## 1 Introduction  
Gallium nitride (GaN) has many attractive properties for high-power electronics and optoelectronic applications due to a wide bandgap of \(\sim\)3.4 eV and a high electron peak velocity of \(\sim\) 3 \(\times\) 10\({}^{7}\) cm/s achieved at high fields [1]. High-electron-mobility transistors (HEMTs) with high density (over 10\({}^{13}\) cm\({}^{-2}\)) two-dimensional electron gas (2 DEG) formed at the interface of AlGaN barrier and GaN channel layers are also resistant to high electric fields without the breakdown of the material [2; 3]. The HEMT, operating as a power amplifier or as a switch, dissipates the electric power in the conductive channel [4] and contacts [5]. Joule heating and the associated temperature increase in the transistor channel and contacts will be more intense at high currents achieved in gated [6; 7] and ungated [5; 8] devices. Self-heating is one of the factors limiting the operation of transistors [9] and plasmonic devices [10]. It can be mitigated by the use of high-thermal-conductivity substrates and the use of GaN-SiC hybrid material developed on SiC substrate without a GaN buffer layer [11]. The voltage drop in the contacts needs to be accounted for when the average electric field in the channel is estimated to be in an ON state of operation of the transistor or especially for the experiments on the ungated devices [12]. At high current densities, the properties of the contacts are important for transistor reliability [13; 14]. Robust electrodes for high-current experiments, especially for ungated or floating gate devices, are also needed [4; 15; 16; 17; 18; 19; 20]. The contact resistance \(R_{\rm c}\) is usually estimated from the standard transfer length model (TLM) measurements [21; 22]. The dependence of the total resistance \(R\) between two electrodes, which is a sum of \(R_{\rm c}\) and resistance of the channel \(R_{\rm ch}\), is measured versus the gap length (channel length) \(L\) between the series of the electrodes of the TLM structure. Due to the absence of a gate in the TLM structure, an electric field profile along the channel is assumed to be homogeneous and current-voltage (I-V) characteristics at low electric fields reveal the ohmicity of the contacts. The structures with H-shaped contacts, where the contact width is wider than the width of the channel, are also employed with the advantage of lower \(R_{\rm c}\)[23, 24]. Therefore, the contact resistance \(R_{\rm c}\) can be obtained from low-field measurements. At higher fields, I-V nonlinearity (the deviation from [PERSON]'s law) of the total resistance is observed and the increase in \(R\) versus the electric field is explained by the increase in the intensity of electron-phonon scattering in GaN [25]. This self-heating-related increase in the total resistance usually is treated as the increase in \(R_{\rm ch}\), while \(R_{\rm c}\) is treated as a constant value, estimated from the TLM measurements at low fields [12, 24]. Technological efforts have been targeted at achieving an \(R_{\rm c}\) value as small as possible, and hopefully, the achieved inequality \(R_{\rm c}\ll R_{\rm ch}\) lets one neglect the role of \(R_{\rm c}\). The \(R_{\rm c}\) dependence on temperature was investigated [26, 27, 28]. However, the \(R_{\rm c}\) dependence on current (and applied electric field) when the temperature of the contacts increases due to Joule heating was not investigated. The current and the electric field in the channel are controlled by the applied voltage to the electrodes, Virtually homogeneous current and electric field are expected in the channel between electrodes (corresponds to \(R_{\rm ch}\)), and non-homogeneous distribution of the current (and the electric field), so-called current crowding phenomena, is assumed under the electrodes (corresponds to \(R_{\rm c}\)) [29, 30, 31]. The qualitatively similar distribution of the current density is considered in different systems, such as organic [32] or graphene transistors [33].