Modeling and Simulation of Power PiN Diodes within SPICE

POLITECNICO DI TORINO Facoltà di 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 advises 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 continuous 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 José Roberto 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 João Batista Vieira Júnior is 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 Validation 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 pointing 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. 2 ) @ A + = J D @ A ', F E C?! % #! 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: J n = qµ n ne + qd n n x = qµ n(ne + kt q J p = qµ p pe qd p p x = qµ p(pe kt q n x ) (1.1) p x ) (1.2) J = J n + J p (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, J n and J p are the current densities of electrons and holes [A/cm 2 ], D n and D p are the diffusion coefficients of electrons and holes [cm 2 /s], µ n and µ p are the mobilities of electrons and holes [cm 2 /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 t = 1 J n q x + (G n R n ) (1.4) p t = 1 J p q x + (G p R p ) (1.5) where (R n G n ) and (R p G p ) 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 U = np n 2 i τ p0 (n + n 1 ) + τ n0 (p + p 1 ) = np 0 + pn 0 + n p τ p0 (n 0 + n + n 1 ) + τ n0 (p 0 + p + p 1 ) (1.6) where n = n 0 + n, p = p 0 + p and p 0 n 0 = n 2 i, n i is the intrinsic density of charge carriers [cm 3 ], n 0 and p 0 are the electrons and holes densities at thermal equilibrium [cm 3 ], n and p are the densities of electrons and holes in excess [cm 3 ], τ n0 and τ p0 are the electrons and holes minority carrier lifetimes in heavily doped P and N type silicon [s], and n 1 and p 1 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: ( ) ( ER E C n 1 = N C exp = n i exp E ) F i E R (1.7) kt kt and p 1 = N V ( ) ( ) EV E R EF i E R exp = n i exp kt kt (1.8) where N C is the effective density of states in conduction band [cm 3 ], N V is the effective density of states in valence band [cm 3 ], E C is the bottom of conduction band [ev], E V is the top of valence band [ev], E F i is the intrinsic Fermi energy level [ev], and E R is the recombination level location [ev]. In addition to the continuity equations, Poisson s equation must be satisfied de dx = ρ ɛ s = q(p n + N D N A ) ɛ s (1.9) where ρ is the space charge density [cm 3 ], ɛ s is the semiconductor permittivity [F/cm], N D is the donor impurity density [cm 3 ], and N A 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/cm 2 to 1000 A/cm 2. 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: n n 0 N D and p p 0 n2 i N D 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 E F i, it means that n 1 and p 1 are of the same order of n i. Even if the energy levels of the recombination centers are not located that close to the E F i, their order are much lower than the order of n and p, and eq. 1.6 becomes [8]: U = n 2 n(τ p0 + τ n0 ) = n τ p0 + τ n0 (1.11) 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: τ hl = τ n0 + τ p0 (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): J n qd n n x qµ n n = J p + qd p n x qµ p n (1.13) From eq. 1.13 and considering the Einstein relation D n = V T µ n and D p = V T µ p (V T = kt/q is the so called thermal voltage [V]): J n J p = 2q n D n D p x (1.14) With respect to the the continuity equations, eq. 1.4 and eq. 1.5 under the assumptions already mentioned, they respectively become: n t n t = 1 q = 1 q J n x n τ hl (1.15) 7 J p x n τ hl (1.16)

1 Physics and Basic Equations of Power PiN Diode The derivative of eq. 1.14 with respect to x takes to 1 D n J n x 1 D p J p x = 2q 2 n x 2 (1.17) Finally, eliminating J n / x and J p / x respectively from eq. 1.15 and eq. 1.16 and substituting into eq. 1.17: Considering eq. 1.10: n t = D a 2 n x 2 n τ hl (1.18) n t = D 2 n a x n (1.19) 2 τ hl 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 D a is the ambipolar diffusion constant [4], [5], [8], [9]: D a = 2D nd p D n + D p (1.20) 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 x = J n x=xl J p x=xl (1.21) x=xl 2qD n 2qD p n = J n x=xr J p x=xr (1.22) x 2qD n 2qD p x=xr where x l is the left border and x r 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 x l = 0 and x r = 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 x l and x r 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, V D, can be divided in three components indicated in Fig. 1.2 as V P + (P + N junction voltage drop), V N + (N N + junction voltage drop), and V M (ohmic voltage drop). 8 2 8 8 2 ) @ A + = J D @ A 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 D a t = 2 n x n 2 L 2 a (1.23) where L a = D a τ hl 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: ( ) ( ) x x n(x) = C 1 cosh + C 2 sinh (1.24) L a and its derivative has the following form: n x = C ( ) 1 x sinh L a L a + C 2 L a L a ( ) x cosh L a (1.25) 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 @ @ ) @ A + = J D @ A @ 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/cm 2 [9], where end region recombination cannot be neglected. So, under the assumptions mentioned above, the following equations are obtained: J p x=xl = J p x= d = J (1.26) J n x=xl = J n x= d = 0 (1.27) J n x=xr = J n x=+d = J (1.28) J p x=xr = J p 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 x = J p x= d = J (1.30) x= d 2qD p 2qD p and n x = J n x=+d = J (1.31) x=+d 2qD n 2qD n Finally, from eq. 1.25, eq. 1.30 and eq. 1.31, constants C 1 and C 2 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: ( ) ( ) x x n(x) = Jτ hl cosh sinh L a L 2qL a ( ) a B ( ) d d (1.32) sinh cosh where L a 10 L a

1.2 Forward conduction B = (D n D p ) (D n + D p ) = (µ n µ p ) (µ n + µ p ) (1.33) 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 V D at the diode is composed of three components V D = V P + + V M + V N + (1.34) In order to relate current density to voltage drop on the diode, it is needed the calculation of diode voltage drop components V P +, V N +, and V M 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): p n p n0 ( ) qvp + = exp kt n n n n 0 = exp p( d)n D n 2 i ( ) qvn + kt = n( d)n D n 2 i n(+d) N D ( ) qvp + = exp kt ( ) qvn + = exp kt (1.35) (1.36) where N D is the epilayer doping, p n is the concentration of holes in the lightly doped epilayer side of the anode junction (x=-d), p n0 is the concentration of holes in the same place at thermal equilibrium, n n is the concentration of electrons in the lightly doped epilayer side of the cathode junction (x=+d), and n n0 is the concentration of electrons in the same place at thermal equilibrium. Thus, V P + = kt [ ] q ln n( d)nd (1.37) n 2 i V N + = kt q ln [ n(+d) N D V P + + V N + = kt [ n(+d)n( d) q ln n 2 i Calculating n( d) and n(+d) through eq. 1.32 and substituting in eq. 1.39 ] ] (1.38) (1.39) V P + + V N + = kt q ln τhl 2 J ( ) 2 n 2 i (2qL 1 d a) 2 ( ) B 2 tanh 2 d tanh 2 L a (1.40) L a 11

1 Physics and Basic Equations of Power PiN Diode Considering eq. 1.34 and substituting in eq. 1.40 ( ) VD V M exp = τ hl 2 J 2 V T n 2 i (2qL 1 a) 2 ( ) B 2 tanh 2 d tanh 2 L a ( d L a ) (1.41) Rearranging eq. 1.41 the relation between density current and voltage drop on the diode is given by: J = 2n i q D ( ) ( ) a d d F VD exp (1.42) L a 2V T with ( ) d F L a = d ( ) ( )] 1 d tanh [1 B 2 tanh 4 2 exp ( L a L a dla V ) M 2V T In order to evaluate V M, we have to integrate the electric field over the epilayer V M = x=+d x= d (1.43) E(x)dx (1.44) 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: J = qµ n ne + qd n n x + qµ pne qd p n x = qne(µ n + µ p ) + qv T n x (µ n µ p ) (1.45) E = J qn(µ n + µ p ) BV T n n x (1.46) 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 ( ) d sinh ( ) V M = V T 8b L a (1 + b) 1 2 ( ) ( 1 arctan B 2 tanh 2 dla B 2 tanh 2 dla ( ) d sinh L a ( ) d 1 + B 2 tanh 2 + B ln L a ( ) d (1.47) 1 B 2 tanh 2 L a 12

1.2 Forward conduction where b = (µ n /µ p ) 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 V M 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 V M 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 (J n x=xl ), and the current density due to the recombination of holes injected in the N + region (J p x=xr ) must be taken into account. With reference to Fig. 1.4 and taking into account the definitions of D a and b, eq. 1.21 and eq. 1.22 can be rewritten as: n x = J n x=0 bj p x=0 (1.48) x=0 qd a (b + 1) n x = J n x=w bj p x=w (1.49) x=w qd a (b + 1) 2 9 ) @ A + = J D @ A 9 2 9 9 F 9 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 x = x=0 qd a (b + 1) + J n x=0 qd a (1.50) n J x = x=w qd a (b + 1) J p x=w qd a (1.51) 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]: ( ) WP + + x [n P +(0) n P 0 +] sinh L n P +(x) n P 0 + = np + ( ) (1.52) WP + sinh L np + where n P + is the density of electrons in the P + region, n P 0 + is the density of electrons in the P + region at thermal equilibrium, W P + is the width of the P + region, and L np + 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: ( ) np + J n x=0 = qd np + (1.53) x where D np + is the diffusion coefficient of electrons in the anode. Using quasiequilibrium at the P + N junction, the injected carrier concentrations on either side of the junction are related by: p P +(0) p(0) = n(0) n P +(0) Under low injection level in the P + region, it is assumed that (1.54) p P +(0) = p P 0 + (1.55) where p P + is the density of holes in the P + region, and p P 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: ( ) VP + n P +(0) = n P 0 + exp (1.56) Using the two expressions above in eq. 1.54: 14 V T

1.2 Forward conduction n(0) p(0) = p P 0 + n P 0 + Using the charge neutrality condition: exp exp ( VP + V T ( VP + V T ) = n 2 i exp ( VP + V T ) (1.57) ) [ ] 2 n(0) = (1.58) Substituting the derivative of eq. 1.52 in eq. 1.53: ( ) VP + qd np + n P 0 + exp V T J n x=0 = ( ) = qd np + n P 0 ( + n(0)2 ) WP + WP + L np + tanh L np + tanh n 2 i L np + Finally, considering the mass action law: n i L np + (1.59) with J n x=0 = q h p n(0) 2 (1.60) h p = L np + tanh D np + ( WP + L np + ) N A (1.61) where h p is the emitter recombination coefficient in the P + region [cm 4 /s] [10]. Substituting eq. 1.60 in eq. 1.50, the boundary condition for the P + N junction is now given by n x = x=0 bj qd a (b + 1) + h p n(0) 2 Da (1.62) 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]: ( ) x WN + [p N +(W ) p N0 +] sinh L pn + p N +(x) p N0 + = ( ) (1.63) W WN + sinh L pn + where p N + is the density of holes in the N + region, p N0 + is the density of holes in the N + region at thermal equilibrium, W N + is the width of the N + region, and L pn + 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: ( ) pn + J p x=w = qd pn + (1.64) x where D pn + 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 N +(W ) p(w ) = n(w ) n N +(W ) (1.65) where n N + is the density of electrons in the N + region. With analogous assumptions made for the P + region: ( ) n(w )p(w ) = n 2 VN + i exp (1.66) Using the charge neutrality condition: exp ( VN + V T V T ) [ ] 2 n(w ) = (1.67) Substituting the derivative of eq. 1.63 in eq. 1.64: ( ) VN + qd pn + p n0 + exp V T J p x=w = ( ) = qd pn + p N0 + n(w )2 ( ) WN + WN + L pn + tanh L pn + tanh n 2 i L pn + Finally, considering the mass action law: n i L pn + (1.68) with J p x=w = q h n n(w ) 2 (1.69) h n = L pn + tanh D pn + ( WN + L pn + ) N D (1.70) where h n is the emitter recombination coefficient in the N + region [cm 4 /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 n x = x=w J qd a (b + 1) h n n(w ) 2 Da (1.71) 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]: ( ) n 2 ie = n 2 Eg i exp (1.72) kt where n ie is the intrinsic carrier concentrarion taking into account bandgap narrowing effect, and E g 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]: ( ) 3q 2 q E g = 2 N I 16πɛ s ɛ s kt (1.73) where N I is the doping concentration, in the case of P + region the acceptor concentration N A, and in the case of N + region the donator concentration N D. 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 h p and h n are increased by the factor exp( E g /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/cm 2 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(v D /V T ), as shown in Fig. 1.5, which was generated with the following physical and geometrical parameters: W = 50 µm, N D = 2 10 14 cm 3, A = 0.04 cm 2, τ hl = 200 ns, h p = h n = 1.5 10 14 cm 4 /s, D n = 34.84 cm 2 /s, D p = 12.82 cm 2 /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: V M = R epi I D 1 n I D 1 I D I D constant 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: V M = R epi I D 1 n I D 1 ID I D I D 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 + @ K? J E + K H H A J, A I E J O )? 9 E J D - @ 4 A C E 4 A? > E = J E - @ 4 A C E 4 A? > E = J E % % # & & # ' ' #. H M = H @ 8 J = C A, H F 8 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 physical and geometrical parameters: W = 100 µm, N D = 2 10 14 cm 3, A = 0.04 cm 2, τ hl = 1 µs, h p = h n = 5 10 14 cm 4 /s, D n = 34.84 cm 2 /s, D p = 12.82 cm 2 /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 N $ " # 9 E J D - @ 4 A C E 4 A? > E = J E - @ 4 A C E 4 A? > E = J E + = H H E A H I +? A J H = J E?!! # # # # # #!! # " " # #, A F J D Figure 1.6. Effect of end region recombination on the carriers concentration in the epilayer under steady state condition for J = 100 [A/cm 2 ]. τ auger = 1 C A n 2 (1.74) where n is the excess carrier concentration in the epilayer, and C A is the Auger recombination coefficient, of the order of 10 31 cm 6 /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: U = n ( 1 = + 1 ) n = n + C A n 3 (1.75) τ eff τ hl τ auger τ hl where τ eff is the effective carrier lifetime, and 1/τ eff = 1/τ hl +1/τ aug. 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: D a 2 n x 2 = n τ hl + C A n 3 (1.76) 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 (L np +, L pn +), and increasing the recombination currents in the end regions (increase of h p and h n ), what means to say, decreasing the emitters efficiency. In conclusion, the resistivity in the epilayer is further increased.! + @ K? J E + K H H A J, A I E J O )? - @ 4 A C E 4 A? > E = J E + = H H E A H? = H H E A H 5? = J J A H E C ) K C A H 4 A? > E = J E 9 E J D - @ 4 A C E 4 A? > E = J E + = H H E A H? = H H E A H 5? = J J A H E C ) K C A H 4 A? > E = J E 9 E J D - @ 4 A C E 4 A? > E = J E 9 J E D + = H H E A H? = H H E A H 5? = J J A H E C 9 E J D ) K C A H 4 A? > E = J E 9 E J D - @ 4 A C E 4 A? > E = J E 9 J E D + = H H E A H? = H H E A H 5? = J J A H E C ) K C A H 4 A? > E = J E % & '! ". H M = H @ 8 J = C A, H F 8 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 helium 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: In this hypotesis the ADE becomes: τ hl = τ hl (x) (1.77) 1 n D a t = 2 n x n 2 L 2 a(x) (1.78) 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.. H M = H @ 4 A? L A H O, E @ A + K H H A J )?, E @ A 8 J = C A 8, E @ A 8 J = C A 6 K H @ E @ J! " # 6 E A I 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: R epi = W q µ n N D [Ω cm 2 ] (1.79) where W is the width of the epilayer, and N D is the epilayer doping. In order to clarify the order of the unmodulated region resistance, let us consider a general 23

J 1 Physics and Basic Equations of Power PiN Diode diode with an epilayer 50 µm wide and with doping N D = 10 14 cm 3. One finds that its resistance is of the order of 10 1 Ω cm 2, which means that for forward density currents equal to 100 A/cm 2, the voltage drop in the unmodulated epilayer should be in the order of 10 1 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. ' 2 % 2 N N N N H - N? A I I? = H H E A H I?? A J H = J E J # I J A = @ O I J = J A N N J " 2 N N H # J J!! J A N? A I I? = H H E A H I * = I A @ F E C F H B E A N N 9 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, L DUT is the parasitic inductance of the diode, L is the inductance of the circuit that can be considered as a constant current source, S 1 is the switch, and V S is the supply voltage. During the first phase (0, t 0 ) 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 t 0 ). The voltage drop on the diode has its steady state value corresponding to the conduction current density through the diode. At the time instant t 0 the switch in the circuit is closed, and the reverse recovery takes place. From t 0 until t 3 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 (t 0, t 3 ) 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 t 1, t 2 and t 3 in Fig. 1.10b. At the time instant t 3, 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 (t 3, t 4 ), which lasts until the instant time when current through the diode reaches its peak negative value. Because of the depleting charge 25