Opportunities for mesoscopics in thermometry and refrigeration: Physics and applications

This review presents an overview of the thermal properties of mesoscopic structures. The discussion is based on the concept of electron energy distribution, and, in particular, on controlling and probing it. The temperature of an electron gas is determined by this distribution: refrigeration is equivalent to narrowing it, and thermometry is probing its convolution with a function characterizing the measuring device. Temperature exists, strictly speaking, only in quasiequilibrium in which the distribution follows the Fermi-Dirac form. Interesting nonequilibrium deviations can occur due to slow relaxation rates of the electrons, e.g., among themselves or with lattice phonons. Observation and applications of nonequilibrium phenomena are also discussed. The focus in this paper is at low temperatures, primarily below $4\mathrm{K}$, where physical phenomena on mesoscopic scales and hybrid combinations of various types of materials, e.g., superconductors, normal metals, insulators, and doped semiconductors, open up a rich variety of device concepts. This review starts with an introduction to theoretical concepts and experimental results on thermal properties of mesoscopic structures. Then thermometry and refrigeration are examined with an emphasis on experiments. An immediate application of solid-state refrigeration and thermometry is in ultrasensitive radiation detection, which is discussed in depth. This review concludes with a summary of pertinent fabrication methods of presented devices.

Figure 1

Schematic picture of the system considered in this review. We describe an electron system in a diffusive wire connected to two normal-metal or superconducting reservoirs via contacts of resistance $R_N$. The reservoirs are further connected to the macroscopic measurement apparatus (see Sec. 2H). The heat flows and thermal conductances between the electron system and external driving forces, the electromagnetic environment, and phonons in the lattice (Secs. 2C, 2D) are indicated with arrows. The description of phonons in the lattice can be further divided in film phonons, substrate phonons, and finally the heat bath on which the substrate resides (Sec. 2G). If the system is used as a radiation detector, it also couples to the radiation field typically via some matching circuit (Sec. 4).

Figure 2

Nonequilibrium quasiparticle energy distribution function in a diffusive normal-metal wire in the absence of inelastic scattering: (a) wire placed between two normal-metal reservoirs and (b) wire placed between a normal-metal $(x=L)$ and a superconducting $(x=0)$ reservoir. In the latter picture, we assume $\hslash D∕L^2⪡eV⪡\Delta$, such that the proximity effect and states above the gap can be neglected. In both pictures, the lattice temperature was fixed to $k_{B}T=eV∕10$.

Figure 3

(Color online) Quasiparticle energy distribution function in the center of a normal-metal wire placed between two normal-metal reservoirs and biased with a voltage $eV=10k_{B}T$. The three lines correspond to three extreme limits: solid line, $L⪡{\ell }_{e\text{−}e},{\ell }_{e\text{−}\mathrm{ph}}$ (nonequilibrium limit); dashed line, ${\ell }_{e\text{−}e}⪡L⪡{\ell }_{e\text{−}\mathrm{ph}}$ (quasiequilibrium limit); and dotted line, ${\ell }_{e\text{−}e},{\ell }_{e\text{−}\mathrm{ph}}⪡L$ (equilibrium limit).

Figure 4

(Color online) Electron cooling at NIS junctions. (a) Sketch of the energy-band diagram of a voltage-biased NIS junction. Upon biasing the structure, the most energetic electrons $\left(e\right)$ can tunnel into the superconductor. As a result the electron gas in the N electrode is cooled. (b) Maximum cooling power surface density ${\dot{Q}}_A$ vs interface transmissivity $\mathcal{T}$ at different temperatures calculated for a NS contact.

Figure 5

(Color online) NS point contact characteristics. (a) Heat current from the N side as a function of voltage for different transparencies $\mathcal{T}$ (from top to bottom, tunneling limit $\mathcal{T}\to 0$, $\mathcal{T}=0.005$, 0.01, 0.05, 0.5, and 1) at $k_{B}T=0.3\Delta$. For $\mathcal{T}\lesssim 0.05$, the heat current is positive, corresponding to cooling. (b) Current-voltage characteristics for the same values of $T$ and $\mathcal{T}$ as in (a) (now $\mathcal{T}$ increases from bottom to top). The first four curves lie essentially on top of each other. The corresponding NIN curves are shown with dashed lines.

Figure 6

(Color online) Nonequilibrium distribution inside the SINIS island, calculated from Eq. (42): (a) No inelastic scattering, ${\mathcal{I}}_{\mathrm{coll}}=0$, $T=0.1T_c$, and depairing strength $\Gamma ={10}^{-4}\Delta$ inside the superconductors. The voltage runs from zero to $V=3\Delta ∕e$ in steps of $\Delta ∕\left(2e\right)$ and spans three regimes: (i) anomalous heating regime ($eV=0–\Delta$, solid lines, widening with an increasing $V$) discussed in the article by 337, (ii) cooling regime ($eV=1.5\Delta$ and $2\Delta$, dashed lines, narrowing with an increasing $V$) and the strong nonequilibrium heating regime ($eV=2.5\Delta$ and $3\Delta$, dotted lines, widening with an increasing $V$). (b) Distribution at $eV=2.5\Delta$ for different strengths of the electron-electron interaction, parametrized by $K_{e\text{−}e}$ from Eq. (16), with $R_D$ replaced by $R_T$ and $E^{*}$ by $\Delta$ as in this case these describe the collision integral.

Figure 7

Supercurrent-induced N-S thermopower ${\alpha }_{\mathrm{NS}}$ for the system depicted in the inset. Solid line: ${\alpha }_{\mathrm{NS}}$ at $T_1\approx T_2$. Dotted line: approximation (47). The correction (48) accounts for most of the difference. Dashed line: ${\alpha }_{\mathrm{NS}}$ at $k_B{T}_1=3.6E_T$ with varying $T_2$. Dash-dotted line: the corresponding approximation. Inset: Andreev interferometer where thermoelectric effects are studied. Two superconducting reservoirs with phase difference $\varphi$ are connected to two normal-metal reservoirs through diffusive normal-metal wires. Adapted from 438.

Figure 8

The suspended silicon-nitride bridge structure on the left, which was used by 389 to measure the quantum of thermal conductance shown in the graph on the right: thermal conductance levels off at the value $16g_0$ at temperatures well below $1\mathrm{K}$. Adapted from 389. Reprinted by permission from Macmillan Publishers Ltd., copyright 2000.

Figure 9

(Color online) Temperature rise according to the presented reservoir heating model. For details see text.

Figure 10

(Color online) NIS thermometer characteristics. (a) Calculated $I\text{−}V$ at various relative temperatures $T∕T_c$; the lower the temperature, the sharper the curves. (b) The corresponding voltage over the junction as a function of temperature when the junction is biased at a constant current. Characteristics at a few values of current are shown.

Figure 11

(Color online) SIS junction as a thermometer. (a) Calculated and (b) measured $I\text{−}V$ curves at a few temperatures. The lowest-temperature curves are almost on top of each other in (a). Supercurrent has been suppressed. Experimental data from 374.

Figure 12

A typical Coulomb blockade thermometer (CBT) sensor for the temperature range $20\mathrm{mK}–1\mathrm{K}$. The structure has been fabricated by electron-beam lithography combined with aluminum and copper vacuum evaporation. Both top view and a view at an oblique angle are shown; the scale indicated refers to the top view. Adapted from 287, with permission of Springer Science and Business Media.

Figure 13

Performance of a CBT thermometer. (a) Typical measurement: $G\left(V\right)∕G_T$ is the differential conductance scaled by its asymptotic value at large positive and negative voltages, plotted as a function of bias voltage $V$. $V_{1∕2}$ indicates the full width at half minimum of the characteristics, which is the main thermometric parameter. The full depth of the line, $\Delta G∕G_T$, is another parameter to determine temperature. (b) The temperature deduced by CBT compared to that obtained by a ${}^3\mathrm{He}$ melting curve thermometer. Saturation of CBT below $20\mathrm{mK}$ indicates typical thermal decoupling between electrons and phonons. From 287, with permission of Springer Science and Business Media.

Figure 14

Crossover from Johnson noise to shot noise when increasing the bias voltage of a tunnel junction. The data shown by open symbols have been measured at $300\mathrm{K}$ with $R_T=0.32\mathrm{G}\Omega$ and data shown by solid symbols at $77\mathrm{K}$ with $R_T=2.7\mathrm{G}\Omega$. Adapted from 45.

Figure 15

Normalized junction noise vs normalized voltage at various temperatures. The residuals from the expected $x\coth \left(x\right)$ law are shown in the bottom half. From 405, copyright 2003, AAAS.

Figure 16

The suspended thermal sensor employed by 51. The ${\mathrm{Nb}}_{x^{\prime }}{\mathrm{N}}_{1-x^{\prime }}$ thermometer can be seen in the lower part of the rectangular silicon membrane. The 450 000 Al superconducting rings are located in the middle part of the membrane; examples of them are shown in (b) and (c). From 51.

Figure 17

(Color online) A micrograph of a “spider-web” bolometer. A neutron-transmutation-doped (NTD) Ge thermistor is located at the center of the web. Courtesy of NASA/JPL-Caltech.

Figure 18

(Color online) Micrograph of an antenna-coupled transition-edge-sensor (TES) bolometer. A dual-polarized double-slot antenna (a) is coupled via microstrip transmission lines (b) with bandpass and low-pass filters (c) to two $\mathrm{Al}∕\mathrm{Ti}$ bilayer TES thermometers located on suspended ${\mathrm{Si}}_3{\mathrm{N}}_4$ membranes (d). Inset: The termination of the microstrip line (e) to a resistor (f), and the $\mathrm{Al}∕\mathrm{Ti}$ TES (g). Courtesy of A. T. Lee, UC Berkeley.

Figure 19

Scanning electron microscope (SEM) micrograph of an isolated ${\mathrm{Si}}_3{\mathrm{N}}_4$ platforms with $\mathrm{Mo}∕\mathrm{Au}$ TESs. As an extreme example of thermal isolation, the meandering ${\mathrm{Si}}_3{\mathrm{N}}_4$ legs have an aspect ratio of $\sim 2800:1$, yielding a thermal conductance $G_{\mathrm{th}}\approx 100\mathrm{fW}∕\mathrm{K}$ at a bath temperature of $210\mathrm{mK}$. Courtesy of M. E. Kenyon, JPL.

Figure 20

(Color online) The energy spectrum of a pulsed $1550\text{−}\mathrm{nm}$ laser measured with an optical TES microcalorimeter. The peaks in the plot correspond to the photon-number state of the incoming pulses. Inset: A micrograph of the devices. Courtesy of A. J. Miller, NIST.

Figure 21

An energy spectrum of the ${}^{55}\mathrm{Mn}K\alpha$ complex obtained with a $\mathrm{Mo}∕\mathrm{Cu}$ TES microcalorimeter. The width of the measured characteristic x-ray lines is a convolution of the intrinsic x-ray linewidth and the Gaussian detector contribution with a FWHM of $2.38\pm 0.11\mathrm{eV}$. The TES is fabricated on a free-standing ${\mathrm{Si}}_3{\mathrm{N}}_4$ membrane. Inset: A SEM micrograph of the TES. The normal-metal bars extending partially across the TES have been experimentally verified to improve the energy resolution of the detector (426). Courtesy of J. Ullom, NIST.

Figure 22

(Color online) A prototype $\mathrm{Ti}∕\mathrm{Au}$ microcalorimeter array. The dark regions within the ${\mathrm{Si}}_3{\mathrm{N}}_4$ membrane are holes in the membrane. Courtesy of SRON-MESA.

Figure 23

(Color online) Basic Peltier thermoelement.

Figure 24

Thermoelectric refrigeration at cryogenic temperatures using cerium hexaboride. Adapted from 164. Reprinted with permission from AIP, copyright 2003.

Figure 25

Schematic diagram of the thermoelectromechanical cooler, time sequences of the pulsed current applied to the device, and two modes of cantilever contact: synchronized and optimized. 292. Reprinted with permission from AIP, copyright 1999.

Figure 26

(Color online) Characteristics of (SI)NIS refrigerators. (a) Calculated cooling power $\dot{Q}$ of a NIS junction vs bias voltage $V$ for several temperatures $T=T_{e,N}=T_{e,S}$. Also shown is the behavior of a NIN junction. (b) $\dot{Q}$ calculated at the optimal bias voltage as a function of temperature, assuming both a temperature-independent energy gap (dash-dotted line) and the real BCS dependence (solid line). $T_{\mathrm{opt}}$ indicates the temperature value that maximizes $\dot{Q}$. (c) Coefficient of performance $\eta$ calculated at the optimal bias voltage vs temperature. (d) Scheme of the energy-band diagram of a voltage-biased SINIS junction. The electric current flows into the normal region through one tunnel junction and out through the other, while the heat current $\dot{Q}$ flows out of the N electrode through both tunnel junctions.

Figure 27

(Color online) SINIS refrigerator. (a) Scanning electron micrograph of a typical SINIS microrefrigerator. The structure was fabricated by standard electron-beam lithography combined with Al and Cu UHV $e$-beam evaporation. The probe junction resistance usually satisfies the condition $R_p⪢R_T$ in order to prevent self-cooling. (b) Sketch of a typical measurement setup for electron cooling. (c) Electron temperature in the N region vs $V_{\mathrm{refr}}$ measured at different bath temperatures. Dots represent experimental data, while solid lines are theoretical fits. Inset: The cooling power (pW) against temperature obtained from the fits (dots). (d) Contour plot of the theoretical electron temperature $T_{e,N}$ vs bias voltage $V$ and $T_{e,S}$ for a SINIS cooler assuming thermal load due to the electron-phonon interaction (see text; note that in a gray-scale version of this contour plot some features may not be correctly presented). (a) Adapted from 337; (c) from 253. Reprinted with permission from AIP, copyright 1996.

Figure 28

(Color online) Optimization of SINIS refrigerators. (a) SEM micrograph of an $\mathrm{Al}∕{\mathrm{Al}}_2{\mathrm{O}}_3∕\mathrm{Cu}$ SINIS microrefrigerator exploiting large-area junctions $(\sim 10\mu {\mathrm{m}}^2)$ with quasiparticle traps. (b) SEM image of a part of a comblike SINIS structure with $10+10$ junctions for cooling and a SINIS thermometer. (c) Electron temperature $T_{e,N}$ in the N region of an $\mathrm{Al}∕{\mathrm{Al}}_2{\mathrm{O}}_3\mathrm{Cu}$ SINIS refrigerator vs $V_{\mathrm{refr}}$ measured at different bath temperatures. The lowest curve (dash-dotted line) shows the anomalous heating effect observable at the lowest temperatures and attributed to the presence of quasiparticle states within the superconducting gap. (d) Theoretical ultimate minimum electron temperature of a SINIS cooler $T_{\min }∕T_c$ at $V\simeq 2\Delta ∕e$ vs $\Gamma ∕\Delta$ in quasiequilibrium. (a) Adapted from 336, reprinted with permission from AIP, copyright 2000; (b) from 265, with permission of Springer Science and Business Media; (c), (d) from 337.

Figure 29

(Color online) ${\mathrm{S}}_1{\mathrm{I}\mathrm{S}}_2\left({\mathrm{I}\mathrm{S}}_1\right)$ refrigerator. (a) Calculated cooling power $\dot{Q}$ of a ${\mathrm{S}}_1{\mathrm{I}\mathrm{S}}_2$ junction vs bias voltage $V$ at $T=T_{e,S1}=T_{e,S2}=0.2{\Delta }_1\left(0\right)∕k_B$ for several ${\Delta }_2∕{\Delta }_1$ ratios. Dotted line represents $\dot{Q}$ when ${\mathrm{S}}_2$ is in the normal state. (b) Measured cooling of a Ti island to and in the superconducting state by quasiparticle tunneling. $R_0$ is related to the electron temperature $T_{e,S2}$ in Ti (see text). Dashed lines: Ti in the normal state. Thin solid lines: Ti cooled from the normal to the superconducting state. Thick solid line: Ti superconducting at $V_{\mathrm{refr}}=0$. (c) Measured minimum electron temperature $\left(T_{\min }\right)$ of Ti vs bath temperature $T_0$. (b), (c) Adapted from 275. Reprinted with permission from AIP, copyright 1999.

Figure 30

(Color online) Semiconductor-superconductor structures for electronic cooling. (a) Optical micrograph of a typical $\mathrm{Al}∕\mathrm{Si}∕\mathrm{Al}$ microrefrigerator and schematics of the measurement setup. The current $I_0$ and voltage $V_{\mathrm{th}}$ are used to determine the electron temperature in the $n^{++}\text{−}\mathrm{Si}$ layer, and the bias voltage $V$ is used to change its value. (b) Cross section of the structure along the line $AB$ in (a). (c) Measured electron temperature $T_{e,N}$ in Si vs bias voltage across the $\mathrm{Al}∕\mathrm{Si}∕\mathrm{Al}$ cooler at different substrate temperatures. Adapted from 376, with permission from Elsevier.

Figure 31

(Color online) Superconductor-ferromagnet refrigerator. (a) Calculated heat current $\dot{Q}$ of a SF junction vs bias voltage $V$ at $T=T_{e,F}=T_{e,S}=0.4\Delta \left(0\right)∕k_B$ for several spin polarizations $\mathcal{P}$. Inset: The same quantity calculated at the optimal bias voltage against temperature for some values of $\mathcal{P}$. (b) Calculated maximum cooling power surface density ${\dot{Q}}_A$ vs $\mathcal{P}$ for various temperatures. Inset: The coefficient of performance $\eta$ calculated at the optimal bias voltage vs temperature for some $\mathcal{P}$ values. Adapted from 139, 140. Reprinted with permission from AIP, copyright 2002.

Figure 32

(Color online) Lattice refrigeration with SINIS. (a) SEM image of a ${\mathrm{Si}}_3{\mathrm{N}}_4$ membrane (in the center) with self-suspended bridges. Two normal-metal cold fingers extending onto the membrane are used to cool down the dielectric platform. The $\mathrm{Al}∕{\mathrm{Al}}_2{\mathrm{O}}_3∕\mathrm{Cu}$ SINIS coolers are on the bulk (far down and top) and the thermometer stands in the middle of the membrane. (b) Maximum temperature decrease $(T_{\min }∕T_0)$ vs bath temperature $T_0$ measured at different positions in the cooler shown in (a). (c) Maximum lattice temperature decrease on the membrane vs bath temperature measured in other samples similar to that shown in (a). In this case, the refrigerators exploited many small-area junctions arranged in parallel in a comblike configuration. (d) SEM micrograph of a NIS refrigerator device with a neutron-transmutation-doped (NTD) Ge resistance thermometer attached on top of it. (c) Adapted from 265, with permission of Springer Science and Business Media; (d) from 79. Reprinted with permission from AIP, copyright 2005.

Figure 33

(Color online) Josephson transistor. (a) Simplified scheme of a SINIS-controlled Josephson transistor. The Josephson current in the ${\mathrm{S}}_J\mathrm{N}{\mathrm{S}}_J$ weak link (along the dashed line) is controlled by applying a bias $V_{\mathrm{SINIS}}$ across the SINIS line connected to the weak link, allowing one to increase or decrease its magnitude with respect to equilibrium. (b) Theoretical normalized critical current $I_J$ vs $V_{\mathrm{SINIS}}$ calculated in the quasiequilibrium limit for several lattice temperatures $T_0$ for a $\mathrm{Nb}∕\mathrm{Cu}∕\mathrm{Nb}$ long junction. Inset: The supercurrent vs electron temperature characteristic calculated at $\varphi =\pi ∕2$. (c) SEM image of an $\mathrm{Al}∕\mathrm{Cu}∕\mathrm{Al}$ Josephson junction including the $\mathrm{Al}∕{\mathrm{Al}}_2{\mathrm{O}}_3∕\mathrm{Cu}$ symmetric SINIS electron cooler. The SNS long weak link is placed in the middle of the structure. Also shown is a scheme of the measurement circuit. (d) Measured critical current $I_J$ vs $V_{\mathrm{SINIS}}$ at three different bath temperatures $T_0$ for the device shown in (c). Inset: The measured differential current gain $G_I$ vs $I_{\mathrm{SINIS}}$ at $T_0=72\mathrm{mK}$. (b) Adapted from Giazotto, Taddei, et al., 2003, reprinted with permission from AIP, copyright 2003; (c), (d) from 375. Reprinted with permission from AIP, copyright 2004.

Figure 34

Refrigeration by combined tunneling and thermionic emission. (a) Schematic diagram of the potential barrier profile $V\left(x\right)$ for $\Phi =1\mathrm{eV}$ and with electrode separation of $60\mathrm{\AA }$. (b) Heat-flow distribution at $T=300\mathrm{K}$. Adapted from 177. Reprinted with permission from AIP, copyright 2001.

Figure 35

Quantum-dot refrigerator. (a) Scheme of a quantum-dot refrigerator. (b) Energy-level diagram of the structure. The reservoir $R$ is cooled as its quasiparticle distribution function is sharpened by resonant tunneling through quantum dots $D_{\mathrm{L}}$ and $D_{\mathrm{R}}$ to the electrodes $V_{\mathrm{L}}$ and $V_{\mathrm{R}}$. From 117.