Dissipation due to bulk localized low-energy modes in strongly disordered superconductors

We develop a theory of the temperature $T$ and frequency $ω$ dependence of ac dissipation in strongly disordered superconductors featuring a pseudogap $Δ_{P}$ in the single-particle spectrum. Our theory applies to the regime $T,,\hbarω\llΔ_{\text{typ}}\llΔ_{P}$, where $Δ_{\text{typ}}$ is the typical superconducting gap. The dissipation is expressed in terms of the quality factor $Q(T,ω)$ of microwave resonators made of these materials. We show that low-$ω$ dissipation is dominated by a new type of bulk localized collective modes. Due to the strongly nonuniform spectral density of these modes, $Q$ decreases sharply with frequency, while its temperature dependence exhibits a two-level-system-like growth as a function of $T$ for $T\ll T_{c}$. Our theory is applicable to InO$_x$, TiN, NbN, and similar strongly disordered materials. We further argue that the experimentally observed behavior of disordered films of granular Aluminum is explained by similar physics, although this case requires a separate theoretical analysis.

💡 Research Summary

The paper presents a comprehensive theoretical framework for the temperature (T) and frequency (ω) dependence of microwave dissipation in strongly disordered superconductors that exhibit a large single‑particle pseudogap ΔP. The authors focus on the regime T, ℏω ≪ Δtyp ≪ ΔP, where Δtyp is the typical superconducting gap, and express the dissipation in terms of the quality factor Q(T, ω) of microwave resonators fabricated from these materials.

The starting point is a pseudospin Hamiltonian that describes pre‑formed Cooper pairs localized on Anderson‑localized electronic states. Random on‑site energies ξj and random pair‑hopping amplitudes Dij define a sparse, locally tree‑like interaction graph with average branching number K. The dimensionless disorder strength κ = Dij/Δ0 (Δ0 is the mean‑field superconducting energy scale) controls the competition between disorder and superconductivity. For κ ≫ 1 the order‑parameter distribution becomes broad and fat‑tailed, while the authors restrict themselves to κ ≪ κ1, where κ1 ≈ exp(1/2λ) and λ is the dimensionless Cooper‑pair coupling constant.

To treat the strongly inhomogeneous system, the authors employ Belief Propagation (BP) on the locally tree‑like graph. BP yields directed “order‑parameter fields” hij on each edge, which satisfy a self‑consistency equation (Eq. 5). These fields encode the local environment of each bond and are mathematically equivalent to the local superconducting gap Δ on that bond. The two‑spin Hamiltonian H⟨ij⟩ (Eq. 4) built from the fields hij and the random energies ξi, ξj is diagonalized for each edge; its eigenvalues Enij and eigenvectors |nij⟩ determine the local current operator I i→j (Eq. 2) and the retarded current correlator Rij(ω) (Eq. 3).

In the low‑frequency limit (ℏω ≪ Δ0) the current response is linear in the phase difference φj − φi (Eq. 7). The real part of the macroscopic conductivity Re σ(ω) can be expressed as an average over the product Im Rij(ω)·(φj − φi)² (Eq. 8). Direct evaluation of this average would require the joint distribution of Im Rij and the phase differences, which is only accessible numerically. Remarkably, the authors find an approximate relation (Eq. 9) that links Re σ to the average Im Rij multiplied by a geometric factor (ri − rj)²/Dω. This relation captures the dominant frequency dependence because Im Rij(ω) is sharply peaked at the minimal transition frequencies Ωij of each bond.

The authors derive an analytic expression for Im Rij(ω) in the limit ω ≪ h (where h is the typical value of the order‑parameter field) (Eq. 12). The result contains two sources of temperature dependence: (i) the thermal occupation factor tanh(ℏω/2T) characteristic of two‑level systems, and (ii) the temperature‑dependent low‑value tail of the order‑parameter distribution P(h). The latter is highly sensitive because the low‑gap tail steepens dramatically as temperature is lowered, leading to a TLS‑like growth of Q(T) at T ≪ Tc.

To obtain quantitative predictions, the authors construct a large‑scale “network model” (NM). The NM proceeds through: (i) generation of a random sparse graph with given K and κ, (ii) solution of the BP self‑consistency equations for hij on each directed edge, (iii) calculation of Rij for each edge using the two‑spin eigenproblem, (iv) solution of Kirchhoff equations for the static phase configuration φi under an imposed macroscopic phase gradient, and (v) evaluation of Re σ via Eq. 8. Repeating this procedure over many disorder realizations yields disorder‑averaged quantities.

Numerical simulations with K = 10, κ = 10, and λ ≈ 0.1373 on graphs of ~10⁶ nodes reproduce the experimentally observed quality‑factor behavior of InOₓ, TiN, and NbN films. The calculated Q(ω) shows a sharp decline as ω increases, matching the measured drop in the frequency range around 3.85 GHz. The temperature dependence at fixed low frequency follows Q(T) ∝ tanh(ℏω/2T), in quantitative agreement with the TLS‑like trends reported in Refs.