Quantum correction method for drift diffusion transportation of gallium nitride power device in advanced packaging and three-dimensional integrated module electrothermal coupling simulation method

Abstract

The invention discloses a quantum correction method for drift diffusion transportation of a gallium nitride power device in advanced packaging and a three-dimensional integrated module electrothermal coupling simulation method. According to the method, a one-dimensional stationary Schrodinger equation and a two-dimensional Poisson equation are solved in a self-consistent mode, sub-band energy levels, corresponding wave functions and potential distribution in the quantum confinement direction are obtained, and the convergence of a self-consistent algorithm is improved by modifying the Poisson equation; quantum correction of the drift diffusion equation is realized by correcting the conduction band; starting from a quantum confinement effect and a drift diffusion transport theory, the quantum effect of two-dimensional electron gas in a triangular potential well formed by a gallium nitride polarization effect and a quantum correction method of a drift diffusion transport model are researched, and the steady-state operating characteristics of an advanced packaging power device containing the quantum effect are evaluated. And an electrothermal coupling simulation method of the device and the three-dimensional integrated module is further researched, the thermal management capability of the three-dimensional integrated module is evaluated, and the method has important application value.

Description

Technical Field

[0001] The present invention relates to the field of quantum effect modeling and simulation of gallium nitride power devices, and relates to a quantum correction method for drift-diffusion transport of gallium nitride power devices in advanced packaging and a three-dimensional integrated module electro-thermal coupling simulation method to evaluate or optimize their working characteristics. Background Art

[0002] With the development of industries such as 5G and new energy vehicles, the demand for high-power, high-frequency, and high-efficiency electronic devices is increasing continuously. Gallium nitride stands out with its advantages such as higher breakdown strength, faster switching speed, higher thermal conductivity, and lower on-resistance. The carrier transport in gallium nitride high electron mobility transistors mainly relies on the two-dimensional electron gas formed at the AlGaN / GaN heterojunction interface. Due to the polarization effect of gallium nitride materials and the polarization charges generated by the mismatch stress during the growth of the aluminum gallium nitride barrier layer, a large number of electrons are trapped on the gallium nitride side at the interface, thus forming a two-dimensional electron gas with a high concentration.

[0003] In the triangular potential well formed by the polarization effect, there are obvious quantum effects. The density distribution of the two-dimensional electron gas can be obtained by self-consistently solving the one-dimensional stationary Schrödinger equation and the Poisson equation. In the channel direction, due to the large size, the drift-diffusion equation can be used to model the transport problem. Therefore, it is necessary to study the Schrödinger-Poisson-drift-diffusion coupling solution method for gallium nitride power devices.

[0004] In recent years, there have been many drift-diffusion simulation methods for quantum corrections in silicon-based MOSFETs. For example, quantum corrections are directly made by solving the Schrödinger equation (C. de Falco, E. Gatti, A. L. Lacaita, and R. Sacco, “Quantum-corrected drift-diffusion models for transport in semiconductor devices,” Journal of Computational Physics, 2005.) and by the density gradient method (M. G. Ancona, “Density-gradient theory: a macroscopic approach to quantum confinement and tunneling in semiconductor devices,” J Comput Electron, 2011.). The quantum correction method based on the density gradient method has also been applied to the simulation of gallium nitride devices (V. Joshi, A. Soni, S. P. Tiwari, and M. Shrivastava, “A Comprehensive Computational Modeling Approach for AlGaN / GaN HEMTs,” IEEE Transactions on Nanotechnology, 2016.). However, there is currently little research on the drift-diffusion simulation of quantum corrections in gallium nitride power devices by directly solving the Schrödinger equation.

[0005] In addition, since power devices will generate serious self-heating effects during operation and affect the surrounding device modules in the integrated module, a good thermal management scheme is very important for device reliability, which requires accurate simulation of the current distribution and heat source distribution inside the device. The present invention provides a quantum correction method for drift-diffusion transport of advanced packaged gallium nitride power devices by self-consistently numerically solving the Schrödinger-Poisson-drift-diffusion equations. This method is applicable to most power devices that rely on two-dimensional electron gas, including gallium nitride high electron mobility transistors. At the same time, a corresponding electrothermal coupling simulation method for a three-dimensional integrated module is given, which can guide the co-optimization of devices and systems. Summary of the Invention

[0006] Aiming at the deficiencies of the prior art, the present invention provides a quantum correction method for drift-diffusion transport of gallium nitride power devices in advanced packaging and an electrothermal coupling simulation method for a three-dimensional integrated module.

[0007] The technical solution adopted by the present invention is as follows:

[0008] A quantum correction method for drift-diffusion transport of gallium nitride power devices in advanced packaging, comprising: 1) In the drift-diffusion transport simulation of gallium nitride power devices, the internal potential energy distribution and quasi-Fermi level distribution of the device under a given drain voltage are obtained by fully coupling the solution of the Poisson equation and the drift-diffusion equation, and used as the initial values for different voltage steps in the drift-diffusion transport simulation;

[0009] 2) According to the device characteristics, a one-dimensional quantum confinement problem and two-dimensional electrostatic and transport problems are modeled, described by the one-dimensional stationary Schrödinger equation and the two-dimensional Poisson equation and drift-diffusion equation respectively, and the one-dimensional stationary Schrödinger equation and the two-dimensional Poisson equation are solved self-consistently by the finite element method until the electric potential converges;

[0010] 3) Quantum correction of the drift-diffusion equation is achieved by modifying the conduction band, and the drift-diffusion equation is solved by the finite volume method; Steps 2) and 3) are iterated until the quasi-Fermi level converges; the electric potential and carrier distribution of the device are obtained; 4) Using the obtained electric potential and carrier distribution, the current density and heat source distribution inside the device are further obtained.

[0011] In the above technical solution, further, the self-consistent solution of the one-dimensional stationary Schrödinger equation by the finite element method specifically includes the following: Take several one-dimensional cross-sections along the quantum confinement direction, interpolate the electric potential obtained from the Poisson equation onto the one-dimensional cross-sections, numerically solve the one-dimensional stationary Schrödinger equation to obtain the subband energy levels and corresponding wave functions in the quantum confinement direction, correct the electron concentration distribution in the two-dimensional electron gas, and calculate the two-dimensional electron density corresponding to different subbands.

[0012] Further, the two-dimensional electron density is multiplied by the normalized wave function of the one-dimensional cross-section to obtain the three-dimensional electron density distribution; Based on the three-dimensional electron density distribution, the Poisson equation is solved, and a coefficient is added in front of the two-dimensional electron gas electron concentration term in the Poisson equation to improve the overall convergence of the Schrödinger-Poisson equation. This coefficient is obtained from the relationship between the old and new electric potentials; The one-dimensional stationary Schrödinger equation and the two-dimensional Poisson equation are iteratively solved until the electron potential energy converges.

[0013] Further, after coupling the solution of the Schrödinger equation and the Poisson equation, the conduction band energy level is corrected based on the above three-dimensional electron density distribution; Then, this conduction band energy level is used as the conduction band energy level in the new drift-diffusion equation, and the equation is solved to update the distribution of the quasi-Fermi level;

[0014] By using the above method, the current density and heat source distribution inside the device can be analyzed. Thus, it is possible to compare and analyze whether to consider the charge distribution in the two-dimensional electron gas with quantum effects and the current density and heat source distribution inside the device, and further evaluate the steady-state working characteristics and self-heating effect of the device.

[0015] In addition, the heat sources of the obtained devices can be further extracted to perform thermal simulation on the three-dimensional integrated module structure, obtain the temperature distribution, and evaluate the thermal management ability of the integrated structure.

[0016] Compared with the prior art, the present invention has the following beneficial effects:

[0017] 1. The present invention provides a quantum correction algorithm for the drift-diffusion transport of advanced packaged gallium nitride power devices. Compared with traditional drift-diffusion transport, this method can simulate the quantum confinement effect of two-dimensional electron gas in the device, obtain more accurate electron density, output the current density distribution and heat source distribution during device operation, which has guiding significance for improving the device thermal distribution and further enhancing the device performance and reliability, and has important application value in the fields of gallium nitride power devices and power device thermal management.

[0018] 2. In the present invention, by adding a coefficient before the two-dimensional electron gas electron concentration term in the Poisson equation, the convergence speed of the self-consistent solution of the Schrödinger-Poisson equation is accelerated.

[0019] 3. In the present invention, the conduction band is corrected based on the two-dimensional electron gas electron concentration solved by the Schrödinger-Poisson equation to achieve quantum correction of the drift-diffusion transport. Compared with the density gradient method, directly using the Schrödinger equation for correction can more accurately simulate the quantum effects in the two-dimensional electron gas.

[0020] 4. The present invention provides a method for electrothermal coupling simulation of a three-dimensional integrated module, realizing multi-physical coupling simulation of micron-scale device simulation and millimeter-scale three-dimensional integrated module, which has important significance for the collaborative optimization of device design and integrated module structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] By reading the following detailed description of non-limiting embodiments with reference to the accompanying drawings, other features, objects, and advantages of the present invention will become more apparent.

[0022] Figure 1 The present invention provides a typical depletion-mode gallium nitride high electron mobility transistor structure, where Figure (a) is the overall structure of the device, and Figure (b) is an enlarged view of the region from the gate to above the buffer layer, where two-dimensional electron gas exists at the junction of the AlGaN spacer and the GaN channel;

[0023] Figure 2 is the overall flow chart of the quantum correction drift-diffusion transport simulation and three-dimensional integrated module electrothermal coupling simulation of the advanced packaged gallium nitride power device provided in the embodiment of the present invention;

[0024] Figure 3 is the solution region of the Schrödinger equation and the region considering quantum effects in the embodiment of the present invention;

[0025] Figure 4 For V in the embodiments of the present invention GS = 0V and V DS = 0V, the conduction band energy level, the three lowest sub - band energy levels, and the normalized wave function of the simulated transistor

[0026] Figure 5 For the electron density distribution of the two - dimensional electron gas without quantum correction, with quantum correction, and after interpolation in the vertical channel direction in the embodiments of the present invention

[0027] Figure 6 For the initially obtained potential energy, the potential energy after Schrödinger - Poisson self - consistent iteration, and the corrected conduction band in the embodiments of the present invention. The dashed line is the quantum potential

[0028] Figure 7 For V in the embodiments of the present invention DS = 8V, the drift - diffusion current density from the heterojunction interface downward (y - axis) along the channel direction (x - axis) (a) without quantum correction (b) with quantum correction

[0029] Figure 8 For the change in the maximum current density in the y - axis direction of the channel before and after quantum correction in the embodiments of the present invention

[0030] Figure 9 For V in the embodiments of the present invention DS = 8V, the temperature distribution of the device

[0031] Figure 10 For the schematic diagram of the three - dimensional integrated module structure in the embodiments of the present invention

[0032] Figure 11 For the temperature distribution of some devices in the three - dimensional integrated module when the bias voltage V DS = 10V in the embodiments of the present invention Detailed implementation manners

[0033] The present invention will be described in detail below in conjunction with specific embodiments. The following embodiments will help those skilled in the art to further understand the present invention, but do not limit the present invention in any form. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present invention, several changes and improvements can still be made. These all belong to the protection scope of the present invention.

[0034] Select a typical gallium nitride power device. Taking the depletion - mode gallium nitride high - mobility transistor as an example, its structure is as Figure 1 shown. A comprehensive and detailed description of the drift - diffusion quantum correction method for this device is given. The drift - diffusion quantum correction method proposed by the present invention is applicable not only to depletion - mode gallium nitride high - mobility transistors, but also to most power devices relying on two - dimensional electron gas for operation.

[0035] According to a specific example of the present invention, a quantum correction method for drift-diffusion transport of gallium nitride power devices in advanced packaging includes the following steps:

[0036] Step 1: Without considering quantum effects, the Poisson equation and the drift-diffusion equation are fully coupled to obtain the potential and quasi-Fermi level distributions inside the device, which are used as the initial values for the quantum-corrected drift-diffusion simulation at different drain voltage steps.

[0037] Step 2: The one-dimensional Schrödinger equation and the two-dimensional Poisson equation are self-consistently solved using the finite element method, where the initial potential value is from Step 1. First, the two-dimensional potential is interpolated onto several one-dimensional cross-sections along the quantum confinement direction. The finite element method is used to solve the one-dimensional Schrödinger equation on the cross-sections to obtain the subbands and the normalized wave functions. For different subbands, the two-dimensional electron density is calculated, multiplied by the one-dimensional probability density, to obtain the three-dimensional electron density distribution. Based on this three-dimensional electron density distribution, the Poisson equation is numerically solved to obtain the updated device potential distribution, and the iteration is solved until the potential converges. In this step, the subband energy levels and the corresponding wave functions in the quantum confinement direction and the potential distribution are obtained. By modifying the Poisson equation, the convergence of the self-consistent algorithm is improved. Among them, the finite element method is used to solve the one-dimensional stationary Schrödinger equation to obtain the subband energy levels and the corresponding wave functions inside the device potential well, which is applicable to any non-uniform one-dimensional grid.

[0038] Step 3: Based on the three-dimensional electron density obtained after the convergence of Step 2, the equivalent conduction band can be calculated as the corrected conduction band to achieve the quantum correction of the drift-diffusion equation. The finite volume method is used to solve the drift-diffusion equation to obtain the updated quasi-Fermi level. Iterate Steps 2 and 3 until the quasi-Fermi level converges.

[0039] Step 4: Evaluate the device operating characteristics. Based on the simulation results of Step 2, the electron density distribution of the two-dimensional electron gas in the channel and the current density and heat source distribution during device operation can be obtained. Further, the output characteristics and the operating temperature distribution of the device can be analyzed. The control equations in the method of the present invention are listed in Table 1.

[0040] Table 1 Control equations

[0041]

[0042]

[0043] The symbols and terms appearing in the text are shown in Table 2.

[0044] Table 2 Symbols and terms appearing in the text

[0045]

[0046]

[0047]

[0048] As Figure 2 shown, a Schrödinger-Poisson-drift diffusion simulation flow chart of a gallium nitride high electron mobility transistor is provided. First, by fully coupling and solving the Poisson equation (Equation 1) and the drift diffusion equations (Equations 2-3), the internal potential energy distribution and the quasi-Fermi level distribution of the device under a given drain voltage are obtained, which are used as the initial values for different voltage steps of the quantum drift diffusion simulation (Step 1). In Figure 3 the region Ω sch where quantum effects need to be considered as shown, several cross-sections perpendicular to the x-axis are taken. The potential energy of the two-dimensional grid is linearly interpolated onto the grid of the cross-section, and the one-dimensional stationary Schrödinger equation (Equation 4) is solved using the finite element method to obtain the subband energy levels and the normalized wave functions. Then, the subband energy levels are linearly interpolated onto the two-dimensional lattice points in the quantum well region Ω qw where the influence of quantum effects on the carrier distribution is significant. At these lattice points, first, the two-dimensional electron density corresponding to each subband (Equation 6) is calculated, and then the two-dimensional electron density is multiplied by the probability density of the lattice point to obtain the three-dimensional electron density (Equation 5). The electron density in this region in the Poisson equation (Equation 7) is replaced by the three-dimensional electron density obtained previously for solution; the two equations are solved iteratively until the electron potential energy converges (Step 2). Before solving the drift diffusion equations, first, the conduction band in the region Ω qw is corrected using the obtained three-dimensional electron density (Equation 8), and then the drift diffusion equations (Equations 2-3) are solved to update the quasi-Fermi levels, where the conduction band E qw in the region Ω c is replaced by E c,qu , and the carrier concentration n is replaced by n qu ; the drift diffusion equations and the Poisson-Schrödinger equations are solved self-consistently until the quasi-Fermi levels converge (Step 3). Further, according to the carrier density distribution and the electric potential distribution of the device, the current density and the heat source distribution during the operation of the device are calculated, and the output characteristics and the temperature distribution during the operation of the device can be analyzed (Step 4).

[0049] For Figure 1 the simulation of the example, the relevant parameters used are listed in Table 3.

[0050] Table 3 Device-related parameters

[0051]

[0052]

[0053] As Figure 4Shown are the three sub-band energy levels with the lowest energy of the transistor and the corresponding normalized wave functions. The quantum confinement effect quantizes the energy bands of the device into discrete energy levels. It can be seen that the position with the highest probability density is near the center of the potential well rather than at the edge. Figure 5 The black line in Figure 5 is the result of the convergence of the Poisson-drift-diffusion equation. The electron density is the highest at the heterojunction interface and then decays rapidly. The light purple line is the electron density distribution obtained by one-dimensional Schrödinger-Poisson. The purple line is the electron density used in two-dimensional Poisson-drift-diffusion after quantum correction. Due to the low electron mobility of aluminum gallium nitride, its quantum effect can be ignored to simplify the model. Figure 6 This is the comparison of the conduction bands before and after correction. The quantum potential is obtained by subtracting the two. It can be seen that the quantum potential fluctuates more significantly in the potential well region where the quantum effect is more obvious. Figure 7 These are several current density curves arranged perpendicular to the channel direction starting from the heterojunction interface along the channel direction. The curves represent the variation of the current density along the channel direction, and the variation of the current density perpendicular to the channel direction can be seen from the overall height change of the curves. Figure 7 (a) is the current density distribution without considering the quantum effect, and (b) is the current density distribution considering the quantum effect. It can be seen that without considering the quantum effect, the current density is concentrated at the interface, while after considering the quantum effect, the maximum value of the current density moves downward. To more clearly compare the differences between the two, Figure 8 The maximum current density from 1 μm to 3 μm in each curve is taken to plot the current density variation curve perpendicular to the channel direction. Figure 9 This is the temperature distribution of the device during operation. The hot spots are concentrated near the drain near the gate.

[0054] Combined with the simulation results of the drift-diffusion transport with quantum correction for GaN devices, it can be seen that there are obvious quantum effects in GaN devices, and there are significant differences between the actual current distribution and the results simulated by only relying on the drift-diffusion transport model.

[0055] According to a specific example of the present invention, the electro-thermal coupling simulation method for a three-dimensional integrated module includes the following steps:

[0056] Step 1: Extract the heat source of the GaN device. Extract the heat source calculated by the device and interpolate it into the grid of the three-dimensional integrated module.

[0057] Step 2: Evaluate the thermal management ability of the three-dimensional integrated module. Introduce the device heat source into the three-dimensional integrated module model, conduct thermal simulation, and calculate the temperature distribution of the overall integrated module during device operation.

[0058] For Figure 10Thermal simulation was carried out on the GaN device three-dimensional integration module described in [reference]. In this example, the structure of the three-dimensional integration module used is a cascaded structure formed by connecting a silicon-based MOSFET and a multi-finger GaN HEMT device up and down. It includes a substrate 3, and the silicon device 1 serves as the gate driver of the GaN HEMT 2. The GaN device is composed of a GaN active region 5 and a SiC substrate 6. Considering the case where only the middle GaN module is working, each module is a multi-finger GaN device, which is equivalent to multiple parallel operations of the above-mentioned GaN devices, and the working voltage is 10V. Figure 11 The corresponding temperature distribution results are shown. The parallel operation of multiple devices leads to a higher central temperature, and at the same time, the heat diffusion along the substrate also affects the surrounding devices.

[0059] An advanced packaging quantum correction method for drift-diffusion transport of gallium nitride power devices and an electro-thermal coupling simulation method for three-dimensional integration modules provided by the present invention have important application values in the fields of gallium nitride power devices and three-dimensional integration module thermal management.

[0060] The specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the above specific embodiments, and those skilled in the art can make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention. Without conflict, the embodiments of the present application and the features in the embodiments can be combined arbitrarily with each other.

Claims

1. A quantum correction method for drift diffusion transport of gallium nitride power devices in advanced packaging, characterized in that: include: 1) In the drift-diffusion transport simulation of GaN power devices, the internal potential energy distribution and quasi-Fermi level distribution of the device under a given drain voltage are obtained by fully coupling the Poisson equation and the drift-diffusion equation, and used as the initial values ​​of different voltage steps of the drift-diffusion transport simulation; 2) Based on the device characteristics, a one-dimensional quantum confinement problem and a two-dimensional electrostatic problem and transport problem are formed, which are described by the one-dimensional steady-state Schrödinger equation and the two-dimensional Poisson equation and the drift-diffusion equation respectively. The finite element method is used to self-consistently solve the one-dimensional steady-state Schrödinger equation and the two-dimensional Poisson equation until the potential converges; 3) A quantum correction of the drift-diffusion equation is realized by modifying the conduction band, and the drift-diffusion equation is solved by the finite volume method; 2) and 3) are iterated until the quasi-Fermi level converges; the potential and carrier distribution of the device are obtained; 4) Using the obtained potential and carrier distribution, the current density and heat source distribution in the device are further obtained.

2. The quantum correction method for drift-diffusion transport simulation of GaN power devices in advanced packaging according to claim 1, characterized in that: The self-consistent solution of the one-dimensional steady-state Schrödinger equation using the finite element method specifically includes the following steps: taking a number of one-dimensional sections along the direction of quantum confinement, interpolating the electric potential obtained by the Poisson equation onto the one-dimensional sections, numerically solving the Schrödinger equation to obtain the subband energy levels and corresponding normalized wave functions on the sections, and calculating the two-dimensional electron density corresponding to different subbands.

3. The quantum correction method for drift-diffusion transport simulation of gallium nitride power devices in advanced packaging according to claim 2, characterized in that: The two-dimensional electron density is multiplied by the normalized wave function of the one-dimensional cutoff to obtain a three-dimensional electron density distribution; based on the three-dimensional electron density distribution, the Poisson equation is solved, and a coefficient is added before the two-dimensional electron gas electron concentration term in the Poisson equation to improve the overall convergence of Schrödinger-Poisson, and the coefficient is obtained from the relationship between the new and old potentials; the one-dimensional steady-state Schrödinger equation and the two-dimensional Poisson equation are iteratively solved until the electron potential energy converges.

4. The quantum correction method for drift-diffusion transport simulation of gallium nitride power devices in advanced packaging according to claim 3, characterized in that: The Poisson equation is: Where, ∈ is the dielectric constant, U(x,y) is the potential at the device coordinate (x,y), q is the charge of the elementary charge, and n qu (x,y) is the electron density in the quantum region, and are the 1 / 2 order Fermi-Dirac integrals obtained from the latest iteration and the previous iteration, respectively, and η new , η old is the corresponding normalized chemical potential, n cl (x, y) is the electron density in the non-quantum region, p(x, y) is the hole density at the device coordinate (x, y), N d (x,y) is the donor concentration at the device coordinate (x,y); σ m (x, y) is the two-dimensional electron density corresponding to the mth self-contained energy level at the device coordinate (x, y), ψ m (x,y) is the normalized wave function of the corresponding one-dimensional section.

5. The quantum correction method for drift-diffusion transport simulation of gallium nitride power devices in advanced packaging according to claim 4, characterized in that: After coupling and solving the Schrödinger equation and the Poisson equation, the conduction band energy level is corrected based on the above three-dimensional electron density distribution; then the conduction band energy level is used as the conduction band energy level in the new drift-diffusion equation, and the equation is solved to update the distribution of the quasi-Fermi level; the corrected conduction band energy level is: Among them, F n is the quasi-electron Fermi level, k B is the Boltzmann constant, T is the temperature, is the inverse function of the 1 / 2 order Fermi-Dirac integral, N 3D is the conduction band effective density of states.

6. A three-dimensional integrated module electrothermal coupling simulation method, characterized in that: Based on the method described in any one of claims 1 to 5, the heat source distribution in the device is obtained, and based on the heat source distribution of the device, the operating temperature distribution of the overall three-dimensional integrated module is calculated to guide or optimize the heat dissipation solution.

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