An analytical model for the current voltage characteristics of GaN-capped AlGaN/GaN and AlInN/GaN HEMTs including thermal and self-heating effects

Received Jan 28, 2019 Revised Oct 26, 2019 Accepted Nov 2, 2019 We present an analytical model for the I-V characteristics of AlGaN/GaN and AlInN/GaN high electron mobility transistors (HEMT). Our study focuses on the influence of a GaN capping layer, and of thermal and self-heating effects. Spontaneous and piezoelectric polarizations at Al(Ga,In)N/GaN and GaN/Al(Ga,In)N interfaces have been incorporated in the analysis. Our model permits to fit several published data. Our results indicate that the GaN cap layer reduces the sheet density of the twodimensional electron gas (2DEG), leading to a decrease of the drain current, and that n-doped GaN cap layer provides a higher sheet density than undoped one. In nGaN/AlInN/GaN HEMTs, the sheet carrier concentration is higher than in nGaN/AlGaN/GaN HEMTs, due to the higher spontaneous polarization charge and conduction band discontinuity at the substrate/barrier layer interface.

Recently, a heavily doped n-GaN cap layer has been employed to improve high-frequency performance and to reduce access and ohmic contact resistances. The beneficial effect of cap layers was demonstrated experimentally by Gessmann et al. [10] for In0.27Ga0.73N/GaN and GaN/Al0.2Ga0.8N structures. Heikman et al. [11] and Asgari et al. [12] have proposed a GaN/Al0.32Ga0.68N/GaN heterostructure deposited on a sapphire substrate, and they show that the sheet carrier density decreases with thicker cap layers while the mobility increases. Similar effects were reported by Tao et al. [13] for an AlN/GaN heterostructure. To maximize the high frequency performance of AlGaN HEMTs, Green et al. [14] incorporated GaN cap layers in GaN/AlGaN/GaN structures by heavily n-doping the upper GaN layer; thus parasitic contact resistances were greatly reduced. Also, the GaN cap layers were shown to be more effective in protecting the samples of AlGaN/GaN HEMTs structures from degradation, compared to in-situ grown Si3N4 cap layers [15]. The ohmic contact performance on GaN capped AlGaN/AlN/GaN HEMTs was improved by Wang et al. [16] by optimizing the Ti/Al/Mo/Au electrodes. Finally, activated p-GaN cap layers have been demonstrated to increase the breakdown characteristics and reduce the leakage current of AlGaN/GaN HEMTs [17].
In this paper, we present an analytical model for the I-V characteristics of these HEMTs with a doped n-GaN cap layer. This study is based on well-established models for the description of the involved phenomena (like polarization-induced sheet charge densities, self-heating effect, drain conductance and so on). Therefore the originality of our work is related to a rather complete modeling of the devices based on classical analytical expressions. As compared to using CAD tools, here the advantage arises from getting analytical expressions, which allow one to enlighten the role of each parameter and to optimize them. We study n + -GaN/AlGaN/GaN and n + -GaN/AlInN/GaN structures and we investigate the effect of the GaN cap layer on threshold voltage, sheet carrier density, current voltage characteristics and drain conductance. Thus our study will allow us to compare AlGaN and AlInN devices with similar structures, as well as the effect of doping the cap layer. The model takes into account the influence of thermal effect over a wide temperature range (300-475 K) and the self-heating effect on the I-V characteristics. Spontaneous and piezoelectric polarization-induced charges have also been considered. Our modeling is validated by comparison with published experimental data.

MODEL FORMULATION 2.1. Description of the studied structure
The studied structures are GaN/Al0.32Ga0.68N/GaN (noted structure A) and GaN/Al0.83In0.17N/GaN (structure B) as depicted in Figure1. They are grown on an undoped GaN thick buffer layer deposited over a SiC, Si or sapphire substrate. The 2DEG channel appears at the interface between the buffer layer and the undoped Al0.32Ga0.68N or Al0.83In0.17N spacer layer. An n-doped AlGaN (AlInN) thick barrier layer provides the 2DEG sheet charge density. It is covered by a second undoped additional layer, and then by the thin GaN cap layer. This last one could be undoped or heavily n + -doped with Si to a concentration equal to 10 20 cm -3 . The second spacer layer prevents impurity scattering from the n + -GaN cap to the n-AlGaN (AlInN) layer [18], which increases the density and mobility of carriers in the channel. The thickness of the different layers are specified on Figure 1. The GaN cap layer on top of the hetero-structure raises the conduction band and produces a large enhancement in effective barrier height Ebeff. This is due to a negative polarization charge (-σ) at the upper hetero-interface, which increases the electric fields in the barrier layer, and hence decreases the 2DEG density, leading to a reduction in the gate leakage current [14].

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When the GaN cap becomes thicker, the charge density in the channel is smaller; the valence-band ∆Ev of the GaN cap shifts upwards, and the valence band eventually reaches the Fermi level EF. At this point, a two-dimensional hole gas (2DHG) is formed at the upper GaN/Al(Ga,In)N interface Figure 2 b-right [11]. The n-doping of the GaN is also advantageous because it reduces the contact resistance [16]. All the main geometrical parameters of the structures are defined and their values are given in Table 1. The electrical parameters of the structure materials are defined and their values are given in Table 2. Finally, the physical parameters that depend on the Al mole fraction m are listed in Table 3, in which their expressions versus m is given.

Threshold voltage
The threshold voltage Vth is defined as the applied gate voltage for which the channel is completely depleted, and is considered as the minimum potential in the channel [23]. The dependence of Vth on temperature and Al mole fraction m (m = 0.32 for structure A, and m = 0.83 for B), including the effects of both spontaneous and piezoelectric charge polarization, is given by the following expressions: without cap [24]: d1 is the distance between the gate and the channel without GaN cap layer, q is the electronic charge, ESchottky represents the Schottky barrier height between the metal and the Al(Ga,In)N layer in bare structures and between the metal and GaN in caped structures, ɛ2(m) and ɛ1 are the dielectric constants of respectively Alm(Ga,In)1-mN and GaN, EF is the Fermi level assuming ∆EF = 0 at 300 K [19], and σ is the polarization-induced charge density at the interface see Figure 3. If σ is positive, free electrons will tend to compensate for the polarization-induced charge and will form a 2DEG. On the other hand, a negative sheet charge density causes an accumulation of holes, leading to a 2DHG at the interface [25]. In our simulation, we use a Ni Schottky barrier contact at the surface.

Charge control model
The sheet carrier density ns is one of the principal entities governing the performance and operation of Al(Ga,In)N/GaN HEMT devices [26]. When a gate voltage Vgs is applied through a Schottky contact, the sheet carrier density ns, under strong inversion region in the channel formed at the AlGa(In)N/GaN interface, is obtained from self-consistent 1-D Poisson's and Schrodinger's equations in the quantum well, assuming a triangular potential profile [27], and considering spontaneous and piezoelectric polarizations [28,29] Here d is the density of state in the conduction band of the 2DEG system, m * is the electron effective mass,   c Vx is the channel potential at any point x along the channel due to the applied drain voltage, Vgs is the gate-to-source bias voltage.

Polarization-induced sheet charge density
The direction of the polarization-induced field formed at the upper GaN/Al(Ga,In)N interface (-σ) is opposite to that in the lower Al(Ga,In)N/GaN one (+σ) Figures 3 and 4. Then, the conduction band is rising as shown in Figure 3. Therefore 2DEG and 2DHG appear at the Al(Ga,In)N/GaN and GaN/Al(Ga,In)N interfaces respectively, as shown in Figure 4. The total polarization P is the sum of the piezoelectric P PZ and spontaneous P SP polarizations, which induce a sheet charge density σ at the hetero-interfaces [19]: At the lower interface of the heterostructures, the total polarization can change abruptly, which creates a fixed polarization charge σ defined by: Similarly, at the upper interface of the hetero-structure, the charge density is: The piezoelectric polarization-induced charge density is:

Current-voltage characteristics
The drain source current in the 2-DEG channel can be expressed from the current density equation presented in [27], where temperature effects are included in the electron mobility and channel charge density terms: where kB is the Boltzmann constant and µ(T,x) is the temperature dependent mobility, whose expression is given in [30][31]: Substituting expressions (3) and (10) in (9), and integrating along the channel leads to [30]: with: Vds is the drain-to-source voltage. For the sake of simplicity, the dependence on T and/or m of the coefficients Ai, µ0 and ɛ has not be written down in the preceding expressions and will not be from now on. The low-field mobility µ0 depends on the electron concentration   , s n T m , which further depends on the doping concentration, on the temperature and on the material quality. The Caughey-Thomas empirical expression gives the electron concentration [31]: , with . max, max, min, 1/ 0 Here i=n for electrons and p for holes, respectively, N is the doping concentration, and To=300 K. The parameters µmax,i , µmin,i , Ng,i , γi, αi and βi are taken from [32] see Table 4.

Drain conductance
The drain conductance is an important microwave parameter that determines the maximum voltage gain delivered by the device. The drain conductance of the HEMT is evaluated as [

Incorporation of self-heating effect (SHE)
Self-heating is one of the critical factors that adversely affects device performance and reliability in high-power and high temperature applications. The temperature increase caused by self-heating reduces the performance of the devices and enhance their degradation. In reality, the self-heating effect (SHE) corresponds to an increase of the crystal temperature due to a high power dissipation in the active zone of AlGaN/GaN HEMT transistors operating at large biases. This enhances the phonon scattering rate, which reduces the mobility and the electron saturation velocity [5]. Consequently, this results in a negative slope of drain current versus drain voltage. Modeling SHE has been address in a few studies, which consider the effect of self-heating and hot electrons on polarization-induced charges and the defect-induced traps at all the device interfaces [28,33]. We include here a simple model of SHE [34], in which the temperature increase results from the dissipation of the power Pdiss in a phenomenological thermal resistance Rth: Thus, the actual working temperature T of the devices is corrected by adding ∆T to the ambient temperature T0: Figure 5 shows the influence of the barrier thickness on the sheet carrier density. For structure A, the sheet density increases rapidly for barriers thinner than 10 nm and then tends to saturate. A similar behavior is observed for structure B, but with a higher saturation value (30×10 12 instead of 15×10 12 cm -2 around 40 nm). Our model is in excellent agreement with experimental data circles in Figure 5 extracted from the literature [11] and [7].

RESULTS AND ANALYSIS s 3.1. Electrical performance of the HEMTs without self-heating effects
The sheet carrier density is plotted in Figure 6 as a function of the GaN cap layer thickness. The sheet carrier density decreases monotonously when the cap layer becomes thicker and remains almost constant for thick cap layers (~3.8×10 12 cm -2 for structure A, and ~5.2×10 12 cm -2 for structure B, over 250 nm). Our model permits to fit well the experimental data published in [11]. Figure 7 shows that both structures with an undoped GaN cap layer exhibit a smaller sheet density than with an n + -doped GaN cap layer. The reduction of 2DEG sheet density may be attributed to the additional negative polarization charges formed at the interface between GaN and Al (Ga,In)N. The sheet carrier density increases slightly with the barrier thickness.On the other hand, the 2DEG sheet density is much higher in structure B than in structure A. This improvement is attributed to a better electron confinement in the channel due to a larger electric field, arising firstly from higher spontaneous polarization charge at the GaN/Al0.83In0.17N/GaN hetero interface, and secondly from the conduction band discontinuity formed at the same interface, which is higher than with GaN/Al0.30Ga0.70N/GaN. In this latter case, our model fits very well with experimental data of reference [35]. Figure 8 shows the variation of the 2DEG sheet density as a function of the barrier thickness for both structures, for different thicknesses dcap of undoped (Fig. 8a) and n + -doped Figure. 8b GaN cap layers. Let us notice that the variation of 2DEG sheet density for various values dcap is significant, especially for undoped GaN cap. The 2DEG sheet density decreases when the undoped GaN cap layer becomes thicker for both structures A and B. This is explained by both the contribution of the heavily doping cap layers in the case of higher thicknesses and by the large spontaneous polarization charge at the GaN/Al0.83In0.17N/GaN hetero interface (structure B).
The variation of threshold voltage Vth as a function of barrier thickness is shown in Figure 9. Vth decreases to more negative values for thicker barrier thicknesses, especially for structure B. Figure 9a shows that the effect of undoped GaN cap is less pronounced than with an n + -doped GaN cap. Our model is in good agreement with experimental data open circles in Figure 9a extracted from [4]. Also Vth decreases to more negative values when the n + -doped GaN cap layer becomes thicker Figure 9b.
The variation of the sheet carrier density ns as a function of gate source voltage Vgs is presented in Figure 10 for different thicknesses of the n + -doped GaN cap layer. ns increases almost linearly with Vgs over a given threshold. Consequently, the depth of the potential well at hetero interface decreases, as well as the 2DEG density [34]. Moreover, the curves shift to more negative Vgs when increasing the n + -doped GaN cap layer thickness, which indicates a simultaneous decrease of the threshold voltage Vth. Here again, our model permits to fit experimental data extracted from [35]. The variation of sheet carrier density ns is plotted in Figure 11 versus the cap layer doping density Ncap. As expected, it increases proportionally to Ncap. Moreover, the variation is stronger when increasing the cap thickness.Typically, with a 5-nm cap thickness, ns in structure B is almost doubled: 1.28×10 13 cm -2 (Ncap = 0) to 2.11×10 13 (Ncap = 2.75×10 20 cm -2 ).This improvement is explained by the additional contribution of heavily doping cap layer of higher thickness and large spontaneous polarization charge in structure B.
As already anticipated from the preceding results, Vth decreases to more negative values when increasing either Ncap or the GaN cap layer thickness Figure 12. For example in structure A, Vth is decreasing from -10.24 V without capping layer to -16.74 V with dcap=5 nm and Ncap = 2.75×10 20 cm -3 . The drain current is diminished in structures covered with an undoped GaN cap layer, as seen in Figure 13. This can be explained by the reduction of the 2DEG sheet density due to the additional negative polarization charges formed at the interface between GaN and Al(Ga,In)N. The drain current is more intense in structure B. Also, it is seen that the drain current increases linearly with small values of Vds and then saturates at higher Vds because of the very high sheet charge density, resulting from large conduction band discontinuity and strong polarization effects. Our model is in good agreement with published experimental results [7,37], even if the doping density Nd in the transistors studied in [7] and [37] is different. The effect of the cap layer on the drain source conductance Figure 14 is rather weak. The drain conductance becomes smaller when increasing the drain bias voltage because, as well, the carrier velocity rises gradually and then saturates.

Thermal and self-heating effects
As already explained, thermal and self-heating effects result in a degradation of the HEMTs performance. Figure 15 presents the Ids-Vds curves of both A and B structures, without a cap layer. Each curve is calculated without and with taking into account thermal and self-heating effect, for three different room temperatures, namely 300 K Figure 15a, 375 K Figure 15b and 425 K Figure 15c. In any case, thermal and self-heating effects decrease strongly the value of the drain-source current. At higher values of the drain voltage, they lead to a negative resistance, which is more important in structure B than in structure A. Increasing the room temperature makes the drain current diminishing, when self-heating is either considered or not for both structures A and B. For example, under a bias Vds = 10 V, Ids decreases from ~728 mA at 300 K to ~663 mA at 425 K in structure B. The effective barrier height Ebeff shown in Figure 2 is derived from the energy band diagram: