Tunable second-order Josephson response in altermagnets

Introduction.—Altermagnism is a newly discovered magnetic phase categorized by spin-group symmetries [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. It is an unconventional collinear antiferromagnet, where opposite spins at different sublattices are related by crystal rotation or reflection symmetry [1, 2], in contrast to classical antiferromagnets, where the spins are related via inversion or lattice translation symmetry. Hence, the combined time-reversal and translation (or inversion) symmetry is broken in altermagnets, leading to anisotropic spin-split energy bands in momentum space. This phenomenon has been predicted in various candidate materials [11, 7, 8, 6, 12, 2, 9, 13, 1, 14, 15, 16], such as RuO₂ [11, 7, 8], MnTe [2, 14], CrSb [2, 15], FeSb₂ [12, 2], Mn₅Si₃ [13], and CrNb₄S₈ [15], and confirmed experimentally in Mn₅Si₃ [17], MnTe [18, 19, 20, 21] and CrSb [22, 23, 24]. One notable feature of altermagnets is spin-momentum locking, protected by spin groups [2, 1, 25]. Similar spin-momentum locking can be achieved by twisting antiferromagnetic bilayers [26, 27, 28, 29]. In these systems, the altermagnetic field strength (AFS) depends crucially on the twist angle.

Recently, the interplay of altermagnetism and superconductivity has triggered intensive interest [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40] due to its potential for uncovering novel phenomena and applications in superconducting spintronics and quantum computing. Particularly, Andreev reflection has been investigated at superconductor/altermagnet interfaces [32, 33, 34] and proposed to function as a memory device [41]. Altermagnetic Josephson junctions (AMJJs), such as superconductor/altermagnet/superconductor junctions, have been predicted to exhibit 0-$\pi$ transitions by modulating the junction length despite the vanishing net magnetization [35, 36, 37]. It has further been suggested that AMJJs support the formation of $\varphi$-junctions [42] and exhibit the superconducting diode effect [43, 44].

In this Letter, we study the effects of external electric or Zeeman fields on the Josephson response of AMJJs as shown in Fig. 1(a). We find that the current-phase relation (CPR) can be either forward or backward-skewed due to a large second-order Josephson response, which can be detected by double SQUID measurements [45]. The skewness changes with AFS parameter $t_{J}$ [Fig. 1(b)]. Near 0-$\pi$ transitions [plateau in Fig. 1(b)], supercurrent reversal emerges as the second-order Josephson response dominates. When an electric field is applied to the altermagnetic region, for significant AFS, the critical current starts to oscillate around zero, leading to $0$-$\pi$ transitions. Interestingly, the magnitude of the critical current can be enhanced [Fig. 1(c)]. Moreover, external electric fields can also modify the CPR skewness [Fig. 1(c)]. Notably, the magnitude of the critical current can be substantially enhanced with increasing field strength for in-plane Zeeman fields [Fig. 1(d)]. Finally, we reveal that the Zeeman field can induce 0-$\pi$ transitions in AMJJs.

Model.—The altermagnet manifests non-relativistic spin splitting while having zero net magnetization, enforced by magnetic rotational symmetry. To model this, we consider the following tight-binding Hamiltonian with $d$-wave symmetry on a square lattice [13, 1]

where ${\bf k}=(k_{x},k_{y})$ is the 2D momentum, $\sigma_{x,y,z}$ and $\sigma_{0}$ are Pauli and identity matrices for spin, respectively. The first term in Eq. (1) represents the normal kinetic energy and the second term parameterized by $t_{J}$ describes the altermagnetic order. The model obeys $\{C_{2}||C_{4z}\}$ spin symmetry, i.e., a combination of four-fold rotation in real space and two-fold rotation in spin space, which enforces $d$-wave planar magnetism in momentum space. Notably, there exist several candidate materials with $d$-wave planar altermagnetism, such as KRu₄O₈, La₂CuO₄ and FeSb₂ [2, 1]. Transforming into real space, the Hamiltonian becomes

where $\psi_{\bf r}=(c_{{\bf r}\uparrow},c_{{\bf r}\downarrow})^{T}$ and $c_{{\bf r}\uparrow}\;(c_{{\bf r}\downarrow})$ is the electron annihilation operator at site $\bf r$ with spin $\uparrow(\downarrow)$, ${\bm{\delta}}_{x}$ (${\bm{\delta}}_{y}$) is the unit vector in the $x$ ($y$) direction, and $T_{x}=t\sigma_{0}/2+t_{J}\sigma_{z}/2$ and $T_{y}=t\sigma_{0}/2-t_{J}\sigma_{z}/2$ are the hopping matrices in the $x$ and $y$ directions, respectively.

We consider planar Josephson junctions constructed by sandwiching the altermagnet with two conventional $s$-wave superconductors, as depicted in Fig. 1(a). The junction length and width are $L$ and $W$, respectively. This setup can be described by the Bogoliubov-de-Gennes Hamiltonian

where the position-dependent pairing potential is defined as $\Delta({\bf r})=\Delta_{0}e^{-i\;\text{sgn}(x)\phi/2}\Theta(|x|-L/2)$ with $\phi$ being the superconducting phase difference between the superconductors, $\Delta_{0}$ the pairing potential, $\Theta(x)$ the Heaviside function, and $\text{sgn}(x)$ the sign function. The chemical potential is $\mu({\bf r})=\mu_{\text{AM}}\Theta(L/2-|x|)+\mu_{\text{S}}\Theta(|x|-L/2)$ with $\mu_{\text{AM}}$ and $\mu_{\text{S}}$ being the chemical potentials in the altermagnetic and superconducting regions, respectively. ${\cal H}({\bf r})$, given by Eq. (2), has finite $t_{J}$ in the altermagnetic region ($|x|<L/2$) while $t_{J}=0$ in the superconducting regions ($|x|\geq L/2$). To obtain the Josephson supercurrent, we adopt the recursive Green’s function approach [46, 47]. For convenience, periodic boundary conditions are applied along the $y$ direction. For concreteness, we set parameters $L=20$, $t=1$, $\mu_{\rm{S}}=\mu_{\rm{AM}}=0.2t$, $\Delta_{0}=0.01t$, and temperature $k_{B}T=0.02\Delta_{0}$ throughout the paper unless specified differently.

Altermagnetism induced tunable CPR skewness.—We first analyze the response of the Josephson current by changing AFS $t_{J}$. Figure 2(a) presents several typical CPRs for various $t_{J}$. All the CPRs are $2\pi$-periodic, i.e., $J_{s}(\phi)=J_{s}(\phi+2\pi)$, and odd in the phase difference $\phi$, i.e., $J_{s}(\phi)=J_{s}(-\phi)$, due to inversion symmetry of the junction. The supercurrent $J_{s}$ generally has either one maximum or minimum in the positive phase region $\phi\in(0,\pi)$, which we define as the critical current $I_{c}$. When the first harmonic dominates (e.g., in case of low-transparent junction interfaces), the supercurrent has a sinusoidal form $J_{s}=I_{c}\sin\phi$. Thus, the maximum or minimum is located at $\phi=\pi/2$. However, when the junction interfaces become more transparent, higher harmonics contribute, leading to skewed CPRs. The supercurrent is essentially determined by the derivative of the free energy $F$ of the junction to the phase difference $\phi$, $J_{s}\propto\partial F/\partial\phi$. For positive $I_{c}$, the ground state (corresponding to the free energy minimum) of the junction occurs at $\phi=0$. In contrast, for negative $I_{c}$, the ground state occurs at $\phi=\pi$. In Fig. 2(b), we calculate $I_{c}$ as a function of $t_{J}$. We observe that $I_{c}$ oscillates around zero as $t_{J}$ increases. Similar results have been reported in a recent preprint exploiting complementary theoretical methods [42]. These oscillations indicate that the AMJJ undergoes 0-$\pi$ transitions [Fig. 2(a)]. They can be attributed to the proximity-induced finite-momentum pairing correlations in the altermagnet [35]. The amplitude of $I_{c}$ first quickly decreases as the altermagnetism is turned on. Then, it oscillates with a reduced amplitude and finally vanishes for $t_{J}>0.9t$. Note that $t_{J}$ is different from material to material and may be varied, e.g., by the twist angle in twisted bilayer platforms [26, 27, 28, 29].

Altermangetism can not only induce 0-$\pi$ transitions but also significantly affect the skewness of CPR where the impact of the second-order Josephson response is vital. The CPR dominated by the first harmonic term has the form $J_{s}=I_{c}\sin\phi$. A skewed CPR implies the presence of pronounced higher harmonic contributing to the supercurrent, resulting in a non-sinusoidal form [48]. The skewness can be characterized by the deviation of the maximum/minimum position $\phi_{m}$ of $J_{s}$ from $\pi/2$, i.e., $\delta\phi\equiv\phi_{m}-\pi/2$. A positive $\delta\phi$ indicates a forward-skewed CPR, while a negative $\delta\phi$ corresponds to a backward-skewed CPR. In Fig. 1(b), we calculate the skewness of CPR as a function of AFS $t_{J}$. Strikingly, the CPR can be either forward- (pink area) or backward-skewed (green area). Examples of CPRs with forward and backward skewness are explicitly shown in Fig. 2(a). When the transparency of the junction changes because of the presence of altermagnetism, the second-order Josephson response can not be ignored, leading to the non-sinusoidal form of Josephson supercurrent $J_{s}=I_{1}\sin(\phi)\pm I_{2}\sin(2\phi)$ with $I_{2}/I_{1}>0$. Note that the skewness of CPR is primarily dictated by the sign of the second harmonic term. Using 0-junction ($I_{1}>0$) as an example, for a negative second harmonic term, the CPR is forward-skewed with $J_{s}=I_{1}\sin(\phi)-I_{2}\sin(2\phi)$ as illustrated in Figs. 2(a) and 2(c) with $t_{J}=0.03t$ and $I_{2}/I_{1}=0.37$. For a positive second harmonic term, the CPR is backward-skewd with $J_{s}=I_{1}\sin(\phi)+I_{2}\sin(2\phi)$ as shown in Figs. 2(a) and 2(d) with $t_{J}=0.44t$ and $I_{2}/I_{1}=0.64$. A similar ratio of $I_{2}/I_{1}$ has recently been observed in planar Josephson junctions based on Al/InAs-quantum wells [49]. The second-order Josephson response in AMJJs can be detected in SQUID devices or in tunnel junctions [45, 50]. More details on the skewness analysis can be found in the Supplemental Material (SM) [51]. Notably, the skewness $\delta\phi$ can be as large as $\pm 0.3\pi$. This particular feature of tunable skewness of CPRs in AMJJs may be exploited to realize a large Josephson diode effect, e.g, in supercurrent interferometers [52]. We also note that multiple maxima may appear in the CPR due to dominant higher harmonics. Figure 2(e) shows the higher harmonic contribution compared to the first harmonic as a function of AFS $t_{J}$ with $I_{n}$ representing the amplitude of $n$th harmonic and $a_{n}\equiv I_{n}/I_{1}$. When higher harmonics are dominant ($a_{n}>1$), as illustrated in the grey shaded area in Fig. 2(e), the AMJJ turns into a $\phi$-Josephson junction. An example for dominant second-order Josephson response $a_{2}>1$ is shown in Fig. 2(f). In these cases, the skewness is not clearly defined. Thus, we exclude them in Fig. 1(b). We emphasize that the tunable second-order Josephson response is important in applications as it enhances the robustness of superconducting quantum bits [45].

Tuning CPR skewness and critical current by gating.—Now, we investigate the influence of an external gate potential $V_{z}$ on the Josephson current. Figure 3(a) shows the critical current $I_{c}$ of the AMJJ as functions of AFS $t_{J}$ and gate potential $V_{z}$ for $\mu_{\text{S}}=\mu_{\text{A}M}$. For small $t_{J}$ (i.e., $t_{J}<0.05t$), $I_{c}$ is always positive with only slight oscillations. These oscillations appear due to Fabry-Pérot interferences. Thus, there is no 0-$\pi$ transition. As $t_{J}$ increases, we observe $0-\pi$ transitions alongside a tilted interference pattern. The larger $t_{J}$, the more inclined the interference pattern becomes, as illustrated in the phase diagram in Fig. 3(a). Hence, $I_{c}$ oscillates around zero with increasing $V_{z}$ for stronger AFS (i.e., $t_{J}>0.1t$). Figure 3(b) presents $I_{c}$ as a function of $V_{z}$ for $t_{J}=0.18t$ and $0.34t$. This plot demonstrates the switching of the AMJJ between 0 and $\pi$ junctions by changing the gate potential, as illustrated in Fig. 3(c). Meanwhile, the magnitude of $I_{c}$ is substantially enhanced for a wide range of gate potential, as shown in Fig. 3(b). Notably, the critical current can be enhanced by a factor 3 before the 0-$\pi$ transition occurs for $V_{z}<0.2t$, as illustrated in Fig. 3(b). The corresponding CPRs are displayed in Fig. 3(c). The observed $I_{c}$ enhancement is primarily governed by finite-momentum pairing, depending on both the gate potential and the AFS [35]. Furthermore, the skewness of CPR can also be manipulated by electric gating. For instance, the CPR changes from forward- to backward-skewed, as shown in Fig. 1(c). Additionally, a 0-junction can be transformed into a $\pi$- or $\phi$-junction by adjusting the gate potential, as illustrated in Figs. 1(c) and 3(d). Gate-tunable $\phi$-junctions, shown in Fig. 3(d), have previously been reported in Ref. [42]. These gate-tunable characteristics of AMJJs enrich their potential applications in superconducting circuits and spintronics.

Enhancement of supercurrent by Zeeman fields.—Now, we discuss the influence of an in-plane magnetic field on the supercurrent in AMJJs, modeled by by the Zeeman term $H_{Z}=B_{y}\cdot\sigma_{y}$. The particular in-plane direction of the magnetic field does not matter qualitatively for our results. Figure 4(a) illustrates the phase diagram of the AMJJ against $t_{J}$ and $B_{y}$. The stripes bend downward as $t_{J}$ grows. This demonstrates the competition between altermagnetism and Zeeman effect. Strikingly, this competition gives rise to the enhancement of the critical current as shown in Figs. 1(d) and 4(b). In absence of altermagnetism ($t_{J}=0$), the critical current $I_{c}$ decreases as the Zeeman field $B_{y}$ increases, illustrated by the blue line in Fig.4(b). However, the critical current can be monotonically enhanced when both altermagnetism and Zeeman field are present, as illustrated in Fig. 1(d) and the orange line in Fig. 4(b). Moreover, the magnitude of the critical current increases monotonically for large AFS as shown in Fig. 4(c) for $t_{J}=0.48t$. Nevertheless, it first increases then decreases for small AFS, illustrated in Fig. 4(c) for $t_{J}=0.16t$. To reach substantial ratios of $B_{y}/{\Delta_{0}}$, we envision that type-II superconductors, such as Nb and MoRe [53, 54], form the AMJJs with an altermagnet with large g-factor as the weak link.

The enhancement of the critical current in the presence of an exchange field is uncommon because the critical current generally decreases with increasing field strength in both ferromagnetic [55] and altermagnetic [Fig. 2(b)] Josephson junctions. Nevertheless, in a superconductor/ferromagnet bilayer separated by an insulating film, the critical current can also be enhanced when the exchange fields in the two ferromagnets are anti-parallel, resulting in no net magnetization [56]. Notably, the mechanism underlying the enhancement of the critical current in AMJJs differs from this scenario, as AMJJs exhibit net magnetization in presence of Zeeman fields. This phenomenon is a characteristic signature of altermagnetic superconducting heterostructures.

Another impact of Zeeman fields on AMJJs is that it induces 0-$\pi$ transitions. $0$-$\pi$ transitions emerge as $B_{y}$ changes as shown in Fig. 4(d). These oscillations stem from the Fermi surface spin splitting by the Zeeman field. For an AFS less than $0.2t$, the AMJJ transforms from 0-junction ($I_{c}>0$) to $\pi$-junction ($I_{c}<0$) for $B_{y}\leq\Delta_{0}$, as shown in Fig. 4(a) and the cyan line in Fig. 4(d). Additionally, it needs larger Zeeman field to induce 0-$\pi$ transition for large AFS as shown in Fig. 4(d).

Conclusion.—We investigate the influence of external electric and Zeeman fields on AMJJs. We reveal that changing the AFS significantly alters the second-order Josephson response, enabling both forward- and backward-skewed CPRs in the junction. Applying either an electric or a Zeeman field can also significantly enhance the magnitude of the critical current across the junction. Remarkably, 0-$\pi$ transitions can be controlled by either electric gating or Zeeman fields. Moreover, we discover that the second-order Josephson response can be manipulated by electric gating.

Note added.—During finalizing the manuscript, we became aware of a related preprint [42], which partially overlaps with our work but puts emphasis on $\varphi$ Josephson junctions.

Supplemental Material

In the following, we derive the real space lattice model and introduce the recursive Green’s function approach. Additionally, we analyse the sknewness change by a simple model.