n−; k〉 =eiχn(k)|n+;−k〉T |n+; k〉 =− eiχn(−k)|n−;−k〉 (2.19)

where n = 1, · · · , N , and T is the TR operator. Now we can consider the difference of shift ofcharge between a Kramers pair. Intuitively it corresponds to shift of spin, but this is not validin the presence of the SOC. To achieve this purpose, let the pseudospin Berry connection beintroduced. TR symmetry (2.19) constrains relation between the pseudospin Berry connections as

A+(−k) =N∑

n=1

〈n+;−k|(−i∂k)|n+;−k〉 =N∑

n=1

〈T n−; k|eiχn(k)(−i∂k)e−iχn(k)|T n−; k〉

=N∑

n=1

[〈n−; k|i∂k|n−; k〉 − ∂kχn(k)] = A−(k)−N∑

n=1

∂kχn(k).

(2.20)


Then the TR polarization can be defined as

Pθ =

∫ π

−π

dk

2π

[A+(k)− A−(k)

]=

∫ π

0

dk

2π

[A+(k) + A+(−k)− A−(k)− A−(−k)

]

=

∫ π

0

dk

2π

N∑n=1

∂k [χn(−k)− χn(k)] .

(2.21)

This can be rewritten in a generic form by introducing a U(2N) matrix, which is related to TRsymmetry,

wmn(k) ≡ 〈m;−k|T |n; k〉 =N⊗

n=1

[0 eiχn(k)

−eiχn(−k) 0

]. (2.22)

We can easily see two useful formulas;

log det w(k) =iN∑

n=1

[χn(k) + χn(−k)]

log Pf w(k∗) =iN∑

n=1

χn(k∗),

(2.23)

where k∗ = 0 and π are TR invariant momenta, where w(k∗) is an antisymmetric matrix charac-terized by Pfaffian. Using (2.23), (2.21) is rewritten as

iπPθ =

∫ π

0

dk

[1

2∂k log det w(k)−

N∑n=1

i∂kχn(k)

]=

1

2log

det w(π)

det w(0)− i

N∑n=1

[χn(π)− χn(0)]

=1

2log

det w(π)

det w(0)− log

Pf w(π)

Pf w(0).

(2.24)

Finally, applying the well-known formula det w = (Pf w)2, we obtain

(−1)Pθ =

√det w(0)

Pf w(0)

√det w(π)

Pf w(π). (2.25)

The blanches of ±√

det w(k) are chosen to evolve continuously along the path of the integralin (2.24). The TR polarization Pθ distinguishes two phases, i.e. the presence or absence of aKramers degenerate state at the end of finite systems, though it is meaningless by itself like thecharge polarization Pρ.

Then again an adiabatic cycle of the Hamiltonian is considered. The Hamiltonian is TRinvariant at t = 0 and T/2, at which Wannier wave functions come in pairs. However, in goingfrom t = 0 to t = T/2, Wannier wave functions may switch partners as shown in Figure 2.10(a), leading to the appearance of an unoccupied state at each end. Thus there must be two-fold


(a) (b)

Figure 2.10: Red and blue lines represent schematic evolution of the centers of a Kramers doublet asa function of t. (a) The non-trivial case with Pθ(T/2)−Pθ(0) = 1. Wannier wave functions switchpartners, resulting in an unoccupied state at each end. (b) The trivial case with Pθ(T/2)−Pθ(0) =0.

degeneracy at each end and four-fold degeneracy in total. On the other hand, there is no degeneracyin the trivial case shown in Figure 2.10 (b). In the result, change of the TR polarization

(−1)ν =4∏

i=1

√det w(Γi)

Pf w(Γi)(2.26)

defines a Z2 topological invariant which characterizes the presence or absence of a Kramers degen-erate state at each end. Here Γi are four TR invariant points given by (t, k) = (0, 0), (0, π), (T/2, 0)and (T/2, π). To apply (2.26), wave functions must be defined continuously on the entire torus,which is guaranteed by the vanishing first Chern number due to TR symmetry.

2.3.3 QSH effect and the TR polarization

To relate two-dimensional QSH insulators to the TR polarization, again we consider a cylinderthreaded by flux Φ. The Z2 number, change of the TR polarization between two TR invarianttime points in terms of one dimension, characterizes in two dimensions how Kramers degeneratestates at TR invariant momenta kx = 0 and π are connected to each other. Kramers pairs mayswitch partners associated with the presence of helical edge modes as shown in Figure 2.11 (b),while may not as in Figure 2.11 (c). The former case corresponds to a QSH insulator and thelatter to an ordinary band insulator. The Z2 number characterizing two-dimensional insulators isgiven by

(−1)ν =4∏

i=1

δi

δi =

√det w(Γi)

Pf w(Γi)

wmn(k) =〈m;−k|T |n; k〉,

(2.27)

where Γi are four TR invariant momenta.The topological number characterizing QSH insulators, the Z2 number, gives only whether the

number of pairs of helical edge modes n is even or odd. This reflects the fact that two pairs


(a) (b) (c)

Figure 2.11: (a) A cylinder threaded by flux Φ. (b, c) Schematic edge modes as a function ofmomentum kx. A shaded area shows bulk states, and red and blue lines indicate evolution ofa Kramers doublet. A black dashed line shows the Fermi energy. (b) The non-trivial case with(−1)ν = −1. Since no degeneracy at kx = 0 is assumed, Kramers degeneracy occurs at kx = π.(c) The trivial case with (−1)ν = 1.

of gapless helical edge modes are generally coupled to open up a gap. To see this, consider thelow-energy effective Hamiltonian describing one edge. For n = 1, the non-interacting part can bewritten as

H0(k) = vFk(ψ†R+ψR+ − ψ†L−ψL−), (2.28)

where ψR+ and ψL− are right- or left-propagating edge modes with up or down pseudospins con-nected by the TR operator. In this case, the single-particle backward scattering term

Hbw(k) = ∆ψ†R+ψL− + h.c. (2.29)

is forbidden due to TR symmetry. On the other hand, for n = 2, the non-interacting part is givenby

H0(k) = vFk(ψ†1R+ψ1R+ − ψ†1L−ψ1L− + ψ†2R+ψ2R+ − ψ†2L−ψ2L−), (2.30)

while a possible mass term is

Hbw(k) = ∆(ψ†1R+ψ2L− − ψ†1L−ψ2R+) + h.c.. (2.31)

Thus even though there exist more than one pair of helical edge modes, every two pairs annihilatesand only the last one pair is guaranteed to be gapless by Kramers theorem.

In three dimensions, there exist eight TR invariant momenta Γn1n2n3 = (n1b1 + n2b2 + n3b3)/2with primitive reciprocal lattice vectors bj and nj = 0, 1. A set of four Z2 numbers (ν0; ν1ν2ν3) isdefined as

(−1)ν0 =∏nj

δn1n2n3

(−1)νk =

nk=1∏nj 6=k

δn1n2n3 .

(2.32)

ν0 is independent of choice of bj, while νk are not. In this sense, ν0 is special, and distinguishesSTIs from WTIs as seen later. Figure 2.12 shows different phases.


(a)

(b)

(c)

(0;001) (0;011) (1;111)

+

-

+ +

+

+

+

+

+

+

+

+

+

- -

-

+

+

+

+ +

+

+

-

Figure 2.12: Different phases characterized by (ν0; ν1ν2ν3). (a) δn1n2n3 at eight TR invariantmomenta Γn1n2n3 . (b) Schematic contour plot of surface modes as a function of kx and ky. Blackand white circles indicate the odd and even TR polarization πn1n2 = δn1n20δn1n21, respectively.Shaded areas represent above the energy shown in (c). (c) Gapless surface modes in the (1; 111)phase.

To understand the physical meaning of (2.32), again imagine a generalized cylinder whichis open along the z-axis and is periodic along the x- and y-axes. It can be threaded by twoindependent fluxes corresponding kx and ky. At each surface TR invariant momenta in the surfaceBrillouin zone, the TR polarization is defined as πn1n2 = δn1n20δn1n21. If the TR polarizationchanges its sign along the path from one surface TR invariant momenta to the other, across thesolid curve in terms of Figure 2.12 (b), there exist gapless surface modes.

For ν0 = 1, one TR polarization differs in sign from the other three as shown in the right panelof Figure 2.12 (b). This means the presence of only one pair of gapless surface modes protectedby TR symmetry. Thus the ν0 = 1 phases are robust against disorder and the weak electroncorrelation, which are referred to as STIs.

On the other hand, for ν0 = 0, two cases are possible. One is that no gapless surface modesexist as shown in the left panel of Figure 2.12 (b). This is an ordinary band insulator. Anotheris that there exist two pairs of gapless surface modes in the middle panel of Figure 2.12 (b). Thiscan be interpreted as a layered QSH insulator. If the number of layers is odd, it is equivalent to atwo-dimensional QSH insulator. However, if the number of layers is even, a weak periodic potentialand disorder can easily open up a gap, leading to an ordinary band insulator, as discussed in twodimensions. Therefore the ν0 = 0 phases are referred to as WTIs.


(a)

(b)

(c)

Figure 2.13: (a) The band structures near Γ point for HgTe and CdTe. (b) Schematic picture ofa quantum well and lowest subbands for different thicknesses. (c) The electronic structure of aquantum well as a function of the thickness of HgTe [9].

2.3.4 Experimental realizations

Up to now, a few specific materials have been confirmed as topological insulators. Various probescan detect helical edge modes, among which ARPES is the most direct. Here we see them inhistorical order, namely a HgTe quantum well [12], a Bi1−xSbx alloy [15], and Bi2Se3 and Bi2Te3

[53, 54, 55].A HgTe quantum well was the first experimental realization of the QSH effect [12], following

a theoretical prediction by Bernevig et al. [11]. The central idea is band inversion tuned by thethickness of HgTe. See Figure 2.13. CdTe has the normal band structure, in which p-like Γ7 andΓ8 are occupied and s-like Γ6 is empty. On the other hand, HgTe has the inverted one due to thelarge SOC, in which Γ7 and Γ6 are occupied and Γ8 is empty. Therefore it is expected that asincreasing the thickness of HgTe, first the band structure is normal like CdTe, then a gap betweenΓ8 and Γ6 closes at some critical thickness d = dc, and finally the band structure gets inverted likeHgTe. k · p model calculation supports this idea and the critical thickness is dc = 6.3nm.

Figure 2.14 (a) and (b) show the device used for detection of helical edge modes. Figure 2.14(c) shows the longitudinal resistance for different sizes. (I) is an ordinary band insulator withouthelical edge modes and has high resistance when the Fermi energy lies in a gap. (II)-(IV) are QSHinsulators with helical edge modes. There exist two channels when the Fermi energy lies in a gap,leading to quantization of the conductance as G = 2e2/h. The conductance of (II) is not quantizedbecause the device size is larger than its inelastic mean free path. These quantitative agreementswith k · p model calculations strongly support the QSH effect.

The second experiment was on a three-dimensional Bi1−xSbx alloy [15], motivated by thoreticalpredictions by two groups [13, 14]. Bismuth is trivial with the Z2 number of (0; 000) while antimonyis non-trivial with the Z2 number of (1; 111). Unfortunately both are semimetal, though Bi1−xSbx

is insulator for 0.07 < x < 0.22 as shown in Figure 2.15 (a). Again we can see that the idea of band


(a)

(b)

(c)

Figure 2.14: (a) Schemaic picture of molecular beam epitaxy (MBE)-grown quantum well struc-ture. (b) The device and helical edge modes. Red and blue arrows indicate up and down spin,respectively. (c) The longitudinal four-terminal resistance R14,23 as a function of the gate voltageat T = 30mK. A black curve (I) is normal (d = 5.5nm < dc) with the size of (20.0 × 13.3)µm2.Blue (II), green (III), and red curves (IV) are inverted (d = 7.3nm > dc) with different sizes. R14,23

is not quantized for (II) with the large size of (20.0× 13.3)µm2, while quantized for (III) and (IV)with the small size of (1.0× 1.0)µm2 and (1.0× 0.5)µm2 [12].


(a) (b)

Figure 2.15: (a) The energy levels of Bi1−xSbx at high symmetric points L, T , and H as a functionof x. (b) The surface band dispersion along Γ-M obtained by using ARPES. A white area indicatethe bulk band dispersion [15].

inversion is active. Hsieh et al. directly observed surface states by using high-resolution ARPESfor x = 0.1 (Figure 2.15 (b)). There exist the odd number of (five) gapless points, which showsthat Bi0.9Sb0.1 is a STI.

Very recently first principles band calculations for Sb2Te3, Sb2Se3, Bi2Te3, and Bi2Se3 wereperformed from the topological viewpoint, which predicted that Sb2Te3, Bi2Te3, and Bi2Se3 areSTIs while Sb2Se3 is not [56]. Most astonishing is that Bi2Se3 has a much larger gap of 0.3eV thanroom temperature. Bi2Se3 and Bi2Te3 were experimentally investigated by scanning tunnelingspectroscopy (STS) [53] and ARPES [54, 55]. Note that the STS measurement was done beforethe proposal of the QSH effect by Kane and Mele [7, 8]. Figure 2.16 clearly shows that the localdensity of states (DOS) is finite not due to impurity or thermal effects, but due to the presence ofsurface states [53]. By using ARPES, a single pair of surface states was observed [55] (not shown).

2.4 Relativistic Mott insulator Sr2IrO4

Recently, a layered perovskite iridate Sr2IrO4 has been focused on as a relativistic Mott insulator.It was an interesting problem why Sr2IrO4 is a Mott insulator [29, 30, 31], although its 4d versionSr2RhO4, with the stronger electron correlation, is a Fermi liquid [32, 33]. Both Sr2IrO4 andSr2RhO4 are distorted by around 10 [29, 57], which was considered to cause band narrowing [30].In fact, lattice distortion of Sr2RhO4 provides good explanation for the missing xy-Fermi surfacesheet [58]. However, it is not reasonable to apply this idea to Sr2IrO4, and we should reexaminethe electronic structure focusing on its large SOC.

Under an octahedral crystalline field, five-fold degenerate d-orbitals split into doubly degenerateeg-orbitals and triply degenerate t2g-orbitals. eg-orbitals are explicitly written as u : 3z2 − r2 andv : x2 − y2, while t2g-orbitals as ξ : yz, η : zx and ζ : xy. Their energy difference is called 10Dq.In the t2g-manifold an angular momentum is not completely quenched, `eff = 1. Therefore, withthe SOC, we obtain states with total angular momenta jeff = 3/2 and 1/2. Now that an Ir4+ ion


(a) (b)

Figure 2.16: (a) Solid and dashed lines indicate experimental and theoretical tunneling differen-tial condactance dI/dVB of Bi2Se3. A bulk gap is −0.35V < VB < 0V. High reproducibilityexcludes the possibility of impurity-induced in-gap states and thermal broadening is negligible atlow temperature of 4.2K. (b) Partial DOS of Se1 obtained by first principle band calculations fora twelve-slab geometry [53].

octahedral

crystalline fieldSOC

Figure 2.17: The electronic structure of an Ir4+ ion under an octahedral crystalline field and withthe SOC.

has five electrons, jeff = 3/2-states are fully occupied and jeff = 1/2-states are half-filled.Kim et al. investigated the electronic structure of Sr2IrO4 by using ARPES, optical conductiv-

ity, X-ray absorption spectroscopy (XAS), and first principles band calculations [34]. The Fermisurface obtained by ARPES in Figure 2.18 (e) cannot be explained without including both theSOC and the electron correlation as shown in Figure 2.18 (a)-(d). Furthermore, the XAS spectracan be well fitted by an assumption of ξ : η : ζ = 1 : 1 : 1 as shown in Figure 2.19 (b), leading tojeff = 1/2 states written as

|+〉 =1√3

(+|ζ ↑〉+ |ξ ↓〉+ i|η ↓〉)

|−〉 =1√3

(−|ζ ↓〉+ |ξ ↑〉 − i|η ↑〉) .(2.33)


Tetragonal and rotational distortions seem to be irrelevant. The SOC and the on-site Coulombinteraction are estimated to 0.5eV by optical conductivity in Figure 2.19 (a), which is consistentwith the electronic structure obtained by LDA+SOC+U.

At the end of this Section, resonant X-ray scattering (RXS) is introduced as more direct probeof relativistic states, which may need some explanations. The RXS process is described by thesecond order perturbation of the electron-photon coupling, and its amplitude is approximatelygiven by

fαβ =∑

n

mω3in

ω

〈i|Rβ|n〉〈n|Rβ|i〉hω − hωin + iΓ/2

. (2.34)

An electron makes a round trip from and to the initial state |i〉 via all possible intermediatestates |n〉, collecting phase factors. That is why RXS can detect phase factors of relativisticstates. On experiments on iridates, L2 (2p1/2 → 5d) and L3 (2p3/2 → 5d) edges are focused on.Straightforward calculations on the Clebsch-Gordan coefficients leads to the important fact thatthe intensities at L2 and L3 edges are equal for non-relativistic s = 1/2-states while L2 edge isforbidden for relativistic jeff = 1/2-states. In the recent RXS study by Kim et al., no intensity wasobserved at L2 edge in contrast to the resonantly enhanced intensity at L3 edge, which confirmedthat jeff = 1/2-states are realized [59].


(a) LDA

(b) LDA+U

(c) LDA+SOC

(d) LDA+SOC+U

(e) ARPES

Figure 2.18: The Fermi surfaces, the band dispersions, and schematic energy diagrams within (a)the local density approximation (LDA), (b) LDA+U with U = 2.0eV, (c) LDA+SOC, and (d)LDA+SOC+U. In the right panel of (b), the unrealistically large U leads to a Mott insulator. (e)ARPES intensity maps at EB = 0.2, 0.3, and 0.4eV. A red square indicates the reduced Brillouinzone neglecting

√2×√2 distortion [34].


Figure 2.19: (a) The optical conductivity at 100K. Peaks A and B are assigned to transitiondenoted in the right panel of Figure 2.18 (d). The O 1s polarization dependent XAS spectra(dotted lines) compared to the expected one (solid lines) under an assumption of ξ : η : ζ = 1 : 1 : 1[34].

Chapter 3

QSH effect in Na2IrO3

The QSH effect was first proposed in the Kane-Mele model based on graphene with the honeycomblattice [7, 8]. The model has the complex and spin-dependent transfer integrals to the NNNsites due to the SOC, in addition to the real and spin-independent ones to the NN sites. Thesecomplex and spin-dependent transfer integrals open a gap in bulk and result in gapless helical edgemodes. However, the SOC in graphene is so small that the QSH cannot be observed. Here thenewly synthesized compound Na2IrO3 with the honeycomb lattice of iridium [60] is investigated.Antiferromagnetism (AFM) due to the electron correlation is also discussed.

3.1 Model construction

Na2IrO3 is composed of (Na1/3Ir2/3)O2 layers and Na layers shown in Figure 3.1. In a (Na1/3Ir2/3)O2

layer, Ir and Na atoms are surrounded by six oxygen atoms, leading to the honeycomb lattice of Iratoms. Na2IrO3 is similar to Sr2IrO4 in that each Ir4+ ion is subject to an octahedral crystallinefield, an effective single-band model can be constructed as basis of jeff = 1/2-states. The procedureis following: (1) The model is completely two-dimensional, neglecting the interlayer coupling.(2)Lattice distortion is neglected, leading to the perfect honeycomb lattice shown in Figure 3.2

(a) (b)

Figure 3.1: (a) The crystal structure of Na2IrO3 drawn with VESTA [61]. Yellow, black and redcircles indicate Na, Ir and oxygen, respectively. (b) A (Na1/3Ir2/3)O2 layer extracted from Na2IrO3.

29

CHAPTER 3. QSH EFFECT IN NA2IRO3 30

(a) (b)

Figure 3.2: (a) Perfect honeycomb lattice. The x, y and z-axes are defined in an octahedron. (b)Effective model. Black solid arrows are translation vectors ~a1 and ~a2. Blue, red and green dottedarrows indicate the obtained transfer integrals it′σx, it′σy and it′σz, respectively.

(a). (3) Oxygen is integrated out since the energy level of p-orbitals (εp) is around 3eV lower thanthat of jeff = 1/2-states (εd) [31]. The transfer integrals to the NN and NNN sites are calculatedwith Slater-Koster parameters [62].

3.1.1 Transfer integrals to the NN sites

First the indirect transfer integrals via oxygen are calculated by the second order perturbation.When an electron moves from Ir1 to Ir2 shown in Figure 3.3 (a), two paths exist, via O1 and O2.However, contributions of these independent paths exactly cancel out as seen later. Therefore weconsider a O1-O2 bond. Slater-Koster parameters [62] using this and next Subsections are listedin Table 3.1. By using this, channels can be restricted like Table 3.2. When jeff = 1/2-states aregenerally expanded as |σ〉 =

∑ms Cσ

ms|ms〉 (σ = ±, m = ξ, η, ζ and s =↑, ↓), contribution of eachstate can be written as

tσ2σ1|A1g =∑

s

〈2σ2|H|A1gs〉〈A1gs|H|1σ1〉εd − EA1g

=(pdπ)2

εd − εp + (ppσ)

∑s

Cσ2ζs∗Cσ1

ζs

tσ2σ1|E1gu =1

2

(pdπ)2

εd − εp + (ppπ)

∑s

(Cσ2

ξs + Cσ2ηs

)∗ (Cσ1

ξs + Cσ1ηs

)

tσ2σ1|E1gv =0

tσ2σ1|E1uu =− 1

2

(pdπ)2

εd − εp − (ppπ)

∑s

(Cσ2

ξs − Cσ2ηs

)∗ (Cσ1

ξs − Cσ1ηs

)

tσ2σ1|E1uv =− (pdπ)2

εd − εp − (ppπ)

∑s

Cσ2ζs∗Cσ1

ζs

tσ2σ1|A1u =0.

(3.1)


O2

O1

Ir1

Ir2

O3

Ir1

Ir2

O5

O1O6

O4

O2Ir0

(a) (b)

Figure 3.3: The transfer integrals to (a) the NN sites. (b) the NNN sites. The numbers are assignedto related Ir and oxygen atoms. Red areas indicate the intermediate clusters of the second orderperturbation.

As a result the indirect transfer integral is given by

tσ2σ1 '(pdπ)2

εd − εp

∑s

(Cσ2

ξs∗Cσ1

ηs + Cσ2ηs∗Cσ1

ξs

)

− (pdπ)2

εd − εp

(ppσ) + (ppπ)

εd − εp

∑s

Cσ2ζs∗Cσ1

ζs −(pdπ)2

εd − εp

(ppπ)

εd − εp

∑s

(Cσ2

ξs∗Cσ1

ξs + Cσ2ηs∗Cσ1

ηs

)

t =− 1

3

(pdπ)2

εd − εp

(ppσ) + 3(ppπ)

εd − εp

.

(3.2)

Here isotropic jeff = 1/2-states like (2.33) are assumed. As for the other directions, we only haveto permute ξ, η and ζ in the first line of (3.2), while the result is unchanged.

In addition to the indirect transfer integrals, the direct transfer integrals are also possible.Again refering to Table 3.1, we obtain

〈2 + |H|1+〉 = 〈2− |H|1−〉 =1

4(ddσ) +

1

3(ddπ) +

5

12(ddδ), (3.3)

while the off-diagonal matrix elements are found to vanish. It should be noted that both theindirect part (3.2) and the direct part (3.3) are real and spin-independent. Hereafter (3.2) iswritten as −t, neglecting (3.3).

3.1.2 Transfer integrals to the NNN sites

The transfer integral to the NNN site shown in Figure 3.3 (b) can be calculated in the same way.Here a cluster composed of six oxygen atoms is considered, leading to eighteen states classified in


Table 3.1: Slater-Koster parameters [62]. l, m and n are the direction cosines along the x, y andz-axes, respectively.

〈x|H|x〉 l2(ppσ) + (1− l2)(ppπ)〈x|H|y〉 lm(ppσ)− lm(ppπ)〈x|H|z〉 ln(ppσ)− ln(ppπ)

〈x|H|ζ〉 √3l2m(pdσ) + m(1− 2l2)(pdπ)

〈x|H|ξ〉 √3lmn(pdσ)− 2lmn(pdπ)

〈x|H|η〉 √3l2n(pdσ) + n(1− 2l2)(pdπ)

〈x|H|v〉 √3l(l2 −m2)(pdσ)/2 + l(1− l2 + m2)(pdπ)

〈y|H|v〉 √3m(l2 −m2)(pdσ)/2−m(1 + l2 −m2)(pdπ)

〈z|H|v〉 √3n(l2 −m2)(pdσ)/2− n(l2 −m2)(pdπ)

〈x|H|u〉 l(2n2 − l2 −m2)(pdσ)/2−√3ln2(pdπ)

〈y|H|u〉 m(2n2 − l2 −m2)(pdσ)/2−√3mn2(pdπ)

〈z|H|u〉 n(2n2 − l2 −m2)(pdσ)/2 +√

3n(l2 + m2)(pdπ)〈ζ|H|ζ〉 3l2m2(ddσ) + (l2 + m2 − 4l2m2)(ddπ) + (n2 + l2m2)(ddδ)〈ζ|H|ξ〉 3lm2n(ddσ) + ln(1− 4m2)(ddπ) + ln(m2 − 1)(ddδ)〈ζ|H|η〉 3l2mn(ddσ) + mn(1− 4l2)(ddπ) + mn(l2 − 1)(ddδ)

Table 3.2: Classification of six states (= two atoms × three p-orbitals).state energy channel|A1g〉 = (|1x〉+ |1y〉 − |2x〉 − |2y〉)/2 εp − (ppσ) ζ → ζ

|E1gu〉 = (|1z〉 − |2z〉)/√2 εp − (ppπ) ξ, η → ξ, η|E1gv〉 = (|1x〉 − |1y〉 − |2x〉+ |2y〉)/2 ζ → ζ

|E1uu〉 = (|1z〉+ |2z〉)/√2 εp + (ppπ) ξ, η → ξ, η|E1uv〉 = (|1x〉 − |1y〉+ |2x〉 − |2y〉)/2 ζ → ζ|A1u〉 = (|1x〉+ |1y〉+ |2x〉+ |2y〉)/2 εp + (ppσ) ζ → ζ


Table 3.3: Classification of eighteen states (= six atoms × three p-orbitals). Here neglecting (ppσ)and (ppπ), all states have the same energy εp.

state channel

|A1g〉 = (|1x〉+ |2y〉+ |3z〉 − |4x〉 − |5y〉 − |6z〉)/√6 ζ → ξ

|Egu〉 = [2|3z〉 − 2|6z〉 − (|1x〉+ |2y〉 − |4x〉 − |5y〉)]/√12 ζ → ξ|Egv〉 = (|1x〉 − |2y〉 − |4x〉+ |5y〉)/2 ζ → ξ|T2gξ〉 = (|2z〉+ |3y〉 − |5z〉 − |6y〉)/2 -|T2gη〉 = (|3x〉+ |1z〉 − |6x〉 − |4z〉)/2 ξ → ζ|T2gζ〉 = (|1y〉+ |2x〉 − |4y〉 − |5x〉)/2 -|T2uξ〉 = (|2x〉 − |3x〉+ |5x〉 − |6x〉)/2 -|T2uη〉 = (|3y〉 − |1y〉+ |6y〉 − |4y〉)/2 -|T2uζ〉 = (|1z〉 − |2z〉+ |4z〉 − |5z〉)/2 -

|T (1)1u α〉 = |1x〉+ |4x〉)/√2 -

|T (1)1u β〉 = (|2y〉+ |5y〉)/√2 ζ → ξ

|T (1)1u γ〉 = (|3z〉+ |6z〉)/√2 -

|T (2)1u α〉 = (|3x〉+ |6x〉+ |2x〉+ |5x〉)/2 -

|T (2)1u β〉 = (|1y〉+ |4y〉+ |3y〉+ |6y〉)/2 -

|T (2)1u γ〉 = (|2z〉+ |5z〉+ |1z〉+ |4z〉)/2 -

|T1gα〉 = (|2z〉 − |3y〉 − |5z〉+ |6y〉)/2 -|T1gβ〉 = (|3x〉 − |1z〉 − |6x〉+ |4z〉)/2 ξ → ζ|T1gγ〉 = (|1y〉 − |2x〉 − |4y〉+ |5x〉)/2 -


Table 3.3. Then contribution of each state is given by

t′σ2σ1|A1g =

2

3

(pdπ)2

εd − εp

∑s

Cσ2ξs∗Cσ1

ζs

t′σ2σ1|T2gη =

1

4

(pdπ)2

εd − εp

∑s

Cσ2ζs∗Cσ1

ξs

t′σ2σ1|T

(1)1u β

=− 1

2

(pdπ)2

εd − εp

∑s

Cσ2ξs∗Cσ1

ζs

t′σ2σ1|T1gβ =− 1

4

(pdπ)2

εd − εp

∑s

Cσ2ζs∗Cσ1

ξs .

(3.4)

On the other hand, Eg-states are exceptional because their energy is lowered by the coupling toeg-orbitals of Ir0. Using 〈Egu|H|u〉 =

√3(pdσ), we can obtain a new eigenstate

|E ′gu〉 =

√√√√1

2

(1 +

x√x2 + 3(pdσ)2

)|Egu〉 −

√√√√1

2

(1− x√

x2 + 3(pdσ)2

)|u〉, (3.5)

with the energy of EE′g = εp + x−√

x2 + 3(pdσ)2, where x ≡ (εd + 10Dq− εp)/2. Thus we obtain

t′σ2σ1|E′gu =

1

2

(1 +

x√x2 + 3(pdσ)2

)(−1

6

)(pdπ)2

εd − εp +√

x2 + 3(pdσ)2 − x

∑s

Cσ2ξs∗Cσ1

ζs . (3.6)

Similarly t′|E′gv = 0 is obtained. The resulting transfer integral is written as

t′σ2σ1=

1

6

(pdπ)2

εd − εp

∑s

Cσ2ξs∗Cσ1

ζs

[1− 1

2

(1 +

x√x2 + 3(pdσ)2

)εd − εp

εd − εp +√

x2 + 3(pdσ)2 − x

]

'1

2

(pdπ)2

εd − εp

∑s

Cσ2ξs∗Cσ1

ζs

[(pdσ)2

(εd − εp)(εd + 10Dq − εp)+

(pdσ)2

(εd + 10Dq − εp)2

]

t′ =1

6

(pdπ)2

εd − εp

[(pdσ)2

(εd − εp)(εd + 10Dq − εp)+

(pdσ)2

(εd + 10Dq − εp)2

]iσy,

(3.7)

where σ’s are Pauli matrices for a pseudospin index (+ or −). Hereafter the cofficient of (3.7) isdenoted as t′. As for the other directions, we can obtain it′σx and it′σz. The point is that thetransfer integrals to the NNN sites are complex and spin-independent similar to Kane-Mele model,leading to the QSH effect as seen in the next Section.


Table 3.4: Classification of H0(~k) by Γa defined by Fu and Kane [13]. P is the space-inversion

operater. k1 = ~k · ~a1 and k2 = ~k · ~a2 run in the range of [−π, π). τ ’s are Pauli matrices forsublattices.

a da(~k) Γa T P1 −t(1 + cos k1 + cos k2) τx ⊗ 1 + +2 −t(− sin k1 + sin k2) τy ⊗ 1 − −3 2t′ sin k1 τz ⊗ σx − −4 2t′ sin k2 τz ⊗ σy − −5 −2t′ sin(k1 + k2) τz ⊗ σz − −

3.2 Paramagnetic phase

From the previous Section, the effective Hamiltonian can be written as

H0 =− t∑

i

(d†i2di1 + d†i+~a12di1 + d†i2di+~a21

)

+ it′∑

i

(d†i+~a11σxdi1 + d†i2σxdi+~a12

)+ it′

∑i

(d†i+~a21σydi1 + d†i2σydi+~a22

)

+ it′∑

i

(d†i1σzdi+~a1+~a21 + d†i+~a1+~a22σzdi2

)+ h.c.,

(3.8)

where d†i1 (di1) is a creation (annihilation) operator at the sublattice 1 of the i-th. unit cell. Thismodel is found to show the QSH effect by the Z2 number and the existence of gapless helical edgemodes.

To calculate the Z2 number, periodic boundary conditions are imposed. ~k-component of theHamiltonian H0(~k) can be written as

H0(~k) =5∑

a=1

da(~k)Γa, (3.9)

where d’s and Γ’s are listed in Table 3.4. The components of the other ten matrices Γab mustvanish due to TR and inversion symmetries. The Z2 number of an inversion symmetric two-bandsystem is determined by the product of δ(Γi) = − sgn d1(Γi) [13]. Explicitly Γi are Γ = [0, 0],M1 = [π, 0], M2 = [0, π] and M3 = [π, π] in terms of [k1, k2]. It is straightforward to see δ(Γ) =δ(M1) = δ(M2) = 1, while δ(M3) = −1. Their product is negative, which means the Z2 number of(−1)ν = −1.

To see gapless helical edge modes directly, the efective Hamiltonian (3.8) is numerically diag-onalized under periodic boundary conditions along one axis and open boundary conditions alonganother. There exist two simple edge geometries, zigzag and armchair. In Figure 3.4, the banddispersion for the zigzag edge geometry is shown. Bulk states are gapped due to the complex andspin-dependent transfer integrals to the NNN sites induced by the SOC, while a pair of helical


-4

-3

-2

-1

0

1

2

3

4

20

ener

gy

wavenumber

(b)(a)

Figure 3.4: (a) The zigzag edge geometry with N = 4 chains. Periodic boundary conditionsare imposed along ~a1 and a black dotted line indicates a supercell. (b) The band dispersion fort = t′ = 1 and N = 50.

edge modes is gapless at a TR invariant point k = π, corresponding to the QSH effect. Note thatfor the armchair edge geometry a pair of helical edge modes is gapless at k = 0.

Now the problem is whether the complex and spin-dependent transfer integrals are peculiar tothe honeycomb lattice. A toy model in the xy-plane can answer this problem. Consider projectedjeff = 1/2-states as

|+′〉 =1√2

(+|ξ ↓〉+ i|η ↓〉)

|−′〉 =1√2

(+|ξ ↑〉 − i|η ↑〉)(3.10)

and a pz-orbital. When one oxygen atom is placed as shown in Figure 3.5 (a), we obtain 〈z|H|±′〉 =±i(pdπ)e∓iθ/

√2 by using Table 3.1. Then the indirect transfer integral between Ir atoms is given by

t = −(pdπ)2e∓2iθ/2(εd − εp), which is complex and spin-dependent. Meanwhile when two oxygenatoms are placed as shown in Figure 3.5 (b), phases of two paths cancel out and the resultingtransfer integral is real and spin-independent like t = −(pdπ)2 cos 2θ/(εd − εp).

From the above discussion, we can derive two essences to design QSH transition metal oxides,i.e. (1) complex and spin-dependent states due to the SOC and (2) lattice asymmetry so thatphases do not cancel out. As for the NNN sites, Ir and Na atoms are asymmetric, leading to thecomplex and spin-dependent transfer integrals. These essences can give us even three-dimensionaltopological insulators.

3.3 Edge AFM phase

In this Section, the effect of the electron correlation is investigated by adding the on-site Coulombinteraction written as

H1 = U∑

i

(ni1+ni1− + ni2+ni2−) , (3.11)


(a) (b)

Figure 3.5: A toy model in the xy-plane. Black and red clouds indicate a projected jeff = 1/2-stateof an Ir atom and a pz-orbital of an oxygen atom, respectively. Two Ir atoms are on the x-axis.(a) When one oxygen atom is placed at an angle of θ to the x-axis, the transfer integral between Iratoms is complex and spin-dependent. (b) When two independent oxygen atoms are symmetricallyplaced, the transfer integral becomes real and spin-independent.

where U is the effective on-site Coulomb interaction in jeff = 1/2-states. In the large U limit, thereal and spin-independent transfer integrals (3.2) result in the AFM exchange interaction betweenthe NN sites,

JNN∑ij

si · sj. (3.12)

Here J = 4t2/U and s is a spin operator. On the other hand, the complex and spin-dependenttransfer integrals (3.7) bring a different type of the exchange interaction between the NNN sites,

− J ′NNN∑

ij

(si · sj − 2sa

i saj

)(a = x, y, z) (3.13)

with J ′ = 4t′2/U . In total, the AFM interaction is effective between the NN sites, while the nearlyFM interaction between the NNN sites. Therefore AFM is preferred on the honeycomb lattice.

However, the paramagnetic phase of Na2IrO3 is a QSH insulator distinct from an ordinary bandinsulator. Now a new mechanism of the AFM transition is proposed based on its topological nature.A QSH system consists of the gapped bulk and the gapless edge protected by TR symmetry. Theformer is difficult to gain the electronic energy and is robust against the AFM order, while thelatter is fragile because it breaks TR symmetry. Thus AFM moments are expected to develop firstin the edge and then spread into the bulk as decreasing temperature.

3.3.1 Mean field approximation

This mechanism can be confirmed within the mean field approximation on a semi-infinite system.The on-site Coulomb interaction can be rewritten as

H1 =1

2U

∑i

(ni1+ + ni1− + ni2+ + ni2−)− 2

3U

∑i

(s2

i1 + s2i2

). (3.14)


-5-4-3-2-1 0 1 2 3 4 5

20en

ergy

wavenumber

(a) (b)

(c) (d)

-5-4-3-2-1 0 1 2 3 4 5

20

ener

gy

wavenumber

Figure 3.6: (a) The edge AFM phase along the X-axis. (b) The bulk AFM phase along the X-axis.Each AFM moment is assumed to have the same length. (c, d) The band dispersions for (a, b)with N = 50 chains and |mn| = 1.

The first term is proportional to the total number of electrons and is constant. Within the meanfield approximation, the quadratic terms of spin operators are approximated as

s2(ln)1 =

[〈sn〉+(s(ln)1 − 〈sn〉

)]2 ' 2〈sn〉 · s(ln)1 − 〈sn〉2

s2(ln)2 =

[−〈sn〉+(s(ln)1 + 〈sn〉

)]2 ' −2〈sn〉 · s(ln)2 − 〈sn〉2,(3.15)

where l = 1, . . . , L is the axis with periodic boundary conditions while n = 1, · · · , N is that withopen boundary conditions. Hence (3.14) can be written as

H1 ' 4

3U

∑

ln

[〈sn〉2 − 〈sn〉 ·(s(ln)1 − s(ln)2

)]=

∑

ln

[3m2

n

U− 2mn ·

(s(ln)1 − s(ln)2

)], (3.16)

where the first constant term of (3.14) is dropped and mn ≡ 2U〈sn〉/3 is introduced. The totalHamiltonian is H = H0 + H1. The free energy is optimized via mn and the most stable spinconfiguration is searched for each temperature. For simplicity spin configurations are restrictedto those shown in Figure 3.6 (a) and (b). Figure 3.6 (c) shows that helical edge modes getgapped associated with spontaneous breaking of TR symmetry. The free energy is calculated forL = N = 100.

Figure 3.7 shows the existence of the edge AFM phase for U = 10.5, namely, the edge AFMtransition occurs at higher temperature than the bulk AFM. It seems that the magnetizationchanges discontinuously at the transition point from the edge AFM phase to the bulk AFM phase,


bulk AFM

edge AFM

QSH

Figure 3.7: The temperature dependence of (the upper panel) difference of the free energy persupercell from the QSH phase and (the lower panel) the magnetization for U = 10.5. As temper-ature decreases, the edge AFM transition occurs at T = 0.96, and then the bulk AFM transitionat T = 0.68.

though it should change continuously when n-dependence of mn is considered. An edge modeitself is the most fluctuating one-dimensional system and generally the mean field approximationis not valid.

Nonetheless the edge AFM phase is possible in terms of the effective theory of one-dimensionalhelical liquids [18]. In fact, TR symmetry allows only two kinds of interactions, i.e. the forwardand umklapp scatterings

Hfw =gfw

∫dxψ†R+(x)ψR+(x)ψ†L−(x)ψL−(x)

Hum =gum

∫dxe−i4kFxψ†R+(x)ψ†R+(x + a)ψL−(x + a)ψL−(x) + h.c.,

(3.17)

where a is the lattice constant. The forward scattering term gives the non-trivial Luttinger pa-rameter K =

√(vF − gfw)/(vF + gfw) but a helical liquid remains gapless. On the other hand,

according to the renormalization group (RG) analysis of the bozonized Hamiltonian, the umklapp

scattering term becomes relevant at K < 1/2, and a gap ∆ ' a−1g1/2−4Kum opens. For gum < 0,

φ ≡ φR−φL is pinned at either 0 or√

π/2, leading to the finite value of the mass order parameterO2 ≡ iηRηL/2πa cos

√4πφ. Since O2 is TR odd, TR symmetry is spontaneously broken at T = 0.

For gum > 0, O1 ≡ iηRηL/2πa sin√

4πφ is the order parameter. Anyway, such Ising-like order isnot stabilized at finite temperatures in one dimension, though a gap remains. In this sense, theedge AFM phase discussed above is a liquid of Ising domain walls.

3.3.2 Fractional charge

In the edge AFM phase, the fractional charge due to the topological nature of a QSH insulatorarises at a domain wall. The idea of the fractional charge goes back to the Su-Schrieffer-Heeger


model [63]. For spinless fermions, a mass kink induces a localized state with zero energy, whichhas charge of ±e/2. However, in realistic systems such as polyacetylene, only the integer chargeis carried because of two spin degrees of freedom. On the other hand, helical liquids have halfdegrees of freedom of spinful TL liquids, i.e. up spins move to the right while down spins to theleft. Thus the fractional charge appears at domain walls of helical liquids [64].

To calculate the fractional charge carried by domain walls, we have to define smooth de-formation from a QSH insulator to the vacuum via the edge AFM order. In five Γ’s, onlyΓ1 ≡ τx ⊗ 1 is TR invariant. Therefore the TR invariant vacuum is described as Hvac(~k) = V Γ1

with V → ∞. On the other hand, the AFM order is odd under TR and inversion symmetries,which is describes as H1 = mxΓ3 + myΓ4 + mzΓ5. Now smooth deformation can be taken asH(~k, θ, ϕ) =

∑5a=1 da(~k, θ, ϕ)Γa with

da(~k, θ, ϕ) =

−t(1 + cos k1 + cos k2) + V (θ)−t(− sin k1 + sin k2)

2t′ sin k1 + mxbulk(1− cos θ)/2 + (mx

edge cos ϕ + mxDW sin ϕ) sin θ

2t′ sin k2 + mybulk(1− cos θ)/2 + (my

edge cos ϕ + myDW sin ϕ) sin θ

−2t′ sin(k1 + k2) + mzbulk(1− cos θ)/2 + (mz

edge cos ϕ + mzDW sin ϕ) sin θ

, (3.18)

where mbulk is the bulk AFM moment, ±medge are the directions of the edge AFM far away fromthe domain wall, and mDW is the direction at the domain wall as shown in Figure 3.8. θ = 0 andπ correspond to the vacuum and the system, and the domain wall is located between ϕ = 0 andπ. The even function V (θ) satisfies V (0) = V , V (π) = 0, and V (π/2) = 3t. The last condition

V (π/2) = 3t is required to guarantee that the level crossing occurs at ~k = 0 in the QSH phase.One choice of V (θ) is given by

V (θ) = 3ta(a + 1)

[1

a− cos θ− 1

a + 1

](3.19)

with a = (1− 6t/V )−1. Note that the parameter V should be taken to V > 3t to make sure thatθ = 0 is the trivial vacuum.

The net charge carried by the domain wall can be calculated by the skyrmion formula [65]

Q =3

8π2εabcde

∫ π

0

dθ

∫ π

0

dϕ

∫d2k

da∂k1db∂k2dc∂θdd∂ϕde

|d(~k, θ, ϕ)|5. (3.20)

For mbulk = 0, the bulk is TR invariant and we obtain Q = ±1/2 as discussed by Qi et al. [65].When the bulk TR symmetry is broken with mbulk 6= 0, the net charge Q deviates from ±e/2.Thus the edge AFM phase discussed in this Section has the topological property different from thebulk AFM phase.

3.4 Experimental proposal

In the realistic three-dimensional compound, the interlayer coupling gives the band dispersionalong k3, leading to two Dirac fermions at k3 = 0 and π. In this case, the backward scattering


Figure 3.8: Schematic picture of the domain wall. Gray and orange areas indicate the vacuum andsystem, corresponding θ = 0 and π, respectively. A red curve shows the charge density around atthe intersection of the edge and the domain wall.

local DOSlocal DOS

energy energy

∆bulk

∆edge

(b) on A and B (c) on C

∆bulk

(a)

ABC

0 0

Figure 3.9: (a) The schematic picture of a plateau (orange) on the surface. A, B and C are typicalpoints outside, inside and just on the plateau edge, respectively. (b, c) The energy dependence ofthe local DOS on (b) A, B and (c) C at different temperatures T À TN (the QSH phase, blue dotline), T > TN (the edge AFM phase, red dash line) and T < TN (the bulk AFM phase, black solidline).

between them is possible via a weak periodic potential and disorder, and a helical liquid becomeslocalized or gapped. In other words, Na2IrO3 is a WTI. However, if we focus on a plateau on thesurface as shown in Figure 3.9 (a), its edge is spatially separated from the edges of underlyinglayers and is robust. The above topological natures can be detected by STS. When a tip is awayfrom the plateau edge, the local DOS shows a large bulk gap ∆bulk for both the QSH and AFMphases as shown in Figure 3.9 (b). On comparison when the tip is just on the plateau edge, thelocal DOS remains finite in the QSH phase at T À TN, reflecting the existence of gapless helicaledge modes. Here TN is the bulk Neel temperature. As temperature decreases, a small edge gap∆edge opens slightly above TN. Even at T = 0, ∆edge remains smaller than ∆bulk as shown in Figure3.6. Such temperature dependence is schenatically shown in Figure 3.9 (c).


Figure 3.10: The Kitaev model implemented in Na2IrO3. Blue, red, and green lines indicate x-,y-, and z-bonds with the superexchange interactions sx

i sxj , sy

i syj , and sz

i szj , respectively.

3.5 Discussion

In our approach, the weak electron correlation is added to the non-interacting Hamiltonian withthe Z2 non-trivial topology. This is the weak coupling limit. On the other hand, the strongcoupling limit is also possible, leading to another interesting phase. Jackeli and Khaliullin derivedthe superexchange Hamiltonian as

H = −JNN∑ij

scis

cj, (3.21)

where we label a ij-bond in the ab-plane perpendicular to the c-axis by a c-bond as shown inFigure 3.10. In their theory, the Hund coupling JH plays a crucial role. In fact, without theHund coupling, J = 4ν2/3 · 4t2/U , with 2ν2 ' JH/U exactly vanishes due to the lattice symmetrydiscussed in Section 3.2. This model is included by the Kitaev model generally written as

H = −NN∑ij

Jcscis

cj, (3.22)

which can be exactly solved, leading to two phases [66]. The phase A, for |Jx| > |Jy| + |Jz| or itscyclic permutations, has a gap asscosiated with Abelian anyons. On the other hand, the phase B,for |Jx| < |Jy|+ |Jz| and its cyclic permutations, is gapless with Dirac cones, but acquires a gap ina magnetic field associated with non-Abelian anyons. Our and their theories are complementary,and it is likely that the realistic compound lies in the intermediate region between two limits, i.e.a QSH insulator and the Kitaev model, leading to a gap and frustration.

Chapter 4

First principles approach to topologicalinsulators

In Chapter 3, we have found that Na2IrO3 is a WTI based on the tight-binding model, though itsvalidity must be examined by more serious calculations. On the other hand, the search for three-dimensional STIs is a challenging problem. To achieve these purposes, first principles calculationsare performed by using the full-potential linearized augmented plane-wave (FP-LAPW) methodimplemented in WIEN2k [67], which is one of the most accurate methods within the densityfunctional theory (DFT) [68, 69]. The generalized gradient approximation (GGA) by Perdew etal. [70] is used to describe the exchange-correlation potential. The SOC can be included by thesecond variational method using scalar-relativistic eigenstates as basis [71].

4.1 Na2IrO3 with the honeycomb lattice

In this Section, the electronic structure of Na2IrO3 is calculated to examine whether several ap-proximations are valid.

4.1.1 Calculations

Since the accurate parameters for the crystal structure of Na2IrO3 have not been known yet, thoseof Na2PtO3 listed in Table 4.1 are used for input. The muffin-tin radii RMT are set to 2.05, 2.10 and1.65a.u. for Ir, Na and oxygen. Wave functions inside atomic spheres are expanded in sphericalharmonics up to `max = 10, while those outside spheres are expanded in plane-waves with thecut-off RMTKmax = 7.0. 125 k-points are taken in the whole Brillouin zone, corresponding to 21k-points in the irreducible Brillouin zone. The convergence criteria are 0.00001Ry for energy and0.0001 for charge.

Figure 4.1 shows the results of the GGA calculations without the SOC. O p-, Ir t2g-, and Ireg-bands are well-separated by around 4.2eV and 3.6eV, respectively, which are almost consistentwith the optical conductivity experiment on Sr2IrO4 [31]. The bands near the Fermi energy EF

43

CHAPTER 4. FIRST PRINCIPLES APPROACH TO TOPOLOGICAL INSULATORS 44

Table 4.1: The atomic positions for Na2PtO3 with the monoclinic space group C2/c (No. 15). Thelattice constant is a = 5.419, b = 9.385 and c = 10.752A and β = 99.67. Note that WIEN2k hasa limitation that in centered monoclinic lattices only B-setting is allowed.

atom x y zPt1 0 0.249 1/4Pt2 0 0.584 1/4Na1 0.261 0.578 0.001Na2 1/4 1/4 0Na3 0 0.918 1/4O1 0.15 0.101 0.148O2 0.095 0.415 0.149O3 0.146 0.735 0.148

are mainly composed of Ir t2g-bands (Figure 4.1 (b)), and the finite DOS at EF shows that thecompound is a metal within the GGA (not including the SOC).

The relativistic calculations including the SOC are shown in Figure 4.3. In Figure 4.3 (b), a dipof the total DOS can be observed at the Fermi energy, in contrast to Figure 4.1 (b). In addition,this double peak is mainly composed of jeff = 1/2-states as shown in Figure 4.3 (e). Note that thedip becomes deeper after performing optimization of atomic positions. All of these supports ourtheory that jeff = 1/2-states on the honeycomb lattice provide a WTI.

4.1.2 Discrepancy between calculations and experiments

Na2IrO3 is experimentally an insulator, while the above calculations predicted a metal close toan insulator. The possible origins of this discrepancy are (1) the drawback of the LDA, and (2)the effect of the electron correlation. The former is that the eigenvalues of Kohn-Sham equationsdo not correspond to the single-particle excitation energies, and the LDA usually underestimatesthe band gap. The latter is very difficult. In cases for the even number of electrons per unitcell, the system is in principle a band insulator in the weak correlation limit, and no distinctionexists between a band and Mott insulators. On the other hand, a topological insulator has bothcharge and spin channels at the edges, while a Mott insulator does not have a charge degree offreedom. Thus we face two challenging problems. One is whether the compound is a topologicalinsulator or a Mott insulator, in other words, where the critical point of the Mott transition is.The other is how the spin channel behaves with the electron correlation. The fractionalized QSH(or quantum spinon Hall) effect proposed by Young et al. [73] may provide a clue, though theirmodel is too artificial. These two problems remain to be solved. It should be emphasized thatsome kinds of frustration are assumed as discussed in Section 3.5. Therefore the Mott transitionis not accompanied with breaking of translational and TR symmetries, otherwise, distinction froma Mott insulator is trivial, i.e. the presence or absence of the magnetic order.


0

5

10

15

20

25

30

35

40

-6 -4 -2 0 2 4

DO

S

energy [eV]

totalIr egIr t2g

O p

0

5

10

15

20

25

30

35

40

-2 -1.5 -1 -0.5 0 0.5

DO

S

energy [eV]

totalIr t2g

O p

(a) (b)

(c) (d)

-6

-4

-2

0

2

4

Y

XLYRZ

ener

gy [e

V]

-2

-1.5

-1

-0.5

0

0.5

Y

XLYRZ

ener

gy [e

V]

Figure 4.1: GGA calculations of (a) the total (black) and projected (red, green, and blue for Ireg, Ir t2g, and O p) DOSs, (b) near EF, and (c) the band structure, (d) near EF. In each panel,the origin of the energy axis is set to EF. See Figure 4.2 (a) for notations of the high symmetrick-points.


(a) (b)

Figure 4.2: Brillouin zone for (a) Na2IrO3 (conventional cell) and (b) Lu2Ir2O7 (primitive cell)drawn with XCrySDen [72].

4.1.3 Extension of the tight binding model

Although the bands near the Fermi energy are composed of jeff = 1/2-states, they appear to bedifferent from those expected from the simple tight binding model. This discrepancy is due to thepossible existence of the interlayer coupling t⊥ and the real and spin-independent transfer integralsto the NNN sites t′0. Especially, since Na2IrO3 is predicted to be a metal within the GGA+SOC,the transfer integrals t′0 are considered to be as large as the transfer integrals t.

First of all, we consider the interlayer coupling based on the crystal structure. Black linesin Figure 4.4 shows the considered interlayer coupling. The bond lengths between the interlayersites are around 5.6A, which are the same as those between the NNN sites (5.4A). The interlayercoupling t⊥ is assumed to be real and spin-independent, which can be given by the direct transferintegrals (3.3). The real and spin-independent transfer integrals to the NNN sites can be given bythe second order perturbations of those to the NN sites, in addition to the direct ones.

Then the three-dimensional 8× 8 Hamiltonian can be written as

H(k) =

[H0(k) H†

⊥(k)H⊥(k) H0(k)

]

H⊥(k) =

[t⊥(1 + eik3) t⊥(eik1 + ei(−k2+k3))t⊥(1 + eik3) t⊥(1 + eik3)

],

(4.1)

where the two-dimensional 4 × 4 Hamiltonian H0(k) is given by (3.9), but d0(k) = 2t′0(cos k1 +cos k2 +cos(k1 +k2)). This effective Hamiltonian can be analytically diagonalized along some highsymmetric k-points as shown in Table 4.2, which makes it possible to estimate fitting parameterst, t′, t′0, and t⊥. Figure 4.5 shows the band dispersions for two typical parameters, (a) is fort′ = 0 and (b) is for t′ 6= 0. The trasnfer integrals it′σa provide no significant change due to thepresence of t⊥, in sharp contrast to the two-dimensional model, which shows that Na2IrO3 is aWTI. Although it can not be called well-fitted, the interlayer coupling t⊥ is estimated to be oneorder of magnitude smaller than t. Discrepancy of the details may be due to lattice distortion,hybridization of jeff = 3/2-states, and the electron correlation which first principles calculationscan include.


-6 -5 -4 -3 -2 -1 0 1 2 3Energy (eV)

0

0.5

1

1.5

2

Jeff=3/2Jeff=1/2

Na2IrO3 (with s-o coupling)

0 5

10 15 20 25 30 35 40 45

-6 -4 -2 0 2 4

DO

S

energy [eV]

totalIr egIr t2g

O p

0

5

10

15

20

25

30

-2 -1.5 -1 -0.5 0 0.5

DO

S

energy [eV]

totalIr t2g

O p

(a) (b)

(c) (d)

-6

-4

-2

0

2

4

Y

XLYRZ

ener

gy [e

V]

-2

-1.5

-1

-0.5

0

0.5

Y

XLYRZ

ener

gy [e

V]

(e) (f)

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Y

XLYRZ

ener

gy [e

V]

Figure 4.3: GGA+SOC calculations of (a) the total (black) and projected (red, green, and bluefor Ir eg, Ir t2g, and O p) DOSs, (b) near EF, and (c) the band structure, (d) near EF. (e) Blackand red lines indicate the projected DOS of jeff = 3/2- and jeff = 1/2-states, respectively. (f) Theband structure near EF after optimization of atomic positions.


Figure 4.4: Three-dimensional lattice of Ir atoms. The primitive translation vectors a1, a2, anda3 are indicated by black arrows. The bond lengths between the NN, NNN (blue, red, and greendotted arrows), and interlayer (black dotted lines) sites are around 3.1, 5.4, and 5.6A

Table 4.2: Eigenvalues of the effective Hamiltonian (4.1) at some high symmetric k-points.k-point eigenvalues degeneracy

Γ 3t + 6t′0, −3t + 6t′0 ± 4t⊥ 4, 2, 2

R ±√

5t2 + 8t′2 + 2t′0 4, 4Y ±3t + 6t′0 4, 4

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Y

XLYRZ

ener

gy [e

V]

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Y

XLYRZ

ener

gy [e

V]

(a) (b)

Figure 4.5: Comparison between first principles calculations (small circle) and the extended tightbinding model (big square) for t = 0.031145, t′0 = −0.013, t⊥ = 0.00628375, and (a) t′ = 0, (b)t′ = 0.01eV.


(a) (b) (c)

Figure 4.6: (a) Pyrochlore lattice drawn with VESTA [61]. Yellow, black and red circles indicateLn, Ir and oxygen, respectively. (b) Network of corner-sharing IrO6 octahedrons. (c) Network ofcorner-sharing OLn4 tetrahedrons.

4.2 Lu2Ir2O7 with the pyrochlore lattice

The pyrochlore lattice is a three-dimensional geometrically frustrated lattice with corner-sharingtetrahedrons as shown in Figure 4.6. Especially pyrochlore iridates Ln2Ir2O7 have attractedmuch interests because they exhibit unique properties originating from geometrical frustration,f -electrons of Ln ions, and the tunable transfer integrals by Ln ions. Some of them are insulators,and good candidates for STIs. To begin with, the previous studies on pyrochlore iridates areshortly explained.

A series of pyrochlore iridates Ln2Ir2O7 shows the metal-insulator changeover with decreasingLn ion radii [74, 75]. Pr2Ir2O7 is known as the frustrated Kondo lattice, in which the magneticlong-range order due to the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction is suppressed bygeometrical frustration and the Kondo effect is stabilized [76]. At low temperature, Pr 4f -momentsare underscreened, leading to a metallic spin liquid and the unconventional anomalous Hall effect[77]. For Ln = Nd, Sm, and Eu, the metal-insulator transition occurs [75]. The compounds withsmaller ions, Ln = Gd, Tb, Dy, Y, Ho, and Yb, are insulators [74], whereas the recent studies byARPES and first principles calculations revealed that Y2Ir2O7 has the small but non-zero DOSat the Fermi energy [78, 79]. This suggests that the experimentally observed insulating phase isdriven by disorder. Therefore the first problem is whether Lu2Ir2O7, with the smallest ion Lu, isan insulator or not. Since Lu3+ has fourteen f -electrons, we do not suffer from the problem of theKondo lattice.

4.2.1 Calculations

The inputs for band calculations are following. The experimental crystal structure listed in Table4.3 is used [80]. The cut-off for plane-waves is set to RMTKmax = 7.0, while that for sphericalharmonics is set to `max = 10. 512 k-points are taken in the whole Brillouin zone, correspondingto 29 k-points in the irreducible Brillouin zone. The convergence criteria are 0.00001Ry for energyand 0.0001 for charge.


Table 4.3: The atomic positions for Lu2Ir2O7 with the cubic space group Fd3m (No. 227, originchoice 2). The lattice constant is a = 10.104A [80]. The muffin tin radii RMT are also listed (inthe atomic unit).

atom x y z RMT

Lu 1/2 1/2 1/2 2.38Ir 0 0 0 2.10O1 0.340 1/8 1/8 1.67O2 3/8 3/8 3/8 1.67

The electronic structure without the SOC is shown in Figure 4.7. Ir eg- and t2g-bands arewell-split, and narrow Lu f -bands lie in wide O p-bands. The bands near EF are mainly composedof Ir t2g-bands as shown in Figure 4.7 (b), and the depressed but finite DOS at EF shows that thecompound is a metal within the GGA.

Figure 4.8 is the results of the GGA+SOC calculations. A main difference from Na2IrO3 isthat t2g-bands are not well-split into jeff = 1/2- and jeff = 3/2-bands. In addition, no dip can beobserved at the Fermi energy EF. As seen in Figure 4.8 (d), the band across EF is degenerateboth with that below EF at Γ point and with that above EF at X point. Thus we conclude thatLu2Ir2O7 is not topological nor ordinary band insulators, though it is expected to be the closestto an insulator among pyrochlore iridates.


0

10

20

30

40

50

60

70

80

-8 -6 -4 -2 0 2 4 6 8

DO

S

energy [eV]

totalLu dLu fIr egIr t2g

O p

0

5

10

15

20

25

30

-2 -1.5 -1 -0.5 0 0.5

DO

S

energy [eV]

totalIr t2g

O p

(a) (b)

(c) (d)

-8

-6

-4

-2

0

2

4

6

8

LWX

ener

gy [e

V]

-2

-1.5

-1

-0.5

0

0.5

LWX

ener

gy [e

V]

Figure 4.7: GGA calculations of (a) the total (black) and projected (red, green, blue, magenta,and cyan for Lu d, Lu f , Ir eg, Ir t2g, and O p) DOSs, (b) near EF, and (c) the band structure, (d)near EF. In each panel, the origin of the energy axis is set to EF. See Figure 4.2 (b) for notationsof the high symmetric k-points.


0

10

20

30

40

50

60

70

80

-8 -6 -4 -2 0 2 4 6 8

DO

S

energy [eV]

totalLu dLu fIr egIr t2g

O p

0 2 4 6 8

10 12 14 16 18 20

-2 -1.5 -1 -0.5 0 0.5

DO

S

energy [eV]

totalIr t2g

O p

(a) (b)

(c) (d)

-8

-6

-4

-2

0

2

4

6

8

LWX

ener

gy [e

V]

-2

-1.5

-1

-0.5

0

0.5

LWX

ener

gy [e

V]

Figure 4.8: GGA+SOC calculations of (a) the total (black) and projected (red, green, blue, ma-genta, and cyan for Lu d, Lu f , Ir eg, Ir t2g, and O p) DOSs, (b) near EF, and (c) the bandstructure, (d) near EF.

Chapter 5

Summary and future problems

To conclude this thesis, 5d transition metal oxides have several unique properties due to the SOCand the electron correlation, and especially were investigated from the topological point of view.

A newly synthesized Ir oxide Na2IrO3, with the layered honeycomb lattice, was found to bea WTI through the study of the effective tight binding model and first principles calculations.In the effective model, the complex and spin-dependent transfer integrals to the NNN sites it′σa

open up a gap, leading to the QSH effect, i.e. the non-trivial Z2 number (−1)ν = −1 and thepresence of helical edge modes. Their origins are the complex and spin-dependent states due tothe SOC (jeff = 1/2-states) and lattice asymmetry so that the effective Aharonov-Bohm phases donot cancel out, which are applicable to the search for STIs in transition metal oxides. The effectivetight binding model can roughly explain first principles band calculations by including the realand spin-independent transfer integrals t′0 and the interlayer coupling t⊥, though discrepancy ofthe details are considered to result from lattice distortion, hybridization of jeff = 3/2-states, andthe electron correlation which the LDA can include. One possible reason of discrepancy betweenfirst principles calculations and experiments is the drawback of the LDA. Another reason is relatedto the electron correlation, which is discussed as future problems later.

In addition, we investigated the interplay between the non-trivial topology and the AFM in-teraction, and proposed the edge AFM phase within the mean field approximation. Helical edgemodes are robust against TR invariant perturbations, while are fragile against TR breaking per-turbations and open up a gap. On the other hand, the gapped bulk is relatively robust than thegapless edge, which leads to the AFM order only at the edge. Interestingly, the fractional charge islocalized at the domain wall of the edge AFM, reflecting the topological nature of QSH insulators.These topological natures can be observed by the STS measurement on the plateau edge. SinceNa2IrO3 is a WTI, usual methods to detect helical edge modes (ARPES and STS) can not detectits topological natures. To avoid the even/odd problem, we can focus on the plateau, whose edgeis spatially separated from the edges of underlying layers. The local DOS at the plateau edge isexpected to be suppressed at an onset temperature higher than the bulk Neel temperature.

The interplay between the non-trivial topology and the electron correlation contains many chal-lenging problems, especially in three dimensions and on frustrated lattices. Topological insulatorshave both charge and spin degrees of freedom at the surface, while in frustrated Mott insulatorsa charge degree of freedom is frozen. Thus topological insulators and Mott insulators can be dis-

53

CHAPTER 5. SUMMARY AND FUTURE PROBLEMS 54

tinguished at the surface, in spite of the even number of electrons per unit cell. It is an importantproblem where the critical point is and how a spin degree of freedom (spinon) behaves. Note thatYoung et al. proposed the fractionalized QSH (or quantum spinon Hall) effect, in which spinonsform edge modes [73].

The Kondo impurity and Kondo lattice in topological metals are another interesting problem.The study of Lu2Ir2O7 was considered as the first step of these problems, in fact, Ln = Yb has onef -hole. We obtained a metal, but instead this problem can be approached by doping of magneticelements to known STIs. The effective theory of one-dimensional helical liquids predicted thatthe conductance G is restored at T = 0 due to the formation of a Kondo singlet in the weakcorrelation region K > 1/4 (K is the Luttinger parameter), while G = 0 in the strong correlationregion K < 1/4 [35]. This prediction makes us understand that helical liquids are completelydifferent from usual TL liquids, though the behavior of two-dimensional helical liquids remains tobe solved. Thus topological transition metal oxides will be developed in various directions.

Acknowledgement

I would like to express my sincere gratitude to my supervisor Professor Naoto Nagaosa for hiskind advices and stimulating discussions. I am grateful to Doctor Hosho Katsura, Doctor Xiao-Liang Qi, and Professor Shou-Cheng Zhang for their collaborations on Chapter 3 and Doctor JanKunes on Chapter 4. I would acknowledge the experimental information by Takagi group, andthe computing resources by Tokura group. I would thank all the former and present members ofNagaosa group. Finally I would appreciate my family for their encouragement and support.

55

Bibliography

[1] K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).

[2] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405(1982).

[3] J. E. Avron, R. Seiler, and B. Simon, Phys. Rev. Lett. 51, 51 (1983).

[4] M. Kohmoto, Ann. Phys. 160, 343 (1985).

[5] Q. Niu, D. J. Thouless, and Y.-S. Wu, Phys. Rev. B 31, 3372 (1985).

[6] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).

[7] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005).

[8] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005).

[9] B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. 96, 106802 (2006).

[10] S. Murakami, Phys. Rev. Lett. 97, 236805 (2006).

[11] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (2006).

[12] M. Konig et al., Science 318, 766 (2007).

[13] L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007).

[14] S. Murakami, New J. Phys. 9, 356 (2007).

[15] D. Hsieh et al., Nature 452, 970 (2008).

[16] C. Liu et al., Phys. Rev. Lett. 100, 236601 (2008).

[17] L. Sheng, D. N. Sheng, C. S. Ting, and F. D. M. Haldane, Phys. Rev. Lett. 95, 136602 (2005).

[18] C. Wu, B. A. Bernevig, and S.-C. Zhang, Phys. Rev. Lett. 96, 106401 (2006).

[19] C. Xu and J. E. Moore, Phys. Rev. B 73, 045322 (2006).

[20] L. Fu and C. L. Kane, Phys. Rev. B 74, 195312 (2006).

56

BIBLIOGRAPHY 57

[21] L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007).

[22] J. E. Moore and L. Balents, Phys. Rev. B 75, 121306(R) (2007).

[23] R. Roy, cond-mat/060753 (unpublished).

[24] M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 79, 1039 (1998).

[25] Y. Tokura and N. Nagaosa, Science 288, 462 (2000).

[26] E. Dagotto, Science 309, 257 (2005).

[27] I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958).

[28] T. Moriya, Phys. Rev. 120, 91 (1960).

[29] M. K. Crawford et al., Phys. Rev. B 49, 9198 (1994).

[30] G. Cao et al., Phys. Rev. B 57, 11039 .

[31] S. J. Moon et al., Phys. Rev. B 74, 113104 (2006).

[32] T. Shimura, M. Itoh, and T. Nakamura, J. Solid State Chem. 98, 198 (1992).

[33] R. S. Perry et al., New J. Phys. 8, 175 (2006).

[34] B. J. Kim et al., Phys. Rev. Lett. 101, 076402 (2008).

[35] J. Maciejko et al., arXiv:0901.1685 (unpublished).

[36] A. Shitade et al., arXiv:0809.1317 (unpublished).

[37] D. J. Thouless, Phys. Rev. B 27, 6083 (1983).

[38] Q. Niu and D. J. Thouless, J. Phys. A 17, 2453 (1984).

[39] M. I. D’yakonov and V. I. Perel, ZhETF Pis. Red. 13, 657 (1971).

[40] S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348 (2003).

[41] J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004).

[42] J. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys. Rev. B 70, 041303(R) (2004).

[43] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004).

[44] J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 94, 047204 (2005).

[45] S. Murakami, Phys. Rev. B 69, 241202(R) (2004).

[46] T. Kimura et al., Phys. Rev. Lett. 98, 156601 (2007).

BIBLIOGRAPHY 58

[47] T. Seki et al., Nat. Mater. 7, 125 (2008).

[48] S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006).

[49] G. Y. Guo, S. Murakami, T.-W. Chen, and N. Nagaosa, Phys. Rev. Lett. 100, 096401 (2008).

[50] H. Kontani et al., J. Phys. Soc. Jpn. 76, 103702 (2007).

[51] T. Tanaka et al., Phys. Rev. B 77, 165117 (2008).

[52] G.-Y. Guo, S. Murakami, and N. Nagaosa, Phys. Rev. Lett. 102, 036401 (2009).

[53] S. Urazhdin et al., Phys. Rev. B 69, 085313 (2004).

[54] H.-J. Noh et al., Europhys. Lett. 81, 57006 (2008).

[55] Y. Xia et al., arXiv:0812.2078 (unpublished).

[56] H. Zhang et al., arXiv:0812.1622 (unpublished).

[57] M. Itoh, T. Shimura, Y. Inaguma, and Y. Morii, J. Solid State Chem. 118, 206 (1995).

[58] B. J. Kim et al., Phys. Rev. Lett. 97, 106401 (2006).

[59] B. J. Kim, private communication.

[60] H. Takagi, private communication.

[61] K. Momma and F. Izumi, J. Appl. Crystallogr. 41, 653 (2008).

[62] J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954).

[63] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).

[64] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Nat. Phys. 4, 273 (2008).

[65] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424 (2008).

[66] A. Kitaev, Ann. Phys. 321, 2 (2006).

[67] P. Blaha et al., An Augmented Plane Wave + Local Orbitals Program for Calculating CrystalProperties (Karlheinz Schwarz, Techn. Universitat Wien, Austria, 2001).

[68] P. Hohenberg and W. Kohn, Phys. Rev 136, B 864 (1964).

[69] W. Kohn and L. J. Sham, Phys. Rev 140, A 1133 (1965).

[70] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).

[71] D. D. Koelling and B. N. Harmon, J. Phys. C 10, 3107 (1977).

BIBLIOGRAPHY 59

[72] A. Kokalj, Comp. Mater. Sci. 28, 155 (2003).

[73] M. W. Young, S.-S. Lee, and C. Kallin, Phys. Rev. B 78, 125316 (2008).

[74] D. Yanagishima and Y. Maeno, J. Phys. Soc. Jpn. 70, 2880 (2001).

[75] K. Matsuhira et al., J. Phys. Soc. Jpn. 76, 043706 (2007).

[76] S. Nakatsuji et al., Phys. Rev. Lett. 96, 087204 (2006).

[77] Y. Machida et al., Phys. Rev. Lett. 98, 057203 (2007).

[78] R. S. Singh, V. R. R. Medicherla, K. Maiti, and E. V. Sampathkumaran, Phys. Rev. B 77,201102(R) (2008).

[79] K. Maiti, arXiv:0901.3847 (unpublished).

[80] N. Taira, M. Wakeshima, and Y. Hinatsu, J. Phys.: Condens. Matter 13, 5527 (2001).

atsuo shitade- quantum spin hall effect in transition metal oxides

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