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Topological Quantum Phase Transition in 5d Transition Metal Oxide Na2IrO3

Choong H. Kim,1Heung Sik Kim,1Hogyun Jeong,2,3Hosub Jin,4and Jaejun Yu1,5, ∗

1Department of Physics and Astronomy and Center for Strongly Correlated Materials Research,

Seoul National University, Seoul 151-747, Korea

2Computational Science and Technology Interdisciplinary Program,

Seoul National University, Seoul 151-747, Korea

3Korea Institute of Science and Technology Information, Daejeon 305-806, Korea

4Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA

5Center for Theoretical Physics, Seoul National University, Seoul 151-747, Korea

(Dated: January 31, 2012)

We predict a quantum phase transition from normal-to-topological insulators in 5d transition

metal oxide Na2IrO3, where the transition can be driven by the change of the long-range hopping

and trigonal crystal field terms. From the first-principles-derived tight-binding Hamiltonian we de-

termine the phase boundary through the parity analysis. In addition, our first-principles calculations

for Na2IrO3 model structures show that the inter-layer distance can be an important parameter for

the existence of a three-dimensional strong topological insulator phase. Na2IrO3 is suggested to

be a candidate material which can have both non-trivial topology of bands and strong electron

correlations.

PACS numbers: 71.70.Ej, 73.20.-r, 73.43.Nq

Topological insulators are newly discovered materials

with a bulk band gap and topologically protected metallic

surface states [1, 2]. The theoretical predictions on Bi-

based topological insulators, such as BixSb1−x, Bi2Se3,

and Bi2Te3[3, 4], have been experimentally realized [5–

8]. The search for topological insulators has been ex-

tended to ternary Heusler [9] and chalcogenides com-

pounds [10], but they are still limited to the either nar-

row or zero gap semiconductors. Recently the layered

honeycomb lattice Na2IrO3[11] and pyrochlore A2Ir2O7

[12, 13] have been suggested as possible topological in-

sulators, though topological insulators with transition

metal d electrons have not been fully investigated. Con-

trary to the s-p electron systems, transition metal ox-

ides with localized d electrons are expected to have both

strong on-site Coulomb interactions and spin-orbit cou-

plings.In particular, 5d transition metal oxides such

as iridates have a relatively weaker Coulomb correlation

competing with spin-orbit coupled band structures.

The interplay between the non-trivial band topology

and the electron correlation effect can be an interest-

ing development in the study of topological insulators.

Ir-based transition metal oxides have shown some no-

ble physics such as jeff= 1/2 insulating state in Sr2IrO4

[14–16], anomalous metal-insulator transition in A2Ir2O7

[17], spin-liquid state in Na4Ir3O8[18], where both spin-

orbit coupling and correlation play important roles.

Na2IrO3[19, 20] has been proposed as a layered quan-

tum spin Hall (QSH) insulator. Assuming the jeff= 1/2

character of the band near the Fermi level [11], a single-

band tight-binding model for the Ir-O layer is mapped

to the Kane-Mele model [1] for the QSH effect. The pro-

posed model Hamiltonian is, however, inconsistent with

∗Corresponding author. Electronic address: jyu@snu.ac.kr

the first-principles band structure result which leads to a

different prediction on the band topology of Na2IrO3[21].

The inconsistency is partly due to the structural param-

eters used for the first-principles calculations. It implies

that its band topology may be sensitive to the structural

variation. Consequently a small change of structure or in-

teraction strength can drive a quantum phase transition,

e.g., the change of its topological character. To under-

stand the basic physics determining the band topology of

Na2IrO3, we need to clarify the topological character of

the spin-orbit coupled ground state and its dependence

on the structure and interaction strength.

In this Letter, we present a quantum phase transition

between topological insulators (TIs) and normal insula-

tors (NIs) in Na2IrO3based on an effective tight-binding

Hamiltonian. We derived the effective Hamiltonian from

the realistic tight-binding description of first-principles

band structures. The electronic structure of the planar

edge-shared IrO6octahedra contains large trigonal crys-

tal field and direct hopping terms as well as a signifi-

cant long range hopping between extended Ir 5d orbitals.

The phase boundary between topological and normal in-

sulating phases is shown to depend on both the trigo-

nal crystal field and the long-range hopping in Na2IrO3.

From the first-principles calculations of model structures,

which simulate the change of tight-binding parameters,

we confirmed that the inter-layer distance can play a cru-

cial role in determination of the three-dimensional strong

TI in Na2IrO3.

We introduce a tight-binding (TB) model for Na2IrO3

from the results of first-principles calculations. (See sup-

plement for details.) The model is based on the Ir t2g

manifold in the two-dimensional honeycomb lattice and

incorporates parameters for an indirect hopping through

oxygen 2p-orbital (tpd), two kinds of direct hopping be-

tween neighboring Ir atoms (tdd1and tdd2), and another

arXiv:1201.5929v1 [cond-mat.mtrl-sci] 28 Jan 2012

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tpd

tdd2

tdd1

(b)

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x

y

z

x

y

z

x

y

z

tn2

tn1

(c)

-2.0

-1.5

-1.0

-0.5

0.0

0.5

Γ

KM

Γ

LDA+SO

TB+SO

FIG. 1. Hopping parameters considered in our tight-binding

model, (a) indirect hopping(tpd) mediated by oxygen 2p

orbital and two kinds of direct hoppings (tdd1and tdd2),

(b) second-nearest-neighbor hopping (tn1), third-nearest-

neighbor hopping (tn2), and (c) band structure of first-

principles (straight) and tight-binding (dashed) calculations

with spin-orbit coupling.

Parameters

tpd

tdd1

tdd2

tn

∆

λ

Character

(pdπ)2/(?d− ?p)

3

4(ddσ) +1

1

2(ddπ) +1

(sdσ)2/(?d− ?s)

Ee?g− Ea1g

Spin-orbit coupling

Value (eV)

0.25

−0.5

0.15

−0.075

0.6

0.5

4(ddδ)

2(ddδ)

TABLE I. Values of tight-binding parameters (in eV) obtained

from the first-principles band structure.

indirect hopping through the sodium 3s-orbital (tn), as

illustrated in Fig. 1(a) and (b). It is noted that the indi-

rect hopping through the sodium 3s-orbital of Fig. 1(b)

makes the second- and third-nearest-neighbor hopping

terms non-negligible and plays a crucial role in determi-

nation of the topology of the band structure. In addi-

tion, we take into account both trigonal crystal field ∆

and spin-orbit coupling (SOC) λ terms. The TB param-

eters are summarized in Table I. The energy unit is in

eV. The values of parameters are fitted to the results of

density-functional-theory (DFT) calculations within the

local density approximation (LDA). For the DFT calcu-

lations, we used the DFT code, OpenMX [22], based on

the linear-combination-of pseudo-atomic-orbitals method

[23].The SOC were included via the relativistic j-

dependent pseudo-potential scheme in the non-collinear

DFT formalism [24–26]. The unit cell we used in the

DFT calculations is a simplified version of the original

δ(Γ)

-1

-1

δ(M1)

-1

1

δ(M2)

-1

1

δ(M3)

-1

1

Z2(ν)

0

1

tn = −0.075

tn = 0

TABLE II. Parity invariants δ(Γi) and Z2 topological invari-

ants (ν) with and without tn as determined from the product

of parity eigenvalues at each time-reversal-invariant-momenta

Γ, M1, M2 and M3.

crystal, where the c-axis periodicity is reduced by chang-

ing the relative stacking of Na layers with respect to Ir

network, and also by neglecting the distortion of oxygen

octahedra. We have checked the band structures with the

different stacking of Na layers and the rotation of oxy-

gen atoms around the axis perpendicular to the plane,

and observed only a small change in the band dispersion

and fitting parameters. Our TB model describes well the

DFT band structure, especially the bands near the Fermi

level (EF), as shown in Fig. 1(c).

It is interesting that the calculated parity invariants

and Z2 topological numbers for our TB model depend

critically on the second- and third-nearest neighbor hop-

ping term tn, as demonstrated in Table II. The topolog-

ical invariants were determined by following the method

proposed by Fu and Kane [27].

the DFT band structure with tn = −0.075 turns out

to be trivial. This result is consistent with the previ-

ous first-principles calculations result [21], contrary to

the quantum spin Hall insulator phase predicted by Shi-

tade et al. [11]. This discrepancy is likely due to an

over simplification of the tight-binding model employed

in Ref. [11]. Their model was based on the assumption

that the low energy degrees of freedom are determined

by the half-filled jeff=1/2 doublets. However, when the

significant trigonal crystal field is introduced in Na2IrO3,

jeff = 1/2 and jeff= 3/2 are not well separated and the

jeff=1/2 doublets no longer serve as a useful basis [21].

The extended nature of 5d-orbital combined with edge-

shared octahedral structure in the Ir2/3Na1/3O2plane of

Na2IrO3is a source of such strong trigonal crystal field.

The topological character of Na2IrO3is quite sensitive

to the magnitude of tn. By turning off the second- and

third-nearest-neighbor hopping, i.e., tn= 0, the system

becomes a non-trivial TI with ν = 1. The tiny differ-

ence in the magnitude of tnis responsible for the inver-

sion of the valance and conduction bands, thereby lead-

ing to the change of the δ(M) sign. It indicates that

tnis a key control parameter for the ’band-inversion’ in

this system. Therefore, the TB model with the nearest-

neighbor hoppings only can not be sufficient for the de-

scription of the topological character of Na2IrO3 even

though the nearest-neighbor TB models can describe rea-

sonably the band dispersions of the jeff = 1/2 states in

Sr2IrO4[14, 16] and the hyper-Kagome Na4Ir3O8[28, 29].

The non-trivial (trivial) Z2 topological number can

also be confirmed by the odd (even) number of pairs of

gapless edge states. To examine the edge states, we con-

The Z2 topology of

Page 3

3

-0.6

-0.4

-0.2

0.0

0.2

0

π/a

2π/a

(a) tn= −0.075

-0.6

-0.4

-0.2

0.0

0.2

0

π/a

2π/a

(b) tn= 0

Energy(eV)

FIG. 2. One-dimensional energy bands for armchair strip in

the (a) NI with tn = −0.075 and (b) TI phase with tn = 0.

-0.20

-0.10

0.00

0.10

0.20

0.30

Γ

KM

Γ

Energy (eV)

tn= −0.000

tn= −0.021

tn= −0.040

tn= −0.075

FIG. 3. TB band structure with the variation of tn. Red

(tn = 0) line belongs to the TI, and blue (tn = −0.040) and

yellow (tn = −0.075) lines belong to the NI. Note that at the

transition point, i.e. tn = −0.021, the dispersion is linear.

structed a TB Hamiltonian for the strip geometry with

two edges in an armchair configuration. Figure 2 shows

the one-dimensional energy bands with tn= −0.075 and

tn = 0. The bulk states and gaps are clearly seen and

there are edge states which transverse the gap. In the

normal insulating phase (ν = 0) the edge states cross the

Fermi energy even number of times as expected from the

trivial Z2topological number.

Transitions between trivial and non-trivial phases can

be tuned by adjusting the key parameters of the band

structure. Murakami and co-workers [30] developed the

low-energy effective theory for phase transition between

TI and NI systems in 2D and discussed the classification

of the possible types of transition. Following their clas-

sification, the gap closing at the time-reversal-invariant-

momenta (k = G/2) occurs in systems with inversion

symmetry. In our model, tncontrols the gap closing at

the M point in the Brillouin zone contrary to the case of

a honeycomb lattice model such as the Kane-Mele model,

where the gap closes at the K and K?points. The low-

energy long-wavelength effective Hamiltonian, which is

expanded to linear order in k around the M point, can

λ/λ0

1

∞

ABCD

-0.00

-0.13

-0.12

-0.00

-1.61

-0.13

0.58

0.00

TABLE III. The expansion coefficients of mass function in

Eq. 2 when λ/λ0 = 0 and ∞. These coefficients at λ/λ0 = 1

and ∆ = 0.6 correctly describes the transition point tn ?

−0.021.

be written as

H(k) = E0+

m

z+ −m

z−

m

−z+

−z− −m

x+ b3k?

,

(1)

where k?= k − M and z±= b1k?

real constant b1, b2and b3[30]. Since the eigenenergies

E = E0±?m2+ z+z−, m = 0 corresponds to the gap

Figure 3 shows the TB band structure along some high-

symmetry line for the several values of tn. In TI region

(tn= 0), the valance and conduction bands are separated

by a finite gap with negative mass. When |tn| increases,

the direct gap at the M point decreases. Finally, the

gap collapses at the transition point tn ? −0.021. As

|tn| passes through the transition point, the gap re-opens

and the system becomes a normal insulator with positive

mass. During this procedure, the order of the bands is

inverted around the M point, which characterizes topo-

logical nature of the system.

To further verify the relation between the mass and

the control parameters, we obtain the expression for the

mass as a function of tn, ∆ and SOC (λ).

y± ib2k?

ywith

closing.

2m = A + B∆ + Ctn1− 2tn2+ D∆tn1.

Here, to clarify the role of second- and third-nearest

hoppings, we refined tninto the second-nearest-neighbor

hopping, tn1, and the third-nearest-neighbor hopping,

tn2. The expansion coefficients A, B, C, and D can be

represented as a function of λ. In the range of λ from

0.8λ0to 1.6λ0, where λ0is the SOC determined by the

LDA calculation, the coefficients can be fitted as follows:

(2)

A = +0.0746 − 0.1438λ

B = −0.0237 − 0.3061λ + 0.2587λ2

C = −2.2178 + 1.2660λ

D = +1.0975 − 0.6837λ.

In the limiting cases of λ/λ0= 1 and ∞ the coefficients

are also shown in Table III. From Eq. (2), we can deter-

mine the Z2topological number of the system for given

tn, ∆ and λ by checking the sign of mass. This mass func-

tion correctly characterizes the topological phase transi-

tion.

Figure 4 shows a λ−tnphase diagram for several val-

ues of ∆ . For a sufficiently large λ, an indirect bulk gap

(3)

Page 4

4

0

1

2

3

4

5

0.000.020.040.06 0.080.10

|tc| ≈ 0.067

λ/λ0

|tn(= tn1,tn2)|

TI

(m < 0)

NI

(m > 0)

Metal

∆ = 0.4

∆ = 0.6

∆ = 0.8

FIG. 4. Phase diagram as a function of tnand SOC for several

values of ∆. Pink circle indicates where the reality exists.

|tc| = 0.062 is the asymptotic line in the case of ∆ = 0.6.

opens and the system becomes an insulator. The insu-

lating region is divided into two phases: TI and NI. The

phase boundary between TI and NI corresponds to the

m = 0 line. There seems to be an asymptotic limit of

tnbeyond which the system stays as a normal insulator

regardless of the strength of the SOC λ. The vertical

dashed line in Fig. 4 is described by

tn=

A

C − 2≈ −0.067 (4)

with ∆=0.6. This implies that an arbitrarily large SOC

cannot guarantee the non-trivial topology even if the up-

per four bands originates from the jeff= 1/2 states.

Another important factor controlling the topological

character is the trigonal crystal field, ∆. In fact both

SOC and trigonal crystal field are two competing pa-

rameters characterizing the bands near the Fermi level.

While SOC prefers the formation of the jeff= 1/2 state,

the large trigonal crystal field becomes an obstacle for

the jeff = 1/2 state. In this system, however, the trig-

onal crystal field seems to play a crucial role. As far as

B < 0 and D > 0 of Eq. (2) as shown in Table III, it is

clear that a large trigonal crystal field is in favor of the

non-trivial topology. This trend in the parameter depen-

dence of the topological character is clearly reflected in

the phase diagram of Fig. 4.

Returning to a realistic system of Na2IrO3materials,

we tried to probe possible TI phases by performing first-

principles calculations. According to the simulated TB

parameters for the TI phase, we choose two represen-

tative structures among the various structures: (i) the

original geometry of Na2IrO3(c/c0= 1.0) and (ii) a vir-

tual structure with the expanded inter-layer distance by

30% (c/c0 = 1.3). Calculated Z2 topological numbers

for each configuration are as listed in Table IV. Here

the c/c0 parameter represents the change of inter-layer

distance relative to the original one. It is remarkable

to find a non-trivial Z2 number for the structure with

c/c0 = 1.3.In other words, the increased inter-layer

δ(Γ)

-1

-1

δ(M)

-1

1

δ(A)

-1

-1

δ(L)

-1

-1

ν;(ν1ν2ν3)

0; (000)

1; (000)

c/c0=1.0

c/c0=1.3

TABLE IV. Parity invariants δ(Γi) and Z2 topological in-

variants calculated from first-principles calculations. We con-

sidered two different structures: (i) the original geometry of

Na2IrO3 (c/c0 = 1.0) and (ii) a virtual structure with the

enlarged inter-layer distance by 30% (c/c0 = 1.3).

distance drives the NI into the TI. To understand the

change of electronic structure between c/c0 = 1.0 and

c/c0= 1.3, we constructed the maximally localized Wan-

nier function (MLWF) [31, 32] and obtained the MLWF

effective Hamiltonian. It is found that the trigonal crys-

tal field enhanced by the increase of inter-layer distance

contributes to the change of Z2 character for Na2IrO3.

When we increase the inter-layer distance, the energy

level of the inter-plane Na 3s state lowers. The second

or higher order hoppings through unoccupied 3s state

of inter-plane Na contributes to the on-site Hamiltonian

matrix elements. Since these matrix elements give rise to

the energy separation between e?

interpreted that the trigonal crystal field contributes to

the band inversion at M point. In fact the band inversion

occurs only at three M points in the kz= 0 plane. How-

ever, as shown in Table IV, the parity invariants δ(Γi)

for c/c0 = 1.3 are quite different for the kz = 0 and

kz = π planes. It indicates that the c-axis hopping is

also important for the existence of the TI phase in the

three-dimensional system of Na2IrO3.

In conclusion we provide a microscopic picture for the

topological phase diagram in Na2IrO3 and identify the

key control parameters based on the effective Hamilto-

nian analysis. We predict that the TI phase of Na2IrO3

can be realized by controlling the long-range hopping and

trigonal crystal field terms. Our first-principles calcula-

tions for the simulated Na2IrO3model structure suggest

that the inter-layer distance can play a crucial role in

determination of a three-dimensional strong TI. In prac-

tice, we propose two ways of driving a transition from

NI to TI: (i) epitaxial strain by the substitution of Na

by other elements such as Li or a film growing on a ap-

propriate substrate, (ii) intercalation of some molecules

for the increase of the inter-layer distance (e.g., is the

water-intercalated Na0.35CoO2·1.3H2O showing the su-

perconductivity [33]). Indeed our first-principles calcula-

tions demonstrate that the topological insulator phase of

Li2IrO3 can be achieved by 2% in-plane lattice strain

[34]. In addition, other structure manipulation tech-

niques developed for superlattice and heterostructure

may be adopted to control the NI-to-TI transition and

to design the topological-insulator-based devices. Recent

experiment observed an antiferromagnetic (AFM) insu-

lating behavior in Na2IrO3[19, 20]. The AFM ordering

in TI phase of Na2IrO3can be a candidate of topological

magnetic insulator [35, 36] or topological Weyl semimetal

gand a1gstates it can be

Page 5

5

[37] below ordering temperature (TN). Above TN, corre-

lations could enhance the SOC effects due to the suppres-

sion of effective bandwidth to stabilize the TI phase [12].

Further there are proposals for the topological Mott insu-

lator having gapless surface spin-only excitations [11, 12].

Na2IrO3is likely to provide a new playground to study

the effect of the correlation in TI.

ACKNOWLEDGMENTS

This work was supported by the NRF through the

ARP (R17-2008-033-01000-0). We also acknowledge the

support from KISTI under the Supercomputing Applica-

tion Support Program.

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