Superconducting proximity effect in three-dimensional topological insulators in the presence of a magnetic field

The proximity-induced pair potential in a topological insulator–superconductor hybrid features an interesting superposition of a conventional spin-singlet component from the superconductor and a spin-triplet one induced by the surface state of the topological insulator. This singlet-triplet superposition can be altered by the presence of a magnetic field. We study the interplay between topological order and superconducting correlations performing a symmetry analysis of the induced pair potential, using Green functions techniques to theoretically describe ballistic junctions between superconductors and topological insulators under magnetic fields. We relate a change in the conductance from a gapped profile into one with a zero-energy peak with the transition into a topologically nontrivial regime where the odd-frequency triplet pairing becomes the dominant component in the pair potential. The nontrivial regime, which provides a signature of odd-frequency triplet superconductivity, is reached for an out-of-plane effective magnetization with strength comparable to the chemical potential of the superconductor or for an in-plane one, parallel to the normal-superconductor interface, with strength of the order of the superconducting gap. Strikingly, in the latter case, a misalignment with the interface yields an asymmetry with the energy in the conductance unless the total contribution of the topological surface state is considered.

Figure 1

(a) Sketch of a normal-superconductor junction on the surface state of a topological insulator (regions $N$ and $S$, respectively) with an intermediate region $I$ modeled as a square potential barrier (see potential profile). The angle of incidence $\theta$ is measured with respect to the $x$ direction. (b) An out-of-plane magnetization $m_z$ is induced in the system by proximity from a ferromagnetic insulator. (c) An external magnetic field, stemming from the vector potential $\mathbf{A}$ (blue arrow) with arbitrary orientation, also generates an effective magnetization (red arrows). (d) Symmetry classification of the anomalous Green function $F_{\sigma {\sigma }^{\prime }}(\omega ,\mathbf{k})$. With respect to the exchange of the time (frequency), spin, and spatial (momentum) coordinates, the Green function can be even (E) or odd (O). The spin part can be divided into a singlet (S) or three triplet (T) components. The Pauli principle allows for four different combinations; using a “frequency/spin/momentum” notation: ESE, OSO, ETO, and OTE.

Figure 2

(a) Evolution of $\rho (E=0)$, angle averaged $|F_0|$, and $F_t$ with the distance inside the superconducting region from the interface at $x=d$ at zero magnetization and $E=0$. (b) Same quantities as before as a function of $m_z$, for $E=0$ and $x=d$. The green solid and dashed lines correspond to the two triplet polarizations $|F_{1,2}|\propto (\uparrow \uparrow \mp \downarrow \downarrow )$, respectively. (c), (d) LDOS as a function of the energy for several values of $m_z$ calculated at the IS interface $x=d$ (c), and inside the superconducting region, where $x\to \infty$ (d). For all plots, we set ${\mu }_{\text{N}}=10\mu , {\mu }_{\text{I}}=20\mu , {\mu }_{\text{S}}=\mu ={10}^3\mathrm{\Delta }$, and $d=0.1(v_F/\mu )$.

Figure 3

(a) Energy-momentum plot of the low-energy bands inside the superconducting region for several values of $m_{\parallel }$. (b) Bulk LDOS of the superconducting region of the NIS junction as a function of the excitation energy $E$ for different $m_{\parallel }$. (c), (d) LDOS at the interface $x=d$ with ${\mu }_{\text{I}}/\mu =10$ and $d=0.1(v_F/\mu )$ as a function of the excitation energy $E$ with different in-plane magnetizations. We show $\alpha /\pi =0,0.5$ for (c) and (d), respectively. For (b)–(d) we set $\mu ={10}^3\mathrm{\Delta }$.

Figure 4

Plots of $\rho (E=0), |F_0(E=0)|$, and $F_t(E=0)$ as a function of $m_{\parallel }$. (a) Bulk results for $x\to \infty$. (b)–(d) Results at the interface with $x=d$ for different orientations of the effective magnetization: $\alpha /\pi =0,0.25,0.5$ for (b)–(d), respectively. The solid and dashed green lines correspond to the polarizations $\uparrow \uparrow -\downarrow \downarrow$ and $\uparrow \uparrow +\downarrow \downarrow$ of the OTE term, respectively. For all plots, ${\mu }_{\text{N}}={\mu }_{\text{S}}=\mu ={10}^3\mathrm{\Delta }, {\mu }_{\text{I}}/\mu =10$, and $d=0.1(v_F/\mu )$.

Figure 5

Conductance as a function of the excitation energy for several effective magnetizations. For each case we include a sketch of the NIS junction displaying the orientation of the effective magnetization as a blue arrow. (a) A perpendicular magnetization is finite in the whole NIS junction with ${\mu }_{\text{N}}=10\mu , {\mu }_{\text{I}}=20\mu$, and ${\mu }_{\text{S}}=\mu$. The thin gray lines are taken at intervals $0.1\mu$ from $m_z=0$. (b), (c) In-plane magnetization finite in the whole junction oriented perpendicular (b) or parallel (c) to the NIS interfaces. For both cases, ${\mu }_{\text{N}}={\mu }_{\text{S}}=\mu$ and ${\mu }_{\text{I}}/\mu =10$. (d) With the same parameters as (b), superposition of the conductance for both surfaces, sketched on the left. The black solid line corresponds to the $m_{\parallel }/\mathrm{\Delta }=1.6$ case of (b) while the dashed one is the conductance of the other surface with the same parameters. In all plots, $d=0.1(v_F/\mu )$ and $\mu ={10}^3\mathrm{\Delta }$.

Figure 6

LDOS inside the superconducting region normalized to the bulk density of states in the normal region ${\rho }_N$ as a function of both the excitation energy and the distance from the interface ($x=d$). (a) Out-of-plane magnetization. From left to right, $m_z/\mu =0,0.9,1.1$. For the three panels, ${\mu }_{\text{N}}=10\mu$ and ${\mu }_{\text{I}}=20\mu$. (b) In-plane magnetization perpendicular to the NIS interface ($\alpha =0$). From left to right, $m_{\parallel }/\mathrm{\Delta }=0.4,1.0,1.6$. (c) In-plane magnetization parallel to the NIS interface ($\alpha =\pi /2$). From left to right, $m_{\parallel }/\mathrm{\Delta }=0.4,1.0,1.6$. For (b) and (c), ${\mu }_{\text{N}}=\mu$ and ${\mu }_{\text{I}}=10\mu$. For all plots, $\mu ={10}^3\mathrm{\Delta }$ and $d=0.1(v_F/\mu )$.