Finite-momentum superconductivity from chiral bands in twisted MoTe₂

Abstract

A recent experiment has reported unconventional superconductivity in twisted bilayer MoTe₂, emerging from a normal state that exhibits a finite anomalous Hall effect – a signature of intrinsic chirality. Motivated by this discovery, we construct a continuum model for twisted MoTe₂ constrained by lattice symmetries from first-principles calculations that captures the moiré-induced inversion symmetry breaking even in the absence of a displacement field. Building on this model, we show that overscreening of the nominally repulsive Coulomb interaction gives rise to finite-momentum superconductivity in this chiral moiré system. Remarkably, the finite-momentum superconducting state can arise solely from internal symmetry breaking of the moiré superlattice, differentiating it from previously studied cases that require external fields. It further features a nonreciprocal quasiparticle dispersion and an intrinsic superconducting diode effect. Our results highlight a novel route to unconventional superconducting states in twisted transition metal dichalcogenides moiré systems, driven entirely by intrinsic symmetry-breaking effects.

A recent experiment has reported an anomalous Hall effect in the normal state of twisted-bilayer MoTe2 - a signature of electronic chirality. Here, based on a new continuum model for this chiral normal state, the authors argue that overscreening of the Coulomb interaction can give rise to a finite-momentum superconducting state that does not rely on external symmetry-breaking fields.

Introduction

Moiré systems based on transition metal dichalcogenides (TMDs) have in recent years emerged as a versatile platform for realizing a wide range of interaction-driven electronic phases^1,2. Examples include correlated insulators^3–8, generalized Wigner crystals^9–11, quantum anomalous Hall states¹², fractional quantum anomalous Hall states^13,14, and fractional quantum spin Hall states¹⁵. Remarkably, superconductivity has also recently been observed in twisted bilayer WSe₂ at twist angles θ = 3.56°¹⁶ and θ = 5°¹⁷, motivating intense theoretical efforts to understand the underlying pairing mechanisms^18–34. In particular, at θ = 5°, the emergence of superconductivity near a van Hove singularity and an antiferromagnetic metal has pointed to spin fluctuations as a possible mediator of pairing between time-reversed states in opposite valleys^18–25.

Very recently, signatures of superconductivity have been observed for another member of moiré TMD family, twisted bilayer MoTe₂ (tMoTe₂) at θ ~ 3.8°³⁵ and have attracted much interest^36–43. Intriguingly, the state above the transition temperature to the zero-resistance state exhibits an anomalous Hall effect and magnetic hysteresis, signaling intrinsic chirality and time-reversal symmetry breaking. Moreover, this time-reversal symmetry breaking coexists with the breaking of inversion symmetry by the moiré pattern⁴⁴ and an applied displacement field. These findings raise many fundamental questions: What microscopic mechanisms can give rise to superconductivity in a twisted TMD system from a parent normal state that is already chiral? What is the nature of the possible superconducting order? And what experimental probes could reveal its symmetry-breaking properties?

In this work, we address these questions by proposing and studying a mechanism for superconductivity in tMoTe₂ based on overscreening of the nominally repulsive Coulomb interaction in this system. Our analysis starts from a new continuum model for the normal state of tMoTe₂ fitted from first-principles calculations, that captures the intrinsic inversion symmetry breaking of the moiré pattern even in the absence of an external displacement field. With this revised continuum model, we show that tMoTe₂ can host a finite-momentum superconducting state, which is distinct from previously studied cases and does not rely on externally applied symmetry-breaking fields. Moreover, we also demonstrate that the superconducting state exhibits a nonreciprocal quasiparticle spectrum, Bogoliubov Fermi surfaces, and an intrinsic superconducting diode effect. We hope that our results will contribute to an understanding on the nature of superconductivity in chiral transition metal dichalcogenides moiré systems and provide a perspective that is complementary to the possibility of anyon superconductivity that was also recently proposed for tMoTe₂^36–39.

Results

Normal-state

The band structure of tMoTe₂ is generally described by a continuum model that captures the low-energy bands near the  ± K valleys^45–49. These valleys are separated by a large momentum transfer and can thus be treated as two decoupled degrees of freedom, labeled by an index τ = ± . The Hamiltonian near the τ-valley takes the form of a 2 × 2 matrix in the space of the two layers, ℓ = t, b,

$$H_{\tau }=\left(\begin{array}{cc}\hfill {\epsilon }_{t,\tau }\left(\widehat{\mathbf{k}}\right)+{\mathrm{\Delta }}_{+}\left(\mathbf{r}\right)+\frac{D}{2}\hfill & \hfill {\mathrm{\Delta }}_T\left(\mathbf{r}\right)\hfill \\ \hfill {\mathrm{\Delta }}_T^{†}\left(\mathbf{r}\right)\hfill & \hfill {\epsilon }_{b,\tau }\left(\widehat{\mathbf{k}}\right)+{\mathrm{\Delta }}_{-}\left(\mathbf{r}\right)-\frac{D}{2}.\hfill \end{array}\right)$$ 1

Here, the diagonal entries describe the kinetic energy and intra-layer moiré potentials, while the off-diagonal entries describe inter-layer tunneling. The displacement field, D, enters as a potential between the layers.

We now describe the terms in Eq. (1) in more detail: First, the kinetic terms, ${\epsilon }_{\ell ,\tau }\left(\widehat{\mathbf{k}}\right)=-{\hslash }^2{\left[\widehat{\mathbf{k}}-\tau {\mathbf{K}}_{\ell }\right]}^2/\left(2m^{*}\right)$, model the lowest bands of the uncoupled layers near their respective Dirac points, K_ℓ. These bands are spin-polarized due to the strong Ising spin-orbit coupling and carry opposite spin-polarization for the two valleys as enforced by time-reversal symmetry. Here, we will adopt the choice ${\mathbf{K}}_t=\left(-1/\sqrt{3},1/2\right)\cdot \left(4\pi /3a_M\right)$ and K_b = (0, 1) ⋅ (4π/3a_M) where a_M ≈ a/θ is the moiré period, a = 3.52 Å is the monolayer lattice constant, and θ = 3.89°.

Second, the intra-layer moiré potentials and inter-layer tunnelings are given by,

$${\mathrm{\Delta }}_{\pm }\left(\mathbf{r}\right)=2V_1\sum _{i=1,3,5}\cos \left({\mathbf{g}}_i^1\cdot \mathbf{r}\pm {\varphi }_1\right)+2V_2\sum _{i=1,3,5}\cos \left({\mathbf{g}}_i^2\cdot \mathbf{r}\right),{\mathrm{\Delta }}_T\left(\mathbf{r}\right)=w_1\sum _{i=1}^3{e}^{-i{\mathbf{q}}_i^1\cdot \mathbf{r}}+w_2\sum _{i=1}^3{e}^{-i{\mathbf{q}}_i^2\cdot \mathbf{r}},$$ 2

where V_1,2 and ϕ are the amplitudes and phase of the moiré potentials, which couple plane-wave states within the same layer via the “first-star" vectors ${\mathbf{g}}_i^1={\mathbf{G}}_i\equiv \left(4\pi /\sqrt{3}{a}_M\right)\cdot \left(\cos \left[\left(i-2\right)\pi /3\right],\sin \left[\left(i-2\right)\pi /3\right]\right)$ and “second-star" vectors ${\mathbf{g}}_i^2={\mathbf{G}}_i+{\mathbf{G}}_{i+1}$. Moreover, w₁ and w₂ are inter-layer tunnelings that couple states in different layers via $\left({\mathbf{q}}_1^1,{\mathbf{q}}_2^1,{\mathbf{q}}_3^1\right)=\left(0,{\mathbf{G}}_2,{\mathbf{G}}_3\right)$ and $\left({\mathbf{q}}_1^2,{\mathbf{q}}_2^2,{\mathbf{q}}_3^2\right)=\left({\mathbf{G}}_2+{\mathbf{G}}_3,{\mathbf{G}}_1,{\mathbf{G}}_4\right)$.

It is important to note that the continuum model construction following Ref. ⁴⁵ preserves an artificial emergent inversion symmetry due to the restriction of interlayer tunneling to real values. In contrast, the actual lattice structure of twisted MoTe₂ retains only a twofold rotational symmetry about the y-axis (C_2y), as evidenced by the small band splitting along the Γ–M direction in the DFT spectrum^44,47,48. To more accurately capture the intrinsic inversion symmetry breaking—an essential ingredient for finite-momentum pairing—we incorporate complex interlayer hopping into our model. This modification allows for a more faithful reproduction of the DFT band structure.

In our following considerations, we will adopt the parameter set (ϕ₁, V₁, w₁, V₂, w₂) = ( − 81. 0°,  8.3meV, − 8.4 e^−i2.4meV, 5meV, 8 e^−i1.69meV), which we have obtained from a fitting of the continuum model bands to the DFT bandstructure⁵⁰.

To illustrate the properties of the continuum model, we show its spectrum along a path in the moiré Brillouin zone (mBZ) in Fig. 1a. The two topmost valence bands, ξ_k,±, are well-isolated from other bands. At D = 0, the bands exhibit a small splitting due to the complex phase of w_1,2, which breaks inversion symmetry even in the absence of an external field. Applying a finite displacement field D ≠ 0 enhances this inversion breaking and thus further increases the band splitting. The corresponding density of states (DOS), shown in Fig. 1b, features a prominent peak whose height is sensitive to both the hole filling and the value of D. Representative Fermi surfaces for selected values of D that “follow" the peak in the DOS are shown in Fig. 1c.

Fig. 1. Normal state of twisted MoTe₂.

a Band structure of tMoTe₂ along a path in the mBZ at θ = 3.89°. A displacement field D = 5 meV induces a sizable spin splitting, with spins (↑, ↓) locked to valleys (τ = ± ). The red dashed line indicates the Fermi level at the hole density, n_h = 0.450. b DOS of the topmost valence band, ξ_k,+ ≡ ξ_k, as a function of its hole density, n_h,+ ≡ n_h and D. c Fermi surfaces for representative values of (D, n_h). From left to right: D = {0, 2, 3, 5} meV, n_h = {0.377,0.387,0.409,0.450}.

We will now propose and discuss a Kohn-Luttinger mechanism [44, 45], where superconductivity emerges from overscreening of the Coulomb interaction.

Superconductivity

Our first goal is to develop a microscopic theory of superconductivity in tMoTe2, assuming that time-reversal symmetry is spontaneously broken in the normal state such that only one valley band, ξ_k,+ ≡ ξ_k, remains “active" and participates in pairing. In particular, we will propose and discuss a mechanism for superconductivity from an overscreening of a nominally repulsive Coulomb interaction^51,52. Such a mechanism has also, more recently, been considered to theoretically explain the emergence of superconductivity in graphene multilayers^53–56.

We begin by defining the bare interaction Hamiltonian, $H_{\mathrm{int}}=\left(1/\left(2\mathrm{\Omega }\right)\right){\sum }_{\mathbf{q}}V\left(\mathbf{q}\right){\widehat{\rho }}_{\mathbf{q}}{\widehat{\rho }}_{-\mathbf{q}}$. Here, Ω is the system area and $V\left(\mathbf{q}\right)=e^2\tanh \left(\mid \mathbf{q}\mid d\right)/\left(2ϵ\mid \mathbf{q}\mid \right)$ is the gate-screened Coulomb potential. We choose d = 10 nm for the distance to the gates and ϵ = 5ϵ₀ for the dielectric permittivity (ϵ₀ is the vacuum permittivity). The bare interaction also includes the density operator projected on the ξ_k-band, ${\widehat{\rho }}_{\mathbf{q}+\mathbf{G}}={\sum }_{\mathbf{k}}{F}_{\mathbf{k},\mathbf{k}+\mathbf{q}}\left(\mathbf{G}\right){\widehat{c}}_{\mathbf{k}}^{†}{\widehat{c}}_{\mathbf{k}+\mathbf{q}}$. Here, ${\widehat{c}}_{\mathbf{k}}$ is an electron annihilation operator in this band and the form factors are $F_{\mathbf{k},\mathbf{k}+\mathbf{q}}\left(\mathbf{G}\right)={\sum }_{\ell ,{\mathbf{G}}^{\prime }}{u}_{\ell ,{\mathbf{G}}^{\prime }}^{*}\left(\mathbf{k}\right)u_{\ell ,{\mathbf{G}}^{\prime }+\mathbf{G}}\left(\mathbf{k}+\mathbf{q}\right)$ with u_ℓ,G(k) the Fourier coefficients of the periodic part of the Bloch wavefunctions, $u_{\mathbf{k}}\left(\mathbf{r}\right)={\sum }_{\mathbf{G}}{u}_{\ell ,\mathbf{G}}\left(\mathbf{k}\right)e^{i\mathbf{G}\cdot \mathbf{r}}∣\ell ⟩$.

As a next step, we incorporate the screening effects at the level of the random–phase approximation (RPA). The resulting RPA–dressed interaction is given by⁵⁷,

$${\left[V_{\mathrm{RPA}}\left(\mathbf{q}\right)\right]}_{\mathbf{G}{\mathbf{G}}^{\prime }}^{-1}=V^{-1}\left(\mathbf{q}+\mathbf{G}\right){\delta }_{\mathbf{G}{\mathbf{G}}^{\prime }}-{\mathrm{\Pi }}_{\mathbf{G}{\mathbf{G}}^{\prime }}\left(\mathbf{q}\right),$$ 3

with ${\mathrm{\Pi }}_{\mathbf{G},{\mathbf{G}}^{\prime }}\left(\mathbf{q}\right)=\left(1/\mathrm{\Omega }\right){\sum }_{\mathbf{k}}{F}_{\mathbf{k},\mathbf{k}+\mathbf{q}}\left(\mathbf{G}\right)F_{\mathbf{k},\mathbf{k}+\mathbf{q}}^{*}\left({\mathbf{G}}^{\prime }\right)\left[n_F\left({\xi }_{\mathbf{k}}\right)-n_F\left({\xi }_{\mathbf{k}+\mathbf{q}}\right)\right]/\left({\xi }_{\mathbf{k}}-{\xi }_{\mathbf{k}+\mathbf{q}}\right)$ being the static polarization and n_F the Fermi-Dirac distribution. We remark that in Eq. (3), only bubble diagrams have been resummed. As such, Eq. (3) only captures RPA-screening effects, but not all correlation effects.

For the superconducting instability, the relevant scattering processes involve two–particle states, $\left\{∣\mathbf{k}+\mathbf{Q}/2⟩\otimes ∣-\mathbf{k}+\mathbf{Q}/2⟩\right\}$, with finite center–of–mass momentum Q. Accordingly, we introduce the effective Hamiltonian,

$$H_{\mathrm{eff}}\left(\mathbf{Q}\right)=\sum _{\mathbf{k}}{\xi }_{\mathbf{k}}{\widehat{c}}_{\mathbf{k}}^{†}{\widehat{c}}_{\mathbf{k}}+\frac{1}{2\mathrm{\Omega }}\sum _{\mathbf{k},{\mathbf{k}}^{\prime }}{g}_{\mathbf{k},{\mathbf{k}}^{\prime },\mathbf{Q}}{\widehat{c}}_{\mathbf{k}+\frac{\mathbf{Q}}{2}}^{†}{\widehat{c}}_{-\mathbf{k}+\frac{\mathbf{Q}}{2}}^{†}{\widehat{c}}_{-{\mathbf{k}}^{\prime }+\frac{\mathbf{Q}}{2}}{\widehat{c}}_{{\mathbf{k}}^{\prime }+\frac{\mathbf{Q}}{2}}$$ 4

where we have defined the effective pairing kernel as, $g_{\mathbf{k},{\mathbf{k}}^{\prime },\mathbf{Q}}={\sum }_{\mathbf{G},{\mathbf{G}}^{\prime }}{F}_{-\mathbf{k}+\mathbf{Q}/2,-{\mathbf{k}}^{\prime }+\mathbf{Q}/2}\left(\mathbf{G}\right){\left[V_{\mathrm{RPA}}\left(\mathbf{k}-{\mathbf{k}}^{\prime }\right)\right]}_{\mathbf{G},{\mathbf{G}}^{\prime }}{F}_{\mathbf{k}+\mathbf{Q}/2,{\mathbf{k}}^{\prime }+\mathbf{Q}/2}\left(-{\mathbf{G}}^{\prime }\right)$.

A mean-field decoupling of Eq. (4) in the Cooper-channel now gives the self–consistency equation for the superconducting order parameter, Δ_k,Q,

$${\mathrm{\Delta }}_{\mathbf{k},\mathbf{Q}}=-\frac{1}{\mathrm{\Omega }}\sum _{{\mathbf{k}}^{\prime }}{g}_{\mathbf{k},{\mathbf{k}}^{\prime },\mathbf{Q}}\frac{{\mathrm{\Delta }}_{{\mathbf{k}}^{\prime },\mathbf{Q}}}{2{\tilde{E}}_{{\mathbf{k}}^{\prime },\mathbf{Q}}}\times \frac{1}{2}\left[\tanh \left(\frac{\beta E_{{\mathbf{k}}^{\prime },\mathbf{Q}}}{2}\right)+\tanh \left(\frac{\beta E_{-{\mathbf{k}}^{\prime },\mathbf{Q}}}{2}\right)\right].$$ 5

where $E_{\mathbf{k},\mathbf{Q}}={\xi }_{\mathbf{k},\mathbf{Q},-}+\sqrt{{\left({\xi }_{\mathbf{k},\mathbf{Q},+}\right)}^2+\mid {\mathrm{\Delta }}_{\mathbf{k},\mathbf{Q}}{\mid }^2}$ is the quasiparticle dispersion and ${\xi }_{\mathbf{k},\mathbf{Q},\pm }=\left({\xi }_{\mathbf{k}+\frac{\mathbf{Q}}{2}}\pm {\xi }_{-\mathbf{k}+\frac{\mathbf{Q}}{2}}\right)/2$ are the symmetrized and antisymmetrized normal-state dispersions. In addition, we have introduced the reduced quasiparticle dispersion, ${\tilde{E}}_{{\mathbf{k}}^{\prime },\mathbf{Q}}=E_{\mathbf{k},\mathbf{Q}}-{\xi }_{\mathbf{k},\mathbf{Q},-}$.

To determine the superconducting order parameter, we now solve Eq. (5) self-consistently at T = 0 by imposing a fixed electron density $n_e={\sum }_{\mathbf{k}}⟨c_{\mathbf{k}}^{†}{c}_{\mathbf{k}}⟩/\mathrm{\Omega }$ and Cooper-pair momentum, Q. The realized order parameter is then found by minimizing the condensation energy E_c(Q) = F(Q) − F_N, where F(Q) is the free energy in the superconducting state,

$$F\left(\mathbf{Q}\right)=-\sum _{\mathbf{k}}\left\{\frac{1}{\beta }\ln \left(1+e^{-\beta E_{\mathbf{k},\mathbf{Q}}}\right)-\frac{1}{2}\left({\xi }_{\mathbf{k},\mathbf{Q},+}-{\tilde{E}}_{\mathbf{k},\mathbf{Q}}\right)\right)-\left(\frac{\mid {{\mathrm{\Delta }}_{\mathbf{k},\mathbf{Q}}\mid }^2}{8{\tilde{E}}_{\mathbf{k},\mathbf{Q}}}\left[\tanh \left(\frac{\beta E_{\mathbf{k},\mathbf{Q}}}{2}\right)+\tanh \left(\frac{\beta E_{-\mathbf{k},\mathbf{Q}}}{2}\right)\right]\right\}$$ 6

and F_N = F(Q, Δ = 0) is the free energy in the normal state. In particular, we remark that the normal state free energy comprises only the non-interacting normal state spectrum.

Our results are shown in Fig. 2. For D = 0 meV, the condensation energy is minimized near (Q_x, Q_y) = (0.1259, 0) nm⁻¹, or at two equivalent momenta related by a C₃-rotation, see Fig. 2a. This behavior is expected because the complex phases in w_1,2 weakly breaks the inversion symmetry of the normal state along the γ − m path (see Fig. 6). As the displacement field D increases, the minima split further but remain related by C₃-symmetry, as shown in Fig. 2b, c. This additional splitting reflects the stronger inversion-symmetry breaking induced by D. Notably, another local minimum emerges near the ${\kappa }^{\prime }$ point of the Brillouin zone. Although this minimum is subleading for small D, it becomes the global minimum for D ≈ 5 meV, as depicted in Fig. 2d. Examples of the corresponding superconducting order parameters at T = 0 are shown in Fig. 3.

Fig. 2. Finite-momentum pairing in twisted MoTe₂.

a Condensation energy, E_c = F − F_N, versus Cooper-pair momentum Q = (Q_x, Q_y) at n_h = 0.377 and D = 0 meV. There is a minimum at Q = (0.1259, 0) nm⁻¹ (marked by white cross) and at the C₃-symmetry-related momenta. b–d Same as a, but for increasing D. The minima split into three C₃-related points, and an additional subleading minimum emerges near the ${\kappa }^{\prime }$-point. At D = 5 meV, the minimum near the ${\kappa }^{\prime }$-point becomes leading.

Fig. 6. Band structure of tMoTe₂ at θ = 3.15° and D = 0 meV.

The black lines represent the bands from the continuum model, while dashed lines corresponds to the DFT spectrum. Blue stars indicate spin-up states and red stars indicate spin-down states.

Fig. 3. Superconducting order parameter.

a Magnitude of the superconducting order parameter Δ_k,Q at D = 0 meV and n_h = 0.377, evaluated at the Cooper-pair momentum Q = (0.1259, 0) nm⁻¹ that minimizes the condensation energy. The Fermi surface is shown as a white line. b Same as a but for D = 5 meV, n_h = 0.450 and corresponding optimal Q = (0.6157, 0.2909) nm⁻¹. c Phase of Δ_k,Q at D = 0 meV, showing a (k_x + ik_y)-like winding. d Same as c but for D = 5 meV.

Finally, we remark that we have also estimated the critical temperature by linearizing the gap equations (see also ref. ⁵⁰). At D = 0 meV, we find T_c ~ 2.7K.

Bogoliubov spectrum

To further characterize the finite-momentum superconducting state, we now assume the system spontaneously selects one of the three C₃-related free-energy minima, Q, and compute the Bogoliubov quasiparticle spectrum.

Two example plots of the Bogoliubov spectrum are shown in Fig. 4. Their properties can be understood from symmetry arguments: In the normal state when D = 0, ξ_k is invariant under C₃ rotations, and has mirror symmetry σ_y along planes at angle Θ ∈ {0°,  60°,  120°} and approximate mirror symmetry σ_x along planes at angles Θ ∈ {30°,  90°,  150°}, where Θ = 0° is the positive k_x-axis. As D increases, the mirror symmetry σ_y is markedly broken, while the spectrum is still approximately invariant under σ_x. Once a superconducting order with a finite Cooper-pair momentum Q develops, the symmetry is further reduced, see Fig. 4. Specifically, the quasiparticle spectrum is only approximately symmetric along a fixed momentum direction, E_k,q ≈ E_−k,q, if σ_xk = − k and σ_xq = q.

Fig. 4. Non-reciprocal quasiparticle dispersion and Bogoliubov Fermi surface.

a Quasiparticle dispersion E_k,Q at D = 0 meV and optimal Cooper-pair momentum Q = (0.1259, 0) nm⁻¹. b Same as (a), but for D = 5 meV and corresponding optimal Q = (0.6157, 0.2909) nm⁻¹. A Bogoliubov Fermi surface appears, highlighted in black. c Dispersion along the dashed line in (a), showing an asymmetry between forward and backward directions. (d) Dispersion along the dashed cut in (c), showing a sign change that leads to the formation of the Bogoliubov Fermi surface.

This property can be directly seen from our numerical results. In Fig. 4b, d, the Cooper-pair momentum Q lies approximately along Θ = 30°. Consequently, the direction Θ = 120° preserves an approximately symmetric Bogoliubov dispersion, whereas all other directions, in particular Θ = 30°, exhibit a marked asymmetry between positive and negative momenta. This asymmetry is a defining characteristic of nonreciprocal superconductivity^58–60.

We can further refine this discussion by comparing the Bogoliubov spectrum at zero and finite displacement field, D. Most notably, we see that the k → − k asymmetry of the Bogoliubov spectrum is present even if D = 0, as can be from Fig. 4a and c. Hence, no external displacement field is required in this case to realize non-reciprocal superconductivity. Instead, the nonreciprocal superconductivity arises directly from the intrinsic inversion and time-reversal symmetry breaking of the material. Moreover, as D increases, the asymmetry in the Bogoliubov spectrum is enhanced, eventually giving rise to a Bogoliubov Fermi surface, as shown in Fig. 4b, d.

These features of the Bogoliubov spectrum have direct experimental implications. The asymmetries in the Bogoliubov dispersion can be probed through angle-resolved conductance measurements in a transparent normal metal-superconductor junction. Specifically, if the normal lead is oriented so that current flows along the direction of the Cooper-pair momentum, Q, then the resulting current-voltage (I-V) characteristic will be asymmetric with respect to V → − V. In comparison, if current flows perpendicular to Q, then the I-V characteristic is symmetric under V → − V. Moreover, the Bogoliubov Fermi surfaces lead to an enhancement of the density of states around zero energy, which can be similarly detected with conductance measurements.

Superconducting diode effect

Lastly, an additional property of the finite-momentum superconducting state in tMoTe₂ is an intrinsic superconducting diode effect. By definition, such a superconducting diode effect manifests as an asymmetry between the critical current in the forward direction, $I_c^{+}\left(\mathrm{\Theta }\right)$, and that in reverse, $I_c^{-}\left(\mathrm{\Theta }\right)$. The degree of asymmetry is quantified by the diode efficiency, $\eta =\frac{\mid I_c^{+}-I_c^{-}\mid }{I_c^{+}+I_c^{-}}$.

Superconducting diodes have been studied in a broad range of symmetry-broken superconducting systems^61–89, including symmetry-broken quantum materials where an intrinsic diode effect can arise without the applications of symmetry-breaking external fields^59,90–97. The diode effect in tMoTe₂ falls into this latter category and, interestingly, requires neither an applied displacement field nor a magnetic field.

Our results are shown in Fig. 5. In particular, as depicted in Fig. 5a and c, there is a finite difference between forward and reverse critical currents and thus a nonzero η even at zero displacement field, D = 0. The emergence of such a finite η can be explained by the breaking of inversion symmetry due to the complex phases of w_1,2 even if D = 0. Moreover, at D = 0, we find that there is  ≲ 0.1 diode efficiency along planes at angle Θ ∈ {0°,  60°,  120°}, as shown in Fig. 5c, while for D = 5 case, the diode efficiency is enhanced and is most prominent along planes at angle Θ ∈ {30°,  90°,  150°}, as shown in Fig. 5d.

Fig. 5. Non-reciprocal critical currents and diode efficiency.

a Angular dependence of the forward and backward critical currents, $I_c^{+}$ and $I_c^{-}$, at D = 0 meV. The asymmetry arises from the inversion breaking due to the complex phase in w₂. (b) Same as (a), but for D = 5 meV. (c) Angular dependence of the diode efficiency, η, at D = 0 meV due to the critical current difference in (a). d Same as c, but for D = 5 meV.

Discussion

In this work, we have developed a theory for superconductivity emerging from chiral bands in tMoTe₂ based on a mechanism from RPA-screened repulsive interaction. In particular, we have found that the resulting superconducting state exhibits a finite Cooper-pair momentum, leading to striking properties such as a nonreciprocal quasiparticle spectrum, Bogoliubov Fermi surfaces, and an intrinsic superconducting diode effect that necessitates neither an applied electric nor magnetic field. We hope that our results contribute to the understanding of symmetry-broken superconducting states in moiré TMD systems and, more broadly, to other platforms^98,99, such as rhombohedral tetralayer graphene, where a superconducting state with a possible spin-polarized chiral superconducting was realized experimentally observed¹⁰⁰ and theoretically studied^101–113.

Methods

DFT bands and continuum model fitting

For lattice relaxation, we follow the method described in Ref. ⁴⁷ To reduce computational cost, we first employ deep potential molecular dynamics (DPMD) to relax the initial rigid structure, where the DPMD force field is trained on data generated by VASP. A van der Waals correction (IVDW = 4) is applied during this stage. Once the structure converges in DPMD, we further relax it using VASP. The relaxation converges within several steps, ensuring that the final structure is stable. During the structural relaxation, we set the plane wave cutoff energy and the energy convergence criterion to 250 eV and 1 × 10⁻⁶ eV, respectively. The structure is fully relaxed when the convergence threshold for the maximum force experienced by each atom is less than 10 meV/Å.

For band structure calculations, the large-scale plane-wave basis first-principle calculations are carried out with Perdew-Burke-Ernzerhof (PBE) functionals using the Vienna Ab initio simulation package (VASP). We choose the projector augmented wave potentials, incorporating 6 electrons for each of the Mo and Te atoms.

The space group of the relaxed structures is P321 (No. 150), whose point group is generated by a two-fold rotational symmetry along y axis (C_2y), and three-fold rotational symmetry along z axis (C_3z). In the crystal momentum space, the C_2y symmetry only protects two-fold degeneracies at the invariant lines or points within the Brillouin Zone, as defined by the relation C_2yk → k. Within this invariant domain, the Hamiltonian commutes with the symmetry operation, allowing it to be block-diagonalized into two distinct sectors, each characterized by unique eigenvalues  ± π. Due to the constraints imposed by the symmetry, a band represented by e^iπ is inherently degenerate with another band represented by e^−iπ, forming a doubly-degenerate band structure. Consequently, the only lines that encapsulate the C_2y symmetries within the two-dimensional Brillouin zone are the ΓK lines (satisfying 2k₁ + k₂ = 0). When considering the C_3z rotational symmetry, the lines that meet the conditions k₁ + 2k₂ = 0 and k₁ − k₂ = 0 also emerge as the symmetry-invariant lines. As a result, bands along the ΓK and MK lines are always doubly degenerate, while a clear splitting is observed along the ΓM line as shown in Fig. 6. To accurately capture this splitting, we introduce complex intralayer hopping in the continuum model. During the parameter fitting process, our attention is directed to the topmost band. The dispersion of the first moiré band, together with the Chern numbers of the top three bands (-1, -1, and 2), are well reproduced. In all our calculations, the parameters were derived by fitting to DFT data at a twist angle of 3.15°. These parameters were then applied to the case of 3.89°, and we confirmed⁴⁷ that the parameters obtained at 3.15° reliably reproduce the DFT band structures across the twist-angle range of 3.15°-4.4°.

Note added. While finalizing this manuscript, we became aware of an independent study⁴⁰ investigating finite-momentum pairing under an applied gating field in tMoTe₂.

Author contributions

Y.C. and C.S. developed the numerical codes for obtaining the free energy, the superconducting order parameter, the Bogoliubov spectrum, and the diode efficiency. C.X. and Y.Z. performed the DFT simulations, the continuum model fitting and calculated the critical temperature. C.S. and Y.Z. conceived the project. C.S. wrote the manuscript with input from all authors.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

The main text data generated in this study have been deposited in the Zenodo database under accession code 10.5281/zenodo.17675801¹¹⁴.

Code availability

The code to generate the figures of the main text is available on Zenodo¹¹⁴.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-025-67836-9.