Thermometry with Subnanometer Resolution in the Electron Microscope Using the Principle of Detailed Balancing

We measure phonon energy gain and loss down to 20 meV in a single nanostructure using an atom-wide monochromatic electron beam. We show that the bulk and surface, energy loss and energy gain processes obey the principle of detailed balancing in nanostructured systems at thermal equilibrium. By plotting the logarithm of the ratio of the loss and gain bulk/surface scattering as a function of the excitation energy, we find a linear behavior, expected from detailed balance arguments. Since that universal linearity scales with the inverse of the nanosystem temperature only, we can measure the temperature of the probed object with precision down to about 1 K without reference to the nanomaterial. We also show that subnanometer spatial resolution (down to similar to 2 angstrom) can be obtained using highly localized acoustic phonon scattering. The surface phonon polariton signal can also be used to measure the temperature near the nanostructure surfaces, but with unavoidable averaging over several nanometers. Comparison between transmission and aloof probe configurations suggests that our method exhibits noninvasive characteristics. Our work demonstrates the validity of the principle of detailed balancing within nanoscale materials at thermal equilibrium, and it describes a transparent method to measure nanoscale temperature, thus representing an advance in the development of a noninvasive method for measurements with angstrom resolution.