content_image/380235.jpg content_image/380234.jpg content_image/380240.jpg content_image/380241.jpg
<img src='content_image/380235.jpg'>

periodically alternating cross-section size. 

## VI. ESTIMATES OF PERSISTENT CURRENT IN RINGS MADE OF REAL BAND INSULATORS 

In the previous section, we have discussed the ring made of the artificial band insulator, namely the 1D GaAs ring with a few conduction electrons subjected to the periodic quantum- dot potential. Now we want to estimate persistent current in rings made of the real crystalline band insulators (with fully occupied valence band and empty conduction band). The problem is that such rings are three-dimensional and the 1D theory from the preceding text is not applicable directly. 

The problem is outlined in figure 5. The top right sketch in the figure is a 2D sketch of the 3D ring (or a hollow 3D cylin- der) of width w , created from the cubic 3D lattice of atoms. Along such ring there is no periodicity with the lattice period a , there is only periodicity with L . However, the theory in Sects. III - V was developed for the 1D rings with periodicity a along the ring circumference. The question is how to extend the 1D theory to be applicable to the ring-shaped 3D cluster of atoms sketched in the figure 5. 

In principle, for L ≫ w it is reasonable to write the 3D 

<img src='content_image/380234.jpg'>

version of equation (24) in the form 

<img src='content_image/380240.jpg'>

with the spectrum of the ring, ε$_{n}$ ( φ, k$_{y}$ = 0 , k$_{z}$ ) , determined as follows: 

<img src='content_image/380241.jpg'>

where ε ( k$_{x}$, k$_{y}$, k$_{z}$ ) is the Bloch energy of the 3D crystal cre- ated by periodic repetition of the linearized 3D cluster (Fig.5). Note that the unit cell of the crystal is the linearized cluster. The Bloch energy ε ( k$_{x}$, k$_{y}$, k$_{z}$ ) should therefore not be con- fused with the Bloch energy of the crystal with unit cell a $^{3}$. We call this approach the model of the linearized 3D cluster. 

The model also allows to rewrite for 3D rings all equations of Sect. III. First of all, since the period a along the ring no longer exists, it has to be replaced by the period L (replace a by L and N by unity). Second, the Bloch energy ε ( k$_{x}$ ) has to be replaced by the Bloch energy ε ( k$_{x}$, k$_{y}$ = 0 , k$_{z}$ ) just in- troduced. Third, the current (35) has to be summed over the occupied states k$_{z}$ . Clearly, the model of the linearized 3D cluster allows to avoid a full 3D treatment of the ring-shaped 3D cluster. However, the model is still cumbersome for prac- tical use because the Bloch spectrum ε ( k$_{x}$, k$_{y}$, k$_{z}$ ) needs to be calculated for a nontrivial artificial crystal resulting fr om the periodic repetition of the linearized cluster. 

A more simple approach is to consider the artificial 3D ring obtained by elastic bending of the bulk crystal, as it is shown 

