Observing temperature fluctuations of a mesoscopic electron system

TL;DR

This paper experimentally observes temperature fluctuations in a mesoscopic electron system, confirming theoretical predictions and revealing how nonequilibrium conditions affect these fluctuations, which are crucial for calorimeter sensitivity.

Contribution

First direct measurement of temperature fluctuations in a nanoscale electron system under equilibrium and nonequilibrium conditions.

Findings

Temperature fluctuations follow fluctuation dissipation theorem at equilibrium.

Nonequilibrium conditions cause nontrivial temperature fluctuation dependence on chemical potential bias.

Results establish fundamental limits for calorimeter energy resolution.

Abstract

Almost a century ago, Johnson and Nyquist presented evidence of fluctuating electrical current and the governing fluctuation dissipation theorem (FDT). Whether, likewise, temperature T can fluctuate is a controversial topic and has led to scientific debates for several decades. To resolve this issue, there was an experiment initially in 1992 where the authors found good agreement between the FDT theory for heat and experiment on a macroscopic sample. A key question is what happens when we consider a nanoscale system with much fewer particles at 100 times lower temperatures. This challenge has not been addressed up to now, due to the demanding experimental requirement on fast and sensitive thermometry on a mesoscopic absorber. Here we observe equilibrium fluctuations of temperature in a canonical system of about 10^8 electrons exchanging energy with phonon bath at a fixed temperature. Moreover, temperature fluctuations under nonequilibrium conditions present a nontrivial dependence on the chemical potential bias of a hot electron source. These fundamental fluctuations of T set the ultimate lower bound of the energy resolution of a calorimeter.

Equations

$$S_T(\omega )=\frac{2k_B{T}^2}{G_{\mathrm{th}}}\frac{1}{1+{\omega }^2{C}^2/G_{\mathrm{th}}^2}.$$

$$S_T(0)=2k_B{T}^2/G_{\mathrm{th}},$$

$$G_{th}=5\Sigma \mathcal{V}{T}^4+\alpha gT.$$

$$\Theta =\sqrt{\frac{2k_{\mathrm{B}}}{5\Sigma \mathcal{V}}}{T}^{-1}(\mathrm{high}T)$$

$$\Theta =\sqrt{\frac{2k_{\mathrm{B}}}{\alpha G_Q}}{T}^{1/2}(\mathrm{low}T),$$

$$S_{\dot{Q}}^{\mathrm{shot}}=\frac{1}{e^2{R}_T}\int dE(E-eV{)}^2{n}_{\mathrm{S}}(E)$$

$$\{f_{\mathrm{N}}(E-eV)[1-f_{\mathrm{S}}(E)]+f_{\mathrm{S}}(E)[1-f_{\mathrm{N}}(E-eV)]\},$$

$$S_{21}=20\lg (v/\tilde{v}),$$

$$\delta S_{21}=\frac{20}{\ln 10}\frac{\delta v}{v},$$

$$\sqrt{⟨\delta S_{21,\mathrm{bg}}^2⟩}=\frac{20}{\ln 10}\frac{\sqrt{⟨\delta v^2⟩}}{\tilde{v}}10^{-⟨S_{21}⟩/20}.$$

Full text

Observing temperature fluctuations of a mesoscopic electron system

Bayan Karimi

QTF Centre of Excellence, Department of Applied Physics, Aalto University School of Science, P.O. Box 13500, 00076 Aalto, Finland

  

Fredrik Brange

Department of Physics and NanoLund, Lund University, Box 188, SE-221 00 Lund, Sweden

  

Peter Samuelsson

Department of Physics and NanoLund, Lund University, Box 188, SE-221 00 Lund, Sweden

  

Jukka P. Pekola

QTF Centre of Excellence, Department of Applied Physics, Aalto University School of Science, P.O. Box 13500, 00076 Aalto, Finland

Almost a century ago, Johnson and Nyquist johnson ; nyquist presented evidence of fluctuating electrical current and the governing fluctuation dissipation theorem (FDT). Whether, likewise, temperature ${T}$ can fluctuate is a controversial topic and has led to scientific debates for several decades lifshitz ; kittel ; mandelbrot . To resolve this issue, there was an experiment initially in 1992 lipa ; lipa2 where the authors found good agreement between the FDT theory for heat and experiment on a macroscopic sample. A key question is what happens when we consider a nanoscale system with much fewer particles at 100 times lower temperatures. This challenge has not been addressed up to now, due to the demanding experimental requirement on fast and sensitive thermometry on a mesoscopic absorber. Here we observe equilibrium fluctuations of temperature in a canonical system of about $10^{8}$ electrons exchanging energy with phonon bath at a fixed temperature. Moreover, temperature fluctuations under nonequilibrium conditions present a nontrivial dependence on the chemical potential bias of a hot electron source. These fundamental fluctuations of $T$ set the ultimate lower bound of the energy resolution of a calorimeter.

Consider fluctuations of a system with coupling to a heat bath at temperature $T$ for which the classical FDT of heat current $\dot{Q}$ holds in form $S_{\dot{Q}}^{\rm eq}=2k_{B}T^{2}G_{{\rm th}}$ in equilibrium. Here $G_{\rm th}$ is the heat conductance to the bath. We can write the energy balance equation $\dot{Q}=Cd{\widetilde{T}}/dt$ for the temperature of the system ${\widetilde{T}}(t)=T+\delta T(t)$ at time $t$, where $C$ denotes the heat capacity. The heat current is composed of its expectation value $-G_{\rm th}\delta T$ and fluctuations $\delta\dot{Q}$ around it. There are two origins of noise in this heat current: 1. there is standard randomness of a transport known for particle current noise (time randomness), and 2. the energies exchanged are also to some extent random leading to enhancement of fluctuations on top of those known for particle current only. We obtain the noise spectrum of temperature of the system by Fourier transformation as $S_{T}(\omega)=\int dte^{i\omega t}\langle\delta T(t)\delta T(0)\rangle$. This yields under steady state conditions

[TABLE]

At low frequencies we have

[TABLE]

and the spectrum has Lorentzian cut-off at $\omega_{c}=G_{{\rm th}}/C$. These results hold also for a system coupled to several equilibrium baths, if one takes $G_{\rm th}$ to represent the sum of all the individual thermal conductances to these baths. For the rms fluctuations we obtain the well-known result $\langle\delta T^{2}\rangle=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}S_{T}(\omega)=k_{B}T^{2}/C$ lifshitz .

In a fermionic system, like the electrons in the calorimeter in the present experiment, temperature is coded in the Fermi distribution $f(\epsilon)=[e^{(\epsilon-\mu)/k_{\rm B}T}+1]^{-1}$ which directly determines the readout signal of our thermometer. Here, $\epsilon$ and $\mu$ denote the single particle energy and chemical potential, respectively. We illustrate the calorimeter xray ; mikko principle of our experiment and setup in Fig. 1 brange . The electron system (absorber), is coupled to the phonon heat bath at constant temperature $T$ via electron-phonon collisions which lead to stochastic exchange of heat, as indicated by the many vertical arrows between the two in Fig. 1a. The arrows from the left depict the electronic injection of heat under nonequilibrium conditions, fluctuating due to the stochastic nature of tunneling. By attaching a fast thermometer to the absorber, one records its time $t$ dependent temperature fluctuations $\delta T(t)$ as shown by a measured time trace. The actual sample (scanning electron micrograph in Fig. 1b) is realised as a $L=1~{}\mu$m long copper normal-metal absorber (brown) connected to three superconducting leads (blue). The right one is a tunnel contact of the thermometer and the other tunnel junction on the left the hot electron injector. The third one pointing down and $50$ nm away from the thermometer, is a direct clean metal-to-metal contact grounded at the sample stage. It provides a fixed chemical potential for the absorber and induces proximity superconductivity to the thermometer facilitating its proper operation. The measuring setup for the thermometer junction shown on the right side of Fig. 1b consists of a parallel on-chip $LC$ resonator, coupled to input $V_{1}$ and output $V_{2}$ rf lines, operating at $f_{0}=620$ MHz which also admits DC biasing at voltage $V_{\rm th}$. The measured signal $S_{21}$ obtained from the ratio of $V_{2}/V_{1}$, yields the conductance of the thermometer junction. It is typically measured at $10$ kHz sampling rate in order to acquire statistics of temporal temperature of the absorber.

In order to calibrate the thermometer we measure $S_{21}$ averaged over typically $1$ s time interval at different bath temperatures of the cryostat, traceable to primary Coulomb blockade thermometry CBT. An example of dependence of thus obtained $\langle S_{21}\rangle$ on $V_{\rm th}$ is shown on a wide bias range in Fig. 2a. The drop of $\langle S_{21}\rangle$ at about $\pm 200~{}\mu$V is due to the superconducting gap $\Delta$ in aluminum. The main feature, the zero bias anomaly (ZBA) at $V_{\rm th}=0$ which is indicated by the central red arrow, presents the basis of our thermometer. This dip originates from proximity induced supercurrent due to the presence of clean contact. Now it is placed $50$ nm away from the tunnel junction which is to be contrasted to $500$ nm in our earlier work bj1 ; this way the sensitivity of the thermometer is enhanced substantially. Quantitatively, the temperature dependence of the transmission $\langle S_{21}\rangle$ at this dip is depicted in Fig. 2b. It manifests approximately linear dependence at sub $200$ mK down to below $20$ mK temperatures, emphasised by the zoom in the inset of this figure. Owing to the competing quasiparticle tunneling, there is eventually back-bending of the characteristics at temperatures above $300$ mK; this leads to temporal loss of sensitivity in this temperature range. Depending on the range of interest, we employ either linear or nonlinear calibration to convert $\langle S_{21}\rangle$ to temperature. As tested and demonstrated in Ref. bj1 , the temperature measured by $\langle S_{21}\rangle$ of the ZBA thermometer in a similar setup is that of the absorber electrons that we indeed want to monitor.

Time domain measurements allow detecting temporal fluctuations of the quantity of interest. In our case we monitor $S_{21}(t)$, yielding the instantaneous temperature of the absorber at $10$ kHz sampling rate over a chosen time interval. We collect data under given conditions typically for up to $1$ hour. As a result we obtain the total fluctuations (variance) $\langle\delta S_{21,{\rm tot}}^{2}\rangle$ in a bandwidth of $\Delta f\approx 10$ kHz. This signal is composed of the amplifier and other instrumental noise $\langle\delta S_{21,{\rm bg}}^{2}\rangle$ (”bg” stands for background), in addition to the noise of interest from the actual sample, $\langle\delta S_{21}^{2}\rangle$=$\langle\delta S_{21,{\rm tot}}^{2}\rangle$-$\langle\delta S_{21,{\rm bg}}^{2}\rangle$. Here we assume uncorrelated noise from the different sources. The way we determine the $\langle\delta S_{21,{\rm bg}}^{2}\rangle$ is explained in the Methods section. Our quantitative results depend critically on the precision of determining this background noise. Taking the linear calibration as in the inset of Fig. 2b, with the responsivity $\mathbb{R}\equiv|d\langle S_{21}\rangle/dT|$, we have for the temperature of the absorber $\langle\delta T^{2}\rangle=\mathbb{R}^{-2}\langle\delta S_{21}^{2}\rangle$. We exhibit in Fig. 3 the central quantity in the experiment, low frequency temperature fluctuations $\sqrt{S_{T}}=\sqrt{\langle\delta T^{2}\rangle/2\Delta f}$ as a function of bath temperature in equilibrium. From now on we denote $\Theta\equiv\sqrt{S_{T}}$ which can also be associated to the noise-equivalent temperature $\mathbb{NET}_{0}$, where with subscript [math] we refer to the fundamental temperature fluctuations discussed here. The data symbols correspond to the averaged bare noise, where the best guess of the background has been subtracted. The shaded area depicts the uncertainty in determining $\Theta$ precisely due to this subtraction. Overall, we observe first increase of $\Theta$ upon lowering $T$ and then gradual turn down of it at the lowest temperatures. The dominant contributions to $G_{th}$ arise from electron-phonon coupling at higher temperatures and radiative heat transfer by thermal photons schmidt towards low $T$ as

[TABLE]

Here $\Sigma$, $\mathcal{V}$ are electron-phonon coupling constant Wellstood and volume of the absorber, respectively. For the photonic contribution schmidt , $G_{Q}=gT$ is the quantum of thermal conductance with $g=\pi k_{\rm B}^{2}/6\hbar$. We assume the coupling coefficient $\alpha$ to have values $\ll 1$. Equation (2) predicts then

[TABLE]

with cross-over between the two regimes at the temperature $T_{\rm co}=(\frac{\alpha g}{10\Sigma\mathcal{V}})^{1/3}$. Using the literature value $\Sigma=2\times 10^{9}$ $\rm WK^{-5}m^{-3}$ Libin , the measured volume $\mathcal{V}=1.0\times 10^{-21}$ $\rm m^{3}$ and an educated guess $\alpha\simeq 10^{-3}$ timofeev according to earlier investigations, we obtain a predicted $\Theta$ versus $T$. Our simple model above predicts a maximum of $\Theta$ at $\sim~{}35$ mK with the value of about $30~{}\mu{\rm K}/\sqrt{\rm Hz}$. This is outside the shaded area of $60-130~{}\mu{\rm K}/\sqrt{\rm Hz}$ of the measured signal in Fig. 3. A possible origin of this discrepancy lies in that we assume the absorber to be in the normal state. However, the clean absorber-superconductor contact leads to a proximity induced superconductivity in the absorber. This suppresses the density of states around the Fermi level, on the scale of the Thouless energy $E_{\rm Th}=\hbar D/L^{2}\sim 10~{}\mu$eV, resulting in a decreased electron-phonon coupling. Here, $D\sim 0.01$ ${\rm m}^{2}/{\rm s}$ is the diffusion constant of the Cu film. As a consequence, for electron temperatures $T\lesssim E_{\rm Th}/k_{\rm B}\sim 100$ mK, the thermal conductance $G_{\text{th}}$ is decreased tero2 and, hence, the temperature noise $\Theta$ is increased. Furthermore, the temperature calibration, i.e., the responsivity $\mathbb{R}$ of the thermometer gets more unreliable towards the lowest temperatures. The overall magnitude at $T>T_{\rm co}$, determined by the electron-phonon heat conductance (no fit parameters) is thus in fair agreement with the experiment within the uncertainty of the measurement as explained. The cross-over, more of phenomenological origin, is also consistent with the experiment with the chosen value of $\alpha=10^{-3}$. Another possible contribution to this cross-over may arise from the fact that the fluctuations $\delta T$ of temperature become comparable to $T$ itself in the lowest temperatures.

Let us next consider the nonequilibrium fluctuations tero ; ciliberto ; jukka2 . In the measurements up to now the injector junction on the left in Fig. 1b has been unbiased in order to ensure equilibrium. By applying a voltage $V$ to it, the system can be driven into nonequilibrium. The well-known influence of such biasing of a superconductor-normal metal junction is that it serves as a local refrigerator of the normal-metal absorber thanks to the energy gap of the superconductor, i.e., it acts as an evaporative cooler Nahum . This effect is manifested in the bias dependence of the average temperature of the absorber, obtained from the values of $\langle S_{21}\rangle$ in Fig. 4a.

Injecting electrons does not only change the average temperature of the absorber but, due to the stochastic nature of tunneling, it leads to noise of heat current as well Dimag . Quantitatively the low frequency heat current noise is given by

[TABLE]

where $n_{S}(E)=|E|/\sqrt{E^{2}-\Delta^{2}}\theta(|E|-\Delta)$ denotes the density of states for superconductor and subscripts N and S stand for normal metal and superconductor, respectively. For typical voltages and temperatures in the regime well below the superconducting gap, the injection noise $\sqrt{S_{\dot{Q}}^{\rm in}}$ is exponentially suppressed brange . In contrast, the equilibrium noise due to phonons, $\sqrt{S_{\dot{Q}}^{\rm eq}}$, is of a roughly constant magnitude $\sim 10^{-20}{\rm W}/\sqrt{\rm Hz}$. Therefore it is not surprising that the temperature noise in Fig. 4b does not change much at sub-gap voltages $V<200~{}\mu$V, in particular as the temperature of the absorber is not changing dramatically in this bias range. For these uncorrelated sources the temperature noise is predicted to obey $S_{T}=(S_{\dot{Q}}^{\rm eq}+S_{\dot{Q}}^{\rm in})/{G_{\rm th}}$. The sudden decrease of temperature noise $\Theta$ at $V>200$ $\mu$V is natural since $G_{\rm th}$ increases rapidly when the absorber heats up in this regime (see Fig. 4a). We consider the sharp peak at the gap (Fig. 4b) to be an artefact arising from unavoidable voltage noise of injector, which directly transforms to temperature noise due to the strong voltage dependence of temperature at this point.

Finally, what is the temperature that fluctuates? In fact, it is the parameter of the distribution of the electrons in the absorber that we monitor. It qualifies as temperature for the following reasons. (i) Number of particles is large, about $10^{8}$. (ii) Due to fast electron-electron internal relaxation over a time scale of $\sim 10^{-9}$ s pothier , the carriers form a local Fermi-Dirac distribution: all other relaxation rates, most notably the electron-phonon time ($\sim 10^{-5}$ s) are much slower viisanen . Furthermore, the temperature of the absorber is spatially uniform, since the heat diffusion time of electrons in the absorber, $\tau_{\rm diff}=\gamma\rho l^{2}/\mathcal{L}_{0}\sim 10^{-10}$ s is very short. Here, $c=\gamma T$, is the specific heat due to conductance electrons with $\gamma\sim 10^{2}$ ${\rm Wm^{-3}K^{-2}}$, $\rho\sim 10^{-8}$ $\Omega$m is the resistivity of the Cu, $l=1~{}\mu$m is the length of the absorber, and $\mathcal{L}_{0}=2.44\times 10^{-8}~{}{\rm W\Omega K^{-2}}$ is the Lorenz number.

Methods

Background measurements

We measure the instrumental noise dominated by that of the low temperature Caltech CITLF2 cryogenic SiGe low noise amplifier $\langle\delta S_{21,{\rm bg}}^{2}\rangle$ by carefully off-tuning the interesting fluctuations from the sample itself. This is achieved by simultaneously (i) biasing the thermometer junction away from the ZBA regime ($V_{\rm th}\simeq 85~{}\mu$V), and (ii) measuring at frequencies either below or above the resonance at $f_{0}$. An example of the corresponding parametric background noise measurement, in form $\sqrt{\langle\delta S_{21,{\rm bg}}^{2}\rangle}$ versus $\langle S_{21}\rangle$ is presented in Fig. 5. We see a typical increase of noise when the attenuation increases towards left. This dependence can be understood quantitatively by assuming constant voltage noise independent of $\langle S_{21}\rangle$. The measured transmission can be written as

[TABLE]

where $v$ is the output of the last stage amplifier, $\widetilde{v}=\sqrt{50~{}{\rm\Omega}\times 1~{}{\rm mW}}\simeq 224$ mV. Noise of $v$ translates then into variations of $S_{21}$ in linear regime as

[TABLE]

and can be written with the help of Eq. (6) for the rms values as

[TABLE]

Based on the fit parameter $a$ in Fig. 5a and the total gain of $60$ dB of the amplifier chain, we find the input voltage noise to be $\sim~{}12$ nV corresponding to the noise temperature of the amplifier of $T_{\rm n}\sim~{}5$ K which is in line with its specifications by the manufacturer.

Figure 5b presents background measurements at frequencies both below and above the resonance over a wide range of attenuation $\langle S_{21}\rangle$. We observe two features that we need to consider when making an accurate evaluation of the $\langle\delta S_{21,{\rm bg}}^{2}\rangle$. First, at large attenuations, due to the fact that the changes are not fully linear in the sense of Eq. (7), the exponential dependence of Eq. (8) is not obeyed strictly. Therefore we resort to polynomial fits in two regimes, to capture the dependence over the full range. Second, there is a weak dependence of the amplifier noise on frequency; thus the data taken below and above the resonance differ from each other slightly. What we do then, e.g., in Fig. 3, is that we take the mean between the two background measurements as the reference and indicate by the shaded area the uncertainty incurred due to the difference between the two extremes. We thus assume that the frequency dependence of the noise is more or less smooth in the narrow range of $\sim~{}10$ MHz around $f_{0}$, and interpolate the data accordingly.

Experimental details

The sample (Fig. 1b) was fabricated on standard oxidized Si substrate using Ge process for achieving robust deposition mask meschke ; pashkin . The electron-beam lithography was used to pattern the structure for three-angle shadow evaporation of metals. First we deposit $20$ nm of Al making the leads followed by oxidation in pure $O_{2}$ (1 min at 1 mbar). Next another Al layer of $20$ nm thickness again provides the clean superconducting contact at the distance of $50$ nm from the thermometer junction, and finally we deposit $35$ nm Cu to form the absorber. The resonator is a spiral on a separate chip made of $100$ nm thick Al by simple one angle evaporation. The heart of the measuring setup is shown in Fig. 1b with inductance $L=100$ nH, $C_{1}=10.3$ fF and $C_{2}=59.3$ fF as coupling capacitors, and $C=0.2$ pF. The rest of the RF circuitry follows closely to what is presented in Ref. viisanen .

Author contributions

The experiment was proposed by J. P. and its realization was conceived by all the authors. B. K. performed the experiment, and designed and fabricated the samples. Data analysis and modeling were performed by B. K. and J. P., with contributions on the noise analysis by F. B. and P. S. The manuscript was written by B. K. and J. P.

acknowledgments

We acknowledge J. T. Peltonen, E. T. Mannila, O-P. Saira and S. Gasparinetti for technical support, W. Belzig, D. Nikolic, I. Khaymovich, and T. Tuukkanen for discussions and tests of thermometry, and M. Campisi and K. Saito for useful discussions. This work was funded through Academy of Finland grants 297240, 312057 and 303677 and from the European Union’s Horizon 2020 research and innovation programme under the European Research Council (ERC) programme and Marie Sklodowska-Curie actions (grant agreements 742559 and 766025). F.B and P.S. were supported by the Swedish VR. We acknowledge the facilities and technical support of Otaniemi research infrastructure for Micro and Nanotechnologies (OtaNano).