Electrothermal modeling for nano AlGaN/GaN HEMTs using dual-phase-lag theory

– The combined dependence of the electronic and thermal characteristics in the AlGaN/GaN HEMTs supported in nano-elctronic devices was studied theoretically and numerically. The Schrödinger-Poisson equations coupled with Dual phase lag (DPL) thermal transfer equation was undertaken. Simultaneous impacts of the conduction band offset and polarization charge between the AlGaN/GaN heterointerface induce the production of the two-dimensional electron gas density (2DEG). The simulation results showed that the 2DEG density at the heterointerface increased with increase of Aluminum fraction. In addition, the simulation results of the thermalization process were found to be in good agreement with the literature. As a result, the maximum heat generation as well the maximum temperature at the heterointerface increased. The obtained result could to be useful in assessing thermal transfer in the AlGaN/GaN HEMTs nano-devices to improve their performance.


Introduction
High Electron Mobility Transistors (HEMTs) power components on Gallium Nitride (GaN) have demonstrated their potential to respond to high frequency applications such as radar, space or telecommunications. Up to now the best performances of HEMTs AlGaN/GaN transistors in terms of current cutoff frequency (ft=454GHz) and in power (fmax=444GHz) have been obtained for 20 nm long gates [1]. The increase in the frequency of components, linked to their miniaturization, is inevitably accompanied by an increase in the power dissipated, causing significant self-heating.
Several ways are being studied to improve the thermal dissipation of transistors in the GaN industry. The substrate used for the growth of the AlGaN/GaN heterostructure must have good electrical resistivity and good thermal conductivity to promote heat dissipation from operating components [2][3][4]. A new and growing voice for improving the cooling of electronic circuits is to integrate micro-channels directly into the circuits allowing the circulation of water and ensuring the cooling of the components [5].
Thus, it has been demonstrated that the management of the thermal dissipation of the components is a major stake in obtaining reliable electronic systems. In order to optimize the thermal management of components, it is important to know the operating temperature of components in real time. The real-time knowledge of this temperature allows control and ensure preventive and non-curative maintenance of electronic systems.
The answer projected by several works considers the temperature as an external parameter as close as possible to their hot point close to the maximum operating temperature [6][7].
Nevertheless it has many shortcomings in real situations. Indeed, the assessment of the temperature depends on the thermal source generation caused by the current density, the electric field due to Joule effect and the electron carrier mobility attributable to the polar optical phonon scattering. Conversely, the temperature evolution affects the electrical characteristics.
Another advance which may be valid is the Fourier thermal transfer theory. This breakthrough has proven to be suitable for the study of thermal conduction in a large number of electronic components [8][9]. Nevertheless, at the nano-electronics components based HEMTs heterostructure, additional complex phenomena are associated with thermal conduction processes [10]. To study thermal conduction in HEMT-based electronic devices at the nanoscale, the dual-phase-lag theory (DPL) was developed to overcome the weakness of the conventional Fourier thermal transfer theory [11][12]. In order to describe the microscopic electron phonon interactions, the Dual-Phase-Lag thermal conduction model taken into account the two relaxation parameters T (the phase lag of the temperature gradient which trap the time delay ensuing from the microstructural interaction effect) and q (the phase lag of the thermal flux that trap the quick transient effect of thermal inertia) model has been developed by Tzou [13][14].
In this work, we consider the combined dependence between the electronic and thermal characteristics on the nano AlxGa1-xN/GaN HEMTs based electronic devices. In section 2, the DPL equation joined with the Schrödinger-Poisson equations is detailed. In section 3, the discussion results of electrical and thermal characteristics are studied with respect to the Aluminum fraction.

Dual phase lag heat transfer equation
The thermal transfer comportment of nano-electronic devices is expressed by the subsequent equation: where , , and are the thermal capacity, the thermal flux and the heat source generation rate, respectively.
The dual phase lag thermal transfer theory associates the thermal flux and the temperature gradient by means of the subsequent law [14]: Equation (2) can be expressed as a Taylor-series expansion. The first order development allows: Replacing the heat flux ( , ) into Eq. (1), we achieve the one-dimensional Dual phase lag equation: here, = ⁄ is thermal diffusivity.
The thermal source generation rate Q is principally caused by the Joule heating.

Schrödinger-Poisson equations
The electronic characteristics of the HEMTs heterostructure is considered by working out both Schrödinger-Poisson equations [19]: where , * , ∆ , , , and are, respectively, the depth direction, the effective mass of electrons, the conduction band energy, the dielectric constant, the wave vector, the energy subband index and the ionized donors density.
The density of the electrons ( ) is expressed as follows: where is the Fermi energy level and fulfills the neutrality equation: where 2 (= 210 12 −2 ) is the density of dopants and is the binding energy of donors (= 30 meV) [20].
The two-dimensional-electron-gas ( 2DEG ) density is calculated in all subbands as follows: The electron effective mass as a function is illustrated by the equation [21]: where 0 , ∆ 0 , Γ and Γ are, respectively, the free electron mass, the spin orbit splitting, the energy associated to the momentum matrix element and the band gap energy. The suggested constants utilized in Equation (15) are summarized in Table 2 for GaN and AlN [21].
The temperature dependency of the dielectric constant ( ) for GaN and AlGaN as approximated by [22]: The sum potential energy ∆ is given as follows: where ∆ , , and (= ) are, respectively, the heterointerface band gap discontinuity, the effective Hartree potential, the exchange-correlation potential and the potential energy attributable to the polarization charge density in the AlGaN/GaN hetrostructure.
The electric field ( ) attributable to the spontaneous (SP) and piezoelectric (PZ) polarizations in the barrier and well are given as follows [23]: where (

1−
) and ( ) correspond to the barrier and well layers.
The polarization charge density at the heterointerfaces is given as follows [24]: The temperature dependencies of the lattice constants are expressed as: where is the coefficient of thermal expansion.
The equilibrium lattice constant ( 0 ) with respect to the aluminum fraction is given as follows [25]: The piezoelectric polarization of The sum of polarization at the heterointerface is evaluated as: The exchange-correlation potential is expressed as [26]: The conduction band offset energy (∆ ) of the AlGaN/GaN heterointerface is expressed as: The temperature dependencies of the band gap energy of the AlN and GaN are given by [27].

Results and discussion
To illustrate this approach, the electrical characteristic of the AlGaN/ GaN HEMT heterostructure on G-face configuration is considered. Figure 1     In Figure 2, the conduction-band edge potential (∆ ) for different values of Aluminum fraction (x=0.2, 0.25 and 0.3) is offered. It is noticeable that the ∆ has at a triangular shape within the AlGaN/GaN heterointerface. It is also shown that the curve becomes deeper with the higher Aluminum fraction. This is chiefly attributable to the internal quantum-confined Stark effect which increases with the increases of Aluminum fraction [26]. Additionally, the disagreement in piezoelectric and spontaneous polarizations between the AlGaN and the GaN layers induced the formation of a 2DEG density in the heterointerface. Furthermore, Figure 2 shows that the conduction-band offset ∆ is enlarged relatively to the Aluminum fraction.
Within the cases considered in this study and at = 100 , the conduction-band offset of the AlGaN/GaN HEMT enlarged from 0.34eV to 0.52eV when x increased from x=0.2 to x=0.3.        Given that the thermal source generation depends on the electrostatic field, the temperature variation grew along with the Aluminum fraction.

Conclusion
The AlGaN/GaN HEMTs could advance high power-efficient nano-electronic devices.
That's why it deserves crucial to expand a model to truthfully explain the thermal transfer impacts in nano-electronics devices, which considerably have an effect on the performance of the electrical characteristics. The present model coupled with the Schrödinger-Poisson and the dual phase lag thermal transfer equations has been evaluated to simulate the AlGaN/GaN HEMTs. The spontaneous and piezoelectric polarization field effect together conduction band offset effect between the AlGaN/GaN heterointerface leads to the formation of the 2DEG density. Moreover, it is shown that the 2DEG density increases with the increase of the Aluminum fraction. In addition, the simulated result of the thermalization process was found to be in good agreement with literature. In view of that, the temperature, the thermal source generation and the thermal flux grow at the heterointerface with the growth of the Aluminum Figure 1 Cross-section illustration of the AlGaN/GaN HEMT on G-face con guration.