Optical conductivity of the Majorana mode at the s- and d-wave topological superconductor edge

Significance

Recent years have seen considerable progress toward realizing non-Abelian particles, fueled by their promised applications to topological quantum devices. A prominent example is the one-dimensional chiral Majorana mode. It opens the possibility of using wave packets propagating at high speed as an alternative to the braiding of zero-dimensional Majorana fermions. While signatures of the latter have been established, a weak spot in detecting chiral Majorana modes lies in reliably capturing quantitative measures such as a quantized conductivity. We here propose using microwave spectroscopy to instead reveal distinct qualitative signatures emerging due to the unique dispersion of the Majorana mode that allows photons to break up Cooper pairs into Majorana fermions propagating along a topological superconductor edge.

Abstract

The Majorana fermion offers fascinating possibilities such as non-Abelian statistics and nonlocal robust qubits, and hunting it is one of the most important topics in current condensed matter physics. Most of the efforts have been focused on the Majorana bound state at zero energy in terms of scanning tunneling spectroscopy searching for the quantized conductance. On the other hand, a chiral Majorana edge channel appears at the surface of a three-dimensional topological insulator when engineering an interface between proximity-induced superconductivity and ferromagnetism. Recent advances in microwave spectroscopy of topological edge states open a new avenue for observing signatures of such Majorana edge states through the local optical conductivity. As a guide to future experiments, we show how the local optical conductivity and density of states present distinct qualitative features depending on the symmetry of the superconductivity, that can be tuned via the magnetization and temperature. In particular, the presence of the Majorana edge state leads to a characteristic nonmonotonic temperature dependence achieved by tuning the magnetization.

Study of Majorana fermions in topological superconductors is an important topic from the viewpoints of fundamental physics and applications to quantum computing (1–4). Zero energy Majorana bound states (5–7) are proposed to act as topologically protected qubits with non-Abelian statistics (8, 9). While significant progress has been made in experimentally establishing their signatures (10–17), another possible Majorana fermion—the propagating one at the edge of the sample in topological superconductors—has remained more elusive (18–20). The one-dimensional nature of these chiral Majorana channels introduces the possibility of using wave packets propagating at high speed as an alternative to the braiding of zero-dimensional Majorana particles (21). One predicted platform is the heterojunction of a ferromagnetic insulator and superconductor on top of a three-dimensional (3D) topological insulator (22–24). The proximity effect of the magnetization and superconductivity to the surface state of the topological insulator results in the chiral Majorana edge channel at the interface (22–26). Even though the Majorana bound state is charge-neutral, it is coupled to the electromagnetic field due to being a composite quasiparticle rather than a fundamental one, contributing to the optical conductivity (27–30). This is particularly promising because the experimental advances in the local probe of microwave spectroscopy (31, 32) have enabled the detection and imaging of the edge channels in the quantum anomalous Hall (33, 34) and Weyl state (35).

In the present paper, we study theoretically the local density of states (LDOS) and local optical conductivity near the interface between proximity-induced ferromagnetism and s-wave or d-wave superconductivity at the surface of a 3D topological insulator. They show a variety of behavior as a function of energy depending on the strength of the magnetic exchange coupling and temperature. Especially the contribution of the nodes in the d-wave superconductors makes the behavior qualitatively distinct from the s-wave case with a full gap. While previous scanning tunneling microscopy (STM) experiments relied on quantitative measures, e.g. a quantized conductance (36–38), that cannot be captured exactly due to measurement noise (19, 20, 39, 40), our results instead provide more robust qualitative features which depend on tuning parameters that can be probed through tunneling spectroscopy and microwave impedance microscopy (MIM) (33).

Results

Model.

We consider the surface of a 3D topological insulator (TI) with spin–orbit parameter A and chemical potential μ ≥ 0 lying in the xy plane. The TI surface can be separated into three regions: For x > 0, the TI surface is in contact with a superconductor (SC) and hosts proximity-induced superconductivity (41–46) with an order parameter (47)

$$\begin{array}{cc}\hfill \mathrm{\Delta }(\theta )& =\left\{\begin{array}{c}{\mathrm{\Delta }}_0 \text{for} s\text{-wave pairing, and}\hfill \\ {\mathrm{\Delta }}_0 cos\{2[\theta -\phi ]\} \text{for} d\text{-wave pairing.}\hfill \end{array}\right.\hfill \end{array}$$ [1]

Above, ${\mathrm{\Delta }}_0$ is the maximum value of the superconducting gap, θ is given by $k_y=k_{\text{F}}sin(\theta )$, where k_F is the Fermi momentum, and φ is the angle of the positive d-wave lobe with respect to the interface normal $\widehat{x}$ (Fig. 1A). In particular, φ = 0 corresponds to $d_{x^2-y^2}$-pairing, and $\phi =\pi /4$ corresponds to $d_{\mathit{xy}}$-wave pairing. For $-d<x<0$, the TI surface has an out-of-plane magnetization m_z induced by an adjacent ferromagnetic insulator (FI). The finite m_z opens an insulating gap in the surface states of the TI. In this region only, we consider μ = 0, yielding a purely decaying wave function. Since the McMillan approach (48) that we will use in the following relies on constructing the Green’s function from propagating wave functions incoming from the left and right, we include a region at $x<-d$, where the TI surface is neither superconducting nor magnetic (${\mathrm{\Delta }}_0=0$, $m_z=0$). This region introduces propagating wave functions incoming from the left that are helpful for correctly constructing the Green’s function (49). However, we focus on the interface between the ferromagnetic and superconducting regions by assuming the length of the ferromagnetic region to be much larger than the decay length of the wave function inside this region ($d\to \infty )$. As schematically illustrated in Fig. 1A, the region where the TI surface is interfaced with a FI forms a 2D quantum anomalous Hall insulator (QAHI), while the region where it is interfaced with a SC forms a 2D topological superconductor (TSC) (50, 51). At the QAHI/TSC interface (x = 0), there is a single chiral Majorana edge channel that will be the focus of this work.

Fig. 1.

Edge states at a QAHI/TSC interface. (A) The interface between a quantum anomalous Hall insulator (QAHI) and a topological superconductor (TSC) can be studied by proximitizing the surface of a three-dimensional topological insulator (TI) to a ferromagnetic insulator (FI) and a superconductor (SC), respectively. At the interface between the 2D QAHI and 2D TSC, there is a single chiral Majorana edge channel (red) decaying over the superconducting coherence length. The optical conductivity of the edge channel can be measured by running a microwave impedance microscopy (MIM) tip (white triangle) across the QAHI/TSC interface from above following the white dotted line (34). We consider a FI with magnetization m_z along $\widehat{z}$, and a TSC with s-wave or d-wave pairing, where the positive d-wave lobe makes an angle φ with respect to the interface normal $\widehat{x}$. Panel (B–J) presents the angle-resolved LDOS $\rho (x=0,\theta ,ϵ)$ normalized by its normal-state value for ${\mathrm{\Delta }}_0=0$ at the QAHI/TSC interface (x = 0) in the case of s-wave (B–D), $d_{x^2-y^2}$-wave (E–G), and $d_{\mathit{xy}}$-wave (H–J) pairing. We consider ${\mathrm{\Delta }}_0/\mu ={10}^{-3}\ll 1$, energies $ϵ+i\delta$, where $\delta /{\mathrm{\Delta }}_0=5·{10}^{-3}$, and different values of $m_z/\mu$, i.e. from Left to Right, $m_z/\mu =0.5$, $1.5$, and $5.0$. The bound state energy dispersion $E_{\text{b}}(\theta )$ (from Eq. 5) is represented by the dotted lines.

The superconducting and magnetic regions of the topological insulator surface can be described by the Hamiltonians $H_{\text{TI}}+H_{\text{SC}}$ and $H_{\text{TI}}+H_{\text{FI}}$, respectively (6, 52), where

$$\begin{array}{l}{H}_{\text{TI}}=-\mu {\displaystyle \sum _{k,\sigma }}{\psi }_{\sigma }^{†}(k){\psi }_{\sigma }(k)\hfill \\ +{\displaystyle \sum _{k,\alpha ,\beta }}{\psi }_{\alpha }^{†}(k){[A(k_x{\sigma }_x+k_y{\sigma }_y)]}_{\alpha ,\beta {\psi }_{\beta }}(k),\hfill \end{array}$$ [2]

$$H_{\text{FI}}={\displaystyle \sum }_{k,\alpha ,\beta }{\psi }_{\alpha }^{†}(k){[m_z{\sigma }_z]}_{\alpha ,\beta }{\psi }_{\beta }(k),$$ [3]

$$H_{\text{SC}}=\frac{1}{2}\sum _{k,\alpha ,\beta }\left\{{\psi }_{\alpha }^{†}(k){[\text{Δ}(\theta )i{\sigma }_y]}_{\alpha ,\beta }{\psi }_{\beta }^{†}(-k)+\text{h}\mathrm{\text{.c}}\text{.}\right\}$$ [4]

as long as we consider positions far away from the interfaces at x = 0 and $x=-d$, where k_x is still a good quantum number. Above, ${\psi }_{\sigma }^{(†)}(\mathit{k})$ annihilates (creates) an electron of momentum $\mathit{k}$ and spin σ, and $({\sigma }_x,{\sigma }_y,{\sigma }_z)$ are the Pauli matrices. To study the edge state at the QAHI/TSC interface, we construct the wave function in all three regions by taking into account all possible scattering processes of electron-like and hole-like particles from the superconducting region (x > 0) to the nonsuperconducting nonmagnetic region ($x<-d$), and vice versa (see Materials and Methods) (49, 53). In the limit $d\to \infty$, the chiral Majorana state at the QAHI/TSC interface (x = 0) follows the bound state energy dispersion (24)

$$\begin{array}{c}\hfill E_{\text{b}}(\theta )=\left\{\begin{array}{c}-\frac{\text{sgn}(m_z)|\mathrm{\Delta }(\theta )|\mu sin(\theta )}{\sqrt{m_z^2{cos}^2(\theta )+{\mu }^2{sin}^2(\theta )}} \text{for} s\text{- and} d_{x^2-y^2}\text{-wave,}\hfill \\ \frac{\text{sgn}(\theta )|\mathrm{\Delta }(\theta )|m_{z}cos(\theta )}{\sqrt{m_z^2{cos}^2(\theta )+{\mu }^2{sin}^2(\theta )}} \text{for} d_{\mathit{xy}}\text{-wave.}\hfill \end{array}\right.\end{array}$$ [5]

The above dispersion describes the topologically nontrivial Majorana edge state. While the interface between a ferromagnet and a $d_{\mathit{xy}}$-wave superconductor holds two spin-degenerate zero energy modes that form Andreev bound states, the TI surface lifts the spin degeneracy leaving a single topologically nontrivial Majorana mode (54). The dispersive nature of the Majorana mode was recently shown to have important implications for the optical conductivity: In the s-wave case, where the bound state dispersion is linear in the momentum k_y at small incidence angles θ, it was shown that a Cooper pair can absorb a photon and break up into two Majorana fermions which momentum k_y along the edge both have the same sign (27). Since a finite $k_y=k_{\text{F}}sin(\theta )$ corresponds to a finite energy $E_{\text{b}}(\theta )$, the zero-temperature optical conductivity peaks at a finite energy, while it is zero at zero energy. Furthermore, it was recently predicted that the chirality of the Majorana edge can be directly probed via circularly polarized light (29). The chirality and dispersive nature makes the Majorana edge mode qualitatively distinct from the nondispersive midgap Andreev bound states at the interface between a $d_{\mathit{xy}}$-wave superconductor and a ferromagnetic insulator (55, 56). We also note that $E_{\text{b}}(\theta )$ is independent of the spin–orbit parameter A, which is true for all of the results presented in this manuscript.

From the wave function of the TSC, we construct the McMillan Green’s function (48, 49, 53). From its retarded part, we numerically calculate the LDOS at the QAHI/TSC interface (x = 0). As shown in Fig. 1B–J, the angle-resolved LDOS has a single edge state that perfectly fits the bound state energy dispersion in Eq. 5. Since the LDOS is evaluated by performing the integral $\rho (x,ϵ)={\int }_{-\pi /2}^{\pi /2}d\theta cos(\theta )\rho (x,\theta ,ϵ)$ over the angle-resolved LDOS, small incidence angles θ give the dominant contributions to the LDOS. In the s-wave (Fig. 1B–D) and $d_{x^2-y^2}$-wave (Fig. 1E–G) TSC, the dispersion of the Majorana edge channel at small incidence angles θ flattens when increasing the magnetization in the QAHI (24, 54). The nodes in the d-wave gap enhance the flatness of the bound state dispersion of the $d_{x^2-y^2}$-wave TSC compared to the s-wave case. Compared to these two cases, the $d_{\mathit{xy}}$-wave TSC (Fig. 1H–J) is distinct in two ways: First, the steepness of the bound state dispersion at small θ instead increases with increasing magnetization (54). Second, the d-wave node is located at θ = 0 and thus gives a large contribution to the LDOS (47, 55). We will show that the enhanced flatness of the bound state dispersion of the d-wave TSC, together with contributions from the nodes in the superconducting gap, give rise to distinct qualitative signatures in the optical conductivity through an enhanced signal below the optical gap.

Magnetization Dependence of the Optical Conductivity.

We use the Kubo formula (57) to numerically evaluate the local optical conductivity at the QAHI/TSC interface from the retarded and advanced McMillan Green’s function (48, 49, 53) of the TSC (Materials and Methods and SI Appendix). We take into account contributions from both the Majorana edge state and energy states above the superconducting gap. This is of particular importance in the d-wave case, where the nodes in the superconducting gap (Fig. 1E–J) contribute to the optical conductivity at energies well below the optical gap $\omega =2{\mathrm{\Delta }}_0$. The optical signal can be measured from above via MIM. During the measurement, the MIM tip is placed above the sample and moved along the x axis across the QAHI/TSC interface as indicated by the white dotted line in Fig. 1A (34). Due to its energy gap, the ferromagnetic insulator is transparent at the energy scale relevant for observing the response of the Majorana edge states. The superconductor can contribute with additional signals due to its finite skin depth, and due to the nodes and sensitivity to impurities in the d-wave case. It is therefore preferable to consider a thin and clean superconducting film. Further details regarding the experimental realization are discussed in SI Appendix. We first consider the local optical conductivity exactly at the QAHI/TSC interface (x = 0) at zero temperature, and address the decay of the edge state inside the TSC, as well as finite temperatures later on.

To understand the results for the optical conductivity, we first consider how the LDOS depends on the magnetization in the QAHI. As already hinted by the angle-resolved LDOS in Fig. 1, the edge state gives rise to peaks in the LDOS for energies $|ϵ|<{\mathrm{\Delta }}_0$. In the s-wave case (Fig. 2A), a single peak with maxima at zero energy develops with increasing magnetization. While in the s-wave case (Fig. 1B–D), the bound state dispersion only crosses zero energy once, the nodes in the d-wave case forces the bound state dispersion to cross zero energy three times (Fig. 1E–J). The increased curvature of the bound state dispersion caused by the d-wave nodes results in two LDOS peaks with maxima at finite energy in the $d_{x^2-y^2}$-wave (Fig. 2D) and $d_{\mathit{xy}}$-wave (Fig. 2G) case. These shift toward lower energies and increase in height with increasing (decreasing) magnetization in the $d_{x^2-y^2}$-wave ($d_{\mathit{xy}}$-wave) case, thus reflecting the behavior of the bound state dispersion. As the bound state dispersion becomes flatter, the contribution from energies $|ϵ|>\mathrm{\Delta }(\theta )$ is suppressed (see Fig. 1D, G, and H compared to Fig. 1B, E, and J, respectively). Thus, as the peaks resulting from the edge state increase in height, the states above the superconducting gap contribute less to the LDOS.

Fig. 2.

Magnetization dependence of the local density of states and optical conductivity. For s-wave (A–C), $d_{x^2-y^2}$-wave (D–F), and $d_{\mathit{xy}}$-wave (G–I) pairing, we consider the local density of states (LDOS) $\rho (x=0,ϵ)$ (A, D, and G) and the real part of the local optical conductivity $\mathcal{R}\text{e}[{\sigma }_{y,y}(x=0,\omega )]$ (B and C), (E and F), and (H and I) normalized by their normal-state values for ${\mathrm{\Delta }}_0=0$ at the QAHI/TSC interface (x = 0). For comparison, the black dashed curves represent the LDOS and local optical conductivity far inside the TSC ($x\gg {\xi }_0$), where ξ₀ is the superconducting coherence length given by $k_{\text{F}}{\xi }_0=\mu /{\mathrm{\Delta }}_0$. For the s-wave TSC, the zero bias peak in the LDOS increases and the peak in the local optical conductivity increase and shift toward zero energy when increasing the magnetization m_z. For the $d_{x^2-y^2}$-wave ($d_{\mathit{xy}}$-wave) TSC, the peaks in the LDOS and optical conductivity increases and shifts toward zero energy with increasing (decreasing) magnetization. We consider ${\mathrm{\Delta }}_0/\mu ={10}^{-3}\ll 1$, and energies $ϵ+i\delta$, where $\delta /{\mathrm{\Delta }}_0=5·{10}^{-3}$.

The edge state similarly results in a peak in the local optical conductivity at a finite energy below the optical gap $0<\omega <2{\mathrm{\Delta }}_0$. This was previously shown to be a distinct qualitative feature resulting from the dispersion of the chiral Majorana mode (27). In the s-wave (Fig. 2 B and C) and $d_{x^2-y^2}$-wave (Fig. 2 E and F) case, this peak increases in height and shifts toward lower energies as the magnetization increases. The peak height increases more rapidly with increasing magnetization in the $d_{x^2-y^2}$-wave case, as explained by the flatness of the dispersion of the edge state. At ω = 0, the small but finite value of the optical conductivity results from having a finite imaginary part of the energy $ϵ+i\delta$, but vanishes in the limit $\delta \to 0$ (SI Appendix) since the Majorana mode can contribute to the optical conductivity only at finite energies (27). Our results for the s-wave case are thus fully consistent with those presented in ref. 27. In the case of $d_{\mathit{xy}}$-wave pairing (Fig. 2 H and I), the optical conductivity is finite at zero energy even in the limit $\delta \to 0$. The finite value in the $d_{\mathit{xy}}$-wave case arises from contributions from the node at θ = 0. Contrary to the s-wave and $d_{x^2-y^2}$-wave cases, the peak value in the $d_{\mathit{xy}}$-wave case decreases and shifts toward higher energies as the magnetization increases due to the opposite behavior of the bound state dispersion in response to an increase in the magnetization m_z in the $d_{\mathit{xy}}$-wave case. Thus, large peaks in the local optical conductivity can be achieved at weaker magnetization.

Optical Conductivity for General Orientations of the d-Wave Nodes.

We next consider a general orientation of the d-wave nodes by studying the dependence of the LDOS and local optical conductivity on the angle φ of the positive d-wave lobe with respect to the interface normal $\mathit{x}$. The LDOS contains two pairs of peaks with maximum value at $\phi =(0,\pi /2)$ and $\phi =\pi /4$, respectively (Fig. 3A–C). At low magnetization, the peaks at $\phi =(0,\pi /2)$ are well separated and the peaks at $\phi =\pi /4$ dominate (A). As we increase the magnetization, the peaks at $\phi =\pi /4$ decrease and shift toward higher energies, while the peaks at $\phi =(0,\pi /2)$ shift toward zero energy and increase in height (C). As a result, the optical conductivity (Fig. 3D–F) has a peak at $\phi =\pi /4$ ($d_{\mathit{xy}}$-wave pairing) at low magnetization (D) and at $\phi =(0,\pi /2)$ ($d_{x^2-y^2}$-wave pairing) at higher magnetization (F), and never for intermediate angles.

Fig. 3.

Dependence of the local density of states and optical conductivity on the orientation of the d-wave nodes. We consider the local density of states (LDOS) $\rho (x=0,ϵ)$ (A–C) and the real part of the local optical conductivity $\mathcal{R}\text{e}[{\sigma }_{y,y}(x=0,\omega )]$ (D–F) normalized by their normal-state values when ${\mathrm{\Delta }}_0=0$ at the QAHI/TSC interface (x = 0) under rotation of the d-wave lobe angle φ with respect to the interface normal $\widehat{x}$ (see schematic). We consider three values of the magnetization m_z corresponding to (Fig. 1E–G) (φ = 0) and (Fig. 1H–J) ($\phi =\pi /4$), ${\mathrm{\Delta }}_0/\mu ={10}^{-3}\ll 1$, and energies $ϵ+i\delta$, where $\delta /{\mathrm{\Delta }}_0=5·{10}^{-3}$. The LDOS and optical conductivity are either maximal at $\phi =(0,\pi /2)$ ($d_{x^2-y^2}$-wave pairing) or at $\phi =\pi /4$ ($d_{\mathit{xy}}$-wave pairing).

Position Dependence of the Optical Conductivity.

So far, we have considered the optical conductivity exactly at the QAHI/TSC interface (x = 0). In experiments, the optical conductivity would however be measured by placing the MIM tip above the sample and moving it along the x axis across the QAHI/TSC interface, as indicated in Fig. 1A (34). We now consider how the optical conductivity varies away from the interface at a fixed magnetization (Fig. 4). In both the s-wave and d-wave TSC, the features below the optical gap ($\omega <2{\mathrm{\Delta }}_0$) decay over a length scale comparable to the superconducting coherence length ξ₀, here given by $k_{\text{F}}{\xi }_0=\mu /{\mathrm{\Delta }}_0$. Above the optical gap ($\omega >2{\mathrm{\Delta }}_0$), the behavior expected for $x\gg {\xi }_0$ is restored when moving away from the interface. In the s-wave case (Fig. 4A), the shifting of the gap edge toward lower energies leads to a characteristic double peak feature at intermediate distances from the interface where the low energy peak is still present and the original optical gap is partly restored. This feature is less prominent in the case of d-wave pairing (Fig. 4 B and C) due to the absence of a hard gap. When considering a finite spot size for the MIM measurement, the local optical conductivity in Fig. 4 is averaged over the spot size. The position-averaged optical conductivity is presented in SI Appendix. We find that while such an averaging reduces the peak associated with the Majorana edge mode compared to its local value at the interface, the peak remains measurable for a realistic spot size. While our model does not allow for determining the z axis dependence of the optical conductivity, the results are not expected to be significantly altered if the Majorana edge mode is not perfectly confined at the TI surface as long as the optical conductivity is measured from above, effectively summing up contributions from different values of z.

Fig. 4.

The optical conductivity as a function of the distance from the interface. We consider the real part of the optical conductivity $\mathcal{R}\text{e}[{\sigma }_{y,y}(x,\omega )]$ normalized by its normal-state value when ${\mathrm{\Delta }}_0=0$ as a function of the distance x from the QAHI/TSC interface for s-wave pairing (A), $d_{x^2-y^2}$-wave pairing (B), and $d_{\mathit{xy}}$-wave pairing (C). The corresponding magnetizations are given by $m_z/\mu =5.0$, $1.5$, and $1.0$, respectively. We consider ${\mathrm{\Delta }}_0/\mu ={10}^{-3}\ll 1$, and energies $ϵ+i\delta$, where $\delta /{\mathrm{\Delta }}_0=5·{10}^{-3}$. The features below the optical gap $\omega <2{\mathrm{\Delta }}_0$ decay monotonically over a length scale comparable to the superconducting coherence length ξ₀.

Temperature Dependence of the Optical Conductivity.

We finally consider how the optical conductivity behaves as the temperature increases (Fig. 5). Considering the limit $\omega \to 0$ (Fig. 5A–C), we find that while the optical conductivity always increases for $k_{\text{B}}T\ll {\mathrm{\Delta }}_0$, the behavior at higher temperatures can be nonmonotonic. For the s-wave and $d_{x^2-y^2}$-wave TSC (A and B), respectively], the nonmonotonic temperature dependence appears when the magnetization increases. For the $d_{\mathit{xy}}$-wave TSC (C), it instead appears when the magnetization decreases. This can be understood by studying the expression for the real part of the local optical conductivity (Eq. 31 in Materials and Methods) in the limit $\omega \to 0$, where it essentially consists of an energy integral over a temperature-dependent factor

$$\begin{array}{c}\hfill {\displaystyle \underset{\omega \to 0}{lim}}\frac{f_{\text{FD}}(ϵ-\omega )-f_{\text{FD}}(ϵ)}{\omega }=\frac{1}{k_{\text{B}}T}{e}^{ϵ/k_{\text{B}}T}{[f_{\text{FD}}(ϵ)]}^2,\end{array}$$ [6]

and a product of Green’s functions. Above, $f_{\text{FD}}(ϵ)$ is the Fermi–Dirac distribution and k_B is the Boltzmann constant. The product of Green’s functions can be assumed temperature independent when the maximum value of the superconducting gap ${\mathrm{\Delta }}_0$ proximity-induced onto the TI surface is much smaller than in the parent SC. In this case, the parent SC, and thus the proximity-induced ${\mathrm{\Delta }}_0$, are nearly unaffected by the $k_{\text{B}}T/{\mathrm{\Delta }}_0$ considered. The Green’s function products are symmetric in the energy ϵ and have coherence peaks at $ϵ=\pm {\mathrm{\Delta }}_0$. At the QAHI/TSC interface, they contain additional features for energies $0<ϵ<{\mathrm{\Delta }}_0$ similar to the peaks in the zero-temperature optical conductivity for $0<\omega <2{\mathrm{\Delta }}_0$ resulting from the edge state (Fig. 2B, E, and H). The temperature-dependent factor in Eq. 6 is peaked at zero energy with height $1/4k_{\text{B}}T$ and width $2\text{ln}(3+2\sqrt{2})k_{\text{B}}T$ at half of the peak height. It thus determines which features of the Green’s function products are included through the broadening of the peak as T increases. A nonmonotonic temperature dependence is possible when the features in the Green’s function products resulting from the edge state enter the peak width of the temperature-dependent factor at small but finite temperatures. A finite magnetization additionally alters the height of the coherence peaks at the QAHI/TSC interface, so that the optical conductivity does not necessarily approach the result for $x\gg {\xi }_0$ at higher temperatures.

Fig. 5.

Temperature dependence of the optical conductivity. We consider how the real part of the local optical conductivity $\mathcal{R}\text{e}[{\sigma }_{y,y}(x=0,\omega )]$ normalized by its normal-state value when ${\mathrm{\Delta }}_0=0$ at the QAHI/TSC interface (x = 0) varies as we increase the temperature T for zero energy (ω = 0) (A–C) and for energies below the optical gap ($\omega <2{\mathrm{\Delta }}_0$) (D–F). From Left to Right, we consider s-wave (A and D), $d_{x^2-y^2}$-wave (B and E), and $d_{\mathit{xy}}$-wave (C and F) pairing. The results in panel (D–F) correspond to magnetizations given by $m_z/\mu =5.0$, $1.5$, and $1.0$, respectively. For comparison, the black dashed curves represent the local optical conductivity far inside the TSC ($x\gg {\xi }_0$). We consider ${\mathrm{\Delta }}_0/\mu ={10}^{-3}\ll 1$, and energies $ϵ+i\delta$, where $\delta /{\mathrm{\Delta }}_0=5·{10}^{-3}$. In the s-wave and $d_{x^2-y^2}$-wave ($d_{\mathit{xy}}$-wave) case, the temperature dependence becomes nonmonotonic when the magnetization increases (decreases).

In Fig. 5D–F, we consider the temperature dependence for all energies ω below the optical gap ($\omega <2{\mathrm{\Delta }}_0$) at a given magnetization. When the behavior of the s-wave TSC (D) is nonmonotonic in the limit $\omega \to 0$, the optical conductivity peaks at finite energy and finite temperature. For the $d_{x^2-y^2}$-wave TSC (E), the behavior is even more nontrivial with a saddle point separating peaks at finite energy and finite temperature. While the optical conductivity of the s-wave and $d_{x^2-y^2}$-wave TSC approaches a small value (zero in the limit $\delta \to 0$, see SI Appendix) when $(k_{\text{B}}T,\omega )\to 0$, the optical conductivity of the $d_{\mathit{xy}}$-wave TSC (F) approaches a finite value due to the nodal states. This causes the features of the $d_{\mathit{xy}}$-wave TSC to be smeared out.

Discussion

The recent advances in microwave impedance microscopy of topological edge states (33, 34) open a promising new avenue for probing the local optical conductivity of the Majorana edge state at the QAHI/TSC interface (27). We have presented a series of features, characteristic of the chiral Majorana mode due to its bound state dispersion, accessible via this technique. The qualitative behavior of the local optical conductivity as a function of energy, magnetization, and temperature depends on the symmetry of the superconducting order parameter through contributions from additional nodal states that enhance the flatness of the bound state dispersion. Qualitatively distinct behavior is expected when comparing the s-wave and $d_{x^2-y^2}$-wave cases to the $d_{\mathit{xy}}$-wave case. When increasing or decreasing the magnetization, respectively, signatures of the Majorana edge state appear in the form of a peak below the optical gap and a nonmonotonic temperature dependence. Distinct and tunable signatures also appear in the LDOS accessible via scanning tunneling spectroscopy measurements (58–61).

While the Majorana edge mode is topologically protected from backward-scattering due to its chiral nature (36), our results are more relevant in experimentally achievable (46) clean systems since the d-wave pairing is not protected by Anderson’s theorem (62) and thus not robust to disorder. The chiral Majorana edge mode considered here is dispersive and one-dimensional and runs along the entire length of the interface. It can therefore be distinguished from trivial Andreev bound states close to impurities in the d-wave superconductor. These may give rise to nondispersive zero-energy states that would show up as a peak in STM measurements near the localized impurity (20), but cannot provide the conductive channel along the interface that gives rise to the finite optical conductivity. The optical conductivity signal can moreover be distinguished from that of flat Andreev bound states at the interface between a $d_{\mathit{xy}}$-wave SC and a FI, since these are nondispersive zero-energy states (54, 55). It is the Majorana edge mode’s dispersive nature that gives rise to the characteristic optical conductivity peak at finite energy (27).

Note that we assume the limit of small tip in LDOS and small spot of the optical excitation in ${\sigma }_{y,y}(\omega )$. The former is usually applicable for the scanning tunneling spectroscopy, while the latter is not trivial. Recent advances of the MIM experiment achieved the ultrahigh spatial resolution of 5 nm (32), but this is still nearly 10 times larger than the lattice constant. Therefore, the momentum transfer associated with the optical transition is of the order of the inverse of 5 nm. However, we are interested in the low energy region below the superconducting gap, and the relevant length scale is the coherence length of the superconductivity, which is larger than 5 nm.

In addition to the position averaging of the optical signal over the spot size, contributions to the optical signal can be picked up from the superconductor deposited on top of the TI surface due to its finite skin depth, and in the d-wave case, due to nodal states and disorder. However, d-wave superconductivity in Bi₂Sr₂CaCu₂O_8−δ has been measured down to the monolayer limit with an approximate thickness of 2 nm (63), well below the zero-frequency zero-temperature skin depth which is of the order of 100 nm. Moreover, taking into account contributions from a disordered d-wave superconductor on top of the TI, the conductance peak resulting from the edge states gives the dominating contribution to the total signal for a superconducting film up to the order of hundred atomic layers, and a measurable signal up to the order of ${10}^4$ atomic layers relevant to, e.g., the experiments in ref. 46. Further details on how our calculation is relevant to experiments are discussed in SI Appendix.

The advantage associated with the qualitative nature of these results lies in their tunability and robustness with respect to measurement noise. Previous measurements using scanning tunneling microscopy relied on quantitative measures through, e.g., a quantized conductance (36–38). Since an exact quantization can only be achieved in theory, it is challenging to distinguish whether an apparent quantization measured in the lab is of trivial or topological origin (19, 39, 40). The rich behavior of the local optical conductivity and density of states presented here—distinct between the different symmetries of the superconducting order parameter—lays a broader foundation to account for nontrivial behavior through magnetization and temperature dependencies specific to the Majorana edge mode.

Materials and Methods

Wave Functions.

sec:MaterialsAndMethodsTo construct the McMillan Green’s function (24, 48, 49, 53, 54), we first construct the ordinary wave functions ${\mathrm{\Psi }}_j(x,y)$ of the four possible scattering processes ($j=1,2,3,4$) of an electron-like and a hole-like particle scattering from the nonsuperconducting nonmagnetic region ($x<-d$) to the superconducting region (x > 0), and vice versa. We also construct the conjugated wave functions ${\tilde{\mathrm{\Psi }}}_j(x,y)$ of the reverse scattering processes. The wave functions are constructed from the eigenvectors obtained by diagonalizing the Hamiltonian in Eqs. 2–4. Their momenta along $\widehat{x}$ are derived from the corresponding eigenenergies. We assume the momentum $k_y=(\mu /A)sin(\theta )$ along the interface to be conserved during the scattering process so that

$$\begin{array}{c}\hfill \stackrel{(\sim )}{\mathrm{\Psi }}(x,y)=\stackrel{(\sim )}{\mathrm{\Psi }}(x)e^{+(-)i{k}_{y}y}\end{array}$$ [7]

In the superconducting region (x > 0), the wave functions are given by

$$\begin{array}{cc}\hfill {\stackrel{(\sim )}{\mathrm{\Psi }}}_j(x)=& {\stackrel{(\sim )}{\mathrm{\Psi }}}_j^{\text{SC,in}}(x)+{\stackrel{(\sim )}{a}}_j{\stackrel{(\sim )}{\mathrm{\Psi }}}_{\text{eR}}^{\text{SC}}{e}^{i{k}_x^{\text{SCe}+(-)}x}+{\stackrel{(\sim )}{b}}_j{\stackrel{(\sim )}{\mathrm{\Psi }}}_{\text{hR}}^{\text{SC}}{e}^{-i{k}_x^{\text{SCh}-(+)}x},\hfill \end{array}$$ [8]

with incoming wave functions

$$\begin{array}{cc}\hfill {\stackrel{(\sim )}{\mathrm{\Psi }}}_1^{\text{SC,in}}(x)={\stackrel{(\sim )}{\mathrm{\Psi }}}_2^{\text{SC,in}}(x)=0,& {\stackrel{(\sim )}{\mathrm{\Psi }}}_3^{\text{SC,in}}(x)={\stackrel{(\sim )}{\mathrm{\Psi }}}_{\text{eL}}^{\text{SC}}{e}^{-i{k}_x^{\text{SCe}-(+)}x},\hfill \\ \hfill & {\stackrel{(\sim )}{\mathrm{\Psi }}}_4^{\text{SC,in}}(x)={\stackrel{(\sim )}{\mathrm{\Psi }}}_{\text{hL}}^{\text{SC}}{e}^{i{k}_x^{\text{SCh}+(-)}x},\hfill \end{array}$$ [9]

where a_j and b_j are coefficients, the wave vectors are given by

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_{\text{eR(L)}}^{\text{SC}}& ={[1+(-)e^{+(-)i\theta }-(+){\mathrm{\Gamma }}_{+(-)}{e}^{+(-)i\theta }{\mathrm{\Gamma }}_{+(-)}]}^T,\hfill \end{array}$$ [10]

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_{\text{hL(R)}}^{\text{SC}}& ={[{\mathrm{\Gamma }}_{+(-)}+(-){\mathrm{\Gamma }}_{+(-)}{e}^{+(-)i\theta }-(+)e^{+(-)i\theta }1]}^T,\hfill \end{array}$$ [11]

$$\begin{array}{cc}\hfill {\tilde{\mathrm{\Psi }}}_{\text{eR(L)}}^{\text{SC}}& ={[1-(+)e^{+(-)i\theta }+(-){\mathrm{\Gamma }}_{-(+)}{e}^{+(-)i\theta }{\mathrm{\Gamma }}_{-(+)}]}^T,\hfill \end{array}$$ [12]

$$\begin{array}{cc}\hfill {\tilde{\mathrm{\Psi }}}_{\text{hL(R)}}^{\text{SC}}& ={[{\mathrm{\Gamma }}_{-(+)}-(+){\mathrm{\Gamma }}_{-(+)}{e}^{+(-)i\theta }+(-)e^{+(-)i\theta }1]}^T,\hfill \end{array}$$ [13]

and

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_{+(-)}& =\frac{{\mathrm{\Delta }}_{0}cos \{2[\theta -(+)\phi ]\}}{ϵ+\sqrt{{ϵ}^2-{|{\mathrm{\Delta }}_{0}cos\{2[\theta -(+)\phi ]\}|}^2}}.\hfill \end{array}$$ [14]

The momenta $k_x^{\text{SCe(h)}\pm }$ of electron-like (hole-like) particles are given by

$$\begin{array}{cc}\hfill A{k}_x^{\text{SCe(h)}\pm }& =\sqrt{{\left\{\mu +(-)\sqrt{{ϵ}^2-{\left|{\mathrm{\Delta }}_{0}cos\left[2\left(\theta \mp \phi \right)\right]\right|}^2}\right\}}^2-A^2{k}_y^2}.\hfill \end{array}$$ [15]

In this region, we have assumed $0<(ϵ,{\mathrm{\Delta }}_0)\ll \mu$. In the ferromagnetic region ($-d<x<0$), the wave functions are given by

$$\begin{array}{cc}\hfill {\stackrel{(\sim )}{\mathrm{\Psi }}}_j^{\text{FI}}& =\left({\stackrel{(\sim )}{c}}_j{\stackrel{(\sim )}{\mathrm{\Psi }}}_{\text{eL}}^{\text{FI}}+{\stackrel{(\sim )}{d}}_j{\stackrel{(\sim )}{\mathrm{\Psi }}}_{\text{hR}}^{\text{FI}}\right)e^{{\kappa }_x^{\text{FI}}x}\hfill \\ \hfill & +\left({\stackrel{(\sim )}{e}}_j{\mathrm{\Psi }}_{\text{eR}}^{\text{FI}}+{\stackrel{(\sim )}{f}}_j{\stackrel{(\sim )}{\mathrm{\Psi }}}_{\text{hL}}^{\text{FI}}\right)e^{-{\kappa }_x^{\text{FI}}x},\hfill \end{array}$$ [16]

where c_j, d_j, e_j, and f_j are coefficients, the wave vectors are given by

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_{\text{eR(L)}}^{\text{FI}}& ={[+(-)i{\gamma }^{+(-)1}\mathrm{1\hspace{0.17em}\hspace{0.17em}0\hspace{0.17em}\hspace{0.17em}0}]}^T,\hfill \end{array}$$ [17]

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_{\text{hL(R)}}^{\text{FI}}& ={[\mathrm{0\hspace{0.17em}\hspace{0.17em}0}-(+)i{\gamma }^{-(+)1}1]}^T,\hfill \end{array}$$ [18]

$$\begin{array}{cc}\hfill {\tilde{\mathrm{\Psi }}}_{\text{eR(L)}}^{\text{FI}}& ={[-(+)i{\gamma }^{+(-)1}\mathrm{1\hspace{0.17em}\hspace{0.17em}0\hspace{0.17em}\hspace{0.17em}0}]}^T,\hfill \end{array}$$ [19]

$$\begin{array}{cc}\hfill {\tilde{\mathrm{\Psi }}}_{\text{hL(R)}}^{\text{FI}}& ={[0\hspace{0.17em}\hspace{0.17em}0\hspace{0.17em}\hspace{0.17em}+(-)i{\gamma }^{-(+)1}1]}^T,\hfill \end{array}$$ [20]

with $\gamma =-A({\kappa }_x^{\text{FI}}-k_y)/m_z$, and the momentum is given by $A{\kappa }_x^{\text{FI}}=\sqrt{m_z^2+{(A{k}_y)}^2}$. In this region, we have assumed μ = 0 and $0<ϵ\ll |m_z|$. In the nonsuperconducting nonmagnetic region ($x<-d$), the wave functions are given by

$$\begin{array}{c}\hfill {\stackrel{(\sim )}{\mathrm{\Psi }}}_j(x)={\stackrel{(\sim )}{\mathrm{\Psi }}}_j^{\text{N,in}}(x)+{\stackrel{(\sim )}{g}}_j{\stackrel{(\sim )}{\mathrm{\Psi }}}_{\text{eL}}^{\text{N}}{e}^{-i{k}_x^{\text{N}}x}+{\stackrel{(\sim )}{h}}_j{\stackrel{(\sim )}{\mathrm{\Psi }}}_{\text{hL}}^{\text{N}}{e}^{i{k}_x^{\text{N}}x},\end{array}$$ [21]

with incoming wave functions

$$\begin{array}{cc}\hfill {\stackrel{(\sim )}{\mathrm{\Psi }}}_1^{\text{N,in}}(x)& ={\stackrel{(\sim )}{\mathrm{\Psi }}}_{\text{eR}}^{\text{N}}{e}^{i{k}_x^{\text{N}}x},\hfill \\ \hfill {\stackrel{(\sim )}{\mathrm{\Psi }}}_2^{\text{N,in}}(x)& ={\stackrel{(\sim )}{\mathrm{\Psi }}}_{\text{hR}}^{\text{N}}{e}^{-i{k}_x^{\text{N}}x},{\stackrel{(\sim )}{\mathrm{\Psi }}}_3^{\text{N,in}}(x)={\stackrel{(\sim )}{\mathrm{\Psi }}}_4^{\text{N,in}}=0,\hfill \end{array}$$ [22]

where g_j and h_j are coefficients, and the wave vectors are obtained by setting ${\mathrm{\Gamma }}_{+(-)}=0$ in Eqs. 10–13. The momentum is given by $A{k}_x^{\text{N}}=\sqrt{{\mu }^2-A^2{k}_y^2}$. The coefficients are evaluated by imposing continuity of the ordinary and conjugated wave function at $x=-d$ and x = 0.

The McMillan Green’s Function.

From the ordinary and conjugated wave functions, we can construct the retarded McMillan Green’s function (48, 49, 53)

$$\begin{array}{c}\hfill G^{\text{R}}(x,x^{\prime })=\left\{\begin{array}{c}{\alpha }_1{\mathrm{\Psi }}_1(x){\tilde{\mathrm{\Psi }}}_3^T(x^{\prime })+{\alpha }_2{\mathrm{\Psi }}_1(x){\tilde{\mathrm{\Psi }}}_4^T(x^{\prime })\hfill \\ \hspace{1em}+{\alpha }_3{\mathrm{\Psi }}_2(x){\tilde{\mathrm{\Psi }}}_3^T(x^{\prime })+{\alpha }_4{\mathrm{\Psi }}_2(x){\tilde{\mathrm{\Psi }}}_4^T(x^{\prime }),x>x^{\prime },\hfill \\ {\beta }_1{\mathrm{\Psi }}_3(x){\tilde{\mathrm{\Psi }}}_1^T(x^{\prime })+{\beta }_2{\mathrm{\Psi }}_4(x){\tilde{\mathrm{\Psi }}}_1^T(x^{\prime })\hfill \\ \hspace{1em}+{\beta }_3{\mathrm{\Psi }}_3(x){\tilde{\mathrm{\Psi }}}_2^T(x^{\prime })+{\beta }_4{\mathrm{\Psi }}_4(x){\tilde{\mathrm{\Psi }}}_2^T(x^{\prime }),x<x^{\prime }.\hfill \end{array}\right.\end{array}$$ [23]

The coefficients α_j and β_j are obtained from the boundary condition

$$\begin{array}{c}\hfill {\displaystyle \underset{\delta \to 0^{+}}{lim}}{G}^{\text{R}}(x^{\prime }+\delta ,x^{\prime })-G^{\text{R}}(x^{\prime }-\delta ,x^{\prime })=-i{\tau }_0{\sigma }_x/A,\end{array}$$ [24]

where τ₀ is the unit matrix in Nambu space. The resulting retarded Green’s function is a $4\times 4$ matrix

$$\begin{array}{cc}& G^{\text{R}}(x,x^{\prime },\theta ,ϵ+i\delta )=\left(\begin{array}{cc}{g}^{\text{R}}(x,x^{\prime },\theta ,ϵ+i\delta )& f^{\text{R}}(x,x^{\prime },\theta ,ϵ+i\delta )\\ \underset{\_}{f^{\text{R}}}(x,x^{\prime },\theta ,ϵ+i\delta )& \underset{\_}{g^{\text{R}}}(x,x^{\prime },\theta ,ϵ+i\delta )\end{array}\right)\hfill \end{array}$$ [25]

written in terms of $2\times 2$ ordinary and anomalous retarded Green’s functions (see SI Appendix for analytic expressions). The incidence angle θ and energy $ϵ+i\delta$, where δ > 0 is a small parameter, was left out until now for simplicity of notation. We neglect terms that oscillate over a length scale much smaller than the superconducting coherence length. Since we assumed ϵ > 0 when constructing the wave functions, negative energies are accessed from the lower elements via the relation

$$\begin{array}{cc}\hfill g^{\text{R}}(x,x^{\prime },\theta ,-ϵ+i\delta )& =-{[\underset{\_}{g^{\text{R}}}(x,x^{\prime },-\theta ,ϵ+i\delta )]}^{\ast },\hfill \end{array}$$ [26]

$$\begin{array}{cc}\hfill f^{\text{R}}(x,x^{\prime },\theta ,-ϵ+i\delta )& =-{[\underset{\_}{f^{\text{R}}}(x,x^{\prime },-\theta ,ϵ+i\delta )]}^{\ast }.\hfill \end{array}$$ [27]

To find the advanced Green’s function, we similarly evaluate

$$\begin{array}{cc}\hfill g_{\alpha ,\beta }^{\text{A}}(x,x^{\prime },\theta ,ϵ-i\delta )& ={\left[g_{\beta ,\alpha }^{\text{R}}(x^{\prime },x,\theta ,ϵ+i\delta )\right]}^{\ast },\hfill \end{array}$$ [28]

$$\begin{array}{cc}\hfill f_{\alpha ,\beta }^{\text{A}}(x,x^{\prime },\theta ,ϵ-i\delta )& =-f_{\beta ,\alpha }^{\text{R}}(x^{\prime },x,-\theta ,-ϵ+i\delta ).\hfill \end{array}$$ [29]

The Local Density of States and Optical Conductivity.

The LDOS is given by

$$\begin{array}{c}\hfill \rho (x,ϵ)=-\frac{1}{\pi }{\int }_{-\pi /2}^{\pi /2}d\theta cos(\theta )\mathfrak{I}\mathrm{m}\{\text{Tr}[g^{\text{R}}(x,x,\theta ,ϵ)]\}.\end{array}$$ [30]

In the normal-state (${\mathrm{\Delta }}_0=0$), $\rho (x,ϵ)=1/A$ is constant. The real part of the optical conductivity (see SI Appendix for its derivation) is given by

$$\begin{array}{c}\mathrm{Re}[{\sigma }_{i,j}(x\ne x^{\prime },\omega +i\delta )]=\hfill \\ \mathrm{Re}[\frac{e^2\mu A}{8\omega }{\displaystyle \sum _{\alpha ,\beta }}{\displaystyle \sum _{{\alpha }^{\prime },{\beta }^{\prime }}}{\sigma }_{\alpha ,\beta }^i{\sigma }_{{\alpha }^{\prime },{\beta }^{\prime }}^j\frac{1}{\pi }{\int }_{-\infty }^{\infty }dϵ[f_{\text{FD}}(ϵ)-f_{\text{FD}}(ϵ-\omega )]\hfill \\ \frac{1}{\pi }{\int }_{-\pi /2}^{\pi /2}d\theta cos(\theta )\frac{1}{\pi }{\int }_{-\pi /2}^{\pi /2}d{\theta }^{\prime }cos({\theta }^{\prime })\hfill \\ \{[g_{\beta ,{\alpha }^{\prime }}^{\text{R}}(x,x^{\prime },\theta ,ϵ)-g_{\beta ,{\alpha }^{\prime }}^{\text{A}}(x,x^{\prime },\theta ,ϵ)]\hfill \\ [g_{{\beta }^{\prime },\alpha }^{\text{R}}(x^{\prime },x,{\theta }^{\prime },ϵ-\omega )-g_{{\beta }^{\prime },\alpha }^{\text{A}}(x^{\prime },x,{\theta }^{\prime },ϵ-\omega )]\hfill \\ +[f_{\beta ,{\beta }^{\prime }}^{\text{R}}(x,x^{\prime },\theta ,ϵ)-f_{\beta ,{\beta }^{\prime }}^{\text{A}}(x,x^{\prime },\theta ,ϵ)]\hfill \\ [f_{{\alpha }^{\prime },\alpha }^{\text{R}}(x^{\prime },x,{\theta }^{\prime },ϵ-\omega )-f_{{\alpha }^{\prime },\alpha }^{\text{A}}(x^{\prime },x,{\theta }^{\prime },ϵ-\omega ){]}^{†}\}],\hfill \end{array}$$ [31]

where we set $x^{\prime }=x+0^{+}$ to evaluate the local value.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix.