Analytical modeling and simulation of lattice-matched Ferro PZT AlGaN/GaN MOSHEMT for high-power and RF/Microwave applications

We present an analytical model for Ferro PZT Al2O3/AlGaN/AlN/GaN MOSHEMT involving the solution of Poisson and Schrödinger equations. This analytical model covers most of the operating regimes of the Ferro PZT MOSHEMT. The two-dimensional electron gas (2-DEG) sheet charge density (ns), threshold voltage (Vth), drain current (Ids), gate capacitance (Cgs and Cgd), and unit gain cutoff frequency(fT) model equations are presented and simulated with MATLAB tool. It is observed that the insertion of the Ferro Pb(Zr, Ti)O3 PZT (lead zirconium titanate) material can improve the device’s performance. The proposed Ferro PZT MOSHEMT model accurately predicts a higher drain current of 1.14 A/mm, a high transconductance of 362 S/mm, a gate-to-source capacitance of 50.99 pF, a gate-to-drain capacitance of 38.25 pF, and high cutoff frequency of 0.033 THz for 20 nm AlGaN barrier layer. The results show good agreement with the TCAD-Atlas simulation and are satisfactory for the different AlGaN barrier layer thicknesses. The generated model and simulation results show the potential of using the Ferro PZT MOSHEMT for high-power and RF/Microwave applications.


Introduction
Gallium nitride-based high electron mobility transistors (HEMTs) exhibit superior performance in terms of wider bandgap (varying from 3.4 to 6.2 eV for AlGaN), highpower, high-temperature, high-voltage, and high-frequency applications [1][2][3] due to their properties such as high mobility (>1500 cm 2 /V.s) [4,5], high saturation velocity, high thermal conductivity, high breakdown field ( ∼10 6 Vcm -1 ), and high two-dimensional electron gas (2-DEG) sheet charge density (10 13 cm -2 ) at the hetero-interface [6][7][8]. However, the major drawbacks of HEMT-based devices are higher gate leakage current [9] and drain current collapse during operation at higher frequencies. Metal oxide semiconductor HEMTs with oxide dielectric show superior performance for reducing gate leakage current. Also, AlGaN/GaN-based MOSHEMT is more suitable for high-power and high-frequency electronic device applications than conventional HEMT-based devices due to their low gate capacitance and low gate leakage current. Today, Al 2 O 3 [9,10], HfO 2 [11,12], SiO 2 [13,14], Si 3 N 4 [14], or La 2 O 3 [13,14] materials are commonly used as the gate dielectric for AlGaN/GaN MOSHEMTs. But, AlGaN/GaN HEMT and MOSHEMT-based devices suffer from high contact and ON-resistance because of metallic ohmic contacts and substantial source-to-drain distance. However, GaN and its alloys have significant chemical and temperature stability, making them ideal for withstanding ferroelectric materials in the high-temperature deposition process. GaNbased high electron mobility transistors (HEMTs) have new opportunities for polarization engineering due to the strong polarization of nitrides and the switchable polarization nature of ferroelectric (FE) material. Hence, there is a possibility of improvements in 2-DEG channel modulation and dynamic threshold voltage control by applying the Ferro polarization engineering techniques. It is perceived that an addition of a Ferro PZT layer in the conventional MOSHEMT-based designs decreases the gate resistance by retaining the gate length [15]. In heterostructures, the difference between spontaneous polarization (P SP ) of two layers can be used to create a high 2-DEG carrier concentration of mobile carriers [16]. The high 2-DEG sheet charge concentration is obtained in Ferro PZT MOSHEMT due to the substantial spontaneous and piezoelectric polarization effect [17]. The sheet carrier concentration and 2-DEG confinement are considerably increased by polarization-induced electric fields. In the interface between GaN and AlGaN, a two-dimensional electron gas (2-DEG) spontaneously forms due to the piezoelectric and spontaneous polarization effect, resulting in depletion mode. In wurtzite group-III nitrides, which progress from GaN over InN to AlN, the spontaneous polarization is incredibly strong at zero strain. This causes the polarization-induced effects in AlGaN/GaN-based devices to rise further. If the polarization-induced charge density is positive, free electrons tend to compensate for the charge, resulting in a 2-DEG with a sheet carrier concentration of n S . Also, the inbuilt polarization field depends on AlGaN barrier layer thickness and aluminum mole content.
Nowadays, ferroelectric material lead zirconium titanate Pb(Zr, Ti)O 3 [17][18][19][20] has been used as a gate dielectric on AlGaN/GaN MOSHEMT to perform E-mode operation due to the strong polarization effect to acquire two-dimensional electron gas with the shifting of threshold voltages. Ferroelectric material lead zirconium titanate has shown a significant polarization impression on Al x Ga 1-x N/GaN MOSHEMT-based devices materializing in 2-DEG [21]. Although PZT and GaN have a substantial lattice mismatch, this can be prevented by utilizing the Al 2 O 3 layer. Also, adding a layer of lead zirconium titanate to the oxide layer gives superior performance to thin-film deposition techniques. Ferroelectric-induced materials show high charge density at the interface in the identical direction of AlGaN [22]. This phenomenon will increase the overall threshold voltage of the Ferro PZT Al 2 O 3 /Al x Ga 1-x N/AlN/GaN MOSHEMT-based device. Hence, the Ferro PZT AlGaN/ GaN MOSHEMT is expected to show high-power-handling capability, higher mobility, higher carrier concentration, and low leakage current compared to conventional HEMT-based devices.
In this work, we have proposed a novel Ferro PZT Al 2 O 3 / Al x Ga 1-x N/AlN/GaN MOSHEMT device and demonstrated the essential device parameters such as threshold voltage (V th ), 2-DEG sheet carrier density (n s ), drain current (I ds ), gate capacitance (C gs and C gd ), and cutoff frequency (f T ) by developing a compact analytical model with the help of Poisson and Schrodinger's equations. This paper is organized as follows. Section 2, "Ferro PZT Al 2 O 3 /AlGaN/AlN/ GaN MOSHEMT device structure and Energy band diagram," describes the device formation and the conduction band profile of the device. In Sect. 3, "Analytical Model Developments for Ferro PZT Al 2 O 3 /AlGaN/AlN/GaN MOSHEMT," we have derived the different expressions of the physics-based compact models with the variation in AlGaN barrier thickness. Different Model results generated using MATLAB are thoroughly investigated and compared with the TCAD-Atlas simulation results in Sect. 4, "Results and discussion". Finally, the paper is summarized in "Conclusion" Sect. 5.

Ferro PZT Al O 3 /AlGaN/AlN/GaN
MOSHEMT device structure and energy band diagram

Device structure
This section presents a cross-sectional view of the Ferro PZT Al 2 O 3 /AlGaN/AlN/GaN MOSHEMT device, as shown in Fig. 1. The different used layer put together from top to bottom approach is metal/Ferro/Al 2 O 3 /AlGaN/AlN/undoped GaN/AlN on a silicon (Si) substrate. The device has Ferro material thickness of (t ferro = 3 nm), Al 2 O 3 oxide thickness of (t ox = 2 nm), AlGaN barrier thickness of (d AlGaN = 13, 17 and 20 nm), thin AlN spacer layer thickness of (d AlN = 1 nm), undoped GaN buffer layer thickness of 1.47 μ m, and the gate contact is Nickel (Ni) with a work function of 5.01 eV. The length and width of the gate contact are 0.6 μ m and 100 μ m, respectively [23]. For reducing the effects of gate leakage current, the dielectric of Al 2 O 3 is placed between the ohmic gate contact and the AlGaN barrier layer [24]. The AlN spacer layer provides an extensive conduction band profile across the GaN nitride channel [25]. The large conduction band offset is generated due to the high bandgap of AlN for the AlGaN barrier layer, which simultaneously generates high carrier concentration and mobility. For achieving stable off-characteristics, the GaN undoped buffer layer is applied to the silicon (Si) substrate. The AlN nucleation layer has been applied to the non-native silicon substrate to lessen stress, dislocation density, and lattice mismatch. The low leakage current is essential for realizing high breakdown fields. The AlN nucleation layer also controls this leakage by reducing the lattice mismatch between GaN and Si substrate. The proposed device is simulated using at TCAD-Atlas [26], and obtained results are compared with the developed physics-based analytical model.

Energy band diagram
Under thermal equilibrium, the energy band bending in any material system balances the differences in applied gate voltage and metal to a semiconductor work function. The energy balancing equation on both sides of the metal/dielectric junction between Fermi energy level (E F ) and vacuum level can be expressed as (2) by the energy band diagram of Ferro PZT Al 2 O 3 /Al x Ga 1-x N/AlN/GaN MOSHEMT, as illustrated in Fig. 2. The initial sub-band and Fermi levels are denoted by E 0 and E F , respectively. The electron volts are used in the aforementioned energy balance equation. Additionally, the AlGaN/GaN interface's energy band discontinuity in the conduction band creates a quasi-triangular quantum well with a highly mobile two-dimensional electron gas (2-DEG). Moreover, the thin AlN spacer layer improves carrier mobility due to reduced alloy disorder scattering. It is visible that the spacer layer AlN generates a large conduction band offset between the AlGaN barrier and the GaN channel layer. This conduction band offset can be represented as follows [27]: The AlGaN/GaN heterojunction's quantum well depth is increased with the insertion of an AlN spacer layer, reducing scattering. As a result, carrier confinement is seen because binary compounds like AlN scatter less than ternary ones. The improvement in carrier mobility in the device is clear from the rise in I DS (drain-to-source current) because the additional AlN spacer layer decreases strain at the interface of AlGaN and GaN layers. So, the conduction band offset between the AlGaN barrier and the GaN channel increases the carrier concentration, mobility, and confinement at the AlN and GaN hetero-interface.

Threshold voltage model for ferro PZT Al 2 O 3 / AlGaN/AlN/GaN MOSHEMT
The aluminum mole content (x) of the AlGaN barrier layer greatly influences the device's behavior. The balanced equation for Ferro PZT Al 2 O 3 /AlGaN/AlN/GaN MOSHEMT according to their energy band diagram can be set up from metal contact to toward of GaN channel. It can be written as follows [28]: where eff (x) is the effective Schottky barrier height of the gate contact from the fermi energy level. For in the case of Al x Ga 1-x N barrier layer, the barrier height can be calculated using Vegard's interpolation formula The generated electric field across the Ferro, oxide, Al x Ga 1-x N, AlN and GaN [23] can be calculated as The spontaneous polarization for Al x Ga 1-x N III-nitride alloy is strain-free and can be formulated as [29] (2) The piezoelectric polarization for the AlGaN barrier layer with induced strain can be given as The piezoelectric polarization of the AlN and GaN can be expressed accurately as (x) is the basal-strain-field (BSF) function of the III-nitride compound material, which is dependent on the equilibrium lattice constant a Al x Ga 1−x N as a function of AlGaN barrier and GaN channel layer strains. (x) is defined as follows: The lattice parameters a GaN and a Al x Ga 1−x N were derived by applying Vegard's interpolation law: The dielectric constant for the AlGaN compound material as a function of aluminum mole content is expressed as are the conduction band offsets between the Al 2 O 3 /Al x Ga 1-x N, AlN/GaN and Al x Ga 1-x N/AlN, respectively. The conduction band offset for AlGaN/AlN hetero-interface is given by [30] where E Al x Ga 1−x N g is the energy bandgap of Al x Ga 1-x N barrier layer as a function of aluminum mole fraction and is given by [31,32] In (17), b and x are the bowing parameter and aluminum mole fraction, respectively, where A=xE g (AlN) and B=(1 − x)E g (GaN) with energy bandgap of E g (GaN) = 3.42 eV , E g (AlN) = 6.2 eV . C ferro and C ox are the Ferro and oxide capacitance, respectively, and are given as follows: where t ferro and t ox are the thickness of the Ferro and oxide layer, respectively, and 0 , ferro and ox are the permittivity of free space, Ferro material, and Al 2 O 3 oxide, respectively. It is evident from (17) that mole fraction x impacts the calculation of the energy band gap in Al x Ga 1-x N compound materials. Further, the spontaneous and piezoelectric polarization charges are also dependent on aluminum mole fraction (x) as per (9) and (10). There is an increase in charge carrier confinement and hence higher two-dimensional electron gas density due to an increase in the Al mole fraction in the AlGaN barrier layer.
The threshold voltage also depends on the aluminum mole content (x); to make this system entirely in the cutoff region, it is necessary to make the channel empty in terms of electrons, for this external potential has to be applied from the gate terminal. Assuming n s = 0 cm −2 in the (2) and solving for the threshold voltage is given by

2-DEG sheet charge carrier density model
The 2-DEG sheet charge carrier density can be measured from the Poisson and Schrodinger's equations in the triangular quantum well and can be expressed as follows [33]: where D is the conduction band density with E 0 = 0 n 2∕3 s and E 1 = 1 n 2∕3 s are the two lowest sub-band energy levels in the triangular quantum well. 0 and 1 are the experimental parameters, E f is the Fermi level for gate voltage V g0 . In (21), the sheet charge density expression is a function of (conduction band density, Fermi Level, Energy Sub-bands, and threshold voltage) for AlGaN/GaN HEMT. But, the incorporation of oxide and Ferro material lowers the parasitic capacitance effect and boosts the cutoff frequency due to the strong and switchable polarization effect. By using Taylor's theorem under total depletion approximation, the 2-DEG of depleted charge can be modeled by Poisson's equation is the total permittivity of the AlGaN barrier, and the AlN spacer layer, V gs , is the gate voltage, V p is the channel potential. For a compact drain current model, n s must be valid in all aspects of 2-DEG and can be given as . In this equation, C g can be calculated as follows [34]:

Drain current model for Ferro PZT Al 2 O 3 /AlGaN/ AlN/GaN MOSHEMT
The expression for the drain current I ds in the triangular quantum well using an analytical model is derived (23) based on the 2-DEG carrier concentration and can be expressed as [7] where 0 , W g , and L g are the low field mobility, gate width, and gate length of the Ferro PZT MOSHEMT, respectively. An analytical mathematical model for measuring the drain current can also be generated as [23] where , with E T critical field. Limits during the integration process can be s p e c i f i e d a s fo l l ows : Table 1 indicates the expressions for the constants k i (i = 1, 2, 3, .., 6) obtained during the integration of (25).

Transconductance model for Ferro PZT Al 2 O 3 / AlGaN/AlN/GaN MOSHEMT
The gate transconductance is an essential parameter for evaluating the RF/microwave performance of the device, Which can be defined as [23] The extracted form of transconductance from (27) where

Capacitance model for Ferro PZT Al 2 O 3 /AlGaN/ AlN/GaN MOSHEMT
The gate-to-source (C gs ) and gate-to-drain (C gd ) capacitance is derived from the partial differentiation of the net gate charges for source and drain terminal voltages. It can be expressed as C gs = Q V gs and C gd = Q V ds . After solving these equations, gate capacitance is obtained as [23] (27)

Unity gain cutoff frequency model for Ferro PZT Al 2 O 3 /AlGaN/AlN/GaN MOSHEMT
The unity gain cutoff frequency (f T ) is the vital figure-ofmerit (FOM) of the AlGaN/GAN-based MOSHEMT performance, and it can be related to transconductance and capacitance as [36] All the statistical model parameters information used for modeling and TCAD-Atlas simulations are listed in Table 2.

Results and discussion
In this section, the comparative analysis of the devel- The physical models FLDMOB and CONMOB, which consider electric field-dependent and concentration-dependent mobility, have been used for simulations. The CVT is a stand-alone model that incorporates all the effects required for simulating carrier mobility, SRH recombination, generation, and Auger recombination. Shockley-Read-Hall  [6] recombination terms have been added to the continuity equation to model traps. The computation of 2-DEG is done using the built-in polarization model during the simulation. The different model parameters used for static and dynamic characteristics calculation are described in Table 2. The proposed analytical model results of I ds −V ds , I ds −V gs , g m −V gs , C gs −V gs , C gd −V gs , and f T −V gs characteristics agree with the TCAD-Atlas simulation results for all the presented AlGaN barrier layer thickness of Ferro PZT Al 2 O 3 /AlGaN/AlN/ GaN MOSHEMT. Figure 3 shows the variation in the 2-DEG sheet charge density (n s ) with AlGaN barrier thickness (22). It also shows the experimental validation of AlGaN/GaN HEMT [16] with the proposed Ferro PZT AlGaN/GaN MOSHEMT. It is observed that the experimental results and the developed model results are quite mismatching as there is no 2-DEG fabrication report of Ferro PZT AlGaN/GaN MOSHEMT. The result shows that the carrier concentration increased as AlGaN barrier thickness varied from 0 to 25 nm, but 2-DEG was almost constant after 25 nm. For AlGaN barrier thickness below 5 nm, the 2-DEG charge carrier concentration is negligible because charges are not present at the surface potential. The insertion of the AlN spacer layer between the AlGaN barrier and GaN channel improves the electron transport properties of the proposed device due to low metal alloy disorder scattering. AlN and GaN-based III-nitride shows high spontaneous polarization fields. The P sp in the AlGaN barrier layer is increased with the increment of aluminum mole fraction. So, when the total polarization of the device is increased at the AlN and GaN interface, it generates a high electric field and increases interface charge densities at the hetero-interface. When the thickness of the AlGaN barrier increases, the distance between the AlGaN barrier and GaN channel becomes more profound for the Fermi level and 2-DEG, forming a large polarization and energy band offset at the hetero-interface. After 25 nm, the carrier concentration is approximately constant due to the maximum accumulation of the charges at the hetero-interface.
The comparative analysis of experimental and model results of 2-DEG sheet charge density with the variation in the AlN spacer layer thickness is shown in Fig. 4. 2-DEG simultaneously increases with the increment of spacer layer thickness due to large total polarization fields. After 2.5 nm of the AlN spacer layer thickness, the 2-DEG remains approximately constant due to maximum carrier confinement at the hetero-interface.
The variation in 2-DEG for gate-to-source voltages for the constant 20 nm AlGaN barrier and 1 nm AlN spacer layer thickness is shown in Fig. 5. The results show the approximate graphical representation with the experimental data. We can conclude that 2-DEG sheet charge density increases with the increment of the gate voltages.
The transfer characteristics curve (I ds −V gs ) of the proposed Ferro PZT Al 2 O 3 /AlGaN/AlN/GaN MOSHEMTs is modeled and simulated with different AlGaN barrier thickness in Fig. 6. The analytical model results are validated compared to the generated atlas simulated results for different AlGaN barrier layer thicknesses of 13 nm, 17 nm, and 20 nm. Both the results show that drain current increases with the rising thickness of the barrier layer. The results show a high drain current density of 1.14 A/mm at the gateto-source voltage of 1 V with the AlGaN barrier thickness of 20 nm.
Device nonlinearity is a significant concern that shows nonlinear characteristics in high-speed and RF devices. A higher transconductance may reduce the nonlinearity features in the devices. The maximum transconductance of 362 mS/mm is obtained for Ferro PZT Al 2 O 3 /AlGaN/AlN/  Fig. 7. These results show that transconductance increases with a barrier layer thickness (28).
Ferro material and Al 2 O 3 oxide reduced the surface trap charges and increased the electric field below the gate contact. This process increases the 2-DEG surface density and gate transconductance of the Ferro PZT MOSHEMT device. The transconductance slope gradually decreased with an increase in gate-to-source voltages after achieving a peak due to saturation. The main reasons behind the high current density and gate transconductance generation are high 2-DEG surface density and mobility. Similarly, to validate the analytical model, for comparison of the I ds -V ds , different gate voltages with varying AlGaN barrier layer thickness are shown in Fig. 9. The proposed model Ferro PZT Al 2 O 3 /AlGaN/AlN/GaN MOSHEMT shows that drain current increases with the gate-to-source voltage variation from (−2 V to 2 V) with the step size of 1 V. The maximum drain current of the  From these characteristics plots, it is clearly understood that an increment in the Al 0.3 Ga 0.7 N barrier layer thickness leads to a significant increase in the drain current of the Ferro PZT MOSHEMT device.
The gate-to-source (C gs ) and gate-to-drain (C gd ) capacitance are referred to as gate channel capacitance. Both C gs and C gd are calculated according to the (29) and (30), respectively, and plotted for gate-to-source voltages for three different AlGaN barrier layer thicknesses as shown in Figs. 10 and 11, respectively.
The gate-to-source capacitance variation in gate-tosource voltages with a constant drain voltage of 10 V is presented. It is observed that there is a substantial dependency of threshold voltage on the AlGaN barrier layer. The C gs are constant for gate voltages between −5 and −3 V and abruptly increase to a specific limit. The C gd (V gs ) plot of Ferro PZT Al 2 O 3 /AlGaN/AlN/GaN MOSHEMT follows the same characteristics as mentioned earlier for C gs .
Finally, these characteristics show that the gate and drain capacitance increases considerably with the reduction in AlGaN barrier layer thickness.
The influence of variation in the Al 0.3 Ga 0.7 N barrier thickness on the cutoff frequency is shown in Fig. 12. (31)   demonstrates that the two primary methods for raising the cutoff frequency are boosting gate transconductance and reducing parasitic capacitance. Since a more significant AlGaN barrier thickness would increase the 2-DEG density by improving the carrier confinement and the number of electrons available to the 2-DEG, there is an enhancement in transconductance. As the growth rate of g m with increased barrier layer thickness is faster than the rising number of parasitic capacitance, f T grows as barrier layer thickness increases.
The results show that the cutoff frequency increases with the Al 0.3 Ga 0.7 N barrier thickness. The maximum cutoff frequency of the Ferro PZT MOSHEMT is 0.033 THz for the AlGaN barrier thickness of 20 nm, 15.45%, and 6.06% higher than 13 nm and 17 nm, respectively. The higher aluminum mole fraction and increment in the AlGaN barrier layer lead to a higher electric field of induced polarization. The electron carrier confinement increases in the channel due to an increment in the Al 0.3 Ga 0.7 N barrier layer thickness. The cutoff frequency increases steadily with the gate bias increasing from (−5 to 1 V) and then becomes stable. Further, if we increase the gate-to-source voltages, the cutoff frequency value will decrease. Table 3 shows the comparative analysis of obtained results after modeling and TCAD-Atlas simulation of Ferro PZT Al 2 O 3 /Al 0.3 Ga 0.7 N/AlN/GaN MOSHEMT device with the different AlGaN barrier layer thickness. These results indicate that the Ferro PZT MOSHEMTs perform better for 20 nm AlGaN barrier thickness.

Conclusion
A physics-based mathematical, analytical model for Ferro PZT Al 2 O 3 /AlGaN/AlN/GaN MOSHEMTs is developed to generate the different static and dynamic characteristics plots. Different model equations are developed for 2-DEG sheet charge density, threshold voltage, drain current, gate transconductance, gate-to-source capacitance, gate-to-drain capacitance, and cutoff frequency. Then, these model equation results are compared with the atlas simulation results for different AlGaN barrier layer thicknesses of 13, 17, and 20 nm. The obtained model results are analyzed and validated with TCAD-Atlas simulations and found satisfactory. The AlGaN barrier thickness of 20 nm shows improved and superior performance in terms of a higher drain current of 1.14 A/mm, high gate transconductance of 362 mS/mm, the gate capacitance of 50.99 pF, drain capacitance of 38.25 pF and the cutoff frequency of 0.0330 THz. Finally, it is concluded that the thickness of 20 nm of the AlGaN barrier is best suitable for high-power and high-frequency applications. The developed model can be used for performance analysis for Ferro PZT AlGaN/GaN-based MOSHEMTs.
Funding The authors have not disclosed any funding.
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