Cooling of electrons via superconducting tunnel junctions and their arrays exhibiting nodal lines

TL;DR

The paper studies theoretically how to cool an electron bath by driving a current through superconducting tunnel junctions with a $\pi$ phase difference and interleaved ferroelectric layers, exploiting nodal-line DOS to maximize entropy exchange. It derives the Gibbs entropy for fermions in the grand canonical ensemble and links low-temperature entropy to the DOS near $E=0$, highlighting the role of nodal lines in cooling efficiency via $S \approx \ln 2 \int_0^{2T} n(E)\,dE$. The authors construct and analyze Hamiltonians for tunnel junctions and multilayer SC/FE/SC structures, showing spectra with nodal lines and a DOS that can be tuned by $\mu$, $t$, $\Delta$, and polarization (aligned or alternating). They discuss practical aspects such as finite peak broadening and provide an estimate for operating time from $Q = F(T_i) - F(T_f)$ to achieve a target temperature, illustrating a feasible path to finely tunable electron cooling in superconducting heterostructures with ferroelectric control.

Abstract

We study theoretically a process of cooling electrons using a superconducting tunnel junction with a $π$ phase difference and a usual insulator or a ferroelectric in-between, and an array of such junctions with ferroelectric layers in-between. These setups have a complex structure of entropy due to nodal lines, where the density of states can be divergent or larger than for a free electron gas at a chemical potential level. We consider a small current running from the bath of electrons through the setup, where electrons have to have higher entropy, and thus remove heat from the bath.

Figures (4)

• Figure 1: (a) The cooling scheme, where the current flows from the electron bath through the setup with a large entropy into the electric circuit. Electrons absorb heat from the bath because they need to increase their entropy. They emit this heat after the region with high entropy. (b) Setup consisting of a tunnel junction with a ferroelectric layer in-between. (c) Setup consisting of an array of tunnel junctions with ferroelectric layers in-between.
• Figure 2: Spectra of (a) the Hamiltonian for a tunnel junction, $H_{\rm TJ}$, and (b) the Hamiltonian for a multilayer structure with alternating polarization of ferroelectric layers, $H_{\rm MLJ}^-$. In (a), $t=\Delta=1$, $\mu=3$, and $m=0.5$. The eigenenergies touch at $E=0$ with their spectra being quadratic around the touching point, i.e., the node. This induces divergence in DOS. As $k=\sqrt{k_x^2+k_y^2}$, the node is actually a nodal circle. In (b), $t_1=0.5$, $t=\sqrt{1-t_1^2}$, $\mu=3$, $\Delta=1$, $m=0.5$. The nodal lines are formed by the second and the third eigenenergies, that span through the whole range of $q_z$. For some other combinations of parameters, they start and end inside of $q_z\in [-\pi,\pi]$. The spectra around these nodal lines are usually not quadratic, except for some special points for certain parameters, therefore the DOS is not overall infinite.
• Figure 3: Density of states of a tunnel junction superconductor-ferroelectric-superconductor (SC/FE/SC, blue lines) with the phase difference $\pi$, and of a multilayer structure of superconducting layers with alternating phases $0$ and $\pi$ and ferroelectric layers with aligned polarization (green dots, $\eta=0.05$), and with alternating polarization (red dots, $\eta=0.01$). In all subplots, $t=\sqrt{1-t_1^2}$, $m=0.5$. We use (a) $\mu=3$, $\Delta=0.5$; (b) $t_1=0.4$, $\Delta=0.5$; (c) $\mu=3$, $t_1=0.4$. In (a), a multilayer structure with alternating polarization gives the most stable values overall. In (b), the behavior of the dependences on $\mu$ is very different for these parameters, but all the curves flatten for larger $\mu$. In (c), the behavior is also very different, the multilayer structure with aligned polarization does not give zero for large $\Delta$ in contrast to the other two setups. We note that the results differ a lot for different parameters, including the amount of peaks/drops, because the spectra change.
• Figure S1: Multilayer structure consisting of superconductors of two kinds (SC1 and SC2) and ferroelectric layers (FE), which can have an aligned polarization (black arrows) or an alternating one (red arrows). The red dashed box shows the unit cell of this structure.