A high-sensitivity resistance bridge for nanoscale thermal microscopy

Measurements of heat flux between micro-objects, in vacuum or in air, are challenging because of their small size and the low thermal conductance of the medium between them. One way to address this issue consists in using a scanning thermal microscope (SThM) equipped with a temperature dependent resistance thermometer. However, this requires an instrument able to both injecting a defined heating Joule power and performing highly-sensitive resistance measurements. Here, we present such an instrument based on a Wheatstone bridge equipped with three Kelvin arms. It can perform resistance measurements in the range from 100 $\Omega$ to 1000 $\Omega$ not only in direct current but also in alternating current regimes at frequencies up to a few tenths of kHz. We first show that measurements of resistance standards are accurate to within one part in $10^4$ with a relative experimental standard deviation which can be as low as one part in $10^8$ for one second measurement. The instrument is then tested with a SThM thermometer. With the support of an electro-thermal model considering thermal time constants of the thermometer, we explain the frequency dependence of detected signals and optimize the measurement protocols of temperature and heat flux. By measuring sub-mK temperature variations, this instrument is then used to determine with a few nanowatts uncertainty the near-field radiative heat flux between a heated glass microsphere and a glass substrate, which is caused by the coupling of surface phonon-polaritons.