Fluctuation-dissipation in thermoelectric sensors

Thermoelectric materials exhibit correlated transport of charge and heat. The Johnson-Nyquist noise formula 4k B T R for the spectral density of voltage fluctuations accounts for fluctuations associated solely with Ohmic dissipation. Applying the fluctuation-dissipation theorem, we generalize the Johnson-Nyquist formula for thermoelectrics, finding an enhanced voltage fluctuation spectral density 4k B T R(1 + Z D T) at frequencies below a thermal cut-off frequency f T , where Z D T is the dimensionless thermoelectric device figure of merit. The origin of the enhancement in voltage noise is thermoelectric coupling of temperature fluctuations. We use a wideband , integrated thermoelectric micro-device to experimentally confirm our findings. Measuring the Z D T enhanced voltage noise, we experimentally resolve temperature fluctuations with a root mean square amplitude of at a mean temperature of 295 K. We find that thermoelectric devices can be used for thermometry with sufficient resolution to measure the fundamental temperature fluctuations described by the fluctuation-dissipation theorem.

The fluctuation-dissipation theorem [1] relates the fluctuations in a system at thermodynamic equilibrium with the coefficient of irreversible dissipation under an externally applied bias field.In an electrical conductor, the fluctuation-dissipation theorem takes the form of the Johnson-Nyquist noise formula for voltage fluctuations V 2 = 4k B T R∆f , where k B is Boltzmann's constant, T is absolute temperature, R is the electrical resistance, and ∆f is the electrical bandwidth [2,3].The resistance R that quantifies the irreversible dissipation of electrical conduction also determines the amplitude of voltage fluctuations at equilibrium.Since it's formal statement, the fluctuation-dissipation theorem has been extended to various physical systems, including the quantum optical regime [4].
Considering fluctuations in an electrical conductor further, fundamental excitations in condensed matter carry not only charge, but other physical quantities such as, for example, heat, spin, and pseudo-spin.The thermoelectric response of a conductor is a result of the correlated transport of charge and heat by fundamental excitations.In the simplest scenario, the Seebeck coefficient S is a measure of the mean entropy carried per unit charge e, and the Peltier coefficient Π = T S is a measure of the mean thermal energy carried per unit charge e [5][6][7].Correlation in the fluctuation of charge and heat transport have been considered theoretically in nanoscale systems [8].
Surprisingly, the fluctuation-dissipation theorem as it applies to a thermoelectric material and the resulting correlations in fluctuation of charge current, heat current, voltage and temperature have not been investigated to date.Temperature fluctuation at equilibrium is wellknown in the theory of statistical mechanics [9], but typically eludes measurement in condensed matter owing to the small absolute scale of temperature fluctuations, which scales inversely with heat capacity.Nonetheless, temperature fluctuations are of prime importance in various phenomena in condensed matter as for example they limit the sensitivity of bolometers [10], generate dynamical phase transitions [11] and are at the origin of non-Gaussian fluctuations in metals [12,13].
In this paper, we apply the fluctuation-dissipation theorem to derive a generalized Johnson-Nyquist formula for thermoelectrics, revealing a modification in the spectral density of fluctuations that is dependent on the dimensionless, material thermoelectric figure of merit zT = ΠS/κρ (not to be confused with device ZT , discussed further below), where κ is the thermal conductivity and ρ is the electrical resistivity.Using integrated, wide-bandwidth, thermoelectric cooling microdevices [14], we measure the spectral density of voltage fluctuations and find agreement with the generalized Johnson-Nyquist formula for thermoelectrics.In the spirit of the fluctuation dissipation theorem, experimental impedance spectroscopy is further used to explicitly test the correspondence between irreversible transport coefficients and equilibrium fluctuations in a thermoelectric material.
Let us consider an electrical current I e and a heat current I Q in a thermoelectric sample of length L and crosssectional area A in the presence of a time-dependent potential difference ∆V and temperature difference ∆T , where ( The dynamical response of the thermoelectric depends upon the extensive heat capacity C Q = ∂U/∂∆T , where U is the heat transported across the length L of the thermoelectric. The transport Eq. ( 1) can be compactly expressed with a generalized potential Ṽ, generalized flux Ĩ, and generalized conductance G, where the Fourier transform x(f ) = e 2πif t x(t)dt is used to work in the frequency domain.
The fluctuation-dissipation theorem applied to the generalized potential Ṽ gives, where the spectral density of fluctuations S α,β (f ) is defined, where x denotes the ensemble average of x, and T = 1/∆f is the reciprocal bandwidth.From an experimental perspective, T is the temporal duration of a measurement [15], while in theoretical analysis T is the period in the periodic model of a stationary physical system [16].
The spectral density S T (f ) = 2T −1 ∆T (+f )∆T (−f ) of the temperature fluctuations across the thermoelectric is, where accordance with the result expected from general considerations of statistical mechanics [9].The spectral density of the voltage fluctuations There are voltage fluctuations of Ohmic origin proportional to the transport coefficient R 0 = L −1 11 , and voltage fluctuations that originate with temperature fluctuations coupled via the thermoelectric coefficient S.An independent analysis showing the coupling between temperature gradient and electric field fluctuations is shown in the Supplemental Material [17].
Combining the results of Eqs.6,7, we arrive at the central theoretical result for voltage fluctuations in a thermoelectric, In the high-frequency limit above the thermal cut-off frequency, f f T , the usual Johnson-Nyquist formula for spectral density of voltage fluctuations applies, S V = 4k B T R 0 .The spectral density of temperature fluctuations diminish S T (f ) ∝ f −2 for f f T .In contrast, in the low-frequency limit below thermal cut-off, f f T , there is an enhancement in spectral density of voltage fluctuation beyond the usual Johnson-Nyquist result, S V = 4k B T R 0 (1 + zT ).The low-frequency spectral density of temperature fluctuations, S T → 4k B T 2 G −1 T , contributes to the observable voltage fluctuations via thermoelectric coupling.The dimensionless zT determines the relative enhancement of voltage noise beyond the Johnson-Nyquist result.Thermoelectric noise enhancement is thus anticipated in materials with large Seebeck coefficient S, large electrical conductivity σ, and small thermal conductivity κ.
To confirm our theoretical findings, we measured the voltage noise spectral density of integrated, wide-bandwidth, thermoelectric cooling micro-devices.The devices consist of a series network of L = 10 µm thick n-type (Bi 2 (Te 0.95 Se 0.05 ) 3 ), abbreviated as BiTeSe) and p-type (pure Te) thermoelectric materials, with Au contacts, arranged in serpentine fashion on a Si substrate, as shown in Fig. 1A.A Π architecture is adopted, Fig. 1B, such that the contra-oriented electrical current I e in each leg results in co-oriented heat current I Q in each leg.A microfabricated structure, Fig. 1C and Fig. 1D, was used to achieve a high thermal cut-off frequency f T ∼ 1 kHz, as confirmed by experiment.The cross-sectional areas of the p-type and n-type legs are A p = 85 µm × 30 µm and A n = 30 µm × 30 µm, respectively.The device systematically studied in our work consisted of 112 series leg pairs.Further details concerning the design and fabrication of wide-bandwidth thermoelectric micro-coolers have been previously reported [14].The spectral density of voltage fluctuations for a thermolectric material in Eq. ( 8) can be extended to a thermoelectric device consisting of a series of N identical leg pairs.
Consider first the voltage fluctuation across a single leg pair in the Π geometry of Fig. 1B.Due to the effective thermal short-circuit between points α and γ from the substrate thermal conductivity, the fluctuation in temperature difference ∆T = T β − T α = T β − T γ across the n-type and p-type legs have a spectral density, where where R i and S i are the electrical resistance and Seebeck coefficient of the i-type leg, i=n,p, and ZT is the thermoelectric device figure of merit, The voltage fluctuation of the thermoelectric microdevice, Eq. ( 10), differs from the voltage fluctuation of a thermoelectric material, Eq. ( 8), albeit sharing a similar form.Note that the device figure of merit ZT is not the sum of the constituent material figures of merit zT , due to the correlated temperature difference ∆T across p-and n-legs [6].The voltage power spectral density S V (f ) was measured using a correlation method, as shown in Fig. 2A.The voltage across two terminals of a series of p/n leg pairs was measured simultaneously with two cascaded voltage amplifiers.Each cascade consisted of two identical voltage pre-amplifiers (LI-75, NF Corp.) with 40 dB total gain.The voltages V 1 (t) and V 2 (t) were digitized (DT-9847, Data Translation) and the cross-spectral density estimated using the cross-periodogram method [15].To suppress the uncorrelated amplifier noise in V 1 (t) and V 2 (t), we averaged 7.5×10 4 periodograms, with 2.15 × 10 5 samples/periodogram acquired at a sampling rate of 2.15 × 10 5 samples/s, for over 20 hours of acquisition time per spectral density measurement.The microdevice was placed in a series of metal boxes for electrical and thermal isolation, open-loop resistive heating was applied to vary the sample substrate temperature T , and a thermistor was used to measure the T , which varied by less than ±1 K over the duration of each measurement.
The spectral density of the voltage across the thermoelectric is reconstructed as The comparatively high thermal cut-off frequency, f T,Π ∼ 1 kHz, is essential to avoid measurements in the extremely low frequency (ELF, f < 30 Hz) band where amplifier 1/f noise is prohibitively large.Frequency independent measurement system response over the bandwidth 200 Hz < f < 20 kHz was confirmed using a resistor of R = 25.9Ω (see Supplemental Material).The gain in the noise measurement system was calibrated with an R = 8.20 Ω resistor at f = 1.7 kHz.
The measured voltage spectral density S V (f ) versus frequency f is shown in Fig. 2B, along with a numerical fit to Eq. (10).Measurements were taken with the thermoelectric substrate temperature T varied from 295 K to 365 K, adjusted with an external resistive heater and measured via thermistor.The dashed horizontal lines denote the Ohmic Johnson-Nyquist noise contribution, 4k B T N (R n + R p ), with the resistance N (R n + R p ) determined from the fit of the measured S V to Eq. (10).There is an enhancement in voltage fluctuations ∆S V = S V −4k B T N (R n +R p ) in the low-frequency limit f f T as compared to the high-frequency limit f f T , indicated with vertical arrows.
The spectral density of temperature fluctuations, S T = ∆S V /S 2 , was determined using the effective, temperature dependent, Seebeck coefficient S = S p − S n inferred from measurements of BiTeSe and Te thin films (see Supplemental Material).For example, at T = 298 K, S p = 194 µVK −1 , S n = −44 µVK −1 and S = 238 µVK −1 .The experimentally determined spectral density of temperature fluctuations, S T , is shown in Fig. 2C versus substrate temperature T , along with a model fit to Eq. ( 9).At T = 295 K, the experimentally resolved amplitude of temperature fluctuations is δT = S 1/2 T = 0.8 µKHz −1/2 , corresponding to a relative fluctuation amplitude δT /T = 3 × 10 −9 Hz −1/2 .In the context of thermometry, our findings show that a thermoelectric micro-device can be used to measure temperature differences as small as the fundamental fluctuations described by the fluctuation-dissipation theorem.
The micro-device figure of merit ZT can be determined from the frequency dependent voltage noise S V , with ZT = lim f →0 ∆S V /(S V − ∆S V ).The figure of merit ZT versus substrate temperature T is shown in Fig. 2D, where ZT was determined from a fit of the experimentally measured S V to Eq. (10).As anticipated, ZT increases with substrate temperature T .The fluctuation dissipation theorem identifies the equivalency of the coefficient for fluctuation at thermal equilibrium and the coefficient for dissipative transport.We therefore investigated this equivalency for thermoelectrics.The transport Eq. ( 1) leads to a frequency dependent electrical impedance, where R = N (R n + R p ) for our thermoelectric microdevice.The impedance of Eq. ( 12) is the basis for impedance spectroscopy of thermoelectrics, which have evolved in complexity [18][19][20][21][22].The thermoelectric cooling device impedance was measured using a lock-in amplifier (MLFI 500kHz, Zurich Instruments) with an AC current bias I e of 100 µA amplitude over a frequency range 100 Hz < f < 100 kHz.The Nyquist plot of measured −Im {Z} versus Re {Z} is shown in Fig. 3A at T =295 K, 329 K, and 365 K.The semi-circle characteristic of first-order response is evident in the Nyquist plot, and a numerical fit to the simple model of Eq.In conclusion, we have demonstrated how the fluctu-ation dissipation theorem can be extended to thermoelectrics, revealing the role of temperature fluctuations in the modification of the renowned Johnson-Nyquist formula.Our work establishes a quantitative understanding of temperature fluctuations in thermoelectrics, defining an important physical limit for micro-scale thermoelectric thermometery in, for example, intracellular thermometry [23].Finally, we note that our results can be generalized to include other physical quantities carried by fundamental excitations in a conductor.For example, spin-orbit coupling can result in fundamental excitations that carry charge and spin, and the correlated transport of charge and spin can in principle lead to a modification of the observable spectral density of voltage fluctuations.Moreover, the spectral density of voltage fluctuations may give insight into the spectral density of spin fluctuations.

FIG. 1 :
FIG. 1: Integrated thermoelectric cooler.A Illustration of the alternating n-type BiTeSe and p-type Te thermoelectric legs with Au electrodes, arranged in a serpentine, series pattern on a substrate.B Diagram demonstrating the direction of heat current IQ downward in both n-type and p-type material as electrical current Ie passes from left to right in the Π architecture of an n/p pair.C Optical microscope image of the fabricated thermoelectric micro-device.D Oblique angle scanning electron microscope image of the fabricated thermoelectric micro-device.
l are the thermal conductances of each leg, l is the length of each leg, and the thermal cut-off frequency 2πf T,Π = (G n + G p )/C Q,Π where C Q,Π is the total heat capacity for establishing a temperature difference ∆T across the leg pair.The voltage fluctuations across the n-leg (α − β) and p-leg (β − γ) that arise from thermoelectric transduction of temperature fluctuations are correlated, and given by (S p − S n ) 2 S T (f ).The voltage fluctuations across distinct leg pairs are not correlated, and thus the expected spectral density of voltage fluctuation across the series of N leg pairs in our thermoelectric device is

FIG. 2 :
FIG. 2:Voltage and temperature fluctuations.A Circuit diagram of the cross-periodogram method used to estimate the spectral density SV (f ) of voltage fluctuations across the thermoelectric micro-device.Independent measurements V1(t) and V2(t) are acquired and digitally Fourier transformed to reconstruct SV from the cross-spectral density.B Voltage spectral density SV (f ) versus frequency f of the thermoelectric micro-device measured at a substrate temperature T from 295 K to 365 K. Solid dark lines show a fit to theoretical spectral density of Eq.(10).The Ohmic Johnson-Nyquist contributions 4kBT N (Rn +Rp) are indicated with dashed lines, and the excess spectral density ∆SV = SV − 4kBT N (Rn + Rp) arising from thermoelectric coupling is indicated with vertical arrows.C The temperature fluctuation spectral density ST (f ) = ∆SV /S 2 versus frequency f , using the independently measured, temperature dependent, effective Seebeck coefficient S. A 40 Hz running average was applied to ST for visual clarity, and a model fit to Eq. (9) is shown.D The thermoelectric micro-device figure of merit ZT = lim f →0 ∆SV /(SV − ∆SV ) versus substrate temperature T .

FIG. 3 :
FIG.3: Impedance and noise spectroscopy.A Nyquist plot of the measured impedance, −Im {Z} versus Re {Z} at T =295 K,329 K, and 365 K, over the frequency 100 Hz < f < 100 kHz.The semi-circle arrows direction indicates an increase in frequency.B A comparison of the measured dissipative impedance Re {Z} (darker circles) and the measured voltage spectral density normalized to impedance units, SV /4kBT , (thin line) versus frequency f .C Model circuit for a thermoelectric, including Ohmic and thermoelectric dissipative elements, Ohmic and thermolectric fluctuation sources, and effective capacitance.

( 12 )
. A direct comparison of the measured dissipative impedance coefficient Re {Z} and the measured normalized voltage fluctuation spectral density S V /4k B T is shown in Fig.3B.Our experiments are in good agreement with the fluctuation dissipation theorem as applied to the effective electronic sector of the transport equations, Re {Z} = S V /4k B T .The thermoelectric coupling thus simultaneously enhances both the spectral density of voltage fluctuations and the electrical impedance above that of the purely Ohmic contribution to each.Voltage fluctuations are enhanced by thermoelectric coupling of temperature fluctuations.Impedance is enhanced by the voltage generated by thermoelectric coupling to the temperature gradient established by the passage of electrical (and thus heat) current in the thermoelectric.Assembling our findings, a simple equivalent electrical transport model for a thermoelectric element is shown in Fig.3C, using simplified notation.The model includes the Ohmic resistance R and associated Johnson-Nyquist fluctuations of spectral density 4k B T R, a resistance ZT • R = T S 2 G −1 T of thermoelectric origin with an associated spectral density of voltage fluctuations 4k B T R • ZT = 4k B S 2 T 2 G −1 T .The frequency dependence of thermal contributions to impedance and voltage fluctuations is accounted for with the inclusion of an effective capacitance C T = C Q T −1 S −2 of thermal origin, with cut-off frequency 2πf T = G T /C T .