Fluctuation-Dissipation in Thermoelectrics

Thermoelectric materials exhibit correlated transport of charge and heat. The Johnson-Nyquist noise formula $4 k_B T R$ for 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 $4 k_B T R (1 + ZT)$ at frequencies below a thermal cut-off frequency $f_T$, where $ZT$ is the dimensionless thermoelectric material figure of merit. The origin of the enhancement in voltage noise is thermoelectric coupling of temperature fluctuations. We use a wideband ($f_T\sim1$ kHz), integrated thermoelectric micro-device to experimentally confirm our findings. Measuring the $ZT$ enhanced voltage noise, we experimentally resolve temperature fluctuations with an amplitude of $0.8~\mu \mathrm{K} \mathrm{Hz}^{-1/2}$ 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.