## 3 Current-Dependent Contact Resistance  
In this work, the contact resistance is defined as:  
\[R_{\mathrm{c}}(I)=R_{\mathrm{c0}}+\Delta R_{\mathrm{c}}(I) \tag{4}\]  
where \(R_{\mathrm{c0}}\) is the low-field (at [PERSON]'s law region) contact resistance and \(\Delta R_{\mathrm{c}}\) represents the change in contact resistance over its low-field value. Because of the dependence of \(R\) on the bias, the \(R(L)\) plot should be taken at a constant bias in contrast to Figure 2, where any bias for different \(L\) channels could be taken until measurements are done in [PERSON]'s region. It is convenient to use a constant current for an \(R(L)\) plot because the current is a good quantity to compare the different length channels; the same current in a different \(L\) implies that the electric field also will be the same in those channels. The data points in Figure 3a are not horizontally aligned because the pulsed  
Figure 3: The dependence of the total resistance \(R\) on the current \(I\) for the different channel length \(L\) (**a**) and on channel length for a different current (**b**). \(L=6\) μm (closed triangles), \(12.5\) μm (open diamonds), \(25\) μm (open triangles), \(45\) μm (open circles) and \(65\) μm (closed rectangles). The measurements are presented for the AlGaN/GaN channel. The resistance values at a given currents (represented by the vertical dashed lines) are obtained from a linear interpolation; for the particular current \(I=126\) mA those are indicated as crosses. The applied maximal voltage for each length channel is indicated in the right side of the figure. Interpolated values of different length channels are depicted in the (**b**) for each current and are used for the estimation of the current dependent contact resistance \(R_{\mathrm{c}}\) and \(L_{\mathrm{x}}\) (see interception with abscissa and ordinate axis in (**b**)). Solid lines are linearly fit with Equation (1) and arrows show the direction of the increase in \(R_{\mathrm{c}}\) and \(L_{\mathrm{x}}\). \(I=24\) mA, \(74\) mA, \(100\) mA, \(126\) mA, \(153\) mA, \(178\) mA, \(203\) mA, \(220\) mA, \(236\) mA, \(257\) mA, \(271\) mA and \(282\) mA. voltage generator is not an ideal current source but is rather a voltage source with some internal resistance. Linear interpolation between experimental points was used so that \(R\) values at constant \(I\) (see vertical lines) could be compared. Crosses in Figure 3a illustrate the procedure for a case when \(I=126\) mA. The \(R(L)\) characteristics obtained in this manner are shown in Figure 3b. It is worth mentioning that the contact resistance was assumed to be the same for all electrodes of the TLM samples. This assumption also allows for the observation of linear \(R(L)\) characteristics shown in Figures 2 and 3b. An increase in \(R_{\rm c}\) and \(L_{\rm x}\) with an increase in \(I\) are observed as indicated in Figure 3b by the vertical arrow and the horizontal arrow, respectively. The \(R_{\rm c}\) and \(\Delta R_{\rm c}\) dependencies on the current density for three samples are depicted in Figure 4a. Here, \(R_{\rm c}\) and \(\Delta R_{\rm c}\) values were multiplied by contact width \(W=250\) and the current \(I\) values were divided by \(W\). Figure 4: The dependence of the contact resistance \(R_{\rm c}\) on the current density \(I/W\)**(a)** and on the dissipated electric power density \(P_{\rm c}\) in contacts **(b)** for different samples; AlGaN/GaN (squares), AlGaN/AlN/GaN (triangles and circles). The dependence of the contact resistance change \(\Delta R_{\rm c}=R_{\rm c}-R_{\rm c0}\) relatively to its low-field value is also presented (right-side axis). Lines are \(\Delta R_{\rm c}\propto I^{5}\)**(a)** and \(\Delta R_{\rm c}\propto P_{\rm c}\)**(b)**, respectively. At low currents, where [PERSON]'s law holds, \(R_{\rm c}\) weakly depends on the current and \(R_{\rm c}(I)\approx R_{\rm c0}\). For smaller \(R_{\rm c0}\) (rectangles), the increase in \(R_{\rm c}\) is slower.