Quantum correction method for drift-diffusion transport in gallium nitride power devices in advanced packaging and electrothermal coupling simulation method for 3D integrated modules

Abstract

This invention discloses a quantum correction method for drift-diffusion transport in gallium nitride (GaN) power devices in advanced packaging and an electrothermal coupling simulation method for three-dimensional integrated modules. The method obtains the sub-band energy levels, corresponding wave functions, and potential distributions along the quantum confinement direction by self-consistently solving the one-dimensional stationary Schrödinger equation and the two-dimensional Poisson equation. The convergence of the self-consistent algorithm is improved by modifying the Poisson equation. Furthermore, quantum correction of the drift-diffusion equation is achieved by modifying the conduction band. Based on quantum confinement effects and drift-diffusion transport theory, this invention studies the quantum effects of the two-dimensional electron gas in the triangular potential well formed by GaN polarization and the quantum correction method for the drift-diffusion transport model. It evaluates the steady-state operating characteristics of advanced packaged power devices containing quantum effects and further investigates the electrothermal coupling simulation method between the device and the three-dimensional integrated module to evaluate the thermal management capabilities of the three-dimensional integrated module, demonstrating significant application value.

Description

Technical Field

[0001] This invention relates to the field of quantum effect modeling and simulation of gallium nitride power devices, and to a quantum correction method for drift diffusion transport of gallium nitride power devices in advanced packaging and a three-dimensional integrated module electrothermal coupling simulation method, in order to evaluate or optimize their operating characteristics. Background Technology

[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 constantly increasing. Gallium nitride (GaN) stands out due to its advantages such as higher breakdown strength, faster switching speed, higher thermal conductivity, and lower on-resistance. In GaN high electron mobility transistors, carrier transport mainly relies on a two-dimensional electron gas formed at the AlGaN / GaN heterojunction interface. Due to the polarization effect of GaN material and the polarization charge generated by the mismatch stress during the growth of the AlGaN barrier layer, a large number of electrons are bound on the GaN side at the interface, thus forming a high-concentration two-dimensional electron gas.

[0003] Within the triangular potential well formed by polarization, significant quantum effects exist. The density distribution of the two-dimensional electron gas can be obtained by self-consistently solving the one-dimensional time-state Schrödinger and Poisson equations. In the channel direction, due to the larger size, the transport problem can be modeled using the drift-diffusion equation. Therefore, it is essential to study the Schrödinger-Poisson-drift-diffusion coupled solution method for gallium nitride power devices.

[0004] In recent years, there have been many studies and reports on drift-diffusion simulation methods for quantum correction of silicon-based MOSFETs, such as quantum correction by directly solving the Schrödinger equation (C. de Falco, E. Gatti, ALLacaita, and R. Sacco, "Quantum-corrected drift-diffusion models for transport in semiconductor devices," Journal of Computational Physics, 2005.) and quantum correction by density gradient method (MG Ancona, "Density-gradient theory: a macroscopic approach to quantum confinement and tunneling in semiconductor devices," J Comput Electron, 2011.). Quantum correction methods based on density gradient method have also been applied to the simulation of gallium nitride devices (V. Joshi, A. Soni, SP Tiwari, and M. Shrivastava, "A Comprehensive Computational Modeling Approach for AlGaN / GaN HEMTs," IEEE Transactions on Nanotechnology, 2016.). However, there is currently almost no research on drift-diffusion simulations of gallium nitride power devices that directly achieve quantum correction by solving the Schrödinger equation.

[0005] Furthermore, since power devices generate significant self-heating effects during operation, which can impact surrounding devices within the integrated module, a robust thermal management scheme is crucial for device reliability. This necessitates accurate simulation of the internal current and heat source distributions of the device. This invention presents a quantum correction method for drift-diffusion transport in advanced packaged gallium nitride (GaN) power devices by self-consistently solving the Schrödinger-Poisson-drift-diffusion equation. This method is applicable to most power devices operating on two-dimensional electron gases, including GaN high-mobility transistors. Simultaneously, an electrothermal coupling simulation method for the corresponding three-dimensional integrated module is provided, guiding the co-optimization of devices and the system. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides a quantum correction method for drift diffusion transport in gallium nitride power devices in advanced packaging, as well as an electrothermal coupling simulation method for three-dimensional integrated modules.

[0007] The technical solution adopted in this invention is as follows:

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

[0009] 2) Based on the device characteristics, a one-dimensional quantum confinement problem and a two-dimensional electrostatic and transport problem are formed. These problems are described by the one-dimensional stationary Schrödinger equation and the two-dimensional Poisson equation and drift-diffusion equation, respectively. The one-dimensional stationary Schrödinger equation and the two-dimensional Poisson equation are solved self-consistently by the finite element method until the potential converges.

[0010] 3) The 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; iterate 2) and 3) until the quasi-Fermi level converges; obtain the device potential and carrier distribution; 4) use the obtained potential and carrier distribution to further obtain the current density and heat source distribution in the device.

[0011] In the above technical solution, the self-consistent solution of the one-dimensional time-limited Schrödinger equation using the finite element method specifically includes the following: taking several one-dimensional intercepts along the quantum confinement direction, interpolating the potential obtained by the Poisson equation onto the one-dimensional intercepts, numerically solving the one-dimensional time-limited Schrödinger equation to obtain the sub-band energy levels and corresponding wave functions in the quantum confinement direction, correcting the electron concentration distribution in the two-dimensional electron gas, and calculating the two-dimensional electron density corresponding to different sub-bands.

[0012] Furthermore, the two-dimensional electron density is multiplied by the normalized wave function of the one-dimensional intercept 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 before the electron concentration term of the two-dimensional electron gas 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 potentials. The one-dimensional stationary Schrödinger equation and the two-dimensional Poisson equation are solved iteratively until the electron potential energy converges.

[0013] Furthermore, after solving the Schrödinger equation and the Poisson equation in a coupled manner, 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] The above method can be used to analyze the current density and heat source distribution within the device. This allows for a comparative analysis of the charge distribution in the two-dimensional electron gas and the current density and heat source distribution within the device, whether or not quantum effects are considered, thereby evaluating the device's steady-state operating characteristics and self-heating effects.

[0015] Furthermore, the obtained heat sources of the devices can be further extracted, and thermal simulation of the three-dimensional integrated module structure can be performed to obtain the temperature distribution and evaluate the thermal management capability of the integrated structure.

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

[0017] 1. This invention provides a quantum correction algorithm for drift-diffusion transport in 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 a more accurate electron density, and output the current density distribution and heat source distribution when the device is working. It has guiding significance for improving the thermal distribution of the device and further improving the device performance and reliability. It has important application value in the fields of gallium nitride power devices and power device thermal management.

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

[0019] 3. In this invention, the electron concentration of the two-dimensional electron gas obtained from the Schrödinger-Poisson equation is corrected to adjust the conduction band, thereby achieving quantum correction of drift-diffusion transport. Compared with the density gradient method, directly using the Schrödinger equation for correction can more accurately simulate quantum effects in a two-dimensional electron gas.

[0020] 4. This invention provides an electrothermal coupling simulation method for three-dimensional integrated modules, realizing multi-physics coupling simulation of micron-level devices and millimeter-level three-dimensional integrated modules, which is of great significance for the collaborative optimization of device design and integrated module structure. Attached Figure Description

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

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

[0023] Figure 2 This is an overall flowchart of the simulation of quantum-corrected drift-diffusion-transport and the electrothermal coupling simulation of the three-dimensional integrated module for advanced packaged gallium nitride power devices provided in this embodiment of the invention;

[0024] Figure 3 This refers to the solution region of the Schrödinger equation and the region considering quantum effects in this embodiment of the invention.

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

[0026] Figure 5 The electron density distribution of a two-dimensional electron gas in the vertical channel direction in this embodiment of the invention is shown before quantum correction, after quantum correction and interpolation;

[0027] Figure 6 The dashed lines represent the initial potential energy obtained in this embodiment of the invention, the potential energy after Schrödinger-Poisson self-consistent iteration, and the corrected conduction band.

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

[0029] Figure 8 This represents the change in maximum current density before and after quantum correction of the y-axis channel in this embodiment of the invention.

[0030] Figure 9 V in the embodiments of the present invention DS Temperature distribution of the device at 8V.

[0031] Figure 10 This is a schematic diagram of the three-dimensional integrated module structure in an embodiment of the present invention.

[0032] Figure 11 In this embodiment of the invention, some components of the three-dimensional integrated module are biased at voltage V. DS Temperature distribution when operating at 10V. Detailed Implementation

[0033] The present invention will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present invention, but do not limit the invention in any way. It should be noted that those skilled in the art can make several changes and improvements without departing from the concept of the present invention. These all fall within the protection scope of the present invention.

[0034] Selecting a typical gallium nitride power device, taking the depletion-mode gallium nitride high-mobility transistor as an example, its structure is as follows: Figure 1 As shown, the drift-diffusion quantum correction method for this device is described in a comprehensive and detailed manner. The drift-diffusion quantum correction method proposed in this invention is not only applicable to depletion-type gallium nitride high mobility transistors, but also applicable to most power devices that rely on two-dimensional electron gas.

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

[0036] Step 1: Without considering quantum effects, solve the Poisson equation and the drift-diffusion equation in a fully coupled manner to obtain the device internal potential and quasi-Fermi level distribution, and use them as the initial values ​​for 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 solved self-consistently using the finite element method (FEM), with the initial potential energy derived from Step 1. First, the two-dimensional potential energy is interpolated onto several one-dimensional intercepts along the quantum confinement direction. The one-dimensional Schrödinger equation on these intercepts is then solved using the FEM to obtain the subbands and normalized wavefunctions. For different subbands, the two-dimensional electron density is calculated and 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 energy distribution. This process is iterative until the potential converges. This step obtains the subband energy levels and corresponding wavefunctions along the quantum confinement direction, as well as the potential distribution. By modifying the Poisson equation, the convergence of the self-consistent algorithm is improved. The use of the finite element method to solve the one-dimensional stationary Schrödinger equation to obtain the subband energy levels and corresponding wavefunctions within the device potential well is applicable to any non-uniform one-dimensional mesh.

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

[0039] Step 4: Evaluate the device's operating characteristics. Based on the simulation results from Step 2, the electron density distribution of the two-dimensional electron gas within the channel, as well as the current density and heat source distribution during device operation, can be obtained. Furthermore, the device's output characteristics and operating temperature distribution can be analyzed. The governing equations in the method of this invention are all listed in Table 1.

[0040] Table 1 Governing equations

[0041]

[0042]

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

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

[0045]

[0046]

[0047]

[0048] like Figure 2 As shown, a flowchart of the Schrödinger-Poisson-drift diffusion simulation for a gallium nitride high-mobility transistor is provided. First, the potential energy distribution and quasi-Fermi level distribution inside the device at a given drain voltage are obtained by fully coupled solution of the Poisson equation (Equation 1) and the drift diffusion equation (Equations 2-3), and these are used as the initial values ​​for different voltage steps in the quantum drift diffusion simulation (Step 1). Figure 3 The region Ω shown needs to consider quantum effects. sch Several intercepts perpendicular to the x-axis are taken, and the potential energy of the two-dimensional grid is linearly interpolated onto the grid of the intercepts. The one-dimensional stationary Schrödinger equation (Equation 4) is solved using the finite element method to obtain the subband energy levels and normalized wave functions. Then, the subband energy levels are linearly interpolated onto the quantum well region Ω, where quantum effects significantly influence carrier distribution. qw On the two-dimensional lattice points. At these lattice points, first calculate the two-dimensional electron density corresponding to each sub-band (Equation 6), then multiply the two-dimensional electron density by the probability density of the lattice point to obtain the three-dimensional electron density (Equation 5). Replace the electron density of this region in the Poisson equation (Equation 7) with the previously obtained three-dimensional electron density for solving; iterate the two equations until the electron potential energy converges (Step 2). Before solving the drift-diffusion equation, first use the obtained three-dimensional electron density to correct the region Ω. qw The conduction band (Equation 8) is then solved, and the drift-diffusion equation (Equation 2-3) is then solved to update the quasi-Fermi level, where the region Ω qw Conduction band E c By E c,qu Instead, the carrier concentration n is changed from n qu The process involves replacing the drift-diffusion equation with a Poisson-Schrödinger self-consistent solution until the quasi-Fermi level converges (step three). Further, based on the device's carrier density and potential distribution, the current density and heat source distribution during device operation are calculated, enabling analysis of the device's output characteristics and operating temperature distribution (step four).

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

[0050] Table 3 Device-related parameters

[0051]

[0052]

[0053] like Figure 4The image shows the three lowest energy sub-band levels of the transistor and their corresponding normalized wave functions. The quantum confinement effect quantizes the device's energy bands into discrete energy levels. It can be seen that the highest probability density is located near the center of the potential well, rather than at the edges. Figure 5 The black line represents the convergence of the Poisson-drift diffusion equation. The electron density is highest at the heterojunction interface and then decreases sharply. The light purple line represents the electron density distribution obtained by the one-dimensional Schrödinger-Poisson equation. The purple line represents the electron density used in the two-dimensional Poisson-drift diffusion after quantum correction. Since the electron mobility of aluminum gallium nitride is low, the influence of its quantum effect can be ignored to simplify the model. Figure 6 This is a comparison of the conduction band before and after the correction. The quantum potential is obtained by subtracting the two. It can be seen that the quantum potential fluctuation is large 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. The curves represent the magnitude of the current density along the channel direction, and the change in the overall height of the curves shows the change in the current density perpendicular to the channel direction. Figure 7 (a) shows the current density distribution without considering quantum effects, and (b) shows the current density distribution considering quantum effects. It can be seen that without considering quantum effects, the current density is concentrated at the interface, while considering quantum effects, the maximum current density shifts downwards. To more clearly compare the differences between the two, Figure 8 The maximum current density from 1μm to 3μm in each curve was plotted as a current density variation curve perpendicular to the channel direction. Figure 9 It is the temperature distribution of the device during operation, with hot spots concentrated near the gate and close to the drain.

[0054] The simulation results of drift-diffusion transport with quantum correction for gallium nitride devices show that there are obvious quantum effects in gallium nitride devices, and the actual current distribution differs significantly from the simulation results based solely on the drift-diffusion transport model.

[0055] According to a specific example of the present invention, the electrothermal 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 from the device and interpolate it to the mesh of the 3D integrated module.

[0057] Step 2: Evaluate the thermal management capabilities of the 3D integrated module. Introduce heat sources into the 3D integrated module model, perform thermal simulation, and calculate the overall temperature distribution of the integrated module during device operation.

[0058] against Figure 10Thermal simulation was performed on the three-dimensional integrated GaN device module described in this example. The three-dimensional integrated module structure used in this example is a cascaded structure formed by connecting silicon-based MOSFETs and multi-finger GaN HEMT devices vertically. This includes a substrate 3, a silicon device 1 serving as the gate driver for the GaN HEMT 2, and GaN devices consisting of GaN active regions 5 and a SiC substrate 6. Considering the case where only the middle GaN module is operational, each module is a multi-finger GaN device, equivalent to multiple GaN devices operating in parallel at a working voltage of 10V. Figure 11 The results show the corresponding temperature distribution. The parallel connection of multiple devices leads to a higher center temperature, and the heat also affects the surrounding devices as it diffuses along the substrate.

[0059] The quantum correction method for drift diffusion transport in advanced packaged gallium nitride power devices and the electrothermal coupling simulation method for three-dimensional integrated modules provided by this invention have important application value in the fields of gallium nitride power devices and thermal management of three-dimensional integrated modules.

[0060] Specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the specific embodiments described above, 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. Unless otherwise specified, the embodiments and features described in this application can be arbitrarily combined with each other.

Claims

1. A quantum correction method for drift-diffusion transport in gallium nitride power devices in advanced packaging, characterized in that, include: 1) In the drift-diffusion-transport simulation of gallium nitride power devices, the potential energy distribution and quasi-Fermi level distribution inside the device under a given drain voltage are obtained by solving the Poisson equation and the drift-diffusion equation in a fully coupled manner, and these are used as the initial values ​​for different voltage steps in the drift-diffusion-transport simulation. 2) Based on the device characteristics, a one-dimensional quantum confinement problem and a two-dimensional electrostatic and transport problem are formed. These problems are described by the one-dimensional stationary Schrödinger equation and the two-dimensional Poisson equation and drift-diffusion equation, respectively. The one-dimensional stationary Schrödinger equation and the two-dimensional Poisson equation are solved self-consistently by the finite element method until the potential converges. 3) Quantum correction of the drift-diffusion equation is achieved by modifying the conduction band, and the drift-diffusion equation is solved using the finite volume method; iterations 2) and 3) are performed until the quasi-Fermi level converges; the device potential and carrier distribution are obtained; among them, after coupled solution of the Schrödinger equation and the Poisson equation, the conduction band level is modified based on the three-dimensional electron density distribution; then, this conduction band level is used as the new conduction band level in the drift-diffusion equation, and the equation is solved to update the distribution of the quasi-Fermi level; the modified conduction band level... for: ， in, For the quasi-electron Fermi level, Boltzmann's constant, For temperature, It is the inverse function of the 1 / 2 order Fermi-Dirac integral. For the effective density of states in the conduction band, The electron density in the quantum region; 4) Using the obtained potential and carrier distribution, the current density and heat source distribution within the device can be further obtained.

2. The quantum correction method for drift diffusion transport in gallium nitride power devices in advanced packaging according to claim 1, characterized in that, The method of using the finite element method to self-consistently solve the one-dimensional time-independent Schrödinger equation specifically includes the following steps: taking several one-dimensional intercepts along the quantum confinement direction, interpolating the potential obtained by the Poisson equation onto the one-dimensional intercepts, numerically solving the Schrödinger equation to obtain the sub-band energy levels and corresponding normalized wave functions on the intercepts, and calculating the two-dimensional electron density corresponding to different sub-bands.

3. The quantum correction method for drift diffusion transport in gallium nitride power devices in advanced packaging according to claim 2, characterized in that, Multiply the two-dimensional electron density by the normalized wave function of the one-dimensional intercept to obtain the three-dimensional electron density distribution; based on the three-dimensional electron density distribution, solve the Poisson equation, and add a coefficient before the electron concentration term of the two-dimensional electron gas 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 potentials; iteratively solve the one-dimensional stationary Schrödinger equation and the two-dimensional Poisson equation until the electron potential energy converges.

4. The quantum correction method for drift diffusion transport in gallium nitride power devices in advanced packaging according to claim 3, characterized in that, The Poisson equation is as follows: ， in, Where is the dielectric constant. Device coordinates The potential at that point, The amount of elementary charge. For the electron density in the quantum region, and Let represent the 1 / 2 order Fermi-Dirac integrals obtained in the latest iteration step and the previous iteration step, respectively. , For the corresponding normalized chemical potential, For the electron density in the non-quantum region, Device coordinates Hole density at that location Device coordinates Donor concentration at the location; ， Device coordinates The two-dimensional electron density corresponding to the m-th self-contained energy level. Let be the normalized wave function of the corresponding one-dimensional intercept.

5. A three-dimensional integrated module electrothermal coupling simulation method, characterized in that, Based on the method described in any one of claims 1-4, the heat source distribution within 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 scheme.

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