Self-Heating of Annealed Ti/Al/Ni/Au Contacts to Two-Dimensional Electron Gas in AlGaN/GaN Heterostructures

Abstract

In this work, we investigated the self-heating effects of annealed Ti/Al/Ni/Au ohmic contacts and two-dimensional electron gas (2DEG) in AlGaN/GaN heterostructures under strong electric field by using the short pulse current–voltage and microwave noise measurement techniques. The experimental results demonstrated that the self-heating phenomena in ohmic contacts with the increase of current may dominate over the electrical performances of 2DEG channel. Moreover, the excess noise temperature of contact resistance was found to be more than four times higher than that of 2DEG channel at the same high current density values. The results reveal the importance of self-heating of the ohmic contacts on the device performance under short-duration (only 100 ns) and low-duty-cycle (only 10⁻⁵) pulse regime.

1. Introduction

Gallium nitride (GaN) has many attractive properties for high-power electronics and optoelectronic applications due to a wide bandgap of ∼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 (2DEG) 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 $R_{\mathrm{c}}$ 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 $R_{\mathrm{c}}$ and resistance of the channel $R_{\mathrm{ch}}$, 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 $R_{\mathrm{c}}$ [23,24]. Therefore, the contact resistance $R_{\mathrm{c}}$ can be obtained from low-field measurements.

At higher fields, I–V nonlinearity (the deviation from Ohm’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 $R_{\mathrm{ch}}$, while $R_{\mathrm{c}}$ is treated as a constant value, estimated from the TLM measurements at low fields [12,24]. Technological efforts have been targeted at achieving an $R_{\mathrm{c}}$ value as small as possible, and hopefully, the achieved inequality $R_{\mathrm{c}}\ll R_{\mathrm{ch}}$ lets one neglect the role of $R_{\mathrm{c}}$. The $R_{\mathrm{c}}$ dependence on temperature was investigated [26,27,28]. However, the $R_{\mathrm{c}}$ 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 $R_{\mathrm{ch}}$), and non-homogeneous distribution of the current (and the electric field), so-called current crowding phenomena, is assumed under the electrodes (corresponds to $R_{\mathrm{c}}$) [29,30,31]. The qualitatively similar distribution of the current density is considered in different systems, such as organic [32] or graphene transistors [33].

At high fields, the electron temperature $T_{\mathrm{e}}$ in GaN is higher than the lattice temperature $T_{\mathrm{L}}$, even for the DC-mode measurements [34]. It is well known that the Joule heating (self-heating)-related increase in $T_{\mathrm{L}}$ is lower in pulsed experiments. Hot-electron temperature is also related to the strong electron–LO-phonon coupling in GaN and the associated hot-phonon effect [25,35]. The experiments on microwave noise thermometry conducted at nearly isothermal lattice temperatures over a wide range of electric fields show that the hot-electron temperature (and electron energy relaxation time) depends on the dissipated power and on the 2DEG density in the GaN-based heterostructures with 2DEG channels [36]. An increase in $T_{\mathrm{L}}$ is related also to the thermal quenching phenomena, when $T_{\mathrm{e}}$ decreases along the voltage pulse [37]. For higher electron density, especially in the bulk, self-heating is stronger; therefore, the usage of short duration pulses is required [38].

In this paper, the self-heating of annealed Ti/Al/Ni/Au contacts to the 2DEG in AlGaN/(AlN)/GaN heterostructures was investigated under pulsed electric fields ranging up to 12 kV/cm and reaching current density values up to ∼1 A/mm. For this reason the pulsed current–voltage (I–V) and microwave noise measurements of the TLM test structures were performed.

The paper is organized as follows. In Section 2, the samples’ growth and ohmic contact fabrication are described, also, the measurement techniques are explained. In Section 3, the estimation of current dependent contact resistance is explained. A significant increase in the $R_{\mathrm{c}}$ with current was found. The change in $R_{\mathrm{c}}$ relative to its low-field value was found to be proportional to the dissipated electric power in $R_{\mathrm{c}}$. In Section 4, the microwave noise measurements are presented and the noise temperatures of $R_{\mathrm{c}}$ and $R_{\mathrm{ch}}$ are extracted, demonstrating much higher excess noise temperature in contacts than in the channel. In Section 5, the increase in $R_{\mathrm{c}}\left(I\right)$ was analyzed in terms of self-heating. It was found that the self-heating of $R_{\mathrm{c}}$ is more intense than that of $R_{\mathrm{ch}}$, resulting in a rapid increase in the ratio $R_{\mathrm{c}}\left(I\right)/R_{\mathrm{ch}}\left(I\right)$ with the current as the main result of this work. Finally, the results are summarized.

2. Samples and Measurement Techniques

Commercial AlGaN/(AlN)/GaN HEMT structures grown along the polar c-axis using the metalorganic chemical vapor deposition (MOCVD) method were selected for this study. The detailed description of the growth and characterization of alike structures can be found elsewhere [39,40]. In particular, layers for the AlGaN/AlN/GaN sample were grown on a 500 μm thick 6H-polytype SiC substrate as follows: a 90 nm thick AlN nucleation layer, a 1300 nmthick high resistivity GaN:C buffer, a 900 nm thick UID GaN layer and a 0.8 nm thick AlN spacer. Then, a 19 nm thick Al${}_{0.25}$Ga${}_{0.75}$N barrier layer was grown capping it with a 2 nm thick GaN and a 1 nm thick SiN${}_{\mathrm{x}}$ passivation layers. A similar structures, but without an AlN spacer (AlGaN/GaN) was developed on a semi-insulating 4H-polytype SiC substrate where a nominally undoped 20 nm thick Al${}_{0.25}$Ga${}_{0.75}$N barrier and 2.5 $\mathsf{\mu }$m thick GaN buffer were capped by a 1.5 nm thick GaN layer. Figure 1b illustrates only top part of the unintentionally doped (UID) GaN layer and AlGaN barrier required for 2DEG formation.

The ohmic contacts were fabricated of a Ti/Al/Ni/Au metal stack with a respective thickness of 30/90/20/100 nm annealed in nitrogen ambient for 30 s at 850 °C. The mesas were formed either by inductively coupled plasma reactive ion etching (ICP-RIE) using Cl plasma or by implantation of 700 keV energy Al ions into a depth of about 650 nm. More details about the device processing have been reported elsewhere [11,41]. The TLMs were used to evaluate the performance of ohmic contacts [22]. Values for the contact resistance $R_{\mathrm{c}}$ and specific contact resistivity ${\rho }_{\mathrm{c}}$ at low electric fields were minimized to 1 $\Omega$mm and 2 $\times {10}^{-3}$ $\Omega$mm${}^2$, respectively, [11,41,42]. The Hall effect experiments in Van der Paw geometry at temperature of 300 K revealed the density $n_{2\mathrm{DEG}}$ and mobility ${\mu }_{2\mathrm{DEG}}$ values of 2DEG in used samples to be of 8.3 × 10${}^{12}$ cm${}^{-2}$ and 9.3 × 10${}^{12}$ cm${}^{-2}$ and of 1.9 × 10${}^3$ cm${}^3$/Vs and 1.9×10${}^3$ cm${}^3$/Vs, respectively.

Usually, GaN device fabrication is completed with the deposition of dielectric layers to passivate the surface states minimizing their influence on the device/transistor performance [43]. However, good performance of unpassivated HEMTs has also been reported [44,45], exposing issues of the selection and usage of the surface passivants which might deteriorate the performance. In this work, the surfaces of all heterostructures after device processing were left unpassivated.

A microscope photograph of the TLM samples, an illustration of the cross-section of AlGaN/GaN heterostructures with two metal contacts annealed on the surface, and the connection between the sample (device under test (DUT)) in the electronic scheme are shown in Figure 1a–c, respectively. The 2DEG channel formed at the interface of the AlGaN and GaN layers is indicated by the blue line. Different length channels are selected for the pulsed I–V and microwave noise measurements by placing the two-pin microwave probe on top of the selected two nearby electrodes. The width W and the length D of rectangular electrodes were 250 $\mathsf{\mu }$m and 100 $\mathsf{\mu }$m, respectively. The distance between electrodes L for different TLM resistor samples were 6, 12.5, 25, 35, 45, 55, and 65 $\mathsf{\mu }$m. We assumed that the length of the 2DEG channel is the same as L and is independent of the applied voltage. The current distribution under the right-hand side and left-hand side electrodes is considered to be the same. The current crowding phenomena is illustrated by the set of arrows (vectors which roughly show the direction and the magnitude of the current flow) parallel and perpendicular to the plane of 2DEG.

The current–voltage characteristics were measured by applying voltage pulses of width ${\tau }_{\mathrm{p}}$ = 100 ns and a duty cycle $\sim {10}^{-5}$. The current–voltage and the hot-electron noise temperature–voltage characteristics were measured by applying voltage pulses to the TLM samples with electric field values reaching up to ∼12 kV/cm.

A voltage pulse generator (Generator AVL-2A-W-B-P-M, Avtech Electrosystems) was connected to the DUT through a coaxial current probe, a coaxial voltage probe, a microwave diplexer, and a microwave probe (Model 40A with a two-pin ground-signal configuration, 100 $\mathsf{\mu }$m pitch, GGB Industries), which was mounted on a high-precision probe station with vibration isolation (EPS150, Cascade Microtech). The coaxial voltage probe was realized as a pick-off tee giving 30 times attenuated replica of the voltage pulse. The coaxial current probe (Model 6027, Pearson Electronics) was inductively coupled giving the voltage replica of the current pulse (conversion ratio 1 Volt/Ampere). Signals from both probes were recorded by a digital oscilloscope (DSO1022A, Keysight Technologies).

As the DUT is biased by the voltage pulse, the excess microwave noise $\Delta T_{\mathrm{n}}=T_{\mathrm{n}}-T_0$ (where $T_{\mathrm{n}}$ is noise temperature and $T_0$ is room temperature) of the DUT appears at the pins of the microwave coaxial probe. The microwave diplexer is used to route the microwave noise of the DUT to a noise analyzer, consisting of low-noise amplifiers, a diode detector, and a box-car averager with a gating function (SR250 with modules 240A, 245, Stanford Research Systems). The diplexer was connected in the close vicinity to the DUT; it was connected straight to the coaxial connector of the coaxial microwave (two-pin) probe. The total resistance of the DUT is a sum of contacts resistances and the resistance of the channel, $R_{\mathrm{c}}+R_{\mathrm{ch}}+R_{\mathrm{c}}$, while $R_{\mathrm{p}}+R_{\mathrm{p}}$ is the resistance of the connections between the voltage probe and the DUT, including the resistance of the microwave probe. The description of experimental setups can be found elsewhere [46,47]. In microwave noise temperature experiments, the module of complex $S_{11}$ parameter (i.e., reflection coefficient in pulsed mode) of the DUT at ∼10 GHz also was measured and used for the proper estimation of the noise temperature [48,49].

Mesa-isolated TLM electrodes (i.e., channel etched) were used for the leakage current test and for standard $S_{11}$ parameter spectra measurements by vector network analyzer operating up to 40 GHz. Extracted parasitic capacitance (parallel to the channel) for the shortest channel was less than ∼50 fF. From $S_{11}$ spectra measurements of normal TLM structures (i.e., channel not etched), a small serial inductance ∼30 pH was extracted.

The results of the low-field total resistance measurements of the three heterostructure samples (with and without an AlN thin barrier layer) in the TLM configuration are shown in Figure 2. TLM measurements were conducted with conventional contact-front (two-terminal) method [22]. The total resistance of the DUT is the sum of the channel resistance and contact resistance (see Figure 1c):

$$R=R_{\mathrm{ch}}+2R_{\mathrm{c}}=(R_{2\mathrm{D}}/W)\times L+2R_{\mathrm{c}}$$

(1)

where the resistance of the 2DEG channel (layer) $R_{\mathrm{ch}}$ is expressed in terms of sheet resistance $R_{2\mathrm{D}}$ of the 2DEG channel between electrodes [22]. If the length L is reduced to zero, the total resistance approaches 2$R_c$. The contact resistance can be expressed as:

$$R_{\mathrm{c}}=R_{\mathrm{c}2\mathrm{D}}{L}_{\mathrm{T}}/W$$

(2)

where $R_{\mathrm{c}2\mathrm{D}}$ is the sheet resistance of the 2DEG channel directly under the electrodes and $L_{\mathrm{T}}$ is a transfer length [50], which is assumed to be $L_{\mathrm{T}}\ll D$ in our work. Measured total resistance of different length TLM samples (symbols in Figure 2) was fitted by Equation (1) (lines) and both resistances, $R_{2\mathrm{D}}$ and $R_{\mathrm{c}}$, were extracted. The interception of lines with the abscissa axis gives $2R_{\mathrm{c}}$ and the interception with ordinate axis is $2L_{\mathrm{x}}$ [50]:

$$L_{\mathrm{x}}=(R_{\mathrm{c}}/R_{2\mathrm{D}})\times W=R_{\mathrm{c}2\mathrm{D}}{L}_{\mathrm{T}}/R_{2\mathrm{D}}$$

(3)

The total resistance dependencies on the current are shown in Figure 3a. At the current region up to ∼50–100 mA, R weakly depends on current, which corresponds to the Ohm’s law region. The corresponding electric field range (i.e., low-field range) is up to 1–2 kV/cm. This also means that the contact resistance and channel resistance are both ohmic at this region. A deviation from Ohm’s law (an increase in R) is observed when the current and electric field become higher than ∼100 mA and ∼2 kV/cm, respectively. Maximal applied voltages for each length channel are indicated in Figure 3a. Those voltages correspond to an electric field in the channels in excess of 10 kV/cm. This nonlinear electron transport phenomena in the AlGaN/GaN channels is explained by the increase in intensity of electron–phonon scattering at higher fields [25]. This is described as a decrease in field-dependent electron drift mobility $\mu \left(E\right)$, which enters a simple relation $R_{2\mathrm{D}}\propto 1/\left(n_{2D}\mu \left(E\right)\right)$ where $n_{2D}$ is the sheet electron density. Unexpectedly, at high currents the total resistance for the shorter channels increases more steeply and thus, the ratio $R\left(L_2\right)/R\left(L_1\right)$ decreases with current when channel lengths $L_2>L_1$ (Figure 3a). Particularly, near Ohm’s region, the ratio for longest to shortest channel is ∼6.5, while at 270 mA it is ∼4.8. According to Equation (1), the change (decrease) in the ratio $R\left(L_2\right)/R\left(L_1\right)=(L_2-L_1)/(2W{R}_{\mathrm{c}}/R_{2\mathrm{D}}+L_1)+1$ with current means that the ratio $R_{\mathrm{c}}/R_{2\mathrm{D}}$ depends (increases) on current. It is worth noting that in most of the experiments, an independent contact resistance on the current density (with a constant value of $R_{\mathrm{c}0}$) is assumed. However, a contact resistance vs. current dependence, $R_{\mathrm{c}}\left(I\right)$, should be considered by using a procedure that is illustrated in Figure 3.

3. Current-Dependent Contact Resistance

In this work, the contact resistance is defined as:

$$R_{\mathrm{c}}\left(I\right)=R_{\mathrm{c}0}+\Delta R_{\mathrm{c}}\left(I\right)$$

(4)

where $R_{\mathrm{c}0}$ is the low-field (at Ohm’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 $R\left(L\right)$ 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 Ohm’s region. It is convenient to use a constant current for an $R\left(L\right)$ 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 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 $R\left(L\right)$ 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 $R\left(L\right)$ characteristics shown in Figure 2 and Figure 3b. An increase in $R_{\mathrm{c}}$ and $L_{\mathrm{x}}$ with an increase in I are observed as indicated in Figure 3b by the vertical arrow and the horizontal arrow, respectively.

The $R_{\mathrm{c}}$ and $\Delta R_{\mathrm{c}}$ dependencies on the current density for three samples are depicted in Figure 4a. Here, $R_{\mathrm{c}}$ and $\Delta R_{\mathrm{c}}$ values were multiplied by contact width W = 250 $\mathsf{\mu }$m and the current I values were divided by W.

At low currents, where Ohm’s law holds, $R_{\mathrm{c}}$ weakly depends on the current and $R_{\mathrm{c}}\left(I\right)\approx R_{\mathrm{c}0}$. For smaller $R_{\mathrm{c}0}$ (rectangles), the increase in $R_{\mathrm{c}}$ is slower. Roughly in the range of the measured currents, $R_{\mathrm{c}}$ increases by more than six times as the current exceeds ∼1 A/mm (closed rectangles). The same increase (approximately six times, circles) for the sample of higher $R_{\mathrm{c}0}$ is reached at ∼0.6 A/mm.

The $R_{\mathrm{c}}\left(I\right)$ dependence is nonlinear and develops a steep increase which approaches and exceeds $\Delta R_{\mathrm{c}}\left(I\right)\propto I^5$ (see lines in Figure 4a) at highest currents. This fast increase in $R_{\mathrm{c}}$ at a high current suggests that, even for very small $R_{\mathrm{c}0}$ values, a noticeable increase in $R_{\mathrm{c}}$ can be expected at high currents. The dependence of $\Delta R_{\mathrm{c}}$ and $R_{\mathrm{c}}$ on the electric power density $P_{\mathrm{c}}=2R_{\mathrm{c}}{I}^2/W$ dissipated in the contact resistances is presented in Figure 4b. It is found that $\Delta R_{\mathrm{c}}\propto P_{\mathrm{c}}$ in the wide range of $P_{\mathrm{c}}$, from ∼0.5–1 W/mm to ≳10 W/mm (see the lines in Figure 4b).

The dependence of $L_{\mathrm{x}}$ on I, which is obtained similarly to the $R_{\mathrm{c}}$ in Figure 3 (but from the interception with the ordinate axis) is depicted in Figure 5. The dependencies are qualitatively similar to those of $R_{\mathrm{c}}$ in Figure 4. Furthermore, the increasing $L_{\mathrm{x}}$ can thus approach or even exceed some channel lengths used in this work (see Figure 5, circles).

4. Noise Performance of ${\mathit{R}}_{\mathbf{c}}\left(\mathit{I}\right)$ and ${\mathit{R}}_{\mathbf{ch}}\left(\mathit{I}\right)$

Next, we proceed with the noise measurements, the results of which are shown in Figure 6. Higher excess noise temperature $\Delta T_{\mathrm{n}}$ is obtained for the shorter channel (Figure 6a, closed triangles).

The excess noise temperature $\Delta T_{\mathrm{n}}$ taken at constant currents for the different length channels is depicted in Figure 6b (symbols). The decrease in $\Delta T_{\mathrm{n}}\left(L\right)$ can be explained by the presence of two distinct noise temperatures $\Delta T_{\mathrm{nc}}$ and $\Delta T_{\mathrm{nch}}$ in the sample where the former is that of the contact resistances $2R_{\mathrm{c}}$ and the latter is that of the channel resistance $R_{\mathrm{ch}}$. It is easy to show that $\Delta T_{\mathrm{n}}$ of the serial connection of $2R_{\mathrm{c}}$ and $R_{\mathrm{ch}}$ is:

$$\Delta T_{\mathrm{n}}=\frac{\Delta T_{\mathrm{nc}}\times 2R_{\mathrm{c}}+\Delta T_{\mathrm{nch}}\times R_{\mathrm{ch}}}{2R_{\mathrm{c}}+R_{\mathrm{ch}}}$$

(5)

It follows that if $\Delta T_{\mathrm{nc}}>\Delta T_{\mathrm{nch}}$, then $\Delta T_{\mathrm{n}}$ decreases from $\Delta T_{\mathrm{nc}}$ to $\Delta T_{\mathrm{nch}}$, as the ratio $R_{\mathrm{c}}/R_{\mathrm{ch}}$ decreases from infinity to zero, while both temperatures $\Delta T_{\mathrm{nc}}$, $\Delta T_{\mathrm{nch}}$ and $R_{\mathrm{c}}$ do not depend on L at a given current and only $R_{\mathrm{ch}}\left(L\right)=(R_{2\mathrm{D}}/W)\times L$ depends on L. Thus, the measured $\Delta T_{\mathrm{n}}\left(L\right)$ dependence can be used to extract the temperatures $\Delta T_{\mathrm{nc}}$ and $\Delta T_{\mathrm{nch}}$ by fitting the experimental data with:

$$\Delta T_{\mathrm{n}}\left(L\right)=\frac{\Delta T_{\mathrm{nc}}\times 2R_{\mathrm{c}}+\Delta T_{\mathrm{nch}}\times (R_{2\mathrm{D}}/W)\times L}{2R_{\mathrm{c}}+(R_{2\mathrm{D}}/W)\times L}$$

(6)

where $R_{\mathrm{c}}$ and $R_{2\mathrm{D}}$ are known (see Figure 3b). The solid curves in Figure 6b are the fit with the $\Delta T_{\mathrm{n}}$($T_{\mathrm{nc}}$, $T_{\mathrm{nch}}$) model (Equation (6)). Formally small signal resistances at ∼10 GHz should be used in Equation (6); however, this simple model gives a good qualitative picture of two different noise temperatures: the noise temperature of $R_{\mathrm{c}}$ is higher than that of the channel resistance. The extracted $\Delta T_{\mathrm{nc}}$ is indicated by the squares in Figure 6a.

5. Discussion

The increase in the contact resistance with the current can be treated in terms of self-heating. The nearly linear dependence of $\Delta R_{\mathrm{c}}$ on dissipated power density $P_{\mathrm{c}}$ is obtained in Figure 4b. In fact, the dissipated power in the contacts controls the change in $R_{\mathrm{c}}$, which is evident from Figure 7, where the relative contact resistance change is plotted. The dependencies of different samples almost coincide when they are plotted versus $P_{\mathrm{c}}$.

Self-heating of the contact resistance also follows from the noise experiment (see Figure 6) under the assumption that noise temperature $T_{\mathrm{nc}}$ is a measure of electron gas temperature $T_{\mathrm{ec}}$ in the contact resistance, i.e. $T_{\mathrm{nc}}\approx T_{\mathrm{ec}}$. The excess noise temperature of contact resistance (squares) approaches ∼700 K.

The assumption $T_{\mathrm{nc}}\approx T_{\mathrm{ec}}$ is realistic because it is based on the experiments with the GaN-based 2DEG channels, where the microwave noise temperature of hot electrons $T_{\mathrm{n}}$ is approximately equal to hot-electron temperature $T_{\mathrm{n}}\approx T_{\mathrm{e}}$ [51].

The dependence of channel resistance change $\Delta R_{2\mathrm{D}}=R_{2\mathrm{D}}-R_{2\mathrm{D}0}$ on dissipated power density $P_{2\mathrm{D}}=R_{2\mathrm{D}}{(I/W)}^2$ in the channel is presented in Figure 8 (closed symbols). It was found, that qualitatively, the change in contact and channel resistances is similar and follows a linear trend at higher power in both cases (see open symbols in Figure 4b and Figure 8, respectively).

The contact resistance $R_{\mathrm{c}}$ can be described as [50]:

$$R_{\mathrm{c}}=\sqrt{{\rho }_{\mathrm{c}}\times R_{\mathrm{c}2\mathrm{D}}}/W$$

(7)

where ${\rho }_{\mathrm{c}}$ is the specific contact resistance. In actuality, ${\rho }_{\mathrm{c}}$ is the resistance from the metal to 2DEG under the electrodes. In Figure 1b, the current flow in ${\rho }_{\mathrm{c}}$ and $R_{\mathrm{c}2\mathrm{D}}$ is shown as vertical and horizontal current vectors (in shaded areas only), respectively.

Therefore, the observed self-heating of $R_{\mathrm{c}}$ is related with the self-heating of both resistances ${\rho }_{\mathrm{c}}$ and $R_{\mathrm{c}2\mathrm{D}}$. The contact-end (three-terminal) measurement method was developed for the estimation of the ${\rho }_{\mathrm{c}}$ and $R_{\mathrm{c}2\mathrm{D}}$ values [22,50]. The contact-front (two-terminal in this work) measurements, which are standard measurements of TLM, assume that $R_{\mathrm{c}2\mathrm{D}}=R_{2\mathrm{D}}$. The specific resistance obtained by contact-front method are denoted here as ${\rho }_{\mathrm{cf}}$. It was found that $R_{\mathrm{c}2\mathrm{D}}\ne R_{2\mathrm{D}}$ and the temperature dependencies of ${\rho }_{\mathrm{c}}$ and ${\rho }_{\mathrm{cf}}$ were qualitatively different [52]. Disregarding those concerns, ${\rho }_{\mathrm{cf}}$ was estimated and the results are shown in the inset of Figure 9. If we assume that the properties of the channel under the electrodes are similar to those between the electrodes ($R_{\mathrm{c}2\mathrm{D}}\approx R_{2\mathrm{D}}$), then the contact-front and contact-end methods give the same results, i.e., ${\rho }_{\mathrm{cf}}\approx {\rho }_{\mathrm{c}}$ and $L_{\mathrm{x}}\approx L_{\mathrm{T}}$.

Reiterating, in the present work, $R_{\mathrm{c}}$, $R_{2\mathrm{D}}$, ${\rho }_{\mathrm{cf}}$ and $L_{\mathrm{x}}$ increase with an increase in the current, while ${\rho }_{\mathrm{c}}$ and $R_{\mathrm{c}2\mathrm{D}}$ are not known with confidence.

We also can consider an unintentionally formed field-plate-like structure near electrode, which, at higher voltages in the channel, could induce a local high-resistance region near the edge of electrode which also would appear as a increase in R. In this case, the open symbols in Figure 4a would correspond to the resistance of the high-resistance region and not to the contact resistance change. However, it would be difficult to explain why this resistance is so different (more than 10 times), as it is compared between three channels at a given current.

An interesting correlation is observed in the dependencies of ${\rho }_{\mathrm{cf}}$ (inset of Figure 9) and $R_{\mathrm{c}}$ (Figure 4) on the current for different samples. In that case the increase in both starts earlier if low-field resistance $R_{\mathrm{c}0}$ is higher. For example, from Figure 4a it follows that the difference of $R_{\mathrm{c}0}$ between two samples (closed squares and circles) is ∼3 times, and that correlates with more than ∼12 times difference in $R_{\mathrm{c}}$ at an elevated current density of ∼700 mA/mm (Figure 4a). This makes sense, because $P_{\mathrm{c}}$ is higher for larger $R_{\mathrm{c}}$ at the same current, resulting in an increase in self-heating.

In other experiments, when the change in sample temperature was realized by direct heating of the sample (instead of Joule heating), two distinct trends of the change in $R_{\mathrm{c}}$ and ${\rho }_{\mathrm{c}}$ versus temperature were found [26,27,28,52,53,54,55,56,57,58,59]. The decrease in $R_{\mathrm{c}}$ was attributed to the metal–semiconductor junction properties and the increase was due to metal-like behavior when electron transport is determined by the “spike mechanism”—metal protrusions directly contacting the 2DEG channel under the electrodes

As it was previously discussed, both $R_{\mathrm{c}}$ and $R_{\mathrm{ch}}$ increase due to self-heating (Figure 4 and Figure 8). This is also observed in Figure 3a, where the dependence of the total resistance on the current is nonlinear. In terms of sheet resistance $R_{2\mathrm{D}}$, an increase is observed in Figure 3b, where the slope (which is proportional to $R_{2\mathrm{D}}$) changes with the current. The increase in the resistance of the 2DEG channel at the interface of AlGaN and GaN is widely acknowledged to be due to electron–phonon scattering, the intensity of which increases with the temperature of the device at a high current. The electron scattering is largely determined by the strong electron-LO–phonon interaction in the GaN and associated hot-phonon effect [25].

It is natural to expect that the 2DEG channel properties of the area beneath the electrodes (which have a sheet resistance of $R_{\mathrm{c}2\mathrm{D}}$) at least in part would preserve some features of the channel between the electrodes (which have a sheet resistance of $R_{2\mathrm{D}}$). Therefore, we compare the self-heating of $R_{\mathrm{c}}$ and $R_{2\mathrm{D}}$ by plotting $R_{\mathrm{c}}\left(I\right)$ versus $R_{2\mathrm{D}}\left(I\right)$ in Figure 9. Here, $R_{\mathrm{c}}$ is multiplied by W as in Figure 4. The channel sheet resistance $R_{2\mathrm{D}}$ is estimated from the $R\left(L\right)$ slope $\Delta R/\Delta L$ (see Figure 3b) and the channel width W as $R_{2\mathrm{D}}=(\Delta R/\Delta L)\times W$. In Figure 9, $R_{\mathrm{c}}$ is nearly proportional to $R_{2\mathrm{D}}$ at lower currents (see solid lines), but then, it starts to increase faster than $R_{2\mathrm{D}}$ (see dashed lines). The threshold-like increase in $R_{\mathrm{c}}$ is resolved, revealing the threshold position for all three samples at value of about ∼600 $\Omega$/sq.

The increase in $R_{\mathrm{c}}$ is not good news for those who fabricate devices with small $R_{\mathrm{c}0}$ and those who believe that it will remain the same small at higher currents (electric fields). The situation is even more dramatic, because the ratio $R_{\mathrm{c}}/R_{2\mathrm{D}}$ (and $R_{\mathrm{c}}/R_{\mathrm{ch}}$) also increases with the current. Indeed, from Figure 9 it can be concluded that above the threshold $R_{\mathrm{c}}$ increases faster than $R_{2\mathrm{D}}$. We recognize that this is the most important result of the present work. This behavior has many practical consequences, because a fast increase in $R_{\mathrm{c}}$ depending on I can deteriorate the performance of electronic devices.

The increase in the ratio $R_{\mathrm{c}}/R_{2\mathrm{D}}$ can be discussed in terms of stronger $R_{\mathrm{c}}$ self-heating compared to that of $R_{\mathrm{ch}}$. Excess noise temperature $\Delta T_{\mathrm{nc}}$ in $R_{\mathrm{c}}$ (Figure 6, squares) is more than four times stronger than $\Delta T_{\mathrm{n}}$ in the total resistance R of 45 $\mathsf{\mu }$m-long channel (circles). The term $\Delta T_{\mathrm{nc}}\times 2R_{\mathrm{c}}$ in Equation (5) is not negligible, even for the 45 $\mathsf{\mu }$m-long channel (it would be negligible for the infinitely long channel only). Therefore, $\Delta T_{\mathrm{n}}$ of a 45 $\mathsf{\mu }$m-long channel reaches ∼130 K at a ∼0.3 A, and it is still slightly higher than its $\Delta T_{\mathrm{ch}}$. Thus, the relation $\Delta T_{\mathrm{nc}}>\Delta T_{\mathrm{n}}\gtrsim \Delta T_{\mathrm{nch}}$ holds. This suggests that the temperature $T_{\mathrm{c}}$ (and thus, the self-heating) of $R_{\mathrm{c}}$ is higher than the temperature $T_{\mathrm{ch}}$ of $R_{\mathrm{ch}}$.

Let us first discuss the option that the self-heating of $R_{\mathrm{c}}$ is caused by the self-heating of $R_{\mathrm{c}2\mathrm{D}}$ (see Equation (7)), while ${\rho }_{\mathrm{c}}$ self-heating is negligible. The current density in $R_{\mathrm{c}2\mathrm{D}}$ is smaller than in $R_{2\mathrm{D}}$, therefore, the stronger self-heating of $R_{\mathrm{c}2\mathrm{D}}$ compared to $R_{2\mathrm{D}}$, might seem surprising. The current profile is illustrated in Figure 1b where the horizontal current vector magnitude gradually decreases as the current branches toward the electrodes. A detailed description of the current dependence on the distance can be found elsewhere [22]. We could suppose, that $R_{\mathrm{c}2\mathrm{D}}>R_{2\mathrm{D}}$ due to lower electron mobility or lower sheet electron density under the electrodes. In this case dissipated power per unit area $P_{2\mathrm{D}}$ at a given current density j will be higher because $P_{2\mathrm{D}}\propto R_{\mathrm{c}2\mathrm{D}}{j}^2$. Higher dissipated power in the contact resistance results in higher self-heating of $R_{\mathrm{c}2\mathrm{D}}$. The next option is the self-heating of ${\rho }_{\mathrm{c}}$. If the metal-like shunt resistances from the metal to the 2DEG channel under the electrodes persists, the electron mobility in these shunt resistances is expected to be poor due to the high density of electrons and associated intensive electron–electron scattering. However, the high electron density in metals is what could make the specific resistance of those metal protrusions low. The current density in metal protrusion is high if its cross section is small, leading to high resulting values of the dissipated power density per unit volume.

The self-heating model in terms of the two temperatures $T_{\mathrm{c}}$ and $T_{\mathrm{ch}}$ is oversimplified. In GaN-based 2DEG channels (applicable to $R_{\mathrm{c}2\mathrm{D}}$ and $R_{2\mathrm{D}}$), excess (hot) electron temperature $\Delta T_{\mathrm{e}}$ is higher than the excess lattice temperature $\Delta T_{\mathrm{L}}$ when the constant electric field is applied [34]. In the pulsed mode, $\Delta T_{\mathrm{L}}$ is lower when compared to the DC mode; therefore, at short voltage pulses a nearly isothermal condition is maintained ($\Delta T_{\mathrm{L}}\ll T_{\mathrm{L}}$), and the self-heating effects related to acoustic phonons are mitigated. Furthermore, in the pulse mode, $\Delta T_{\mathrm{e}}$ is higher due to the thermal quenching effect when the hot-electron temperature decreases along with the voltage pulse due to the electron mobility decrease [37]; the shorter the pulse, the weaker the thermal quenching. Mainly because of lower $\Delta T_{\mathrm{L}}$ in the pulsed mode, higher electric fields and currents can usually be reached without the thermal breakdown of the channel. It is also important to mention, that even in the pulsed mode, at higher electric fields the LO-phonon-related self-heating is present. For 2DEG densities, typical for GaN-based heterostructures, the hot-phonon effect is important, as non-equilibrium LO-phonons (hot-LO-phonons) strongly impact the hot-electron transport and temperature [25,35]. At high fields under conditions of strong hot-electron–hot-phonon coupling, when the system is strongly non-equilibrium, the hot-electron temperature $T_{\mathrm{e}}$ is close to the effective hot-LO-phonon temperature $T_{\mathrm{LO}}$. In this case, different temperatures $T_{\mathrm{e}}$, $T_{\mathrm{LO}}$ and $T_{\mathrm{L}}$ are introduced, for which the relation $T_{\mathrm{e}}\gtrsim T_{\mathrm{LO}}\gg T_{\mathrm{L}}$ holds [51]. The hot-electron excess temperature $\Delta T_{\mathrm{e}}$ is controlled by electron energy relaxation time ${\tau }_{\mathrm{en}}$, $\Delta T_{\mathrm{e}}\propto {\tau }_{\mathrm{en}}{P}_{\mathrm{e}}\propto {\tau }_{\mathrm{en}}{P}_{\mathrm{V}}/n\propto {\tau }_{\mathrm{en}}\rho j^2/n$, where $P_{\mathrm{e}}$ is the dissipated power per electron and n is the electron density [60]. The dependence of excess noise temperature on dissipated power per electron in the 45 $\mathsf{\mu }\mathrm{m}$ long channel $P_{\mathrm{e}}=U_{\mathrm{a}}I/N$ is shown in the inset of Figure 8 (open circles), where N is electron number in the channel available from the Hall measurements. The line in the inset is plotted for constant ${\tau }_{\mathrm{en}}$ in the channel. At a higher power per electron, ${\tau }_{\mathrm{en}}$ decreases due to enhanced electron–LO-phonon scattering, and experimental points (circles) deviate from that line. If the electron mobility in the 2DEG channel under the electrodes is lower than in the 2DEG channel between electrodes (corresponding to $R_{\mathrm{c}2\mathrm{D}}$ and $R_{2\mathrm{D}}$, respectively), then at a given current, $\Delta T_{\mathrm{e}}$ in the former channel will be higher. Indeed, at a given current, the electric field (and thus, $P_{\mathrm{e}}$) will be higher for the 2DEG channel with a lower electron mobility. A higher $P_{\mathrm{e}}$ results in a higher $\Delta T_{\mathrm{e}}$ (see inset of Figure 8). For the metal-like conductor, the hot-electron effect is not well expressed due to poor electron mobility; therefore, $T_{\mathrm{e}}\approx T_{\mathrm{L}}$. In this case, $\Delta T_{\mathrm{e}}\approx \Delta T_{\mathrm{L}}\propto (1/C_{\mathrm{V}})\rho j^2$, where $C_{\mathrm{V}}$ is the heat capacity per unit volume. As such, qualitatively both cases (GaN-based channels and metal-like conductors) are similar in that $\Delta T_{\mathrm{e}}\propto \rho j^2$. This is the reason why it is difficult to differentiate between the self-heating of resistances $R_{\mathrm{c}2\mathrm{D}}$ and ${\rho }_{\mathrm{c}}$ from the temperature–current measurements. Assuming that $R_{\mathrm{c}2\mathrm{D}}\approx R_{2\mathrm{D}}$ and ${\rho }_{cf}\approx {\rho }_c$, the data in the inset of Figure 9 indicate the strong self-heating of ${\rho }_c$.

To conclude, we investigated the pulsed current–voltage (I–V) and hot-electron microwave noise (at 10 GHz frequency) characteristics of the TLM samples composed of AlGaN/(AlN)/GaN heterostructures with annealed Ti/Al/Ni/Au ohmic contacts to the 2DEG channel. From the I–V measurements, the increase in the contact resistance $R_{\mathrm{c}}$ dependent on the current was found for all samples, with a change up to six times as the current density approached ∼0.6 A/mm. The comparison of current dependence $R_{\mathrm{c}}$ and the channel resistance $R_{\mathrm{ch}}$ revealed that $R_{\mathrm{c}}$ is more sensitive to the current change. It was found that the change in $R_{\mathrm{c}}$ from its low-field value is proportional to the electric power dissipated in the contacts. The pulsed microwave noise measurements further confirmed that the noise temperature of the contacts is notably higher when compared to the noise temperature of the channel. The increase in the contact resistance (and temperature) was interpreted as a result of self-heating.