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A Pedagogical Review of Topological Insulators

Yi-Chun Chen

Topological insulators, although insulated in the bulk, own conducting states on the edge andlead to edge spin currents. The strange behavior of topological insulators is the result of topologicalindex in energy bands and time-reversal symmetry. In this article, we will review topological insula-tors from quantum Hall effect, which first connected topology and energy bands, then generalize itto quantum spin Hall effect, which stands for the phenomena on topological insulators, and demon-strate the topological Z2 numbers. Finally, Graphene and HgTe/CdTe quantum well are introducedas models, especially the latter one was confirmed as topological insulators by experiments. Forthe more development such as 3D topological insulators, topological field theory and topologicalsuperconductors, we put them in the end as an extension.

* This review paper is the final project of the course Theory of Solids.

INTRODUCTION

Topological insulators were first predicted by Kaneand Mele in 2005 [1] [2]. They predicted that whenthe spin-orbit interactions are taken into account in

graphene, the resulting Hamiltonian will behave liketwo copies of Hamiltonian in Quantum Hall Effect(QHE) for each spin. They termed such a conse-quence as Quantum Spin Hall Effect (QSHE). Justlike QHE, QSHE also provides the gap-less conduct-ing states on the boundaries while the bulk remainsinsulated. However, different from QHE, QSHE pro-vides spin current rather than charge current. It isworthy to emphasize that such spin current is robustdue to the so-called Z2 topological invariant. Actu-ally, it is such Z2 invariant distinguish the topologi-cal insulators from other usual insulators. Althoughgraphenes are proposed as topological insulators by

Kane and Mele in 2005, it never comes true becausethe spin-orbit interactions in carbon atoms are weak.Hence people look up to the heavy elements that ownthe strong spin-orbit interactions. Bernevig, Hughesand Zhang (BHZ) considered the HgTe/CdTe quan-tum well and provided the theoretical prediction [3].In such quantum well, HgTe is sandwiched betweenlayers of CdTe. When the thickness of HgTe is lagerthan 6.3nm, the s-type and p-type bands will inversein HgTe but remain same in CdTe. BHZ showedthat the inversion of the bands leads to non-trivialZ2 invariant and the quantum well can be treatedas topological insulator. Within one year after the

BHZ proposal, the Wurzburg group, led by LaurensMolenkamp, made the devices and performed the ex-periment [4], showing the first observation of topo-logical insulator. In this paper, we give a review ofthe development introduced above. We first intro-duce QHE and the connection between topology andphysics. Then we talk about topological numberZ2in topological insulators and its models, including thegraphene model and quantum well model with experi-mental observations. This article focuses on 2D topo-logical insulator. The 3D case and other developments

will be discussed in the conclusion and extension part.

SHORT REVIEW OF QHE

QHE (we focus on Integer QHE here) was discov-

ered by v. Klitzing etal.in 1980. At low temperatureand under high magnetic fields, the Hall conductivityxy will be quantized

xy=e2

hn (1)

,where n is an integer. Traditionally, people use theconcept of Landau level to understand QHE as fol-lowing: under magnetic field, the classical circularmotions of electrons could be treated as simple har-monic oscillation with frequency c = eB/meC andthe energy will be quantized as En = c(n+ 1/2).If some Landau levels are occupied and the rest areempty, than we can roughly view Landau levels as aband structure and the treat it as a insulator.Near the boundary, the confinement potential leadsthe bands no longer flat and there will be some edgecurrent accounted for (1). However, for usually in-sulators, there is no such edge current. So what isthe difference between ordinary insulators and QHEinsulators? Thouless, Kohmoto, Nightingale and denNijs (TKNN) gave the answer in 1982 [5]. In theirbreakthrough paper, they pointed out that the differ-ence is caused by topology defined on reciprocal space.They defined TKNN topological invariant (or Chern

invariant in math) as

n= 1

2

BZ

f d2k (2)

,where integral is over the first Brillouin zone andf is the Berry flux which is defined as the curl ofBerry connection A = iuk|k|uk, i.e., f = Aand the uk is the Bloch wave function. The conceptof Berry connection and flux was first proposed byMichael Berry in 1984 [6]. The TKNN invariant isthe total Berry flux in the first Brillouin zone. The

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integer property of the integral (2) is actually analo-gous to Gauss-Bonnet theorem in mathematics : theintegral of curvature of whole surface divided by 2will be an integer. For example, such integral for aball is 2 and 0 for a donut. Obviously, (2) only de-fined to one band and we can easily extend to wholeband structure. That is N=

all bands n. What does

(2) have something to do with the Hall conductivity

? By linear respond theorem,

xy = ie2

hV m2

Ea

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Figure 2(a), which has one Kramer pair and there aregap-less bands, then it will satisfy the Kramer theo-rem. But the gapped band structure in Figure 2(b)is impossible (because bands are not continuous anymore) and violates the Kramers theorem. On theother hand, both of the 2-Kramer-pair band struc-tures like Figure 2(c) and 2(d) are allowable. So wecan deform the band structure from 4-degenerate(c)

to gapped structure(d).

Figure2: Illustration of Kramer pairs band. Red andBlue denote opposite spins. Quoted from [8]

That is to say, the only way to transform the bandto gapped configuration is even-number Kramer pairs.Now we have topologically different indexes to classifythe insulators: odd or even number of Kramer pairs,i.e., Z2 index. Thus for insulators whose number ofKramer pairs is odd, there must be a gap-less config-uration on the edge, because the bandgap like Figure2(b) will not happen. Nevertheless, for the insulators

whose number of Kramer pairs is even, it will accountfor those ordinary insulators which have no gap-lessedge states. To get such index, Kane and Mele gavea mathematical formalism [2]. They defined

P(k) = P F(u(k)||u(k)) (5)

In math, the determinant of a skew-symmetric ma-trix can always be written as the square of a polyno-mial in the matrix entries. This polynomial is calledthe Pfaffian of the matrix [10]. Then they define thetopological index as

= 1

2ic

dkklog[P(k) + i] (6)

,where integral routeCis around the half of first Brol-louin zone. We call asZ2 index. will evaluate thewinding number of phase of P(k) around the loop C,or equivalently, the number of pairs of complex zero ofP(k), which will correspond to the number of Kramerpairs. There are other ways to calculate the Z2 index[2][11][12]. For example,

= 1

2

(hBZ)

dkA

hBZ

d2kf

mod2 (7)

,which connect the Berry phase and topological indexagain while the integration is over the half of the firstBrillouin zone. For the crystal with inversion symme-try, there is a shortcut to get the Z2 index [12]. Forthose time-reversal invariant point i in the first Bril-louin zone, we can define 2m(i) =1 as the parityeigenvalue of the 2mth occupied band at i, whichshares the same eigenvalue for Kramer partner. Then

define

i=N

m=1

2m(i) (8)

Then theZ2 invariant= 0 or 1 can be defined as

(1) =i

i (9)

QSHE ON GRAPHENE

Now we look at the QSHE on the Graphene. It wasfirst pointed out by Kane and Mele [1]. In the sim-ple picture, spin operator Sz commutes with the spin-orbit-interaction Hamiltonian, so the Hamiltonian de-couples into two Hamiltonian for each spin. The re-sult is just two copies of Haldane model [9]. The Hallconductivity in this case is zero because the the di-rections of charge currents are opposite for differentspins. But there is a quantized spin Hall conductiv-itysxy by defining J

x J

x =

sxyEy. There are also

edge gap-less states for QSHE. The above descriptionis based on the conservation of spin Sz. However, the

conservation of spinSz is not necessary for the robustedge gap-less states. Kane and Mele [1] pointed outthat the edge states are protected by the Kramer de-generacy as we mentioned above. Interestingly, theQSH edge states make the up-spins propagate in onedirection but down-spins in the other and such phe-nomena are termed helical.

QSHE ON QUANTUM WELL

To enhance the strength of spin-orbit interaction intopological insulator, it is better to look for the heavy

elements. Bernevig, Hughes and Zhang (BHZ) consid-ered the HgTe/CdTe quantum well structure [3]. Forboth HgTe and CdTe, the important states near Fermilevel are those near point in the Brillouin zone. InCdTe, the band structure is similar to semiconductorswith s-type conduction band and p-type valence band.The separation between two bands are large (1.6 eV).But in HgTe, due to the stronger spin-orbit interac-tion, the p-like band is lower than s-like band withseparation 300 meV. There is something interestingwhen HgTe is sandwiched by CdTe. If the thickness

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of HgTe is larger thandc=6.3 nm, then two bands willbehave like what they should be in HgTe. That makesbands inverted from CdTe to HgTe. If the thicknessof HgTe is less than 6.3 nm , there is no such inver-sion. BHZ showed the inversion of the bands withincreasing thickness is a signal of the transformationfrom ordinary insulator to topological insulator. Theproof could be easily done through (9). Because s-

band and p-band have different parities, the inversionof two bands leads to (-1) to the value of R.H.S. of (9)andZ2 changes from 0 to 1.

Within a year of the BHZ proposal, the Wurzburggroup, led by Laurens Molenkamp, made the devicesand performed the experiment [4]. The experiment re-sult is Figure (3). The experiment measured the resis-tance of several samples. The black curve(Sample(i))is a narrow(d < dc) quantum well, which has a highresistance when Fermi energy is between the gap.Sample(ii-iv) are wider quantum wells(d > dc). Sam-ple(iii) and (iv) show the conductivity 2e2/h. The

length of (iii) and (iv) are the same (L= 1 m) butdifferent in w (0.5 and 1 m), indicating that theedge current is in L direction. Sample(ii) with longerL=20 m shows the finite temperature scattering ef-fect. This experiment result showed the QSHE edgecurrent conductance and proved that the wider quan-tum wells with inverted band are topological insula-tors.

EXTENSION AND CONCLUSION

We mainly focus on the early development in this re-view. The theoretical description here is called Topo-logical band theory(TBT). TBT was extended to 3Dcase by three different groups [12][13][14]. It turns outthere are 16 topologically distinct states for 3D ratherthan only two for 2D. On the other hand, TBT isonly valid for noninteracting systems. QHZ[11] devel-oped the topological field theory and showed thatthese QSH states are stable even under the stronginteractions. After the discovery of topological insu-lators, the topologically classification was generalizedto superconductors and superfluids [15][16][17][18]. Sotopological insulators are not only fascinating in its

own field but also stimulate the theoretical develop-ment in other fields. There are a number of greatreviews, even written by the men who discovered thetopological insulators [7][19]. In our review, we givea pedagogical introduction from the relation betweentopology and band theory in QHE, then talking aboutQSHE in graphene and HgTe/CgTe quantum well, fi-nally showing the experimental observation of topo-logical insulators. This review is for the course The-ory of Solid in NTU physics. Here is my acknowl-edgement to the instructor Pro.Ying-Jer Kao.

Figure3: The observation of QSHE. The longitu-dinal 4-terminal resistance of several samples fornormal(d = 5.5nm < dc)(i) and inverted(d =7.3nm > dc)(ii-iv) QW structure as a function of gate

voltage. They are measured for B=0 and T=30mk.The up-left inset is the sample. The right insetshows the comparison between T=30mk(green) andT=1.8k(black) of sample(iii) on a linear scale. Quotedfrom [4].

[1] Kane, C. L., and E. J. Mele, 2005a, Phys. Rev. Lett.95, 226801

[2] Kane, C. L., and E. J. Mele, 2005b, Phys. Rev. Lett.95, 146802

[3] Bernevig, B. A., T. A. Hughes, and S. C. Zhang, 2006,

Science 314, 1757[4] Knig, M., S. Wiedmann, C. Brne, A. Roth, H. Buh-mann, L.W. Molenkamp, X. L. Qi, and S. C. Zhang,2007, Science 318,766

[5] Thouless,D. J., M. Kohmoto, M.P. Nightingale andM. den Nijs, 1982, Phys. Rev. Lett. 98, 010506

[6] Berry, M.V., 1984, Proc. R. Soc. London, Ser.A392,45.

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

[8] A.Strom. PhD thesis, Gothenburg, 2011[9] Haldane,F.D.M., 1988, Phys.Rev. Lett. 61,2015

[10] http://en.wikipedia.org/wiki/Pfaffian[11] X. L. Qi, T. L. Hughes and S. C. Zhang, Phys. Rev.

B, 78, 195424, 2008.

[12] L. Fu and C. L. Kane, Phys. Rev. B, 76, 045302, 2007.[13] Fu, L., C. L. Kane and E. J. Mele, 2007, Phys. Rev.

Lett. 98,106803[14] Moore, J.E. and L. Balents, 2007, Phys. Rev. B

75,121306[15] Roy, R.,2008, arXiv:0803,2868[16] Schnyder, A. P., S. Ryu, A. Furusaki and S. Das

Sarma, 2010, Phys. Rev. Lett. 194, 040502.[17] Kitaev, A., 2009, AIP conf. Proc. 1134, 22[18] X. L. Qi, T. L. Hughes, S. Raghu and S. C. Zhang,

2009, Phys. Rev. Lett. 102, 187001.[19] X. L. Qi and S. C. Zhang, Rev. Mod. Phys. 83, 1057