Development of high frequency and wide bandwidth Johnson noise thermometry

We develop a high frequency, wide bandwidth radiometer operating at room temperature, which augments the traditional technique of Johnson noise thermometry for nanoscale thermal transport studies. Employing low noise amplifiers and an analog multiplier operating at 2~GHz, auto- and cross-correlated Johnson noise measurements are performed in the temperature range of 3 to 300~K, achieving a sensitivity of 5.5~mK (110 ppm) in 1 second of integration time. This setup allows us to measure the thermal conductance of a boron nitride encapsulated monolayer graphene device over a wide temperature range. Our data shows a high power law (T$^{\sim4}$) deviation from the Wiedemann-Franz law above T$\sim$100~K.

As modern electronics continue to miniaturize and increase in operation frequency, heat dissipation has begun to bottleneck their performance [1].Understanding thermal transport in these complex devices requires new thermometry techniques capable of dealing with the challenges unique to nanoscale systems [2].Diminishing system sizes call for measurements to be noninvasive to avoid thermal agitation of minute heat capacities.As weak electron-phonon coupling can result in different electronic and lattice temperatures [3,4], separate probes must be used to measure each temperature independently.Various mesoscopic experimental techniques, such as resistive thermometry, normal-insulatorsuperconductor thermometry, Coulomb blockade thermometry, and shot noise thermometry [5][6][7], have been developed to meet these requirements.However, a versatile thermometry that works in a wide range of temperatures and experimental conditions (such as under strong magnetic field) has yet to be developed at the nanoscale.
Fundamentally based upon the fluctuation-dissipation theorem [8], Johnson noise thermometry (JNT) is a primary thermometry having a straight forward interpretation, independent of the material details, that stands out as a natural candidate for nanoscale thermal measurements.Temperature is measured by passively monitoring fluctuations of the conducting components within the nanoscale device without current excitations.Difficulties in measuring low level voltage noise [5] has limited the applications of JNT to metrology [9,10] and extreme environments such as nuclear reactors [11] and ultralow temperatures [5].Recently, considerable progress has been made in the measurement of noise on mesoscopic devices by applying radiometry techniques [4,7,12,13], opening up the possibility of using JNT to study a wide range of mesoscopic phenomena.In this letter, we incorporate room temperature low noise amplifiers and an analog linear multiplier operating at 2 GHz to develop JNT for nanoscale devices.A precision of 0.01% is achieved on auto-and cross-correlated noise measurements in 1 sec-ond of integration time.We demonstrate the capability of this setup by measuring the electronic thermal conductance of a mesoscopic graphene device over a temperature range of 3-300 K.
Johnson noise results from the spontaneous thermal fluctuation of charges in dissipative electrical elements at finite temperature.The time averaged mean-square voltage, V 2 , across a resistance, R, is described by the Nyquist theorem, V 2 = 4Rk B T e ∆f where k B is the Boltzmann constant, T e is the electron temperature, and ∆f is the equivalent noise bandwidth of the system.The Nyquist theorem holds in the limit of the photon energy being much smaller than the thermal energy, such that hf ≪ k B T e , where h is the Planck constant and f is the frequency.The full quantum mechanical description is given by [14] where Z is the complex impedance of the dissipative element.The first term in Eqn.(1) is the zero point motion of the photon field while the second term is the onedimensional blackbody radiation that our radiometer detects [12].The spectrum is frequency independent until rolling off above k B T e /h.At 1 K, this roll off is centered at 20 GHz, which sets the upper bound of the JNT operating frequency.
In typical mesoscale conducting samples, a characteristic value of the channel resistance is on the order of h/e 2 ∼25 kΩ, much larger than the 50 Ω characteristic impedance typical of high frequency, low noise amplifiers (LNA).To account for this mismatch, the Nyquist theorem can be rewritten to describe the Johnson noise power absorbed by an amplifier with characteristic impedance Z 0 , as . For large sample impedance, an LC tank circuit can be employed to couple the noise power into the amplifier [13,15].the LNA for Johnson noise amplification.The signalto-noise ratio of the noise measurement is mostly determined by this front-end LNA [16].The SiGe LNA (Caltech CITLF3) used in this report has a room temperature noise figure, in the frequency range of 0.01 to 2 GHz, of about 0.64 dB, corresponding to a noise temperature of 46 K.A low pass filter and homodyne mixer define a bandwidth of 328 MHz centered at 1 GHz.The center frequency should be high enough to avoid 1/f fluctuations and allow wide bandwidth noise measurements, while low enough to avoid stray capacitance in the device and the high frequency Johnson noise roll off.An analog linear multiplier and RF power splitter detects the noise power.Operating from dc to 2 GHz, the multiplier, Analog Devices ADL5931, serves as a square law detector with 30 dB dynamic range.The noise power is modulated by a microwave switch at 13.7 Hz and subsequently measured by a lock-in amplifier.
By varying the temperature T of the resistive load attached to a cold finger in a cryostat, the auto-correlated Johnson noise data is collected.As shown in the inset of Fig. 2, the signal is linear in device temperature and offset by the constant amplifier noise over the measured temperature range of 3-300 K. Calibration is done through the Nyquist equation resulting in the main panel of Fig. 2. The autocorrelation data is best described by the equation: S V = Gk B (T e + T N ) where G is the proportional gain factor set by the LNA amplification together with the insertion loss of microwave components and T N is the total system noise temperature.As T e tends to zero, Johnson noise subsides but the system noise remains.We find that our auto-correlation setup has a T N of 68 K, consistent with the LNA specification at room temperature.We note that this can be further reduced by lowering the LNA operating temperature.
Dissipation between the resistive load and the LNA, such as coaxial attenuation and contact resistance, can contaminate thermal transport measurements [9,17].Johnson noise from the sample is added to the unwanted Johnson noise from these lossy components.Crosscorrelation techniques can mitigate this problem by amplifying the Johnson noise signal of interest independently via two separate measurement lines [17][18][19][20] and discarding uncorrelated noise between the two channels.Previously, cross-correlation measurements were limited to frequencies below a few MHz due to the practical implementation of multipliers and digital processing speeds [18,20,21].The 2 GHz analog multiplier and SiGe LNA, combined with the lock-in amplifier modulation scheme described in Fig. 1(b), measure the correlated noise between the two channels, rejecting a large portion of the uncorrelated amplifier noise.Residual cross talk [17,21] between the two channels offsets the data by 2.6 K.
Johnson noise is a stochastic process with statistical uncertainties determined by the integration time and measurement bandwidth.The noise of our Johnson noise thermometer can be characterized by repeating a measurement multiple times and studying how it fluctuates about the mean.Fig. 3(a) compares two histograms, both containing 20,000 autocorrelation measurements at 50 K with 50 ms integration time but using two different bandwidths: 28 and 328 MHz.Wider bandwidth reduces the statistical uncertainties in Johnson noise thermometry, leading to the better performance demonstrated at 328 MHz.Fig. 3(b-c) plots the standard deviation of 1000 temperature measurements for 28 MHz and 328 MHz correlation bandwidth, respectively, as a function of integration time.In both autoand cross-correlation experiments, the sensitivity follows an inverse square root power law described by the Rice relation [9,22]: where δT is the measurement uncertainty, τ is the integration time and T sys characterizes the statistical noise in the system.While cross-correlation reduces the noise offset shown in Fig. 2 compared with autocorrelation, their statistical fluctuations are comparable as T sys is dominated by the amplifier noise.We attain 5.5 mK uncertainties on a 50 K signal (0.01% precision) in 1 second of integration time.
Finally, the capability of our JNT operating on nanoscale devices is demonstrated by measuring the electronic thermal conductance of a graphene device at a wide range of bath temperatures: 3 K, 30 K, and 300 K. Monolayer graphene is mechanically exfoliated, encapsulated in hexagonal boron nitride, and contacted along its 1-dimensional edge [23] to form the 2 µm × 6 µm channel shown in the insert of Fig. 4(b).A typical two-terminal channel resistance R of this device varies between 150-800 Ω depending on the back gate voltage.As maximal noise power is collected when the device is impedance matched to the measurement chain, an LC tank circuit is placed on chip to transform the graphene to 50 Ω [13,15] within the measurement bandwidth.The matching network, shown in Fig. 4(a), defines a bandwidth of 25 MHz centered at 133 MHz.For the results shown here, the graphene device is measured away from the charge neutrality point with hole density n ≈ 3.2×10 11 cm −2 , where R ≈208 Ω.The JNT is calibrated to a given sample using the autocorrelation setup shown in Fig. 1(a), following the procedure outlined for the results in Fig. 2. Fig. 4(a) shows the electronic thermal conductance measurement schematic.A sinusoidal heating current is injected into the graphene device at 13.7 Hz resulting in a cosinusoidal heating at 27.4 Hz.Fig. 4(b) shows the electron temperature, T e , as a function of the heating current I in the channel.The thermal conductance between the electronic system and the bath, G th , can be found from the relation between the heat flux injected into the graphene via Joule heating, Q = I 2 R and the rise in electron temperature, ∆T e , through Fourier's law: Q ≡ G th ∆T e .Fig. 4(b) shows the mean electron temperature as a function of heating current for three different bath temperatures.Solid lines are quadratic fits with the only fitting parameter being G th .This application demonstrates the versatility of our developed JNT over two orders of magnitude in temperature with wide variations in channel impedance.This high sensitivity noise measurement setup is also adaptable to study shot noise [24], electron-phonon coupling [4], single photon detection [13,25], and other correlation phenomena in mesoscopic devices, such as the Hanbury-Brown and Twiss effect [26].
In summary, we have developed a high frequency, wide bandwidth radiometer capable of both auto-and crosscorrelated Johnson noise thermometry at gigahertz frequencies.Using room temperature SiGe LNAs and an analog RF linear multiplier, we demonstrated fast and precise temperature measurements reaching sensitivity of 5.5 mK (0.01%) in 1 s of integration time.We tested our nanothermometer by measuring thermal transport in hexagonal boron nitride encapsulated, monolayer graphene from 3 to 300 K.
We would like to thank S. Weinreb for insightful discussion and fabricating the SiGe LNA; B. Blake and C. Ryan for digitizer drivers used for comparison to analog measurements; J. Ravichandran and B. Kaye for inspiration and critiques.KCF and TAO acknowledge Raytheon BBN Technologies support on this work.JC acknowledges the support of the FAME Center, sponsored by SRC MARCO and DARPA.XL is supported by DOE (DE-FG02-05ER46215).PK acknowledges a partial support from the Basic Science Research Program (2014R1A1A1004632) through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning.

Fig. 1 (FIG. 1 .
FIG. 1. Simplified electrical schematic for the auto-(a) and cross-correlation (b) Johnson noise thermometry setups.The SiGe low noise amplifiers (LNA) have a noise temperature of ∼50 K. Bandwidth is defined by a homodyne mixer and lowpass filter.Linear multiplication is performed at frequencies up to 2 GHz by an RF multiplier (ADL5931) that acts as a square-law detector (a) or a cross-correlator (b).A microwave switch chops the signal at 13.7 Hz, modulating the signal away from dc.

FIG. 2 .
FIG.2.Johnson noise of a 50 Ω resistive load as measured by auto-and cross-correlation setups, Fig.1 (a) and (b), respectively.The inset shows the lock-in amplifier outputs as a function of bath temperature.These signals are converted to noise power in the main panel using the Nyquist equation.The solid lines are linear fits, where the auto-and cross-correlation data exhibit an offset of 68 K and 2.6 K, respectively, due to amplifier noise (see text).

FIG. 3 .
FIG. 3. (a) Histograms of 20,000 auto-correlation temperature measurements for 28 and 328 MHz bandwidth using 50 ms integration time.Histogram peaks are normalized to 1 for clarity.Standard deviation of 1000 temperature measurements as a function of integration time for (b) 328 MHz and (c) 28 MHz bandwidth.All data is taken on a 50 K resistive load with uncertainties approaching 100 ppm.

2 FIG. 4 .
FIG. 4. (a) Simplified electrical schematic of the thermal conductance measurement.Graphene is impedance matched to ∼50 Ω in the measurement bandwidth by an on chip, LC tank circuit.Low frequency heating current is injected through the bias tee while high frequency Johnson noise is monitored similar to Fig. 1(a).(b) Electronic thermal conductance of a hexagonal boron nitride encapsulated, monolayer graphene device reported for three base temperatures.Solid lines are quadratic fits with G th as the only fitting parameter.Inset shows an image of the 2-terminal device.