XML Generation Notice This is an automatically generated XML version of the publication: "Fluctuation-dissipation in thermoelectric sensors - https://doi.org/10.34657/11372", originally published under the license CC BY 4.0 International. The XML version was generated on 28-08-2026 by Technische Informationsbibliothek (TIB). In spite of high standards in the development of the script, we cannot outrule the possibility of mistakes in the process of file conversion. The authors are not responsible for possibly existing mistakes or differences in this XML-version. The link above directs you to the authorised version of the publication. If you spot mistakes in the XML-version, please contact us at oarepository@tib.eu. This XML-Version of the publication "Fluctuation-dissipation in thermoelectric sensors - https://doi.org/10.34657/11372" is provided "as is", without any guarantee (“ausdrücklich oder stillschweigend”). TIB does not take responsibility for potential mistakes, missing passages and consequences resulting from this use. Hinweis zur XML-Generierung Dies ist eine durch die Technische Informationsbibliothek – TIB am 28-08-2026 automatisiert erstellte XML-Version der Publikation "Fluctuation-dissipation in thermoelectric sensors - https://doi.org/10.34657/11372", die unter der Lizenz CC BY 4.0 International veröffentlicht wurde. Trotz sorgfältiger Entwicklung des Skripts können Fehler im Konvertierungsprozess nicht ausgeschlossen werden. Die Autor*innen der Originalpublikation sind für etwaige Fehler/Differenzen in der XML-Version nicht verantwortlich. Die oben verlinkte Datei stellt die autoritative Version der Publikation dar. Hinweise auf Fehler on der XML-Version nehmen wir unter oarepository@tib.eu gern entgegen. Die vorliegende XML-Version der Publikation "https://doi.org/10.1209/0295-5075/acb009" wird "wie es ist" bereitgestellt, ohne jegliche Gewährleistung oder Garantie, sei es ausdrücklich oder stillschweigend. Die TIB übernimmt keine Verantwortung für etwaige Fehler, Auslassungen oder Folgen, die aus der Nutzung dieser Datei resultieren. xml_created 28-08-2026 dc.identifier.uri https://doi.org/10.1209/0295-5075/acb009 dc.rights.license CC BY 4.0 International dc.relation.doi https://doi.org/10.34657/11372 Fluctuation-dissipation in thermoelectric sensors 19 01 2023 Fluctuation-dissipation in thermoelectric sensors, Tran, NDutt, APulumati, NReith, HHu, ADumont, ANielsch, KTremblay, A.-MSchierning, GReulet, BSzkopek, T, , (2023-01-19). 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. Introduction. -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 RΔf , where k B is Boltzmann's constant, T is absolute temperature, R is the electrical resistance, and Δf 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. Owing to its generality, the fluctuation-dissipation theorem has been applied to various physical systems, including quantum optics [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 (a) E-mail: thomas.szkopek@mcgill.ca (corresponding author) 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]. Correlations in the fluctuation of charge and heat transport have been considered theoretically in nanoscale systems [8][9][10][11], and the role of charge fluctuations in non-linear thermoelectric response of superconducting junctions has been investigated [12,13]. 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 [14,15], but typically eludes measurement in condensed matter owing to the small absolute scale of temperature fluctuations, which scales inversely with heat capacity. The first experimental measurement of spontaneous temperature fluctuations was observed in a paramagnetic salt in cryogenic conditions by magnetometry [16]. Temperature fluctuations are of prime importance in various phenomena in condensed matter as for example they limit the sensitivity of bolometers [17], generate dynamical phase transitions [18] and are at the origin of non-Gaussian fluctuations in metals [19,20]. Micro-scale and nano-scale thermoelectric thermometry is of increasing interest for scanning thermal microscopy [21], cellular-level thermometry [22][23][24][25], semiconductor circuit thermometry [26], and wearable sensors [27,28], further motivating the understanding of fluctuations in thermoelectrics. 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 Z D T , see eq. ( 11)), where κ is the thermal conductivity and ρ is the electrical resistivity. Using integrated, wide-bandwidth, thermoelectric micro-devices [29] in a Λ leg-pair configuration and operated with a thermally floating terminal suitable for thermometry, 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. Results . - Fluctuation-dissipation analysis.We first consider fluctuations in a uniform thermoelectric material, before considering a thermoelectric device. The relationship between electric current I e , heat current I Q in a thermoelectric material of length L and cross-sectional area A in the presence of a time-dependent potential difference ΔV and temperature difference ΔT across the length L,I e = L 11 ΔV + L 12 ΔT T + C e ∂ ∂t ΔV, I Q = L 21 ΔV + L 22 ΔT T + C Q ∂ ∂t ΔT,(1)where L ij = (A/L)L ij areσ = L 11 , S = − 1 T L 12 L 11 , κ = 1 T L 22 L 11 − L 12 L 21 L 11 . (2)The dynamical response of the thermoelectric depends upon the extensive heat capacity C Q = ∂U/∂ΔT , where U is the heat transported across the length L of the thermoelectric. For ΔT T , where the temperature varies linearly across the length L of the thermoelectric, the heat capacity C Q = c q AL/2, where c q is the volumetric specific heat and the factor of 1/2 accounts for the mean increase in temperature of ΔT /2 over the body of the thermoelectric. Electrical capacitance is given by C e = ∂Q/∂ΔV , where Q is the charge separation across the thermoelectric. Our analysis and subsequent experiments concern frequencies below the electrical cut-off frequency, 2πf e = L 11 /C e , and thus the displacement current C e ∂ΔV /∂t will be neglected from herein.The transport equation ( 1) can be compactly expressed with a generalized potential Ṽ, generalized flux Ĩ, and generalized conductance G,Ĩ = G Ṽ, I e (f ) I Q (f ) = L 11 L 12 L 21 L 22 + i2πf T C Q ΔV (f ) ΔT (f )/T , (3)where the Fourier transform x(f ) = e 2πif t x(t)dt is used to work in the frequency domain, and the limit of frequency below electrical cut-off, f f e is considered. The fluctuation-dissipation theorem applied to the generalized potential Ṽ givesS α,β = 4k B T Re G −1 α,β ,(4)where the spectral density of fluctuations S α,β (f ) is defined,S α,β (f ) = 2 T Ṽα (+f ) Ṽβ (−f ) ,(5)where x denotes the ensemble average of x, and T = 1/Δf is the reciprocal bandwidth. From an experimental perspective, T is the temporal duration of a measurement [30], while in theoretical analysis T is the period in the periodic model of a stationary physical system [31]. A field theoretical analysis, presented in the Supplementary Material Supplementarymaterial.pdf (SM)1 , arrives at a similar result. The spectral density S T (f ) = 2T −1 ΔT (+f )ΔT (−f ) of the temperature fluctuations across the thermoelectric isS T (f ) = 4k B T 3 Re[G −1 22 ] = 4k B T 2 G −1 T 1 + (f /f T ) 2 , (6)where G T = κA/L is the thermal conductance and 2πf T = G T /C Q is the thermal cut-off frequency. The total integrated temperature fluctuation, (ΔT) 2 = ∞ 0 S T df = k B T 2 /C Q , inaccordance with the result expected from general considerations of statistical mechanics [14].The spectral density of the voltage fluctuations S V (f ) = 2T −1 ΔV (+f )ΔV (−f ) across the thermoelectric isS V (f ) = 4k B T Re[G −1 11 ] = 4k B T R 0 + S 2 S T (f ). (7)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. A similar relation for the coupling between temperature gradient and electric field fluctuations is derived in the SM.Combining the results of eqs. ( 6), (7), we arrive at the central theoretical result for voltage fluctuations in a thermoelectric,S V (f ) = 4k B T R 0 1 + zT 1 + (f /f T ) 2 . (8)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 fluctu- ations diminish S T (f ) ∝ f −2 for f f T . In con- trast, Integrated thermoelectric micro-devices.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. 1(A). A Λ architecture is adopted, fig. 1(B), 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. 1(C) and fig. 1(D), 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. 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 Λ geometry of fig. 1(B). The points α and γ are fixed at a temperature T determined by the substrate, while the point β is free to fluctuate to a temperature T + ΔT . The fluctuations in temperature difference ΔT across the n-type and p-type legs involve both electrons and phonons and have a spectral densityS T (f ) = 4k B T 2 (G n + G p ) −1 1 + (f /f T,Λ ) 2 , (9)where (10) where R i and S i are the electrical resistance and Seebeck coefficient of the i-type leg, i = n, p, and Z D T is the thermoelectric device figure of merit,G n = κ n A n /l, G p = κ p A p /S V (f ) = 4k B T N(R n + R p ) 1 + Z D T 1 + (f /f T,Λ ) 2 ,Z D T = T (S p − S n ) 2 (R n + R p )(G n + G p ) . (11)See the SM for more detail on the derivation of eq. ( 10).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 Z D T is not the sum of the constituent material figures of merit zT , due to the correlated temperature difference ΔT across p-and n-legs [6].The origin of the device Z D T figure of merit in the increase of spectral density of voltage fluctuations can be understood as follows. At frequencies below thermal cut-off, f f T,π , the spectral density of temperature fluctuations, S T = 4k B T 2 (G n + G p ) −1 , is inversely proportional to device thermal conductance. The Seebeck effect leads to an additional voltage spectral density ΔS V = (S p − S n ) 2 S T , proportional to the square of device Seebeck coefficient. Since the Ohmic contribution to voltage fluctuations is proportional to the device electrical resistance, S V,Ohmic = 4k B T (R n + R p ), the relative size of the thermoelectric contribution to the Ohmic contribution, ΔS V /S V,Ohmic = Z D T . In other words, Z D T is the dimensionless parameter that defines the ratio of thermoelectric to Ohmic contributions of voltage fluctuations. Voltage power spectral density.The spectral density of the voltage across the thermoelectric is reconstructed using a voltage cross correlation method as shown in fig. 2(A). The comparatively high thermal cutoff frequency, f T,Λ ∼ 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 = 25.9 Ω (see the SM). 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 ) vs. frequency f is shown in fig. 2(B), 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 ΔS 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 increase in low-frequency voltage fluctuations cannot be accounted for by 1/f noise, which has a much stronger frequency dependence than that experimentally observed here. Moreover, the model fit permits extraction of various physical parameters such as the heat capacity of a leg-pair, C Q,Λ = 55.2 nJ/K at 295 K, which is in good agreement with the theoretically predicted value of 52.4 nJ/K (see the SM).Temperature power spectral density. The spectral density of temperature fluctuations, S T = ΔS V /S 2 , was determined using the effective, temperature-dependent, Seebeck coefficient S = S p − S n . The Seebeck coefficients of BiTeSe and Te were measured vs. temperature (see the SM). For example, at T = 298 K, S p = 194 μ V K −1 , S n = −44 μ V K −1 and S = 238 μ V K −1 . The experimentally determined spectral density of temperature fluctuations, S T , is shown in fig. 2(C) vs. 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 μ KHz −1/2 , corresponding to a relative fluctuation amplitude δT /T = 3 × 10 −9 Hz −1/2 . We have thus demonstrated that the remarkably small size of temperature fluctuations in our micro-device can be measured using the Seebeck effect. In the general 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 Z D T can be determined from the frequency-dependent voltage noise S V , with 2(D), where Z D T was determined from a fit of the experimentally measured S V to eq. ( 10). As anticipated, Z D T increases with substrate temperature T .Z D T = lim f →0 ΔS V /(S V − ΔS V ). The figure of merit Z D T vs. substrate temperature T is shown in fig. Impedance spectroscopy.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 equation ( 1) leads to a frequency-dependent electrical impedance,Z(f ) = ΔV I e = R 1 + Z D T 1 + if /f T,Λ ,(12)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 [32][33][34][35][36]. The Nyquist plot of measured −Im {Z} vs. Re {Z} is shown in fig. 3(A) 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. ( 12).Discussion. -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. 3(B). 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. T R • Z D T = 4k B S 2 T 2 G −1T . The frequency dependence of thermal contributions to impedance and voltage fluctuations is accounted for with the inclusion of an effective capacitanceC T = C Q T −1 S −2 of thermal origin, with cut-off frequency 2πf T = G T /C T .Our model is readily extended to a variety of other contexts involving thermoelectrics. Changes to the thermal environment can be incorporated by modifying the thermal circuit. For example, loading the top conductor of a Λ leg-pair introduces an additional heat capacity and a thermal conductance to the environment. The effect of increasing heat capacity is to reduce the thermal cut-off frequency. The effect of increased thermal conductance is to reduce the effective Z D T , but care must be taken to account for time-dependent heat diffusion through the thermal load, as observed by impedance spectroscopy of thermoelectric devices [34][35][36]. By the general arguments presented in this work, the power spectral density of voltage fluctuations will be modified by thermal loading in analogous fashion to the modification of the dissipative component of impedance. Finally, we note that our model can be easily extended to include the electrical environment of a thermoelectric device, including capacitive and inductive effects. defines an important physical limit to temperature measurement by microscale thermoelectric thermometry, and is expected to be of importance to applications such as 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. Materials and methods. -The thermoelectric micro-coolers were fabricated using standard photolithography and a modified electrochemical deposition technique, details can be found in [29]. The voltage power spectral density S V (f ) was measured using a correlation method, as shown in fig. 2(A). 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 crossspectral density estimated using the cross-periodogram method [30]. 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 20hours of acquisition time per spectral density measurement. The micro-device 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 mean temperature T , which varied by less than ±1 K over the duration of each measurement.The Seebeck coefficient of both the n-type and p-type thermoelectric legs was inferred from measurements of BiTeSe and Te thin films, with further details provided in the SM. The thermoelectric device impedance was measured using a lock-in amplifier (MLFI 500 kHz, Zurich Instruments) with an AC current bias I e of 100 μA amplitude over a frequency range 100 Hz < f < 100 kHz. * * * A-MST, BR and TS acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) under grants RGPIN-2016-04400, RGPIN-2018-04851, RGPIN-2019-05312, the Canada First Research Excellence Fund, and the Canada Excellence Research Chairs program. ASD, HR and GS acknowledge funding from the DFG (Deutsche Forschungsgemeinschaft) under grant No. SCHI1010/11-1, and NBP and HR acknowledge funding under grant No. RE3973/1-1. ASD, NBP, HR, KN and GS fabricated the integrated thermoelectric micro-device and measured the Seebeck coefficient of the n-type and p-type thermoelectric material. The theoretical derivations were done by NAMT, AMST, BR, and TS. NAMT acquired all the noise measurements with the assistance from AH and AD. The impedance spectroscopy was done by NAMT. NAMT and TS prepared the manuscript with contributions from all co-authors. We have no competing interest to declare. 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. 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 Λ 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,Λ = (G n +G p )/C Q,Λ where C Q,Λ is the total heat capacity for establishing a temperature difference ΔT 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 ) vs. frequency f of the thermoelectric microdevice 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 ΔSV = SV − 4kBT N(Rn + Rp) arising from thermoelectric coupling is indicated with vertical arrows. (C) The temperature fluctuation spectral density ST (f ) = ΔSV /S 2 vs. 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 ZDT = lim f →0 ΔSV /(SV − ΔSV ) vs. substrate temperature T . Fig. 3 : Fig. 3: Impedance and noise spectroscopy. (A) Nyquist plot of the measured impedance, −Im {Z} vs. Re {Z} at T = 295 K, 329 K, and 365 K, over the frequency 100 Hz < f < 100 kHz. The semi-circle arrow direction indicates an increase in frequency. (B) A comparison of the measured dissipative impedance Re {Z} (dark circles) and the measured voltage spectral density normalized to impedance units, SV /4kB T , (coloured lines) vs. frequency f . (C) Model circuit for a thermoelectric, including Ohmic and thermoelectric dissipative elements, Ohmic and thermolectric fluctuation sources, and effective capacitance. Conclusions. - In conclusion, we have demonstrated how the fluctuation-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 model for the temperature fluctuations and voltage fluctuations inherent to a thermoelectric device as a consequence of the fluctuation-dissipation theorem. The temperature fluctuation δT = S 1/2 T Data availability statement:The data that support the findings of this study are available upon reasonable request from the authors. , Callen, HWelton, TPhys. Rev, 83, 34 (1951). , Johnson, JPhys. Rev, 32, 97 (1928). , Nyquist, HPhys. Rev, 32, 110 (1928). , Quantum Noise, , (2004). , Solid State Physics, , (1976). , CRC Handbook of Thermoelectrics, , (1995). , Goupil, CSeifert, WZabrocki, KMüller, ESnyder, GEntropy, 13, 1481 (2011). , Crépieux, AMichelini, FJ. Phys.: Condens. Matter, 27, 15302 (2014). , Dashti, NMisiorny, MSamuelsson, PSplettstoesser, JPhys. Rev. Appl, 10, 24007 (2018). , Karimi, BBrange, FSamuelson, PPekola, JNat. Commun, 11, 1 (2020). , Crépieux, APhys. Rev. B, 103, 45427 (2021). , Marchegiani, GBraggio, AGiazotto, FAppl. Phys. Lett, 117, 212601 (2020). , Marchegiani, GBraggio, AGiazotto, FPhys. Rev. Lett, 124, 106801 (2020). , Course of Theoretical Physics: Statistical Physics Part I, , (1981). , Puglisi, ASarracino, AVulpiani, APhys. Rep, 709, 1 (2017). , Chui, TSwanson, DAdriaans, MNissen, JLipa, JPhys. Rev. Lett, 69, 3005 (1992). , Richards, PJ. Appl. Phys, 76, 1 (1994). , Spahr, KGraveline, JLupien, CAprili, MReulet, BPhys. Rev. B, 102, 100504 (2020). , Nagaev, KPhys. Rev. B, 66, 75334 (2002). , Pinsolle, EHoule, SLupien, CReulet, BPhys. Rev. Lett, 121, 27702 (2018). , Zhang, YZhu, WHui, FLanza, MBorca-Tasciuc, TRojo, MuñozAdv. Funct. Mater, 30, 1900892 (2020). , Shrestha, RChoi, T. -YChang, WKim, DSensors, , 8826 (2011). , Tian, WWang, CWang, JChen, QSun, JLi, CWang, XGu, NNanotechnology, , 355501 (2015). , Bai, TGu, NSmall, 12, 4590 (2016). , Rajagopal, MValavala, KGelda, DMa, JSinha, SSens. Actuators A: Phys, 272, 253 (2018). , Harzheim, AKönemann, FGotsmann, BVan Der Zant, HGehring, PAdv. Funct. Mater, 30, (2020). , Zhu, PWang, YWang, YMao, HZhang, QDeng, YAdv. Energy Mater, 10, (2020). , Tian, BLiu, ZWang, CLiu, YZhang, ZLin, QJiang, ZRev. Sci. Instrum, 91, 45004 (2020). , Li, GFernandez, JRamos, DBarati, VPérez, NSoldatov, IReith, HSchierning, GNielsch, KNat. Electron, 1, 555 (2018). , Spectral Analysis and Time Series: Probability and Mathematical Statistics, , (1981). , , , (1996). , Harman, TJ. Appl. Phys, 29, 1373 (1958). , Downey, AHogan, TCook, BRev. Sci. Instrum, 78, 93904 (2007). , García-Cañadas, JMin, GJ. Appl. Phys, 116, 174510 (2014). , , 6, 35008 (2016). , Hasegawa, YHomma, ROhtsuka, MJ. Electron. Mater, 45, 1886 (2016).