Modeling and Simulation of Power PiN Diodes within SPICE · Modeling and Simulation of Power PiN Diodes within SPICE Gustavo Buiatti Direttore del corso di dottorato Prof. Carlo Naldi

POLITECNICO DI TORINO

Facolta di IngegneriaCorso di Dottorato in Dispositivi Elettronici

Tesi di Dottorato

Modeling and Simulation of PowerPiN 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 humansupport. 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 contributionsimprove the quality of the results.I wish to kindly thank Professor Jose Roberto Camacho, from Federal University ofUberlandia, Brazil, for his support, ideas and fruitful discussions during my researchperiod in that institution and even in Italy. Professor Joao Batista Vieira Junior isalso acknowledged, especially for the support on his Power Electronics Laboratoryin the Federal University of Uberlandia, 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 timewe have been in Italy. Without you I would never finish this work. I’ll be alwaysgrateful for your love. To you, I dedicate this thesis.

1

Table of contents

1 Physics and Basic Equations of Power PiN Diode 31.1 The Ambipolar Diffusion Equation (ADE) . . . . . . . . . . . . . . . 51.2 Forward conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 The stationary forward behavior of the PiN diode . . . . . . . 91.2.2 End region recombination effect . . . . . . . . . . . . . . . . . 131.2.3 Carrier-carrier scattering . . . . . . . . . . . . . . . . . . . . . 181.2.4 Auger recombination . . . . . . . . . . . . . . . . . . . . . . . 191.2.5 Lifetime control . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3 Forward recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.4 Reverse recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Power PiN Diode Models for Circuit Simulations 302.1 An overview of PiN diode modeling . . . . . . . . . . . . . . . . . . . 312.2 Circuit simulator and model implementation . . . . . . . . . . . . . . 352.3 PiN diode models: different approaches to solve the ADE . . . . . . . 35

2.3.1 Analytical model: Laplace transform for solving the ADE . . . 362.3.2 Analytical model: Asymptotic Waveform Evaluation for solv-

ing the ADE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.3.3 Analytical model: Fourier based-solution to the ADE . . . . . 402.3.4 Hybrid model: Finite Element Method for solving the ADE . 442.3.5 Hybrid model: Finite Difference Method for solving the ADE . 48

3 Finite Difference Based Power PiN Diodes Modeling and Valida-tion 503.1 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.1 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . 533.3.2 Finite Difference Modeling of the Base Region . . . . . . . . . 54

3.4 The complete diode model . . . . . . . . . . . . . . . . . . . . . . . . 573.4.1 Voltage drop on the junctions . . . . . . . . . . . . . . . . . . 583.4.2 Voltage drop on the epilayer . . . . . . . . . . . . . . . . . . . 59

I

TABLE OF CONTENTS

3.4.3 Voltage drop on the space-charge regions . . . . . . . . . . . . 603.5 Model implementation within SPICE . . . . . . . . . . . . . . . . . . 613.6 Model results and Validation . . . . . . . . . . . . . . . . . . . . . . . 65

3.6.1 Comparison with the FEM based diode model . . . . . . . . . 653.6.2 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 areclosely related to these discrete devices performance. Their study, comprehensionand performance improvement is of major importance for the development of efficientpower electronics equipments.

The effort needed for assembling and experimenting a power electronics con-verter, even taking into account the simplest topology existent, takes to a strongmotivation in the search for tools that in a simple and reliable way can simulatethe operation of the semiconductors involved in the circuit, dependent on the vari-ous parameters of the load and control circuit, and that, in such a way, allows thecomparison 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 implementationof the final design. The limited use of prototypes results in a reduction of the timeneeded 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. Thesimulation of converters, with the inclusion of detailed characteristics of the bipolarpower semiconductors devices, by means of using a personal computer, allows anaccurate understanding of the design and increases the possibility of a working firstprototype 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 semi-conductor is taken into account, and we focus our attention on the semiconductorsbehavior considering this particular simplified situation. In the second option thewhole converter is simulated, but making use of simplified models for the semicon-ductors involved in the design. Both solutions are usually incompatible in terms ofintegration. Therefore, the development of designs in the field of power electronicscan be beneficiated if somehow we can simulate both the macroscopic aspect of theconverter, and the microscopic aspect of the commutations of the semiconductorsinvolved in the circuit.

Unfortunately, the models available in commercial circuit simulators are notsuited to model the actual behavior of bipolar power semiconductors, which are notsuited either for a study that intends to be physics-based.

1

So, the main goals of this work were to create a mathematical model applicablefor bipolar power semiconductors, that must be physics-based, capable to be imple-mented in any commercial circuit simulator and to reproduce reliable and accurateresults.

Many power devices have been proposed in the past years but the need of bet-ter models is always present since circuit models need to adapt to the demand ofadvanced CAD tools and to the increased computation power available.

Another reason for the development of new device models is that they are theresult of the trade-off between contradicting requirements such as low computationcomplexity and accuracy. A better trade-off between these requirements is alwaysdesirable and pushes the development of more efficient device models.

In this thesis the results obtained during the study and research period at hePolitecnico di Torino are reported. The attention is focused on the power PiNdiode, a power device as simple as essential in power systems, with emphasis tothe development of compact circuit models of this device, ideally suited for circuitsimulation.

Main characteristic of the model presented in this work are the low computationalpower needed and the accurate modeling of static characteristic and of forward andreverse recovery effects. The model is implemented for simulation and comparisonwith experimental data in Pspice simulator. However, the model can be handled byany other SPICE-based simulator.

The thesis is structured as follows. Chapter 1 is devoted to a general introduc-tion to the physics of power PiN diodes, aimed to highlight the main static anddynamic effects of the same, and to provide the background underlying the designand optimization of such devices.

In chapter 2 the main topics on power diode modeling are introduced, and dif-ferent techniques and models are presented in order to compare the same and clarifythis issue.

Chapter 3 focuses on a novel approach for modeling power PiN diodes. Thecomplete diode model is described and introduced, followed by its implementationwithin the Pspice circuit simulator. Circuit simulations of practical power circuitsare reported, and the model is validated against experimental and simulations usingdifferent diode models.

Finally, in chapter 4 the final conclusions are presented.

2

Chapter 1

Physics and Basic Equations ofPower PiN Diode

The PiN diode was one of the first semiconductor devices developed for power cir-cuit applications. It is the simplest semiconductor device present in every powerelectronics converter, as can be seen by the structure presented in Fig. 1.1. ThePiN diode is basically composed of three regions: the cathode, the epilayer and theanode. The cathode is a wide highly N doped region; the epilayer is a lightly Ndoped region, epitaxially grown over the cathode; the anode is a highly P dopedregion placed at the top of the epilayer.

The main difference between signal diodes (low power PN diodes) and powerPiN diodes is this additional sandwiched region, the epilayer, which allows the PiNdiode to block large negative voltages depending on its width and low doping. Thepresence of this region also has important effects on the diode’s direct characteristicand dynamic behavior.

Regarding the direct characteristic, the presence of the epilayer (which behavesas 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, holesand electrons injected from diode end regions (anode and cathode), and the resis-tance of the epilayer becomes very small, allowing the diode to carry a high currentdensity with limited voltage drop. If not flooded by the carriers, the epilayer ishighly resistive. So, the resistance of the epilayer depends on the carriers distribu-tion 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 ofthe epilayer is modulated according to its carriers distribution. Thus, when a PiNdiode is switched on with a high di/dt, it takes some time to reach the stationaryflooded state of the epilayer, and the voltage drop at a given current will initially behigher. 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 exceedits breakdown voltage.

N +P + N -

A n o d e C a t h o d e

1 0 1 3

1 0 1 5

1 0 1 7

1 0 1 9

Dopin

g [cm

-3 ]

Figure 1.1. PiN diode structure and doping example.

The second drawback appears when the diode is turned off, because the excesscarriers in the epilayer cannot disappear immediately, but it takes some time forthem to recombine and to be extracted. So, the device is not able to reach theblocking state if carriers stored in the epilayer have not been extracted. This resultsin the presence of a reverse current until the epilayer is free of excess carriers. Thiseffect is generally called reverse recovery and has unpleasant effects such as increaseof dynamic losses (the current also flows through the switches used in the circuit,adding to power dissipation and degrading their reliability), electromagnetic inter-ference (EMI), and limitation of maximum working frequency due to the increasedturn-off time.

All the effects mentioned above may be modeled through the basic equations ofthe 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 intodevice physics have resulted in a mathematical model of the operation of semicon-ductor devices. This model consists of a set of fundamental equations which linktogether the electrostatic potential and the carrier densities, with suitable boundaryconditions. These equations consist of Poisson’s equation, the continuity equationsand the transport equations. Poisson’s equation relates variations in electrostaticpotential to local charge densities. The continuity equations describe the way thatthe electron and hole densities evolve as a result of transport processes, generationand recombination processes. Poisson’s equation and the continuity equations havebeen derived from Maxwell’s laws.

The first set of equations considered here constitute the transport equations, alsocalled the current density equations:

Jn = qµnnE + qDn∂n

∂x= qµn(nE +

kT

q

∂n

∂x) (1.1)

Jp = qµppE − qDp∂p

∂x= qµp(pE − kT

q

∂p

∂x) (1.2)

J = Jn + Jp (1.3)

where eq. 1.1 is the expression for the electron current density, eq. 1.2 for thehole current density, and eq. 1.3 for the total conduction current density. In theequations above J is the total current density, Jn and Jp are the current densitiesof electrons and holes [A/cm2], Dn and Dp are the diffusion coefficients of electronsand holes [cm2/s], µn and µp are the mobilities of electrons and holes [cm2/V·s], nand p are the electrons and holes densities [cm−3], E is the electric field [V/cm], q isthe magnitude of electronic charge [C], k is the Boltzmann constant [J/K], and T isthe absolute temperature [K].

The second set of equations is composed by the continuity equations:

∂n

∂t=

1

q

∂Jn

∂x+ (Gn −Rn) (1.4)

∂p

∂t= −1

q

∂Jp

∂x+ (Gp −Rp) (1.5)

where (Rn − Gn) and (Rp − Gp) are the rate of recombination, also called U. Thestatistics of the recombination of electrons and holes in semiconductors via recom-bination centers, or the rate of recombination U, is given by the Shockley-Read-Hallequation [1], [3], [4], [5], considering a single level recombination center:

5


U =np− n2

i

τp0(n + n1) + τn0(p + p1)=

∆np0 + ∆pn0 + ∆n∆p

τp0(n0 + ∆n + n1) + τn0(p0 + ∆p + p1)(1.6)

where n = n0 + ∆n, p = p0 + ∆p and p0n0 = n2i , ni is the intrinsic density of charge

carriers [cm−3], n0 and p0 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], τn0 andτp0 are the electrons and holes minority carrier lifetimes in heavily doped P and Ntype silicon [s], and n1 and p1 are the equilibrium electron and hole densities whenthe Fermi level position coincides with the recombination level position in the bandgap [cm−3]. The two last ones are given by the expressions:

n1 = NC exp

(ER − EC

kT

)= ni exp

(−EFi − ER

kT

)(1.7)

and

p1 = NV exp

(EV − ER

kT

)= ni exp

(EFi − ER

kT

)(1.8)

where NC is the effective density of states in conduction band [cm−3], NV is theeffective density of states in valence band [cm−3], EC is the bottom of conductionband [eV], EV is the top of valence band [eV], EFi 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

dEdx

=ρ

εs

=q(p− n + ND −NA)

εs

(1.9)

where ρ is the space charge density [cm−3], εs is the semiconductor permittivity[F/cm], ND is the donor impurity density [cm−3], and NA is the acceptor impuritydensity [cm−3].

In principle, the equations above with appropriate boundary conditions have aunique solution. Because of the complexity of this set of equations, in most cases theequations are simplified with physical approximations before a solution is attempted,or they are solved using numerical methods as is the case of semiconductors devicesimulators [6], [7].

Power semiconductors are designed for high current densities. Power diodes areoften rated for current densities from 100 A/cm2 to 1000 A/cm2. For high currentdensities, electrons and holes concentrations in the epilayer are much higher thanbackground carrier concentration, since the epilayer is a lightly doped region, thatis the device works in the high injection level condition [4], [5], [8].

When the high injection level condition holds, hole and electrons concentra-tion 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 carriersconcentration at thermal equilibrium. Considering a type-N semiconductor underhigh injection condition:

∆n À n0 ≈ ND and ∆p À p0 ≈ n2i

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 theintrinsic Fermi level EFi, it means that n1 and p1 are of the same order of ni. Evenif the energy levels of the recombination centers are not located that close to theEFi, their order are much lower than the order of ∆n and ∆p, and eq. 1.6 becomes[8]:

U =∆n2

∆n(τp0 + τn0)=

∆n

τp0 + τn0

(1.11)

where the carrier lifetime is equal for both electrons and holes, and is equal tothe sum of low injection level electron and hole lifetimes. The same is called highinjection level lifetime:

τhl = τn0 + τp0 (1.12)

Always under the assumptions of high injection level and quasi-neutrality in theepilayer and eliminating the electric field in the current density equations (eq. 1.1and eq. 1.2):

Jn − qDn∂∆n

∂xqµn∆n

=Jp + qDp

∂∆n

∂xqµp∆n

(1.13)

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]):

Jn

Dn

− Jp

Dp

= 2q∂∆n

∂x(1.14)

With respect to the the continuity equations, eq. 1.4 and eq. 1.5 under the assump-tions already mentioned, they respectively become:

∂∆n

∂t=

1

q

∂Jn

∂x− ∆n

τhl

(1.15)

∂∆n

∂t= −1

q

∂Jp

∂x− ∆n

τhl

(1.16)

7


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

1

Dn

∂Jn

∂x− 1

Dp

∂Jp

∂x= 2q

∂2∆n

∂x2(1.17)

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

∂∆n

∂t= Da

∂2∆n

∂x2− ∆n

τhl

(1.18)

Considering eq. 1.10:

∂n

∂t= Da

∂2n

∂x2− n

τhl

(1.19)

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

Da =2DnDp

Dn + Dp

(1.20)

Eq. 1.19, the ADE, models the transient behavior in the epilayer and mustbe solved considering the boundary conditions obtained from eq. 1.14, taking intoaccount eq. 1.10:

∂n

∂x

∣∣∣∣x=xl

=Jn |x=xl

2qDn

− Jp |x=xl

2qDp

(1.21)

∂n

∂x

∣∣∣∣x=xr

=Jn |x=xr

2qDn

− Jp |x=xr

2qDp

(1.22)

where xl is the left border and xr is the right border of the region flooded with freecarriers.

Considering the width of the epilayer equal to W, the borders of the floodedregion are xl = 0 and xr = W if the carriers concentrations are higher than thedoping of the epilayer. During reverse recovery, the carrier concentration on theborders becomes equal to the doping of the epilayer, and the borders start movingmeaning that xl and xr do not coincide anymore with the physical borders of theepilayer (diode junctions).

1.2 Forward conduction

In this section PiN diode forward conduction basic equations are presented. Understeady 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 recombinationof minority carriers injected in the highly doped end regions. It is assumed that theequations 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 andAuger recombination will be neglected. Furthermore, high injection carrier lifetimewill 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), andVM (ohmic voltage drop).

N +P + N -


V P+ V N

+V M

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

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

1

Da

∂n

∂t=

∂2n

∂x2− n

L2a

(1.23)

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, inorder to simplify the solution. The solution for eq. 1.23 has the general form:

n(x) = C1 cosh

(x

La

)+ C2 sinh

(x

La

)(1.24)

and its derivative has the following form:

∂n

∂x=

C1

La

sinh

(x

La

)+

C2

La

cosh

(x

La

)(1.25)

The assumption that end region recombination is negligible means that end re-gions doping is much higher than epilayer doping, and therefore, the current densityis determined by recombination in the epilayer. So, end regions have unity injectionefficiency, and it may be assumed as a very good approximation that at the bordersof the highly doped regions, P+ and N+, the total current conduction is carried outonly by holes and electrons respectively. However, it is useful for analytical purpose

9


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

n ( - d ) n ( + d )

0Figure 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 higherthan 20 A/cm2 [9], where end region recombination cannot be neglected. So, underthe 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 fol-lowing equations are obtained substituting eq. 1.26 - 1.29, in eq. 1.21 and eq. 1.22respectively:

∂n

∂x

∣∣∣∣x=−d

= −Jp |x=−d

2qDp

= − J

2qDp

(1.30)

and

∂n

∂x

∣∣∣∣x=+d

=Jn |x=+d

2qDn

=J

2qDn

(1.31)

Finally, from eq. 1.25, eq. 1.30 and eq. 1.31, constants C1 and C2 are evaluatedand then substituted in eq. 1.24 leading to the solution, that is the concentrationdistribution inside the epilayer in the forward state related to the total currentdensity through the diode:

n(x) =Jτhl

2qLa

cosh

(x

La

)

sinh

(d

La

) −B ·sinh

(x

La

)

cosh

(d

La

)

(1.32)

where

10


B =(Dn −Dp)

(Dn + Dp)=

(µn − µp)

(µn + µp)(1.33)

is a measure for the inequality of the mobilities. If the mobilities are equal, only thefirst 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 threecomponents

VD = VP+ + VM + VN+ (1.34)

In order to relate current density to voltage drop on the diode, it is needed thecalculation of diode voltage drop components VP+ , VN+ , and VM as a function ofcurrent density.

Following the mass action law and theory of PN junction, it is possible to relateon-state voltage drop to carrier concentration in both junctions (junction law):

pn−

pn0−= exp

(qVP+

kT

)⇒ p(−d)ND

n2i

=n(−d)ND

n2i

= exp

(qVP+

kT

)(1.35)

nn−

nn−0

= exp

(qVN+

kT

)⇒ n(+d)

ND

= exp

(qVN+

kT

)(1.36)

where ND is the epilayer doping, pn− is the concentration of holes in the lightlydoped epilayer side of the anode junction (x=-d), pn0− is the concentration of holesin the same place at thermal equilibrium, nn− is the concentration of electrons inthe lightly doped epilayer side of the cathode junction (x=+d), and nn0− is theconcentration of electrons in the same place at thermal equilibrium.Thus,

VP+ =kT

qln

[n(−d)ND

n2i

](1.37)

VN+ =kT

qln

[n(+d)

ND

](1.38)

VP+ + VN+ =kT

qln

[n(+d)n(−d)

n2i

](1.39)

Calculating n(−d) and n(+d) through eq. 1.32 and substituting in eq. 1.39

VP+ + VN+ =kT

qln

τ 2hlJ

2

n2i (2qLa)2

1

tanh2

(d

La

) −B2 tanh2

(d

La

)

(1.40)

11


Considering eq. 1.34 and substituting in eq. 1.40

exp

(VD − VM

VT

)=

τ 2hlJ

2

n2i (2qLa)2

1

tanh2

(d

La

) −B2 tanh2

(d

La

) (1.41)

Rearranging eq. 1.41 the relation between density current and voltage drop onthe diode is given by:

J = 2niqDa

dF

(d

La

)exp

(VD

2VT

)(1.42)

with

F

(d

La

)=

d

La

tanh

(d

La

)[1−B2 tanh4

(d

La

)]− 12

exp

(− VM

2VT

)(1.43)

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

VM =

∫ x=+d

x=−d

E(x)dx (1.44)

The electric field is found by adding eq. 1.1 and eq. 1.2 to obtain the total currentdensity, and by resolving for E always under the high level injection condition:

J = qµnnE + qDn∂n

∂x+ qµpnE − qDp

∂n

∂x= qnE(µn + µp) + qVT

∂n

∂x(µn − µp) (1.45)

E =J

qn(µn + µp)− BVT

n

∂n

∂x(1.46)

Finally, considering eq. 1.32 and its derivative, and substituting them into eq. 1.46and then integrating the same (eq. 1.44), it is found that

VM = VT

8b

(1 + b)2

sinh

(d

La

)

√1−B2 tanh2

(d

La

) · arctan

(√1−B2 tanh2

(d

La

)·

· sinh

(d

La

)+ B · ln

1 + B2 tanh2

(d

La

)

1−B2 tanh2

(d

La

)

(1.47)

12


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 currentdensity. This is because an increase of current density proportionally increases car-rier concentration in the epilayer and hence provides a reduction of epilayer resistiv-ity which is proportional to current density, and the two effects cancel each other. Ifcarrier-carrier scattering are considered, for example, this is not true anymore, andVM 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 inthe P+ and in the N+ regions, was negligible. This assumption is quite restrictive,and useful only for analytical purpose. Since the minority carrier lifetime decreasesrapidly with increasing doping level, recombination in the end regions adds addi-tional components to the forward current, and so, at high current densities, thecurrent 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 theN+ 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

∂x

∣∣∣∣x=0

=Jn|x=0 − bJp|x=0

qDa(b + 1)(1.48)

∂n

∂x

∣∣∣∣x=W

=Jn|x=W − bJp|x=W

qDa(b + 1)(1.49)

A n o d e C a t h o d e0 W

n ( 0 ) n ( W )n P + ( 0 ) p N + ( W )

- W P + W N +

Figure 1.4. On-state carrier concentration in the epilayer and heavily dopedregions.

Considering eq. 1.3, eq. 1.48 and eq. 1.49 respectively become:

∂n

∂x

∣∣∣∣x=0

=−bJ

qDa(b + 1)+

Jn|x=0

qDa

(1.50)

∂n

∂x

∣∣∣∣x=W

=J

qDa(b + 1)− Jp|x=W

qDa

(1.51)

13


The current density due to injected carriers in the end regions can be derivedfrom low level injection theory because minority carrier density is far smaller thanthe high doping levels of the end regions.

With respect to the P+N− junction, it is assumed that the same is abrupt andthat the P+ region is uniformly doped. Considering the low injection level conditionof carriers injected into the anode, and the fact that all the injected electrons havealready recombined in the ohmic contact, since there are no excess electrons in theohmic contact, that is there is no voltage drop in the same, it is found through thecontinuity equation that in the neutral region [5], [10]:

nP+(x)− nP0+ =

[nP+(0)− nP0+ ] sinh

(WP+ + x

LnP+

)

sinh

(WP+

LnP+

) (1.52)

where nP+ is the density of electrons in the P+ region, nP0+ is the density of electronsin the P+ region at thermal equilibrium, WP+ is the width of the P+ region, andLnP+ is the minority carrier diffusion length in the P+ region. Eq. 1.52 aboverepresents the expression for the excess electrons injected in the anode.

Considering the fact that the density current of electrons in the anode may beconsidered only to the diffusion component, since the electric field is approximatelyzero and the electrons concentration is too low in the P+ region, meaning that theelectrons drift current is neglected:

Jn|x=0 = qDnP+

(∂nP+

∂x

)(1.53)

where DnP+ is the diffusion coefficient of electrons in the anode. Using quasi-equilibrium at the P+N− junction, the injected carrier concentrations on eitherside of the junction are related by:

pP+(0)

p(0)=

n(0)

nP+(0)(1.54)

Under low injection level in the P+ region, it is assumed that

pP+(0) = pP0+ (1.55)

where pP+ is the density of holes in the P+ region, and pP0+ is the density of holes inthe P+ region at thermal equilibrium. The injected electron concentration is relatedto the voltage across the anode junction:

nP+(0) = nP0+ exp

(VP+

VT

)(1.56)

Using the two expressions above in eq. 1.54:

14


n(0) p(0) = pP0+ nP0+ exp

(VP+

VT

)= n2

i exp

(VP+

VT

)(1.57)

Using the charge neutrality condition:

exp

(VP+

VT

)=

[n(0)

ni

]2

(1.58)

Substituting the derivative of eq. 1.52 in eq. 1.53:

Jn|x=0 =

qDnP+ nP0+ exp

(VP+

VT

)

LnP+ tanh

(WP+

LnP+

) =qDnP+ nP0+ n(0)2

LnP+ tanh

(WP+

LnP+

)n2

i

(1.59)

Finally, considering the mass action law:

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

with

hp =DnP+

LnP+ tanh

(WP+

LnP+

)NA

(1.61)

where hp is the emitter recombination coefficient in the P+ region [cm4/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

qDa(b + 1)+

hp 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 abruptand that the N+ region is uniformly doped. Considering the low injection levelcondition of carriers injected into the cathode, and the fact that all the injected holeshave already recombined in the ohmic contact, it is found through the continuityequation that in the neutral region [5], [10]:

pN+(x)− pN0+ =

[pN+(W )− pN0+ ] sinh

(x − WN+

LpN+

)

sinh

(W −WN+

LpN+

) (1.63)

where pN+ is the density of holes in the N+ region, pN0+ is the density of holes inthe N+ region at thermal equilibrium, WN+ is the width of the N+ region, and LpN+

15


is the minority carrier diffusion length in the N+ region. Eq. 1.63 above representsthe expression for the excess holes injected in the cathode.

Considering the fact that the density current of holes in the cathode may beconsidered only to the diffusion component, since the electric field is approximatelyzero and the holes concentration is too low in the N+ region, meaning that the holesdrift current is neglected:

Jp|x=W = qDpN+

(∂pN+

∂x

)(1.64)

where DpN+ is the diffusion coefficient of holes in the cathode. Using quasi-equilibriumat the N−N+ junction, the injected carrier concentrations on either side of the junc-tion are related by:

pN+(W )

p(W )=

n(W )

nN+(W )(1.65)

where nN+ is the density of electrons in the N+ region. With analogous assumptionsmade for the P+ region:

n(W )p(W ) = n2i exp

(VN+

VT

)(1.66)

Using the charge neutrality condition:

exp

(VN+

VT

)=

[n(W )

ni

]2

(1.67)

Substituting the derivative of eq. 1.63 in eq. 1.64:

Jp|x=W =

qDpN+ pn0+ exp

(VN+

VT

)

LpN+ tanh

(WN+

LpN+

) =qDpN+ pN0+ n(W )2

LpN+ tanh

(WN+

LpN+

)n2

i

(1.68)

Finally, considering the mass action law:

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

with

hn =DpN+

LpN+ tanh

(WN+

LpN+

)ND

(1.70)

where hn is the emitter recombination coefficient in the N+ region [cm4/s] [10].Substituting eq. 1.69 in eq. 1.51, the boundary condition for the N−N+ junction

is now given by

16


∂n

∂x

∣∣∣∣x=W

=J

qDa(b + 1)− hn n(W )2

Da(1.71)

which takes into account the recombination effect of carriers injected into the cath-ode.

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 theP+ and N+ regions the intrinsic carrier concentration arises in the same [5], [11]:

n2ie = n2

i · exp

(∆Eg

kT

)(1.72)

where nie is the intrinsic carrier concentrarion taking into account bandgap nar-rowing effect, and ∆Eg is the band gap narrowing due to the combined effects ofimpurity band formation, band tailing and screening , which is calculated by thefollowing formula [5]:

∆Eg =

(3q2

16πεs

) √q2 NI

εskT(1.73)

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 isobserved that the energy bandgap reduces with increasing doping concentration,and intrinsic carrier concentration increases on the other hand. Thus, consideringthe bandgap narrowing effects in the highly doped end regions of PiN diodes, it isfound that the emitter recombination coefficients hp and hn are increased by thefactor 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, consideringbandgap narrowing effect: PiN diode current conduction is dominated from epilayerrecombination for low current densities, while end region recombination dominatescurrent flow for high current conditions. In fact, for low current densitis, n(0) andn(W ) are small, currents 1.59 and 1.68 are negligible, epilayer carrier concentrationincreases linearly with current density following equation 1.32, and current increasesexponentially with forward voltage drop. If the ADE is solved numerically with theboundary conditions taking into account the end region recombination effect, for lowcurrent densities until 20 A/cm2 it is actually found that the carrier concentrationand also the voltage drop in the diode are almost the same for both solutions, withand without end region recombination, reinforcing the conclusion that the currentdensity 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 ofthe carrier concentration and the diode enters in the working region where endregion recombination rules current conduction. The forward voltage drop in this

17


case will increase more rapidly with increasing current density than exp(VD/VT ), asshown in Fig. 1.5, which was generated with the following physical and geometricalparameters: W = 50 µm, ND = 2·1014 cm−3, A = 0.04 cm2, τhl = 200 ns, hp = hn =1.5·10−14 cm4/s, Dn = 34.84 cm2/s, Dp = 12.82 cm2/s, T = 300 K. The differencebetween the two curves is due to the fact that in the case in which the end regionrecombination is neglected, the voltage drop in the junctions rules the exponentialbehavior of the current density (eq. 1.37 - 1.41). In fact, the carriers concentrationincreases linearly with the current density and the ohmic voltage drop in the epilayerdoes not depend on the current density. It can be explained by the fact that theresistance in the epilayer is inversely proportional to the carriers concentration,meaning that:

VM = Repi · ID ∝ 1

n· ID ∝ 1

ID

· ID ≈ constant

When end region recombination becomes dominant, at higher current densities, thecurrent increases with the square of carriers concentration. The voltage drop in thejunctions 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 junctionsare lower due to the reduced emitters efficiency. It can be observed in Fig. 1.6, whichwas generated using the same parameters of Fig. 1.5. However, the voltage dropin the epilayer is not independent of the conduction current anymore but increaseswith the square root of same:

VM = Repi · ID ∝ 1

n· ID ∝ 1√

ID

· ID ∝√

ID

In this case, the ohmic voltage drop in the epilayer has a significant contributionto the voltage drop, which for high current densities becomes much greater than thecase 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 onlyphenomenon responsible for the deviation of the forward voltage drop characteristicsfrom an exponential behavior, as predicted by eq. 1.42. Two additional phenomenaimpact the current conduction characteristics, being the carrier-carrier scatteringthe first one to be considered here.

Carrier-carrier scattering occurs in the epilayer at high current densities dueto the simultaneous presence of a high concentration of both electrons and holes.The greater probability of mutual Coulombic scattering causes a reduction in themobility and diffusion length for both carriers [12]. The reduction in diffusion lengthwith increasing current density produces a decrease in the conductivity modulationin the central portion of the epilayer, which in turn, combined with the reduction

18


0 . 7 0 . 7 5 0 . 8 0 . 8 5 0 . 9 0 . 9 5 11 0 1

1 0 2

1 0 3Co

nductio

n Curr

ent De

nsity

[A/cm

2 ]

F o r w a r d V o l t a g e D r o p [ V ]

W i t h E n d R e g i o n R e c o m b i n a t i o nN o E n d R e g i o n R e c o m b i n a t i o n

Figure 1.5. Effect of end region recombination on the forward conduction char-acteristics of a PiN diode.

of the mobilities, results in the increase of the epilayer resistivity with consequentincrease of diode voltage drop (see Fig. 1.7, which was generated with the followingphysical and geometrical parameters: W = 100 µm, ND = 2·1014 cm−3, A = 0.04cm2, τhl = 1 µs, hp = hn = 5·10−14 cm4/s, Dn = 34.84 cm2/s, Dp = 12.82 cm2/s, T= 300 K).

1.2.4 Auger recombination

The second phenomenon that also impacts the current conduction characteristics isthe Auger recombination [13].

The Auger recombination process occurs by the transfer of the energy releasedby the recombination of an electron-hole pair to a third particle that can be eitheran electron or a hole. This process becomes significant in heavily doped P and Ntype silicon, such as the end regions of power PiN diodes, and is also an importanteffect in determining recombination rates in lightly doped regions operating at highinjection levels during forward conduction, the epilayer of PiN diodes, because ofthe high concentration of holes and electrons injected into this region. In the case ofAuger recombination occurring at high injection levels, the Auger lifetime is givenby [5]:

19


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 01

1 . 5

2

2 . 5

3

3 . 5

4

4 . 5

5 x 1 0 1 6

D e p t h [ m m ]

Carrie

rs Conc

entrat

ion [c

m-3]

W i t h E n d R e g i o n R e c o m b i n a t i o nN o E n d R e g i o n R e c o m b i n a t i o n

Figure 1.6. Effect of end region recombination on the carriers concentration inthe epilayer under steady state condition for J = 100 [A/cm2].

τauger =1

CA · n2(1.74)

where n is the excess carrier concentration in the epilayer, and CA is the Augerrecombination coefficient, of the order of 10−31 cm6/s. There are similar expressionsfor the end regions, considering the majority carrier concentration.

It can be observed from eq. 1.74 that the Auger recombination lifetime decreaseswith the injected carrier concentration in the epilayer, and it begins to affect thecarriers distribution in the same. In addition to the Shockley-Read-Hall recombi-nation described by eq. 1.11, the rate of recombination must include the Augerrecombination process:

U =n

τeff

=

(1

τhl

+1

τauger

)n =

n

τhl

+ CA · n3 (1.75)

where τeff is the effective carrier lifetime, and 1/τeff = 1/τhl +1/τaug. The inclusionof the Auger recombination term reduces the effective carrier lifetime of holes andelectrons with increasing current density, and so the diffusion length, decreasing theconductivity modulation in the central portion of the epilayer, that is decreasing thestorage charge in the epilayer, and resulting in the increase of the epilayer resistivity

20


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:

Da∂2n

∂x2=

n

τhl

+ CA · n3 (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 de-crease of the minority carrier diffusion lengths (LnP+ , LpN+), and increasing therecombination currents in the end regions (increase of hp and hn), what means tosay, decreasing the emitters efficiency. In conclusion, the resistivity in the epilayeris further increased.

0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 1 . 3 1 . 41 0 1

1 0 2

1 0 3

Condu

ction C

urrent

Densi

ty [A

/cm2 ]

F o r w a r d V o l t a g e D r o p [ V ]

- N o E n d R e g i o n R e c o m b i n a t i o n- N o C a r r i e r - c a r r i e r S c a t t e r i n g- N o A u g e r R e c o m b i n a t i o n

- W i t h E n d R e g i o n R e c o m b i n a t i o n- N o C a r r i e r - c a r r i e r S c a t t e r i n g- N o A u g e r R e c o m b i n a t i o n

- W i t h E n d R e g i o n R e c o m b i n a t i o n- W t i h C a r r i e r - c a r r i e r S c a t t e r i n g- N o A u g e r R e c o m b i n a t i o n

- W i t h E n d R e g i o n R e c o m b i n a t i o n- W t i h C a r r i e r - c a r r i e r S c a t t e r i n g- W i t h A u g e r R e c o m b i n a t i o n

Figure 1.7. Effect of end region recombination, carrier-carrier scattering, andAuger 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 PiNdiode. Reduction of on state losses or increase of diode speed are achieved throughmodifications of carrier lifetime.

21


Commercially available diodes often use lifetime control techniques to reduce car-rier lifetime in the whole epilayer. Well known lifetime control techniques are goldand platinum doping and electron irradiation, which result in lifetime profiles ap-proximately uniform in the considered region. This is due to the fact that the metalsdiffuse so rapidly through the silicon that they are normally uniformly distributedacross the epilayer. Likewise, electron bombardment also creates recombinationcenters uniformly throughout the device structure. The first mentioned method in-volves the thermal diffusion of an impurity that exhibits deep levels in the energygap of silicon (gold or platinum). The second method is based upon the creationof lattice damage in the form of vacancies and interstitial atoms by bombardmentof the silicon wafers with high energy particles. Both methods are characterized bythe introduction of recombination centers in silicon, reducing carrier lifetime in thelightly doped region and providing decrease of turn-off time [11], [14]. A drawback ofthese techniques is the increase of on-state voltage drop. Moreover, the introductionof deep level recombination levels increases leakage current and results in a strongerinfluence 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 diodesoftness with a little worsening of on-state voltage drop. This technique results ina better trade-off curve than achieved with lifetime killing in the whole epilayerregion. Recent investigations show that the optimal position for the low-lifetimeregion is at the beginning of the epilayer on the anode side, while the optimal widthof the low lifetime region depends on the amount of lifetime reduction, while it isless dependent on the operating current of the device. Diode design using a reducedlifetime region not placed near the anode junction provides a worse behavior withrespect to lifetime killing in the whole epilayer [17].

During the development of the equations regarding the behavior of PiN diode, ithas been always assumed that the high-injection lifetime in the epilayer is constantand given by eq. 1.12. This assumption was made in order to simplify the modeland equations. Actually, lifetime depends on the injection-level and also on thecapture cross sections of the recombination centers. Further, when spatially selectivetechniques are able to control carrier lifetime locally, the same becomes a functionof the position:

τhl = τhl(x) (1.77)

In this hypotesis the ADE becomes:

1

Da

∂n

∂t=

∂2n

∂x2− n

L2a(x)

(1.78)

which has no closed form solution. In this case eq. 1.78 must be solved numericallyusing appropriate techniques.

22

1.3– Forward recovery

1.3 Forward recovery

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

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 00

1

2

3

4

5

6

T i m e [ n s ]

Diode

Voltag

e [V]

F o r w a r d R e c o v e r y

D i o d e C u r r e n t [ 1 0 0 A / c m 2 ]

D i o d e V o l t a g e

T u r n - o n d i / d t

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:

Repi =W

q µn ND

[Ω · cm2] (1.79)

where W is the width of the epilayer, and ND is the epilayer doping. In orderto clarify the order of the unmodulated region resistance, let us consider a general

23


diode with an epilayer 50 µm wide and with doping ND = 1014 cm−3. One finds thatits resistance is of the order of 10−1 Ω · cm2, which means that for forward densitycurrents equal to 100 A/cm2, the voltage drop in the unmodulated epilayer shouldbe in the order of 101 V.

This example makes clear that if a PiN diode is forced in the conducting statewith a high di/dt, meaning that the current is increasing in a faster rate than therate of carriers being injected into the epilayer, transient voltage drop will be muchgreater than steady stage voltage drop. This is due to the fact that during thefirst instants, when the epilayer is not modulated, its resistance is very high. Thisvoltage overshoot due to the fast switch from the blocking state to the conductionstate through the forcing of a direct current, is called forward recovery. The voltagepeak increases with increasing di/dt, and its value depends on how high the currenthas risen before conductivity modulation is fully effective.

In Fig. 1.8 an example of PiN diode forward recovery is shown, which thesimulated diode is the same used for generating Fig. 1.5. It can be observed thatdiode voltage reaches about 5 V while steady state on-state voltage drop is about1 V. Fig. 1.9 shows the behavior of excess carriers in the epilayer when the diodeis turned on from zero current. It can be observed that excess carriers are initiallyinjected 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 steadystate value.

N +P + N -

1 0 1 3

1 0 1 5

1 0 1 7

1 0 1 9

x = 0 x = W

B a s e d o p i n g p r o f i l e

E x c e s s c a r r i e r s c o n c e n t r a t i o nJ JJ P | x = x l

J P | x = x rJ n | x = x l

J n | x = x r

t 1 t 3

t 5 = s t e a d y s t a t e

t 2

t 4

t 0 = n o e x c e s s c a r r i e r s

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

24

1.4– Reverse recovery

1.4 Reverse recovery

A major limitation to the performance of PiN diodes at high frequencies is theloss that occurs during switching from the on-state to the off-state, which have asignificant effect on the maximum operating frequency. During the reverse recovery,the charge stored in the epilayer during forward conduction must be removed. As canbe seen in Fig. 1.10a, a large reverse transient current occurs in PiN diodes duringreverse recovery. Since the voltage across the diode is also large following the peakin the reverse current, a large power dissipation occurs in the diode. In addition, thepeak reverse current adds to the average current flowing through the switches thatare controlling the current flow in the circuit. This not only produces an increasein the power dissipation in the switches, but also creates a high internal stressdegrading their reliability. Moreover, reverse recovery also causes EMI phenomena.

In the following the different phases of the reverse recovery are described, regard-ing Fig. 1.10. The widely used diode test circuit in Fig. 1.11 is used for a betterunderstanding 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 canbe considered as a constant current source, S1 is the switch, and VS is the supplyvoltage.

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

At the time instant t0 the switch in the circuit is closed, and the reverse recoverytakes place. From t0 until t3 the current through the diode is determined by theexternal circuit conditions and decreases with a constant di/dt, the so called turn-offdi/dt. Hence, the charge profile in the epilayer during this phase is such that it isable to support an increase in current, in the reverse direction. As far as the diode isable to support this increasing current at a certain di/dt, there will be just a smallforward voltage drop across the diode, which is determined primarily by the chargeprofile within the epilayer. The diode is still forward biased. During this secondphase (t0, t3) the injected carriers in the epilayer are extracted from the diode, bydiffusion and recombination, and there is a change in the slope of the injected carrierprofile near the two junctions. This slope changes its sign due to the reversal in thecurrent 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 outas reverse current, the carriers concentration at the P+N− junction reaches the levelsof 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 thebeginning of the third phase (t3, t4), which lasts until the instant time when currentthrough the diode reaches its peak negative value. Because of the depleting charge

25


T u r n - o f f d i / d t

R e v e r s eR e c o v e r y d i / d t- 5 0

- 4 0

- 3 0

- 2 0

- 1 0

0

1 0

2 0

3 0 t 0Dio

de Cu

rrent

Densi

ty [ A

/cm2 ]

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 0T i m e [ n s ]

- 1 4 0

- 1 2 0

- 1 0 0

- 8 0

- 6 0

- 4 0

- 2 0

0

2 0

Diode V

oltage [

V]

t 1

t 2

t 3t 4

t 5

t 6

N +P + N -

1 0 1 3

1 0 1 5

1 0 1 7

1 0 1 9

x = 0 x = W

B a s e d o p i n g p r o f i l e

E x c e s s c a r r i e r s c o n c e n t r a t i o nJ JJ P | x = x l

J P | x = x rJ n | x = x l

J n | x = x rt 0t 1t 3

t 4 t 5

( b )

( a )

t 2

t 6

Figure 1.10. PiN diode reverse recovery: (a) Reverse recovery current waveform,(b) Dynamics of carrier concentration in the epilayer during reverse recovery

26


S 1

LL D U T

D U T

V S

Figure 1.11. Circuit used to emulate diode switching

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

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

27


initially taking over the L inductor current and then drawing reverse current out ofthe diode, as the space charge layer is established in the diode the reverse voltagesettles at the supply voltage with no significant overshoot. We call the attention tothe fact that the reverse diode current adds to the total current carried by S1 andcauses a transient peak, as already mentioned.

The last phase of the recovery starts at time t5 and lasts until the moment inwhich the current reaches its saturation value. If there is a residual amount of excesscharge present in the epilayer from this instant time recombination dominates theexcess carrier absorption, resulting in a slow tail in the current waveform (see currentwaveform and excess charge during time sample t6 in Fig. 1.10). This last phaseof the recovery is very critical, and some considerations must be done. The firstconsideration is with respect to the applied reverse voltage. If the applied reversebias voltage is small, the space charge region will extend only slightly inside theepilayer. As a result, there will still be a lot of excess carriers remaining in theepilayer. 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 sameconditions, the recombination dominated regime will be quite prominent causinga significant tail near the end of the reverse recovery process [20]. This kind ofrecovery is the so called soft recovery, and a behavior like this is desirable for powerelectronics applications.

The second and more serious consideration regards to the fact that at timeinstant t5, it is possible that the depletion regions can advance through the wholeepilayer and the current that is still through the diode cannot be supported by anyexcess charge. This is the classic snappy recovery. Depending on the rate thatexcess carriers are extracted from the epilayer, the current goes rapidly to zerowith very high reverse recovery di/dt because a stronger depletion from both sideshappens before the current is ceased, resulting in oscillation. The snappy recoveryis detrimental to the diode, as it increases the chance of its destruction due to theexcessive electric field strength. Furthermore, the large-amplitude high-frequencyoscillations cause excessive amounts of electromagnetic interference (EMI).

When the switching frequency of a power circuit increases, the turn-off di/dtmust be increased. It has been found that this causes an increase in both the peakreverse recovery current and the ensuing di/dt, which in turn results in less recoverytime. If the reverse recovery di/dt is large, an increase in the breakdown voltageof all the circuit components becomes essential, since the reverse recovery di/dtflows through parasitic inductances in the circuit causing the already mentionedvoltage overshoot on the diode. Raising the breakdown voltage capability causesan increase in the forward voltage drop of power switches, which degrades circuitefficiency. Consequently, much of the recent work on PiN diodes has been focusedupon improving the reverse recovery characteristics.

However, a trade-off between the switching speed and the forward voltage dropis essential during PiN design. This trade-off is dependent upon a number of factors

28


such as the epilayer width, the recombination center position in the energy gap, thedistribution of the deep level impurities, and the doping profile. In section 1.2.2 itwas found that end region recombination results in an increase of voltage drop withrespect to eq. 1.42, and it could be assumed that a careful design should reducethis effect (increasing the emitter efficiency, that is reducing hp and hn). It wouldassure that also under high current conditions, end region recombination is smallwith respect to epilayer recombination. This assumption is not correct. Actualdevices tend to increase end region recombination effects, that is, to reduce endregion emitter efficiency (increasing hp and/or hn), since the increase of end regionrecombination results in an improvement of dynamic behavior [11], [18], [19]. Thisimprovement 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 peakand faster reverse recovery, which is desirable in order to reduce switching powerlosses. It can be achieved through techniques like the lifetime control techniquesdescribed in section 1.2.5. From eq. 1.61 and eq. 1.70 it can be observed that inorder to increase hp and hn, that is to reduce the emitters efficiency, there are twoother design techniques: the reduction of end region doping (increase of minoritycarrier equilibrium concentration p+

N0 and n+P0) [11], [18], [19], and the reduction

of end region thickness (WP+ , WN+). It can be noticed that the reduction of endregion 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 forCircuit Simulations

As computer aided simulation (CAD) has become essential for power electronicscircuit development, special compact power PiN diode models must be available inthe circuit simulators used for electronic circuit analysis. In this chapter differentPiN diode models present in literature are introduced and discussed.

The main difficulty in designing models for PiN diodes, and also for other bipolarpower devices, is in the distributed nature of the charge transport in the semicon-ductor, which is governed by the ambipolar diffusion equation (ADE). Therefore, thecentral part of this chapter describes problems and different solutions in treatmentof the ADE.

The different modelling methods are compared as to their compromise betweenconvenience, accuracy, numerical efficiency and accuracy in implementing physicaleffects. Many of the existing PiN diode models are listed, and their main featuresare 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 powerelectronics researchers put more attention to the switching losses. An accurate modelof power PiN diodes is very useful for computer aided simulation (CAD), since it isvery helpful for evaluating the loss distributions of power electronics converters andmaking design and optimization.

The most fundamental aspects that distinguish among different power diodemodels is the model formulation technique and concept employed. The differentmodels can be classified either as micromodels, or as macromodels. Micromodelsare closely based on the internal device physics and yields good accuracy over a widerange of operating conditions. Because device physics unavoidably require mathe-matical equations, micromodels are also known as mathematical models. Macromod-els reproduce the external behavior of the device largely by using empirical tech-niques without considering its geometrical nature and its internal physical processes.This external behavior is usually modeled by means of simple data-fitting empiri-cal equations, lookup tables, or an electrical subcircuit of common components toemulate known experimental data [21], [22], [23]. Macromodels are limited in termsof accuracy and flexibility an they will not be treated here. They are not derivedfrom the fundamental device equations, and as a consequence they are valid only ina narrow range of operating conditions, requiring non physical parameters. On theother hand, micromodels are generally more computationally efficient, more accu-rate, and more related to the device structure and fabrication process. Micromodelsare classified as numerical models, analytical models, and hybrid models.

Numerical micromodels use the partial differential equation set of the semicon-ductor physics (eq. 1.1 to eq. 1.9). Since this set of nonlinear partial differentialequations has no closed mathematical solution, they have to be solved numericallyby inserting a geometric mesh and solving the equations step by step by the finite-element or finite-difference methods. These equations describe the physical phenom-ena within the semiconductor, consisting of carrier drift and diffusion components,carrier generation and recombination effects, and the relationship between spacecharge and electric field. The solution of the whole system of basic equations isdone 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, ow-ing 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 onesingle semiconductor device is present. Although this may suffice during the devicedesign and technological development stages, such techniques are unable altogetherto cope with the simulation of realistic power circuits and are therefore of little usein their design, where compact models must be used. From the engineering pointof view, the degree of accuracy that is achieved by an exact numerical model is notalways 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 compactmodels may suffice, being obtained through some assumptions and simplificationsin the semiconductor physics as done in chapter 1.

Analytical micromodels rely on a set of mathematical functions to describe thedevice’s terminal characteristics without resorting to numerical methods. An exam-ple is the standard diode model packaged in SPICE. Standard SPICE-type modelswere designed for microelectronic devices, and they poorly describe the dynamicand static behavior of power devices [24]. The standard SPICE models use a simplecharge control method and lack the important physical effects in power devices. Thesoft recovery of the power diode cannot be simulated by the SPICE diode model,leading to erroneous predictions of switching power dissipation. Another drawbackof 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 thefailures of the SPICE model, some important effects must be considered when deal-ing with PiN diodes: mainly the conductivity modulation and the charge storage inthe epilayer.

The resistance of the epilayer is variable and its dependence on voltage or currentcan be highly nonlinear. In the PiN diode, this low-doped layer is swamped byelectrons and holes when the device is in its on state. The density of the injectedcharge carriers can be much higher than the level of the doping concentration, andthe resistivity of the region is significantly reduced. The resistance of a region withthe boundaries xl and xr and the area A is given by:

Repi =

∫ xr

xl

dx

qA(µn n + µp p)=

∫ xr

xl

dx

qA(µn + µp)p(2.1)

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 distributedhomogeneously, and their density depends on position; the mobilities also cannotbe regarded as constants, since it depends on the carrier concentration in the epi-layer. During transient operation, the variation of the resistivity does not follow thechanging current instantaneously − this effect can influence the switching behav-ior (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 solutionof the time-dependent charge densities is found, the calculation of the resistanceremains difficult since the integration in eq. 2.1 is not possible without simplifica-tions. However, accurate solutions for the resistance can be achieved through somesimplifications.

The charge carriers, which are stored in the lightly doped region of the PiNdiode during the conduction mode, must be extracted before the device can reachits blocking state. This effect causes switching delays and switching energy losses.Standard device models for circuit simulation use a quasi-static description of thecharge 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. Thismethod is completely insufficient for power devices as said before. A real dynamicdescription derived from the basic physical equations is required instead. The chargestored in the epilayer of the power PiN diode varies, under transient operation, withboth time and position. This variation is determined by the ambipolar diffusionequation (ADE):

∂p

∂t= Da

∂2p

∂x2− p

τhl

(2.2)

where p(x,t) is the density of the charge carriers, τhl is the high-injection level carrierlifetime, and Da is the ambipolar diffusion coefficient. This equation is valid in thecase of high-level injection when hole and electron densities are approximately equal.

Unfortunately, an exact analytical solution of the ADE is not possible in thegeneral case. However, there are different analytical approaches to solve the ADEas the Laplace transformation method [26], [27], [28], the Asymptotic WaveformEvaluation 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 an-alytical models are based on the modification of the standard diode model to allowdynamic diode characteristics to be predicted [35], [36], [37], [38], as the lumped-charge approach in which the excess carrier distribution profile is discretized intoseveral critical regions, each containing a lumped charge node to represent dynamiccharge variation [39], [40], [41], [42]. Like numerical modeling, the limits to simula-tion accuracy are more due to the accuracy of the input parameters rather than dueto the models themselves. The computational overheads of analytical models are farlower than those of numerical models. In addition, there is a large pool of popularcommercial simulators, such as SPICE and Saber, the solver algorithms of whichhave been evolved to solve these types of models most efficiently. Having powerdevice models in the libraries of these simulators allows the latter to function asgeneral purpose power electronics circuits CAD tools. Analytical models are, thus,very appropriate for simulation of power electronic circuits over a large number ofswitching cycles.

A third type of micromodel formulation technique is to use a combination ofnumerical and analytical models, and they are called the hybrid models. The moti-vation behind such hybrid model arises from the fact that certain physical phenom-ena in power devices are very difficult to simulate realistically using only analyticalequations, particularly the charge storage effects in the epilayer. The basic idea ofthis method is to use a fast numerical algorithm that solves the semiconductor equa-tions in the epilayer only. Analytical equations are applied to the rest of the devicestructure. This procedure has the advantage that a high accuracy of the chargecarrier behavior may be simulated without the long execution time associated withfully numerical models. The semiconductor equations in the epilayer, that is the

33


ADE, are solved using the finite element method [43], [44], [45] or the finite differ-ence method [46], [47], [48], [49] and will be discussed with more details in section2.3.

Another approach is the approximate solution, where the model equations arebased on the device physics, but since exact solutions are not possible or restrictedto a few special cases, appropriate mathematical representations are found to ap-proximate the solution. These approaches are purely empirical in many cases, butit is also possible to show that some functions come close to an exact solution un-der certain constraints of the boundary conditions. The approximations appliedto 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 fromtheoretical considerations or numerical calculations (device simulators [6], [7]).

Several papers can be found discussing the different techniques of modeling powerdiodes 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 marketas SPICE-based circuit simulators, such as Pspice [58], and Saber [59]. These sim-ulation programs differ in particular in the possibilities and methods they providefor the implementation of new device models.

In SPICE-based simulators, the insertion of mathematical equations (differentialequations and implicit functions) is based on subcircuits of user-defined controlled E-type voltage sources and G-type current sources, combined with passive componentslike resistors, capacitors, inductances, active components like conventional diodes,MOSFETs, bipolar transistors, and general elements like controlled switches, currentand voltage sources.

On the other hand, Saber offers the individual definition of device models bydescribing them in mathematical form in a special description language (MAST) orby writing a program in a general-purpose programming language (C or Fortran).Hybrid micromodels involve many mathematical equations to solve the numericalportion of the model, and simulators that provide powerful simulation languagessuch as Saber are often used for this type of models. However, subcircuit form ofimplementation is becoming very popular as the capabilities of the personal com-puter 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 introducedand briefly discussed. First, some analytical models are presented, and then somehybrid 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], wherethe mobilities are dependent of carrier concentration and temperature, the othercompact 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 ofinjection;

• The carriers density in the epilayer is much larger than the doping density(high injection).

35


2.3.1 Analytical model: Laplace transform for solving theADE

This approach was applied by Strollo [26], [27], [28], where the epilayer is representedwith a two-port network obtained by solving the ADE with the Laplace transformmethod, 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 firststep is to convert the ADE into the s-domain with respect to time (capital lettersare used to indicate L-transformed quantities):

Da∂2P (x,s)

∂x2=

(1

τhl

+ s

)P (x,s) (2.3)

From the general solution of eq. 2.3 and substituting P(0,s) and P(W,s) on it, theresulting equations can be interpreted as the equations of a two-port network asshown in Fig. 2.1 in which node voltages correspond to carrier concentration whileinput 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 repre-sented by two nonlinear current generators, GL and GR, controlled by the currentthrough the diode, p(0) and p(W).

p ( 0 ) p ( W )

G L G R

L m R m

Z m

Y p Y p

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

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

e l e m e n t

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 withthe first two terms of its Taylor series and in this way is represented by the seriesof 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 areobtained by using the Pade approximation. After some algebraic manipulations,the approximate representation for the admittance Yp is shown in Fig. 2.1. All thepassive components in the two-port network are dependent of the epilayer widthW, which is kept unchangeable during the simulations. More accurate results canbe obtained by the insertion of more elements like the one in Fig. 2.1 connected inseries [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 approximationsused 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 equivalentcircuit representing the solution of the ADE, the voltage drops in the junctionsare calculated through eq. 1.37 and eq. 1.38. The ohmic component is evaluatedtaking into account carrier-carrier scattering through a series resistance, and by anaverage value of carrier concentration for each element of the epilayer, where thereis a factor that must be obtained by curve fitting through the voltage overshootforward recovery measurements. In this approach only the voltage drop due to thespace-charge region in the left junction (P+N− junction) is considered, since it istrue that almost all the diode voltage drops in this region. The width of the leftspace-charge region is obtained by following the analytical approach proposed in[50], leading to a complex subcircuit where an ideal switch and other controlledgenerators using small look-up tables are present (these generators are used as flagsto indicate the beginning of the reverse-recovery phase and also to clamp to zerothe voltages in the network representing the epilayer), since negative values have nophysical sense. This complex implementation results in convergence problems formore than one switch simulation. The voltage drop in the space-charge region iscalculated by:

VSC =

(I

2Aευs

+qND

2ε

)x2

SC (2.4)

where xSC is the the width of the left space-charge region and υs is the hole saturateddrift 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 thesubcircuit are directly obtained from the geometrical and physical parameters of thedevice (epilayer doping and width, high injection lifetime in the same, width anddoping of the emitters).

37


2.3.2 Analytical model: Asymptotic Waveform Evaluationfor solving the ADE

This approach was also applied by Strollo [29], [30] and is an extension of the previousmodel, since the starting point of the model is to solve the ADE using the Laplacetransform. However, in this approach instead of using the boundary condition inthe N−N+ junction (eq. 1.71), the following condition is imposed:

∂p

∂x(xm,t) = 0 (2.5)

where W/2 < xm < W, and to simplify the modeling, xm is assumed to be aconstant abscissa. This is only an approximation however. Strickly speaking, xm isa constant only in the idealized condition of equal electron and hole mobility andequal 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, anddoes not significantly affect the accuracy of the overall SPICE model. The boundarycondition in the P+N− junction (eq. 1.62) is kept unchanged, but some additionalvariables are introduced in the same. Finally, the solution of the ADE through theLaplace transform with respect to time with the new boundary conditions leads to(capital letters are used to indicate L-transformed quantities):

I1(s) = Q0(s)1

τhl

La

xm

√1 + sτhl tanh

(xm

La

√1 + sτhl

)(2.6)

Thus, a continued-fraction expression in the s-domain of the carrier distributionin the epilayer is obtained (following the Asymptotic Waveform Evaluation theory[61]), and by truncating the following continued-fraction expansion, lumped RCrepresentations of the epilayer are easily obtained (as seen in Fig. 2.2):

I1(s)

Q0(s)=

1

Z0 +1

3

T0

+1

5Z0 +1

7

T0

+ . . .

(2.7)

where:

Z0 =τhl

1 + sτhl

(2.8)

T0 =x2

m

Da

(2.9)

where i1 is the component of the total current due to the carrier injection in theepilayer, and q0 is the amount of excess charge in the left junction.

38


+

t h l 5 t h l 9 t h l

1 1 / 5 1 / 9T 03

T 07

T 01 1

q 0

i 1

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

The model was implemented in the Pspice simulator as a subcircuit, and it wasshown 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-fractionexpansion (first and second order respectively) while more accurate models can beobtained from higher order approximations of the continued-fraction expansion.

This model takes into account emitter recombination effect in the highly dopedend regions, conductivity modulation in the epilayer, carrier-carrier scattering andthe moving-boundaries effect during reverse recovery. It requires a total of 13 inputparameters, with 8 more coefficients in addition to the standard SPICE diode pa-rameters. All the parameters must be extracted by fitting DC measurements (mostof the parameters), and from the forward and reverse recovery results. It is donethrough a stochastic global optimization algorithm in an automatic fashion [62],[63]. However, the implementation of latter seems to be very complex and not thatclear. The model shows good convergence properties and fast simulation times, andseems to apply to PiN diode at all conditions.

39


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 throughthe Fourier based-solution. Many other authors adopted this kind of solution forsolving the ADE, including temperature dependence in their models [64], [65], [66].Again the solution can be implemented by an electrical analogy. The original workdeveloped by Leturcq [31], where the high-level lifetime of carriers is consideredconstant in the whole epilayer, is treated here.

It was shown in [67] that a discrete cosine Fourier transform of p(x,t):

p(x,t) = V0(t) +∞∑

k=1

Vk(t) cos

[kπ(x− xl)

xr − xl

](2.10)

allows the ADE (eq. 2.2) to be converted into an infinite system of first order lineardifferential equations for the series coefficients V0 . . . Vk:

(x2 − x1)

(∂V0

∂t+

V0

τhl

)= Da

[∂p

∂x

∣∣∣∣xr

− ∂p

∂x

∣∣∣∣xl

]− I0 (2.11)

with:

I0 =∞∑

n=1

Vn

(∂xl

∂t− (−1)n ∂xr

∂t

)(2.12)

(xr − xl)

2

(∂Vk

∂t+ Vk

[1

τhl

+Dak

2π2

(xr − xl)2

])= Da

[(−1)k ∂p

∂x

∣∣∣∣xr

− ∂p

∂x

∣∣∣∣xl

]−Ik (2.13)

with:

Ik =Vk

4

∂(xl − xr)

∂t+

∞∑

n=1, n6=k

n2Vn

n2 − k2

(∂x1

∂t− (−1)k+n ∂x2

∂t

)(2.14)

This set of equations can further be represented, by way of a simple electricalanalogy, in the form of two RC lines wich correspond to even and odd values ofk, the voltages across the successive cells representing the Fourier series coefficientsVk(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):

for k = 0,

C0 = xr − xl

R0 =τhl

xr − xl

(2.15)

40


I e v e n

R 0

C 0

I 0

R 2

C 2

I 2

E V E N L I N ER a d d ( e v e n )

V 0 ( t ) V 2 ( t )

I o d d

R 1

C 1

I 1

R 3

C 3

I 3

O D D L I N ER a d d ( o d d )

V 1 ( t ) V 3 ( t )

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

for k 6= 0,

Ck =xr − xl

2

Rk =2

xr − xl

1

1

τhl

+k2π2Da

(xr − xl)2

(2.16)

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

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

Ieven = Da

[∂p

∂x

∣∣∣∣xr

− ∂p

∂x

∣∣∣∣xl

](2.17)

41


Iodd = −Da

[∂p

∂x

∣∣∣∣xr

+∂p

∂x

∣∣∣∣xl

](2.18)

Remembering the boundary conditions:

∂p

∂x

∣∣∣∣xl

=1

2qA

(In |xl

Dn

− Ip |xl

Dp

)(2.19)

∂p

∂x

∣∣∣∣xr

=1

2qA

(In |xr

Dn

− Ip |xr

Dp

)(2.20)

When the boundaries of the excess charge start moving, the following conditionmust 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 theends of the carrier storage region are fixed (xl=0 and xr=W). At reverse recoverythe boundaries become mobile as cleared out zones appear on. In this case, theboundary 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 typeshown in Fig. 2.6, the ideal diodes D enabling the continuity of the representation tobe maintained between the fixed and mobile boundary cases. However, the resistorsRxl

and Rxr must be adjusted in order to acquire the latter conditions, meaningthat they are additional heuristic parameters in the model, added to the physicaland geometrical parameters of the diode. It was also found that these resistors arealso 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-thtime-constant is much smaller than the characteristic times involved in the currentand voltage waveforms. The truncate error can be greatly reduced by terminatingthe lines by additional series resistances corresponding to the cumulated resistancevalues 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 theepilayer.

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

42


during reverse recovery. This component is added to the components due to therecombination in the highly doped end regions (In1 and Ip2), and they are involvedin 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.

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

I p 1

W - x r+I I p 2

I n 2

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 theRC 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 netsare all non linear elements, that depend on the width of the mentioned region.This fact takes to convergence problems, and higher number of elements neededfor implementing the RC net elements. For implementing a variable capacitor ina circuital way in some SPICE versions as IsSPice, as an example, four elementsare needed since these versions do not support the derivative command DDT as isthe case of the Pspice version, and the higher number of elements impacts on thecomputation 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 ef-fect. 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 dopingof the emitters) and the heuristic parameters already mentioned.

43


2.3.4 Hybrid model: Finite Element Method for solving theADE

In this model, Araujo solves 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 givesthe time and space charge carriers distribution in the epilayer. The implementationof the obtained model, in a general circuit simulator, is made by means of an elec-trical 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 ofa set of RC nets and current sources.

e l e m e n t

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

C s

R s

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

C s

R s

G L G R

1 2 n + 1n

Figure 2.6. RC equivalent circuit representation of the x -solution of the ADEthrough 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:

Π =

∫

V

[1

2

(∂p(x,t)

∂t

)2

+p(x,t)2

2Daτhl

]dV +

∫

V

[p(x,t)

Da

∂p(x,t)

∂t

]dV −

−∫

S

[(f(t) + g(t))p(x,t)] dS (2.22)

where f(t) is the left boundary condition (eq. 2.19), and g(t) is the right boundarycondition (eq. 2.20). The minimization of the functional above (eq. 2.22) is equiva-lent to the solution of the ADE (eq. 2.2) and its boundary conditions (eq. 2.19 andeq. 2.20).

Assuming that the solution of the unknown function, p(x,t), is approached by asum of elementar functions, the finite elements, of the form [68], [69]:

44


[p(e)] = [N(x)] [p(t)] (2.23)

being each element:

p(e) = N e1 (x) + . . . + N e

j (x) + . . . + N ek(x) (2.24)

and after substitution in eq. 2.22 and its minimization (δΠ = 0) we get the followingmatrix equation:

[M ]

[∂p(t)

∂t

]+ [K][p(t)] + [F (t)] = [0] (2.25)

where

[G(x)] =e=n∑e=1

[Ge(x)] (2.26)

[M(x)] =e=n∑e=1

[Me(x)] (2.27)

F (t) =e=n∑e=1

[Fe(x)] (2.28)

being each of the elementar matrices

Me =

∫

Ve

1

Da

[N(x)]T [N(x)] dV (2.29)

Ge =

∫

Ve

[B(x)]T [B(x)] dV +

∫

Ve

[N(x)]T [N(x)]

Daτhl

dV (2.30)

Fe = −∫

S(e)l

f(t)[N(x)]T dS −∫

S(e)r

g(t)[N(x)]T dS (2.31)

and

[B] =

[∂N e

1 (x)

∂x. . .

∂N ek(x)

∂x

](2.32)

[N ] = [N e1 (x) . . . N e

k(x)] (2.33)

This system of ODEs can now be coupled with differential equations for therest of the circuit and conveniently solved with any of the methods for this kindof equations [70], [71]. So, the obtained system of ODEs can be interpreted as acombination of RC nets and current sources that can be solved with the aid of ageneral circuit simulator as SPICE based simulators.

45


Once the type of elements is chosen, N and B are defined, the integrals in2.29, 2.30 and 2.31 can be solved, and the RC net constructed. It is used forimplementation a simplex approach for the elements, so we will have two nodes foreach element and, in local coordinates:

[N(s)] =

[1− s

∆xe

s

∆xe

](2.34)

[p(t)] =

[pe(t)

pe+1(t)

](2.35)

So

[B] =

[− 1

∆xe

1

∆xe

](2.36)

the product [B]T [B]:

[B]T [B] =

(1

∆xe

)2

−(

1

∆xe

)2

−(

1

∆xe

)2 (1

∆xe

)2

(2.37)

and [N(s)]T [N(s)]:

[N(s)]T [N(s)] =

(1− s

∆xe

)2s

∆xe

− s2

∆x2e

s

∆xe

− s2

∆x2e

(1− s

∆xe

)2

(2.38)

the integrals:

Me =Ae∆xe

6Da

[2 1

1 2

](2.39)

Ge =Ae

∆xe

[1 −1

−1 1

]+

Ae∆xe

6Daτhl

[2 1

1 2

](2.40)

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 elementarea.

46


The obtained matrices for each element are equivalent to the highlighted elementcircuit in Fig. 2.6, and the whole epilayer is represented by the series connectionof n single elements. The voltages at the nodes, V1(t), Vn+1(t), are equivalent tothe concentration, p(x,t), along the epilayer. The number of nodes is that of theelements 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 andGR = −g(t)An+1. the values of the other components needed for the implementationof the model are, according to Fig. 2.6:

Rp1 = Rp2 =2Daτhl

Ae∆xe

(2.42)

Rs =6Daτhl∆xe

Ae[6Daτhl −∆x2e]

(2.43)

Cp1 = Cp2 =Ae∆xe

2Da

(2.44)

Cs = =Ae∆xe

2Da

(2.45)

Again, we can note that the values of the elementar components of the RC net areall variable ones during recovery because the width of the epilayer, and so of theelements, varies in time.

This model takes into account emitter recombination effect in the end regions,conductivity modulation and moving boundaries effect. This work was originallyimplemented by the authors in the IsSpice software. It is assumed in this modelthat during the reverse recovery all voltage drops in the left space region. Thesame scheme used by Leturq [31] for calculating the borders of the epilayer due tothe moving boundaries effect is used in this model. As in the case of the previousmodel, the implementation of the whole model takes to the need of many nonlinearelements, as the case of the implementation of the variable capacitor, resultingin a large number of components and increased simulation times and convergenceproblems. We have implemented this model in the Pspice simulator as well, and theresults were not encouraging since there were lots of convergence problems duringthe simulation of very simple test circuits. Input parameters of the subcircuit, whenimplemented in circuital way in SPICE based simulators, are directly obtained fromthe geometrical and physical parameters of the device (epilayer doping and width,high injection lifetime, width and doping of the emitters) and the heuristic parametermentioned in the previous model, but in this case regarding only the circuit usedfor calculating the left border.

47


2.3.5 Hybrid model: Finite Difference Method for solvingthe ADE

The most accurate solutions of the ADE, regarding compact diode models, areobtained through numerical solutions of the same [54], which are based on thediscretization of the considered region into a finite number of mesh points. Twomethods can be distinguished. The first one was applied in the previous hybridmodel, and is known as the Finite Element Method (FEM). The second one is theFinite Difference Method (FDM), which has been applied most often. If the FDMis used, the derivatives in the ADE and in the boundary conditions are expressedby differences which may have the form:

∂p

∂x

∣∣∣∣i

=−pi+2 + 4pi+1 − 3pi

2∆x(2.46)

∂2p

∂x2

∣∣∣∣i

=pi+1 − 2pi + pi−1

∆x2(2.47)

where index “i” indicates the mesh-point number. Eq. 2.46 is an example of forwarddifference for the derivative in the space. However, there are other ways of repre-senting it, and also the backward and central differences. These differences can beobtained through Taylor series, and are different depending on the truncated termconsidered in the series [70], [72]. Time is also discretized:

∂p

∂t

∣∣∣∣j

=pj+1 − pj

∆t(2.48)

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 thiswork, the so called Enthalpy Method is used for solving the problem of the movingboundaries. When the region flooded with excess carriers detaches from the diodejunctions, the boundary conditions should be modified in order to account for themoving 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 followingproperties:

• u(x)=p(x)-kJR (where k is a constant parameter and JR is the reverse recoverycurrent density) within the carriers storage region;

• ∂2u

∂x2= 0 outside the carriers storage region.

The carrier concentration p outside the storage region is forced to zero. By usingthis approach it is possible to maintain the same boundary conditions for the whole

48


reverse recovery transient and directly obtain the position of the space charge movingboundaries. 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 theboundary conditions of the ADE are calculated from terminal current densities, thecircuit 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-carrierscattering and Auger recombination can be easily included, as is the case of themodel proposed by Vogler [49], [60]. Thereby, temperature dependent algebraicexpressions for scattering and recombination processes replace widely employed pa-rameters such as high injection lifetime, mean carrier concentration and averagemobility. In addition, the algorithm of these models can easily be understood andmodified, while the simple simulator equations take to no convergence problems andto an accurate solution of the ADE.

Another FDM based model is the model proposed by Goebel [48], [73], in whichconductivity modulation and the field dependence of carriers mobilities are included,while emitter recombination effects are not included. The latter was then includedin [74].

The only model similar to the FDM based models implemented in a circuital waywas 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 fixedmesh of the one-dimensional epilayer as explained above.

In all FDM based models, the input parameters are directly obtained from thegeometrical and physical parameters of the device. In particular, the model pre-sented by Profumo [47] takes into account only the space charge region in the leftjunction, 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 methodof finite differences. It can be regarded as a simplification to the greatest possibleextent with a minimal number of nodes. But in a lumped model, the average chargedensities of the sections instead of the densities at the nodes are inserted into theequations.

In the following, a novel Finite Difference based model of PiN diodes is presented.

49

Chapter 3

Finite Difference Based Power PiNDiodes Modeling and Validation

A physics-based model for PiN power diodes is developed and implemented as aSPICE subcircuit. The starting point of the model is an equivalent circuit represen-tation 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 highlydoped end regions, conductivity modulation in the epilayer, carrier-carrier scatteringand the moving boundaries effect during reverse recovery, showing good convergenceproperties and fast simulation times. Comparisons between the results of the SPICEmodel 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 mobilityDa ambipolar diffusion coefficient in the epilayer (cm2/s)Dn electron diffusion coefficient in the epilayer (cm2/s)Dp hole diffusion coefficient in the epilayer (cm2/s)ε dielectric constant of silicon (F/cm)hp emitter recombination coefficient in the highly doped P region (cm4/s)hn emitter recombination coefficient in the highly doped N region (cm4/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)Jn electron current density (A/cm2)Jp hole current density (A/cm2)k Boltzman constant (J/K)µ0 sum of electron and hole mobilities in the epilayer (cm2/V·s)µn electron mobility in the epilayer (cm2/V·s)µp hole mobility in the epilayer (cm2/V·s)ND epilayer doping (cm3)ni intrinsic carrier concentration (cm−3)p(x,t) hole concentration in the epilayer (cm−3)p0 coefficient describing carrier-carrier scattering in the base (cm3)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 semi-conductors. In the early design stages, these devices are often considered as binaryon-off switches, allowing vary fast computation of the circuit [75]. However, as thecurrent trend is towards reducing power converter size and increasing switching fre-quencies, the binary on-off representation of semiconductor devices has to be revised.This very simple device representation cannot take into account for the non idealbehaviors 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-diffusionsemiconductor equations, using the finite element or the finite difference methodfor example [6], [7]. In this way, device behavior can be modelled very preciselyin two or even three dimensions. Unfortunately a very precise physical model withhigh numerical complexity involves a very long computational time, making thisapproach more suited to individual device design and optimization, since its use iscomputationally prohibitive at system level modelling.

A physical compact modelling approach as the ones presented in the last chapterlies between these two extremes, and it seems to represent the ideal solution for powersemiconductor device representation within circuit simulators. This physic approachis based on certain mathematical simplifications of the fundamental semiconductorcharge 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 num-ber 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 thespace-discretization of the ADE in the base region provides the equivalent circuitmodel for the carriers concentration in the epilayer. The carriers concentration isthen related to the diode voltage by the junction and ohmic relationships underforward bias, and the Poisson equation under reverse bias operation, which are theanalytical contributions of this hybrid model. This results in an easy-to-implementdiode model as a SPICE subcircuit, which takes into account the emitter recombina-tion effects in the highly doped end regions, carrier-carrier scattering, conductivitymodulation, and the dynamic of the space-charge voltage build-up (moving bound-aries effect during reverse recovery). It is worth to outline that the FDM approachto 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 nonlinear elements, especially those based on SPICE. In this work, the commerciallyavailable circuit simulator Pspice was used [58], and the model was implementedas a Pspice subcircuit. This simulator was chosen since it is the standard referencecircuit simulator in the world [76]. The subcircuit is much simpler than some otherproposed subcircuit models described in the last chapter, resulting in faster simula-tion and improved convergence properties. Parameters for the model are all relatedto the physical and geometrical properties of the device.

Pspice model simulations are compared with experimental results and simu-lations using another compact diode model present in literature, and either withexperimental characterizations or with SILVACO mixed-mode module simulations[7] of a commercial PiN power diode [19]. A good agreement is obtained, with muchsmaller computation times of Pspice simulations than SILVACO ones. Finally, sim-ulation 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

N +P + N -I D

x = 0 x = WA n o d e C a t h o d e

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 highinjection, the carrier distribution is governed by the ADE [9]:

∂p

∂t= Da

∂2p

∂x2− p

τhl

(3.1)

where Da = 2DnDp/(Dn + Dp) is the ambipolar diffusion coefficient and τhl is thehigh-level lifetime. The boundary conditions for eq. 3.1 at x = xl and x = xr canbe written as:

∂p

∂x(xl,t) =

1

2qA

(In(x = xl)

Dn

− Ip(x = xl)

Dp

)(3.2)

∂p

∂x(xr,t) =

1

2qA

(In(x = xr)

Dn

− Ip(x = xr)

Dp

)(3.3)

53


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

ID = In + Ip + Idep (3.4)

In(x = xl) = q hp A p2(x=xl)

(3.5)

Ip(x = xr) = q hn A p2(x=xr) (3.6)

Idep = −q ND A∂xl

∂t(3.7)

where ID is the total diode current, Idep is an additional current component duringreverse recovery, due to the charge variations in the space-charge region (this compo-nent charges and discharges the anode-base depletion capacitor), hp and hn accountfor emitter recombination effects, and xl and xr are the left and right borders of theregion 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 respectto 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 areapplied in the first and last nodes respectively, while central differences are appliedfor the other internal nodes of the discretized region. It results in a set of algebraicequations, with n + 1 equations and n + 1 variables.

Instead of using also backward and forward differences as in [48], [49], this modelonly uses central differences. Thus, for the n elements in which the base region isdivided, we have n + 1 nodes, and for the ith node we have [72]:

∂p

∂x

∣∣∣∣i

=pi+1 − pi−1

2∆x(3.8)

∂2p

∂x2

∣∣∣∣i

=pi+1 − 2pi + pi−1

∆x2. (3.9)

Note that using only central differences, we have n + 3 nodes instead of the n + 1nodes resulting when using central, forward and backward differences. However, byusing central differences in the boundary conditions as well, we obtain an explicitexpression 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 thecorresponding finite difference eq. 3.9 in eq. 3.1 we obtain the general expressionfor the ith node

∂p

∂t

∣∣∣∣i

= Da

(pi+1 − 2pi + pi−1

∆x2

)− pi

τhl

. (3.10)

For the 1st and n + 1th node we have the following boundary conditions at x = xl

and x = xr:

∂p

∂x

∣∣∣∣i=1

=p2 − p0

2∆x=

1

2qA

(In(x = xl)

Dn

− Ip(x = xl)

Dp

)(3.11)

∂p

∂x

∣∣∣∣i=n+1

=pn+2 − pn

2∆x=

1

2qA

(In(x = xr)

Dn

− Ip(x = xr)

Dp

)(3.12)

Substituting eqs. 3.4-3.6 in eq. 3.11 and eq. 3.12 above:

∂p

∂x

∣∣∣∣i=1

=p2 − p0

2∆x=

1

2qA

(qAhpp

21

Dn

− ID − Idep − qAhpp21

Dp

)(3.13)

∂p

∂x

∣∣∣∣i=n+1

=pn+2 − pn

2∆x=

1

2qA

(ID − Idep − qAhnp2

n+1

Dn

− qAhnp2n+1

Dp

)(3.14)

Substituting p0 from (3.13) and pn+2 from (3.14) into the expressions equivalentto 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 systemas follows

[M ]

[∂p

∂t

]+ [K][p] + [F ] = 0 (3.15)

where the n + 1 x n + 1 symmetric matrices [M] and [K], and the n + 1 x 1 vector[F], are given as

M =

0.5 0 · · · 0 0

0 1 · · · 0 0...

......

......

0 0 · · · 1 0

0 0 · · · 0 0.5

(3.16)

55


K =

Da

∆x2+

1

2τhl

− Da

∆x2· · · 0 0

− Da

∆x2

2Da

∆x2+

1

τhl

. . . 0 0

......

......

...

0 0 · · · 2Da

∆x2+

1

τhl

− Da

∆x2

0 0 · · · − Da

∆x2

Da

∆x2+

1

2τhl

(3.17)

F =

Da

2qA∆x

(qAhpp

21

Dn

− ID − Idep − qAhpp21

Dp

)

0

...

0

− Da

2qA∆x

(ID − Idep − qAhnp2

n+1

Dn

− qAhnp2n+1

Dp

)

(3.18)

The ODEs system (3.15) can be interpreted as a combination of RC nets andcurrent sources, meaning that an equivalent electrical circuit of the system above isobtained, using Kirchhoff’s Current Law, which can be solved with the aid of thePspice circuit simulator. So, system (3.15) is equivalent to the system:

[C]

[∂V

∂t

]+ [G][V ] + [I] = 0 (3.19)

which in turn corresponds to the circuit in Fig. 3.2, the node voltages Vi(t) beingequivalent to the concentration p(xi,t) along the base. The number of nodes is thenumber 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

I1 =Da

2qA∆x

(ID − Idep − qAhpV

21

Dp

− qAhpV21

Dn

)(3.20)

and

In+1 =Da

2qA∆x

(ID − Idep − qAhnV 2

n+1

Dn

− qAhnV2n+1

Dp

). (3.21)

56

3.4– The complete diode model

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

G 1 2

G n C n

G n n + 1

G n + 1 C n + 1

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

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

Ri, i+1 =∆x2

Da

(3.22)

During reverse recovery, moving boundary effect is taken into account by allowingan adaptive definition of the space-charge region width, which in turn changes thewidth-dependent series resistances in the RC network modeling the base regionabove. The other components, the shunt components of the RC net, they are allconstant and have well defined values dependent upon the physical characteristicsof the diode, as follows:

for i = 1, n+1,

C1 = Cn+1 = 0.5

R1 = Rn+1 = 2τhl

(3.23)

for i 6= 1, n+1,

Ci = 1

Ri = τhl

(3.24)

With respect to other hybrid models [43], [44], [45], the proposed approach offersa significant reduction of circuit complexity in terms of number of components as wellas of the single component complexity. It results in an easy and fast implementationof the model. The accuracy of the model will depend on the number of elementsused.

3.4 The complete diode model

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

57


The complete compact model for the PiN diode is shown in Fig. 3.3, and includesthe internal voltages and currents. The total voltage across the diode is the sum ofvarious 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 dropsacross 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 withexcess carriers (carrier storage region), where conductivity modulation takes placewhenever the excess carriers are present. The total diode current consists of bothhole and electrons components within the carrier storage region, and the depletioncapacitance current Idep also contributes to the total diode current. The currents Ipl,Inl, Ipr, Inr represent the individual hole and electron currents entering and leavingthe 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.

V r e s V l e f t V r i g h t

p l p r

I p l

I n l I n r

I p r

I d e p

+ - + - I D V j l V j r

B a s e r e g i o n ( E p i l a y e r )


I D x l x r

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

3.4.1 Voltage drop on the junctions

The voltage drops in the junctions are calculated through the equations obtainedby the junction law under forward bias, as demonstrated in chapter 1. The voltagedrop in the left and right junctions are calculated from p1 and pn+1, and in theequivalent RC network from V1 and Vn+1, according to:

Vjl = VT · ln(

p1 ·ND

n2i

)= VT · ln

(V1 ·ND

n2i

)(3.26)

58

3.4– The complete diode model

and

Vjr = VT · ln(

pn+1

ND

)= VT · ln

(Vn+1

ND

)(3.27)

where ND is the base doping concentration [cm−3], ni is the intrinsic carrier concen-tration [cm−3], and VT is the equivalent thermal voltage [V].

We call the attention that these equations are valid while there are excess carriersin the physical junctions of the diode, being zero otherwise. During the implemen-tation of these equations, special care must be taken in order to avoid using negativevalues 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 sumof an ohmic component, due to the drift current, and of a Dember component, dueto 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 theohmic voltage drop Vres. The ohmic voltage drop across the carrier storage regionis found by integrating the electric field E responsible for driving the drift currentsin this region:

Vres =

∫ x=xr

x=xl

Edx (3.28)

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:

Vres =ID

qA

∫ x=xr

x=xl

dx

p(x,t)(µn + µp) + µnND

(3.30)

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

Vres =ID

qA

n∑i=1

∆x

Vi(x,t)(µn + µp) + µnND

(3.31)

For the sake of simplicity, it is assumed that the concentration between the twonodes of an element to be equal to

pi =pi + pi+1

2(3.32)

59


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

µn + µp =p0 · µ0

p0 + p(3.33)

where µ0 is considered as the sum of the mobilities in the base region and p0 is asuitable 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 conditionthe 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 thesezones must also be taken into account as parasitic resistances, and other parame-ters of the diode must be supplied, as the doping concentration, width of the sameregions, and mobilities of holes and electrons in these regions.

3.4.3 Voltage drop on the space-charge regions

The quasi-neutrality condition in the base layer is not satisfied during the final stageof reverse-recovery transient, when the two space-charge regions build-up. Thisphase begins when the concentration of carriers in the left junction drops to zero;from this time instant on the quasi-neutrality region is confined between two movingboundaries, advancing from both left and right junctions. In these conditions almostall the diode voltage drops on the left space-charge region [9], [28]. By assuming asaturated drift velocity condition in the above region, this voltage drops can easilybe expressed as follows [66], [79]:

Vleft =q

2εs

(ND +

| ID |qAvs

)x2

l (3.34)

Vright =q

2εs

(ND +

| ID |qAvs

)(W − xr)

2 (3.35)

where xl is the width of the left junction space-charge region, (W−xr) is the width ofthe right junction space-charge region, and vS is the hole and electron saturated driftvelocity. Here, the space-charge includes the ionized impurities and the density offree carriers injected in the depleted regions. In this model, as most of the depletedvoltage drop is present in the left junction, in order to simplify the model just thisone is taken into account, following eq. 3.34.

The great challenge when modeling PiN diode models based in the solution ofthe ADE regards the moving boundaries effect, that is, the calculation of the left andright borders position during reverse recovery. One possibility is through the widelyused feedback scheme proposed the first time by Leturq [31], and adopted by manyother authors [31], [34], [43], [64], [65], [66]. The feedback scheme introduced in thelast chapter, makes possible the calculation of xl and xr from the voltages across the

60

3.5– Model implementation within SPICE

depletion layers (Vleft and Vright) on each side of the carrier storage region. Theseare derived from the carrier densities at the boundaries between the depletion layersand carrier storage region, using a high-gain feedback loop as shown in Fig. 3.4,where K is the high-gain. The effect of the feedback is to produce a depletion layervoltage when the boundary carrier density falls below zero. The high gain ensuresthat the boundaries xl and xr move to maintain a negligible error in the boundarycarrier densities. The depletion layers may form on each side of the carrier storageregion.

Another option would be using the standard equation representing the exponen-tial relationship between current and voltage of a diode in the steady state condition:

i(t) =IS(exp(Vj)/(N VT )− 1)√

1 + IS(exp(Vj)/(N VT )− 1)

IKF

(3.36)

where IS, IKF and N are model parameters, and Vj is the voltage drop on thetwo P+N− and N−N+ junctions. So, the current is by analogy substituted bythe concentration of carriers in the left junction, and the junction voltage is easilyrelated to the carrier concentration in the left junction [29], [30]. In this way, whenthe depletion voltage starts building up in the diode left junction, the concentrationis imposed to be approximately zero, that is to the saturation value that is relatedwith the saturation reverse current of the standard diode IS, and it is possible tocalculate the position of the left border through eq. 3.34, since the voltage Vleft

is known. The continuity of the equations are maintained by the standard diodeequation. Analyzing both implementations, it may be concluded that both of themwork in a very similar way. An advantage of the latter is that the heuristic gain of thefeedback scheme is eliminated, and the parameters needed for the implementationof the model are all related to the physical and geometrical properties of the PiNdiode, plus the model parameters that must be fitted through static measurements.Both the sub-models were implemented and simulated with the proposed model.

3.5 Model implementation within SPICE

In the following, the complete diode model will be presented as a Pspice subcircuit[58], without the need of modifying the code of the simulator. The subcircuit inputparameters are all directly related to the geometrical and physical parameters ofthe diode, as shown in Table 3.1. If the feedback scheme proposed by Leturq [31] isused, an additional parameter K is present, while for the proposed use of a standarddiode as explained in the last section [30], there will be three more parameters (IS,IKF , N) that must be fitted togheter with the p0 coefficient, accounting for carrier-carrier scattering effects. The fitting must be done only with respect to static I-V

61


K

K R C N E T W O R K

p 1 , p 2 , . . . p n , p n + 1

x l

x r

x l x r p x l

p x r I n l I n r

I p l I p r

I d e p

Figure 3.4. Feedback scheme for the calculation of xl and xr.

measurements, and it can be done in a fast and easy way, using MATLAB forexample [78].

Table 3.1. Input parameters of the Pspice model.Notation Parameter Units

ND Base doping cm−3

W Base width cmτhl high injection lifetime sWP P+ emitter depth cmNP P+ peak doping cm−3

WN N+ emitter depth cmNN N+ peak doping cm−3

A device area cm2

p0 coefficient describing carrier-carrier scattering in the base cm−3

n number of elements of the discretized epilayer

Regarding the discretization of the epilayer, it is suggested that readers shoulddiscretize the base in at least 10 elements for obtaining accurate results. Accuracy ofthe model depends on the number of elements, however it was found that discretizingthe epilayer in more than 10 elements does not improve that much the results, andsimilar results are obtained for both cases. Fig. 3.5 shows the basic elements of the

62

3.5– Model implementation within SPICE

power PiN diode subcircuit, considering the feedback scheme. The base region ofthe device is represented by the network of Fig. 3.2, based on the FDM. In Fig. 3.5the voltages at the nodes 1, 2, 3, 4 to 11 are proportional to p(0), p(W/n), p(2W/n),p(3W/n) to p(W) respectively. Equally spaced elements W/n in the discretized baseregion are considered.

G L

R P 1 C P 1 C P 2 C P 1 0 C P 1 1R P 2 R P 1 0 R P 1 1

G R r e c

G R S 1 G R S 1 0 1 2 1 0 1 1. . .G L r e c G R

+ E X L R X L

5 0

+

V d i o d e E d i o d e

A n o d e C a t h o d e 1 0 0 2 0 3 0

Figure 3.5. Schematic of PiN diode Pspice subcircuit using the feedback scheme.

The device is represented by a controlled voltage source Ediode that representsthe sum of the 4 components of voltage described in the last section. The voltagegenerator Vdiode is actually a short circuit and is used only to sense the currentflowing into the diode. As shown in Fig. 3.5, the series resistors of the networkof Fig. 3.2 are represented by a voltage-controlled current generator in order torepresent a variable resistor that also depends on the width of the elements. Thesewidths are calculated with the help of a subcircuit representing the feedback schemewhere the Pspice command LIMIT is used, and voltage on it represents the positionof the left boundary. During forward conduction the concentration of carriers in thefirst node is positive, and so the position of the boundary is x = xL=0. So, therelationship R = V/I = f(V ) = (Vi − Vi+1) ·Da/∆x2 is implemented, where ∆x issubstituted by the expression (W−V (50))/n where V(50) represents the voltage inthe voltage generator, used for calculating the position of the left boundary (xL), Wis the physical length of the base region and n is the number of elements in whichthe base region is discretized. The contribution of VJL and VJR are proportional tothe voltages V(1) and V(n+1) respectively, according to equations (3.26) and (3.27).The third contribution to the diode voltage, Vres, is dependent on the voltages inthe nodes 1 to 11 and on the diode current, according to equation (3.31). It wasfound that if not all the nodes are considered in the calculation of this contributionof voltage, faster simulation times are obtained. Finally, the last contribution takesinto account the voltage drop on the left space-charge region and is function of thediode current and the width xL of the left space-charge region, according to equation(3.34). The value of xL is supplied in the form of a voltage that is ”zero” until thetime when p(0), that means V(1) in the RC network, tends to become negative(from the time instant in which xL is different of zero, the carriers in the base region

63


are confined in a region that is wide (W - xL), and from this time the width of theelements starts to decrease). The complete model uses much less components (morecompact model), and is much more simple and with better convergence properties,due to the fact that there are no discontinuities in the present model, than themodel presented by Araujo in [43]. This is not only due to the simpler solution ofthe ADE, but also by the way in which the model is implemented within the Pspicesimulator, as will be seen in section 3.6.1. A constant of normalization kn for theimplementation into Pspice must be used with the RC network representing thesolution of the ADE, due to the fact that the SPICE based simulators do not workwith voltages higher than 10 GV, and the values of the single values will change.The complete subcircuit code is shown in Appendix A, with reference to Fig. 3.5.

E 0 R P 1 C P 1 C P 2 C P 1 0 C P 1 1R P 2 R P 1 0 R P 1 1

G R r e cG R S 1 G R S 1 0 1 2 1 0 1 1

. . .G L r e c G R

+ E J +V d i o d eE r e s

A n o d e C a t h o d e 1 0 0 2 0 3 0 4 0

V S 1 5 0 6 0G p i n

+

V S 2

D 1

Figure 3.6. Schematic of PiN diode Pspice subcircuit using the standard diode.

The second way of implementation works in a similar way, and is shown in Fig.3.6. Instead of using the boundary conditions in the same way of eq. 3.2 and eq.3.3, eq. 1.48 and eq. 1.49 are used for obtaining the symmetrical system of ODEs,meaning that the depletion current component is not taken into account. As canbe seen in Fig. 3.6, instead of having a current source in the first node, we have avoltage source that is linked to the current flowing through the standard diode D1.The current through the diode by analogy is equivalent to the carriers concentrationin the first node, and the voltage drop on diode D1 represents the voltage dropin both junctions. When the concentration tends to become negative, there is anegative voltage drop in the junctions, that means the voltage drop in the depletedregion. The whole system of ODEs is divided by an additional constant kidiode, inorder to avoid convergence problems. The reason for doing that is the same for theuse of kn. The complete subcircuit code is shown in Appendix B, with reference toFig. 3.6.

64

3.6– Model results and Validation

3.6 Model results and Validation

Finally, Pspice simulations through the proposed compact diode model are com-pared with experimental results and simulations using another compact diode modelpresent in literature, and either with experimental characterizations or with SIL-VACO mixed-mode module simulations [7] of a commercial PiN power diode [19].

3.6.1 Comparison with the FEM based diode model

In order to compare the results of the proposed model with the model based inthe Finite Element Method proposed by Araujo [43], [44], [45], the first one wasimplemented exactly in the same way that the last one, but substituting the RCnetwork of the FEM based model representing the solution of the ADE, by the RCnetwork of the proposed model. The circuit representing the complete diode modelis shown in Fig. 3.7. Only 4 elements for the discretization of the base region isshown in order to illustrate the same, however, authors suggest that readers shoulddiscretize the base in at least 10 elements for obtaining accurate results, as alreadymentioned. The feedback scheme proposed by Leturq [31] was adopted in bothmodels, and only the voltage drop in the left space-charge region is considered here,according to the equation below, which does not take into account the injection ofholes in the depleted region:

Vleft =qNDx2

l

2εs

(3.37)

where εs is the dieletric constant of silicon [F/cm], and xl is the left boundary of thebase region [cm].

G 1

R P 1 C P 1 C P 2 C P 3 C P 4 C P 5

R W 1 R W 2 R W 3 R W 4

I W


1 1 2 2 2 3 3 3 4 4 4 5 5

R P 2 R P 3 R P 4 R P 5

G 5

E j l E j r E r e s E S C V d i o d e

V 1 2 V 2 3 E 1 2 E 2 3

1 0 2 0 3 0 4 0 5 0 6 0

E r e s 1 E r e s 2 E r e s 3 E r e s 4

V W7 0 8 0 9 0 1 0 0

G S C2 1

2 2

V 3 4 E 3 4 E 4 5 V 4 5

Figure 3.7. Circuit representation of the diode model for 4 elements, implementedexactly as the FEM diode model.

In order to verify the effectiveness of the proposed Pspice model, several simula-tions have been run, and the results are presented and compared with experimental

65


measurements and simulations using the FEM based model. Physical parametersof the diode under test are: W = 90 µm, τhl = 10 µs, ND = 1014 cm−3, A = 4mm2, Dn = 25 cm2/s, Dp = 10 cm2/s and hp = hn = 1.5 · 10−14 cm4/s. Carrier-carrier scattering and Auger recombination effects were not taken into account herefor the sake of comparison with the FEM-based model, which has not used theseseffects either. The test circuit used to measure and simulate the reverse recovery ispresented in Fig. 3.8, while the circuit operating conditions and the most relevantperformance parameters are reported in Table 3.2. The performance parametersare highlighted in Fig. 3.9. An ideal switch was used in the simulations. The re-sults of two different tests are highlighted in Fig. 3.10 and in Fig. 3.11, showingexcellent agreement either with experiments or the FEM-based model. Finally, inTable 3.3 the relative percentual errors of the FDM-based model with respect to theFEM-based model in [44] are presented.

L D U T

L

D U TV C C

I D

D A U XV G

Figure 3.8. Test circuit.

As highlighted from the presented comparisons, the proposed FDM-based modelyields results in good agreement with the FEM-based model. However it has to bestressed that the advantages of the proposed diode model based on FDM are thefaster simulations and the use of fewer components modeling the base region. Thecomponents used in the model are simpler, resulting in less convergence problems,at least in the sense that only the series resistances depend on the widths of theelements. In contrast, in the FEM-based model in [44], capacitors and shunt resistorsare also dependent on the element width (all the elements are miplemented as nonlinear elements), and for each element there are more resistors and capacitors thanin the proposed model. In fact, the FEM-based model uses 3 capacitors for eachelement, whereas in the FDM-based model there are just n + 1 capacitors for nelements. In the past, it was used to say that numerical models were very CPU

66


0

s w i t c h i s c l o s e d : t 0

Diode

Curre

nt [ A

]

T i m e [ n s ]

0

Diode V

oltage [

V]

I r mt I r mt V r m

t s

V r m

I D

d I D / d t

Figure 3.9. Definition of the values associated with the characteristic stages of adiode under reverse recovery condition.

time consuming and used to require a relatively complex interface between circuitsimulator and ADE solver [28]. This is no longer true with the present model, owingto the electrical circuit analogy, which allows it to reduce the simulation to a Pspicesubcircuit analysis. This work was published in [80].

67


Table 3.2. Results from measurements and simulations within Pspice (FDM andFEM)

Method Irm(A) tIrm(ns) Vrm(V ) tV rm(ns) ts(ns)

Test 1 (ID = 12A,∂ID

∂t= 50 A

µs,VCC = 200V,LDUT = 50nH)

Meas. 24 604 217 625 396FEM 27 548 216 606 448FDM 25.3 540 216 600 474


∂t= 100 A


Meas. 32 406 221 458 292FEM 35 369 229 415 321FDM 34.1 350 229 374 316


∂t= 200 A


Meas. 41 258 238 300 171FEM 43 226 250 254 199FDM 43.5 226 247 278 207


∂t= 100 A


Meas. 28 324 119 396 246FEM 33 339 126 353 302FDM 32.3 341 127.4 373 322


∂t= 100 A


Meas. 32 400 296 483 250FEM 33 373 343 447 301FDM 33.3 345 340.4 451 313

Table 3.3. Relative percentual errorsFDM Irm tIrm Vrm tV rm tsTest Error Error Error Error Error

1 6 1.5 0 1 5.52 2.5 5 0 9.8 1.53 1.2 0 1.2 8.6 3.84 2.1 0.6 1 5.4 6.25 1 7.5 0.8 0.9 3.8

68


0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5T i m e ( m s )

0- 1 0 0- 2 0 0- 3 0 0- 4 0 0- 5 0 0- 6 0 0- 7 0 0

1 0 0

Diode Voltage (V)01 02 03 04 0

- 4 0- 3 0- 2 0- 1 0

D io de C

u r re nt ( A

)

0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5T i m e ( m s )

0- 1 0 0- 2 0 0- 3 0 0- 4 0 0- 5 0 0- 6 0 0- 7 0 0

1 0 0

Diode Voltage (V)

01 02 03 04 0

- 4 0- 3 0- 2 0- 1 0

D io de C

u r re nt ( A

)

0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5T i m e ( m s )

0- 1 0 0- 2 0 0- 3 0 0- 4 0 0- 5 0 0- 6 0 0- 7 0 0

1 0 0

Diode Voltage (V)

01 02 03 04 0

- 4 0- 3 0- 2 0- 1 0

D io de C

u r re nt ( A

)

( b )

( a )

( c )

Figure 3.10. Reverse recovery transient - Test 2: (a) Experimental results, (b)Simulation results using the FEM-based model, (c) Simulation results using the

proposed model.

69


0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2T i m e ( m s )

0- 1 0 0- 2 0 0- 3 0 0- 4 0 0- 5 0 0- 6 0 0- 7 0 0

1 0 0

01 02 03 04 0

- 4 0- 3 0- 2 0- 1 0

Diode Voltage (V)D io de C

u r re nt ( A

)

01 02 03 04 0

- 4 0- 3 0- 2 0- 1 0

D io de C

u r re nt ( A

)

0- 1 0 0- 2 0 0- 3 0 0- 4 0 0- 5 0 0- 6 0 0- 7 0 0

1 0 0

Diode Voltage (V)

T i m e ( m s )0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2

T i m e ( m s )

0- 1 0 0- 2 0 0- 3 0 0- 4 0 0- 5 0 0- 6 0 0- 7 0 0

1 0 0

Diode Voltage (V)

01 02 03 04 0

- 4 0- 3 0- 2 0- 1 0

D io de C

u r re nt ( A

)

0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2

( a )

( b )

( c )

Figure 3.11. Reverse recovery transient - Test 5: (a) Experimental results, (b)Simulation results using the FEM-based model, (c) Simulation results using the

proposed model.

70


3.6.2 Simulation of Commercial Fast Recovery Diodes

The present model has been applied to the analysis of IR Si-based FREDs (fast re-covery diodes) and validated against experimental characterization data. The simu-lated PiN diodes has a 50 µm long base, with doping level in the order of 10−14 cm−3.The devices have a specified current rating of 8A. Two devices have been investi-gated, having identical physical structure but different carrier lifetimes: τhl = 200ns (device D1) and τhl = 40 ns (device D2). The base region of the simulated diodeswas discretized in 10 elements, and once more the feedback scheme is employed.Further details on the device structure may be found in [19]. Simulations were alsocarried out by exploiting the SILVACO mixed-mode module [7], where the device isdescribed through the standard drift-diffusion physics based transport model. TheSILVACO numerical model includes mobility dependence on doping, carrier-carrierscattering, temperature, and electric field, bandgap narrowing, Schockley-Read-Hallrecombination and Auger recombination. The actual doping profile is analyzed,while an abrupt approximation is used in the Pspice model. Material parametersare the same in both simulators. A comparison between simulated and measuredforward I-V characteristic of the D1 and D2 diodes is presented in Fig. 3.12 andFig. 3.13, showing good agreement, which is possible due to the use of the mobilitymodel taking into account scattering effects.

0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2 2 . 2 2 . 4 2 . 61 0 - 1

1 0 0

1 0 1

1 0 2

D i o d e V o l t a g e [ V ]

Diode C

urrent

[A]

E x p e r i m e n t a lP s p i c eS i l v a c o

Figure 3.12. Measured and simulated forward I-V characteristics of Diode D1.

Fig. 3.14 and Fig. 3.15 shows forward recovery simulations for both diodes,

71


0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 51 0 - 1

1 0 0

1 0 1

1 0 2

D i o d e V o l t a g e [ V ]

Diode C

urrent

[A]

E x p e r i m e n t a lP s p i c eS i l v a c o

Figure 3.13. Measured and simulated forward I-V characteristics of Diode D2.

obtained by supplying the device with a current ramp having a rise time of 100 nsand a steady-state value Imax (see Fig. 1.8). Three cases have been considered:Imax = 10A, Imax = 5A and Imax = 1A. Results of Fig. 3.14 and Fig. 3.15show that the diode voltage behavior given by Pspice is in good agreement with theones of SILVACO. Note the forward voltage overshoot, due to the time needed forinjected carriers to fill-up the base, creating a conductivity-modulated region.

Concerning the recovery behavior, an ad hoc boost-like circuit has been employedfor diode characterization (Fig. 3.16) in order to ensure a complete knowledge ofall the circuit components (LDUT , LD and LS are the parasitic inductors) . As faras the Pspice and SILVACO simulations, since the circuit is operated in continu-ous current mode, the input inductor LF and the output capacitor CR have beenreplaced by a constant current source (IF ) and a constant voltage source (VR), re-spectively. A STANDARD SPICE LEVEL 1 model for the IRFP460 MOSFET hasbeen implemented. Reverse recovery simulation results obtained with the Pspicemodel are reported from Fig. 3.17 to Fig. 3.26 for both diodes with different di/dtratings, in order to investigate its influence on reverse recovery behavior. For thecase which the di/dt is 100 A/µs, the parasitic inductances are: LDUT = 40 nH, LD=30 nH and LS= 13 nH. For the other analyzed di/dt ratings, the values are: LDUT =50 nH, LD= 5 nH and LS= 13 nH. The comparison against either experiments orSILVACO-based simulations shows very good agreement, and it can be observedthat the peak reverse current and the recovery time obtained with the Pspice model

72


0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 00

0 . 5

1

1 . 5

2

2 . 5

3

3 . 5

4

4 . 5

T i m e [ n s ]

Diode

Voltag

e [V]

P s p i c eS i l v a c oI m a x = 1 0 A

I m a x = 5 A

I m a x = 1 A

Figure 3.14. Forward recovery simulations of diode D1: comparison betweenPspice model and SILVACO numerical results.

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 00

0 . 5

1

1 . 5

2

2 . 5

3

3 . 5

4

4 . 5

T i m e [ n s ]

Diode

Voltag

e [V]

P s p i c eS i l v a c o

I m a x = 1 0 A

I m a x = 5 A

I m a x = 1 A

Figure 3.15. Forward recovery simulations of diode D2: comparison betweenPspice model and SILVACO numerical results.

are close the SILVACO and experimental results in all the simulated cases. It mustbe highlighted, since the values of peak reverse current and peak overvoltage are the

73


v G

L DD U TR G

V RL S

V i n

L F L D U T

C R

I F

Figure 3.16. Circuit schematic of the boost-like converter.

most significant factors in predicting power losses and EMI phenomena.Simulation times are much lower than the simulation times of the SILVACO

simulator, as show in Table 3.4. This table reports the CPU time required by Pspice,on a 3 GHz, 512 Mb RAM personal computer. The values in Table 3.4 regard thediode D1. However, the simulation times regarding diode D2, and also for all thedifferent conditions in the reverse recovery simulations, are similar. Simulationsthrough the SILVACO simulator are of the order of 10 minutes or more for the samesimulation parameters. It was found that if just half nodes of the discretized epilayerare taken into account for calculating the voltage drop in the base, the simulationtimes during reverse recovery are much lower, and the same accuracy is obtained.On the other hand, during forward recovery the accuracy is not kept the same as inthe case which all the nodes are considered for evaluating the voltage drop in theepilayer, while the simulation times are much lower as in the other case. This workwas published in [81].

Table 3.4. Pspice simulation times, in seconds.Maximum step size 10 ns 1 ns

Forward recovery (1 µs simul.) 0.69 6.16Reverse recovery (1 µs simul.) 5.42 12.08

DC sweep (100 points) 0.53

74


0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0- 1 2

- 1 0

- 8

- 6

- 4

- 2

0

2

4

6

8

1 0

T i m e [ n s ]

Diode

Curre

nt [A

]P s p i c eS i l v a c oM e a s u r e m e n t s

Figure 3.17. Measured and simulated current recovery waveforms at di/dt =200A/µs, 8A, 200V - Diode D1.

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0- 2 0 0

- 1 8 0

- 1 6 0

- 1 4 0

- 1 2 0

- 1 0 0

- 8 0

- 6 0

- 4 0

- 2 0

0

T i m e [ n s ]

Diode

Voltag

e [V]

P s p i c e S i l v a c oM e a s u r e m e n t s

Figure 3.18. Measured and simulated voltage recovery waveforms at di/dt =200A/µs, 8A, 200V - Diode D1.

75


0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0- 5

0

5

1 0

T i m e [ n s ]

Diode

Curre

nt [A



0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0- 2 0 0

- 1 8 0

- 1 6 0

- 1 4 0

- 1 2 0

- 1 0 0

- 8 0

- 6 0

- 4 0

- 2 0

0

T i m e [ n s ]

Diode

Voltag

e [V]

P s p i c eS i l v a c oM e a s u r e m e n t s


76


0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0- 1 0

- 8

- 6

- 4

- 2

0

2

4

6

8

1 0

T i m e [ n s ]

Diode

Curre

nt [A



0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0- 2 0 0

- 1 8 0

- 1 6 0

- 1 4 0

- 1 2 0

- 1 0 0

- 8 0

- 6 0

- 4 0

- 2 0

0

T i m e [ n s ]

Diode

Voltage

[V]


-


77


0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0- 4

- 2

0

2

4

6

8

1 0

T i m e [ n s ]

Diode

Curre

nt [A



0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0- 2 0 0

- 1 8 0

- 1 6 0

- 1 4 0

- 1 2 0

- 1 0 0

- 8 0

- 6 0

- 4 0

- 2 0

0

Diode

Voltag

e [V]



78


0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0- 2 0

- 1 5

- 1 0

- 5

0

5

1 0

T i m e [ n s ]

Diode

Curre

nt [A



0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0

- 2 0 0- 1 8 0- 1 6 0- 1 4 0- 1 2 0- 1 0 0- 8 0- 6 0- 4 0

- 2 00

T i m e [ n s ]

Diode

Voltag

e [V]



79


3.7 Simulation of Switched Mode Power Supplies

Power supply design has today challenging goals in terms of high efficiency andelectrical performance, together with low weight, size and cost. In order to achievesuch goals, a reliable and accurate CAD simulation and design of the SMPS isneeded [82], allowing for the choice of the best suited diode to be used in the DC-DC converter and, when available, in the active Power Factor Correction (PFC)stage. Owing to the good accuracy combined with short simulation time and robustconvergence, the proposed PiN diode model may be very useful from this point ofview, since it enables the reliable and fast evaluation of power losses in the switchingand conduction state.

After the validation of the commercial diodes presented in the previous section,we illustrate the effectiveness and robustness of the presented diode model by ex-ploiting it in the analysis of complex realistic circuits. We consider two SMPSs [83],[84]; the first one (Fig. 3.27) uses a dc-dc buck converter with the following char-acteristics: Pout = 800 W, Voutput = 90 V dc, Vinput = 127 V ac rms at 50 Hz andswitching frequency of 100 kHz. The second one (Fig. 3.28) uses a dc-dc flybackconverter, operating in continuous current mode, with the following characteristics:Pout = 100 W, Voutput = 5 V dc, Vinput = 220 V ac rms at 50Hz and switchingfrequency of 100 kHz. Both circuits exploit as switching devices the IRFP460 MOS-FET and the diode D1 analyzed in the previous section. The SMPSs were simulatedusing a SPICE LEVEL 1 model for the IRFP460 MOSFET, and taking into accountthe parasitic inductance of the diode.

L B

R L

OA

D

C B C F

L DUT

DUT R G

IRFP460

Figure 3.27. SMPS based on buck topology.

Fig. 3.29 and Fig. 3.30 report the current waveforms across the diode D1 andthe switch for the buck SMPS. It is observed that the reverse recovery current of thediode shows up in the MOSFET drain current. This current will cause significantpower dissipation in the MOSFET, along with increased EMI. In order to improve

80

3.7– Simulation of Switched Mode Power Supplies

C F

R G

IRFP460

L DUT DUT

C B

R L

OA

D

L P L S

Figure 3.28. SMPS based on flyback topology.

the efficiency of the SMPS, one can trade-off between the peak reverse current andthe recovery time of the diode or limit over-currents and EMI generation by meansof snubber circuits, which on the other hand cause unwanted RC power loss. Duringthe design stage, besides the usual parameters involved in the design of a SMPS, thedi/dt controlling the diode turn-off should also be taken into account, since higherdi/dt causes higher reverse current peak.

Fig. 3.31 and Fig. 3.32 show the voltage across the diode and the switch,highlighting the voltage overshoot during the diode turn-on (forward recovery) thatalso appears across the MOSFET drain-source prior to the blocking state. Suchadditional voltage should also carefully considered during the design stage, in orderto choose a suitable switch with proper voltage specification and to properly designa limiting snubber circuit. Finally, Fig. 3.33 to Fig. 3.36 report the simulationresults for the SMPS using the dc-dc flyback converter. This work was published in[85].

81


1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0- 2 0- 1 5- 1 0- 505

1 01 52 02 53 0

Curre

nt [A]

D i o d e C u r r e n tS w i t c h C u r r e n t

D i o d e P e a k R e c o v e r y C u r r e n t

T i m e [ m s ]Figure 3.29. Buck based SMPS: simulated diode and switch currents during

commutation.

0 0 . 5 1 . 0 1 . 5 2 . 0- 2 0- 1 5- 1 0- 505

1 01 52 02 53 03 5

T i m e [ m s ]

Curre

nt [A]

D i o d e C u r r e n tS w i t c h C u r r e n t


Figure 3.30. Buck based SMPS: detail of the diode and switch currents during asingle commutation.

82


1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0- 2 0 0- 1 5 0- 1 0 0- 5 00

5 01 0 01 5 02 0 02 5 03 0 0

T i m e [ m s ]

Volta

ge [V

]D i o d e V o l t a g e S w i t c h V o l t a g e

Figure 3.31. Buck based SMPS: simulated diode and switch voltages duringcommutation.

0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0- 2 0 0- 1 5 0- 1 0 0- 5 00

5 01 0 01 5 02 0 02 5 0

T i m e [ m s ]

Volta

ge [V

]

D i o d e V o l t a g eS w i t c h V o l t a g e

D i o d e V o l t a g e O v e r s h o o t

Figure 3.32. Buck based SMPS: detail of the diode and switch voltages during asingle commutation.

83


- 2

0

2

4

Switch C

urren

t [A]

1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 - 2 0

0

2 0

4 0

Diod

e Curr

ent [A

]


T i m e [ m s ]Figure 3.33. Flyback based SMPS: simulated diode and switch currents during

commutation.

- 2

0

2

4

Switch C

urren

t [A]

0 0 . 5 1 1 . 5 2 - 2 0

0

2 0

4 0Diod

e Curr

ent [A

]


T i m e [ m s ]Figure 3.34. Flyback based SMPS: detail of the diode and switch currents during

a single commutation.

84


0 1 0 2 0 3 0 4 0 5 0- 6 0 0

- 4 0 0

- 2 0 0

0

2 0 0

4 0 0

6 0 0

T i m e [ m s ]

Switch

Volta

ge [V

]

S w i t c h V o l t a g eD i o d e V o l t a g e

- 6 0

- 4 0

- 2 0

0

2 0

4 0

6 0

Diode

Volta

ge [V

]

Figure 3.35. Flyback based SMPS: simulated diode and switch voltages duringcommutation.

0 0 . 5 1 . 0 1 . 5- 3 0 0- 2 0 0- 1 0 0

01 0 02 0 03 0 04 0 05 0 06 0 0

T i m e [ m s ]

Switch V

oltag

e [V]

S w i t c h V o l t a g eD i o d e V o l t a g e

- 3 0- 2 0- 1 00

1 02 03 04 05 06 0

Diode

Volt

age [

V]

D i o d e V o l t a g e O v e r s h o o t

Figure 3.36. Flyback based SMPS: detail of the diode and switch voltages duringa single commutation.

85

Chapter 4

Conclusions

In this tesis we have presented a novel approach for modelling power PiN diodes,based on an equivalent circuit representation of the epilayer, developed from theFinite Difference solution of the Ambipolar Diffusion Equation. This model fullydescribes the distributed nonlinear behavior of carriers within the epilayer, and iseasily implemented as a SPICE subcircuit which takes into account the most impor-tant physical effects such as emitter recombination effects, carrier-carrier scatteringeffects and the dynamic of the space-charge voltage build-up.

Comparisons of results of the proposed diode model with experimental resultsand FEM-based model simulations reported in literature, and also of commerciallyavailable fast recovery diodes and simulations carried on a physics based mixed-modesimulator were done, and very good agreement has been obtained.

The model is characterized by its simplicity and good simulation times andconvergence properties, suitable for the simulation of practical power circuits such asSwitched Mode Power Supplies (SMPSs). We have reported examples of applicationto the simulation of SMPSs. In this way it is possible to evaluate the power lossesand the efficiency of the power supplies already during the design stage by means ofCAD tools. The proposed modelling technique can be applied to any device regionin high injection condition and therefore is suitable for the modeling of many powerdevices beside PiN diode, like BJTs and IGBTs.

86

Appendix A

Pspice subcircuit listing - feedbackscheme

.SUBCKT PiNdiode 100 30*PiN DIODE model through Finite Differences, using the feedback scheme.PARAM Da=2*Dn*Dp/(Dn+Dp), Dn=34.84, Dp=12.82, W=50e-4, Nd=2e14+ thl=200n, n=10, hp=1e-15, hn=1e-15, A=0.04, q=1.6e-19+ un=Dn/25.9m, up=Dp/25.9m, ni=1.45e10.PARAM VT=25.9m, po=7.4e16, uo=un+up, K=1e2, kn=1e16+ eps=8.854e-14, epssi=11.7, vs=1e7.FUNC pos(x) limit(x,0,1e3)

GRS1 1 2 VALUE=(V(1)-V(2))*Da/(((W-V(50))/n)ˆ2)GRS2 2 3 VALUE=(V(2)-V(3))*Da/(((W-V(50))/n)ˆ2)GRS3 3 4 VALUE=(V(3)-V(4))*Da/(((W-V(50))/n)ˆ2)GRS4 4 5 VALUE=(V(4)-V(5))*Da/(((W-V(50))/n)ˆ2)GRS5 5 6 VALUE=(V(5)-V(6))*Da/(((W-V(50))/n)ˆ2)GRS6 6 7 VALUE=(V(6)-V(7))*Da/(((W-V(50))/n)ˆ2)GRS7 7 8 VALUE=(V(7)-V(8))*Da/(((W-V(50))/n)ˆ2)GRS8 8 9 VALUE=(V(8)-V(9))*Da/(((W-V(50))/n)ˆ2)GRS9 9 10 VALUE=(V(9)-V(10))*Da/(((W-V(50))/n)ˆ2)

RP1 1 0 2*thlRP2 2 0 thlRP3 3 0 thlRP4 4 0 thlRP5 5 0 thlRP6 6 0 thlRP7 7 0 thlRP8 8 0 thl

87

A – Pspice subcircuit listing - feedback scheme

RP9 9 0 thlRP10 10 0 thlRP11 11 0 2*thl

CP1 1 0 0.5CP2 2 0 1CP3 3 0 1CP4 4 0 1CP5 5 0 1CP6 6 0 1CP7 7 0 1CP8 8 0 1CP9 9 0 1CP10 10 0 1CP11 11 0 0.5

Vdiode 100 20 DC 0

Ediode 20 30 VALUE=-q/(2*eps*epssi)*(Nd+abs(I(Vdiode))/(q*A*vs))*(V(50))ˆ2++ VT*pos(log((V(1)*kn*Nd)/(niˆ2)))+I(Vdiode)*((W-V(50))/n)/(q*A)*(1/((pos(V(1))++ pos(V(2)))*kn*(po*uo/((pos(V(1))++ pos(V(2)))*kn+2*po))+Nd*un)+1/((pos(V(2))+pos(V(3)))*kn*(po*uo/((pos(V(2))++ pos(V(3)))*kn+2*po))+Nd*un)+1/((pos(V(3))+pos(V(4)))*kn*(po*uo/((pos(V(3))++ pos(V(4)))*kn+2*po))+Nd*un)+1/((pos(V(4))+pos(V(5)))*kn*(po*uo/((pos(V(4))++ pos(V(5)))*kn+2*po))+Nd*un)+1/((pos(V(5))+pos(V(6)))*kn*(po*uo/((pos(V(5))++ pos(V(6)))*kn+2*po))+Nd*un)+1/((pos(V(6))+pos(V(7)))*kn*(po*uo/((pos(V(6))++ pos(V(7)))*kn+2*po))+Nd*un)+1/((pos(V(7))+pos(V(8)))*kn*(po*uo/((pos(V(7))++ pos(V(8)))*kn+2*po))+Nd*un)+1/((pos(V(8))+pos(V(9)))*kn*(po*uo/((pos(V(8))++ pos(V(9)))*kn+2*po))+Nd*un)+1/((pos(V(9))+pos(V(10)))*kn*(po*uo/((pos(V(9))++ pos(V(10)))*kn+2*po))+Nd*un)+1/((pos(V(10))+pos(V(11)))*kn*(po*uo/((pos+ (V(10))+pos(V(11222)))*kn+2*po))+Nd*un))+VT*pos(log((V(11)*kn)/(Nd)))

GL 0 1 VALUE=Da/(2*q*A*kn*(W-V(50))/n)*(I(Vdiode)+q*Nd*A*DDT(V(50))-+(q*(knˆ2)*A*hp*((pos(V(1)))ˆ2)))/DpGLrec 1 0 VALUE=Da/(2*q*A*kn*(W-V(50))/n)*(q*(knˆ2)*A*hp*+((pos(V(1)))ˆ2))/Dn

GR 0 11 VALUE=Da/(2*q*A*kn*(W-V(50))/n)*(I(Vdiode)+q*Nd*A*DDT(V(50))-+(q*(knˆ2)*A*hn*((pos(V(11)))ˆ2)))/DnGRrec 11 0 VALUE=Da/(2*q*A*kn*(W-V(50))/n)*(q*(knˆ2)*A*hn*+((pos(V(11)))ˆ2))/Dp

88

EXL 50 0 VALUE=limit(-K*V(1),0,1e2)RXL 50 0 1

.ENDS PiNdiode

89

Appendix B

Pspice subcircuit listing - standarddiode

.SUBCKT Pin 100 40*PiN DIODE model through Finite Differences, using the standard diode.PARAM Da=2*Dn*Dp/(Dn+Dp), Dn=34.84, Dp=12.82, W=50e-4,+ Nd=2e14, thl=200n, n=10, hp=1e-15, hn=1e-15, A=0.04,+q=1.6e-19, un=Dn/25.9m, up=Dp/25.9m, ni=1.45e10.PARAM po=7.4e16, uo=un+up, h=W/n;.PARAM b=Dn/Dp, kn=1e16, kpoisson=1/(7.66e-8*Nd), kidiode=b/((1+b)*q*A).FUNC pos(x) limit(x,0,1e3)

Ej 50 0 VALUE=V(20,40)vs1 50 60 DC 0D1 60 0 Djunction.MODEL Djunction D(IS=1.35e-8, N=1.45, IKF=0.32)

Eres 100 20 VALUE=I(Vdiode)*(W-sqrt(pos(-V(20,40))*kpoisson))/n/(q*A)*+ (1/((pos(V(1))+pos(V(2)))*kn*(po*uo/((pos(V(1))++ pos(V(2)))*kn+2*po))+Nd*un)+1/((pos(V(2))+pos(V(3)))*kn*(po*uo/((pos(V(2))++ pos(V(3)))*kn+2*po))+Nd*un)+1/((pos(V(3))+pos(V(4)))*kn*(po*uo/((pos(V(3))++ pos(V(4)))*kn+2*po))+Nd*un)+1/((pos(V(4))+pos(V(5)))*kn*(po*uo/((pos(V(4))++ pos(V(5)))*kn+2*po))+Nd*un)+1/((pos(V(5))+pos(V(6)))*kn*(po*uo/((pos(V(5))++ pos(V(6)))*kn+2*po))+Nd*un)+1/((pos(V(6))+pos(V(7)))*kn*(po*uo/((pos(V(6))++ pos(V(7)))*kn+2*po))+Nd*un)+1/((pos(V(7))+pos(V(8)))*kn*(po*uo/((pos(V(7))++ pos(V(8)))*kn+2*po))+Nd*un)+1/((pos(V(8))+pos(V(9)))*kn*(po*uo/((pos(V(8))++ pos(V(9)))*kn+2*po))+Nd*un)+1/((pos(V(9))+pos(V(10)))*kn*(po*uo/((pos(V(9))++ pos(V(10)))*kn+2*po))+Nd*un)+1/((pos(V(10))+pos(V(11)))*kn*(po*uo/((pos+ (V(10))+pos(V(11222)))*kn+2*po))+Nd*un))

90

Gpin 20 30 VALUE=I(VS2)*(W-sqrt(pos(-V(20,40))*kpoisson))/nVdiode 30 40 DC 0

E0 70 0 VALUE=I(VS1) VS2 70 1 DC 0

GLrec 1 0 VALUE=(knˆ2)*hp*((pos(I(VS1)))ˆ2)/(h*kidiodo)GRS1 1 2 VALUE=(V(1)-V(2))*Da*kn/kidiode/+((W-sqrt(pos(-V(20,40))*kpoisson))/n)ˆ2)GRS2 2 3 VALUE=(V(2)-V(3))*Da*kn/kidiode/+((W-sqrt(pos(-V(20,40))*kpoisson))/n)ˆ2)GRS3 3 4 VALUE=(V(3)-V(4))*Da*kn/kidiode/+((W-sqrt(pos(-V(20,40))*kpoisson))/n)ˆ2)GRS4 4 5 VALUE=(V(4)-V(5))*Da*kn/kidiode/+((W-sqrt(pos(-V(20,40))*kpoisson))/n)ˆ2)GRS5 5 6 VALUE=(V(5)-V(6))*Da*kn/kidiode/+((W-sqrt(pos(-V(20,40))*kpoisson))/n)ˆ2)GRS6 6 7 VALUE=(V(6)-V(7))*Da*kn/kidiode/+((W-sqrt(pos(-V(20,40))*kpoisson))/n)ˆ2)GRS7 7 8 VALUE=(V(7)-V(8))*Da*kn/kidiode/+((W-sqrt(pos(-V(20,40))*kpoisson))/n)ˆ2)GRS8 8 9 VALUE=(V(8)-V(9))*Da*kn/kidiode/+((W-sqrt(pos(-V(20,40))*kpoisson))/n)ˆ2)GRS9 9 10 VALUE=(V(9)-V(10))*Da*kn/kidiode/+((W-sqrt(pos(-V(20,40))*kpoisson))/n)ˆ2)

RP1 1 0 2*thl*kidiode/knRP2 2 0 thl*kidiode/knRP3 3 0 thl*kidiode/knRP4 4 0 thl*kidiode/knRP5 5 0 thl*kidiode/knRP6 6 0 thl*kidiode/knRP7 7 0 thl*kidiode/knRP8 8 0 thl*kidiode/knRP9 9 0 thl*kidiode/knRP10 10 0 thl*kidiode/knRP11 11 0 2*thl*kidiode/kn

CP1 1 0 .5*kn/kidiodeCP2 2 0 1*kn/kidiodeCP3 3 0 1*kn/kidiodeCP4 4 0 1*kn/kidiodeCP5 5 0 1*kn/kidiode

91

B – Pspice subcircuit listing - standard diode

CP6 6 0 1*kn/kidiodeCP7 7 0 1*kn/kidiodeCP8 8 0 1*kn/kidiodeCP9 9 0 1*kn/kidiodeCP10 10 0 1*kn/kidiodeCP11 11 0 .5*kn/kidiode

GRrec 11 0 VALUE=(knˆ2)*hn*((pos(V(11)))ˆ2)/(w*kidiode)GR 0 11 VALUE=I(VS2)/b

.ENDS Pin

92

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Modeling and Simulation of Power PiN Diodes within SPICE · Modeling and Simulation of Power PiN Diodes within SPICE Gustavo Buiatti Direttore del corso di dottorato Prof. Carlo Naldi

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