Edge-induced pairing states in a Josephson junction through a spin-polarized quantum anomalous Hall insulator

First, we consider how the chiral edge mode is affected by the superconducting proximity effect. The chiral edge mode is known to be robust against the proximity effect in the sense that the linear dispersion cannot be gapped, which can be seen in the spectrum of the chiral edge mode. Fig. 2 (a) represents the change of the energy spectrum of a S/QAHI/S Josephson junction as a function of the phase difference between the left and right superconductors.

The energy spectrum within the pair potential $|\Delta_{\text{R(L)}}|$ is equally spaced, and looks almost insensitive to the phase difference. Since the chiral edge mode extends along the whole boundary of the QAHI [Fig. 2 (c)], the inter-level spacing is determined by the perimeter of the QAHI. This result is in stark contrast to the energy spectrum of the Josephson junction via a QSHI (S/QSHI/S) [Fig. 2 (b)], in which the proximity effect induces a gap of $\sim 0.4|\Delta|$, and the energy spectrum of the bound states is largely affected by the change of the phase difference. The ingap modes within the induced gap are bound to the top and bottom boundaries where the proximity effect is absent [Fig. 2 (d)], and thus the interlevel-spacing is determined by the length of the QSHI (or equivalently, the distance between two superconductors).

From the perspective of the Andreev reflection, the chirality of a QAHI edge prohibits a hole to be reflected locally, that is, a spin-up electron flowing along the bottom boundary is reflected as a spin-up hole on the top boundary [Fig. 2 (c)], which is known as the crossed Andreev reflection. On the other hand, since a QSHI has spin-up and spin-down edge modes flowing in the opposite direction, the Andreev reflection occurs in the ordinary way, that is, separately in the bottom and top boundaries [Fig. 2 (d)], which is consistent with the gapped interface by the induced pair potential.

When the equal-spin crossed Andreev reflection occurs in a QAHI, triplet pairs are induced along the boundary. The amount of the pairing states is measured by the anomalous part of the Green’s function calculated by the inverse of $\omega-\mathcal{H}\pm i\delta$, where $\delta=|\Delta_{\text{R/L}}|/100$ is used in the following. The anomalous Green’s function is exponentially small inside the gapped bulk of the QAHI, and the only non-vanishing components in the QAHI are those between an spin-up electron and hole along the boundary. In the following, only the $s$-orbital part is focused since two orbitals are qualitatively equivalent. The equal-spin triplet anomalous Green’s function is decomposed into symmetric and anti-symmetric parts with respect to the sign change of the frequency $\omega\to-\omega$ and the permutation of the positions as [4]

where $F^{\text{R/A}}_{\bm{r}\bm{r}^{\prime}}$ is the retarded (advanced) anomalous Green’s function between two points $\bm{r}$ and $\bm{r}^{\prime}$. The real-, imaginary-part, and the absolute value of even-parity components at the same site $F^{\text{ESE/OTE}}_{\bm{r}\bm{r}}$ and odd-parity components between neighboring sites $F^{\text{OSO/ETO}}_{\bm{r}\bm{r}+\hat{e}_{x}}$ along the junction boundary are shown for the chemical potential $\mu/|\Delta|=0$ and $0.6$, and the frequency $\omega/|\Delta|=0.6$ in Fig. 3 (a).

As can be seen from the figure, the absolute value is almost constant while the real- and imaginary part of the anomalous Green’s function have spatial oscillation. The periodicity of the oscillation coincides with $2\pi/2k_{F}(=\pi v/\mu)$, where $v$ is the Fermi velocity of the chiral edge mode and in our model the same as the hopping amplitude $t_{s}=t_{p}=1$. This is an evidence that the Cooper pair has momentum $2k_{F}$, that is, two electrons with both the Fermi momentum $k_{F}$ are paired, which has been pointed out in [47] for singlet pairings in a $\nu=2$ spin-degenerate quantum Hall system. The finite momentum pairing is common to pairings in S/ferromagnet/S junctions [67, 68], while the alternation of 0 and $\pi$ junctions by the length of the weak link cannot be seen in our case. The presence of both even- and odd-frequency components is consistent with the breaking of translational symmetry by the finite-momentum pairing, which transforms the ETO into OTE component and vice versa.

For the case of the S/QSHI/S junction, the anomalous Green’s function along the boundary is also shown in Fig. 3 (b). There are four non-vanishing components away from the interface, that is, ESE and OSO components, and opposite-spin ETO and OTE components. The absolute value shows a stationary wave between two superconductors, and its period is approximately inversely proportional to the frequency $\omega$, but not dependent on the Fermi momentum as in the case of the S/QAHI/S junction. This result is consistent with the fact that electrons in the QSHI boundary make pairs in the conventional way, that is, a spin-up electron with the momentum $k_{F}$ makes a pair with a spin-down electron with the momentum $-k_{F}$.

Since the crossed Andreev reflection at the QAHI/S interface occurs between the top and bottom boundaries [Fig. 1 (b)], electrons and holes have a correlation across the boundaries in a nonlocal way. The nonlocal anomalous Green’s function in a S/QAHI/S junction between a fixed point $\bm{r}_{0}$ on the bottom boundary and a point $\bm{r}$ along the top and bottom boundaries is shown in Fig. 4 (a) for $\mu/|\Delta|=0$ and $\omega/|\Delta|=0.6$.

The anomalous Green’s function is finite even at the opposite boundary ($\bm{r}$ on the top boundary). When $\mu=0$, the anomalous Green’s function $F_{\bm{r}\bm{r}_{0}}$ is a periodic function of $\bm{r}$, whose period is inversely proportional to the frequency $\omega$. We have also examined that the nonlocal correlations persist in the presence of disorder due to the robustness of the chiral edge mode (not shown in this paper).

Fig. 4 (b) is the nonlocal anomalous Green’s function in a S/QSHI/S junction for $\omega/|\Delta|=0.2$, which is below the induced gap ($\sim 0.4|\Delta|$) at the QSHI/S interface, and $\mu/|\Delta|=0.6$. The anomalous Green’s function extends within the same boundary since the top and bottom boundaries are separated by the gapped interface regions. This result is consistent with an intuitive picture given in Fig. 2 (c) and (d).

Next, we consider the DC Josephson effect induced by the phase difference $\varphi$ between the pair potential of two superconductors. When the absolute value of the pair potential is the same ($|\Delta_{\text{R}}|=|\Delta_{\text{L}}|$), the Josephson current is estimated from the free energy $F$ of the junction by $I=(2e/\hbar)dF/d\varphi$ [69], and in the limit of vanishing temperature $T\to 0$, the expression is simplified as $I=(2e/\hbar)dE/d\varphi$, where $E=\sum_{\epsilon_{n}<0}\epsilon_{n}$ is the ground-state energy.

When the Josephson current is carried by the equal-spin triplet pairs, the equilibrium phase difference where the ground state energy is minimized is continuously dependent on the angle formed by the direction of the exchange field of the bulk, left, and right interfaces [Fig.5 (a)] [65, 11, 66, 14, 16]. The equilibrium phase difference is either 0 or $\pi$ when the three directions are coplanar, and it is continuously changed by $2\pi$ when the direction of the exchange field of one of the interface is rotated around the bulk one. The azimuth-angle dependence of the equilibrium phase difference in junctions with a QAHI length $L=12$ and $30$ is shown in Figs. 5 (b).

For $L=30$, the ground state energy and the Josephson current as a function of the phase difference $\varphi$ are shown in Figs. 5 (c) and (d), respectively, for the azimuth angle of the left interface $\phi_{L}=0,\pi/4,\pi/2,3\pi/4,$ and $\pi$. The current-phase relation is almost always sinusoidal: $I=I_{c}\sin(\varphi-\varphi_{0})$, where $I_{c}$ is the critical current and $\varphi_{0}$ is the equilibrium phase difference [70, 10]. Exceptional cases occur when accidental near-zero-energy modes cross $\epsilon=0$ during the change of the phase difference from $\varphi=0$ to $2\pi$. However, for a long junction these exceptions can be neglected since the variation of each eigenmode near $\epsilon=0$ is far smaller than the inter-level spacing [see Fig. 2 (a)]. Thus the following calculations are performed under the assumption of the sinusoidal current-phase relation to reduce the computational cost. When $L=30$, the equilibrium phase $\varphi_{0}$ varies continuously from $-\pi$ to $\pi$ during the change of the azimuth angle $\phi_{\text{L}}$ of the left interface from $0$ to $2\pi$. On the other hand, for $L=12$, the equilibrium phase difference is insensitive to the direction of the the interface exchange field. This result indicates that pairing states carrying the Josephson current are changed from opposite-spin components [ESE, OSO, ETO ($S=0$), OTE ($S=0$)] to equal-spin ones [ETO ($S=1$) and OTE ($S=1$)] between $L=12$ and $L=30$. From this result, a short junction (in our setting, $L<20$) is considered to behave as an ordinary S/insulator/S junction where the chiral edge mode doesn’t work as a conducting channel.

The QAHI length dependence of the critical current and the equilibrium phase difference is shown in Fig. 6 (a) and (b), respectively.

Around a length $L\sim 20$, an exponential decay implying that the QAHI behaves as an insulator is changed to an algebraic decay implying that the chiral edge mode carries the Josephson current. This crossover can also be seen from the equilibrium phase in Fig. 6 (b), where a short junction is insensitive to the relative direction of the exchange field between two interfaces while the equilibrium phase in a long junction is changed from $\pi$ to 0 by rotating the azimuth angle of the left interface by $\pi$ (from parallel to symmetric configuration). The critical current for the parallel configuration shows a dip around $L=20$ where the transition from 0 to $\pi$ junction occurs. When the exchange field is uniform throughout the QAHI, triplet pairing states are not induced and thus the critical current decays exponentially even in a long junction ($L>20$) [the red curve in Fig. 6 (a)].

The length and chemical-potential dependence of the critical current is shown in Fig. 6 (c) and (d) for the S/QAHI/S junction and (e) for the S/QSHI/S junction, respectively. In contrast to the S/QSHI/S junction, a periodic wavy pattern of a S/QAHI/S junction at $\mu>0$ and $L>20$ is attributed to the finite-size effect, where the critical current shows a cusp when the chemical potential $\mu$ agrees with a discrete spectrum of the chiral edge mode [Fig. 2 (a)]. Notice that, the periodicity of the cusp pattern in Fig. 6 (d) is considered to represent the oscillation corresponding to the momentum $k_{F}$ (not $2k_{F}$) as a function of the perimeter of the QAHI. The current phase relation in the S/QSHI/S junction is not sinusoidal due to the discrete ingap spectrum sensitive to the phase difference [Fig. 2 (b)].

Finally, we consider the disorder effect on the DC Josephson effect. Here, we consider on-site potential disorder given in the Nambu space by

where $\bm{r}$ is the position in the QAHI$,\delta\epsilon_{\bm{r}}$ is uniformly distributed within $[-W,W]$, and each matrix element implicitly accompanies the identity matrices of the spin and orbital degrees of freedom. The critical current and the equilibrium phase difference in the presence of disorder of the strength $W$ are shown in Fig. 7 (a) and (b), respectively.

It is known that within a disorder strength $W\in[0,2|\Delta|]$ studied here, the bulk is still in a QAHI phase and thus the chiral edge mode persists. The results indicates that the both critical current $I_{c}$ and the equilibrium phase difference $\varphi_{0}$ are sensitive to disorder configurations, which can also be seen in rapidly decaying critical current of the configuration-averaged current-phase relation $\text{max}[\overline{I_{c}\sin(\varphi-\varphi_{0})}]$ [the blue curve in Fig. 7 (a)], while the average of the critical current $\bar{I}_{c}$ is less sensitive to disorder [the red curve in Fig. 7 (a)].

Although disorder is an irrelevant perturbation to the chiral edge mode, it can change local parameters such as the Fermi velocity and the chemical potential. When the chemical potential has asymmetry between the top and bottom boundaries, the equilibrium phase difference can be shifted [54]. This property will be studied analytically in Sec. V.3. It will be shown that the configuration-dependent equilibrium phase difference is attributed to asymmetry of the potential on top and bottom boundaries. Notice that in order for the Josephson current through robust edge modes to be sensitive to disorder configurations, it is essential that the chiral edge mode extends across the boundaries [Fig. 2(c)], since the shift of the equilibrium phase difference is resulting from the interference of two conduction modes of the Josephson current: one is by an electron on the bottom boundary and a hole on the top boundary and the other is by a similar mode with the electron and hole inverted. On the other hand, less sensitive averaged critical current [the red curve in Fig. 7 (a)] is considered to be resulting from the robustness of the chiral edge mode (the critical current through a diffusive normal metal typically decays as $I_{c}\propto W^{-4}$, since when the Thouless energy $E_{\text{Th}}$ is smaller than the pair potential $|\Delta|$, the critical current obeys $I_{c}R_{\text{N}}\sim E_{\text{Th}}$ [71], where the normal-metal resistance and the Thouless energy are estimated as $R_{\text{N}}\propto\tau^{-1}$ and $E_{\text{Th}}\propto\tau$, and the relaxation time as $\tau\propto W^{-2}$ by the Born approximation [72]). While the averaged critical current decreases as the disorder becomes stronger within $W\in[0,2|\Delta|(=t_{s})]$, it increases as the disorder strength approaches the critical value $W\sim 4t_{s}$, over which the bulk becomes a (non-topological) Anderson insulator.

Fig. 7 (c) shows the disorder effect of the critical current of the S/QSHI/S junction. Since the Andreev reflection in the S/QSHI/S junction occurs locally and thus there is no interference, the phase difference at an equilibrium is always 0. Also the critical current is insensitive to both disorder strength and disorder configurations.

First, consider the anomalous Green’s function between two sites $j,j^{\prime}$ on the chiral edge. When $j$ and $j^{\prime}$ are away from the contact points, only spin-up components propagate. The leading order terms of the anomalous Green’s function representing the equal-spin Andreev reflection in the perturbation series with respect to the tunneling amplitude $t_{\text{tunnel}}$ are given by

where $C_{\text{R/L}}=t_{\text{tunnel}}^{4}|\Delta_{\text{R/L}}|V\tau(1-\tau^{2})\sin\theta_{\text{R/L}}/t_{\text{SC}}^{2}$ (see derivation in Appendix B). Some notices on (18) are in order: the anomalous Green’s function depends on (i) the tunneling of the evanescent minority spin ($\propto\tau$), (ii) the reflection of the spin-down electrons and holes from a superconductor back to the same superconductor ($\propto[g^{\text{edge}}_{00}(i\omega_{n})]_{22}-[g^{\text{edge}}_{00}(i\omega_{n})]_{44}=-2V/(\omega_{n}^{2}+V^{2})$), and also (iii) the polar angle of the tunneling matrix $T_{\text{R/L}}$ through $\sin\theta_{\text{R/L}}$. These facts indicate that the equal-spin Andreev reflection is resulting from the combination of spin-flipping normal reflection and the opposite-spin (ordinary) Andreev reflection. In addition, (iv) the anomalous Green’s function vanishes in the limit $\omega_{n}\to 0$, which is consistent with vanishing Andreev reflection in the same limit of the Josephson junction through half metals [16]. (v) The azimuth angle $\phi_{\text{R/L}}$ works equivalently as the phase of the pair potential $\Delta_{\text{R/L}}$ [65, 11, 66, 14, 16]. (vi) Since

the anomalous Green’s function with a fixed interval $j-j^{\prime}$ behaves as $e^{2i\mu j/v}$, which agrees with a numerically found $2k_{F}(=2\mu/v)$ oscillation in Sec. III.1.

The ETO and OTE components can be directly derived by substituting $i\omega_{n}\to\pm(\omega+i\delta)$ and using the definition (4), part of which resulting from the right superconductor is given by

Here, the interval $a$ appears in the expression as $\chi^{\text{ETO}}(a)=ie^{i\mu a/v}\sin[\omega a/v]$ and $\chi^{\text{OTE}}(a)=e^{i\mu a/v}\cos[\omega a/v]$, which agree with the periodicity of the nonlocal anomalous correlation in Sec. III.2. From the expression of $\chi^{\text{ETO/OTE}}(a)$, large OTE indicates small ETO and vice versa [6]. Both the ETO and OTE components have singularities at $\omega=2\pi vs/N\mp\mu(s\in\mathbb{Z})$, which corresponds to the spectra of the chiral edge electron (hole).

Next, consider the Josephson current through the chiral edge mode (see derivation in Appendix C). The Josephson current is an equilibrium current, and thus is derived from the equilibrium Green’s function as

where $\mathcal{T}_{\text{R(L)}}$ is defined in (38). Notice that $0$ and $N/2$ in the subscript of the Green’s functions in (21) are the coordinate of the right and left point contacts on the chiral edge, respectively, and $1$ ($-1$) is the leftmost (rightmost) site of the right (left) superconductor.

Intuitively, the Josephson current is carried by electrons moving in one direction and holes moving in the opposite direction, the number of which is balanced to be in an equilibrium by the Andreev reflection at two superconductors [75]. In our model, electrons on the bottom (top) boundary and holes on the top (bottom) boundaries are related by the equal-spin Andreev reflection by the right and left superconductors, which is described by the two terms in (18) by assigning $j$ and $j^{\prime}$ on the bottom and top (bottom and top) boundaries, respectively. The leading-order contribution to the Josephson current is thus the combination of the leading contribution of the anomalous Green’s function $[G_{N/2,0}^{(2)}(i\omega_{n})]_{13(31)}$ given in (18) and and its counterpart $[G_{0,N/2}^{(2)}(i\omega_{n})]_{31(13)}$ The expression of the Josephson current is shown in (48).

In the high-temperature limit ($2\pi v/N\ll\beta^{-1}$), the summation over the Matsubara frequency $\omega_{n}$ is approximated by the summation over $\omega_{0(1)}=\pm\pi\beta^{-1}$, which gives

where

On the other hand, in the low-temperature limit ($2\pi v/N\gg\beta^{-1}$) and when the inter-level spacing of the chiral edge mode is much smaller than the pair potential and the gap of the minority spin ($2\pi v/N\ll|\Delta_{R/L}|,V$) the Josephson current is given by

where $F(a)$ is a dimensionless periodic function [$F(a+1)=F(a)$] defined in (50), and it gives the maximum value of 1 when the chemical potential lies exactly at the discrete spectrum of the chiral edge mode $\mu=2\pi vs/N(s\in\mathbb{Z})$ (see Fig. 9). The Josephson current is dependent on the relative azimuth angle $\phi_{R}-\phi_{L}$ of the tunneling matrix in addition to the phase difference of the pair potential of two superconductors, which shares the same property as the Josephson current through half metals [65, 11, 66, 14, 16]. The phase (23) is the consequence of the interference mentioned in Sec. III.4. The critical current decays as $1/N^{3}$, which agrees with [56].

Next, we consider a case where the geometry of the junction has asymmetry with respect to the permutation of the top and bottom boundaries [Fig. 8 (b)] (see details in Appendix D). A similar topic has been studied from the perspective of symmetry of a spin-unpolarized QAHI in [54].

Let the length and the chemical potential of the top (bottom) boundary be denoted by $N_{1}$ and $\mu_{1}$ ($N_{2}$ and $\mu_{2}$). The plane wave $e^{ikx}$ is an eigenstate with the eigenenergy $vk-\mu_{1}$ ($vk-\mu_{2}$) on the top (bottom) boundary. Eigenstates in the whole chiral edge are thus determined by the scattering problem, in which plane waves with the same eigenenergy are connected by boundary conditions. The boundaries of the two regions are the two contact points with superconductors. Let the perimeter of the chiral edge be denoted by $N(=N_{1}+N_{2})$. The eigenenergies are $\epsilon=2\pi vs/N-\tilde{\mu}(s\in\mathbb{Z})$, where $\tilde{\mu}=(\mu_{1}N_{1}+\mu_{2}N_{2})/N$, and the corresponding momentum is $k_{1s}=2\pi s/N+(\mu_{1}-\mu_{2})N_{2}/Nv$ and $k_{2s}=2\pi s/N+(\mu_{2}-\mu_{1})N_{1}/Nv$ on the top and bottom boundaries, respectively. The unperturbed Green’s function of the chiral edge without tunneling can be derived in the same way as before. The resulting expression of the Josephson current is obtained from that for the symmetric case (48), (22), and (24) by changing the chemical potential as $\mu\to\tilde{\mu}$ and $\delta\varphi$ defined in (23) by

The dependence of the length difference appears in the second term of the right-hand side of (25) and that of the chemical-potential difference in the third term.

According to (25), even when the length of the two boundaries is the same ($N_{1}=N_{2}$), a chemical-potential difference of the interlevel spacing of the chiral edge mode ($\mu_{1}-\mu_{2}=2\pi v/N$) can change a 0 junction into a $\pi$ junction or vice versa. Therefore, although the expression in this section is applicable only to systems without disorder, it can be deduced that strong disorder-configuration dependence of the equilibrium phase in Sec. III.4 is the consequence of asymmetric profile of the disorder potential.