Origin of Excess Low-Energy States in a Disordered Superconductor in a Zeeman Field

Tunneling density of states measurements of disordered superconducting Al films in high Zeeman fields reveal a significant population of subgap states which cannot be explained by standard BCS theory. We provide a natural explanation of these excess states in terms of a novel disordered Larkin-Ovchinnikov phase that occurs near the spin-paramagnetic transition at the Chandrasekhar-Clogston critical field. The disordered Larkin-Ovchinnikov superconductor is characterized by a pairing amplitude that changes sign at domain walls. These domain walls carry magnetization and support Andreev bound states that lead to distinct spectral signatures at low energy.

Figure 1

(a) Tunneling conductance $G(V)$ normalized by normal-state conductance $G_n\sim (1\mathrm{k}\Omega {)}^{-1}$ for a 24 Å superconducting Al film in a 4.75 T parallel field at 100 mK (symbols, experiment; curve, homogeneous theory). (b) Zero-bias tunneling conductance $G(0)$ at 60 mK as a function of parallel field $H$. Between $H_0\sim 2.8\mathrm{T}$ and $H_{c\parallel }\sim 6.1\mathrm{T}$, the homogeneous theory (blue curve) significantly underestimates the number of states near the Fermi energy, and even when the temperature is artificially increased (red curve) it is unable to describe the broad tail in $G(0)$. We ascribe the discrepancy to a disordered LO phase. (Inset) Tunnel conductance as a function of $H_{\perp }=4.5\sin (\theta )$ where $\theta$ is the tilt angle $\theta$. The solid lines are a linear least-squares fit to the data. The sharp $\mathsf{V}$-shaped minimum allows us to accurately determine parallel alignment.

Figure 2

Root-mean-square pairing amplitude ${\Delta }_{\mathrm{rms}}$, average magnetization $m_{\mathrm{avg}}$, and Fermi-level density of states $N(0)$ as functions of Zeeman field $h$, in units of the hopping amplitude $t$ [see Eq. (1)]. For $h_{c1}<h<h_{c2}$ there is a disordered LO state with coexistent pairing and magnetization, in which the gap is partially filled in. The results are obtained by using Bogoliubov–de Gennes simulations on a $36\times 36$ Hubbard model at weak disorder $W=1t$ (well below the critical disorder [38] for the destruction of superconductivity $W_c\sim 3t$), nonzero chemical potential $\mu =-0.25t$ to avoid perfect nesting effects at half-filling, low temperature $T=0.1t$, and a relatively large attraction $|U|=4t$ so that the coherence length is less than the system size. $h=\frac{1}{2}g{\mu }_{B}H$, where $g\approx 2$ is the $g$ factor, ${\mu }_B$ is the Bohr magneton, and $H$ is the parallel field.

Figure 3

The first two columns show spatial maps of the local pairing amplitude $\Delta$ and the magnetization $m$. The third column show the DOSs of up and down electrons $N_{\sigma }(E)$. The last column shows the total DOS $N(E)$. For intermediate fields (e.g., $h/t=0.95$ and $h/t=1.2$) the system exhibits disordered Larkin-Ovchinnikov states with domain walls at which $m$ is finite, $\Delta$ changes sign, and the DOS becomes finite at low energy. Other parameters are as in Fig. 2.

Figure 4

(a) Combined plot of $m(\mathbf{r})$ and $\Delta (\mathbf{r})$ for $h/t=1$ (other parameters as in Fig. 2). Red (blue) indicates regions where $\Delta (\mathbf{r})$ is large and positive (negative). Brown regions, where the magnetization $m(\mathbf{r})$ is large, occur at domain walls where $\Delta$ changes sign. White regions are hills or valleys of the disorder potential corresponding to empty sites or localized pairs that participate in neither superconductivity nor magnetism. (b) and (c) show oscillations of $\Delta$ along the vertical dashed line in (a). (d) and (e) show the correspondence between magnetization $m(\mathbf{r})$ and low-energy spectral weight $I(\mathbf{r})={\int }_{-0.1t}^{0.1t}dE{N}_{\mathbf{r}}(E)$.