Persistent currents of noninteracting electrons in one-, two-, and three-dimensional thin rings

We thoroughly study the persistent current of noninteracting electrons in one-, two-, and three-dimensional thin rings. We find that the results for noninteracting electrons are more relevant for individual mesoscopic rings than hitherto appreciated. The current is averaged over all configurations of the disorder, whose amount is varied from zero up to the diffusive limit, keeping the product of the Fermi wave number and the ring’s circumference constant. Results are given as functions of disorder and aspect ratios of the ring. The magnitude of the disorder-averaged current may be larger than the root-mean-square fluctuations of the current from sample to sample even when the mean-free path is smaller, but not too small, than the circumference of the ring. Then a measurement of the persistent current of a typical sample will be dominated by the magnitude of the disorder-averaged current.

Figure 1

The disorder-averaged PC depends on $k_{F}L$, $N_z$, and $N_r$ in a nontrivial fashion. Here we plot $⟨I_{m=1}^{2\text{D}}⟩$, see Eq. (13), for $L/\ell =5$ and for $N_z=70$ (solid line) and $N_z=100$ (dashed line). The typical magnitude of the disorder-averaged current, given by Eq. (19), for the above two values of $N_z$ is $0.25I_0$ and $0.30I_0$, respectively.

Figure 2

The PC of a 3D ring in the uncorrelated-channel regime. The typical magnitudes of the first harmonic (solid line) and the second harmonic (dashed-dotted line) are plotted in units of $I_0\sqrt{N_{\text{tot}}}$, using Eq. (23). In the inset ${\left(\overline{{⟨I_{m=1}^{3\text{D}}⟩}^2}\right)}^{1/2}/I_0$ (solid line) is obtained by substituting $N_z=N_r=20$ in Eq. (23). For a later discussion, the rms fluctuations of the PC with respect to the disorder, $\delta I/I_0$ (dashed line) are plotted using Eq. (33). Here ${\left(\overline{{⟨I_{m=1}⟩}^2}\right)}^{1/2}=\delta I$, when $L/\ell =8.5$, in agreement with Eq. (34). The horizontal axis of the inset begins at $L/\ell =3$ since Eq. (33) is valid only in the diffusive limit.

Figure 3

The disorder-averaged PC of a 2D diffusive cylinder in the correlated-channel regime (solid line) is plotted as a function of $\ell /L$. We replace $\text{sin}\left(k_{F}L\right)$ by $1/\sqrt{2}$ in Eq. (30) to obtain the typical magnitude, and use $N_z={10}^3$ and $k_{F}L=5\times {10}^3$. The rms fluctuations (dashed line), see Eqs. (31, 33), equals $⟨I_{m=1}^{2\text{D}}⟩$, for the above parameters, at $L/\ell \simeq 10$, in agreement with Eq. (35). Both $⟨I_{m=1}^{2\text{D}}⟩$ and $\delta I$ are given in units of $I_0$.

Figure 4

The energy levels of a single channel are plotted as a function of the flux. The consecutive energy levels for a given positive flux and longitudinal indices $-n$, $n$, and $-n-1$, are marked by full circles. The bottom level corresponds to $n=0$. The random choice of $\mu$ in the interval $\left[E_{q,s,-n}\left(\varphi \right),E_{q,s,-n-1}\left(\varphi \right)\right]$ yields an odd number of occupied levels when $\mu >E_{q,s,n}\left(\varphi \right)$ and an even number of occupied levels when $\mu <E_{q,s,n}\left(\varphi \right)$. The former regime is marked by the bold line in the figure. Here, without loss of generality, we take $n>0$.