# Thermodynamics of a Phase-Driven Proximity Josephson Junction

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Thermodynamics of Hybrid Systems

#### 2.1. Model

#### 2.2. Hybrid Junction as Thermodynamic System

#### 2.3. Proximity Induced Minigap

#### 2.4. Kulik-Omel’yanchuk Theory

#### 2.5. Total Entropy

## 3. Thermodynamic Processes

#### 3.1. Isothermal Process

#### 3.2. Isophasic Process and Heat Capacity

#### 3.3. Isentropic Process

## 4. Thermodynamic Cycles

#### 4.1. Josephson-Otto Cycle

**Isentropic $\mathbf{1}\to \mathbf{2}$**. All thermal valves are closed to make the system thermally isolated. The system is driven from the state $({\phi}_{1}=0,{T}_{1}={T}_{R})$ to $({\phi}_{2}=\pi ,{T}_{2})$, where ${T}_{2}={T}_{f}(\phi =\pi ,{T}_{1})$. In this process the universe spends a work $|{W}_{12}|$ (${W}_{12}<0$ according to the convention defined in Section 3). $|{W}_{12}|$ is represented by the green area in Figure 8b. No heat is exchanged, ${Q}_{12}=0$.**Isophasic $\mathbf{2}\to \mathbf{3}$**. By opening the thermal valve ${v}_{L}\phantom{\rule{0.166667em}{0ex}}$, the system goes from the state $(\phi =\pi ,{T}_{2})$ to $(\phi =\pi ,{T}_{3}={T}_{L})$. The system releases heat $|{Q}_{23}|$ to the left reservoir (magenta area in Figure 8a). No work is performed, ${W}_{23}=0$.**Isentropic $\mathbf{3}\to \mathbf{4}$**. All thermal valves are again closed to make the system thermally isolated. The system is driven from the state $({\phi}_{3}=\pi ,{T}_{3}={T}_{L})$ to $({\phi}_{4}=0,{T}_{4})$. By construction, if ${T}_{2}>{T}_{L}$ then it is ${T}_{4}<{T}_{R}$. In this process the system returns a work ${W}_{34}$ (${W}_{34}>0$ according to our convention), represented by the sum of the green and blue areas in Figure 8b. No heat is exchanged, ${Q}_{34}=0$.**Isophasic $\mathbf{4}\to \mathbf{1}$**. By opening the thermal valve ${v}_{R}\phantom{\rule{0.166667em}{0ex}}$, the system goes from the state $(\phi =0,{T}_{4})$ to $(\phi =0,{T}_{1}={T}_{R})$. The system absorbs heat ${Q}_{41}$ from the reservoir at ${T}_{R}$ (magenta+pink area in Figure 8a). No work is performed, ${W}_{41}=0$.

**Isentropic $\mathbf{1}\to \mathbf{2}$**. All thermal valves are closed to make the system thermally isolated. The system is driven from the state at the ambient temperature $({\phi}_{1}=0,{T}_{1}={T}_{R})$ to $({\phi}_{2}=\pi ,{T}_{2})$, where ${T}_{2}={T}_{f}(\phi =\pi ,{T}_{1})$. In this process, the universe spends a work $|{W}_{12}|$ (${W}_{12}<0$ for of Section 3). No heat is exchanged, ${Q}_{12}=0$.**Isophasic $\mathbf{2}\to \mathbf{3}$**. By opening the thermal valve ${v}_{L}\phantom{\rule{0.166667em}{0ex}}$, the system goes from the state $(\phi =\pi ,{T}_{2})$ to $(\phi =\pi ,{T}_{3}={T}_{L})$, removing the heat ${Q}_{23}$ from the CS (magenta area in Figure 9b). No work is performed, ${W}_{23}=0$.**Isentropic $\mathbf{3}\to \mathbf{4}$**. All thermal valves are closed. The system is driven from the state $({\phi}_{3}=\pi ,{T}_{3}={T}_{L})$ to $({\phi}_{4}=0,{T}_{4})$. Now, ${T}_{4}>{T}_{R}$. In this process, the system returns a work ${W}_{34}$. No heat is exchanged, ${Q}_{34}=0$.**Isophasic $\mathbf{4}\to \mathbf{1}$**. By opening the thermal valve ${v}_{R}\phantom{\rule{0.166667em}{0ex}}$, the system goes from the state $(\phi =0,{T}_{4})$ to $(\phi =0,{T}_{1}={T}_{R})$. The system releases heat ${Q}_{41}$ to the reservoir at ${T}_{R}$, since ${T}_{4}>{T}_{R}$, which correspond to the magenta+pink area in Figure 9b. The temperature ${T}_{4}$ plays an analogous role of the hot heat exchanger that is present in the refrigerators. No work is performed, ${W}_{41}=0$.

#### 4.2. Josephson-Stirling Cycle

**Isothermal $\mathbf{1}\to \mathbf{2}$**. The thermal valves ${v}_{R}$ is open and ${v}_{L}$ is closed, so that the system is in thermal contact with the right reservoir. The system is driven from the state $({\phi}_{1}=0,{T}_{1}={T}_{R})$ to $({\phi}_{2}=\pi ,{T}_{2}={T}_{R})$. Here a work is spent $|{W}_{12}|$ represented by the green area in Figure 12b. The heat ${Q}_{12}$ is absorbed from the reservoir, represented by the green + dark purple area in Figure 12a.**Isophasic $\mathbf{2}\to \mathbf{3}$**. By closing ${v}_{R}$ and opening ${v}_{L}\phantom{\rule{0.166667em}{0ex}}$, the system goes from the state $(\phi =\pi ,{T}_{2})$ to $(\phi =\pi ,{T}_{3}={T}_{L})$. The system releases heat ${Q}_{23}$ to the left reservoir, represented by the light purple + dark purple area. No work is performed, ${W}_{23}=0$.**Isothermal $\mathbf{3}\to \mathbf{4}$**. The valves are kept in the same state: ${v}_{R}$ open and ${v}_{L}$ closed. The system is driven from the state $({\phi}_{3}=\pi ,{T}_{3}={T}_{L})$ to $({\phi}_{4}=0,{T}_{4}={T}_{L})$. In this process the system returns a work ${W}_{34}$ represented by the sum of the green and blue areas in Figure 12b. The heat $|{Q}_{34}|$ is released to the left reservoir, represented by the blue area in Figure 12a.**Isophasic $\mathbf{4}\to \mathbf{1}$**. By closing ${v}_{L}$ and opening ${v}_{R}$, the system goes from the state $(\phi =0,{T}_{4})$ to $(\phi =0,{T}_{1}={T}_{R})$. The system absorbs the heat ${Q}_{41}$ from the reservoir at ${T}_{R}$, given by the sum of the areas in blue, red and light purple in Figure 12a. No work is performed, ${W}_{41}=0$.

## 5. Experimental Feasibility

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Thermodynamics Close to the Critical Temperature

**Figure A1.**(

**a**) Entropy scheme close to the critical temperature. $\tilde{S}$ is equal to S where a phase transition is imposed at ${T}_{c2}$, calculated with the method in the text. ${S}_{N}\left(T\right)$ is the normal metal entropy. (

**b**) Dependence of the critical temperature of the system ${T}_{c2}$ at $\phi =\pi $, normalized to the bulk critical temperature ${T}_{c}$, versus $\alpha $.

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**Figure 1.**(

**a**) Sketch of the SNS proximized system. It consists of superconducting ring, ${L}_{S}$ long, pierced by a magnetic flux $\mathsf{\Phi}$. The ring is interrupted by a normal metal weak link. The electron system of the whole device is thermally and electrically isolated and at temperature T. The system is connected to a thermal reservoir at temperature $\overline{T}$ through a heat valve v. (

**b**) Magnification of the SNS junction. The normal metal weak, ${L}_{N}$ long, is in clean electric contact with the superconducting leads. ${A}_{j},{\mathcal{N}}_{j}$ are respectively the cross-section and the DoS at Fermi energy of the $j=N$ or S metal. The phase drop $\phi $ of the superconducting order parameter takes place across the junction.

**Figure 2.**Color plots of the quasi-particle local normalized Density of States (DoS) N in a Superconductor/Normal metal/Superconductor (SNS) junction, versus energy $\epsilon $ and position x, for $\phi =0,\pi /3,2\pi /3,\pi $. The dashed lines separate the S regions (on the sides) to the N region (in the center), as shown by the junction sketch. The phase dependence of the DoS is mirrored in a phase-dependence of the junction entropy S. The numerical calculation has been obtained within the quasi-classical methods of Reference [84] with $a=10,l=0.1,\Delta (T\to 0)={\Delta}_{0}$.

**Figure 3.**Characteristics of the KO theory, reported versus phase $\phi $ for chosen temperatures T in legend. (

**a**) Supercurrent $I(\phi ,T)$, in Equation (11). The dotted curve at $T=0$ is given by Equation (12). (

**b**) Electric Energy $\mathcal{E}(\phi ,T)$, in Equation (14). The dotted curve at $T=0$ is given by Equation (15). (

**c**) Entropy variation $\delta S(\phi ,T)$, in Equation (5).

**Figure 4.**Total entropy S of the system for $\alpha =0.6$. (

**a**) S versus temperature T for chosen phases $\phi $ in legend. The case $\phi =0$ correspond to the BCS entropy ${S}_{0}\left(T\right)$ in Equation (10). (

**b**) Magnification of panel (

**a**) around $T=0.2{T}_{c}$, highlighting the passage from a exponential suppressed behavior at $\phi =0$ to a linear behavior at $\phi =\pi $. The dashed curve is the analytical low-temperature in expression (23), (24).

**Figure 5.**Isophasic heat capacity properties. (

**a**) Map of the isophasic heat capacity $C(\phi ,T)$. (

**b**) Cuts from panel (

**a**) for chosen phases in legend. The dashed line shows the low temperature expression (41). (

**c**) Cuts from panel (

**a**) for chosen temperatures in legend.

**Figure 6.**Isentropic processes properties. (

**a**) Colormap of the temperature decrease ${T}_{f}({T}_{i},\phi )/{T}_{i}$) for an isentropic process from initial temperature ${T}_{i}$ at $\phi =0$ to $\phi $. (

**b**) Cuts from panel (

**a**) for the chosen temperatures in legend. (

**c**) Temperature decrease ${T}_{f}/{T}_{i}$ for an isentropic process from $(\phi =0,{T}_{i})$ to $(\phi =\pi ,{T}_{f})$ for different values of $\alpha $. (

**d**) Isentropic current phase relation (red solid curve) across the state $(\phi =0,{T}_{i}=0.6{T}_{c})$, for $\alpha =0.6$. For comparison, the dashed curves report two isothermal current phase relations at $T=0.6{T}_{c}$ and $T=0.51{T}_{c}$ (see legend).

**Figure 7.**Sketch of the system connected to two reservoirs, identified as Left Reservoir (L) and Right Reservoir (R), through two heat valves ${v}_{L},{v}_{R}$ respectively. Thermodynamic cycles can be implemented varying configurations between different temperatures ${T}_{L}$ and ${T}_{R}$, achieving also opposite operational modes such as engine or refrigerator configurations (see text).

**Figure 8.**Otto cycle scheme. The example considers an engine from a hot reservoir at $0.6{T}_{c}$, cold reservoir at $0.2{T}_{c}$ and $\alpha =0.6$. (

**a**) Scheme in the $(T,S)$ plane. The colored areas help for the discussion in the text of the heat exchanges. (

**b**) Scheme in $(\phi ,I)$ plane. Of the four processes of the Otto cycle, only the two isentropic are visible, since the two isophasics are collapsed at the points $(\phi =0,I=0)$ and $(\phi =\pi ,I=0)$. The colored areas help for the discussion in the text of the work exchanges. For completeness, the dotted curves represent partial isothermal CPRs at the labelled temperature in the plot.

**Figure 9.**Particular cases of the Otto cycle on ${T}_{L},{T}_{R}$. (

**a**) Approaching the degenerate case of ${T}_{f}\left({T}_{R}\right)={T}_{L}$. (

**b**) Otto cycle as refrigerator for ${T}_{f}\left({T}_{R}\right)<{T}_{L}$.

**Figure 10.**(

**a**) Work released in a Josephson-Otto cycle as a function of $({T}_{L},{T}_{R})$. The dashed red curve, given by Equation (53), reports $W=0$ and separates the region where the cycle operates as engine or refrigerator. (

**b**) Heat absorbed in a Josephson-Otto cycle. As an engine, the heat ${Q}_{R}$ from the Hot reservoir is represented by the R reservoir. As a refrigerator, the heat ${Q}_{L}$ from the CS is represented by the L reservoir. The dash-dotted line represents the thermal equilibrium ${T}_{L}={T}_{R}$, below which the system is a cold pump. (

**c**) Cuts of the work in panel (

**a**) versus the Hot Reservoir temperature ${T}_{R}$ for fixed temperatures ${T}_{L}$ of the Cold Reservoir. (

**d**) Cuts of the absorbed heat versus the CS temperature ${T}_{L}$ for fixed temperatures ${T}_{R}$ of the Heat Sink. The black solid curve reports the absorbed heat at ${T}_{L}={T}_{R}$. The violet dash-dotted curve reports the analytical result of Equation (55). The curves have been obtained with $\alpha =0.6$.

**Figure 11.**Efficiency and COP of the Otto machine. (

**a**) Color plot of $\eta $ and $\mathrm{COP}$ versus $({T}_{L},{T}_{R})$, with different color palettes. The gray region represents the state where the cooled subsystem temperature is above the heat sink temperature. (

**b**) Cuts of Otto cycle efficiency $\eta $ versus ${T}_{R}$ for chosen ${T}_{L}$ in legend. The dot-dashed line reports the Carnot limit to efficiency. The curves end at the Otto characteristic curve, Equation (53), where the efficiency reaches the Carnot limit. (

**c**) Cuts of Otto cycle COP versus ${T}_{L}$ for chosen ${T}_{R}$ in legend. The dot-dashed line report the Carnot limit to COP. The curves are limited on the right by the thermal equilibrium state ${T}_{L}={T}_{R}$; on the right, the curves are limited by the Otto cycle characteristic curve. On this curve, the COP reaches the COP Carnot limit.

**Figure 12.**Josephson-Stirling cycle scheme. The plotted example concerns an engine between a hot reservoir ${T}_{R}=0.6{T}_{c}$ and a cold reservoir ${T}_{L}=0.3{T}_{c}$ and $\alpha =0.6$. (

**a**) Scheme in the $(T,S)$ plane. The colored areas help for the discussion in the text about the exchanged heats. (

**b**) Scheme in $(\phi ,\mathcal{I})$ plane. Of the four processes of the Josephson-Stirling cycle, only the two isothermals are visible, since the two isophasics are collapsed at the points $(\phi =0,I=0)$ and $(\phi =0,I=0)$. The colored areas help for the discussion in the text about the exchanged works.

**Figure 13.**Particular examples of the Josephson-Stirling cycle for ${T}_{R}<{T}_{L}$. (

**a**) Stirling inverse cycle working as refrigerator. The heat absorbed from the R reservoir in the process $\mathbf{1}\to \mathbf{2}$, represented by the area defined by the related green arrow, is bigger than the heat released to R reservoir in the process $\mathbf{4}\to \mathbf{1}$, represented by the area defined by the related red arrow. (

**b**) Stirling inverse cycle working as Joule pump, exploiting work to release heat to both reservoirs.

**Figure 14.**(

**a**) Work released in a Stirling cycle as a function of $({T}_{L},{T}_{R})$. The dashed curve $W=0$ correspond to the thermal equilibrium curve ${T}_{L}={T}_{R}$ and separates the region where the cycle operates as engine or refrigerator. Moreover, the curves ${Q}_{R}=0$ and ${Q}_{L}=0$ further distinguish regions where the cycle is a Joule Pump (JP) or a Cold Pump. (

**b**) Heat absorbed in a Stirling cycle. In both engine and refrigerator modes, the heat ${Q}_{R}$ is absorbed from the R reservoir that plays the role of Hot Reservoir or CS in the respective regions. (

**c**) Cuts of the work in panel (

**a**) versus the Hot Reservoir temperature ${T}_{R}$ for fixed temperatures ${T}_{L}$ of the Cold Reservoir. The black dashed line reports expression (60). (

**d**) Cuts of the absorbed heat ${Q}_{R}$ versus the CS temperature ${T}_{R}$ for fixed temperatures ${T}_{L}$ of the Heat Sink. The black solid curve reports the absorbed heat at ${T}_{L}={T}_{R}$. The curves have been obtained with $\alpha =0.6$.

**Figure 15.**Efficiency and COP of the Stirling machine. (

**a**) Color plot of $\eta $ and $\mathrm{COP}$ versus $({T}_{L},{T}_{R})$, with different color palettes. The gray region represents where the cycle is a Joule pump or Cold pump. (

**b**) Cuts of Stirling cycle efficiency $\eta $ versus ${T}_{R}$ for chosen ${T}_{L}$ in legend. The dot-dashed line reports the Carnot limit to efficiency. The curves end at ${T}_{R}={T}_{L}$. (

**c**) Cuts of Stirling cycle COP versus ${T}_{R}$ for chosen ${T}_{L}$ in legend. The dot-dashed line reports the Carnot limit to the COP. The curves go to infinity on the right at the thermal equilibrium state ${T}_{L}={T}_{R}$; on the left, the curves are limited by the Stirling characteristic curve.

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**MDPI and ACS Style**

Vischi, F.; Carrega, M.; Braggio, A.; Virtanen, P.; Giazotto, F. Thermodynamics of a Phase-Driven Proximity Josephson Junction. *Entropy* **2019**, *21*, 1005.
https://doi.org/10.3390/e21101005

**AMA Style**

Vischi F, Carrega M, Braggio A, Virtanen P, Giazotto F. Thermodynamics of a Phase-Driven Proximity Josephson Junction. *Entropy*. 2019; 21(10):1005.
https://doi.org/10.3390/e21101005

**Chicago/Turabian Style**

Vischi, Francesco, Matteo Carrega, Alessandro Braggio, Pauli Virtanen, and Francesco Giazotto. 2019. "Thermodynamics of a Phase-Driven Proximity Josephson Junction" *Entropy* 21, no. 10: 1005.
https://doi.org/10.3390/e21101005