Field-Effect Josephson Diode
via Asymmetric Spin-Momentum-Locking States

To begin with, we employ a representative 2-component ASMLS Hamiltonian $h_{0}$, with a Zeeman field $m$ along the spin-polarized axis $\hat{z}$ and electrical potential $V$ (Appendix A.1):

where $v_{0}$ is the Fermi velocity, $\sigma_{z}$ and $\sigma_{0}$ are the $z$-Pauli matrix and identity matrix in spin space. To induce asymmetry in the spin-momentum locking, the $tk_{x}\sigma_{0}$ term ($0<t<1$) tilts the dispersion [Fig. 1(b)], resulting in distinct spin-dependent counter-propagating velocities, i.e. $v_{e}^{\sigma}=v_{0}(\sigma+t)$ where $\sigma=\pm 1$ denotes the two opposite spins.

Systems described by Eq. (2) are suitable candidates for FEJD. The spin-momentum locking from the $k_{x}\sigma_{z}$ term breaks inversion symmetry while the asymmetry induced by $tk_{x}$ breaks TRS, which is usually achieved through the Zeeman field $m$. The simultaneous breaking of both symmetries indicates the possibility of realizing the Josephson diode without a Zeeman field (Appendix A.2).

The center-of-mass momentum of paired electrons in ASMLS is crucial to achieve FEJDs. This is modeled through the Bogoliubov-de Gennes (BdG) Hamiltonian [95] (Appendix A.3)

in Nambu space, where $\Delta$ is the proximity-induced $s$-wave SC order parameter. A non-vanishing center-of-mass momentum of paired electrons [60]

results from the net Fermi momentum between holes $k_{h}^{-\sigma}$ and electrons $k_{e}^{\sigma}$ [Fig. 1(b)] with $k_{e(h)}^{\sigma}=+(-)(V-\sigma m)/[\hbar v_{0}(\sigma+t)]$ obtained by the electron and hole block dispersion in Eq. (3).

The momentum $k_{C}$ shifts the SC-gap center from $E=E_{F}=0$ to a finite energy, known as the Doppler energy shift [60] (Appendix A.4),

in the unit of effective SC gap $\Delta_{\text{eff}}=\sqrt{1-t^{2}}\Delta$. As a result, the energy of right-propagating (left-propagating) spin-up (spin-down) states increases (decreases), manifesting as spin-dependent SC gaps $\Delta_{\sigma}=\Delta_{\text{eff}}(1-\sigma D)$ below $E_{F}=0$ [Fig. 1(c)].

From Eq. (4) and (5), the joint effect of the asymmetry and gate voltage, $tV$, is seen to play the same role as the Zeeman field $m$, leading to gate-tunable center-of-mass momentum of paired electrons $k_{C}$.

Having explained how gate-tunable center-of-mass momentum of paired electrons $k_{\text{C}}$ can be achieved, we next describe how they give rise to FEJDs with gate-controlled diode efficiency $\eta$.

Our prototypical FEJD setup, based on Eq. (3), is schematically illustrated in Fig. 1(a) with super gates ($V_{\text{SC}}$) applied to the left ($x<0$) and right ($x>d$) SC leads and a tunneling gate ($V_{\text{TG}}$) applied to the middle non-SC region of length $d$ [40]. The SC gap and electrical potential profiles are $\Delta(x)=\Delta\left[e^{i\varphi_{L}}\Theta(-x)+e^{i\varphi_{R}}\Theta(x-d)\right]$ and $V(x)=V_{\text{SC}}\Theta(x^{2}-xd)+V_{\text{TG}}\Theta(xd-x^{2})$ with macroscopic SC phases $\varphi_{\text{L/R}}$ in the left/right SC leads.

Here the supercurrent is carried by ASMLSs [Fig. 1(a)]. Because of spin-momentum locking, the spin-up (spin-down) electrons are right-propagating (left-propagating), and undergo spin-flip Andreev reflections at the interfaces with the SC regions. Considering the scattering matrix at these interfaces (Appendix B), the asymmetric BdG dispersion [Fig. 1(c)] in the SC regions results in asymmetric Andreev reflection probabilities $R_{A,\pm}(E)\not=$ $R_{A,\pm}(-E)$ for a quasiparticle of energy $E$, as shown in Fig. 1(d), where

In the $\left|\frac{E}{\Delta_{\text{eff}}}\mp D\right|<1$ case, $R_{A,\pm}=1$ identically. This asymmetry crucially arises from the nonzero Doppler energy shift $D$ and gives rise to spin-dependent Andreev bound states (ABSs) spectra, which, in the short junction limit of $d\ll\hbar v_{0}/\Delta$, (Appendix B.3) is

as plotted in Fig. 1(e), where $\varphi=\varphi_{R}-\varphi_{L}$ is the phase difference between two SC leads. Importantly, gapless ABSs occurs when $\varphi=\varphi_{c}=2\cos^{-1}D$.

Through the scattering matrix formalism [96, 97], (Appendix C), we obtain the zero-temperature supercurrent,

where

with the ground-state parity $P(\varphi)=sign\{\cos(\varphi/2)-Dsign[\sin(\varphi/2)]\}$ , is the contribution from the ABS and

arise from the continuum states. Particularly, when $m=0$, $I_{\text{cont}}=etV_{\text{SC}}/(\pi\hbar)$ is solely gate-induced. For $|D|\geq 1$, both contributions become inseparable since $\varphi_{c}$ disappears, and the continuous expression is $\Gamma(\varphi)=(2/\pi)\left\{|D|-\sqrt{D^{2}-1}-\sin(\varphi/2)\tan^{-1}\left[\frac{\sin(\varphi/2)}{\sqrt{D^{2}-1}}\right]\right\}$.

As evident in Fig. 1(f), $I(\varphi)$ exhibits a sharp discontinuity at $\varphi=\varphi_{c}$ for $D<1$. This has significant implications for the dissimilarity of the direction-selective critical Josephson currents, which are defined as the extrema $I(\varphi)$ i.e. $I_{c+}=\max_{\varphi}I(\varphi)$ and $I_{c-}=|\min_{\varphi}I(\varphi)|$. In particular, for $0<D<1$, the current reverses at $\varphi=\varphi_{c}$ and $I_{c+}=I(\varphi_{c})$, $I_{c-}=-I(\pi)$. Intuitively, we expect the ABSs contributions to $I_{c\pm}$ by the right-moving spin-up elections and the left-moving counterpart to be different because of the different gaps experienced, i.e. $I_{\text{ABS}}(\varphi_{c})\propto\Delta_{+}<|I_{\text{ABS}}(\pi)|\propto\Delta_{-}$. Indeed, in the $D\ll 1$ limit, the $I_{\text{cont}}$ contribution is seen to enhance (suppress) the right-moving (left-moving) current, leading to $I_{c\pm}=e\Delta_{\text{eff}}/(2\hbar)(1\pm 2D/\pi)$.

All in all, by substituting $I_{c\pm}$ into Eq. (1), we obtain the diode efficiency

which solely depends on $D$ as shown in Fig. 1(g), alongside results at nonzero temperatures $T$. Notably, for $T=0$, the diode efficiency reaches a theoretical maximum of $|\eta|\sim 40.5\%$ at $|D|$ $=4\pi/(4+\pi^{2})\approx 0.906$, which is comparable to the value obtained by applying a magnetic field [60, 67]. However, it gradually vanishes as $\left|D\right|$ gets larger, since in the $|D|\gg 1$ limit, the extrema of the current $\lim_{\left|D\right|\gg 1}I(\varphi)=e\Delta_{\text{eff}}/(hD)\cos\varphi$ becomes symmetric, as shown in Fig. 1(f).

The exclusive functional dependence of the diode efficiency $\eta$ on the Doppler energy shift $D=(tV_{\text{SC}}+m)/\Delta_{\text{eff}}$ [Eq. (5)] implies that the gate control voltage $V_{\text{SC}}$ plays, in the presence of asymmetry $t$, an indistinguishable role from that of the Zeeman field $m$ [60]. Hence, the diode efficiency can be completely gate-controlled; for instance, in the $m=0$ and $D\ll 1$ limit, $\eta\propto D\propto tV_{\text{SC}}$ (Appendix C.2).

Before diving into the field-effect behavior of the Josephson diode, we first illustrate the existence of asymmetric helical edge states in QSHI without TRS. A general ansatz of QSHI Hamiltonian is [82, 84, 83, 101, 84, 83]

where $\bm{k}=(k_{x},k_{y})$, $k_{\pm}=k_{x}\pm ik_{y}$, $M_{\gamma}=M_{\gamma}-B_{\gamma}k_{+}k_{-}$, and $M_{\pm}=M_{\alpha}\pm M_{\beta}$. This Hamiltonian is written in the basis $\{|s,\uparrow\rangle,|p,\uparrow\rangle,|s,\downarrow\rangle,|p,\downarrow\rangle\}^{T}$, where $s$ ($p$) denotes the sublattice/orbital component and $\uparrow$ ($\downarrow$) denotes the spin component of the bulk states which are distinct from the spin alignments of the edge states. The first term of Eq. (17) is the Bernevig-Hughes-Zhang Hamiltonian [101] of a QSHI with TRS which is broken by the second term induced by high-frequency circularly polarized light [82] or magnetic doping [84, 83].

It is possible to maintain the gapless edge states in a TRS-broken QSHI if the system is invariant under the up-down parity symmetry $\hat{S}=\sigma_{z}\tau_{z}$, i.e. $[\hat{S},H_{\text{QSHI}}]=0$. With this symmetry, the Hamiltonian Eq. (17) transferred into a block-diagonalized form corresponding to bispinors with only two upper or lower nonzero components that describe a quantum anomalous Hall insulator whose topology is determined by the Chern number of each block as $C_{\pm}=\pm[sign(M_{\gamma}+M_{\alpha})+sign(B_{\gamma})]/2$. The spin-Chern number of the TRS-broken QSHI is thus well-defined as $C_{s}=(C_{+}-C_{-})/2$. The gapless edge perseveres even when TRS is broken. With a non-zero $C_{s}$, the asymmetric helical edge states [82] is described by

where $\sigma$ represents the spin orientation of the edge state, s$=$t$\equiv+1$ (s$=$b$\equiv-1$) represents the top (bottom)-edge, and $V_{\text{SG}}^{\text{s}}$ is the SG applied locally on each edge.

By specifying $h_{0}(k_{x})$ in Eq. (3) as Eq. (23), it is expected that the top (t) and bottom (b) edges of a QSHI each constitute an individual FEJD, with gates $V_{\text{SC}}^{\text{s}=\text{t/b}}$ and Zeeman fields $m^{\text{s}}=\pm m$ [82]. It is straightforward to apply Eqs. (9-12) to calculate each edge current $I^{\text{s}}(\varphi)$ and the diode efficiency $\eta^{\text{s}}$ (Appendix D.2).

A QSHI is an ideal SC quantum interference device where the supercurrent through a device results from the interference between two ASMLS currents. As a result, the advantage of this topological implementation is to enhance $\eta$ through interference [61] between two topological edge currents. The combined efficiency is (Appendix D.3)

with $0\leq D^{\text{b}}<1$ and $Z_{\pm}=\frac{\pi}{2}-\cot^{-1}(\sqrt{(D^{\text{t}})^{2}-1})\pm\sqrt{1-(D^{\text{b}})^{2}}[\frac{\pi}{2}+\tan^{-1}(\sqrt{\frac{1-(D^{\text{b}})^{2}}{(D^{\text{t}})^{2}-1}})]$ shows that when $|D^{\text{t}}|\gtrsim 1$ and $|D^{\text{b}}|\lesssim 1$, the diode efficiency is enhanced up to $90\%$, much higher than the one in individual edge [Fig. 2(a)]. Notably, this enhancement is unattainable through a magnetic field [87, 88, 89], which cancels [89] continuum-state currents with $D^{\text{t}}=D^{\text{b}}$ , thereby leading to a vanishing diode effect.

In a finite-length junction, $d>\hbar v_{0}\Delta$, a phase difference between two interfering ASMLS is accumulated [82], which gives rise to $I^{\text{tot}}=\sum_{\text{s}}I^{\text{s}}(\varphi+\varphi_{\text{TG}}^{\text{s}})$ with $\varphi_{\text{TG}}^{\text{s}}=2tV_{\text{TG}}^{s}d[\hbar v_{F}(1-t^{2})]^{-1}$ and $\varphi_{\text{TG}}^{\text{t}}(V^{\text{t}}_{\text{TG},0})=2\pi$. As a result, the phase difference between two interfering currents is controlled by the TGs and the critical current is the consequence of constructive or destructive interference leading to a TG-controlled oscillating diode efficiency [Fig. 2(b)].

The result in a finite-length junction is numerically obtained through the lattice Green’s function technique [102, 103, 82]. We set $M_{\gamma}=2B_{\gamma}$ ($B_{\gamma}=1$). The width of the junction is $W=30a$ ($a\equiv 1$ is the lattice constant) which is much larger than the characteristic lengths of the edge states ($\sim 5a$), to avoid the interactions between opposite edges. In the TRS-broken term, $M_{\alpha}=0.1M_{\gamma}$ to keep the system supporting ASMLSs with $t=0.1$ while $M_{\beta}=0$ is set for simplification. In the SC regions, the SC gap is $\Delta(T)=\Delta(0)\tanh(1.74\sqrt{T_{c}/T-1})$, where $T_{c}$ is the critical temperature and $\Delta(0)=0.01M_{\gamma}$ is the zero-temperature SC gap. The N-SC interfaces are assumed to be transparent. SGs and TGs are applied within the areas of a width comparable to the characteristic lengths of the edge states (Appendix D.4). When manipulating the gates, the Fermi levels are restricted within the bulk gap to ensure that the supercurrent is solely contributed by the edge states. A unit gate is defined as $V_{TG,0}^{\text{s}}=\Delta(T)(1-t^{2})/t$ corresponding to the phase $\varphi_{g0}^{\text{s}}=2tV_{TG,0}^{\text{s}}d_{0}[\hbar v_{F}(1-t^{2})]^{-1}=\pi$ with $d_{0}\approx 80a$. Only the cases with the gate at the top edge are studied while the gate at the bottom edge is turned off ($V_{\text{TG}}^{\text{b}}=0$).

Apart from the asymmetry in helical edge states, the TRS-broken term in Eq. (17) also includes (i) the $M_{\beta}$ term, which is not related to the band topology but induces a spin-dependent shifting to the energy spectrum like a Zeeman field [104], and (ii) the $M_{\alpha}$ term, which modifies the bulk band gaps and can finally, drive the system to be in a QAHI phase when $|M_{\alpha}|>M_{\gamma}$. Since only the case with helical edge state is concerned here, the effect of the QAHI phase is beyond our discussion, and $|M_{\alpha}|<M_{\gamma}$ is set.

Our numerical results exhibit that effect (i) leads to a gate-free diode effect on each edge [Fig. 3(a)], while in the whole system, the diode effect vanishes because of the compensation between two edges [Fig. 3(b)]. The reason is the Doppler energy shift depends on $M_{\beta}$ via $m_{\text{s}}=$s$M_{\beta}$. As a result, when $V_{\text{SC}}^{\text{s}}=0$, we obtain $D^{\text{t}}=D^{\text{b}}$, and thus $\eta^{\text{t}}=-\eta^{\text{b}}$ and $\eta^{\text{tot}}=0$. Effect (ii) is dual. First, the Josephson diode vanishes when the system is driven to be a QAHI, which is beyond this work due to the destruction of helical edge states. Besides, we observe that edge-state SC gap $\Delta_{\text{edge}}$ depends on both the bulk-state SC gap $\Delta_{\text{bulk}}=\Delta(T)$ as expected and bulk-state band gap controlled by $M_{\alpha}$ which is reported in Ref. [92]. Since an increasing $M_{\alpha}$ reduces $\Delta_{\text{edge}}$, a lower gate voltage is required to obtain the maximum value of the diode efficiency on each edge as shown in Fig. 3(c).

It is possible to induce a gap in the edge spectrum when the TRS is broken. This couples the left- and right-mover in the edge leading to a back-scattering that smoothens the skewed CPR and thus reduces the diode efficiency in each edge. However, because of the interference, the total diode efficiency can be enhanced (Appendix D.4).