Thermoelectric properties of magic angle twisted bilayer graphene-superconductor hetero-junction: effect of valley polarization and trigonal warping

Abstract

We theoretically investigate the thermoelectric properties (electronic contribution) of a normal-superconductor (NS) hybrid junction, where the normal region consists of magic-angle twisted bilayer graphene (MATBG). The superconducting region is characterized by a common $s$-wave superconductor closely proximitized to the MATBG. We compute various thermoelectric coefficients, including thermal conductance, thermopower, and the figure of merit ($zT$), using the scattering matrix formalism. These results are further supported by calculations based on a lattice-regularized version of the effective Hamiltonian. Additionally, we explore the impact of trigonal warping and valley polarization on the thermoelectric coefficients. Notably, we find a significant variation in $zT$ as a function of these parameters, reaching values as high as 2.5. Interestingly, we observe a violation of the Wiedemann-Franz law near the charge neutrality point with the superconducting correlation, indicating that MATBG electrons behave as slow Dirac fermions in this regime. This observation is further confirmed by the damped oscillatory behavior of the thermal conductance as a function of the barrier strength when an insulating barrier is modelled at the interface of the NS junction. Beyond theoretical insights, our findings suggest new possibilities for thermoelectric applications using MATBG based NS junctions.

Figures (8)

• Figure 1: In panel (a), we illustrate the band dispersion of MATBG derived from the effective two-orbital tight-binding model with the chosen parameters $t_{1} = 0.331$ meV, $t_{2} = -0.01$ meV, and $\tilde{t}_{2} = 0.097$ meV. In panel (b), we depict a schematic diagram of our setup, where the left region of the hybrid geometry represents the MATBG (labelled as normal "N"), while the right region denotes the proximity induced MATBG superconductor with $s$-wave pairing (labeled as superconductor "S"). An insulating barrier (MATBG with different Fermi energy) of width $d$ (labeled as insulator "I") is placed at the NS interface. The thermal baths are connected to the left and right sides of the hybrid structure and are depicted in blue and red, maintaining temperatures $T$ and $T + \Delta T$, respectively, along with an applied voltage bias $V$.
• Figure 2: In Panels (a), (b), (c), and (d), we depict various thermoelectric coefficients: thermal conductance ($\kappa$) in units of $k_{B}^{2}T/h$, thermopower ($\mathcal{S}$) in units of $k_{B}/e$, Lorentz number ($\mathcal{L}/\mathcal{L}_{0}$), and figure of merit ($zT$), respectively, as a function of temperature choosing different values of the chemical potential, using the continuum model analysis. We choose the model parameters as $\lambda_{0} = 125$ in $\Delta \cdot nm^{2}$ unit, $\lambda_{1} = 50$ in $\Delta \cdot nm^{3}$ unit, $\Delta_{vp} = 0$ and $\mu_{s} = 200 \Delta$.
• Figure 3: In panels (a), (b), (c), and (d), we illustrate various thermoelectric coefficients: thermal conductance ($\kappa$) in units of $k_{B}^{2}T/h$, thermopower ($\mathcal{S}$) in units of $k_{B}/e$, Lorentz number ($\mathcal{L}/\mathcal{L}_{0}$), and figure of merit ($zT$), respectively, as a function of temperature for different values of the chemical potential, using the lattice model simulation for a finite system size with $l = 200$ lattice sites (measured in units of the lattice constant). We consider the other model parameters to be same as mentioned for the continuum model (see Fig. \ref{['fig2']}).
• Figure 4: Panels (a), (b), and (c) demonstrate the density plots for $zT$ in the $\lambda_1 - \Delta_{vp}$ plane choosing three different sets of temperature and $\mu_N$ values: (i) $T = 0.2T_{c}$ and $\mu_{N} = 10 \Delta$, (ii) $T = 0.5T_{c}$ and $\mu_{N} = 30 \Delta$, and (iii) $T = 0.99T_{c}$ and $\mu_{N} = 50 \Delta$, respectively. In contrast, panel (d) exhibits the density plot for $zT$ in the $\lambda_1 - T/T_c$ plane with $\Delta_{vp} = 0$ and $\mu_{N} = 30\Delta$. Here, $\lambda_1$ in $\Delta \cdot nm^{3}$ unit. We consider the lattice model with a finite size of $l = 200$ sites (measured in units of the lattice constant) and other model parameters remain same as mentioned for the continuum model in Fig. \ref{['fig2']}.
• Figure 5: In panel (a), we illustrate the thermal conductance $\kappa$ (in units of $k_{B}^{2}T/h$) as a function of barrier strength ($V_0$) for different values of $\mu_N$, calculated from the continuum model Hamiltonian. In panel (b), we depict the same quantity calculated employing the lattice model for a system with size $l = 200$ sites (measured in units of the lattice constant) with $T = 0.8T_{c}$ and $d = 10$ nm. For both the continuum and lattice model, the system parameters are chosen to be the same as mentioned in Fig. \ref{['fig2']} and Fig. \ref{['fig4']}.