Brownian refrigeration by hybrid tunnel junctions

Voltage fluctuations generated in a hot resistor can cause extraction of heat from a colder normal metal electrode of a hybrid tunnel junction between a normal metal and a superconductor. We extend the analysis presented in Phys. Rev. Lett. 98, 210604 (2007) of this heat rectifying system, bearing resemblance to a Maxwell’s demon. Explicit analytic calculations show that the entropy of the total system is always increasing. We then consider a single-electron transistor configuration with two hybrid junctions in series, and show how the cooling is influenced by charging effects. We analyze also the cooling effect from nonequilibrium fluctuations instead of thermal noise, focusing on the shot noise generated in another tunnel junction. We conclude by discussing limitations for an experimental observation of the effect.

Figure 1

(a) Illustration of an environment-assisted tunneling event in an NIS junction. Voltage fluctuations generated by the electromagnetic environment of the junction provide the energy $E^{\prime }-E$ for the electron at $E$ to tunnel to available states at $E^{\prime }$ on the superconductor side above the energy gap. For positive $E^{\prime }-E$, $P(E^{\prime }-E)$ is the probability density for the quasiparticle to emit energy $E^{\prime }-E$ to the environment, whereas $P(E-E^{\prime })$ describes the probability to absorb the energy $E^{\prime }-E$. Removal of high-energy electrons above the Fermi energy $E_{\mathrm{F}}$ ($E⩾0$, marked with the dotted line) results in refrigeration of the N island. $V$ denotes a constant dc bias voltage across the junction. For the Brownian refrigeration effect, $V=0$, and only a fluctuating voltage over the junction is present. (b) Electrical diagram of the resistor (resistance $R$, temperature $T_{\mathrm{R}}$) and the NIS junction (normal-state tunnel resistance $R_{\mathrm{T}}$; temperatures $T_{\mathrm{N}}$ and $T_{\mathrm{S}}$ for the normal metal N and superconductor S, respectively). The parallel capacitance $C$ includes the junction capacitance and a possible shunt capacitor. The N side of the junction can be connected to the resistor via a superconducting line with a direct NS contact. (c) Thermal diagram of the system. The NIS tunnel junction acts as a Brownian refrigerator (BR) between the normal metal island and the superconducting electrode. ${\dot{Q}}_{\mathrm{N}}$, ${\dot{Q}}_{\mathrm{S}}$, and ${\dot{Q}}_{\mathrm{R}}$ denote heat flows in the system. Heat is carried by tunneling electrons in ${\dot{Q}}_{\mathrm{N}}$ and ${\dot{Q}}_{\mathrm{S}}$, whereas the resistor is coupled to the NIS junction only via voltage fluctuations, and the heat exchange can be described in terms of photonic coupling. $P_{\mathrm{ext}}$ denotes the externally applied power needed to raise $T_{\mathrm{R}}$ over $T_{\mathrm{N}}$. The electrons in the resistor, superconductor, and the normal metal island are assumed to be thermally coupled to the lattice phonons, described as a heat bath at temperature $T_0$.

Figure 2

Cooling power ${\dot{Q}}_{\mathrm{N}}$ from Eq. (7) for $R=10R_{\mathrm{K}}$ (dashed lines) and $R=0.5R_{\mathrm{K}}$ (solid lines) at various values of $\Delta /E_{\mathrm{C}}$. Notice that for $\Delta >E_{\mathrm{C}}$ better cooling power is obtained with large $R$, whereas for $\Delta \lesssim E_{\mathrm{C}}$ the larger cooling power is found with $R=0.5R_{\mathrm{K}}$. The $R=10R_{\mathrm{K}}$ curves fall below the $R=0.5R_{\mathrm{K}}$ ones around $\Delta \approx 1.5E_{\mathrm{C}}$. The dash-dotted lines show the analytical approximation discussed in Appendix , valid for $R\gg R_{\mathrm{K}}$ and $k_{\mathrm{B}}{T}_{\mathrm{N}}\ll \sqrt{2E_{\mathrm{C}}{k}_{\mathrm{B}}{T}_{\mathrm{R}}}$, and capturing most of the cooling effect.

Figure 3

Maximum cooling power ${\dot{Q}}_{\mathrm{N}}^{\mathrm{opt}}$ (top) and the corresponding optimum resistor temperature $T_{\mathrm{R}}^{\mathrm{opt}}$ (bottom) as a function of $R$ and $E_{\mathrm{C}}$ at $k_{\mathrm{B}}{T}_{\mathrm{N}}=k_{\mathrm{B}}{T}_{\mathrm{S}}=0.1\Delta$. At fixed $R$, $T_{\mathrm{R}}^{\mathrm{opt}}$ increases approximately linearly as a function of $\Delta /E_{\mathrm{C}}$, and starts to become independent of $R$ at $R\gtrsim R_{\mathrm{K}}$.

Figure 4

Cooling efficiency $\eta ={\dot{Q}}_{\mathrm{N}}/{\dot{Q}}_{\mathrm{R}}$ at the optimum resistor temperature $T_{\mathrm{R}}^{\mathrm{opt}}$ as a function of $T_{\mathrm{N}}=T_{\mathrm{S}}$. The curves from top to bottom correspond to $\Delta /E_{\mathrm{C}}=20,10,5,2,\mathrm{and}1$, while $R/R_{\mathrm{K}}=10$ remains fixed.

Figure 5

$\left(\mathrm{a}\right)$Hybrid single-electron transistor in the presence of an environment, modeled as an impedance $Z\left(\omega \right)$ in series with the bias voltage source $V$. A gate voltage $V_g$ is coupled capacitively to the N island via $C_{\mathrm{g}}$. The arrows define tunneling rates and heat fluxes for each of the two NIS junctions, with tunneling resistance $R_{\mathrm{T},i}$ and capacitance $C_i$ $(i=1,2)$. $\left(\mathrm{b}\right)$Transformation of the environment seen from junction 1 into an effective single junction circuit.

Figure 6

Noise-induced cooling power $\dot{Q}$ in a SINIS structure with $\Delta /E_{\Sigma }=2$. Solid lines correspond to $n_{\mathrm{g}}=0.5$ (“gate open”), and dashed lines to $n_{\mathrm{g}}=0$ (“gate closed”). Dash-dotted lines indicate the cooling power of a single NIS junction, showing the influence of the effective circuit parameters $C^{*}$ (in general, optimum cooling shifts toward larger $T_{\mathrm{R}}$ in the SINIS) and $R^{*}$ (cooling power per junction is reduced in the SINIS for large $\Delta /E_{\Sigma }$ and increased for small $\Delta /E_{\Sigma }$).

Figure 7

Gate modulation of the maximum cooling power for four different charging energies at $R^{*}/R_{\mathrm{K}}=1$ and $k_{\mathrm{B}}{T}_{\mathrm{N}}=k_{\mathrm{B}}{T}_{\mathrm{S}}=0.1\Delta$.

Figure 8

Models for non-ohmic junction environments. $\left(\mathrm{a}\right)$Junction environment formed by an inductance $L$ in series with the resistance $R$. $\left(\mathrm{b}\right)$Symmetric distributed model showing the transformation to two standard transmission lines.

Figure 9

Series inductance: $\left(\mathrm{a}\right)$$P\left(E\right)$ at $T_{\mathrm{R}}=5T_{\mathrm{N}}$ for ten evenly spaced values of $Q$ between 0 and 1, with the thick black line denoting the case $Q=0$ (pure RC circuit). $\left(\mathrm{b}\right)$Cooling power ${\dot{Q}}_{\mathrm{N}}$ as a function of $T_{\mathrm{R}}$ at $R=R_{\mathrm{K}}$, $\Delta /E_{\mathrm{C}}=5$, and $k_{\mathrm{B}}{T}_{\mathrm{N}}/\Delta =0.1$ for the same values of $Q$ as in $\left(\mathrm{a}\right)$.

Figure 10

RLC transmission line: $\left(\mathrm{a}\right)$$P\left(E\right)$ at $T_{\mathrm{R}}=5T_{\mathrm{N}}$, and $\left(\mathrm{b}\right)$cooling power ${\dot{Q}}_{\mathrm{N}}$ out of the N electrode for an NIS junction coupled to an RLC transmission line. The different curves correspond to the indicated values of the line length $l$ at the fixed inductance and capacitance per unit length $C_0$ and $L_0$. The solid black line shows ${\dot{Q}}_{\mathrm{N}}$ for a lumped RC environment with $R=2R_{\mathrm{K}}$.

Figure 11

LC transmission line: $\left(\mathrm{a}\right)$$P\left(E\right)$ at $T_{\mathrm{R}}=5T_{\mathrm{N}}$, and $\left(\mathrm{b}\right)$cooling power ${\dot{Q}}_{\mathrm{N}}$ out of the N electrode for an NIS junction coupled to an LC transmission line terminated by a lumped resistor $R$. The different curves correspond to the indicated values of the line length $l$ at the fixed inductance and capacitance per unit length $C_0$ and $L_0$. The thick black line shows ${\dot{Q}}_{\mathrm{N}}$ for a lumped RC environment.

Figure 12

Circuit for studying shot-noise-induced cooling in a hybrid tunnel junction: an NIS junction of resistance $R_{\mathrm{T}}$ and capacitance $C$ is coupled capacitively through $C_{\mathrm{C}}$ to two voltage-biased NIN junctions A and B that generate shot noise. The noise is described by the current fluctuations $\delta I_i$, and the junction parameters include the resistance $R_i$, capacitance $C_i$, bias voltage $V_i$, and temperature $T_i$ $(i=\mathrm{A},\mathrm{B})$.

Figure 13

Examples of the shot-noise-induced cooling power ${\dot{Q}}_{\mathrm{N}}$ for various noise source resistances $R_{\mathrm{A}}=R_{\mathrm{B}}$, as a function of the average current $I_{\mathrm{N}}$. Black curves correspond to $k_{\mathrm{B}}T=0.12\Delta$, and gray ones to $k_{\mathrm{B}}T=0.1\Delta$. Other parameters were kept fixed at $C=2\mathit{fF}$, $C_{\mathrm{A}}=C_{\mathrm{B}}=0.5\mathit{fF}$, and $C_{\mathrm{C}}=10\mathit{fF}$.

Figure 14

Practical coupling scheme for the noise-generating resistor of resistance $R$ and the NIS junction of capacitance $C$ and tunneling resistance $R_{\mathrm{T}}$. See the text for details.

Figure 15

Photon absorption power $P_{\mathrm{ph}}$ due to a finite $R_{\mathrm{N}}$ compared to the cooling power ${\dot{Q}}_{\mathrm{N}}$ (gray dashed line). The curves from bottom to top were calculated with the indicated values of $C_{\mathrm{C}}^{\mathrm{ph}}$ ranging from 1 to $100\mathit{fF}$, whereas other parameters were fixed to the shown values.

Figure 16

Heat balance of the Brownian NIS refrigerator. $\left(\mathrm{a}\right)$Contour plot of the temperature drop $T_{\mathrm{N}}-T_{\mathrm{N},0}$ of the N island as a function of the environment (resistor) and bath temperatures $T_{\mathrm{R}}$ and $T_0$, obtained as a solution of the steady-state heat balance equation of the N island, Eq. (36). The temperature $T_{\mathrm{N}}$ is compared to $T_{\mathrm{N},0}$, defined as $T_{\mathrm{N}}$ at $T_{\mathrm{R}}=T_0$. The calculation assumes $\Delta =200\mu \mathit{eV}$, $R=R_{\mathrm{K}}$, $\Delta /E_{\mathrm{C}}=5$, $R_{\mathrm{T}}=20\mathrm{k}\Omega$, ${\Omega }_{\mathrm{N}}=2\times {10}^{-21}{\mathrm{m}}^3$, ${\Sigma }_{\mathrm{N}}=2\times {10}^9\mathrm{W}{\mathrm{K}}^{-5}{\mathrm{m}}^{-3}$, and $R_{\mathrm{N}}=5\Omega$. In addition, we assume bias resistors with $R_{\mathrm{B}}=R$, and that all three resistors are heated uniformly to $T_{\mathrm{R}}$. The coupling capacitance $C_{\mathrm{C}}$ is set to $100C$, and we include the constant parasitic heating $P_0=1\mathit{fW}$, the cooling power ${\dot{Q}}_{\mathrm{therm}}$ of an NIS junction thermometer of resistance $R_{\mathrm{T},\mathrm{therm}}=20R_{\mathrm{T}}$ biased at $eV=0.6\Delta$ [Eq. (1) with $P\left(E\right)=\delta \left(E\right)$], and finally the N island photon absorption $P_{\mathrm{ph}}$ from $R_{\mathrm{eff}}$ via an effective capacitance $C_{\mathrm{C}}^{\mathrm{ph}}=2\mathit{fF}$. The value of $T_{\mathrm{R}}$ at each $T_0$ that results in the minimum $T_{\mathrm{N}}$ is indicated by the thick black line, while the black dot shows the point of optimum $T_{\mathrm{R}}$ and $T_0$. $\left(\mathrm{b}\right)$Influence of various parameters on the minimum temperature of the island as a function of $T_0$. The reference curve corresponds to the optimum line in $\left(\mathrm{a}\right)$.