Valley- and spin-dependent quantum Hall states in bilayer silicene
Thi-Nga Do, Godfrey Gumbs, Po-Hsin Shih, Danhong Huang, Ming-Fa Lin
VValley- and spin-dependent quantum Hall states in bilayer silicene
Thi-Nga Do ∗ , Godfrey Gumbs , † , Po-Hsin Shih , Danhong Huang , Ming-Fa Lin Department of Physics, National Cheng Kung University, Taiwan 701 Department of Physics and Astronomy,Hunter College of the City University of New York,695 Park Avenue, New York, New York 10065, USA Donostia International Physics Center (DIPC),P de Manuel Lardizabal, 4, 20018 San Sebastian, Basque Country, Spain US Air Force Research Laboratory, Space Vehicles Directorate,Kirtland Air Force Base, New Mexico 87117, USA (Dated: October 3, 2019)
Abstract
The Hall conductivity σ xy of many condensed matter systems presents a step structure whena uniform perpendicular magnetic field is applied. We report the quantum Hall effect in buckledAB-bottom-top bilayer silicene and its robust dependence on the electronic valley and spin-orbitcoupling. With the unique multi-valley electronic structure and the lack of spin degeneracy, thequantization of the Hall conductivity in this system is unlike the conventional sequence as reportedfor graphene. Furthermore, the conductivity plateaux take different step values for conduction(2 e /h ) and valence (6 e /h ) bands since their originating valleys present inequivalent degeneracy.We also report the emergence of fractions under significant effect of a uniform external electricfield on the quantum Hall step structure by the separation of orbital distributions and the mixingof Landau levels from distinct valleys. The valley- and spin-dependent quantum Hall conductivityarises from the interplay of lattice geometry, atomic interaction, spin-orbit coupling, and externalmagnetic and electric fields. ∗ Corresponding author:
E-mail : [email protected] † Corresponding author:
E-mail : [email protected] a r X i v : . [ phy s i c s . c o m p - ph ] O c t . INTRODUCTION The quantum Hall effect (QHE) for two-dimensional (2D) systems in the presence ofa perpendicular magnetic field has attracted tremendous attention from both condensedmatter physics theoreticians and experimentalists. The explanations for the QHE have beenbased on specific behaviors of the single-particle Landau level (LL) states arising under amagnetic field. The conductivity step structure could be understood in terms of the electronsfilling Landau levels. The highly precise quantization of the integer QHE was demonstratedto be a consequence of gauge invariance and the existence of a mobility gap . More explicitly,they were explained by a topological invariant called the Chern number .In this work, we report the interesting valley- and spin- dependent QHE of buckled bi-layer silicene based on a robust connection with the lattice geometry. We accomplish thisby numerically calculating the Hall conductivity for AB-bottom-top (bt) stacking when aperpendicular magnetic field is applied. The special lattice configuration of the system in-duces complex interlayer atomic interactions and significant spin-orbit couplings (SOCs).We demonstrate below that the interplay between these intrinsic characteristics and anexternal perpendicular magnetic field leads to the extraordinary quantization of LLs. Con-sequently, the quantum Hall conductivity (QHC) exhibits integer step structures with dif-ferent sequences, depending on the initiating valley of LLs. Such unusual quantization ofthe QHE could be remarkably enriched by the application of a perpendicular electric field,specifically the emergence of fractions by the separation of orbital distributions and theodd plateau sequence due to mixing of Landau levels from distinct valleys. These diversemagneto-transport properties have never been reported in the literature for condensed mat-ter systems . This demonstrates the essential role played by the crystal structure in theQHE. Our theoretical predictions open the door to new possibilities in understanding thenature of quantum Hall conductivity quantization in 2D materials.Silicene has been successfully synthesized on different substrates . This novel 2D ma-terial has also been the subject of numerous theoretical studies due to its exotic electronicstructure and promising applications in silicon-based electronic technology . Layered sil-icene was predicted to show interesting physical features beyond its monolayer counterparts,such as topological and superconducting properties . So far, high-angle annular dark fieldscanning transmission electron microscopy has verified bilayer silicene with various stack-2ng positions of the A and B atoms of the underlying lattice (see Fig. 1(a)) and bucklingorderings . Among four types of bilayer systems, AB-bt has proven to be the most stablestacking . It possesses a sizable band gap between the oscillatory energy dispersion whichyields high potential for semiconductor applications and interesting magnetic quantizationbehaviors . Here, we show that the combined influence of lattice geometry, atomic in-teractions, SOCs, and external magnetic and electric fields on the Hall conductivity of thissystem gives rise to the integer and fractional QHE with peculiar sequences of conductivityplateaux. II. METHOD
Within linear response theory, we calculated the Hall conductivity from the dynamicKubo formula in the form of σ xy = ie ¯ hS (cid:80) α (cid:80) β (cid:54) = α ( f α − f β ) (cid:104) α | ˙u x | β (cid:105)(cid:104) β | ˙u y | α (cid:105) ( E α − E β ) +Γ . (1)The Landau energies ( E α,β ) and wave functions ( | α, β (cid:105) ) of the initial and final states inthe inter-Landau level (LL) transitions are evaluated from the tight-binding Hamiltonian ,which can be written as H = (cid:88) m,l ( (cid:15) lm + U lm ) c † lmα c lmα + (cid:88) m,j,α,l,l (cid:48) t ll (cid:48) mj c † lmα c l (cid:48) jα + i √ (cid:88) (cid:104)(cid:104) m,j (cid:105)(cid:105) ,α,β,l λ SOCl γ l v mj c † lmα σ zαβ c ljβ − i (cid:88) (cid:104)(cid:104) m,j (cid:105)(cid:105) ,α,β,l λ Rl γ l u mj c † lmα ( (cid:126)σ × ˆ d mj ) zαβ c ljβ . (2)Here, c lmα ( c † lmα ) is the annihilation (creation) operator, (cid:15) lm ( A l , B l ) the site energies, and U lm ( A l , B l ) the height-dependent Coulomb potential energies. The hopping terms, t ll (cid:48) mj , andSOCs, λ SOC , and λ R , , are chosen in order to reproduce the band structure from the first-principle result . The matrix elements of velocity operators, ˙u x,y , directly determine theavailable inter-LL transitions. They can be efficiently computed from the gradient approx-imation, as successfully done for many condensed matter systems, such as carbon-relatedmaterials . 3 II. RESULTS AND DISCUSSION
It is worthwhile noting that AB-bt bilayer silicene possesses a buckled lattice arrangementand a slightly mixed sp -sp chemical bonding. Furthermore, the two silicene sheets haveopposite buckled orderings (Fig. 1(a)), leading to significant inter-layer atomic interactionsand layer-dependence of the SOCs which generate its special band structure. There are twopairs of energy bands, for which the low-lying pair plays the main role in many unusualphysical properties of the system. The low-energy conduction and valence bands are mainlydetermined by the 3 p z orbitals, exhibiting a sizable band gap and the evidently asymmetricbehavior near the Fermi energy E F = 0, as presented in Fig. 1(b). The conduction bandoriginates from the K ( K (cid:48) ) valley while the valence band starts from a specific point midwaybetween K ( K (cid:48) ) and Γ , which we refer to as the T ( T (cid:48) ) valley. The main features of theenergy dispersion are remarkably modified by an external electric field, referreing to theblue curves in Fig. 1(b). In particular, the field reduces the band gap, separates the orbitaldistributions for each band, and enhances the oscillation at each valley. In general, bothconduction and valence bands present peculiar oscillatory and strong anisotropic properties, FIG. 1: (color online) (a) The side view of the atomic structure for AB-bt bilayer silicene and (b)its band structures for zero and finite perpendicular electric fields. The constant-energy diagramsare presented for (c) the conduction and (d) the valence subbands. E F (the red curves), which are respectively around the K and T valleys, show non-circular energy contours. The anisotropy becomes more obviousfor higher conduction and deeper valence energy bands. The conduction (valence) statesonly come to exist near the T ( K ) valley for sufficiently high (deep) energy. The uniquefeatures of the band structure and its sensitive behavior in an electric field give rise to theextraordinary QHE quantization, as we discuss below. FIG. 2: (color online) The (a) conduction Landau level energy spectrum and (b) wave functionson the dominant sublattices of n c = 0 LLs for B = 40 T. The blue and red curves represent,respectively, the spin-up and spin-down dominated LLs. Similar plots for the valence band arepresented in (c) and (d). The distinctive lattice geometry produces eight non-equivalent sublattices of four orbitalswith two spin states, which play a decisive role in the unconventional QHE in the presence ofa magnetic field. The remarkable difference between the A and B atoms due to their chemicalenvironment raises the notion of dominant sublattices. Our numerical calculations show thatthe spatial distribution of quantized LLs originating from the low-lying energy bands havemuch larger amplitude on the B sublattices than the A ones. Quite the opposite is true for5Ls quantized from the outer pair. Consequently, the B sublattices mainly contribute to thetransitions between LLs near E F . Both the conduction and valence LLs can be classified intofour separate subgroups based on the behavior of their spatial distribution on the dominant B sublattices with two spin states. For each subgroup, the wave functions are dominated byone sublattice among B ↑ , B ↓ , B ↑ and B ↓ , as illustrated for the four n c = 0 and four n v = 0 LLs in Figs. 2(a)-(d). Moreover, LLs are split into spin-up (blue) and spin-down (red)states because of the significant SOC in bilayer silicene. FIG. 3: (color online) The (a) Fermi energy-dependent Hall conductivities for bilayer AB-bt silicenewhen a perpendicular magnetic field B z = 40 T is applied. A closer examination of the valenceand conduction band conductivities is shown in (b) and (c), respectively. The degeneracy of quantized LLs is of special importance in accounting for the height6f quantum Hall steps. Here, we show that it is associated with the unique valley andspin properties. Similar to other honeycomb lattice systems such as graphene , LLsoriginating from the K and K (cid:48) valleys are degenerate. This is also true for LLs quantizedfrom the T and T (cid:48) valleys due to special valley symmetry characteristics. Interestingly, thenumber of T and T (cid:48) points is thrice that for the K and K (cid:48) points in each first Brillouinzone. Therefore, the spin-split LLs at T ( T (cid:48) ) valleys are six-fold degenerate while they aredoubly degenerate for the K ( K (cid:48) ) valleys. Moreover, the low-lying LLs are initiated fromeither the K ( K (cid:48) ) point for the conduction band or T ( T (cid:48) ) for the valence spectrum. Thesepeculiar quantization phenomena are directly reflected in the quantum Hall conductivity, aswe discuss in detail below.We now turn our attention to an interpretation of the QHE of AB-bt bilayer silicene inthe presence of a magnetic field. The Fermi energy-dependent QHC is quantized as integermultiples of e /h for both the low-lying valence ( σ xy = 6 e /h ) and conduction ( σ xy = 2 e /h )LLs, as depicted in Figs. 3(a) through 3(c). The step structure could be understood throughinter-LL transitions, as clearly indicated in the formula of Hall conductivity in Eq. (1) interms of the velocity matrix elements. During the variation of the Fermi level ( E F ), a plateauappears whenever one LL becomes occupied. On the other hand, the QHC steps of 6 e /h and 2 e /h are associated with the six-fold degenerate valence LLs and doubly degenerateconduction ones, respectively. As an exception, the overlap of valence LLs with ∆ n = 1leads to several double steps where the two plateaus differ by 12 instead of 6 in the unitof e /h . The QHC steps in bilayer silicene is in great contrast with the conventional 4 e /h in bilayer graphene due to the ± p ˆ z symmetry and spin degree of freedom induced four-folddegenerate LLs .The Hall conductivity at low Fermi energy yields both integer and fractional steps whena uniform perpendicular electric field is applied to the system. The principal reason is that,a finite E dramatically changes the main features of the LLs, especially the degeneratedegree of freedom and the originated electronic valleys. As a matter of fact, the field inducesseparation of LLs dominated by the B and B sublattices, as shown in Fig. 4(a) for the E -dependent LLs. It is crucial to note that, the fractional QHE we refer to is in the sense thatwe take the unit to be ge /h , where g is the LL degeneracy degree of freedom. We observedthe unconventional QHC quantization sequences of ( m − / e /h and ( m (cid:48) / e /h ( m = 0,-1, -2,...; m (cid:48) = 1, 2, 3,...) for valence and conduction spectra, respectively, referring to Figs.7 IG. 4: (color online) The (a) dependence of LL energies in a uniform perpendicular electric fieldfor B z = 40 T. The solid and dashed curves denote the LLs originating from the K and T valleys,respectively. (b) and (c) show the Fermi energy-dependent Hall conductivities for the valence andconduction spectra for a finite electric field E = 100 meV. K valley and conduction LLsat the T valley in the low energy range enriches the conductivity spectra with more plateaus,as shown in Fig. 4(b) for the valence band. This creates an unique sequence of QHC withthe steps at -8/3, -11/3, -14/3, -15/3... which has never been reported in the literature forany other 2D materials. Critical E ’s may significantly change the characteristics of thesystem, as revealed in the E -dependent LL spectrum. As a result, the existence of zeroconductivity is strongly dependent on the E strength through the band gap closing and8pening behaviors. The crossing phenomenon of the conduction and valence LLs at K ( E =130 meV) and T ( E = 153 meV) valleys leads to the absence of zero conductivity, similar inbehavior to that in monolayer graphene . The E -controlled QHC in AB-bt bilayer silicenemay have high potential in Si -based electronic device applications. IV. CONCLUDING REMARKS
In summary, we have investigated the QHE in bilayer silicene and explained its naturebased on its intrinsic material properties as well as the influence of an external field. Thelattice geometry, atomic interaction, SOCs, and external magnetic and electric fields areresponsible for the extraordinary integer and fractional QHC for the AB-bt system. Thequantization of QHC can be manipulated by an external electric field through the separationof orbital distributions and the mixing of Landau levels from distinct valleys. Our discoveryopens up a possible new physical mechanism which serves to account for the unusual sequenceof the step structure. The calculated results, obtained from the efficient combination of theKubo formula and the generalized tight-binding model, can be useful for comparison intransport experiments with silicene.
Acknowledgments
The authors thank the Ministry of Science and Technology of Taiwan (R.O.C.) for fi-nancial support under Grant R. B. Laughlin, Phys. Rev. B , 5632 (1981). D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. , 405(1982). J. Bellissard, A. van Elst, and H. Schulz-Baldes, J. Math. Phys. (N.Y.) , 5373 (1994). V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett. , 146801 (2005). R. Ma, Eur. Phys. J. B , 6 (2013). L. Zhang, Y. Zhang, J. Camacho, M. Khodas, and I. Zaliznyak, Nat. Phys. , 953 (2011). D. S. Lee, C. Riedl, T. Beringer, A. H. Castro Neto, K von Klitzing, U. Starke et al., Phys.Rev. Lett. , 216602 (2011). X. Y. Zhou, R. Zhang, J. P. Sun, Y. L. Zou, D. Zhang, W. K. Lou et al., Sci. Rep. , 12295(2015). B. Feng, Z. Ding, S. Meng, Y. Yao, X. He, P. Cheng, L. Chen, and K. Wu, Nano Lett. , 3507(2012). P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, B.Ealet, and G. Le Lay, Phys. Rev. Lett. , 155501 (2012). L. Tao, E. Cinquanta, D. Chiappe, C. Grazianetti, M. Fanciulli, M. Dubey, A. Molle, and D.Akinwande, Nature Nanotechnology , 227 (2015). R. Yaokawa, T. Ohsuna, T. Morishita, Y. Hayasaka, M. J. S. Spencer, and H. Nakano, NatureCommunications , 10657 (2016). X. Wang and Z. Wu, Phys. Chem. Chem. Phys. , 2148 (2017). J. E. Padilha and R. B. Pontes, J. Phys. Chem. C , 3818 (2015). S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, Phys. Rev. Lett. , 236804(2009). F. Liu, C.-C. Liu, K. Wu, F. Yang, and Y. Yao, Phys. Rev. Lett. , 066804 (2013). H. Fu, J. Zhang, Z. Ding, H. Li, and S. Meng, Appl. Phys. Lett. , 131904 (2014). T.-N. Do, P.-H. Shih, G. Gumbs, D. Huang, C.-W. Chiu, and M.-F. Lin, Phys. Rev. B ,125416 (2018). P. Dutta, S. K. Maiti, and S. N. Karmakar, Journal of Applied Physics , 044306 (2012). M. F. Lin and K. W. K Shung, Phys. Rev. B , 17744 (1994). M. S. Dresselhaus and G. Dresselhaus, Adv. In Phys. , 186 (2002). N. B. Brandt, S. M. Chudinov, and Y. G. Ponomarev, Semimetals 1: Graphite and Its Com-pounds (North-Holland, Amsterdam, 1988) J. H. Ho, Y. H. Lai, C. P. Chang, and M. F. Lin, Diamond and Related Materials , 374(2009). K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S.V. Dubonos, and A. A. Firsov, Nature , 197 (2005)., 197 (2005).