2, ST Microelectronics, Crolles, , France
3, CMI, Brno, , Czechia
4, IES, Madrid, , Spain
Scanning thermal microscopy (SThM) is a key thermal characterization tool, where a micro to nanometer-scale probe tip measures the temperature close to a surface. The spatial and temperature resolutions of this technique are limited in particular by the size of the tip, the sample properties, and the various tip-sample heat transfer mechanisms  depending on the operation conditions. In this communication, we present a study of heat transfer between a sample surface and a SThM probe based on resistive thermometry: the probe electrical resistance depends on the probe temperature. Such probe operates either in thermometry (passive) mode with a minimal Joule self-heating to enable sample surface temperature measurement or in thermal-property measurement (active) active mode where a significant Joule heating is needed to allow thermal power flowing into the sample, depending on its thermal conductivity.
We first study the hot tip-cold sample (active mode) heat exchange as a function of the tip-sample distance . When the tip is far from the surface (>100 micrometers), it is cooled by heat convection and there is no exchange with the sample. At lower distances, the diffusive heat exchange starts. The exact geometry is required to compute accurately the transfer, which can be done by means of Finite Element Modeling (FEM). In the sub-micrometer distance regime, air conduction cannot be modeled by usual FEM because the air mean free path (~60 nm) is on the order of the average distance between the hot object and the sample. In our simulations, we account for this deviation to the diffusive regime by adding thermal resistances on surfaces, which correspond to the ballistic transfer limit in air. This allows us to reproduce numerically the ballistic leveling off seen experimentally. Depending on the probe considered, a 1D approximation  or improved considerations based on the Boltzmann transport equation for air molecules are to be taken into account.
When the tip and the sample are in contact, heat transfer is enhanced, which is described by the contact thermal conductance. It includes contributions from the solid-solid mechanical contact, the water meniscus around the mechanical contact and the constrictions in the heat flux path. In the best cases, this thermal contact is responsible for 5% of the total tip temperature decrease due to the tip-sample exchange. We discuss the impact of various tips with curvature radii ranging from a micron down to 10 nm, especially the ballistic thermal dissipation in the sample, and the impact of vacuum-condition operation, where air transfer is removed. In addition, the modest impact of the applied force on the heat exchange is observed.
In a second step, we consider SThM thermometry (passive mode). There are two key phenomena to account for. First, the probe temperature is not equal to that of the sample, as the cantilever base acts as a heat sinks which imposes a temperature distribution in the tip. Second, since the tip acts partially as a heat sink which is placed close to the sample, it modifies the sample temperature distribution. Accounting for this effect can be crucial, as highlighted by experiments performed with samples heated by means of Joule-heated electrically-resistive serpentines.
We believe that this work, where both the understanding of the physical mechanisms responsible for thermal transport between the tip and the samples and precise knowledge of the energy balance of the tip-sample systems are targeted, provides a decisive step for determining quantitative data from the experiments.
 S. Gomès et al., Phys. Status Solidi A 212, No. 3, 477–494 (2015)
 A.M. Massoud et al., submitted
 L. Shi et al., ASME J. Heat Transfer 124, 329-337 (2001)
The support of projects TIPTOP (ANR-16-CE09-0023), DEMO-NFR-TPV (ANR-16-CE05-0013), QUANTIHEAT (FP7-2012 604668), EFINED (H2020-FETOPEN-1-2016-2017 766853), THERMOS (INSA-BQR-2017) is acknowledged.