A modified finite difference model to the reverse recovery of silicon PIN diodes
Introduction
The PIN diodes are widely used either as independent power devices or a part in insulated gate bipolar transistors [1], [2], [3], [4]. The main technical goal of fast PIN diode is to perform fast and soft reverse recovery [1], [2]. As PIN diodes usually work in high injection levels, so the ambipolar diffusion equation (ADE) with the moving boundaries needs to solve in a reverse recovery simulation [3], [4], [5], [6]. To meet this goal, different approaches have been proposed such as analytical approximation [4], finite difference (FD) [5], [6], finite element (FEM) [7], Laplace transform [8], Fourier expansion (FE) [9], [10], and numerical methods [11], [12]. The FD and FEM methods are often implemented in SPICE simulators, while the FE methods are usually implemented in MATLAB Simulink. Different approximations have been made in each of these methods, so even with the same input parameters, the simulated reverse recovery current and voltage (IV) profiles may be different for different simulation methods. However, many papers show the excellent agreements between their ADE-based simulation results and the reference results, which either from the experimental results or the results based on the mixed circuit and technology computer aided design (TCAD) simulations. If the reference results are experimental results, the agreement may be achieved by curve fitting and the parameter adjustment. If the reference results are TCAD simulation results with a set of input parameter values, the curve fitting may extract a different set of parameter values for an ADE-based simulation. Even the excellent agreement may be true for one PIN device structure, but is not guaranteed to be true for other PIN device structures. The fair comparison is to have a same set of parameter values for both TCAD-based and ADE-based simulations. Currently few papers have performed such comparisons among different ADE-based methods.
For FE methods, the implementation of MATLAB Simulink has both advantages and disadvantages. There are at least two advantages: there are many reliable and easy-to-use MATLAB functions or algorithms; and a system is easily built by connecting different Simulink blocks. One disadvantage is that in most cases the final diagram of the Simulink blocks is too big and has complex signal flows. As only a limited number of Fourier expansion coefficients are taken, the Gibbs’ phenomenon may happen. Usually a triangular window function is applied to Fourier coefficients to reduce the spatial oscillations in the carrier density [10]. However, it may change the total carrier number and cause the loss of the total carrier number. The same question may also take place in other methods such as FD and FEM methods. So it is worthwhile to examine whether an ADE-based algorithm can keep the carrier conservation.
During a reverse recovery, it is observed that the carrier densities decrease in both P+N− and N−N+ junction sides [9]. In a usual FE method a Fourier expansion form of the carrier density is assumed in the central N− region. Often a small number of Fourier expansion coefficients in this method are taken, the depletion region is observed to appear only in the P+N− junction side during a reverse recovery. Our Silvaco Atlas TCAD simulation shows that a small depletion region may appear in the N−N+ junction side during a reverse recovery. One may wonder if the Fourier series is truncated to keep more expansion coefficients, this phenomenon is still observable in an FE simulation. In addition, increasing the number of Fourier expansion coefficients in FE method, the edges of the carrier density in the un-depleted N− region become steeper, the usual feedback control of boundaries of the un-depleted N− region may have problems.
For numerical methods, Goebel et al.. adopted the first-order forward Euler method to solve the ADE [11], [12]. As the forward Euler method is not as stable as the backward Euler (BE) method for the stiff differential equations, the computation time step should take a very small value. The advantage is that the FE does not need to solve complex nonlinear equations as the BE does. Yamashita et al. compute the evolution of carriers in the N− region of a PIN diode using the Green’s function of the basic diffusion equation [13]. Their computation is shown very fast and the resultant current–voltage (IV) profiles are shown to agree very well with those from TCAD simulations. However, in their model an electron and a hole have the same carrier density and move together, so the same electron and hole fluxes are obtained in either side of the un-depleted N− region. It is totally different from the ambipolar diffusion during a reverse recovery process, in which holes mainly move toward to P+ region and electrons move to the N+ region at a reverse bias. For the FD method, Buiatti et al. discretize the second-order partial space differentiation by a FD form with a time-dependent constant spatial step in the ADE, then convert the resultant equations to a circuit form and get the reverse recovery IV profiles through a fast SPICE simulation [5], [6]. In mathematics, a second-order partial space differentiation should be discretized by a FD form with a fixed constant space step. Thus, the FD implementation by Buiatti et al. may give some errors.
Another important problem is the validity of the boundary conditions at two boundaries of the un-depleted N− region from the ambipolar diffusion approximation (ADA), in which some further approximations should be made to compute the current components such as electron, hole and displacement current in the depletion layer of P+N− or N−N+ junction. The reason is that in the ADA the electron, hole continuity equation and Poisson equation are not solved self-consistently, we cannot determine all current components including electron, hole and displacement current when the boundary carrier density is approximated as zero for either reverse-biased junction.
In this paper, we compare FE and FD methods through MATLAB programming. We use a first-order BE method to solve both the ADE and the total carrier-current density equation. In order to examine whether both carrier density and current density follow the ADE and the equation of carrier conservation, the focus is given to solving the nonlinear equations of carrier density and boundary current density iteratively using three different approaches. The values of ∂p/∂x at two boundaries of the un-depleted N− region determined from the boundary electron and hole current component in the ADA and the FD approximation based on the carrier density profile may not equal. We compare them in detail in this paper. We take two sets (a slow set and a fast one) of the carrier concentration dependent Shockley-Read-Hall (SRH) recombination life time to study their effect on the validity of the boundary conditions at two boundaries of the un-depleted N− region in the ADA. The result from the Silvaco Atlas TCAD simulation is taken as the reference.
Section snippets
Ambipolar diffusion equation
Fig. 1 shows a schematic diagram of a PIN diode and the corresponding doping profile. The N− base has a width of W, extending from the P+N− junction (x = 0) to the junction N−N+ (x = W). Under high injection, the carrier distribution p(x,t) is governed by the following ADE [9], [10]:
Here Da = 2DnDp/(Dn + Dp) is the abmbipolar diffusion coefficient and τhl is the high-level injection life time. The notation xl is used to describe the thickness of the depletion
Results and discussion
Fig. 3a presents the reverse recovery current–voltage (IV) curves computed using Silvaco Atlas TCAD, FE and FD method and the slow-set of consrh lifetime parameters τn0 and τp0. The initial current is set to 50 A. The value of M is set to M = 25 in the FE and FD computation. In Fig. 3a we denote IV curves as FE25 and FD25 and Atlas. For FE25 and FD25, the x2-dependent Vd1 model (10a) is used. The feedback control approach (29), (37) are used for the FE and FD method, respectively. It is found
Conclusions
In summary, we have used Silvaco Atlas TCAD device-circuit mixed simulation and ADE-based FE and FD methods to simulate the reverse recovery processes of silicon PIN diodes. We use MATLAB programming solve the ADE with the moving boundaries. We find that the result from the ADE-based FE method agrees very well with that from the Atlas simulation, while the result from the ADE-based FD method is much different. The reason is attributed to approximating the second-order partial space
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Manhong Zhang received the M.S. degree in condensed matter physics from University of Science and Technology of China, and Ph.D. degree in condensed matter physics from Institute of Physics, Chinese Academy of sciences, in 1997. He received another Ph.D. degree in solid electronics from University of Texas at Austin, Texas, USA, in 2008. From 2009 to 2013 he was with Institute of Microelectronics, Chinese Academy of Sciences. During the period his research was focused on memory devices. Since
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Manhong Zhang received the M.S. degree in condensed matter physics from University of Science and Technology of China, and Ph.D. degree in condensed matter physics from Institute of Physics, Chinese Academy of sciences, in 1997. He received another Ph.D. degree in solid electronics from University of Texas at Austin, Texas, USA, in 2008. From 2009 to 2013 he was with Institute of Microelectronics, Chinese Academy of Sciences. During the period his research was focused on memory devices. Since 2014, he has been a Professor in the School of Electrical and Electronics Engineering, North China Electric Power University, China. His research interests include simulation of power devices and CMOS analog circuit design.