Impact of Nonlocal Electrodynamics on the Flux Noise and Inductance of Superconducting Wires

We present exact numerical calculations of the supercurrent density, inductance, and impurity-induced flux noise of cylindrical superconducting wires in the nonlocal Pippard regime, which occurs when the Pippard coherence length is greater than the London penetration depth. In this regime, the supercurrent density displays a peak away from the surface and changes sign inside the superconductor, signaling a breakdown of the usual approximation of local London electrodynamics with a renormalized penetration depth. Our calculations show that the internal inductance and the bulk flux noise power are enhanced in nonlocal superconductors. In contrast, the kinetic inductance is reduced and the surface flux noise remains the same. As a result, impurity spins in the bulk may dominate the flux noise in superconducting qubits in the Pippard regime, such as the ones using aluminum superconductors with a large electron mean free path.

Figure 1

An infinite wire with cylindrical geometry. The current is assumed to flow along $+\hat{\mathit{z}}$.

Figure 2

A comparison between the local and nonlocal current density and vector potential for $R=1000$ nm, $\lambda =70$ nm, and $\xi =200$ nm. Note that $J$ and $A$ are normalized by their local values so that in the local regime they fall on the same curve. The impact of nonlocality is evident in two features: $J$ acquires a peak away from the surface of the wire and it changes sign inside the wire. Both $A$ and the magnetic field $B_{\theta }=-{\mathrm{\partial }}_{\rho }A$ also change sign inside the wire, due to self-induced overscreening.

Figure 3

The (a) vector potential and (b) current density for $R=1000$ nm and different values of the mean free path $l$. We use parameters appropriate for aluminum, ${\lambda }_L=16$ nm and ${\xi }_0=1600$ nm, with $\xi ={\xi }_{0}l/({\xi }_0+l)$ and $\lambda ={\lambda }_L\sqrt{{\xi }_0/\xi }$ changing with $l$. The plots are normalized by the local surface values $A_{\mathrm{local}}(R)$ and $J_{\mathrm{local}}(R)$ calculated from Eqs. (6) and (7), with parameter $\lambda =203$ nm fixed for all curves (this is the same $\lambda$ used in the nonlocal $l=10$ nm case). Note that the surface magnetic field $B_{\theta }(R)$ and the total current $I$ is the same for all curves. A plot of the magnetic field $B_{\theta }(\rho )=-{\mathrm{\partial }}_{\rho }A$ is shown in Fig. 10.

Figure 4

The vector potential calculated using the local approximation with ${\lambda }_R=0.85({\lambda }^2\xi {)}^{1/3}$ compared to the exact solution of the Fredholm integral equation, for $\lambda =70$ nm and $\xi =500$ nm.

Figure 5

The approximate current density obtained by numerical evaluation of the Pippard relation given in Eq. (1), using the approximate $A(\rho )$ of Fig. 4.

Figure 6

The kinetic inductance per unit length as a function of the wire radius, for the local case (dashed) and for nonlocal cases with varying $l$. The parameters $\lambda$ and $\xi$ are the same as used in Fig. 3.

Figure 7

The internal inductance per unit length as a function of the wire radius, for the local case (dashed) and for nonlocal cases with varying $l$. The parameters $\lambda$ and $\xi$ are the same as used in Fig. 3.

Figure 8

The ratio of the kinetic and internal inductances as a function of the wire radius, for the local case (dashed) and for nonlocal cases with varying $l$. The parameters $\lambda$ and $\xi$ are the same as used in Fig. 3. In the local regime, the kinetic inductance is always greater than the internal inductance; in contrast, the nonlocal regime shows the opposite behavior.

Figure 9

The total inductance as the sum of the kinetic and internal inductances, as a function of the wire radius, for the local (dashed) and for nonlocal cases with varying $l$. The parameters $\lambda$ and $\xi$ are the same as used in Fig. 3.

Figure 10

The flux vector as a function of the impurity spin radial location $\rho$ inside the wire, for the same parameters as used in Fig. 3. The flux vector determines the flux coupled to the wire according to $\mathrm{\Phi }=-{\mathit{F}}_i\cdot {\mathit{s}}_i$, where ${\mathit{s}}_i$ is the impurity spin operator.

Figure 11

The calculated flux noise power due to the presence of spin impurities in the cylindrical wire, for the local and nonlocal cases, with $\lambda$ and $\xi$ as in Fig. 3. We separate the flux noise into contributions from spins at the surface (spin density ${\sigma }_2=5\times {10}^{12}{\mathrm{cm}}^{-2}$) and in the bulk (spin density ${\sigma }_3=1.3\times {10}^{18}{\mathrm{cm}}^{-3}$).