Application of the Mattis-Bardeen theory in strongly disordered superconductors

The low-energy optical conductivity of conventional superconductors is usually well described by Mattis-Bardeen (MB) theory, which predicts the onset of absorption above an energy corresponding to twice the superconducing (SC) gap parameter $\mathrm{\Delta }$. Recent experiments on strongly disordered superconductors have challenged the application of the MB formulas due to the occurrence of additional spectral weight at low energies below $2\mathrm{\Delta }$. Here we identify three crucial items that have to be included in the analysis of optical-conductivity data for these systems: (a) the correct identification of the optical threshold in the Mattis-Bardeen theory and its relation with the gap value extracted from the measured density of states, (b) the gauge-invariant evaluation of the current-current response function needed to account for the optical absorption by SC collective modes, and (c) the inclusion into the MB formula of the energy dependence of the density of states present already above $T_c$. By computing the optical conductivity in the disordered attractive Hubbard model, we analyze the relevance of all these items, and we provide a compelling scheme for the analysis and interpretation of the optical data in real materials.

Figure 1

Summary of the main spectral properties of strongly disordered superconductors. (a) Disorder induces a strong variation of the local SC order parameter ${\mathrm{\Delta }}_i$, which is captured already at the level of the Bogoliubov–de Gennes solutions for the disordered attractive Hubbard model [28, 29, 32, 33]. Experimentally, the variations occur as fluctuations of the coherence-peak height in the local DOS probed by STS [23, 25, 28]. (b) Sketch of the typical average tunneling DOS, as found numerically (see Fig. 2 below) and experimentally [20, 21, 22, 23, 24, 25, 26, 27]. The typical SC coherence peaks are strongly suppressed in intensity, and they are located at an energy scale $E_{\text{peak}}$ somehow larger than the minimum excitation energy $E_{\text{gap}}$. (c) Optical conductivity of disordered SC. Here the solid lines denote the SC Nambu Green function, the dashed lines the e.m. field, and the wavy line the collective modes computed at RPA level, which build up the vertex corrections. The BCS approximation is equivalent to compute the “bare bubble” diagram, which accounts only for the breaking of Cooper pairs by the incoming e.m. field. Even when the SC order parameter is inhomogeneous, as in panel (a), it leads to an optical conductivity with a hard threshold at $2E_{\text{gap}}$, that reproduces for intermediate disorder the usual Mattis-Bardeen results. The RPA vertex corrections account for the exchange of collective excitations between the Cooper pairs broken apart by the e.m. field. When these are included the optical conductivity acquires an extra contribution which piles up below $2E_{\text{gap}}$, stealing spectral weight from the superfluid peak at zero frequency, denoted here by an arrow. The black dashed line indicates that the precise form of ${\sigma }_{1s}$ as $\omega \to 0$ depends on the disorder level.

Figure 2

Main panel: Density of states in the normal (blue) and superconducting state (black). The normal DOS is obtained from an average over 150 disorder configurations ($U/t=2, V_0/t=2$) of $24\times 24$ systems. The SC DOS (solid, black) is the average over 10 disorder configurations from $52\times 52$ systems, whereas the black dashed curve is the DOS for a particular disorder realization. The eigenvalues of the latter system are indicated by bars on the frequency axis. Vertical lines indicate the position of the lowest eigenvalue (solid, red) and the “coherence” peak feature (dashed, green). Inset: same for a system with disorder in the local interaction $U_i=U_0\pm \delta u_i$ with $U_0/t=2$ and $\delta u_i$ is taken from a flat distribution between $-0.5t\le \delta u_i\le +0.5t$. The higher energy peak of the double peak feature is a remainder of the van Hove singularity. In all cases the charge density is $n=0.875$.

Figure 3

Optical conductivity at $V_0/t=1$ (panels a, b) and at $V_0/t=2$ (right panels c, d). The remaining parameters are: $U/t=2, n=0.875, 24\times 24$ lattice. Panels (a) and (c) show the real (solid, black) and imaginary (dotted, maroon) part of the optical conductivity in the SC state and ${\sigma }_1$ in the normal state (dashed, blue) evaluated as an average over 50 disorder configurations within the gauge-invariant RPA approach. For comparison, the real part of the conductivity in the SC state within the BCS approximation is also shown by the red solid line. The vertical green and red lines denote $2E_{\text{peak}}$ and $2E_{\text{gap}}$, respectively, as extracted from the DOS. The DOS for the case $V_0/t=2$ is the one shown in Fig. (2), while the one for $V_0/t=1$ is shown explicitly as an inset of panel (a). Panels (b) and (d) show the ratio between the real part of the optical conductivity computed in the SC and normal state. The red curve reports the corresponding result in the BCS limit and the dashed line is the Mattis-Bardeen fit done using $2E_{\text{gap}}$ as the optical threshold. For comparison, panel (d) also shows experimental data from Ref. [14] for the most disordered sample (${\mathrm{T}}_c=3.8K$) and we have scaled the energy by $t=0.8$ THz. Notice that for the BCS case also ${\sigma }_{1,n}$ has been computed at the bare-bubble level, leading to a normal-state conductivity slightly different from the gauge-invariant result reported in the upper panels.

Figure 4

Ratio between optical conductivities in the superconducting and normal state for the Hubbard model at $V_0/t=2$ [as in Figs. 2 and 3] in the BCS approximation (red line), compared to the prediction of the standard MB theory (dashed black line) and of the modified MB formula Eq. (12) (brown symbols). Inset: fit of the numerical results for the normal-state DOS of the Hubbard model. Near the Fermi level we used the approximated formula Eq. (20) with $\alpha =0.0156/t,\beta =0.1095/t,\mathrm{\Omega }=0.1968t$, while away from it we used a linear extrapolation at large energies.

Figure 5

Real (a) and imaginary (b) part of the optical conductivity for a NbN sample with $T_c=6.2$ K. Data at $T=7$ K and $T=1.45$ K are shown by squares (blue) and circles (red); the corresponding fits from the modified MB theory Eq. (12) are the solid lines, while the dashed black line is the standard MB formula with a constant DOS. In both cases we used $\mathrm{\Delta }=0.2$ THz. The left inset of panel (a) and the inset (b) shows the ratio between the $T=1.45$ and $T=7$ K data (symbols) compared with conventional and modified Mattis-Bardeen theory. Notice that while in the ratio ${\sigma }_{1s}/{\sigma }_{1n}$ the effect of the DOS disappears, making the standard MB formula apparently good, the ratio $\sigma 2s/\sigma 1n$ cannot be well reproduced by the standard MB approach. The right inset of panel shows the DOS used to reproduce the normal-state conductivity, with parameters $\alpha =1.32, \beta =0.37, \mathrm{\Omega }=0.58$ THz in Eq. (20). Experimental data by courtesy of the authors of Ref. [14].

Figure 6

Same as Fig. 5 but for a NbN sample with $T_c=9.2$ K. Parameters for the DOS Eq. (20): $\alpha =2.24, \beta =0.24, \mathrm{\Omega }=0.75$ THz. Superconducting gap parameter: $\mathrm{\Delta }=0.35$ THz. Experimental data by courtesy of the authors of Ref. [14].