First-principles calculation of electronic transport in low-dimensional disordered superconductors

We present a novel formulation to calculate transport through disordered superconductors connected between two metallic leads. An exact analytical expression for the current is derived and applied to a superconducting sample described by the negative-$U$ Hubbard model. A Monte Carlo algorithm that includes thermal phase and amplitude fluctuations of the superconducting order parameter is employed, and a new efficient algorithm is described. This improved routine allows access to relatively large systems, which we demonstrate by applying it to several cases, including superconductor-normal interfaces and Josephson junctions. Moreover, we can link the phenomenological parameters describing these effects to the underlying microscopic variables. The effects of decoherence and dephasing are shown to be included in the formulation, which allows the unambiguous characterization of the Kosterlitz-Thouless transition in two-dimensional systems and the calculation of the finite resistance due to vortex excitations in quasi-one-dimensional systems. Effects of magnetic fields can be easily included in the formalism, and are demonstrated for the Little-Parks effect in superconducting cylinders. Furthermore, the formalism enables us to map the local super and normal currents, and the accompanying electrical potentials, which we use to pinpoint and visualize the emergence of resistance across the superconductor-insulator transition.

Figure 1

Schematic of the experimental setup within the negative-$U$ Hubbard model. The left- and right-hand metallic leads are shown in blue, from which electrons can tunnel through the barriers shown by the gray links into the central, possibly disordered SC region which is shown in red.

Figure 2

Variation of conductance $G$ with the average superconducting order parameter $U\langle c_{\uparrow }{c}_{\downarrow }\rangle$ as the interaction strength is increased. The normal current is shown in blue and the supercurrent (Andreev) in red. The numerical results are shown with error bars, and BTK theory by the solid line. The green vertical line denotes the chemical potential difference across the sample. The upper plot (a) is at “zero” temperature, with no thermal or quantum fluctuations, and the lower (b) at $T=0.01t$, showing a marked difference between the mean-field BTK formula and the numerical data that include thermal fluctuations.

Figure 3

Upper: Setup to model the Josephson junction. Traversing the center of the SC region is a Josephson junction (opaque cuboid). The junction is modeled by the reduction of the matrix hopping elements to $t^{\prime }=0.005$ (brown interconnects) compared to $t=1$ in the superconductor. The two metallic leads are shown on the far left and right (in blue), and the lead-superconductor tunneling barrier by the gray cuboids. Lower: The variation of conductance with temperature for the Josephson junction. Results of the numerical computation (points) and of the theoretical model (solid lines) are shown for a weak, intermediate, and strong coupling between the two superconductors.

Figure 4

Upper: Relative fall in current in a one-dimensional sample due to the introduction of self-energy at $k_{\text{B}}T=0.1\mu$. The black points show the numerical results, and the solid red line highlights the expected theoretical variation with length,[37] where $J_0$ is the current that flows when impeded solely by the contact resistance. Middle: Relative change ($\Delta J\equiv J_0-J$) in current as a function of temperature in a one-dimensional normal phase sample. The black points show the numerical results, and the red solid line the expected model variation. The two green dashed lines show the $\exp (-\mu /k_{\text{B}}T)$ and $T^2$ behavior. Lower: Changing current in the presence of a SC phase in a two-dimensional sample. The vertical green dashed lines show the BKT and normal phase transitions.

Figure 5

Fall in conductance with length in a noninteracting one-dimensional wire with disorder $W=0.2t$ at two different temperatures. The straight (red) trend lines show a linear drop off in conductance with length, and the curved (green) an exponential decay.

Figure 6

(a) Variation of resistance with temperature for two different values of disorder calculated numerically (points). The red solid line shows the theoretical low-temperature behavior, and the blue dotted line the theoretical high-temperature behavior. The dashed vertical green lines show the BKT $T_{\text{KT}}$ and mean-field $T_{\text{c}}$ temperatures. The insets show the variation of the superconducting order parameter $U\langle c_{\uparrow }{c}_{\downarrow }\rangle$ renormalized by its $T=0$ value. (b) Numerical results (black points) and the deduced linear length dependence (red solid line) of the resistance for three different temperatures (i) $T<T_{\text{KT}}$, (ii) $T_{\text{KT}}<T<T_{\text{c}}$, and (iii) $T>T_{\text{c}}$. The $L\to 0$ values correspond to the contact resistance, which is eliminated in the values plotted in (a).

Figure 7

Upper: Schematic of the cylindrical wire within the negative-$U$ Hubbard model. The left- and right-hand metallic leads are shown in blue, from which electrons can tunnel through the gray toroids into the central SC region, which is shown in red. The magnetic flux threading the cylinder is shown as a green arrow. Lower: Variation of conductivity with flux. The computational points are $(+)$ for $T=0.1t$, where the data reasonably fit the mean-field Little-Parks model (red solid line), and $(\times )$ for $T=0.03t$, well below the mean-field transition temperature $T_c\simeq 0.09t$, where the numerics is best fit by the sinusoidal blue dashed line.

Figure 8

(a) Average fractional error in conservation of current ${\sum }_{i=1}^N\left|\Delta J_i\right|/J_{i}N$ on each site against the fraction of total states $K/N$ included in the calculation of the current. (b) The changing conductance (black line with arrow toward left) of a Josephson junction with width $L_{\text{norm}}$ of central normal region (left axis), and the fraction of the normal current $J_{\text{norm}}/J_{\text{total}}$ (blue dashed line with arrow toward right) flowing through the central region (right axis).

Figure 9

The upper panels in each figure show the potential difference $V\left(x\right)$ across the sample with total potential drop $V$. The lower panels show current maps for (a) short and (b) long normal intermediate regions. Supercurrent is shown by cyan darts and normal current by violet pointers; the arrow length corresponds to current magnitude and orientation to the direction of current flow. Color density corresponds to the order parameter $U\langle c_{\uparrow }{c}_{\downarrow }\rangle$, which has peak value $U{\langle c_{\uparrow }{c}_{\downarrow }\rangle }_0$.

Figure 10

(a) The fall in conductance across the superconductor-insulator transition. Current maps on tuning temperature from (b) a superconductor at $T\approx 0.14T_{\text{c}}$ through to (d) an insulator at $T\approx 2.3T_{\text{c}}$. At $T\approx T_{\text{c}}$ the superconductor-insulator transition takes place. Supercurrent is shown by cyan darts and normal current by violet pointers, arrow length corresponds to current magnitude and orientation to the direction of current flow. Color density corresponds to the order parameter $U\langle c_{\uparrow }{c}_{\downarrow }\rangle$. Lines of equal chemical potential are shown in white. In the current map (b) three points of interest are labeled: (1) the normal state, (2) the superconductor state, and (3) Josephson tunneling.

Figure 11

(a) Estimate of the current with number of Monte Carlo iterations, $i$, out of a total number $I=1000$. The primary $y$ axis shows the best estimate of the current (blue). The secondary $y$ axis shows the estimated standard deviation in this estimate (green) and idealized improvement in the accuracy (red). (b) Distribution of 50 separate current estimates at $T=0$ (red) and $T=0.2T_{\text{c}}$ (green) with best-fit Gaussian distributions. (c) and (d) Time $\tau$ to perform a run on a $32\times 32$ system renormalized by the time ${\tau }_0$ for a $M=512$, $N=1$ system. In (c) the change with varying the system size $N$, where the blue line is for the standard $O\left(N^4\right)$ method of finding all of the energy eigenvalues, the green is the $O\left(N^2\right)$ standard Chebyshev expansion method,[47] and the blue is the $O\left(N^{1.9}\right)$ extended Chebyshev approach. In (d) the two Chebyshev expansion method approaches are compared by varying the expansion order $M$; the upper (green) line is the standard $O\left(M\right)$ approach,[47] and the lower (red) line is the new $O\left(M^{2/3}\right)$ algorithm. In (e) we compare the conductivity at the superconductor-insulator transition studied in Fig. 10 with and without density fluctuations.