Modeling and Simulation of Power Pin Diodes Within SPICE

POLITECNICO DI TORINO

Facolt`adi Ingegneria Corso di Dottorato in Dispositivi Elettronici

Tesi di Dottorato

Modeling and Simulation of Power PiN Diodes within SPICE

Gustavo Buiatti

Direttore del corso di dottorato Prof. Carlo Naldi

Tutore: Prof. Giovanni Ghione

Febbraio 2006 Acknowledgements

I wish to thank my Ph.D. thesis advisor, Professor Giovanni Ghione for his ad- vises and comments throughout the whole research activity, and also for his human support. I also wish to thank Federica Cappelluti for supporting my work with con- tinuous helpful suggestions and discussions, and in the preparation of this Thesis. Their scientific methodology have been a reference for me and their contributions improve the quality of the results. I wish to kindly thank Professor JoséRoberto Camacho, from Federal University of Uberlândia,Brazil, for his support, ideas and fruitful discussions during my research period in that institution and even in Italy. Professor JoãoBatista Vieira Júnioris also acknowledged, especially for the support on his Power Electronics Laboratory in the Federal University of Uberlândia,Brazil. A final and very special thought goes to my wife, Natalia, and my daughter, Gabriela. Thank you for your encouragement, patience and constant support during the time we have been in Italy. Without you I would never finish this work. I’ll be always grateful for your love. To you, I dedicate this thesis.

1 Table of contents

1 Physics and Basic Equations of Power PiN Diode 3 1.1 The Ambipolar Diﬀusion Equation (ADE) ...... 5 1.2 Forward conduction ...... 8 1.2.1 The stationary forward behavior of the PiN diode ...... 9 1.2.2 End region recombination eﬀect ...... 13 1.2.3 Carrier-carrier scattering ...... 18 1.2.4 Auger recombination ...... 19 1.2.5 Lifetime control ...... 21 1.3 Forward recovery ...... 23 1.4 Reverse recovery ...... 25

2 Power PiN Diode Models for Circuit Simulations 30 2.1 An overview of PiN diode modeling ...... 31 2.2 Circuit simulator and model implementation ...... 35 2.3 PiN diode models: diﬀerent approaches to solve the ADE ...... 35 2.3.1 Analytical model: Laplace transform for solving the ADE . . . 36 2.3.2 Analytical model: Asymptotic Waveform Evaluation for solving the ADE ...... 38 2.3.3 Analytical model: Fourier based-solution to the ADE . . . . . 40 2.3.4 Hybrid model: Finite Element Method for solving the ADE . 44 2.3.5 Hybrid model: Finite Diﬀerence Method for solving the ADE . 48

3 Finite Diﬀerence Based Power PiN Diodes Modeling and Valida- tion 50 3.1 Nomenclature ...... 51 3.2 Introduction ...... 52 3.3 Model description ...... 53 3.3.1 Fundamental Equations ...... 53 3.3.2 Finite Diﬀerence Modeling of the Base Region ...... 54 3.4 The complete diode model ...... 57 3.4.1 Voltage drop on the junctions ...... 58 3.4.2 Voltage drop on the epilayer ...... 59

I TABLE OF CONTENTS

3.4.3 Voltage drop on the space-charge regions ...... 60 3.5 Model implementation within SPICE ...... 61 3.6 Model results and Validation ...... 65 3.6.1 Comparison with the FEM based diode model ...... 65 3.6.2 Simulation of Commercial Fast Recovery Diodes ...... 71 3.7 Simulation of Switched Mode Power Supplies ...... 80

4 Conclusions 86

A Pspice subcircuit listing - feedback scheme 87

B Pspice subcircuit listing - standard diode 90

Bibliography 93

II Introduction

Power devices are very important for power electronics systems since the latter are closely related to these discrete devices performance. Their study, comprehension and performance improvement is of major importance for the development of efficient power electronics equipments. The effort needed for assembling and experimenting a power electronics converter, even taking into account the simplest topology existent, takes to a strong motivation in the search for tools that in a simple and reliable way can simulate the operation of the semiconductors involved in the circuit, dependent on the various parameters of the load and control circuit, and that, in such a way, allows the comparison for different options of control and topology of conversion. Time to market is an important target for modern, highly competitive industry. A widely used method to reduce time to market is the use of computer aided design (CAD) tools which reduce the number of prototypes needed for the implementation of the final design. The limited use of prototypes results in a reduction of the time needed to obtain the final product with a consequent saving of design cost. In the the power electronics field, circuit simulation is the favorite CAD tool. The simulation of converters, with the inclusion of detailed characteristics of the bipolar power semiconductors devices, by means of using a personal computer, allows an accurate understanding of the design and increases the possibility of a working first prototype close to the final product. We usually have two different solutions for dealing with this simulation problem. The first one simulates a very simple circuit where the whole physics of the semiconductor is taken into account, and we focus our attention on the semiconductors behavior considering this particular simplified situation. In the second option the whole converter is simulated, but making use of simplified models for the semiconductors involved in the design. Both solutions are usually incompatible in terms of integration. Therefore, the development of designs in the field of power electronics can be beneficiated if somehow we can simulate both the macroscopic aspect of the converter, and the microscopic aspect of the commutations of the semiconductors involved in the circuit. Unfortunately, the models available in commercial circuit simulators are not suited to model the actual behavior of bipolar power semiconductors, which are not suited either for a study that intends to be physics-based.

1 So, the main goals of this work were to create a mathematical model applicable for bipolar power semiconductors, that must be physics-based, capable to be implemented in any commercial circuit simulator and to reproduce reliable and accurate results. Many power devices have been proposed in the past years but the need of better models is always present since circuit models need to adapt to the demand of advanced CAD tools and to the increased computation power available. Another reason for the development of new device models is that they are the result of the trade-off between contradicting requirements such as low computation complexity and accuracy. A better trade-off between these requirements is always desirable and pushes the development of more efficient device models. In this thesis the results obtained during the study and research period at he Politecnico di Torino are reported. The attention is focused on the power PiN diode, a power device as simple as essential in power systems, with emphasis to the development of compact circuit models of this device, ideally suited for circuit simulation. Main characteristic of the model presented in this work are the low computational power needed and the accurate modeling of static characteristic and of forward and reverse recovery effects. The model is implemented for simulation and comparison with experimental data in Pspice simulator. However, the model can be handled by any other SPICE-based simulator. The thesis is structured as follows. Chapter 1 is devoted to a general introduction to the physics of power PiN diodes, aimed to highlight the main static and dynamic effects of the same, and to provide the background underlying the design and optimization of such devices. In chapter 2 the main topics on power diode modeling are introduced, and different techniques and models are presented in order to compare the same and clarify this issue. Chapter 3 focuses on a novel approach for modeling power PiN diodes. The complete diode model is described and introduced, followed by its implementation within the Pspice circuit simulator. Circuit simulations of practical power circuits are reported, and the model is validated against experimental and simulations using different diode models. Finally, in chapter 4 the final conclusions are presented.

2 Chapter 1

Physics and Basic Equations of Power PiN Diode

The PiN diode was one of the first semiconductor devices developed for power circuit applications. It is the simplest semiconductor device present in every power electronics converter, as can be seen by the structure presented in Fig. 1.1. The PiN diode is basically composed of three regions: the cathode, the epilayer and the anode. The cathode is a wide highly N doped region; the epilayer is a lightly N doped region, epitaxially grown over the cathode; the anode is a highly P doped region placed at the top of the epilayer. The main difference between signal diodes (low power PN diodes) and power PiN diodes is this additional sandwiched region, the epilayer, which allows the PiN diode to block large negative voltages depending on its width and low doping. The presence of this region also has important effects on the diode’s direct characteristic and dynamic behavior. Regarding the direct characteristic, the presence of the epilayer (which behaves as a series resistance) increases on-state voltage drop with respect to signal diodes. With respect to the dynamic behavior, two important drawbacks are worth point- ing out. During forward conduction the epilayer is flooded with charge carriers, holes and electrons injected from diode end regions (anode and cathode), and the resistance of the epilayer becomes very small, allowing the diode to carry a high current density with limited voltage drop. If not flooded by the carriers, the epilayer is highly resistive. So, the resistance of the epilayer depends on the carriers distribution in the same, which in turn depends on the current density through the diode. This is the so called conductivity modulation, what means that the resistance of the epilayer is modulated according to its carriers distribution. Thus, when a PiN diode is switched on with a high di/dt, it takes some time to reach the stationary flooded state of the epilayer, and the voltage drop at a given current will initially be higher. This effect results in a voltage overshoot, generally called forward recovery, increasing dynamic losses. Further, it can be a problem in power circuits since this

3 1 – Physics and Basic Equations of Power PiN Diode voltage peak may appear across the switch used as the active element and exceed its breakdown voltage.

P + N - N + A n o d e C a t h o d e

1 0 1 9 ] - 3 1 0 1 7

D1 o 0 p1 i 5 n g [ c m

1 0 1 3 Figure 1.1. PiN diode structure and doping example.

The second drawback appears when the diode is turned off, because the excess carriers in the epilayer cannot disappear immediately, but it takes some time for them to recombine and to be extracted. So, the device is not able to reach the blocking state if carriers stored in the epilayer have not been extracted. This results in the presence of a reverse current until the epilayer is free of excess carriers. This effect is generally called reverse recovery and has unpleasant effects such as increase of dynamic losses (the current also flows through the switches used in the circuit, adding to power dissipation and degrading their reliability), electromagnetic interference (EMI), and limitation of maximum working frequency due to the increased turn-off time. All the effects mentioned above may be modeled through the basic equations of the device, obtained by the physics of the same, as follows in the next sections.

4 1.1– The Ambipolar Diﬀusion Equation (ADE)

1.1 The Ambipolar Diﬀusion Equation (ADE)

Semiconductor devices are characterized an modeled by some basic equations [1], [2], obtained through the solid state device physics. Many years of research into device physics have resulted in a mathematical model of the operation of semiconductor devices. This model consists of a set of fundamental equations which link together the electrostatic potential and the carrier densities, with suitable boundary conditions. These equations consist of Poisson’s equation, the continuity equations and the transport equations. Poisson’s equation relates variations in electrostatic potential to local charge densities. The continuity equations describe the way that the electron and hole densities evolve as a result of transport processes, generation and recombination processes. Poisson’s equation and the continuity equations have been derived from Maxwell’s laws. The ﬁrst set of equations considered here constitute the transport equations, also called the current density equations:

∂n kT ∂n J = qµ nE + qD = qµ (nE + ) (1.1) n n n ∂x n q ∂x

∂p kT ∂p J = qµ pE − qD = qµ (pE − ) (1.2) p p p ∂x p q ∂x

J = Jn + Jp (1.3) where eq. 1.1 is the expression for the electron current density, eq. 1.2 for the hole current density, and eq. 1.3 for the total conduction current density. In the equations above J is the total current density, Jn and Jp are the current densities 2 of electrons and holes [A/cm ], Dn and Dp are the diffusion coefficients of electrons 2 2 and holes [cm /s], µn and µp are the mobilities of electrons and holes [cm /V·s], n and p are the electrons and holes densities [cm−3], E is the electric field [V/cm], q is the magnitude of electronic charge [C], k is the Boltzmann constant [J/K], and T is the absolute temperature [K]. The second set of equations is composed by the continuity equations:

∂n 1 ∂J = n + (G − R ) (1.4) ∂t q ∂x n n

∂p 1 ∂J = − p + (G − R ) (1.5) ∂t q ∂x p p where (Rn − Gn) and (Rp − Gp) are the rate of recombination, also called U. The statistics of the recombination of electrons and holes in semiconductors via recombination centers, or the rate of recombination U, is given by the Shockley-Read-Hall equation [1], [3], [4], [5], considering a single level recombination center:

5 1 – Physics and Basic Equations of Power PiN Diode

np − n2 ∆np + ∆pn + ∆n∆p U = i = 0 0 (1.6) τp0(n + n1) + τn0(p + p1) τp0(n0 + ∆n + n1) + τn0(p0 + ∆p + p1)

2 where n = n0 + ∆n, p = p0 + ∆p and p0n0 = ni , ni is the intrinsic density of charge −3 carriers [cm ], n0 and p0 are the electrons and holes densities at thermal equilibrium −3 −3 [cm ], ∆n and ∆p are the densities of electrons and holes in excess [cm ], τn0 and τp0 are the electrons and holes minority carrier lifetimes in heavily doped P and N type silicon [s], and n1 and p1 are the equilibrium electron and hole densities when the Fermi level position coincides with the recombination level position in the band gap [cm−3]. The two last ones are given by the expressions: µ ¶ µ ¶ E − E E − E n = N exp R C = n exp − F i R (1.7) 1 C kT i kT and µ ¶ µ ¶ E − E E − E p = N exp V R = n exp F i R (1.8) 1 V kT i kT −3 where NC is the effective density of states in conduction band [cm ], NV is the −3 effective density of states in valence band [cm ], EC is the bottom of conduction band [eV], EV is the top of valence band [eV], EF i is the intrinsic Fermi energy level [eV], and ER is the recombination level location [eV]. In addition to the continuity equations, Poisson’s equation must be satisfied

dE ρ q(p − n + N − N ) = = D A (1.9) dx ²s ²s −3 where ρ is the space charge density [cm ], ²s is the semiconductor permittivity −3 [F/cm], ND is the donor impurity density [cm ], and NA is the acceptor impurity density [cm−3]. In principle, the equations above with appropriate boundary conditions have a unique solution. Because of the complexity of this set of equations, in most cases the equations are simpliﬁed with physical approximations before a solution is attempted, or they are solved using numerical methods as is the case of semiconductors device simulators [6], [7]. Power semiconductors are designed for high current densities. Power diodes are often rated for current densities from 100 A/cm2 to 1000 A/cm2. For high current densities, electrons and holes concentrations in the epilayer are much higher than background carrier concentration, since the epilayer is a lightly doped region, that is the device works in the high injection level condition [4], [5], [8]. When the high injection level condition holds, hole and electrons concentration in excess are approximately equal in the whole epilayer, in order to hold the

6 1.1– The Ambipolar Diﬀusion Equation (ADE) quasi-neutrality condition, since they are much higher than the majority carriers concentration at thermal equilibrium. Considering a type-N semiconductor under high injection condition:

2 ni ∆n À n0 ≈ ND and ∆p À p0 ≈ ND

∆n ≈ n ≈ ∆p ≈ p (1.10) Taking into account the high injection level condition in the epilayer (eq. 1.10) and considering that the energy levels of the recombination centers are near the intrinsic Fermi level EF i, it means that n1 and p1 are of the same order of ni. Even if the energy levels of the recombination centers are not located that close to the EF i, their order are much lower than the order of ∆n and ∆p, and eq. 1.6 becomes [8]:

∆n2 ∆n U = = (1.11) ∆n(τp0 + τn0) τp0 + τn0 where the carrier lifetime is equal for both electrons and holes, and is equal to the sum of low injection level electron and hole lifetimes. The same is called high injection level lifetime:

τhl = τn0 + τp0 (1.12) Always under the assumptions of high injection level and quasi-neutrality in the epilayer and eliminating the electric ﬁeld in the current density equations (eq. 1.1 and eq. 1.2):

∂∆n ∂∆n Jn − qDn Jp + qDp ∂x = ∂x (1.13) qµn∆n qµp∆n

From eq. 1.13 and considering the Einstein relation Dn = VT µn and Dp = VT µp (VT = kT/q is the so called thermal voltage [V]): J J ∂∆n n − p = 2q (1.14) Dn Dp ∂x With respect to the the continuity equations, eq. 1.4 and eq. 1.5 under the assumptions already mentioned, they respectively become:

∂∆n 1 ∂J ∆n = n − (1.15) ∂t q ∂x τhl ∂∆n 1 ∂J ∆n = − p − (1.16) ∂t q ∂x τhl

7 1 – Physics and Basic Equations of Power PiN Diode

The derivative of eq. 1.14 with respect to x takes to

2 1 ∂Jn 1 ∂Jp ∂ ∆n − = 2q 2 (1.17) Dn ∂x Dp ∂x ∂x

Finally, eliminating ∂Jn/∂x and ∂Jp/∂x respectively from eq. 1.15 and eq. 1.16 and substituting into eq. 1.17:

∂∆n ∂2∆n ∆n = Da 2 − (1.18) ∂t ∂x τhl Considering eq. 1.10:

∂n ∂2n n = Da 2 − (1.19) ∂t ∂x τhl where eq. 1.19 is the continuity equation valid for both electrons and holes in the epilayer, called Ambipolar Diﬀusion Equation (ADE) and which rules the free carrier distribution in the same, where n(x) is the epilayer carrier concentration and Da is the ambipolar diﬀusion constant [4], [5], [8], [9]:

2DnDp Da = (1.20) Dn + Dp Eq. 1.19, the ADE, models the transient behavior in the epilayer and must be solved considering the boundary conditions obtained from eq. 1.14, taking into account eq. 1.10: ¯ ∂n¯ J | J | ¯ n x=xl p x=xl ¯ = − (1.21) ∂x x=x 2qDn 2qDp ¯ l ∂n¯ J | J | ¯ = n x=xr − p x=xr (1.22) ∂x¯ 2qD 2qD x=xr n p where xl is the left border and xr is the right border of the region ﬂooded with free carriers. Considering the width of the epilayer equal to W, the borders of the ﬂooded region are xl = 0 and xr = W if the carriers concentrations are higher than the doping of the epilayer. During reverse recovery, the carrier concentration on the borders becomes equal to the doping of the epilayer, and the borders start moving meaning that xl and xr do not coincide anymore with the physical borders of the epilayer (diode junctions).

1.2 Forward conduction

In this section PiN diode forward conduction basic equations are presented. Under steady state conditions, the current ﬂow in the PiN diode can be accounted for by the

8 1.2– Forward conduction recombination of electrons and holes in the epilayer, and also by the recombination of minority carriers injected in the highly doped end regions. It is assumed that the equations that rule PN junctions are known to the reader.

1.2.1 The stationary forward behavior of the PiN diode In the following analysis end region recombination, carrier-carrier scattering and Auger recombination will be neglected. Furthermore, high injection carrier lifetime will be supposed constant in the whole epilayer [4], [5], [8], [9]. On-state voltage drop, VD, can be divided in three components indicated in Fig. + − − + 1.2 as VP + (P N junction voltage drop), VN + (N N junction voltage drop), and VM (ohmic voltage drop).

+ + V P V M V N

P + N - N + A n o d e C a t h o d e

Figure 1.2. Forward voltage drop components in a PiN diode.

The ADE (eq. 1.19) can be rewritten in the following way:

1 ∂n ∂2n n = 2 − 2 (1.23) Da ∂t ∂x La √ where La = Daτhl is the ambipolar diﬀusion length [cm or µm]. For the steady state conditions the time dependence in eq. 1.23 may be omitted, and the ADE must be solved in the epilayer with coordinates shown in Fig. 1.3, in order to simplify the solution. The solution for eq. 1.23 has the general form: µ ¶ µ ¶ x x n(x) = C1 cosh + C2 sinh (1.24) La La and its derivative has the following form: µ ¶ µ ¶ ∂n C x C x = 1 sinh + 2 cosh (1.25) ∂x La La La La The assumption that end region recombination is negligible means that end regions doping is much higher than epilayer doping, and therefore, the current density is determined by recombination in the epilayer. So, end regions have unity injection eﬃciency, and it may be assumed as a very good approximation that at the borders of the highly doped regions, P + and N +, the total current conduction is carried out only by holes and electrons respectively. However, it is useful for analytical purpose

9 1 – Physics and Basic Equations of Power PiN Diode

n ( - d ) n ( + d ) A n o d e C a t h o d e - d0 + d Figure 1.3. Example of carrier concentration shape during forward conduction. Boundary conditions and axes for the solution of the ADE are highlighted too. only, as will be seen in the next subsection for the cases with current densities higher than 20 A/cm2 [9], where end region recombination cannot be neglected. So, under the assumptions mentioned above, the following equations are obtained:

Jp |x=xl = Jp |x=−d= J (1.26)

Jn |x=xl = Jn |x=−d= 0 (1.27)

Jn |x=xr = Jn |x=+d= J (1.28)

Jp |x=xr = Jp |x=+d= 0 (1.29) In order to obtain the boundary conditions needed to solve the ADE, the following equations are obtained substituting eq. 1.26 - 1.29, in eq. 1.21 and eq. 1.22 respectively: ¯ ¯ ∂n¯ Jp |x=−d J ¯ = − = − (1.30) ∂x x=−d 2qDp 2qDp and ¯ ¯ ∂n¯ Jn |x=+d J ¯ = = (1.31) ∂x x=+d 2qDn 2qDn

Finally, from eq. 1.25, eq. 1.30 and eq. 1.31, constants C1 and C2 are evaluated and then substituted in eq. 1.24 leading to the solution, that is the concentration distribution inside the epilayer in the forward state related to the total current density through the diode:  µ ¶ µ ¶  x x cosh sinh Jτ  L L  n(x) = hl  µ a ¶ − B · µ a ¶ (1.32) 2qL  d d  a sinh cosh La La where

10 1.2– Forward conduction

(D − D ) (µ − µ ) B = n p = n p (1.33) (Dn + Dp) (µn + µp) is a measure for the inequality of the mobilities. If the mobilities are equal, only the ﬁrst symmetrical term within the brackets is left in eq. 1.32. As said before, the total voltage drop VD at the diode is composed of three components

VD = VP + + VM + VN + (1.34) In order to relate current density to voltage drop on the diode, it is needed the calculation of diode voltage drop components VP + , VN + , and VM as a function of current density. Following the mass action law and theory of PN junction, it is possible to relate on-state voltage drop to carrier concentration in both junctions (junction law):

µ ¶ µ ¶ pn− qVP + p(−d)ND n(−d)ND qVP + = exp ⇒ 2 = 2 = exp (1.35) pn0− kT ni ni kT µ ¶ µ ¶ n − qV + n(+d) qV + n = exp N ⇒ = exp N (1.36) nn−0 kT ND kT where ND is the epilayer doping, pn− is the concentration of holes in the lightly doped epilayer side of the anode junction (x=-d), pn0− is the concentration of holes in the same place at thermal equilibrium, nn− is the concentration of electrons in the lightly doped epilayer side of the cathode junction (x=+d), and nn0− is the concentration of electrons in the same place at thermal equilibrium. Thus, · ¸ kT n(−d)ND VP + = ln 2 (1.37) q ni · ¸ kT n(+d) VN + = ln (1.38) q ND · ¸ kT n(+d)n(−d) VP + + VN + = ln 2 (1.39) q ni Calculating n(−d) and n(+d) through eq. 1.32 and substituting in eq. 1.39

   µ ¶  2 2   kT τhlJ 1 2 2 d V + + V + = ln   µ ¶ − B tanh  (1.40) P N q n2(2qL )2  d L  i a tanh2 a La

11 1 – Physics and Basic Equations of Power PiN Diode

Considering eq. 1.34 and substituting in eq. 1.40   µ ¶ µ ¶ V − V τ 2 J 2  1 d  exp D M = hl  µ ¶ − B2 tanh2  (1.41) V n2(2qL )2  d L  T i a tanh2 a La Rearranging eq. 1.41 the relation between density current and voltage drop on the diode is given by: µ ¶ µ ¶ Da d VD J = 2niq F exp (1.42) d La 2VT with

µ ¶ µ ¶ · µ ¶¸− 1 µ ¶ d d d d 2 V F = tanh 1 − B2 tanh4 exp − M (1.43) La La La La 2VT

In order to evaluate VM , we have to integrate the electric ﬁeld over the epilayer

Z x=+d VM = E(x)dx (1.44) x=−d The electric ﬁeld is found by adding eq. 1.1 and eq. 1.2 to obtain the total current density, and by resolving for E always under the high level injection condition:

∂n ∂n ∂n J = qµ nE + qD + qµ nE − qD = qnE(µ + µ ) + qV (µ − µ ) (1.45) n n ∂x p p ∂x n p T ∂x n p J BV ∂n E = − T (1.46) qn(µn + µp) n ∂x Finally, considering eq. 1.32 and its derivative, and substituting them into eq. 1.46 and then integrating the same (eq. 1.44), it is found that

 µ ¶ d  sinh Ãs µ ¶  8b L d V = V  s a · arctan 1 − B2 tanh2 · M T  2 µ ¶ (1 + b) d La 1 − B2 tanh2 La  µ ¶ d µ ¶ 1 + B2 tanh2 d  L  · sinh + B · ln  µ a ¶ (1.47) L  d  a 1 − B2 tanh2 La

12 1.2– Forward conduction

where b = (µn/µp) is the ratio between electron and hole mobility. In conclusion, the PiN diode characteristic is obtained by solving eq. 1.42, taking into account eq. 1.43 and eq. 1.47. Note that VM is not dependent on the current density. This is because an increase of current density proportionally increases carrier concentration in the epilayer and hence provides a reduction of epilayer resistivity which is proportional to current density, and the two eﬀects cancel each other. If carrier-carrier scattering are considered, for example, this is not true anymore, and VM is dependent on the injection level, and so, on the current density.

1.2.2 End region recombination eﬀect Previous analysis supposed that end region recombination, that is recombination in the P + and in the N + regions, was negligible. This assumption is quite restrictive, and useful only for analytical purpose. Since the minority carrier lifetime decreases rapidly with increasing doping level, recombination in the end regions adds additional components to the forward current, and so, at high current densities, the current density due to the recombination of electrons injected in the P + region

(Jn|x=xl ), and the current density due to the recombination of holes injected in the + N region (Jp|x=xr ) must be taken into account. With reference to Fig. 1.4 and taking into account the deﬁnitions of Da and b, eq. 1.21 and eq. 1.22 can be rewritten as: ¯ ¯ ∂n¯ Jn|x=0 − bJp|x=0 ¯ = (1.48) ∂x x=0 qDa(b + 1) ¯ ¯ ∂n¯ Jn|x=W − bJp|x=W ¯ = (1.49) ∂x x=W qDa(b + 1)

n ( 0 ) n ( W ) + + n P ( 0 ) p N (W) A n o d e C a t h o d e + + -W P 0 W W N Figure 1.4. On-state carrier concentration in the epilayer and heavily doped regions.

Considering eq. 1.3, eq. 1.48 and eq. 1.49 respectively become: ¯ ¯ ∂n¯ −bJ Jn|x=0 ¯ = + (1.50) ∂x x=0 qDa(b + 1) qDa ¯ ¯ ∂n¯ J Jp|x=W ¯ = − (1.51) ∂x x=W qDa(b + 1) qDa

13 1 – Physics and Basic Equations of Power PiN Diode

The current density due to injected carriers in the end regions can be derived from low level injection theory because minority carrier density is far smaller than the high doping levels of the end regions. With respect to the P +N − junction, it is assumed that the same is abrupt and that the P + region is uniformly doped. Considering the low injection level condition of carriers injected into the anode, and the fact that all the injected electrons have already recombined in the ohmic contact, since there are no excess electrons in the ohmic contact, that is there is no voltage drop in the same, it is found through the continuity equation that in the neutral region [5], [10]: µ ¶ WP + + x [nP + (0) − nP 0+ ] sinh LnP + nP + (x) − nP 0+ = µ ¶ (1.52) W + sinh P LnP + + where nP + is the density of electrons in the P region, nP 0+ is the density of electrons + + in the P region at thermal equilibrium, WP + is the width of the P region, and + LnP + is the minority carrier diffusion length in the P region. Eq. 1.52 above represents the expression for the excess electrons injected in the anode. Considering the fact that the density current of electrons in the anode may be considered only to the diffusion component, since the electric field is approximately zero and the electrons concentration is too low in the P + region, meaning that the electrons drift current is neglected: µ ¶ ∂nP + J | = qD + (1.53) n x=0 nP ∂x where DnP + is the diffusion coefficient of electrons in the anode. Using quasi- equilibrium at the P +N − junction, the injected carrier concentrations on either side of the junction are related by:

p + (0) n(0) P = (1.54) p(0) nP + (0) Under low injection level in the P + region, it is assumed that

pP + (0) = pP 0+ (1.55) + where pP + is the density of holes in the P region, and pP 0+ is the density of holes in the P + region at thermal equilibrium. The injected electron concentration is related to the voltage across the anode junction: µ ¶ VP + nP + (0) = nP 0+ exp (1.56) VT Using the two expressions above in eq. 1.54:

14 1.2– Forward conduction

µ ¶ µ ¶ VP + 2 VP + n(0) p(0) = pP 0+ nP 0+ exp = ni exp (1.57) VT VT Using the charge neutrality condition:

µ ¶ · ¸2 V + n(0) exp P = (1.58) VT ni Substituting the derivative of eq. 1.52 in eq. 1.53: µ ¶ VP + qDnP + nP 0+ exp 2 VT qDnP + nP 0+ n(0) Jn|x=0 = µ ¶ = µ ¶ (1.59) WP + WP + 2 LnP + tanh LnP + tanh ni LnP + LnP + Finally, considering the mass action law:

2 Jn|x=0 = q hp n(0) (1.60) with

DnP + hp = µ ¶ (1.61) WP + LnP + tanh NA LnP + + 4 where hp is the emitter recombination coeﬃcient in the P region [cm /s] [10]. Substituting eq. 1.60 in eq. 1.50, the boundary condition for the P +N − junction is now given by ¯ ¯ 2 ∂n¯ −bJ hp n(0) ¯ = + (1.62) ∂x x=0 qDa(b + 1) Da which takes into account the recombination eﬀect of carriers injected into the anode. With respect to the N −N + junction, it is also assumed that the same is abrupt and that the N + region is uniformly doped. Considering the low injection level condition of carriers injected into the cathode, and the fact that all the injected holes have already recombined in the ohmic contact, it is found through the continuity equation that in the neutral region [5], [10]: µ ¶ x − WN + [pN + (W ) − pN0+ ] sinh LpN + pN + (x) − pN0+ = µ ¶ (1.63) W − W + sinh N LpN + + where pN + is the density of holes in the N region, pN0+ is the density of holes in + + the N region at thermal equilibrium, WN + is the width of the N region, and LpN +

15 1 – Physics and Basic Equations of Power PiN Diode is the minority carrier diffusion length in the N + region. Eq. 1.63 above represents the expression for the excess holes injected in the cathode. Considering the fact that the density current of holes in the cathode may be considered only to the diffusion component, since the electric field is approximately zero and the holes concentration is too low in the N + region, meaning that the holes drift current is neglected: µ ¶ ∂pN + J | = qD + (1.64) p x=W pN ∂x where DpN + is the diffusion coefficient of holes in the cathode. Using quasi-equilibrium at the N −N + junction, the injected carrier concentrations on either side of the junction are related by:

p + (W ) n(W ) N = (1.65) p(W ) nN + (W ) + where nN + is the density of electrons in the N region. With analogous assumptions made for the P + region: µ ¶ 2 VN + n(W )p(W ) = ni exp (1.66) VT Using the charge neutrality condition:

µ ¶ · ¸2 V + n(W ) exp N = (1.67) VT ni Substituting the derivative of eq. 1.63 in eq. 1.64: µ ¶ VN + qDpN + pn0+ exp 2 VT qDpN + pN0+ n(W ) Jp|x=W = µ ¶ = µ ¶ (1.68) WN + WN + 2 LpN + tanh LpN + tanh ni LpN + LpN + Finally, considering the mass action law:

2 Jp|x=W = q hn n(W ) (1.69) with

DpN + hn = µ ¶ (1.70) WN + LpN + tanh ND LpN + + 4 where hn is the emitter recombination coeﬃcient in the N region [cm /s] [10]. Substituting eq. 1.69 in eq. 1.51, the boundary condition for the N −N + junction is now given by

16 1.2– Forward conduction

¯ ¯ 2 ∂n¯ J hn n(W ) ¯ = − (1.71) ∂x x=W qDa(b + 1) Da which takes into account the recombination effect of carriers injected into the cathode. Band gap narrowing effects are caused by an alteration of the band structure, that is a variation of the energy band gap of silicon, due to high doping levels [5], [11]. If band gap narrowing effects are considered, depending on the doping of the P + and N + regions the intrinsic carrier concentration arises in the same [5], [11]: µ ¶ ∆E n2 = n2 · exp g (1.72) ie i kT where nie is the intrinsic carrier concentrarion taking into account bandgap narrowing effect, and ∆Eg is the band gap narrowing due to the combined effects of impurity band formation, band tailing and screening , which is calculated by the following formula [5]: µ ¶ s 2 2 3q q NI ∆Eg = (1.73) 16π²s ²skT + where NI is the doping concentration, in the case of P region the acceptor con- + centration NA, and in the case of N region the donator concentration ND. It is observed that the energy bandgap reduces with increasing doping concentration, and intrinsic carrier concentration increases on the other hand. Thus, considering the bandgap narrowing effects in the highly doped end regions of PiN diodes, it is found that the emitter recombination coefficients hp and hn are increased by the factor exp(∆Eg/kT ), meaning that the efficiency of both emitters is reduced. Some important conclusions can be made from eq. 1.59 and eq. 1.68, considering bandgap narrowing effect: PiN diode current conduction is dominated from epilayer recombination for low current densities, while end region recombination dominates current flow for high current conditions. In fact, for low current densitis, n(0) and n(W ) are small, currents 1.59 and 1.68 are negligible, epilayer carrier concentration increases linearly with current density following equation 1.32, and current increases exponentially with forward voltage drop. If the ADE is solved numerically with the boundary conditions taking into account the end region recombination effect, for low current densities until 20 A/cm2 it is actually found that the carrier concentration and also the voltage drop in the diode are almost the same for both solutions, with and without end region recombination, reinforcing the conclusion that the current density is dominated by recombination in the epilayer. For higher current densities, currents 1.59 and 1.68 become dominant since they increase with the square of the carrier concentration and the diode enters in the working region where end region recombination rules current conduction. The forward voltage drop in this

17 1 – Physics and Basic Equations of Power PiN Diode

case will increase more rapidly with increasing current density than exp(VD/VT ), as shown in Fig. 1.5, which was generated with the following physical and geometrical 14 −3 2 parameters: W = 50 µm, ND = 2·10 cm , A = 0.04 cm , τhl = 200 ns, hp = hn = −14 4 2 2 1.5·10 cm /s, Dn = 34.84 cm /s, Dp = 12.82 cm /s, T = 300 K. The diﬀerence between the two curves is due to the fact that in the case in which the end region recombination is neglected, the voltage drop in the junctions rules the exponential behavior of the current density (eq. 1.37 - 1.41). In fact, the carriers concentration increases linearly with the current density and the ohmic voltage drop in the epilayer does not depend on the current density. It can be explained by the fact that the resistance in the epilayer is inversely proportional to the carriers concentration, meaning that:

1 1 VM = Repi · ID ∝ · ID ∝ · ID ≈ constant n ID When end region recombination becomes dominant, at higher current densities, the current increases with the square of carriers concentration. The voltage drop in the junctions keeps increasing exponentially with the current density (eq. 1.37 - 1.41), but in this case with lower values, since the carriers concentrations in the junctions are lower due to the reduced emitters eﬃciency. It can be observed in Fig. 1.6, which was generated using the same parameters of Fig. 1.5. However, the voltage drop in the epilayer is not independent of the conduction current anymore but increases with the square root of same:

1 1 p VM = Repi · ID ∝ · ID ∝ √ · ID ∝ ID n ID In this case, the ohmic voltage drop in the epilayer has a signiﬁcant contribution to the voltage drop, which for high current densities becomes much greater than the case with unity emitter eﬃciency (see Fig. 1.5).

1.2.3 Carrier-carrier scattering At high current densities, the recombination in the end regions is not the only phenomenon responsible for the deviation of the forward voltage drop characteristics from an exponential behavior, as predicted by eq. 1.42. Two additional phenomena impact the current conduction characteristics, being the carrier-carrier scattering the first one to be considered here. Carrier-carrier scattering occurs in the epilayer at high current densities due to the simultaneous presence of a high concentration of both electrons and holes. The greater probability of mutual Coulombic scattering causes a reduction in the mobility and diffusion length for both carriers [12]. The reduction in diffusion length with increasing current density produces a decrease in the conductivity modulation in the central portion of the epilayer, which in turn, combined with the reduction

18 1.2– Forward conduction

3 1 0 ] 2

2 1 0 C onduction C urrent D ensity [A /cm W ith End R egion R ecom bination N o End R egion R ecom bination

1 1 0 0 . 7 0 . 7 5 0 . 8 0 . 8 5 0 . 9 0 . 9 5 1 Forw ard V oltage D rop [V ] Figure 1.5. Eﬀect of end region recombination on the forward conduction characteristics of a PiN diode. of the mobilities, results in the increase of the epilayer resistivity with consequent increase of diode voltage drop (see Fig. 1.7, which was generated with the following 14 −3 physical and geometrical parameters: W = 100 µm, ND = 2·10 cm , A = 0.04 2 −14 4 2 2 cm , τhl = 1 µs, hp = hn = 5·10 cm /s, Dn = 34.84 cm /s, Dp = 12.82 cm /s, T = 300 K).

1.2.4 Auger recombination The second phenomenon that also impacts the current conduction characteristics is the Auger recombination [13]. The Auger recombination process occurs by the transfer of the energy released by the recombination of an electron-hole pair to a third particle that can be either an electron or a hole. This process becomes signiﬁcant in heavily doped P and N type silicon, such as the end regions of power PiN diodes, and is also an important eﬀect in determining recombination rates in lightly doped regions operating at high injection levels during forward conduction, the epilayer of PiN diodes, because of the high concentration of holes and electrons injected into this region. In the case of Auger recombination occurring at high injection levels, the Auger lifetime is given by [5]:

19 1 – Physics and Basic Equations of Power PiN Diode

x 1 0 1 6 5

W ith End R egion R ecom bination 4 . 5 N o End R egion R ecom bination

4 ] - 3 3 . 5

2 . 5

C arriers2 C oncentration [cm

1 . 5

1 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 D e p t h [ m m ] Figure 1.6. Eﬀect of end region recombination on the carriers concentration in the epilayer under steady state condition for J = 100 [A/cm2].

1 τauger = 2 (1.74) CA · n where n is the excess carrier concentration in the epilayer, and CA is the Auger recombination coefficient, of the order of 10−31 cm6/s. There are similar expressions for the end regions, considering the majority carrier concentration. It can be observed from eq. 1.74 that the Auger recombination lifetime decreases with the injected carrier concentration in the epilayer, and it begins to affect the carriers distribution in the same. In addition to the Shockley-Read-Hall recombination described by eq. 1.11, the rate of recombination must include the Auger recombination process: µ ¶ n 1 1 n 3 U = = + n = + CA · n (1.75) τeff τhl τauger τhl where τeff is the effective carrier lifetime, and 1/τeff = 1/τhl +1/τaug. The inclusion of the Auger recombination term reduces the effective carrier lifetime of holes and electrons with increasing current density, and so the diffusion length, decreasing the conductivity modulation in the central portion of the epilayer, that is decreasing the storage charge in the epilayer, and resulting in the increase of the epilayer resistivity

20 1.2– Forward conduction with further increase of diode voltage drop (see Fig. 1.7). Consequently, the ADE (eq. 1.19) in the steady state condition must be rewritten in the form:

2 ∂ n n 3 Da 2 = + CA · n (1.76) ∂x τhl in order to take into account this phenomenon. With respect to the Auger recombination in the end regions, due to the high majority carrier density, it alters the minority carrier lifetimes resulting in the decrease of the minority carrier diﬀusion lengths (LnP + ,LpN + ), and increasing the recombination currents in the end regions (increase of hp and hn), what means to say, decreasing the emitters eﬃciency. In conclusion, the resistivity in the epilayer is further increased.

1 0 3

- N o End R egion R ecom bination - N o C arrier-carrier ] - W ith End R egion 2 S cattering R ecom bination - N o A u g e r - W tih C arrier-carrier R ecom bination S cattering - N o A u g e r R ecom bination

1 0 2 - W ith End R egion R ecom bination - W tih C arrier-carrier S cattering -W ith A uger - W ith End R egion R ecom bination R ecom bination

C onduction C urrent D ensity- N[A o /cmC arrier-carrier S cattering - N o A u g e r R ecom bination

1 0 1 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 1 . 3 1 . 4

Forw ard V oltage D rop [V ] Figure 1.7. Eﬀect of end region recombination, carrier-carrier scattering, and Auger recombination on the forward conduction characteristics of PiN diode.

1.2.5 Lifetime control Epilayer lifetime value is one of the most important design parameters for a PiN diode. Reduction of on state losses or increase of diode speed are achieved through modiﬁcations of carrier lifetime.

21 1 – Physics and Basic Equations of Power PiN Diode

Commercially available diodes often use lifetime control techniques to reduce carrier lifetime in the whole epilayer. Well known lifetime control techniques are gold and platinum doping and electron irradiation, which result in lifetime profiles approximately uniform in the considered region. This is due to the fact that the metals diffuse so rapidly through the silicon that they are normally uniformly distributed across the epilayer. Likewise, electron bombardment also creates recombination centers uniformly throughout the device structure. The first mentioned method involves the thermal diffusion of an impurity that exhibits deep levels in the energy gap of silicon (gold or platinum). The second method is based upon the creation of lattice damage in the form of vacancies and interstitial atoms by bombardment of the silicon wafers with high energy particles. Both methods are characterized by the introduction of recombination centers in silicon, reducing carrier lifetime in the lightly doped region and providing decrease of turn-off time [11], [14]. A drawback of these techniques is the increase of on-state voltage drop. Moreover, the introduction of deep level recombination levels increases leakage current and results in a stronger influence of the temperature on the diode performance. Another technique is the local lifetime control, mainly based on proton or he- lium irradiation [11], [15], [16], which reduces the turn-off time and increases diode softness with a little worsening of on-state voltage drop. This technique results in a better trade-off curve than achieved with lifetime killing in the whole epilayer region. Recent investigations show that the optimal position for the low-lifetime region is at the beginning of the epilayer on the anode side, while the optimal width of the low lifetime region depends on the amount of lifetime reduction, while it is less dependent on the operating current of the device. Diode design using a reduced lifetime region not placed near the anode junction provides a worse behavior with respect to lifetime killing in the whole epilayer [17]. During the development of the equations regarding the behavior of PiN diode, it has been always assumed that the high-injection lifetime in the epilayer is constant and given by eq. 1.12. This assumption was made in order to simplify the model and equations. Actually, lifetime depends on the injection-level and also on the capture cross sections of the recombination centers. Further, when spatially selective techniques are able to control carrier lifetime locally, the same becomes a function of the position:

τhl = τhl(x) (1.77) In this hypotesis the ADE becomes:

1 ∂n ∂2n n = 2 − 2 (1.78) Da ∂t ∂x La(x) which has no closed form solution. In this case eq. 1.78 must be solved numerically using appropriate techniques.

22 1.3– Forward recovery

1.3 Forward recovery

The lightly doped epilayer allows PiN diodes to support large reverse voltages, and has an important role during commutation between conducting state and blocking state, and vice-versa. It was shown in section 1.1 that the presence of epilayer during forward conduction increases on-state voltage drop with respect to signal diodes, since the epilayer behaves such as a variable series resistance connected to the diode. This resistance increases with the current density, considering the phenomena described in the last section, such as end region recombination, and so the voltage drop on the epilayer. The voltage drop due to the epilayer region is more or less in the range from 0.1 V to 1 V. Anyway, the presence of the carriers in the epilayer is the main reason that makes possible to the PiN diodes conducting high current densities.

Forw ard R ecovery 6

4 D iode C urrent [100 A /cm 2 ]

3 D iode V oltage D iode V oltage [V ] 2

1 Turn-on di/dt

0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 T i m e [ n s ]

Figure 1.8. PiN diode forward recovery

For the sake of illustrating what would happen if the epilayer was unmodulated, the resistance of the epilayer is evaluated without the carriers injected in the same:

W 2 Repi = [Ω · cm ] (1.79) q µn ND where W is the width of the epilayer, and ND is the epilayer doping. In order to clarify the order of the unmodulated region resistance, let us consider a general

23 1 – Physics and Basic Equations of Power PiN Diode

14 −3 diode with an epilayer 50 µm wide and with doping ND = 10 cm . One finds that its resistance is of the order of 10−1 Ω · cm2, which means that for forward density currents equal to 100 A/cm2, the voltage drop in the unmodulated epilayer should be in the order of 101 V. This example makes clear that if a PiN diode is forced in the conducting state with a high di/dt, meaning that the current is increasing in a faster rate than the rate of carriers being injected into the epilayer, transient voltage drop will be much greater than steady stage voltage drop. This is due to the fact that during the first instants, when the epilayer is not modulated, its resistance is very high. This voltage overshoot due to the fast switch from the blocking state to the conduction state through the forcing of a direct current, is called forward recovery. The voltage peak increases with increasing di/dt, and its value depends on how high the current has risen before conductivity modulation is fully effective. In Fig. 1.8 an example of PiN diode forward recovery is shown, which the simulated diode is the same used for generating Fig. 1.5. It can be observed that diode voltage reaches about 5 V while steady state on-state voltage drop is about 1 V. Fig. 1.9 shows the behavior of excess carriers in the epilayer when the diode is turned on from zero current. It can be observed that excess carriers are initially injected into the regions closest to the P +N − and N −N + junctions. From there, they diffuse into the center of the epilayer, and its resistance diminishes to its steady state value.

P + N - N + 1 0 1 9

J J P | x = x l J n | x = x r J E xcess carriers concentration 1 0 1 7 t 5 = steady state

J n | x = x l t 4 J P | x = x r 1 0 1 5 t 3 t 1 t 2

t 0 = no excess carriers 1 0 1 3 B ase doping profile x = 0 x = W

Figure 1.9. Excess carrier concentration proﬁles during the turn on process in PiN diode

24 1.4– Reverse recovery

1.4 Reverse recovery

A major limitation to the performance of PiN diodes at high frequencies is the loss that occurs during switching from the on-state to the off-state, which have a significant effect on the maximum operating frequency. During the reverse recovery, the charge stored in the epilayer during forward conduction must be removed. As can be seen in Fig. 1.10a, a large reverse transient current occurs in PiN diodes during reverse recovery. Since the voltage across the diode is also large following the peak in the reverse current, a large power dissipation occurs in the diode. In addition, the peak reverse current adds to the average current flowing through the switches that are controlling the current flow in the circuit. This not only produces an increase in the power dissipation in the switches, but also creates a high internal stress degrading their reliability. Moreover, reverse recovery also causes EMI phenomena. In the following the different phases of the reverse recovery are described, regarding Fig. 1.10. The widely used diode test circuit in Fig. 1.11 is used for a better understanding of the switching process, where DUT is the diode under test, LDUT is the parasitic inductance of the diode, L is the inductance of the circuit that can be considered as a constant current source, S1 is the switch, and VS is the supply voltage.

During the ﬁrst phase (0, t0) the switch in the circuit is open. The diode is in the forward conduction state, and the injected carriers are almost symmetrically distributed along the epilayer (see Fig. 1.10b, sample time t0). The voltage drop on the diode has its steady state value corresponding to the conduction current density through the diode.

At the time instant t0 the switch in the circuit is closed, and the reverse recovery takes place. From t0 until t3 the current through the diode is determined by the external circuit conditions and decreases with a constant di/dt, the so called turn-off di/dt. Hence, the charge profile in the epilayer during this phase is such that it is able to support an increase in current, in the reverse direction. As far as the diode is able to support this increasing current at a certain di/dt, there will be just a small forward voltage drop across the diode, which is determined primarily by the charge profile within the epilayer. The diode is still forward biased. During this second phase (t0, t3) the injected carriers in the epilayer are extracted from the diode, by diffusion and recombination, and there is a change in the slope of the injected carrier profile near the two junctions. This slope changes its sign due to the reversal in the current direction, as can be observed by time samples t1, t2 and t3 in Fig. 1.10b. At the time instant t3, when sufficient charge has recombined, or has diffused out as reverse current, the carriers concentration at the P +N − junction reaches the levels of thermodynamic equilibrium, allowing the formation of a space charge region. So, the voltage drop on the diode becomes negative and starts to increase. This is the beginning of the third phase (t3, t4), which lasts until the instant time when current through the diode reaches its peak negative value. Because of the depleting charge

25 1 – Physics and Basic Equations of Power PiN Diode

3 0 2 0 t 0

2 0 0

1 0 - 2 0 ] 2 T urn-off di/dt t 1 0 - 4 0

t 6 - 1 0 - 6 0 t 2 t 5 - 2 0 - 8 0 D iode V oltage [V ]

D- iode 3 0 C urrent D ensity [ A /cm - 1 0 0

t 3 R e v e r s e - 4 0 - 1 2 0 R e c o v e r y t 4 d i / d t - 5 0 - 1 4 0 0 20 40 60 80 100 120 140 160 180 200 T i m e [ n s ] ( a )

P + N - N + 1 0 1 9

J J P | x = x l J n | x = x r J E xcess carriers concentration 1 0 1 7 t 0 t 1 t 2 J n | x = x l J P | x = x r t 3 1 0 1 5 t 4 t 5 t 6

1 0 1 3 B ase doping profile x = 0 x = W ( b ) Figure 1.10. PiN diode reverse recovery: (a) Reverse recovery current waveform, (b) Dynamics of carrier concentration in the epilayer during reverse recovery

26 1.4– Reverse recovery

S 1

V S

L DUT L

DUT

Figure 1.11. Circuit used to emulate diode switching

profile, after t3 the diode will be unable to support an increase in the current as determined by the circuit di/dt. However, it must be recognized that the diode may be able to support an increase in the current if the magnitude of the di/dt is reduced. At time t3, the diode starts determining the circuit boundary conditions, being controlled by the diffusion and recombination processes in the epilayer, and it is the voltage across the diode, rather than the current, that is determined by the external circuit [11], [20]. The actual time difference between the voltage becoming negative and the diode current reaching its peak negative value, that is the duration of the third phase, depends very much on the circuit conditions as well as on the diode characteristics. It can also be deduced that the diode current waveform will become more rounded near its peak negative value, because di/dt through the diode will first decrease and then change sign, at time t4, when the charge carrier profile in the diode is no longer able to support any further increase in the current in the reverse direction.

During the fourth phase (t4,t5), after the di/dt has changed its sign, the depletion regions are advancing from both borders of the epilayer and the resulting reduced charge proﬁle is just able to support lower currents. The reverse recovery di/dt during this phase induces an overshoot of the reverse voltage as the energy stored in the parasitic inductance LDUT, always present in practical circuits, is transferred into the diode (see Fig. 1.10b). This is undesirable, and parasitic inductances must be minimized by good circuit designers. This phase lasts until time t5, when the diode is blocking the whole applied reverse voltage, and the reverse diode voltage reaches its peak value. If the turn on of S1 is controlled so that the current rises gradually,

27 1 – Physics and Basic Equations of Power PiN Diode initially taking over the L inductor current and then drawing reverse current out of the diode, as the space charge layer is established in the diode the reverse voltage settles at the supply voltage with no signiﬁcant overshoot. We call the attention to the fact that the reverse diode current adds to the total current carried by S1 and causes a transient peak, as already mentioned.

The last phase of the recovery starts at time t5 and lasts until the moment in which the current reaches its saturation value. If there is a residual amount of excess charge present in the epilayer from this instant time recombination dominates the excess carrier absorption, resulting in a slow tail in the current waveform (see current waveform and excess charge during time sample t6 in Fig. 1.10). This last phase of the recovery is very critical, and some considerations must be done. The first consideration is with respect to the applied reverse voltage. If the applied reverse bias voltage is small, the space charge region will extend only slightly inside the epilayer. As a result, there will still be a lot of excess carriers remaining in the epilayer. These excess carriers can be removed only through recombination. Hence, if the applied reverse bias voltage is less than a second one operating in the same conditions, the recombination dominated regime will be quite prominent causing a significant tail near the end of the reverse recovery process [20]. This kind of recovery is the so called soft recovery, and a behavior like this is desirable for power electronics applications. The second and more serious consideration regards to the fact that at time instant t5, it is possible that the depletion regions can advance through the whole epilayer and the current that is still through the diode cannot be supported by any excess charge. This is the classic snappy recovery. Depending on the rate that excess carriers are extracted from the epilayer, the current goes rapidly to zero with very high reverse recovery di/dt because a stronger depletion from both sides happens before the current is ceased, resulting in oscillation. The snappy recovery is detrimental to the diode, as it increases the chance of its destruction due to the excessive electric field strength. Furthermore, the large-amplitude high-frequency oscillations cause excessive amounts of electromagnetic interference (EMI). When the switching frequency of a power circuit increases, the turn-off di/dt must be increased. It has been found that this causes an increase in both the peak reverse recovery current and the ensuing di/dt, which in turn results in less recovery time. If the reverse recovery di/dt is large, an increase in the breakdown voltage of all the circuit components becomes essential, since the reverse recovery di/dt flows through parasitic inductances in the circuit causing the already mentioned voltage overshoot on the diode. Raising the breakdown voltage capability causes an increase in the forward voltage drop of power switches, which degrades circuit efficiency. Consequently, much of the recent work on PiN diodes has been focused upon improving the reverse recovery characteristics. However, a trade-off between the switching speed and the forward voltage drop is essential during PiN design. This trade-off is dependent upon a number of factors

28 1.4– Reverse recovery such as the epilayer width, the recombination center position in the energy gap, the distribution of the deep level impurities, and the doping profile. In section 1.2.2 it was found that end region recombination results in an increase of voltage drop with respect to eq. 1.42, and it could be assumed that a careful design should reduce this effect (increasing the emitter efficiency, that is reducing hp and hn). It would assure that also under high current conditions, end region recombination is small with respect to epilayer recombination. This assumption is not correct. Actual devices tend to increase end region recombination effects, that is, to reduce end region emitter efficiency (increasing hp and/or hn), since the increase of end region recombination results in an improvement of dynamic behavior [11], [18], [19]. This improvement is due to the fact that the reduced carriers concentration in the epilayer (see Fig. 1.6) takes to a faster extraction of the carriers during reverse recovery, considering the same operation conditions, resulting in less reverse current peak and faster reverse recovery, which is desirable in order to reduce switching power losses. It can be achieved through techniques like the lifetime control techniques described in section 1.2.5. From eq. 1.61 and eq. 1.70 it can be observed that in order to increase hp and hn, that is to reduce the emitters efficiency, there are two other design techniques: the reduction of end region doping (increase of minority + + carrier equilibrium concentration pN0 and nP 0) [11], [18], [19], and the reduction of end region thickness (WP + ,WN + ). It can be noticed that the reduction of end region doping is always effective, as the case of the weak anode diode [11], [18], [19], while the reduction of end region thickness has a relevant effect only if WP + ,WN + are smaller than minority carrier diffusion length (LnP + ,LpN + ).

29 Chapter 2

Power PiN Diode Models for Circuit Simulations

As computer aided simulation (CAD) has become essential for power electronics circuit development, special compact power PiN diode models must be available in the circuit simulators used for electronic circuit analysis. In this chapter different PiN diode models present in literature are introduced and discussed. The main difficulty in designing models for PiN diodes, and also for other bipolar power devices, is in the distributed nature of the charge transport in the semiconductor, which is governed by the ambipolar diffusion equation (ADE). Therefore, the central part of this chapter describes problems and different solutions in treatment of the ADE. The different modelling methods are compared as to their compromise between convenience, accuracy, numerical efficiency and accuracy in implementing physical effects. Many of the existing PiN diode models are listed, and their main features are identified and compared.

30 2.1– An overview of PiN diode modeling

2.1 An overview of PiN diode modeling

As the operating frequency of power electronics converters becomes higher, the power electronics researchers put more attention to the switching losses. An accurate model of power PiN diodes is very useful for computer aided simulation (CAD), since it is very helpful for evaluating the loss distributions of power electronics converters and making design and optimization. The most fundamental aspects that distinguish among different power diode models is the model formulation technique and concept employed. The different models can be classified either as micromodels, or as macromodels. Micromodels are closely based on the internal device physics and yields good accuracy over a wide range of operating conditions. Because device physics unavoidably require mathematical equations, micromodels are also known as mathematical models. Macromod- els reproduce the external behavior of the device largely by using empirical techniques without considering its geometrical nature and its internal physical processes. This external behavior is usually modeled by means of simple data-fitting empirical equations, lookup tables, or an electrical subcircuit of common components to emulate known experimental data [21], [22], [23]. Macromodels are limited in terms of accuracy and flexibility an they will not be treated here. They are not derived from the fundamental device equations, and as a consequence they are valid only in a narrow range of operating conditions, requiring non physical parameters. On the other hand, micromodels are generally more computationally efficient, more accurate, and more related to the device structure and fabrication process. Micromodels are classified as numerical models, analytical models, and hybrid models. Numerical micromodels use the partial differential equation set of the semiconductor physics (eq. 1.1 to eq. 1.9). Since this set of nonlinear partial differential equations has no closed mathematical solution, they have to be solved numerically by inserting a geometric mesh and solving the equations step by step by the finite- element or finite-difference methods. These equations describe the physical phenomena within the semiconductor, consisting of carrier drift and diffusion components, carrier generation and recombination effects, and the relationship between space charge and electric field. The solution of the whole system of basic equations is done without simplifications, and it is commonly used by device simulators [6], [7]. This sort of modeling provides a rigorous picture of the device behavior but, owing to their numerical intensity, convergence properties, and very long CPU times, they are limited to the analysis of simplified test circuits, where in practice just one single semiconductor device is present. Although this may suffice during the device design and technological development stages, such techniques are unable altogether to cope with the simulation of realistic power circuits and are therefore of little use in their design, where compact models must be used. From the engineering point of view, the degree of accuracy that is achieved by an exact numerical model is not always necessary or even justified, in particular, if the input data, such as the doping

31 2 – Power PiN Diode Models for Circuit Simulations profile, is only known with a limited accuracy. In such cases, simplified compact models may suffice, being obtained through some assumptions and simplifications in the semiconductor physics as done in chapter 1. Analytical micromodels rely on a set of mathematical functions to describe the device’s terminal characteristics without resorting to numerical methods. An example is the standard diode model packaged in SPICE. Standard SPICE-type models were designed for microelectronic devices, and they poorly describe the dynamic and static behavior of power devices [24]. The standard SPICE models use a simple charge control method and lack the important physical effects in power devices. The soft recovery of the power diode cannot be simulated by the SPICE diode model, leading to erroneous predictions of switching power dissipation. Another drawback of the SPICE diode model is its inability to simulate the forward recovery, and so, they are not suitable for power electronics applications [25]. In order to cover the failures of the SPICE model, some important effects must be considered when dealing with PiN diodes: mainly the conductivity modulation and the charge storage in the epilayer. The resistance of the epilayer is variable and its dependence on voltage or current can be highly nonlinear. In the PiN diode, this low-doped layer is swamped by electrons and holes when the device is in its on state. The density of the injected charge carriers can be much higher than the level of the doping concentration, and the resistivity of the region is significantly reduced. The resistance of a region with the boundaries xl and xr and the area A is given by: Z Z xr dx xr dx Repi = = (2.1) xl qA(µn n + µp p) xl qA(µn + µp)p where n and p are the densities of electrons and holes, respectively, and µn and µp are the mobilities of the charge carriers. The charge carriers are not distributed homogeneously, and their density depends on position; the mobilities also cannot be regarded as constants, since it depends on the carrier concentration in the epilayer. During transient operation, the variation of the resistivity does not follow the changing current instantaneously − this effect can influence the switching behavior (e.g., forward recovery of power diodes), and in order to take it into account, a dynamic description of the charge distribution is necessary. Even if a solution of the time-dependent charge densities is found, the calculation of the resistance remains difficult since the integration in eq. 2.1 is not possible without simplifications. However, accurate solutions for the resistance can be achieved through some simplifications. The charge carriers, which are stored in the lightly doped region of the PiN diode during the conduction mode, must be extracted before the device can reach its blocking state. This effect causes switching delays and switching energy losses. Standard device models for circuit simulation use a quasi-static description of the charge carriers, as the diode SPICE model. It means that the charge distribution

32 2.1– An overview of PiN diode modeling is always a function of the instantaneous voltages at the device terminals. This method is completely insuﬃcient for power devices as said before. A real dynamic description derived from the basic physical equations is required instead. The charge stored in the epilayer of the power PiN diode varies, under transient operation, with both time and position. This variation is determined by the ambipolar diﬀusion equation (ADE):

∂p ∂2p p = Da 2 − (2.2) ∂t ∂x τhl where p(x,t) is the density of the charge carriers, τhl is the high-injection level carrier lifetime, and Da is the ambipolar diffusion coefficient. This equation is valid in the case of high-level injection when hole and electron densities are approximately equal. Unfortunately, an exact analytical solution of the ADE is not possible in the general case. However, there are different analytical approaches to solve the ADE as the Laplace transformation method [26], [27], [28], the Asymptotic Waveform Evaluation method [29], [30], and by using Fourier based-solution [31], [32], [33], [34]. These approaches will be discussed with more details in section 2.3. Other analytical models are based on the modification of the standard diode model to allow dynamic diode characteristics to be predicted [35], [36], [37], [38], as the lumped- charge approach in which the excess carrier distribution profile is discretized into several critical regions, each containing a lumped charge node to represent dynamic charge variation [39], [40], [41], [42]. Like numerical modeling, the limits to simulation accuracy are more due to the accuracy of the input parameters rather than due to the models themselves. The computational overheads of analytical models are far lower than those of numerical models. In addition, there is a large pool of popular commercial simulators, such as SPICE and Saber, the solver algorithms of which have been evolved to solve these types of models most efficiently. Having power device models in the libraries of these simulators allows the latter to function as general purpose power electronics circuits CAD tools. Analytical models are, thus, very appropriate for simulation of power electronic circuits over a large number of switching cycles. A third type of micromodel formulation technique is to use a combination of numerical and analytical models, and they are called the hybrid models. The motivation behind such hybrid model arises from the fact that certain physical phenomena in power devices are very difficult to simulate realistically using only analytical equations, particularly the charge storage effects in the epilayer. The basic idea of this method is to use a fast numerical algorithm that solves the semiconductor equations in the epilayer only. Analytical equations are applied to the rest of the device structure. This procedure has the advantage that a high accuracy of the charge carrier behavior may be simulated without the long execution time associated with fully numerical models. The semiconductor equations in the epilayer, that is the

33 2 – Power PiN Diode Models for Circuit Simulations

ADE, are solved using the finite element method [43], [44], [45] or the finite difference method [46], [47], [48], [49] and will be discussed with more details in section 2.3. Another approach is the approximate solution, where the model equations are based on the device physics, but since exact solutions are not possible or restricted to a few special cases, appropriate mathematical representations are found to approximate the solution. These approaches are purely empirical in many cases, but it is also possible to show that some functions come close to an exact solution under certain constraints of the boundary conditions. The approximations applied to the time-dependent charge carrier distribution can be simple geometrical curves (e.g., straight lines and sine functions), which imitate the shape of the distribution [50], [51], [52]. The knowledge of how the shape must look like is obtained from theoretical considerations or numerical calculations (device simulators [6], [7]). Several papers can be found discussing the different techniques of modeling power diodes and other power semiconductor devices [53], [54], [55], [56], [57]. In this thesis, special attention is dedicated to the micromodels based on the solution of the ADE.

34 2.2– Circuit simulator and model implementation

2.2 Circuit simulator and model implementation

There is a variety of commercial circuit simulation programs available on the market as SPICE-based circuit simulators, such as Pspice [58], and Saber [59]. These simulation programs differ in particular in the possibilities and methods they provide for the implementation of new device models. In SPICE-based simulators, the insertion of mathematical equations (differential equations and implicit functions) is based on subcircuits of user-defined controlled E- type voltage sources and G-type current sources, combined with passive components like resistors, capacitors, inductances, active components like conventional diodes, MOSFETs, bipolar transistors, and general elements like controlled switches, current and voltage sources. On the other hand, Saber offers the individual definition of device models by describing them in mathematical form in a special description language (MAST) or by writing a program in a general-purpose programming language (C or Fortran). Hybrid micromodels involve many mathematical equations to solve the numerical portion of the model, and simulators that provide powerful simulation languages such as Saber are often used for this type of models. However, subcircuit form of implementation is becoming very popular as the capabilities of the personal computer processors are increased.

2.3 PiN diode models: diﬀerent approaches to solve the ADE

In this section some compact PiN diode models present in literature are introduced and brieﬂy discussed. First, some analytical models are presented, and then some hybrid models are described. With the exception of the model presented in [34], where lifetime is variable in the epilayer, and the one presented in [49], [60], where the mobilities are dependent of carrier concentration and temperature, the other compact models analyzed in literature have the following assumptions: • The problem is treated as one-dimensional in the space; • The temperature T is constant; • The P+N− and N−N+ junctions are abrupt; • The doping in the base is constant (N or P type);

• The lifetime τhl and the mobilities in the base are constant and independ of injection; • The carriers density in the epilayer is much larger than the doping density (high injection).

35 2 – Power PiN Diode Models for Circuit Simulations

2.3.1 Analytical model: Laplace transform for solving the ADE This approach was applied by Strollo [26], [27], [28], where the epilayer is represented with a two-port network obtained by solving the ADE with the Laplace transform method, and by approximating the solution in the s-domain with rational functions. The model was implemented as a subcircuit in the Pspice simulator, and the ﬁrst step is to convert the ADE into the s-domain with respect to time (capital letters are used to indicate L-transformed quantities): µ ¶ ∂2P (x,s) 1 Da 2 = + s P (x,s) (2.3) ∂x τhl From the general solution of eq. 2.3 and substituting P(0,s) and P(W,s) on it, the resulting equations can be interpreted as the equations of a two-port network as shown in Fig. 2.1 in which node voltages correspond to carrier concentration while input and output currents correspond to the x-derivative of carrier concentration. The nonlinear boundary conditions of the ADE (eq. 1.62 and eq. 1.71) are represented by two nonlinear current generators, GL and GR, controlled by the current through the diode, p(0) and p(W).

e l e m e n t

Z m p ( 0 ) p ( W )

L m R m

Y p Y p C p 1 C p 2 C p 1 C p 2

G L G R

R p 1 R p 2 R p 1 R p 2

Figure 2.1. Two-port network describing the epilayer of PiN diode.

To obtain a lumped equivalent circuit model, Zm and Yp, that are function of “s”, are approximated with rational expressions. Function Zm is approximated with the ﬁrst two terms of its Taylor series and in this way is represented by the series of a resistance Rm and of an inductance Lm. On the other hand function Yp is

36 2.3– PiN diode models: different approaches to solve the ADE approximated by using a rational function where the coefficients of the same are obtained by using the Padéapproximation. After some algebraic manipulations, the approximate representation for the admittance Yp is shown in Fig. 2.1. All the passive components in the two-port network are dependent of the epilayer width W, which is kept unchangeable during the simulations. More accurate results can be obtained by the insertion of more elements like the one in Fig. 2.1 connected in series [27], in order to obtain the carrier concentration in other parts of the epilayer. However, it affects the simulation time. Owing to the accuracy of the approximations used for Zm and Yp it is possible to divide the epilayer in very few subregions. In order to obtain the complete diode model, after obtaining the equivalent circuit representing the solution of the ADE, the voltage drops in the junctions are calculated through eq. 1.37 and eq. 1.38. The ohmic component is evaluated taking into account carrier-carrier scattering through a series resistance, and by an average value of carrier concentration for each element of the epilayer, where there is a factor that must be obtained by curve fitting through the voltage overshoot forward recovery measurements. In this approach only the voltage drop due to the space-charge region in the left junction (P+N− junction) is considered, since it is true that almost all the diode voltage drops in this region. The width of the left space-charge region is obtained by following the analytical approach proposed in [50], leading to a complex subcircuit where an ideal switch and other controlled generators using small look-up tables are present (these generators are used as flags to indicate the beginning of the reverse-recovery phase and also to clamp to zero the voltages in the network representing the epilayer), since negative values have no physical sense. This complex implementation results in convergence problems for more than one switch simulation. The voltage drop in the space-charge region is calculated by: µ ¶ I qND 2 VSC = + xSC (2.4) 2A²υs 2² where xSC is the the width of the left space-charge region and υs is the hole saturated drift velocity. This model takes into account emitter recombination effect in the end regions, conductivity modulation, carrier-carrier scattering and the dynamic of the space- charge voltage build-up through an analytical approach. Input parameters of the subcircuit are directly obtained from the geometrical and physical parameters of the device (epilayer doping and width, high injection lifetime in the same, width and doping of the emitters).

37 2 – Power PiN Diode Models for Circuit Simulations

2.3.2 Analytical model: Asymptotic Waveform Evaluation for solving the ADE This approach was also applied by Strollo [29], [30] and is an extension of the previous model, since the starting point of the model is to solve the ADE using the Laplace transform. However, in this approach instead of using the boundary condition in the N−N+ junction (eq. 1.71), the following condition is imposed: ∂p (x ,t) = 0 (2.5) ∂x m where W/2 < xm < W, and to simplify the modeling, xm is assumed to be a constant abscissa. This is only an approximation however. Strickly speaking, xm is a constant only in the idealized condition of equal electron and hole mobility and + + equal P and N emitter efficiency, where xm=W/2 for symmetry [39], [40], [41], [42]. Anyway, the use of eq. 2.5 results in a big simplification of the model, and does not significantly affect the accuracy of the overall SPICE model. The boundary condition in the P+N− junction (eq. 1.62) is kept unchanged, but some additional variables are introduced in the same. Finally, the solution of the ADE through the Laplace transform with respect to time with the new boundary conditions leads to (capital letters are used to indicate L-transformed quantities): µ ¶ 1 La √ xm √ I1(s) = Q0(s) 1 + sτhl tanh 1 + sτhl (2.6) τhl xm La Thus, a continued-fraction expression in the s-domain of the carrier distribution in the epilayer is obtained (following the Asymptotic Waveform Evaluation theory [61]), and by truncating the following continued-fraction expansion, lumped RC representations of the epilayer are easily obtained (as seen in Fig. 2.2):

I (s) 1 1 = (2.7) Q (s) 1 0 Z + 0 3 1 + T 1 0 5Z + 0 7 + ... T0 where:

τhl Z0 = (2.8) 1 + sτhl

2 xm T0 = (2.9) Da where i1 is the component of the total current due to the carrier injection in the epilayer, and q0 is the amount of excess charge in the left junction.

38 2.3– PiN diode models: diﬀerent approaches to solve the ADE

t t t h l 5 h l 9 h l i 1

+ 1 1 / 5 1 / 9 q 0 T 0 T 0 T 0 3 7 1 1

Figure 2.2. Electrical network representing the continued-fraction expansion of eq. 2.7.

The model was implemented in the Pspice simulator as a subcircuit, and it was shown that the quasi-static model in SPICE [24] and the lumped-charge model [39], [40], [41], [42] can be obtained as low-order approximations of the continued-fraction expansion (first and second order respectively) while more accurate models can be obtained from higher order approximations of the continued-fraction expansion. This model takes into account emitter recombination effect in the highly doped end regions, conductivity modulation in the epilayer, carrier-carrier scattering and the moving-boundaries effect during reverse recovery. It requires a total of 13 input parameters, with 8 more coefficients in addition to the standard SPICE diode parameters. All the parameters must be extracted by fitting DC measurements (most of the parameters), and from the forward and reverse recovery results. It is done through a stochastic global optimization algorithm in an automatic fashion [62], [63]. However, the implementation of latter seems to be very complex and not that clear. The model shows good convergence properties and fast simulation times, and seems to apply to PiN diode at all conditions.

39 2 – Power PiN Diode Models for Circuit Simulations

2.3.3 Analytical model: Fourier based-solution to the ADE In this model, introduced the ﬁrst time by Leturcq [31] and then extended by Bryant [34] in order to include variable lifetime in the epilayer, the ADE is solved through the Fourier based-solution. Many other authors adopted this kind of solution for solving the ADE, including temperature dependence in their models [64], [65], [66]. Again the solution can be implemented by an electrical analogy. The original work developed by Leturcq [31], where the high-level lifetime of carriers is considered constant in the whole epilayer, is treated here. It was shown in [67] that a discrete cosine Fourier transform of p(x,t):

X∞ · ¸ kπ(x − xl) p(x,t) = V0(t) + Vk(t) cos (2.10) xr − xl k=1 allows the ADE (eq. 2.2) to be converted into an infinite system of first order linear differential equations for the series coefficients V0 ... Vk: µ ¶ " ¯ ¯ # ∂V V ∂p ¯ ∂p ¯ (x − x ) 0 + 0 = D ¯ − ¯ − I (2.11) 2 1 ∂t τ a ∂x¯ ∂x¯ 0 hl xr xl with: µ ¶ X∞ ∂x ∂x I = V l − (−1)n r (2.12) 0 n ∂t ∂t n=1

µ · ¸¶ " ¯ ¯ # (x − x ) ∂V 1 D k2π2 ∂p ¯ ∂p ¯ r l k + V + a = D (−1)k ¯ − ¯ −I (2.13) 2 ∂t k τ (x − x )2 a ∂x¯ ∂x¯ k hl r l xr xl with: µ ¶ V ∂(x − x ) X∞ n2V ∂x ∂x I = k l r + n 1 − (−1)k+n 2 (2.14) k 4 ∂t n2 − k2 ∂t ∂t n=1, n6=k This set of equations can further be represented, by way of a simple electrical analogy, in the form of two RC lines wich correspond to even and odd values of k, the voltages across the successive cells representing the Fourier series coeﬃcients Vk(t) (see Fig. 2.3). The R and C circuit element values are functions of the storage zone thickness, that is the region in the epilayer ﬂooded with free carriers (xr-xl): ( C0 = xr − xl for k = 0, τhl (2.15) R0 = xr − xl

40 2.3– PiN diode models: diﬀerent approaches to solve the ADE

V 0 ( t ) V 2 ( t )

R R 0 2 R a d d ( e v e n ) E V E N L I N E

C 0 C 2

I e v e n I 0 I 2

V 1 ( t ) V 3 ( t )

R R 1 3 R a d d ( o d d ) O D D L I N E

C 1 C 3

I o d d I 1 I 3

Figure 2.3. RC line representation of the x-solution of the ADE through Fourier series.

 x − x  C = r l  k 2 2 1 for k 6= 0, R = (2.16)  k x − x 1 k2π2D  r l a + 2 τhl (xr − xl)

The I0 ... Ik sources, as given by eq. 2.12 and eq. 2.14, account for the boundary shifts (in case of ﬁxed boundaries, Ik ≡ 0).

These RC lines are driven by current sources Ieven and Iodd which are deﬁned by the boundary conditions (eq. 1.21 and eq. 1.22).

" ¯ ¯ # ∂p ¯ ∂p ¯ I = D ¯ − ¯ (2.17) even a ∂x¯ ∂x¯ xr xl

41 2 – Power PiN Diode Models for Circuit Simulations

" ¯ ¯ # ∂p ¯ ∂p ¯ I = −D ¯ + ¯ (2.18) odd a ∂x¯ ∂x¯ xr xl Remembering the boundary conditions: ¯ µ ¶ ∂p ¯ 1 I | I | ¯ n xl p xl ¯ = − (2.19) ∂x x 2qA Dn Dp ¯ l µ ¶ ∂p ¯ 1 I | I | ¯ = n xr − p xr (2.20) ∂x¯ 2qA D D xr n p

When the boundaries of the excess charge start moving, the following condition must be added:

p(xl,t) ≈ 0 and/or p(xr,t) ≈ 0 (2.21)

So, the calculation of the carriers distribution in the epilayer is explicit when the ends of the carrier storage region are ﬁxed (xl=0 and xr=W). At reverse recovery the boundaries become mobile as cleared out zones appear on. In this case, the boundary abscissa values xl and xr must be controlled so as to maintain p(xl,t) and/or p(xr,t) ≈ 0 in the calculation. This is obtained by simple circuits of the type shown in Fig. 2.6, the ideal diodes D enabling the continuity of the representation to be maintained between the ﬁxed and mobile boundary cases. However, the resistors

Rxl and Rxr must be adjusted in order to acquire the latter conditions, meaning that they are additional heuristic parameters in the model, added to the physical and geometrical parameters of the diode. It was also found that these resistors are also related to the value of the leakage current of the diode when reverse biased. The number of cells to be retained in the RC lines must be such that the k-th time-constant is much smaller than the characteristic times involved in the current and voltage waveforms. The truncate error can be greatly reduced by terminating the lines by additional series resistances corresponding to the cumulated resistance values of the missing cells, as shown in Fig. 2.3.

Figure 2.4. Circuits to calculate the position of the left and right borders in the epilayer.

In this model, the displacement current is taken into account. This current adds to the carrier current components due to the variations of the space-charge regions

42 2.3– PiN diode models: diﬀerent approaches to solve the ADE during reverse recovery. This component is added to the components due to the recombination in the highly doped end regions (In1 and Ip2), and they are involved in the calculation of the diﬀusion currents Ip1 and In2. It is done by way of analogy, in that of the space-charge width xSC , according to the subcircuits of Fig. 2.5, always using ideal diodes D, combined with the subcircuit of Fig. 2.4.

I p 1 + I n 1 I d i s I x s c x l

I n 2 + I p 2 I - W x r

Figure 2.5. Sub-circuits for calculations of Ip1, In2 and xSC .

A drawback of the models based in this approach is that all the elements of the RC nets are dependent on the width of the flooded region with excess charge carriers. So, due to the moving boundaries effect, the capacitors and resistors in the RC nets are all non linear elements, that depend on the width of the mentioned region. This fact takes to convergence problems, and higher number of elements needed for implementing the RC net elements. For implementing a variable capacitor in a circuital way in some SPICE versions as IsSPice, as an example, four elements are needed since these versions do not support the derivative command DDT as is the case of the Pspice version, and the higher number of elements impacts on the computation time and in the convergence properties of the simulation. This model takes into account emitter recombination effect in the end regions, conductivity modulation, carrier-carrier scattering and the moving boundaries effect. Input parameters of the subcircuit, when implemented as a SPICE subcircuit, are directly obtained from the geometrical and physical parameters of the device (epilayer doping and width, high injection lifetime in the same, width and doping of the emitters) and the heuristic parameters already mentioned.

43 2 – Power PiN Diode Models for Circuit Simulations

2.3.4 Hybrid model: Finite Element Method for solving the ADE In this model, Ara´ujosolves the ADE through a variational formulation with poste- rior solution using an one dimensional Finite Element approximation [43], [44], [45]. This results in a set of Ordinary Diﬀerential Equations (ODEs), whose solution gives the time and space charge carriers distribution in the epilayer. The implementation of the obtained model, in a general circuit simulator, is made by means of an electrical analogy with the resulting system of ODEs, whose matrices are symmetric. In fact the solution of this system is equivalent to the solution of a circuit made of a set of RC nets and current sources. e l e m e n t

R s R s 1 2n n + 1 C s C s

G L G R

R p 1 C p 1 C p 2 R p 2 R p 1 C p 1 C p 2 R p 2

Figure 2.6. RC equivalent circuit representation of the x-solution of the ADE through the Finite Element Method.

The departure point of the model is a variational formulation of the ADE (eq. 2.2) subjected to the boundary conditions (eq. 2.19 and eq. 2.20), namely:

Z " µ ¶ # Z · ¸ 1 ∂p(x,t) 2 p(x,t)2 p(x,t) ∂p(x,t) Π = + dV + dV − V 2 ∂t 2Daτhl V Da ∂t Z − [(f(t) + g(t))p(x,t)] dS (2.22) S where f(t) is the left boundary condition (eq. 2.19), and g(t) is the right boundary condition (eq. 2.20). The minimization of the functional above (eq. 2.22) is equivalent to the solution of the ADE (eq. 2.2) and its boundary conditions (eq. 2.19 and eq. 2.20). Assuming that the solution of the unknown function, p(x,t), is approached by a sum of elementar functions, the ﬁnite elements, of the form [68], [69]:

44 2.3– PiN diode models: diﬀerent approaches to solve the ADE

[p(e)] = [N(x)] [p(t)] (2.23) being each element:

e e e p(e) = N1 (x) + ... + Nj (x) + ... + Nk (x) (2.24) and after substitution in eq. 2.22 and its minimization (δΠ = 0) we get the following matrix equation: · ¸ ∂p(t) [M] + [K][p(t)] + [F (t)] = [0] (2.25) ∂t where Xe=n [G(x)] = [Ge(x)] (2.26) e=1 Xe=n [M(x)] = [Me(x)] (2.27) e=1 Xe=n F (t) = [Fe(x)] (2.28) e=1 being each of the elementar matrices Z 1 T Me = [N(x)] [N(x)] dV (2.29) Ve Da

Z Z T T [N(x)] [N(x)] Ge = [B(x)] [B(x)] dV + dV (2.30) Ve Ve Daτhl Z Z T T Fe = − f(t)[N(x)] dS − g(t)[N(x)] dS (2.31) (e) (e) Sl Sr and · ¸ ∂N e(x) ∂N e(x) [B] = 1 ... k (2.32) ∂x ∂x

e e [N] = [N1 (x) ...Nk (x)] (2.33) This system of ODEs can now be coupled with diﬀerential equations for the rest of the circuit and conveniently solved with any of the methods for this kind of equations [70], [71]. So, the obtained system of ODEs can be interpreted as a combination of RC nets and current sources that can be solved with the aid of a general circuit simulator as SPICE based simulators.

45 2 – Power PiN Diode Models for Circuit Simulations

Once the type of elements is chosen, N and B are deﬁned, the integrals in 2.29, 2.30 and 2.31 can be solved, and the RC net constructed. It is used for implementation a simplex approach for the elements, so we will have two nodes for each element and, in local coordinates: · ¸ s s [N(s)] = 1 − (2.34) ∆xe ∆xe " # pe(t) [p(t)] = (2.35) pe+1(t) So · ¸ 1 1 [B] = − (2.36) ∆xe ∆xe the product [B]T [B]:  µ ¶ µ ¶  1 2 1 2  −   ∆x ∆x  [B]T [B] =  µ e ¶ µ ¶e  (2.37)  1 2 1 2  − ∆xe ∆xe and [N(s)]T [N(s)]:  µ ¶  s 2 s s2  1 − −   ∆x ∆x ∆x2  T  e e e  [N(s)] [N(s)] =  µ ¶  (2.38)  s s2 s 2  − 2 1 − ∆xe ∆xe ∆xe the integrals: " # Ae∆xe 2 1 Me = (2.39) 6Da 1 2 " # " # Ae 1 −1 Ae∆xe 2 1 Ge = + (2.40) ∆xe −1 1 6Daτhl 1 2

Fe(t) = −f(t) A1 − g(t) An+1 (2.41) where ∆xe is the width of each element within the epilayer, and Ae is the element area.

46 2.3– PiN diode models: diﬀerent approaches to solve the ADE

The obtained matrices for each element are equivalent to the highlighted element circuit in Fig. 2.6, and the whole epilayer is represented by the series connection of n single elements. The voltages at the nodes, V1(t),Vn+1(t), are equivalent to the concentration, p(x,t), along the epilayer. The number of nodes is that of the elements plus one. The ﬁrst node and the last node will have, accordingly with eq. 2.31, an additional current source whose values are, respectively, GL = −f(t)A1 and GR = −g(t)An+1. the values of the other components needed for the implementation of the model are, according to Fig. 2.6:

2Daτhl Rp1 = Rp2 = (2.42) Ae∆xe

6Daτhl∆xe Rs = 2 (2.43) Ae[6Daτhl − ∆xe]

Ae∆xe Cp1 = Cp2 = (2.44) 2Da

Ae∆xe Cs = = (2.45) 2Da Again, we can note that the values of the elementar components of the RC net are all variable ones during recovery because the width of the epilayer, and so of the elements, varies in time. This model takes into account emitter recombination effect in the end regions, conductivity modulation and moving boundaries effect. This work was originally implemented by the authors in the IsSpice software. It is assumed in this model that during the reverse recovery all voltage drops in the left space region. The same scheme used by Leturq [31] for calculating the borders of the epilayer due to the moving boundaries effect is used in this model. As in the case of the previous model, the implementation of the whole model takes to the need of many nonlinear elements, as the case of the implementation of the variable capacitor, resulting in a large number of components and increased simulation times and convergence problems. We have implemented this model in the Pspice simulator as well, and the results were not encouraging since there were lots of convergence problems during the simulation of very simple test circuits. Input parameters of the subcircuit, when implemented in circuital way in SPICE based simulators, are directly obtained from the geometrical and physical parameters of the device (epilayer doping and width, high injection lifetime, width and doping of the emitters) and the heuristic parameter mentioned in the previous model, but in this case regarding only the circuit used for calculating the left border.

47 2 – Power PiN Diode Models for Circuit Simulations

2.3.5 Hybrid model: Finite Difference Method for solving the ADE The most accurate solutions of the ADE, regarding compact diode models, are obtained through numerical solutions of the same [54], which are based on the discretization of the considered region into a finite number of mesh points. Two methods can be distinguished. The first one was applied in the previous hybrid model, and is known as the Finite Element Method (FEM). The second one is the Finite Difference Method (FDM), which has been applied most often. If the FDM is used, the derivatives in the ADE and in the boundary conditions are expressed by differences which may have the form: ¯ ¯ ∂p ¯ −pi+2 + 4pi+1 − 3pi ¯ = (2.46) ∂x i 2∆x ¯ ∂2p ¯ p − 2p + p ¯ = i+1 i i−1 (2.47) 2 ¯ 2 ∂x i ∆x where index “i” indicates the mesh-point number. Eq. 2.46 is an example of forward difference for the derivative in the space. However, there are other ways of representing it, and also the backward and central differences. These differences can be obtained through Taylor series, and are different depending on the truncated term considered in the series [70], [72]. Time is also discretized: ¯ ¯ ∂p¯ pj+1 − pj ¯ = (2.48) ∂t j ∆t and an algebraic equation system results. Several authors have adopted the FDM for solving the ADE. The Finite Differ- ence Method was first developed into a power diode model by Berz [46]. In this work, the so called Enthalpy Method is used for solving the problem of the moving boundaries. When the region flooded with excess carriers detaches from the diode junctions, the boundary conditions should be modified in order to account for the moving boundaries effect as the space charge regions build up from both junctions. This problem is solved by introducing an auxiliary variable u which has the following properties:

• u(x)=p(x)-kJ R (where k is a constant parameter and JR is the reverse recovery current density) within the carriers storage region; ∂2u • = 0 outside the carriers storage region. ∂x2 The carrier concentration p outside the storage region is forced to zero. By using this approach it is possible to maintain the same boundary conditions for the whole

48 2.3– PiN diode models: different approaches to solve the ADE reverse recovery transient and directly obtain the position of the space charge moving boundaries. The mesh in which the epilayer is discretized is kept unchanged. Most of the models using the FDM are implemented in the form of subroutines, such as when the SABER simulator is used [48], [49]. In these models, since the boundary conditions of the ADE are calculated from terminal current densities, the circuit simulator delivers current to the model and receives voltage from the model. This is the concept of hybrid models, consisting of the numerical and analytical part. Thus, the model appears to the simulator as a current controlled voltage source. In models using the finite difference method, local effects such as carrier-carrier scattering and Auger recombination can be easily included, as is the case of the model proposed by Vogler [49], [60]. Thereby, temperature dependent algebraic expressions for scattering and recombination processes replace widely employed parameters such as high injection lifetime, mean carrier concentration and average mobility. In addition, the algorithm of these models can easily be understood and modified, while the simple simulator equations take to no convergence problems and to an accurate solution of the ADE. Another FDM based model is the model proposed by Goebel [48], [73], in which conductivity modulation and the field dependence of carriers mobilities are included, while emitter recombination effects are not included. The latter was then included in [74]. The only model similar to the FDM based models implemented in a circuital way was proposed by Profumo [47]. In this work, implemented in the Pspice simulator, Analog Behavioral Modeling (ABMs) blocks were used for implementing the model. The model was implemented in the same way done by Berz [46], considering a fixed mesh of the one-dimensional epilayer as explained above. In all FDM based models, the input parameters are directly obtained from the geometrical and physical parameters of the device. In particular, the model presented by Profumo [47] takes into account only the space charge region in the left junction, while the other models consider the space charge region in both junctions. The lumped-charge approach in [39], [40], [41], [42], looks similar to the method of finite differences. It can be regarded as a simplification to the greatest possible extent with a minimal number of nodes. But in a lumped model, the average charge densities of the sections instead of the densities at the nodes are inserted into the equations. In the following, a novel Finite Difference based model of PiN diodes is presented.

49 Chapter 3

Finite Diﬀerence Based Power PiN Diodes Modeling and Validation

A physics-based model for PiN power diodes is developed and implemented as a SPICE subcircuit. The starting point of the model is an equivalent circuit representation of the epilayer, obtained by solving the Ambipolar Diffusion Equation (ADE) with the Finite Difference Method. The proposed model takes into account emitter recombination in the highly doped end regions, conductivity modulation in the epilayer, carrier-carrier scattering and the moving boundaries effect during reverse recovery, showing good convergence properties and fast simulation times. Comparisons between the results of the SPICE model and both numerical device simulations and experimental results are presented, in order to validate the proposed model.

50 3.1– Nomenclature

3.1 Nomenclature

A device area (cm2) b ration between electron and hole mobility 2 Da ambipolar diffusion coefficient in the epilayer (cm /s) 2 Dn electron diffusion coefficient in the epilayer (cm /s) 2 Dp hole diffusion coefficient in the epilayer (cm /s) ² dielectric constant of silicon (F/cm) 4 hp emitter recombination coefficient in the highly doped P region (cm /s) 4 hn emitter recombination coefficient in the highly doped N region (cm /s) ID total diode current (A) Idep current component due to the depletion region (A) In electron current (A) Inl electron current in the left border of the carrier storage region (A) Inr electron current in the right border of the carrier storage region (A) Ip hole current (A) Ipl hole current in the left border of the carrier storage region (A) Ipr hole current in the right border of the carrier storage region (A) 2 Jn electron current density (A/cm ) 2 Jp hole current density (A/cm ) k Boltzman constant (J/K) 2 µ0 sum of electron and hole mobilities in the epilayer (cm /V·s) 2 µn electron mobility in the epilayer (cm /V·s) 2 µp hole mobility in the epilayer (cm /V·s) 3 ND epilayer doping (cm ) −3 ni intrinsic carrier concentration (cm ) p(x,t) hole concentration in the epilayer (cm−3) 3 p0 coefficient describing carrier-carrier scattering in the base (cm ) q electronic charge (C) Repi resistance of the epilayer (Ω) T Temperature (K) τhl high injection lifetime in the epilayer (s) VD total voltage drop in the diode (V) + − Vjl voltage drop in the P N diode junction (V) − + Vjr voltage drop in the N N diode junction (V) Vleft space charge voltage drop in the left junction (V) Vres resistive voltage drop in the epilayer (V) Vright space charge voltage drop in the right junction (V) vs saturated hole drift velocity (cm/s) VT thermal voltage (V)

51 3 – Finite Diﬀerence Based Power PiN Diodes Modeling and Validation

∆x width of each element of the discretized epilayer (cm) xl left boundary of the epilayer (cm) xr right boundary of the epilayer (cm) W width of the epilayer (cm)

3.2 Introduction

In power converter systems, the role of switching is dedicated to the power semiconductors. In the early design stages, these devices are often considered as binary on-off switches, allowing vary fast computation of the circuit [75]. However, as the current trend is towards reducing power converter size and increasing switching frequencies, the binary on-off representation of semiconductor devices has to be revised. This very simple device representation cannot take into account for the non ideal behaviors of the semiconductor device, especially during switching transients. One approach is to employ a physically-based model to represent the device. This method is based on describing and solving numerically the basic drift-diffusion semiconductor equations, using the finite element or the finite difference method for example [6], [7]. In this way, device behavior can be modelled very precisely in two or even three dimensions. Unfortunately a very precise physical model with high numerical complexity involves a very long computational time, making this approach more suited to individual device design and optimization, since its use is computationally prohibitive at system level modelling. A physical compact modelling approach as the ones presented in the last chapter lies between these two extremes, and it seems to represent the ideal solution for power semiconductor device representation within circuit simulators. This physic approach is based on certain mathematical simplifications of the fundamental semiconductor charge transport equations, resulting in the Ambipolar Diffusion Equation (ADE). The model presented in this chapter exploits the Finite Difference Method (FDM) for the discretization of the lightly doped base region, the epilayer, in a finite number of mesh points (nodes). This is the numerical contribution of this hybrid model. An electrical analogy with the set of ordinary differential equations obtained by the space-discretization of the ADE in the base region provides the equivalent circuit model for the carriers concentration in the epilayer. The carriers concentration is then related to the diode voltage by the junction and ohmic relationships under forward bias, and the Poisson equation under reverse bias operation, which are the analytical contributions of this hybrid model. This results in an easy-to-implement diode model as a SPICE subcircuit, which takes into account the emitter recombination effects in the highly doped end regions, carrier-carrier scattering, conductivity modulation, and the dynamic of the space-charge voltage build-up (moving boundaries effect during reverse recovery). It is worth to outline that the FDM approach to the ADE solution yields the time-varying free carrier distributions in the epilayer, thus allowing for a better comprehension of the device dynamic behavior. The model

52 3.3– Model description can be implemented in any commercially available circuit simulator allowing for non linear elements, especially those based on SPICE. In this work, the commercially available circuit simulator Pspice was used [58], and the model was implemented as a Pspice subcircuit. This simulator was chosen since it is the standard reference circuit simulator in the world [76]. The subcircuit is much simpler than some other proposed subcircuit models described in the last chapter, resulting in faster simulation and improved convergence properties. Parameters for the model are all related to the physical and geometrical properties of the device. Pspice model simulations are compared with experimental results and simulations using another compact diode model present in literature, and either with experimental characterizations or with SILVACO mixed-mode module simulations [7] of a commercial PiN power diode [19]. A good agreement is obtained, with much smaller computation times of Pspice simulations than SILVACO ones. Finally, simulation examples of practical Switched-Mode Power Supplies (SMPS) are provided, in order to demonstrate the model eﬀectiveness, speed and convergence properties.

3.3 Model description

3.3.1 Fundamental Equations

I D P + N - N + A n o d e C a t h o d e x = 0 x = W

Figure 3.1. Structure of PiN power diode.

Let us consider a device with a base width W, extending from the P+-N− junction − + (x = xl = 0) to the N -N junction (x = xr = W ), see Fig. 3.1. Under high injection, the carrier distribution is governed by the ADE [9]:

∂p ∂2p p = Da 2 − (3.1) ∂t ∂x τhl where Da = 2DnDp/(Dn + Dp) is the ambipolar diﬀusion coeﬃcient and τhl is the high-level lifetime. The boundary conditions for eq. 3.1 at x = xl and x = xr can be written as: µ ¶ ∂p 1 In(x = xl) Ip(x = xl) (xl,t) = − (3.2) ∂x 2qA Dn Dp µ ¶ ∂p 1 In(x = xr) Ip(x = xr) (xr,t) = − (3.3) ∂x 2qA Dn Dp

53 3 – Finite Diﬀerence Based Power PiN Diodes Modeling and Validation

In eq. 3.2 and eq. 3.3, In is the electron current, Ip is the hole current, A the device area, q the electronic charge, Dn and Dp are the diﬀusion coeﬃcients for electrons and holes respectively. During evaluation of the equations above, the following equations must also be taken into account:

ID = In + Ip + Idep (3.4)

I (x = x ) = q h A p2 (3.5) n l p (x=xl)

2 Ip(x = xr) = q hn A p(x=xr) (3.6)

∂x I = −q N A l (3.7) dep D ∂t where ID is the total diode current, Idep is an additional current component during reverse recovery, due to the charge variations in the space-charge region (this component charges and discharges the anode-base depletion capacitor), hp and hn account for emitter recombination eﬀects, and xl and xr are the left and right borders of the region ﬂooded with excess carriers.

3.3.2 Finite Difference Modeling of the Base Region In this model, Finite Difference substitutions in eq. 3.1 are done only with respect to space, meaning that the time derivative in eq. 3.1 is kept unchanged. In [48], [49], the time derivative is also discretized, forward and backward differences are applied in the first and last nodes respectively, while central differences are applied for the other internal nodes of the discretized region. It results in a set of algebraic equations, with n + 1 equations and n + 1 variables. Instead of using also backward and forward differences as in [48], [49], this model only uses central differences. Thus, for the n elements in which the base region is divided, we have n + 1 nodes, and for the ith node we have [72]: ¯ ¯ ∂p ¯ pi+1 − pi−1 ¯ = (3.8) ∂x i 2∆x ¯ ∂2p ¯ p − 2p + p ¯ = i+1 i i−1 . (3.9) 2 ¯ 2 ∂x i ∆x Note that using only central differences, we have n + 3 nodes instead of the n + 1 nodes resulting when using central, forward and backward differences. However, by using central differences in the boundary conditions as well, we obtain an explicit expression for the nodes 0 and n + 2. Consequently, we come back to a set of

54 3.3– Model description n + 1 equations in n + 1 variables (concentration in the nodes). Substituting the corresponding ﬁnite diﬀerence eq. 3.9 in eq. 3.1 we obtain the general expression for the ith node ¯ µ ¶ ∂p¯ p − 2p + p p ¯ = D i+1 i i−1 − i . (3.10) ¯ a 2 ∂t i ∆x τhl

st th For the 1 and n + 1 node we have the following boundary conditions at x = xl and x = xr: ¯ µ ¶ ¯ ∂p ¯ p2 − p0 1 In(x = xl) Ip(x = xl) ¯ = = − (3.11) ∂x i=1 2∆x 2qA Dn Dp ¯ µ ¶ ¯ ∂p ¯ pn+2 − pn 1 In(x = xr) Ip(x = xr) ¯ = = − (3.12) ∂x i=n+1 2∆x 2qA Dn Dp Substituting eqs. 3.4-3.6 in eq. 3.11 and eq. 3.12 above: ¯ µ ¶ ¯ 2 2 ∂p ¯ p2 − p0 1 qAhpp1 ID − Idep − qAhpp1 ¯ = = − (3.13) ∂x i=1 2∆x 2qA Dn Dp

¯ µ ¶ ¯ 2 2 ∂p ¯ pn+2 − pn 1 ID − Idep − qAhnpn+1 qAhnpn+1 ¯ = = − (3.14) ∂x i=n+1 2∆x 2qA Dn Dp

Substituting p0 from (3.13) and pn+2 from (3.14) into the expressions equivalent to the 1st and n + 1th nodes in (3.10), and rearranging the ﬁrst and the last (n + 1th) rows of the system of ODEs, dividing them by 2, we ﬁnally get a symmetric system as follows · ¸ ∂p [M] + [K][p] + [F ] = 0 (3.15) ∂t where the n + 1 x n + 1 symmetric matrices [M] and [K], and the n + 1 x 1 vector [F], are given as   0.5 0 ··· 0 0      0 1 ··· 0 0    M =  . . . . .  (3.16)  . . . . .     0 0 ··· 1 0  0 0 ··· 0 0.5

55 3 – Finite Diﬀerence Based Power PiN Diodes Modeling and Validation

  Da 1 Da  2 + − 2 ··· 0 0   ∆x 2τhl ∆x     Da 2Da 1   − + ... 0 0   ∆x2 ∆x2 τ   hl    K =  . . . . .  (3.17)  . . . . .     2D 1 D   0 0 ··· a + − a   2 2   ∆x τhl ∆x   Da Da 1  0 0 · · · − 2 2 + ∆x ∆x 2τhl   µ 2 2 ¶ Da qAhpp1 ID − Idep − qAhpp1  −   2qA∆x Dn Dp         0        F =  .  (3.18)  .         0     µ ¶   D I − I − qAh p2 qAh p2  − a D dep n n+1 − n n+1 2qA∆x Dn Dp The ODEs system (3.15) can be interpreted as a combination of RC nets and current sources, meaning that an equivalent electrical circuit of the system above is obtained, using Kirchhoﬀ’s Current Law, which can be solved with the aid of the Pspice circuit simulator. So, system (3.15) is equivalent to the system: · ¸ ∂V [C] + [G][V ] + [I] = 0 (3.19) ∂t which in turn corresponds to the circuit in Fig. 3.2, the node voltages Vi(t) being equivalent to the concentration p(xi,t) along the base. The number of nodes is the number of elements plus one. The ﬁrst and last node (n + 1th node) will have, according to (3.19), additional current sources whose values are respectively

µ 2 2 ¶ Da ID − Idep − qAhpV1 qAhpV1 I1 = − (3.20) 2qA∆x Dp Dn and µ 2 2 ¶ Da ID − Idep − qAhnVn+1 qAhnVn+1 In+1 = − . (3.21) 2qA∆x Dn Dp

56 3.4– The complete diode model

G G n o d e 1 1 2 n o d e 2 n o d e n n n + 1 n o d e n + 1 I 1 I n + 1 G C G C C G C 1 1 2 2 G n n n + 1 n + 1

Figure 3.2. Equivalent circuit representing the lightly doped base region of the PiN diode model.

Regarding the other components of the RC net representing the epilayer, we have the following relationships. The series resistors between two adjacent nodes i and i+1 have all the same value,

∆x2 Ri, i+1 = (3.22) Da During reverse recovery, moving boundary effect is taken into account by allowing an adaptive definition of the space-charge region width, which in turn changes the width-dependent series resistances in the RC network modeling the base region above. The other components, the shunt components of the RC net, they are all constant and have well defined values dependent upon the physical characteristics of the diode, as follows:   C1 = Cn+1 = 0.5 for i = 1, n+1, (3.23)  R1 = Rn+1 = 2τhl   Ci = 1 for i 6= 1, n+1, (3.24)  Ri = τhl With respect to other hybrid models [43], [44], [45], the proposed approach offers a significant reduction of circuit complexity in terms of number of components as well as of the single component complexity. It results in an easy and fast implementation of the model. The accuracy of the model will depend on the number of elements used.

3.4 The complete diode model

In order to obtain the complete hybrid diode model, other sub-models, the analytical part of the model, must be coupled to the solution of the ADE. These sub-models are derived from the junction and ohmic relationships under forward bias, and the Poisson equation under reverse bias operation.

57 3 – Finite Diﬀerence Based Power PiN Diodes Modeling and Validation

The complete compact model for the PiN diode is shown in Fig. 3.3, and includes the internal voltages and currents. The total voltage across the diode is the sum of various components as shown in Fig. 3.3:

VD = Vjl + Vleft + Vres + Vjr + Vright = Vanode − Vcathode (3.25)

The voltages Vjl and Vleft represent the junction and depletion voltage drops + − − + across the P N junction. Similarly, Vjr and Vright account for the N N junction. In addition, there is the ohmic voltage drop Vres across the ﬂooded region with excess carriers (carrier storage region), where conductivity modulation takes place whenever the excess carriers are present. The total diode current consists of both hole and electrons components within the carrier storage region, and the depletion capacitance current Idep also contributes to the total diode current. The currents Ipl, Inl, Ipr, Inr represent the individual hole and electron currents entering and leaving the carrier storage region at the anode and cathode junctions within the base region. Thus, the diode equivalent circuit appears as a current-controlled voltage source.

B ase region (Epilayer)

I p l I p r p l p r

I I I V j l n l n r V j r I D D + - x l x r + - A n o d e C a t h o d e V l e f t V r e s V r i g h t

I d e p

Figure 3.3. PiN diode model: all voltages and currents incorporated into model are shown.

3.4.1 Voltage drop on the junctions The voltage drops in the junctions are calculated through the equations obtained by the junction law under forward bias, as demonstrated in chapter 1. The voltage drop in the left and right junctions are calculated from p1 and pn+1, and in the equivalent RC network from V1 and Vn+1, according to: µ ¶ µ ¶ p1 · ND V1 · ND Vjl = VT · ln 2 = VT · ln 2 (3.26) ni ni

58 3.4– The complete diode model and µ ¶ µ ¶ pn+1 Vn+1 Vjr = VT · ln = VT · ln (3.27) ND ND −3 where ND is the base doping concentration [cm ], ni is the intrinsic carrier concen- −3 tration [cm ], and VT is the equivalent thermal voltage [V].

We call the attention that these equations are valid while there are excess carriers in the physical junctions of the diode, being zero otherwise. During the implementation of these equations, special care must be taken in order to avoid using negative values of the carriers concentration in the same, since during the reverse recovery, the carrier concentrations in the ﬁrst and last nodes will assume very small values, in order to approximate their values to zero.

3.4.2 Voltage drop on the epilayer The voltage drop on the conductivity modulated based region is given by the sum of an ohmic component, due to the drift current, and of a Dember component, due to the diﬀusion current. The latter arises from unequal electron and hole mobilities [9] and will be neglected in the following, since it is typically much smaller than the ohmic voltage drop Vres. The ohmic voltage drop across the carrier storage region is found by integrating the electric ﬁeld E responsible for driving the drift currents in this region:

Z x=xr Vres = Edx (3.28) x=xl where E is found from the total hole and electron drift current in the plasma:

ID = In + Ip = qA(µn(p + ND) + µpp)E (3.29) Thus, from eq. 3.28 and 3.29, the ohmic voltage drop is found to be:

Z x=xr ID dx Vres = (3.30) qA x=xl p(x,t)(µn + µp) + µnND

In order to solve eq. 3.30 and determine Vres, the discretized epilayer is used:

I Xn ∆x V = D (3.31) res qA V (x,t)(µ + µ ) + µ N i=1 i n p n D For the sake of simplicity, it is assumed that the concentration between the two nodes of an element to be equal to p + p p = i i+1 (3.32) i 2 59 3 – Finite Diﬀerence Based Power PiN Diodes Modeling and Validation

We can include the mobility degradation eﬀect, due to the electron-hole scattering, by assuming [29], [30], [77]:

p0 · µ0 µn + µp = (3.33) p0 + p where µ0 is considered as the sum of the mobilities in the base region and p0 is a suitable constant in the order of the carriers concentration in the base under high- injection condition. By this assumption, we can note that in low-injection condition the sum of the mobilities is not disturbed by electron-hole scattering eﬀects. De- pending on the width of the high-doped end regions of the diode, resistance in these zones must also be taken into account as parasitic resistances, and other parameters of the diode must be supplied, as the doping concentration, width of the same regions, and mobilities of holes and electrons in these regions.

3.4.3 Voltage drop on the space-charge regions The quasi-neutrality condition i