Optical Signatures and Quantum Geometry in Proximity-Induced Topological Superconductors

Abstract

Proximity-induced superconductivity at topological insulator-superconductor (TI-SC) interfaces offers a promising route to topological superconductivity with Majorana boundary modes. However, probing the interfacial superconductivity at buried interfaces is challenging with conventional surface methods. Here, we present a theoretical study of the longitudinal optical response of a TI-SC heterostructure, focusing on the complex interface sheet conductance as a direct and layer-selective probe of the interfacial superconducting gap. Within a minimal TI--SC model, we demonstrate that proximity-induced superconductivity at the buried interface generates a two-dimensional topological superconducting phase supporting Majorana edge modes. Using a Bogoliubov-de Gennes slab model and the Kubo formalism, we compute the optical conductance and introduce a thickness-extrapolation protocol that isolates the interface contribution only. The resulting interface conductance exhibits a robust, thickness-independent coherence peak at an energy set by the proximity-induced gap, distinguishable from both the parent superconductor's pair-breaking feature and the ungapped Dirac cone on the top surface. We further demonstrate that the low-frequency spectral weight of this interface resonance obeys a quantum-metric sum rule, quantitatively linking the optical response to the quantum geometry of the proximitized interfacial state. Our results propose terahertz/infrared spectroscopy of the interfacial sheet conductance as a non-invasive diagnostic of Majorana-hosting TI--SC interfaces.

Figures (8)

• Figure 1: Schematics of the TI-SC heterostructure and its optical and quantum-geometric signatures.(a) Isolated topological insulator (TI) with a gapped bulk and a gapless Dirac surface state. The chemical potential $\mu$ (orange) intersects the Dirac cone, producing a Pauli-blocked, step-like in-plane optical conductance $G_{xx}(\omega)$ with onset at $2\left|\mu\right|$. (b) Isolated conventional $s$-wave BCS superconductor (SC) with pairing amplitude $\Delta$. Its optical response is gapped below $2\Delta$ and exhibits a coherence peak at the gap edge. (c) TI-SC heterostructure obtained by placing the TI (blue) on top of the SC (green). The interface region (red) becomes a two-dimensional TSC that supports Majorana edge modes (cyan). Proximity pairing opens an induced gap $2\Delta_{\text{ind}}$ in the TI surface Dirac cone. The resulting heterointerface conductance $G_{xx}(\omega)$ shows a coherence peak at $2\Delta_{\text{ind}}$, a second peak at $2\Delta$ from the bulk SC, and a Pauli-blocking contribution from the ungapped top-surface Dirac cone. (d) Quantum-geometric characterization and optical-conductance sum rule. The gray ring in momentum space marks the region where the quantum metric $g_{xx}(\mathbf{k})$ of the proximitized Dirac surface state is strongly enhanced due to the proximity-induced gap. The associated quantum weight $K_{\mu \mu } \propto \int_{\text{BZ}} d^{2}\mathbf{k} g_{\mu\mu}(\mathbf{k})$ is connected to the generalized optical weight through the sum rule $\int_{0}^{\infty} d\omega \text{Re}[G_{\mu\mu}(\omega)]/\omega \propto K_{\mu\mu}$. As a result, the dimensionless ratio $\chi$ between these two quantities remains essentially unchanged under variations of key parameters, such as the chemical potential and the proximity-coupling strength.
• Figure 2: Topological characteristics and band structures of the TI-SC heterostructure.(a) Parity eigenvalues of the occupied bands at the time-reversal-invariant momenta (TRIM) for the tight-binding TI model. A single band inversion at $\Gamma$ identifies the bulk as a TI. (b) Surface band structure of a TI slab finite along $z$ and periodic in the in-plane directions. The resulting Dirac surface state at $\bar{\Gamma}$ is highlighted in blue. (c) Bulk band structure of the $s$-wave SC, which is fully gapped and topologically trivial. (d) Band structure of the TI-SC heterostructure near the $\bar{\Gamma}$ point for vanishing interface hybridization $\lambda = 0$. The TI Dirac surface state (blue) remains gapless and decoupled from the SC bands. (e) Band structure of the heterostructure for finite interface coupling $\lambda = 3$. Hybridization with the SC induces superconducting pairing in the TI surface Dirac cone, opening a proximity-induced gap (red) and producing a fully gapped interface spectrum consistent with the effective BdG surface theory in Eq. \ref{['eq:TI_SW']}. Representative parameters are $m_0=-120$ meV, $t_0=48$ meV, $\alpha=40$ meV, $m=-45$ meV, $t=30$ meV, $\Delta=2$ meV, $\mu=1.5$ meV, with slab thicknesses $l_T = 8$ for the TI and $l_S = 8$ for the SC.
• Figure 3: Optical response of a topological-insulator slab and extraction of the surface sheet conductance.(a,d) Real and imaginary parts of the longitudinal conductance $G^{\text{TI}}_{xx}(\omega,l_T)$ (normalized by $G_{0}=e^{2}/h$) for several TI slab thicknesses $l_T$. The spectra show a thickness-dependent Drude contribution and interband background, together with a Pauli-blocked onset at $\omega = 2|\mu|$ associated with the Dirac surface state. (b,e) Thickness-independent sheet conductance $G^{\text{TI}}_{xx,s}(\omega)/G_{0}$ obtained by extrapolating $G^{\text{TI}}_{xx}(\omega,l_T)/G_{0}$ to zero thickness. (c,f) Interband surface sheet conductance $G^{\text{TI}}_{xx,p}(\omega)/G_{0}$ obtained from the projected single-particle eigenstates onto the outermost TI layers (top and bottom surfaces), thereby isolating a surface-only optical channel. The close correspondence between panels (b) and (c) confirms that the Dirac surface response can be regarded as a two-dimensional conducting sheet superimposed on a bulk background that scales with $l_T$. Model parameters are the same as in Fig. \ref{['fig2:SetUp']}.
• Figure 4: Optical response of the TI-SC heterostructure and isolation of the interface sheet conductance.(a,e) Real and imaginary parts of the total in-plane sheet conductance $G^{\text{H}}_{xx}(\omega,l_{T})/G_{0}$ of the TI-SC heterostructure for several TI thicknesses $l_{T}$. The spectra contain the TI bulk Drude response and Dirac surface contribution together with the SC superfluid and quasiparticle terms, and they display a pronounced feature near $\omega \simeq 2\Delta_{\text{ind}}$ associated with the proximitized interface states (highlighted in the insets, red circles). (b,f) Thickness-independent sheet component $G^{\text{H}}_{xx,s}(\omega)/G_{0}$ obtained by extrapolating $G^{\text{H}}_{xx}(\omega,l_{T})/G_{0}$ as $l_{T}$ goes to zero, thereby removing the TI bulk background that scales with thickness and retaining the combined sheet response of the top TI surface, the buried interface, and the SC slab. (c,g) Normalized surface sheet conductance $G^{\text{H}}_{xx,t}(\omega)/G_{0}$ after subtracting the calibrated SC sheet and bulk contributions, so that the remaining response arises from the top TI surface and the buried interface. (d,h) Interface-only sheet conductance $G^{\text{I}}_{xx}(\omega)/G_{0}$ obtained by further subtracting the known TI top-surface spectrum from the surface term in (c,g). This response consists of a single, sharply defined resonance at $\omega \simeq 2\Delta_{\text{ind}}$, identifying the induced coherence peak as an intrinsic property of the proximitized Dirac interface states rather than of TI or SC bulk channels. Model parameters are the same as in Fig. \ref{['fig2:SetUp']}.
• Figure 5: Chemical-potential dependence of the interface optical response.(a,b) Real and imaginary parts of the total in-plane conductance $G^{\text{H}}_{xx}(\omega,\mu)/G_{0}$ for several chemical potentials $\mu$. The dominant feature at $\omega \simeq 2\Delta$ reflects the parent $s$-wave condensate, while the inset in (a) resolves the additional low-energy resonance associated with the induced interface gap (red circle). (c,d) Interface-only sheet conductance $G^{\text{I}}_{xx,s}(\omega,\mu)/G_{0}$ obtained by subtracting the calibrated TI-surface and bulk-SC backgrounds from the total response. The coherence peak near $2\Delta_{\text{ind}}$ shifts and reshapes with $\mu$, revealing the sensitivity of the induced pairing on the Dirac surface state to carrier density. All other parameters are as in Fig. \ref{['fig2:SetUp']}.