Bloch-Wannier theory of persistent current in a ring made of the band insulator: Exact result for one-dimensional lattice and estimates for realistic lattices arXiv:1001.0496v2[cond-mat.mes-hall]27 Sep 2011 A. Mošková, M. Moško, ∗ and J. Tóbik Institute of Electrical Engineering, Slovak Academy of Sciences, 841 04 Bratislava, Slovakia (Dated: September 29, 2018) It is known that a mesoscopic normal-metal ring pierced by magnetic flux supports persistent current. It was proposed recently that the persistent current exists also in a ring made of a band insulator, where it is carried by electrons in the fully occupied valence band. In this work, persistent currents in rings made of band insulators are analyzed theoretically. We first formulate a recipe whic h determines the Bloch states of a one-dimensional (1D) ring from the Bloch states of an infinite 1D crystal creat ed by the periodic repetition of the ring. Using the recipe, we derive an expression for the persistent current in a 1D ring made of an insulator with an arbitrary valence band ϵ ( k ) . To find an exact result for a specific insulator, we consider a 1D ring represented by a periodic lattice of N identical sites with a single value of the on-site energy. If the Bloch states in the ring are expanded over a complete set of N on-site Wannier functions, the discrete on-site energy splits into the energy band. At full filling, the band emulates the valence band of the bandinsulator and the ring is insulating. It carries a persistent current equal to the product of N and the derivative of the on-site energy with respect to the magnetic flux. This current is not zero if one takes into account that th e on-site Wannier function and consequently the on-site energy of each ring site depend on magnetic flux. To de rive the current analytically, we expand all N Wannier functions of the ring over the infinite basis of Wanni er functions of the constituting infinite 1D crystal and eventually determine the crystal Wannier functions by a method of localized atomic orbitals. In terms of the crystal Wannier functions, the current at full filling arise s because the electron in the ring with N lattice sites is allowed to make a single hop from site i to its periodic replicas i ± N . In terms of the ring Wannier functions, the same current is due to the flux dependence of the Wannier fu nctions basis, and the longest allowed hop is from i to i ± Int( N/ 2) only. Finally, we estimate the persistent current at full fil ling in a few realistic systems. In particular, the rings made of real band insulators (GaAs, Ge, InAs) with a fully filled valence band and empty conduction band are studied. The current decays with the ring length exponentially due to the exponential decay of the Wannier functions. In spite of that, it can be of measurable size. PACS numbers: 73.23.-b, 73.61.Ey It is known that a conducting ring pierced by a constant magnetic flux can support persistent electron current circu lat- ing around the ring [1]. The persistent current is well known to exist in a superconducting ring, and also in a normal-metal ring as long as it is mesoscopic (of size comparable or smaller than the electron coherence length). Early work dealing with persistent current and flux quantization in superconductin g rings [2] also contains results relevant to the normal metal rings. Later it was mentioned [3] that one can have circulating currents for free electrons in ballistic normal-metal rings, and the reference [4] proposed that persistent currents should exist in the disordered normal-metal rings. The persistent currents were indeed observed in the disordered normal-metal rings [5–9] as well as in ballistic conducting rings [10, 11]. Deta ils are reviewed in various papers [12–14]. One of us recently proposed [15] that persistent current ex- ists also in a ring made of a band insulator, where it is carried exclusively by Bloch electrons in the fully occupied valence band. The effect is interesting for two reasons [15]. First, it seems to be the only effect manifesting electron transport in a fully occupied valence band. In a standard conductance mea- surement, the band insulator is biased by two metallic elec- trodes and one observes the tunneling of conduction electrons from metal to metal through the energy gap of the insulator. One does not observe the transport of valence band electrons in the insulator. Second, in the fully occupied valence band, Bloch electrons cannot be scattered by phonons or other elec- trons. Therefore, the electron coherence length should be huge even at room temperature if the valence band is separated from the conduction band by a large energy gap. The persis- tent current in the insulating ring should thus be observable up to high temperatures as a temperature-independent effect as long as excitations into the (empty) conduction band are negligible. By contrast, persistent currents in metallic rings are observable only at low temperatures. [9]. We report on a study of persistent currents in rings made of band insulators using the formalism of the Bloch and Wannier functions. It was partly motivated by a pioneering analysis [16] of persistent currents in a 1D ring-shaped periodic lattice composed of the lattice sites with a single value of the on-site energy. The authors derived persistent current I for spinless electrons at zero temperature assuming that they move along the lattice by means of the nearest-neighbor-site hopping. The result reads for odd N$_{e}$ and I = − 4 π Γ$_{1}$ Nφ$_{0}$ sin π $_{N}$N$_{e}$ sin π N sin [ 2 π N φ φ$_{0}$ ] , − φ$_{0}$ 2 ≤ φ < φ$_{0}$ 2 , (1) I = − 4 π Γ$_{1}$ Nφ$_{0}$ sin π $_{N}$N$_{e}$ sin π N sin [ π N ( 2 φ φ$_{0}$ − 1 )] , 0 ≤ φ < φ$_{0}$ , (2) I. INTRODUCTION