Topological Insulators on the Ruby Lattice with Rashba Spin-Orbit Coupling

Transcription

1 Commun. Theor. Phys. 60 (2013) Vol. 60, No. 1, July 15, 2013 Topological Insulators on the Ruby Lattice with Rashba Spin-Orbit Coupling HOU Jing-Min ( ) and WANG Guo-Xiang ( ) Department of Physics, Southeast University, Nanjing , China (Received October 22, 2012; revised manuscript received March 11, 2013) Abstract We investigate a tight-binding model of the ruby lattice with Rashba spin-orbit coupling. We calculate the band structure of the lattice and evaluate the Z 2 topological indices. According to the Z 2 topological indices and the band structure, we present the phase diagrams of the lattice with different filling fractions. We find that topological insulators occur in some range of parameters at 1/6, 1/3, 1/2, 2/3 and 5/6 filling fractions. We analyze and discuss the characteristics of these topological insulators and their edge states. PACS numbers: f, Fd, r, b Key words: topological insulators, ruby lattice, rashba spin-coupling 1 Introduction Recently, the study of topological phases has become an exciting area of research in condensed matter physics. [1 2] Topological matters are classified according to topological invariants rather than symmetries. The integer quantum Hall states are classified by the TKNN number, [3] which is nonzero when time-reversal symmetry is broken. Time-reversal invariant band insulators can be classified by a Z 2 topological index. [4 5] In two dimensions, topological insulators and normal band insulator are characterized by the Z 2 topological index ν = 1 and ν = 0, respectively. Time-reversal invariant band insulators can be generalized to three dimensions and classified according to four Z 2 topological indices (ν 0 ; ν 1 ν 2 ν 3 ) with ν i = 0, 1. [6 8] Topological insulators have a bulk gap and topologically protected gapless helical edge states or helical surface states when placed next to a vacuum or an ordinary band insulator because topological invariants cannot change as long as a material remains insulating. The remarkable metallic boundaries of topological insulators may result in new spintronic or magnetoelectric devices and a new architecture for topological quantum bits. Theoretical studies have demonstrated that several tight-binding models with the spin-orbit coupling, such as honeycomb, [4] diamond, [6] kagome, [9] checkerboard, [10] decorated honeycomb, [11] Lieb and perovskite, [12] squareoctagon, [13] ruby, [14] pyrochlore, [15] and octahedrondecorated cubic [16] lattices support two-dimensional or three-dimensional topological insulators. For real materials, HgTe quantum wells were first found to support two-dimensional topological insulators. [17 18] Fu and Kane firstly predicted that Bi 1 x Sb x supports a three-dimensional topological insulator, [19] which was conformed experimentally by Hsieh et al. in [20] Later, Bi 2 Se 3 was discovered to support a threedimensional insulator experimentally as a second generation material, [21] which also was supported by theoretical calculations. [21 22] Additionally, Ref. [22] also predicted that Bi 2 Te 3 and Sb 2 Te 3 are second generation materials supporting three-dimensional topological insulators. The later experimental studies on Bi 2 Te [23 25] [25] 3 and Sb 2 Te 3 identified their topological band structures. Recently, several ternary compounds are identified as topological insulators. [26 30] Very Recently, based on the firstprinciples calculations, Miao et al. find that, due to the effect of the intrinsic polarization fields, GaN/InN/GaN quantum well undergo inverted band transition and become a topological insulator. [31] This is an important progress because semiconductor systems have advantages of integration in practical devices. The above theoretical models and real materials considered the intrinsic spin-orbit coupling, which is a mirrorsymmetric interaction. In this communication, we shall demonstrate that a model with the Rashba spin-orbit coupling, a mirror-symmetry-broken interaction, can also support topological insulators. Specifically, we shall consider the ruby lattice as shown in Fig. 1 with the Rashba spin-orbit coupling and study its topological insulators. The ruby lattice with the mirror-symmetric intrinsic spinorbit coupling were firstly investigated by Hu et al. [14] 2 Model We consider the ruby lattice as shown in Fig. 1. The ruby lattice consists of six triangular sublattices denoted as A, B, C, D, E, and F, respectively. Every six lattice sites from sublattices A, B, C, D, E, and F, respectively, make up a hexagon. The hexagons are separated by triangles and squares. Here, we assume that the distance between two nearest-neighbor lattice sites is d and the lattice constant of all sublattices is a = (1 + 3)d. With the tight-binding approximation, we can write the second Supported by the National Natural Science Foundation of China under Grant Nos and jmhou@seu.edu.cn c 2013 Chinese Physical Society and IOP Publishing Ltd

2 130 Communications in Theoretical Physics Vol. 60 quantized Hamiltonian of the lattice as follows, H 0 = t c iσ c jσ t 1 c iσ c jσ, (1) i,j,σ [i,j],σ where c iσ is the annihilation operator destructing an electron with spin σ on the site r i of the ruby lattice, i, j represents nearest-neighbor hopping within the same hexagon with amplitude t and [i, j] denotes nearest-neighbor hopping between two different hexagons with amplitude t 1. In momentum space, Hamiltonian (1) can be represented by H 0 = Ψ k H0 kψ k k with Ψ k = (c Ak, c Bk, c Ck, c Dk, c Ek, c F k, c Ak, c Bk, c Ck, c Dk, c Ek, c F k ) T, which are ordered according to the sequence denoted in Fig. 1 for every spin state. Here, Hk 0 takes the following form, 0 t t 1 e ik 0 t 1 e iky t t 0 t t 1 e iky 0 t 1 e ik+ ( ) 1 0 Hk 0 t 1 e ik t 0 t t 1 e ik+ 0 =, (2) t 1 e iky t 0 t t 1 e ik t 1 e iky 0 t 1 e ik+ t 0 t t t 1 e ik+ 0 t 1 e ik t 0 where k ± = ( 3k x ± k y )/2, and the 2 2 matrix is the unit matrix in spin space. Since H 0 is spin-decoupling, Hk 0 is block-diagonal, i.e. two blocks representing spin-up and spin-down electrons are the same. of bands and the fourth and fifth pairs of bands touch at K point, near which Dirac cones occurs. Fig. 1 Schematic diagram of the ruby lattice, which consists of six sublattices denoted by A, B, C, D, E, and F, respectively. We evaluate the eigenenergies of Hamiltonian (2). The Brillouin zone for t 1 = t is shown in Fig. 2(a), which is similar to that of honeycomb lattice. Figure 2(b) shows the corresponding spectrum, which contains twelve bands which come from the six sites in every unit cell for spinup and spin-down electrons. Due to spin decoupling, these bands are double degenerate and divided into six pairs. For this special case, all the bands are connected or touched with other bands. Thus, the system is metal for all filling fractions but 1 and 0 filling fractions in this case. The second third and fourth pairs of bands touch together at Γ point and the fifth and sixth pairs of bands touch at the same point. Along the Γ M line in momentum space, the third and fourth pairs of bands are degenerate and they separate near M point. the first and second pairs Fig. 2 (Color online) (a) The Brillouin zone of the ruby lattice. (b) Band structures of the ruby lattice with t 1 = t, λ R = 0, λ CDW = 0. Here, the horizontal axis represents the wave vectors along the path in the first Brillouin zone indicated by the red lines in (a). Now, in order to find topological insulators on the ruby lattice, we proceed to introduce Rashba spin-orbit coupling between nearest-neighbor sites as follows, H R = iλ R c iα (σ αβ d ij ) z c jβ, (3) {ij},αβ

3 No. 1 Communications in Theoretical Physics 131 where {i, j} represents two nearest-neighbor sites i, j that belong to the same hexagon or the two different hexagons, and λ R is the amplitude of Rashba spin-orbit coupling of the two nearest-neighbor sites. σ = (σ x, σ y, σ z ) is the vector of Pauli spin matrices. d ij are the nearest neighbor bond unit vector traversed between sites i and j. In momentum space, the Hamiltonian for Rashba spin-orbit coupling (3) can be expressed as with H R = k Ψ k HR k Ψ k Since H R k with Ψ k = (c Ak, c Bk, c Ck, c Dk, c Ek, c F k, c Ak, c Bk, c Ck, c Dk, c Ek, c F k ) T. does not decouple for the two spin projections, it is a matrix as follows, R = H R k = iλ R ( 0 R R 0 ), (4) 0 1 e i(k +5π/6) 0 e i(ky π/2) e i2π/3 1 0 e iπ/3 e i(ky π/2) 0 e i(k+ π/6) e i(k +π/6) e i2π/3 0 e i2π/3 e i(k+ π/6) 0 0 e i(ky π/2) e iπ/3 0 1 e i(k +π/6) e i(ky π/2) 0 e i(k+ 5π/6) 1 0 e i2π/3 e iπ/3 e i(k+ 5π/6) 0 e i(k +5π/6) e iπ/3 0, (5) where k ± = ( 3k x ± k y )/2. In our model we also consider the effect of a charge-density-wave (CDW) on-site potential, which can be written as H CDW = λ CDW ξ i c iσ c iσ, (6) where ξ i is +1 for sublattices A, C, E, and 1 for sublattices B, D, F. We can rewrite this term as H CDW = k Ψ k HCDW k Ψ k. Here the single-particle expression Hk CDW has the form as follows, ( ) 1 0 Hk CDW = λ CDW , (7) where the 2 2 matrix is the unit matrix in spin space. In momentum space, the total single particle Hamiltonian is H k = Hk 0 + HR k + HCDW k. The bands and eigenstates can be obtained by exactly diagonalizing H k. The classification of two-dimensional topological insulators is presented in Ref. [6]. For a time-reversalsymmetry system, the energy eigenstates must come in pairs due to Kramer s theorem. The Bloch functions satisfy ψ n ( k) = Θ ψ n (k), (8) where Θ = iσ 2 K is the time-reversal operator with σ 2 being the spin operator and K being the complex conjugate operator. Thus, one only need to consider Bloch functions in a half of the Brillouin zone, say B. The Bloch functions in another half of the Brillouin zone can be fixed by time-reversal transformation. The Z 2 invariant characterizing two-dimensional topological insulators can be i,σ expressed as [32] ν = 1 [ ] dk A(k) d 2 kf(k) mod 2, (9) 2π B B where A(k) = i n ψ n(k) k ψ n (k) is the Berry connection and F(k) = k A(k) z is the Berry curvature. Fu and Kane have found a simple method to identify the Z 2 invariants for the system with the presence of inversion symmetry. [19] Our model is not inversion-symmetric due to the existence of the Rashba spin-orbit coupling and the CDW on-site potentials. Fortunately, Fukui and Hatsugai provide an n-field method to evaluate the Z 2 invariant in the systems without inversion symmetry. [33] This method enables one to implement numerical calculations of the Z 2 invariant in a lattice Brillouin zone. We will evaluate the Z 2 invariant in our system with this method. For convenience, we discretize a cell in the reciprocal lattice. For the case with M filled bands, we write the eigenstates for the same k l as a form of multiplet as Ψ(k l ) = ( ψ 1 (k l ),..., ψ M (k l ) ). A link variable is defined as, U µ (k l ) = N 1 µ det[ψ (k l )Ψ(k l + ˆb µ )], (10) where N µ (k l ) det[ψ (k l )Ψ(k l + ˆb µ )] with ˆb µ is the unit vector on the mesh of the discrete Brillouin zone. The discrete Berry connection is introduced as, A µ (k l ) = i ln U µ (k l ), (11) and the corresponding discrete Berry curvature is defined as, F(k l ) = i ln[u 1 (k l )U 2 (k l + ˆd 1 )U1 1 (k l + ˆd 2 )U2 1 (k l)].(12) An integer field n(k l ) for the Brillouin zone is defined as, n(k l ) = 1 2π [ 1A 2 (k l ) 2 A 1 (k l ) F(k l )]. (13) The Z 2 topological invariant is given by the sum of the n-field in half of the Brillouin zone as ν = k l B n(k l) mod 2.

4 132 Communications in Theoretical Physics 3 Topological Insulators In this section, we will show that the ruby lattice with Rashba spin-orbit coupling supports topological insulators. In order to illustrate our results clearly, we first consider a specific case with t1 = t, λr = 2t and λcdw = 0.5t. For this specific case, we calculate the bands and the edge bands with a zigzag boundary as shown in Figs. 3(a) and 3(b), respectively. From Fig. 3(a), we can find that the degeneracy of spin-down and spin-up states is lifted but at some special points such as Γ, M. There are gaps between two neighbor pairs of bands except between the first and second pairs of bands. Thus, for this special case, the system is an insulator for 1/3, 1/2, 2/3, 5/6 filling fractions and a Vol. 60 metal for 1/6 filling fraction. In this special case, the system is a topological insulator for 1/3, 1/2, 5/6 filling fractions and a normal insulator for 2/3 filling fraction, which can be verified from the edge states as shown in Fig. 3(b). There is a single time-reversed pair of gapless edge states on each edge that traverse the bulk gap between the second and third pairs of bands. This imply that the system is a topological insulator for 1/3 filling fraction. For 1/2 and 5/6 filling fractions, the similar gapless edge states traverse the gap near the Fermi level can be found. However, the edge states do not fully traverse the gap between the fourth and fifth pairs of bands, which imply that the system is a normal insulator for 2/3 filling fraction. Fig. 3 (Color online) (a) Band structures and (b) edge states of the ruby lattice with t1 = t, λr = 2t and λcdw = 0.5t. In (b), the red and green lines represent the edge states located at two different side of the ruby lattice. Fig. 4 (Color online) The n-field configuration of the ruby lattice with t1 = t, λr = 2t and λcdw = 0.5t for (a) 1/3 filling, (b) 1/2 filling, (c) 2/3 filing, and (d) 5/6 filling, respectively. Here, the shaded area indicate the half of the Brillouin zone B. Here, the red and green circles denote n = 1 and n = 1, respectively, and the blank lattice cell denotes n = 0.

5 No. 1 Communications in Theoretical Physics In order to exactly identify the topological insulators, we apply the n-field method to calculate the Z2 topological invariant. Figures 4(a) 4(d) show the n-field configuration of the ruby lattice with t1 = t, λr = 2t and λcdw = 0.5t for 1/3, 1/2, 2/3 and 5/6 filling fractions, respectively. From Fig. 4(a), we find that the sum of n in the half of Brillouin zone B is odd, i.e. the system with with t1 = t, λr = 2t and λcdw = 0.5t for 1/3 filling fraction is a topological insulator. Figures 4(b) and 4(d) show 133 that the sum of n in the half of Brillouin zone B are also odd, so the system with t1 = t, λr = 2t and λcdw = 0.5t for 1/2 and 5/6 filling fractions are topological insulators. However, from Fig. 4(c), we find that the sum of n in the half of Brillouin zone B is even, so we conclude that the system with t1 = t, λr = 2t, and λcdw = 0.5t for 2/3 filling fraction is a normal insulator. From the analyses of the n-field configuration and the edge states, we obtain the same results. Fig. 5 (Color online) Phase diagrams of the ruby lattice with λcdw = 0 for (a) 1/6 filling, (b) 1/3 filling, (c) 1/2 filling, (d) 2/3 filling, and (e) 5/6 filling. Here, the blue color denotes a trivial band insulator; the white color denotes topological insulators; the green color denotes a metal phase. For completeness, we calculate the bands and the Z2 topological invariants for the cases with various parameters. According to the results, we draw a series of phase diagrams as shown in Figs. 5 and 6. Here, Figs. 5 and 6 show the λr t1 phase diagrams of the ruby lattice with λcdw = 0 and λcdw = 0.5t for various filling fractions, respectively. From these phase diagrams, we arrive at some results. First, the ruby lattice with Rashba spinorbit coupling supports topological insulators for 1/6, 1/3, 1/2, 2/3, and 5/6 filling fractions. Besides topological insulators, there exist normal insulator and metal phases for various filling fractions. Secondly, the system is a normal insulator or a metal for all filling fractions while λr vanishes, that is to say, the Rashba spin-orbit coupling is the key factor for the presence of topological insulators. Thirdly, by comparing Figs. 5 and 6, we can see that the CDW on-site potential makes the normal and topological insulator regimes enlarged. In addition, we need to ex- plain that the system is a normal insulator at the point with t1 = 0 and λr = 0 for all filling fractions, which is easily understood for when t1 and λr approach to zero the ruby lattice becomes separated hexagons. It is obvious that the system is a normal insulator at the point t1 = 0 and λr = 0 for 1/6, 1/2 and 5/6 filling fractions. However, for 1/3 and 2/3 filling fractions, the system is a metal near the point t1 = 0 and λr = 0. For t1 = 0 and λr = 0, the second and third pairs of bands are degenerate and become flat bands, which means that electrons are localized. Similarly, the third and fourth pairs of bands are degenerate flat bands. In other words, the system with t1 = 0 and λr = 0 for 1/3 and 2/3 filling fractions is a normal insulator. However, a tiny change from t1 = 0 and λr = 0 for parameters t1 and λr makes the degenerate flat bands become two pairs of dispersive bands that are crossover each other, then the lattice with two or four pairs of bands occupied becomes a metal.

6 Communications in Theoretical Physics 134 Vol. 60 Fig. 6 (Color online) Phase diagrams of the ruby lattice with λcdw = 0.5t for (a) 1/6 filling, (b) 1/3 filling, (c) 1/2 filling, (d) 2/3 filling, and (e) 5/6 filling. Here, the blue color denotes a trivial band insulator; the white color denotes topological insulators; the green color denotes a metal phase. 4 Conclusion In summary, we have shown that the ruby lattice with Rashba spin-orbit coupling supports topological insulators for 1/6, 1/3, 1/2, 2/3, and 5/6 filling fractions. We have calculated the band structure and edge band structure for the tight-binding model of the ruby lattice with Rashba spin-orbit coupling for a special case. We have evaluated the Z2 topological invariants with the n-field method for various filling fractions and drawn the phase diagrams. We have analyzed and discussed the characters of the band structures and the edge states of different References [1] M.Z. Hasan and C.L. Kane, Rev. Mod. Phys. 82 (2010) [2] X.L. Qi and S.C. Zhang, Rev. Mod. Phys. 83 (2011) [3] D.J. Thouless, M. Kohmoto, M.P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49 (1982) 405. [4] C.L. Kane and E.J. Mele, Phys. Rev. Lett. 95 (2005) [5] C.L. Kane and E.J. Mele, Phys. Rev. Lett. 95 (2005) [6] L. Fu, C.L. Kane, and E.J. Mele, Phys. Rev. Lett. 98 (2007) [7] J.E. Moore and L. Balents, Phys. Rev. 75 (2007) [8] R. Roy, Phys. Rev. B 79 (2009) [9] H.M. Guo and M. Franz, Phys. Rev. B 80 (2009) phases. The model might as well be built from optical lattices due to their diversity and controllability.[34] The hopping parameters t and t1 can be tuned by adjusting the depth of optical lattices and the CDW potential can be realized by adjusting the configuration and frequency of lasers that make up optical lattices. Here, we mainly investigated the non-interacting systems. If the interactions between atoms is considered, the phase diagram will become complicated and more phases, such as topological Mott insulators, will appear.[35] [10] K. Sun, H. Yao, E. Fradkin, and S.A. Kivelson, Phys. Rev. Lett. 103 (2009) [11] A. Ru egg, J. Wen, and G.A. Fiete, Phys. Rev. B 81 (2010) [12] C. Weeks and M. Franz, Phys. Rev. B 82 (2010) [13] M. Kargarian and G.A. Fiete, Phys. Rev. B 82 (2010) [14] X. Hu, M. Kargarian, and G.A. Fiete, Phys. Rev. B 84 (2011) [15] H.M. Guo and M. Franz, Phys. Rev. Lett. 103 (2009) [16] J.M. Hou, W.X. Zhang, and G.X. Wang, Phys. Rev. B 84 (2011) [17] B.A. Bernevig, T.L. Hughes, and S.C. Zhang, Science 314 (2006) 1757.

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