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in figure 6. In this case we have instead of the formula (94) the formula 

where the energy spectrum ε$_{n}$ ( φ, k$_{y}$, k$_{z}$ ) is given by equation 

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with ε ( k$_{x}$, k$_{y}$, k$_{z}$ ) being the Bloch energy in the bulk crystal before it is bent. We call this approach the model of the elas- tically bent 3D crystal. In this model there is an artificial peri- odicity with period a along the ring. In spite of this, the model can provide a reasonable quantitative information on how the persistent current in the 3D ring depends on the realistic band structure of the constituting bulk crystal. In the model, the per- sistent current at full filling is given by equations (47) and(46) modified as follows. The Bloch energy ε ( k$_{x}$ ) in the Fourrier transformation (46) has to be replaced by the Bloch energy ε ( k$_{x}$, k$_{y}$, k$_{z}$ ) and the current (47) should be summed over the occupied states k$_{y}$ and k$_{z}$ . Practical calculations are relatively straightforward, because the Bloch spectrum of the bulk crys- tal, ε ( k$_{x}$, k$_{y}$, k$_{z}$ ) , is known for many insulators. 

In what follows we apply the model of the elastically bent 3D crystal to the rings with thickness equal to the unit cell of the bulk crystal. The energy dispersion of the ring, ε$_{n}$ ( φ, k$_{y}$ = 0 , k$_{z}$ = 0) , is extracted from the bulk crystal dispersion ε ( k$_{x}$, k$_{y}$ = 0 , k$_{z}$ = 0) , calculated by microscopic methods [28, 29]. The obtained persistent current should be viewed as a minimum value estimate, because the thickness of realistic rings is much larger than the unit cell. 

Using standard methods [28, 29], in the figure 7 we nu- merically reproduce the well-known band structure of the band insulators like the GaAs, Ge, and InAs. Specifically, 

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the figure summarizes the ϵ$_{k}$ curves of the light-hole band, heavy-hole band and lowest band, calculated for the line ( L [ − π/a, − π/a, − π/a ] , Γ[0 , 0 , 0] , L [ π/a, π/a, π/a ] ) in the first Brillouin zone. We set each of these ϵ ( k ) curves numer- ically into the Fourier transformation (46) and we calculate for each band the hopping amplitude Γ$_{N}$ . The resulting Γ$_{N}$ dependencies are shown in the figure 7. 

By means of the figure 7 we estimate the persistent current as follows. We form the ring by elastic bending of the crystal. We choose the ring orientation for which the k states on the line ( L [ − π/a, − π/a, − π/a ] , Γ[0 , 0 , 0] , L [ π/a, π/a, π/a ] ) are directed along the ring circumference. If the ring is one-dimensional (with thickness equal to a single unit cell of the crystal), the persistent current is given by the 1D formula (47) with the hopping amplitude Γ$_{N}$ given by numerical values in the figure 7. As expected, Γ$_{N}$ decays with increasing N exponentially, however, for the light-hole band the decay is much slower than for other two bands. Therefore it is sufficient to calculate the persistent curre nt in the light-hole band. The results are shown in the figure 7. We recall that these results represent the persistent current carried by electrons in the fully occupied valence band. The following features are worth to point out. 

The persistent current in the insulating ring decreases with 

