научная статья по теме EDGE STATES AND TOPOLOGICAL PROPERTIES OF ELECTRONS ON THE BISMUTH ON SILICON SURFACE WITH GIANT SPIN-ORBIT COUPLING Физика

Текст научной статьи на тему «EDGE STATES AND TOPOLOGICAL PROPERTIES OF ELECTRONS ON THE BISMUTH ON SILICON SURFACE WITH GIANT SPIN-ORBIT COUPLING»

EDGE STATES AND TOPOLOGICAL PROPERTIES OF ELECTRONS ON THE BISMUTH ON SILICON SURFACE WITH GIANT SPIN-ORBIT COUPLING

D. V. Khomitsky* A. A. Chubanov

University of Nizhny Novgorod 603950, Nizhny Novgorod. Russia

Received August 22, 2013

We derive a model of localized edge states in a finite-width strip for the two-dimensional electron gas formed in the hybrid system of a bismuth monolayer deposited on the silicon interface and described by the nearly free electron model with giant spin-orbit splitting. The edge states have the energy dispersion in the bulk energy gap with a Dirac-like linear dependence on the quasimomentum and the spin polarization coupled to the direction of propagation, demonstrating the properties of a topological insulator. The topological stability of edge states is confirmed by the calculations of the invariant taken from the structure of the Pfaffian for the time reversal operator for the filled bulk bands in the surface Brillouin zone, which is shown to have a stable number of zeros with the variations of material parameters. The proposed properties of the edge states may support future advances in experimental and technological applications of this new material in nanoelectronics and spintronics.

DOI: 10.7868/S0044451014030148

1. INTRODUCTION

During the last decade, an increasing attention is given to a new class of structures called topological insulators (TIs) with promising characteristics as regards both fundamental aspects of their physics and possible applications in nanoelectronics, spintronics, and fabrication of new magnetic, optical, and information processing devices fl 5]. The principal features of TIs include the presence of time-reversal invariance in the system where the propagating edge states may exist, being localized near the boundary of the host material and having the dispersion relation that is linear near the origin of their quasimomentum (Dirac-like structure), corresponding to energies belonging to the insulating gap of the bulk material. The spin of such states is firmly attached to the direction of propagation along the edge, making them protected against backscatter-ing due to the time-reversal invariance, which leads to effective cancelation of two scattered states with opposite possible directions of the spin flip that accompanies such backscattcring. The existence of such edge states has been shown in numerous theoretical models

E-mail: khomitsky'fi'phys.unn.ru

of TIs, and also in the experiments. The materials included graphene [6], HgTe/CdTe quantum wells [7 10], bismuth thin films [11], quantum wires [12], nanocon-tacts or bilayers [13], the LiAuSe and KHgSb compounds [14], and general two-dimensional (2D) models of paramagnetic semiconductors [15], silicene [16,17], and topological nodal semimetals [18]. Another 2D TI has been predicted in the inverted type-II semiconductor InAs/'GaSb quantum well [19] and observed experimentally in the contribution of edge modes to the electron transport [20]. Many studies have also been devoted to the general properties of 2D and 3D models of TIs with certain symmetries [17,21 27], where four topological invariants have been found in 3D TIs instead of a single Z2 invariant in 2D TIs [1,2,22].

Recently a general group-theory analysis has been made for the links between the geometry of the Bravais lattice and the properties of TIs [28]. We note that the symmetry arguments always played a significant role in classifying the systems as trivial or topologically protected against external perturbations [6,21,28 31]. The time-reversal property of spin-1/2 particles in such systems can be described by the presence of time-reversal invariance (without magnetic impurities or an external magnetic field) and the absence of the spin rotation invariance. Here, the time-reversal operator is given by

O = i(TyK, where K is the complex conjugation operator and <7y is the second Pauli matrix. According to the general symmetry considerations [30, 31], this means the class-All symmetry for the Hamiltonian for which the so-called Z-2 topological order is possible for 2D and 3D systems, forming the basis for the TI properties.

The studies of 3D materials were mostly focused on Bi2Se3, Bi2Te3, orBi2Te2Se [1,2,32 34], where also the edge states were constructed explicitly in several models of finite-size geometry [35,36]. Another important issue is the effect of impurities and disorder on the band structure and topological stability in TIs. It is known that TIs are robust against weak disorder or the potentials produced by nonmagnetic impurities [6,21,37], while the presence of magnetic impurities may lead to hybridization of the insulator atomic orbitals and magnetic material orbitals, producing a strong modification to the metallic or nonmetallic nature of the states and their spin polarization [38]. Even for nonmagnetic impurities, it has been shown recently that the formation of impurity bands within the energy gap at strong doping of the bulk material may lead to their mixing with the edge states of a TI, modifying their structure, although preserving the Z-2 order and topological stability [39].

It can be seen that although the features of TIs are very general and describe a truly novel state of matter, the number of different materials demonstrating these features is currently quite limited. It is therefore of interest to find new materials and compounds where possible manifestations of TIs may be present, for both fundamental aspects and applied purposes. It is also necessary to understand which properties of edge states are common for different systems, and which are special, and how all of them are related to the bulk quantum states in a specific model.

Here, we consider a model of edge states and relate their properties to topological characteristics of the host material for a new candidate to the class of TIs: the 2D electron g cts hi ct material with strong spin orbit coupling (SOC) formed at the interface of a monolayer of bismuth deposited on silicon. This material is characterized by a giant SOC splitting which, was also predicted or observed experimentally in a number of metal films or the combined materials of the "metal on semiconductor" type [40 46], and recently described theoretically [46, 47]. Its huge spin splitting together with the hexagonal type of the lattice creates a certain potential of manifestation of TI properties, since the spin-resolved bands may evolve into spin-resolved edge states, and the hexagonal type of the lattice is favorable for the TI to exist [28].

The properties of the 2D electron gas at the Bi/'Si interface have been studied experimentally with the help of angle-resolved photoeniission spectroscopy (ARPES) [40 44,46,48 50], applied also to other materials. It was found that this material represents an example of the currently widely studied class of materials with a large (up to 0.2 0.4 eV) SOC spin splitting of their energy bands, which can be formed in various compound materials or heterostructures of the "metal on semiconductor" type. It has been known for many years that SOC plays an important role in the formation of TI properties [51], including the localization effects of the Rasliba SOC combined with electron electron interaction [52], the Dirac-cone surface states in Bi2Se3 [53] and Bi2Te3.Se3_3. [54], PbSb2Tc4 or Pb2Bi2Te2S3 [55], and Bii_3.Sb3. [56], topological phases [57, 58] and the quantum spin Hall phase in a honeycomb lattice [59], the ultracold Fermi gases [60], the spin Hall effect in graphcnc [61], and the Kondo insulator effects [62, 63].

Various materials with a strong SOC have been the subject of intensive recent studies, including the structures of Bi deposited on Si Ge substrates [64], the Pb on Si structure [43], the trilayer Bi Ag Si system [42], the structures with a monolayer of Pb atoms covering the Ge surface [65] or the Pb on G© structures [661. We also mention new types of triple bulk compounds with strong SOC like GeBi2Te4 [67], BiTel, or other bismuth tellurohalides [68 70] or the recently discussed Bii4Rh3I0 material [71].

In this paper, we adopt the nearly free model of two-dimensional bulk states in Bi/'Si, developed earlier [46] and applied in the extended form in our previous paper for the description of spin polarization, charge conductance, and optical properties of this promising material [47], for the calculation of ID edge states of electrons on the Bi/'Si interface in a finite strip geometry. We obtain both the explicit form of edge wavefunc-tions and the edge energy spectrum, calculate their spin polarization, and relate the possible topological stability of their properties to the Z-2 topological invariant studied by analyzing the behavior of the matrix elements of the time-reversal operator in the Brillouin zone [1,2,5,6,21,23]. The results of our paper are of interest for expanding the knowledge of new materials with the topologically protected properties where the SOC plays a significant role, making them suitable for further applications in spintronics as stable current-carrying and spin-carrying channels.

The paper is organized as following. In Sec. 2, we briefly describe the nearly free electron (NFE) model of 2D bulk states at a Bi/'Si interface, and derive a

model for ID edge states for the electrons in a finite strip geometry. We calculate their spectrum, wavefunc-tions, and spin polarization. In Sec. 3, we reinforce our findings on the edge state stability by considering the topological band properties of 2D bulk states in Bi/Si, and find the results supporting the presence of the TI phase. Our conclusions are given is Sec. 4.

2. MODEL FOR BULK STATES AND EDGE STATES

2.1. Bulk states

Our model for ID edge states is based on the 2D NFE model for bulk states of the 2D electron gas formed at the interface of the trimer Bi/Si(lll) structure [46] developed for the description of the spectrum near the M point of the

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