Momentum-Resolved Spectroscopy of Superconductivity with the Quantum Twisting Microscope

Abstract

We develop a theoretical framework for probing superconductivity with momentum resolution using the quantum twisting microscope (QTM), a planar tunneling device where a graphene tip is rotated relative to a two-dimensional sample. Due to in-plane momentum conservation, the QTM directly measures the superconducting spectral function along well-defined trajectories in momentum space. The relative intensities of electron and hole excitations encode the Bogoliubov coherence factors, revealing the momentum dependence of the pairing magnitude. Three $C_{3z}$-related tunneling channels enable direct detection of rotational symmetry breaking, as well as nodal points in the superconducting order parameter. We apply our framework to superconductivity within the Bistritzer-MacDonald model of noninteracting electrons and the Topological Heavy Fermion model, which accounts for electron-electron interactions. Together, these capabilities establish the QTM as a direct probe of the pairing symmetry and microscopic origin of superconductivity in two-dimensional materials.

Figures (12)

• Figure 1: (a) Schematic diagram of the QTM junction (adapted from theoryphonons), consisting of tip (blue circles) and sample (red circles) layers with independently tunable gate and bias voltages. The top layer can be rotated relative to the bottom in a controlled manner. (b) Band alignment of a monolayer graphene tip with a parabolic sample band. A superconducting gap is introduced in the sample; the color intensity reflects the electron spectral weight. The chemical potentials of tip and sample ($\mu_T,\mu_S$), the electrostatic shift $\phi$, applied bias $V_b$, and the corresponding tunneling window (gray rectangle) are indicated. (c) Momentum-space trajectory of the tip Dirac point across the extended moiré-Brillouin zone of MATBG, traversing all high-symmetry points. The symmetry points $\kappa$,$\kappa'$, and $\gamma$ are marked.
• Figure 2: (a–c) Superconducting pairing magnitude in the mBZ for $s$-wave (constant), $p_{y}$-wave, and $p_{x}$-wave pairings, each shown together with the three line scans related by $120^{\circ}$ rotations. We adopt the following pairing function: $\Delta_{p_y}=\Delta_0\left[\cos(\sqrt{3} \tilde{k}_x/2)\sin(\tilde{k}_y/2)+\sin(\tilde{k}_y)\right]$ and $\Delta_{p_x}=\Delta_0\sin(\sqrt{3}\tilde{k}_x/2)\cos(\tilde{k}_y/2)$, with $\tilde{k}_{x/y}=2\pi \frac{k_{x/y}}{|k_{M,y}|}$ and $k_{M,y}$ the size of the y-component of the reciprocal moiré vector schafferpairing. The small Fermi pockets around the $\gamma$ point are shown. (d–f) Corresponding $I"$ spectra for BM bands after introducing superconducting pairing. A zoomed-in region around the Fermi surface of the sample is shown (see inset in (d)). For $s$-wave pairing, the three traces following the Bogoliubov excitations coincide, while for $p_{x}$ and $p_{y}$ pairings broken $C_{3z}$ symmetry causes a splitting of the traces and the appearance of two distinct gaps; in (f) the positive and negative lines touch at $V_b=0$, thus one of the gaps closes, producing a nodal point. The small white region at $V_b=0$ is caused by the transition of $I"$ from negative to positive sign. Calculations are performed using pairing magnitude $\Delta_{0}=0.4~\,meV$, and quasi-particle lifetime broadening $\Gamma_{SC}=0.04~\,meV$.
• Figure 3: (a) Cuts of $|I"|$ for $s$-wave pairing at selected twist angles $\theta$, showing coherence peaks whose intensities are proportional to the Bogoliubov coherence factors (marked by red dots). (b) Bogoliubov excitations dispersions $E_{\mathbf{k}}$ extracted from the spectra in (a), together with the pairing magnitude $\Delta$ obtained from the combination of $E_{\mathbf{k}}$ and the peak intensity ratio, $R$, see discussion around \ref{['eq:R']}. (c) Same as (a) but for $p_{y}$-wave pairing, where two pairs of symmetric peaks are resolved. (d) Quasiparticle dispersions and extracted pairing for the $p_{y}$ pairing, illustrating how the pairing magnitude varies with $\theta$ (corresponding to momentum $\mathbf{K}_\theta$). The extracted pairing magnitude changes by approximately $50\%$ along the trajectory.
• Figure 4: (a) Band structure of MATBG at $\nu=-2$, obtained within the heavy-fermion model using a one-shot mean-field treatment of interactions based on the K-IVC parent state. The colored dots indicate the $f$-electron weight in the eigenstates. For simplicity, the band structure is shown in the flat and chiral limit, setting the first-order correction of the $f$-$c$ coupling to zero ($v'_\star = 0$), and neglecting the mass splitting ($M = 0$). Possible alignment of the Fermi energy is illustrated (dashed black), assuming that the doping is small enough not to modify the band structure. (b)-(c) QTM $I"$ spectra for superconducting states. In (b) the Fermi energy lies within the $f$-electron flat band, opening a gap near the $\kappa$, $m$, and $\kappa'$ points (corresponding to tip rotations $-\theta_{\mathrm{tbg}}$, $-\theta_{\mathrm{tbg}}/2$, and $0$). In (c) the Fermi energy lies within the $c$-electron dispersive band, opening a gap around the $\gamma$ point (tip rotation $\theta_{\mathrm{tbg}}$). Calculations are performed assuming $w_{aa}/w_{ab}=0.8$, $\theta_{\text{tbg}}=1.05^\circ$, $\ \Gamma_{SC}=0.04~meV,\ T=0.2~K$. We choose $\Delta_f=0.4~meV,\Delta_c=0.8~meV$ as the pairing magnitudes in the $f$ and $c$ regions, respectively. The color intensity of the feature around $\gamma$ is multiplied by $3$ to account for degeneracy.
• Figure 5: Comparison of QTM spectra for an isotropic superconducting gap in MATBG between two measurement modes. (a) $d^2I/dV_b^2$ spectrum, where $V_b$ tunes the electrostatic potential $\phi$, while $\mu_S$ and $\mu_T$ are fixed. (b) $dI/dV_b$ spectrum, where $V_b$ tunes $\mu_T$, while $\mu_S$ and $\phi$ are fixed.