content_image/410859.jpg
<img src='content_image/410859.jpg'>

the increasing ring length exponentially, but a careful choice of the insulating material allows to achieve the persistent cur- rent of measurable value for a technologically realizable ring size. Indeed, the maximum ring lengths considered in the fig- ure 8 are close to the value ∼ 120 nm, in principle achiev- able by modern nanotechnology. Concerning the amplitude of the persistent current, the value ∼ 0 . 0003 nA estimated for the InAs ring of length ∼ 120 nm is very small, but in prin- ciple detectable say by the experimental technique of Refs. [9, 25, 26]. The considered insulators are ordered as GaAs, Ge and InAs for the persistent currents ordered increasingly. This suggests a simple rule: the smaller the light-hole effec- tive mass the larger the persistent current. 

We conclude with a few remarks showing that the above 1D estimates hold roughly also for the 3D rings. Similar calcu- lations as in the figure 7 were performed for the k values on the line ( X [ − 2 π/a, 0 , 0] , Γ[0 , 0 , 0] , X [2 π/a, 0 , 0] ). We have found the energies ϵ ( k ) and amplitudes Γ$_{N}$ (not shown) very similar to those in the figure 7, with the resulting persisten t currents being of the same order of magnitude as those in the figure 8. Moreover, we have observed similar results when testing the k values on the lines ( K, Γ , K ) and ( W, Γ , W ). The model of the elastically bent crystal thus provides roughly the same persistent current independently on the fact which crys- talline axis is chosen to be directed along the ring circum- ference. This suggests that anisotropy of the valence band with respect to the ⃗ k direction does affect the resulting persis- tent current dramatically. Therefore, the estimated values of the current should hold reasonably also for the realistic ring- shaped crystals similar to the ring sketched in figure 5. 

## VII. SUMMARY AND CONCLUDING REMARKS 

We have analyzed theoretically persistent currents in rings made of band insulators. We started by formulating a recipe which determines the Bloch states of a one-dimensional (1D) ring from the Bloch states of an infinite 1D crystal created by the periodic repetition of the ring. Using the recipe, we have derived 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 have con- sidered a 1D ring represented by a periodic lattice of N identi- cal 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 valenc e band of the band insulator and the ring is insulating. 

We have found that the ring 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 because the on-site Wannier function and consequently the on-site energy of each ring site depend on magnetic flux. 

Further, we have expanded all N Wannier functions of the ring over the basis of Wannier functions of the constituting infinite 1D crystal. In terms of the crystal Wannier function s, the current at full filling arises because the electron in thering with N lattice sites is allowed to make a single hop from site i to its periodic replicas i ± N . Alternatively, in terms of the ring Wannier functions the same current is due to the flux dependence of the Wannier functions basis, and the longest allowed hop is from i to i ± Int( N/ 2) only. We have eventually derived the crystal Wannier functions in 

the LCAO approximation, and have expressed the persistent current at full filling by means of an exact formula. The valid - ity of the formula was verified by comparison with the numer- ical LCAO approach. We have presented quantitative results for a 1D ring made of an artificial band insulator, namely for a GaAs ring subjected to a periodic potential emulating the insulating lattice. 

Moreover, we have provided estimates for the rings made of real band insulators (GaAs, Ge, InAs) with a fully filled valence band and empty conduction band. The current at full filling decays with the ring length exponentially due to the exponential decay of the Wannier function tails. In spite of that, it can be of measurable size. 

We have ignored the effect of nonzero temperature. We expect the persistent currents in rings made of the band in- sulators to be temperature-independent as long the thermal energy k$_{B}$T is much smaller than the energy gap separating the valence band from the conduction band. For example, the energy gap in the InAs crystal is ∼ 400 meV. Hence, the zero- temperature values of the persistent current, predicted for the insulating InAs ring, could persist even at room temperature. 

We have also ignored the electron-electron interaction. The many-body studies [30, 31] suggest that in the 1D ring consid- ered in this paper the repulsive electron-electron interaction 

