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All injection level power PiN diode model including temperature dependence
Solid-State Electronics 51 (2007) 719–725 www.elsevier.com/locate/sse
All injection level power PiN diode model including temperature dependence Nebojsa Jankovic b
a,*
, Tatjana Pesic a, Petar Igic
b
a Faculty of Electronic Engineering Nis, University of Nis, Aleksandra Medvedeva 14, 18000 Nis, Serbia School of Engineering, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, United Kingdom
Received 22 August 2006; received in revised form 19 January 2007; accepted 3 February 2007 Available online 26 March 2007
The review of this paper was arranged by Prof. S. Cristoloveanu
Abstract A novel power PiN diode model is derived based on the generalised all-injection level minority carrier drift–diffusion theory. An equivalent lossy transmission lines describing the carriers transport trough arbitrarily doped emitter and base quasi-neutral regions are defined. The extended electro-thermal diode model including temperature dependences of carrier transport parameters is also described and implemented in circuit simulator PSPICE. By tuning a small set of model parameters, an excellent agreement of modelling results with numerical simulations of realistic power PiN diode is obtained for different switching conditions and temperatures. 2007 Elsevier Ltd. All rights reserved. Keywords: PiN; Power; Diode; Model; Electro-thermal
1. Introduction Power PiN diodes are relatively simple devices but modeling of their transient behavior is a complex task. If the standard SPICE diode model is used to simulate a power diode, neither the static nor dynamic behavior can be obtained correctly [1]. Inaccuracy occurs predominantly due to the inability of conventional diode models to incorporate the dynamic charge storage and the diode conductivity modulation effects at high-level injection conditions. In order to evaluate carrier distribution in high level injection, many physics-based power PiN diode models [2–13] rely on solving the ambipolar diffusion equation in lightly doped N base layer. These models inherently omit, however, the low-level injection conditions that are important during the diode turn-off or the operation at *
Corresponding author. Tel.: +381 18 529 338. E-mail address: [email protected] (N. Jankovic).
0038-1101/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.sse.2007.02.018
small biasing voltages. Hence, in order to improve the overall modeling accuracy, the ambipolar approach is often combined with the classic low-level injection diode equations [2,8,9]. In this paper, we describe, for the first time, a novel power PiN diode model based on the theory of all-injection level minority carrier transport trough an arbitrarily doped quasi-neutral regions (QNRs) [14]. In contrast to ambipolar approach, the new diode model exhibits a natural transition from high- to low-level injection operating conditions. It is achieved by implementation of the equivalent diode subcircuit that virtually solves a coupled minority-carrier drift–diffusion and continuity equations [14,20]. An electro-thermal model of power PiN diode is also developed and implemented in PSPICE simulator. The model accuracy is verified trough comparisons with the numerical simulations of PiN power diode having the net doping profile obtained from process simulator ATHENA [15] as shown in Fig. 1.
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N. Jankovic et al. / Solid-State Electronics 51 (2007) 719–725
Using the new spatially dependant variables u and J defined as:
19
10
+
P
u¼
−3
Concentration (cm )
18
10
17
10
+
N buffer 16
10
P
15
10
−
N base
10
ð6Þ
we can transform (1) to:
13
10
2
4
42
44
46
48
50
Depth (μm)
Fig. 1. One-dimensional net doping profile of studied power PiN diode.
2.1. Basic theory For the start, one can assume a p-type QNR that is arbitrarily doped as NA(x). Then, the one-dimensional drift– diffusion isothermal minority-carrier transport equations at all injection levels are [14]: J n ¼ qnln K þ kT ln
on ; ox
on 1 oJ n ¼ R; ot q ox
ð1Þ ð2Þ
where Jn and n are the electron current and concentration densities, respectively. All other symbols have their usual meaning. The well-known boundary conditions of the system (1) and (2) are: ueb uð0Þ ¼ exp 1; ð3Þ VT J ðwÞ qnðW Þvs ðW Þ;
ou p ¼ J: ox kT ln n2ie
ð8Þ
In addition, using an approximation on/ot op/ot in combination with (6), we can also transform (2) as:
2. The PiN diode model
ð4Þ
where ueb is the diode external biasing, and vs(W) and n(W) are the minority carrier effective velocity and concentration, respectively, taken at the QNR’s end (x = W). Also, VT(=kT/q) denotes the thermal voltage. The total carrier recombination R appearing in Eq. (2) represents the sum of Shockley–Read–Hall RSRH and Auger recombination RA as [16]: R ¼ RSRH þ RA ¼
and J ¼ J n ;
and the approximate expression for spatially varied built-in electric field K(x) in QNR region [17]: kT d pðxÞ ln 2 KðxÞ ; ð7Þ q dx nie ðxÞ
−
14
pn n2ie ; p 0 n0
pn n2ie þ ðC n n þ C p pÞðpn n2ie Þ; sp0 ðn þ nie Þ þ sn0 ðp þ nie Þ ð5Þ
where sn0 and sp0 are the electron and holes doping-dependant minority carrier lifetimes at low injection level, nie is the effective intrinsic carrier concentration incorporating band-gap-narrowing effects and Cn and Cp are Auger constants [16].
oJ n2 ou ¼ qR þ q ie ox p þ n ot
ð9Þ
The Eqs. (8) and (9) are analogues to well-known telegraphers’ equations usually appearing in transmission line theory in the form: ou 0 0 o ¼ R þL J; ð10Þ ox ot oJ o ¼ G0 þ C 0 u; ð11Þ ox ot They describe the equivalent non-linear inhomogeneous lossy transmission line (TL) with spatial and voltage dependant line parameters R 0 , L 0 , G 0 , C 0 . They are easily defined from (8) and (9) as: R0 ¼
p ; kT ln n2ie
and 0
G ¼
qn2ie
C0 ¼ q
n2ie ; L0 ¼ 0 pþn
1 Cnn þ Cpp þ ; sp0 ðn þ nie Þ þ sn0 ðp þ nie Þ
ð12Þ
ð13Þ
Consequently, u(0) becomes the input biasing voltage of the equivalent TL with loading impedance Zl derived from (4) as: Zl ¼
uðW Þ pðW Þ : J ðW Þ qn2ie vs ðW Þ
ð14Þ
The expressions (10)–(14) implies that an arbitrarily doped QNR can be described by the equivalent lossy TL represented with a cascade of finite number (N) of RC segments of the length w(=W/N). Fig. 2 shows an example of QNR segmentation method in case of N = 3. Each kth segment (k = 1:N) is taken as uniformly doped with doping concentration Nk = NA((k 1/2) Æ w) corresponding to the segment position. The non-linear spatially and voltage dependent lumped elements Rk, Gk, Ck and Zl of a single kth segment are expressed with analytical formulas (A6), (A7) derived in the Appendix A.
N. Jankovic et al. / Solid-State Electronics 51 (2007) 719–725
Fig. 2. Illustration of arbitrarily doped QNR represented by the equivalent TL with three RC segments.
2.2. The circuit model Fig. 3 shows the circuit implementation of PiN power diode model. A core model is shown in Fig. 3a. It consists of a voltage variable junction capacitor C(ueb), a variable resistor Rd (ueb), and the current-controlled current source id. The capacitance model has the same form as in the conventional SPICE diode model [18]. The current-controlled current source id in Fig. 3a mirrors the current flowing trough the voltage source f(ueb) of the auxiliary sub-circuits shown in Fig. 3b and c. In this work, the QNRs of P+/P emitter and the N+/N base are represented with N = 2 and N = 4 RC segments, respectively. In spite of choosing a relatively small N, the good modeling accuracy can be achieved as will be shown in Section 5. Note that the model accuracy can be improved by using TLs with more RC segments (higher N) [14,20], but at the expense of prolonged simulation time and increased sub-circuit complexity. The conductivity modulation effect [19] is included in the model by defining the voltage dependant non-linear resistor Rd(ueb) as: Rd ðueb Þ ¼ Rd0 þ RMF expðukl =NFÞ for ueb > 0;
ð15Þ
721
where Rd0 is the unmodulated diode resistance, while the second term represents the conductivity modulated base resistance with RMF and NF as fitting parameters. The variable ukl is the voltage of the last node of the Nbase TL (the node between the third and the fourth RC segment of the base in Fig. 3b), which exactly follows the dynamics of N base charging when the diode is switched into a forward-bias condition. The efficiency of (15) is demonstrated in Fig. 4 showing the forward recovery waveforms obtained with the new diode model. In these simulations, the diode was supplied with ideal current ramp having ia,max = 1.1 A as shown in the inset of Fig. 4. Three cases of ramp rise times have been considered: 0.2 A/lm, 0.1 A/lm and 0.07 A/lm. Results shown in Fig. 4 confirm that the new diode model including (15) is able to predict the forward voltage overshoots and, moreover, the experimentally confirmed effect [4–6] of lowering the voltage peaks with extending the rise time of current pulse. The effect of moving boundary condition [19] is also included in the new diode model trough the dependence of N base QNR width Wb on the inverse biasing voltage ueb (<0). Namely, in the model, the width w of all RC-segments modeling the transport trough N base layer is taken in (A7) as: w¼
pffiffiffiffiffiffiffiffiffi
W ðjueb jÞ W b ¼ 1 EARLY jueb j N N
if ueb < 0; ð16Þ
where EARLY is another model parameter. The expression (16) is based on the depletion approximation theory [16] assuming that the N+/N junction (Fig. 1) is super-abrupt. The base lifetime s10 parameter appearing in (A5) is highly important for accurate modeling of the power PiN diodes. During the PiN diode manufacturing, a special fabrication step for adjusting s10 in the N base layer is usually performed [19] in order to obtain a device faster response. Consequently, a new lifetime parameter s is introduced as: s ¼ TAU s10
ð17Þ
8
Anode Voltage (V)
7
1.1A
0.2 A/μs
6 5
i a (t)
0.1A/μs
4
0
0.07A/μs
3 2 1 0
0
2
4
6
8
10
Time (μs) Fig. 3. (a) The PiN diode circuit model; (b) the auxiliary TL sub-circuit (c) the basic RC segment.
Fig. 4. Forward recovery waveforms of PiN power diode supplied by an ideal current ram (shown in the inset) simulated with the new model for different pulse rise times.
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N. Jankovic et al. / Solid-State Electronics 51 (2007) 719–725
1.5
0
0
0.6
-100
0.3 -150 0.0 TAU=1 TAU=0.1 TAU=10 EARLY=0.45
-0.3 -0.6 5.00
5.01
-200
5.03
5.04
5.05
5.06
-1000
The model Silvaco R=1Ω
5.0
5.1
50
1.2
1.2
0
1.0
-50
0.8
Anode current (A)
-150
0.3
-200
0.0
-250
-0.3 -0.6
EARLY=0.45 EARLY=0.045 EARLY=0.9 TAU=10
-0.9 5.00
5.01
5.02
-300
Anode voltage (V)
-100
-350 -400
5.03
5.04
5.05
Anode current (A)
1.5
0.6
5.2
5.3
5.4
5.5
5.6
Time (μs)
Time (μs)
0.9
L=100μH
L=10μH
-500
-1500 4.9
-250
5.02
Anode voltage (A)
L=1μH
-50
0.9
Anode voltage (V)
Anode current (A)
1.2
The model Silvaco R=1 Ω L=100μH
0.6 0.4 L=10μH
0.2 0.0
L=1μH
-0.2 -0.4 -0.6 4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
Time (μs)
5.06
Time (μs) Fig. 5. Reverse recovery voltage and current waveforms of PiN diode simulated for various (a) TAU and (b) EARLY parameters.
It replaces s10 of (A7) only in those RC-cells representing the N base layer. The use of TAU parameter allows a better model calibration by separate fine tuning of the minority hole lifetime in the N base layer.
1.2 slope ~ 1/Vt
-1
Anode current (A)
1.0 0.8
Anode current (A)
10
0.6
-3
10
-5
10
SILVACO The model
-7
10
T=300K
-9
10 0.25
0.50
0.75
1.00
1.25
Fig. 7. The reverse recovery waveforms of (a) the anode voltages and (b) the anode currents simulated for different values for L and for constant R = 1 X in the test circuit shown in the inset.
The sensitivity of simulated diode reverse recovery waveforms on TAU and EARLY parameters is illustrated in Fig. 5. In these simulations, the test circuit shown in the inset of Fig. 7a has been analyzed with L = 1 lH and R = 1 X . Results shown in Fig. 5a predict correctly that the reverse recovery current peak decreases with decreasing TAU (e.g., imposing a shorter lifetime in the N base layer), whereas the voltage peak is weakly sensitive on the same parameter. In contrast, the results in Fig. 5b show that the variation of EARLY parameter influences both voltage and current peaks in a complex manner. The results presented in Fig. 5 confirm the potentials of TAU and EARLY parameters for successful calibration of the new diode model.
1.50
Anode voltage (V)
0.4
3. Electro-thermal model
T=398K 0.2 0.0 0.0
The model Silvaco
0.5
T=300K 1.0
1.5
2.0
2.5
Anode voltage (V) Fig. 6. Simulated forward DC characteristics of PiN power diode operating at different temperatures. The figure in inset shows a log-scale of anode current where the transition from low-to-high injection operation is clearly observed at around 0.6 V.
The electro-thermal PiN diode model is based on temperature dependences of C1–C9 parameters calculated trough (A7), (A9) and (A10), assuming that the impurity atoms are fully ionized for T > 120 K, hence ND 5 f(T). In addition, the increase of s10 in low-doped N base layer [21] followed by simultaneous decrease of l0n,p [22] with temperature increase are included with semi-empirical expressions of s and Rd as:
N. Jankovic et al. / Solid-State Electronics 51 (2007) 719–725
ð18Þ ð19Þ
where b and NRES are the fitting parameters, and T0 is the referent temperature (K). 4. Parameter extraction The electro-thermal power PiN diode model described in this paper requires seven parameters Rd0, NF, RMF, NRES, TAU, b and EARLY, since the constancies C1– C9 can be calculated from (A7) using a known diode doping profile and geometry. In addition, a standard SPICE diode capacitance model [18] is employed in this work with well-known parameters CJ0, VJ, and M. They are extracted in this work from diode C–V characteristics obtained form AC simulations with ATLAS. The (TAU, b) and (Rd0, NRES) parameters are obtained from tuning the model with simulated DC forward characteristics at high and low current regions, respectively. The base resistance parameters NF and RMF are determined from simulated forward recovery waveforms. The EARLY parameter is extracted from the resistive and/or inductive turn-off diode characteristics obtained with the mixed-mode numerical simulations. For final calibration, however, TAU and EARLY have to be fine tuned simultaneously (for the reason illustrated in Fig. 4), which eventually yield to a minimal error of modeling both the DC and the transient diode characteristics. The extracted parameter values with their notation and meanings are listed in Table 1 shown in the Appendix B. 5. Modeling results and numerical simulations The electrical characteristics of 25 lm2 area power PiN diode described by its one-dimensional net doping profile in Fig. 1 are simulated using the test circuit shown in the inset of Fig. 7a. First, the DC characteristics are obtained for L = 0 and R = 1 X at different temperatures using the mixed-mode numerical device simulator ATLAS [15]. They are used to extract TAU, b, Rd0, NF, NF, RMF, and NRES parameters. Note that C1–C9 parameters are precalculated from (A7) using the physical constants given in the Appendix A. Fig. 6 shows a good match between the model and ATLAS results of simulated DC characteristics obtained for both high and low level injection regions. The later is better observed when plotting the anode current in log-scale as shown in the inset of Fig. 6. It shows that the new model yields a smooth transition from low to high injection operation depicted by the change of curve grading from 1 to 0.5 at around 0.6 V. It stems from the all-injection level TL modeling approach presented that solves virtually a coupled minority-carrier drift–diffusion and continuity equations in the diode QNR regions [14,20].
Following the DC calibration, the diode reverse recovery waveforms are then compared with ATLAS simulations as shown in Figs. 7 and 8. An excellent match between TL model and ATLAS results is obtained for different L and R values set in the test circuit. The extended electro-thermal model including temperature effects is also validated by simulating instantaneous power dissipation P(t, T) = ia(t, T) Æ ua(t, T) developed during the diode reverse recovery. The test is repeated for two different temperatures, T = 300 K and T = 398 K. An excellent agreement is obtained between the new model and ATLAS results as shown in Fig. 9. Finally, it is worth to point out that the emitter recombination effect is inherently included in the diode model trough the equivalent TL modeling of the emitter QNR. For example, the one would obtain a substantially different reverse recovery waveforms (not shown in Fig. 7 and 8.) departing from ATLAS results if the two RC segments of P+/P emitter region (see Fig. 3b) are omitted in the diode model. It confirms that the minority carrier’s distribution in the P+/P emitter QNR has substantial influence on the diode transient behavior, which has also been emphasized in Refs. [24–26].
50 0
Anode voltage (A)
b T sðT Þ ¼ TAU s10 ; T0 NRES T Rd0 ðT Þ ¼ Rd0 ; T0
R=0.01Ω
-50 -100
R=1Ω
-150 -200 -250
R=0.1Ω
L=1μH
-300
The model Silvaco
-350 4.90
4.95
5.00
5.05
5.10
5.15
5.20
Time (μs) 6
The model Silvaco L=1μH
R=0.01Ω
5
Anode current (A)
723
4
R=0.1Ω
3 2 1
R=1Ω
0 -1 -2 4.90
4.95
5.00
5.05
5.10
5.15
5.20
Time (μs) Fig. 8. The reverse recovery waveforms of (a) the anode voltages and (b) the anode currents simulated for different values for R and for constant L = 1 lH.
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N. Jankovic et al. / Solid-State Electronics 51 (2007) 719–725
where p(u, x), n(u, x) and NA(x) are the spatially and voltage dependent hole and electron densities and the doping concentration, respectively. Combining (A1) with (6) yields: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# " N A ðxÞ 4uðxÞn2ie ðxÞ 1 þ 1 þ nðu; xÞ ¼ ; ðA2Þ 2 N 2A ðxÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# " N A ðxÞ 4uðxÞn2ie ðxÞ 1þ 1þ pðu; xÞ ¼ : ðA3Þ 2 N 2A ðxÞ
70 The model Silvaco
Power dissipation (W)
60
R=1Ω L=1μH
50 40 T=300K
30 20
T=398K
10 0 4.90 4.95 5.00 5.05 5.10 5.15 5.20 5.25 5.30
Time (μs) Fig. 9. The instantaneous power dissipation P(t, T) = ia(t, T) Æ ua(t, T) developed during the diode reverse recovery transient time simulated at different temperatures (R = 1X, L = 1 lH).
6. Conclusions Novel power PiN diode model based on the generalised all-injection level minority carrier drift–diffusion theory is presented in this paper. The equivalent lossy transmission lines describing the carriers’ transport trough arbitrarily doped emitter and base quasi-neutral regions are defined. The effect of the moving boundary condition and the emitter recombination effect are included in the model. The extended electro-thermal diode model including the temperature as parameter is also described and implemented in circuit simulator PSPICE. By tuning a small set of model parameters, a very good agreement between modelling results and numerical simulation results of the realistic power PiN diode is obtained for different switching conditions and temperatures. Apart from thermal equilibrium simulations, it is also doable to employ this new diode model using a standard approach with auxiliary thermal analog sub-circuit [27] to predict the device’s dynamic self-heating. However, due to the complex temperature dependences of the model parameters C1–C9 expressed by (A7)–(A10), the present form of electro-thermal diode model has exhibited the substantial convergence problems during self-heating simulations with SPICE. Hence, in order to obtain the convergence-free self-heating simulations, the authors have been developing an improved version of this electro-thermal diode model with optimized expressions for C1–C9 temperature dependences, which will be discussed elsewhere.
where l0 is the doping-dependant low-field mobility, K is the effective electric field and vsat is the carrier drift saturation velocity. The doping-dependant carrier lifetime is expressed as [23]: s10 s0n;p ¼ ; ðA5Þ 1 þ NN AR where s10 = 5 · 107 s and NR = 5 · 1016 cm3. Finally, by replacing p(u,x), n(u,x) in (12)–(14) and (A8) in (A4), one can derive the analytical expressions for Ck, Gk, Rk, and Zl as: C2 C k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ C 1 u2k Gk ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ C 5 ð1 þ 1 þ C 1u2k Þ C 3 ð1 þ C 1 u2k Þ þ C 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ C 6 ð1 þ 1 þ C 1 u2k Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rk ¼ C 7 ð1 þ 1 þ C 1 u2k Þ Z l ¼ C 9 ð1 þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 u2k1 u2kþ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ C 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ C 1 u2k ð1 þ 1 þ C 1 u2k Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ C 1 u2nþ1 Þ
ðA6Þ where u2k1, u2k+1 and u2k are the input, the output and the middle node voltage, respectively, of the kth RC segment. The associated constancies C1–C9 are expressed as: C1 ¼
4n2ie ; N 2k
C4 ¼
2s10 ; qnie w
C2 ¼
qn2ie w ; Nk
C5 ¼
C3 ¼
s10 N k ; qn2ie w
qn2ie C n N k w ; 2
qn2ie C p N k w N kw ; C7 ¼ ; 2 2qV T n2ie l0 2 l0 C 1 V T N k¼N : ; C9 ¼ C8 ¼ vsat 2w 2qn2ie vs ðW Þ
C6 ¼
Appendix A The derivations of analytical expressions for Rk, Gk, Ck and Zl are described in what follows. One can start with quasi-neutrality approximation [17]: pðu; xÞ N A ðxÞ þ nðu; xÞ
The high-field carrier mobilities are calculated by following formula [22]: l0n;p ln;p ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA4Þ
ffi; l0n;0p K 2 1 þ vsat
ðA1Þ
ðA7Þ
Note that the effective electric field Kk appearing in each kth segment is approximately derived from (7) and (A3) as:
N. Jankovic et al. / Solid-State Electronics 51 (2007) 719–725 Table 1 The PiN power diode model parameters Parameter
Description
Value 2
Diode surface area (cm ) Unmodulated total resistance (X) Grading factor for conductivity modulated base resistance Zero-bias conductivity modulated base resistance (X) Exponent for diode resistance temperature dependence Adjustment factor for carrier lifetime in lowdoped base region Exponent for carrier lifetime temperature dependence in low-doped base region Pre-factor for moving boundary condition effect Zero-bias junction capacitance (F/cm2) Junction potential (V) Grading coefficient for junction capacitance
AREA Rd0 NF RMF NRES TAU b EARLY CJ0 VJ M
Kk
25 · 108 107 2.5 3 · 105 0.95 0.1 8 0.05 2 · 107 1.5 0.5
kT 1 C1 u2k1 u2kþ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; q ð1 þ 1 þ C 1 uÞ 2 1 þ C 1 u w ðA8Þ
and it appears in the mobility Eq. (A4). The temperature and doping dependence of parameters l0n,p and n2ie are calculated from published semi-empirical expressions [22–24] as: l0n ¼ 88 T 0:57 n þ
7:4 108 T 2:33 n ; 0:146 1 þ N D =ð1:26 1017 T 2:4 n Þ 0:88 T n
l0p ¼ 54:3 T n0:57 1:36 108 T 2:23 n 0:146 1 þ N D =ð2:35 1017 T 2:4 n Þ 0:88 T n qV g0 qDV g0 n2ie ¼ C n T 3 exp exp ; kT kT 8 9 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 < N = N A A þ ln þ C c1 ; DV g0 ¼ V ref ln : N ref ; N ref þ
ðA9Þ
ðA10Þ
where Tn = T/300, vsat = 1.1 · 107 cm/s, Cn = 10.38 · 1032 cm6 K3, Vg0 = 1.206 V, Vref = 0.009 V, Nref = 1.3 · 1017 cm3, Cc1 = 0.5. Appendix B See Table 1. References [1] Massmoudi N, M’bairi D, Allard B, Morel H. On the validity of the standard SPICE model of the diode for simulation in power electronics. IEEE Trans Ind Electron 2001;48:864–7. [2] Vogler T, Schroder D. A new and accurate circuit-modeling approach for the power-diode. In: Proceedings of power electronics specialists conference, vol. 2; 1992. p. 870–6. [3] Strollo AGM. A new SPICE subcircuit model of power p-i-n diode. IEEE Trans Power Electron 1994;9:553–9.
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[4] Strollo AGM, Spirito P. A new PiN diode modelling approach for power electronic PSPICE simulations. In: Proceedings of power electronics specialists conference, vol. 1; 1994. p. 52–8. [5] Strollo AGM. A new SPICE model of power P-I-N diode based on asymptotic waveform evaluation. IEEE Trans Power Electron 1997;12:12–20. [6] Maxim A, Maxim G. A novel power PiN diode behavioral SPICE macromodel including the forward and reverse recoveries and the self-heating process. In: Proceedings of applied power electronics conference and exposition, vol. 2; 2000. p. 1088–94. [7] Kolessar R. Physical study of the power diode turn-on process. In: Proceedings of industry applications conference, vol. 5; 2000. p. 2934–40. [8] Kraus R, Hoffmann K, Mattausch HJ. A precise model for the transient characteristics of power diodes. In: Proceedings of 23rd power electronics specialists conference 1992 (PESC ’92 ); 1992. p. 863–9. [9] Lauritzen PO, Ma CL. A simple diode model with reverse recovery. IEEE Trans Power Electron 1991;6:188–91. [10] Igic PM, Mawby PA, Towers MS, Batcup S. New physically-based PiN diode compact model for circuit modeling applications. IEE Proc Circuit Des Syst 2002;149:257–63. [11] Kang X, Caiafa A, Santi E, Hudgins JL, Palmer PR. Parameter extraction for a power diode circuit simulator model including temperature dependent effects. In: Proceedings of applied power electronics conference and exposition, vol. 1; 2002. p. 452–8. [12] Buiatti GM, Cappelluti F, Ghione G. Power PiN diode model for PSPICE simulations. In: Proceedings of Applied Power Electronics Conference and Exposition, vol. 3; 2005. p. 1911–6. [13] Bryant AT, Palmer PR, Santi E, Hudgins JL. A compact diode model for the simulation of fast power diodes including the effects of avalanche and carrier lifetime zoning. In: Proceedings of IEEE power electronics specialists conference; 2005. p. 2042–8. [14] Jankovic N, Pesic T, Karamarkovic J. 1D physical non-quasi-static BJT circuit model based on the equivalent transmission line analysis. J Comput Electron 2004;3:13–24. [15] SILVACO User’s Manuel, SILVACO International Inc., 2000. [16] Sze SM. Physics of semiconductor devices. New York: Wiley & Sons; 1981. [17] Chyan Y, Chang C, Sze SM, Lin M, Liao K, Reif R. Analytical modeling of polycrystalline silicon emitter bipolar transistors under high-level injection. Solid-State Electron. 1994;37:1521–9. [18] HSPICE Manuel, Meta-software Inc., 1990. [19] Baliga BJ. Modern Power Devices. New York: Wiley; 1987. [20] Jankovic N, Pesic T, Karamarkovic J. 1D physically based nonquasi-static analog behavioural BJT model for SPICE. In: Proceedings of IEEE international conference on microelectronics, vol. 2; 2002. p. 463–8. [21] Hefner Jr AR. A dynamic electro-thermal model for the IGBT. IEEE Trans Ind Appl 1994;30:394–405. [22] Arora ND, Hauser JR, Roulston DJ. Electron and Hole Mobilities in Silicon as a Function of Concentration and Temperature. IEEE Trans Electron Dev 1982;29:292–5. [23] Jacoboni C, Canali C, Ottaviani G, Alberigi Quaranta A. A Review of Some Charge Transport Properties of Silicon. Solid-State Electron 1977;20:77–89. [24] Strollo AGM. Calculation of power diode reverse-recovery time for SPICE simulations. Electron Lett 1994;30:1109–10. [25] Mantooth HA, Duliere JL. A unified diode model for circuit simulation. IEEE Trans Power Electron 1997;12(5):816–23. [26] McNutt TR, Hefner AR, Mantooth HA, Duliere J, Berning DW, Singh R. Silicon carbide PiN and merged PiN Schottky power diode models implemented in the Saber circuit simulator. IEEE Trans Power Electron 2004;19(3):573–81. [27] Maxim A, Maxim G. A novel power PiN diode behavioral SPICE macromodel including the forward and reverse recoveries and the selfheating process. In: 15th annual IEEE applied power electronics conference and exposition; 2000. p. 1088–94.
All injection level power PiN diode model including temperature dependence Nebojsa Jankovic b
a,*
, Tatjana Pesic a, Petar Igic
b
a Faculty of Electronic Engineering Nis, University of Nis, Aleksandra Medvedeva 14, 18000 Nis, Serbia School of Engineering, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, United Kingdom
Received 22 August 2006; received in revised form 19 January 2007; accepted 3 February 2007 Available online 26 March 2007
The review of this paper was arranged by Prof. S. Cristoloveanu
Abstract A novel power PiN diode model is derived based on the generalised all-injection level minority carrier drift–diffusion theory. An equivalent lossy transmission lines describing the carriers transport trough arbitrarily doped emitter and base quasi-neutral regions are defined. The extended electro-thermal diode model including temperature dependences of carrier transport parameters is also described and implemented in circuit simulator PSPICE. By tuning a small set of model parameters, an excellent agreement of modelling results with numerical simulations of realistic power PiN diode is obtained for different switching conditions and temperatures. 2007 Elsevier Ltd. All rights reserved. Keywords: PiN; Power; Diode; Model; Electro-thermal
1. Introduction Power PiN diodes are relatively simple devices but modeling of their transient behavior is a complex task. If the standard SPICE diode model is used to simulate a power diode, neither the static nor dynamic behavior can be obtained correctly [1]. Inaccuracy occurs predominantly due to the inability of conventional diode models to incorporate the dynamic charge storage and the diode conductivity modulation effects at high-level injection conditions. In order to evaluate carrier distribution in high level injection, many physics-based power PiN diode models [2–13] rely on solving the ambipolar diffusion equation in lightly doped N base layer. These models inherently omit, however, the low-level injection conditions that are important during the diode turn-off or the operation at *
Corresponding author. Tel.: +381 18 529 338. E-mail address: [email protected] (N. Jankovic).
0038-1101/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.sse.2007.02.018
small biasing voltages. Hence, in order to improve the overall modeling accuracy, the ambipolar approach is often combined with the classic low-level injection diode equations [2,8,9]. In this paper, we describe, for the first time, a novel power PiN diode model based on the theory of all-injection level minority carrier transport trough an arbitrarily doped quasi-neutral regions (QNRs) [14]. In contrast to ambipolar approach, the new diode model exhibits a natural transition from high- to low-level injection operating conditions. It is achieved by implementation of the equivalent diode subcircuit that virtually solves a coupled minority-carrier drift–diffusion and continuity equations [14,20]. An electro-thermal model of power PiN diode is also developed and implemented in PSPICE simulator. The model accuracy is verified trough comparisons with the numerical simulations of PiN power diode having the net doping profile obtained from process simulator ATHENA [15] as shown in Fig. 1.
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Using the new spatially dependant variables u and J defined as:
19
10
+
P
u¼
−3
Concentration (cm )
18
10
17
10
+
N buffer 16
10
P
15
10
−
N base
10
ð6Þ
we can transform (1) to:
13
10
2
4
42
44
46
48
50
Depth (μm)
Fig. 1. One-dimensional net doping profile of studied power PiN diode.
2.1. Basic theory For the start, one can assume a p-type QNR that is arbitrarily doped as NA(x). Then, the one-dimensional drift– diffusion isothermal minority-carrier transport equations at all injection levels are [14]: J n ¼ qnln K þ kT ln
on ; ox
on 1 oJ n ¼ R; ot q ox
ð1Þ ð2Þ
where Jn and n are the electron current and concentration densities, respectively. All other symbols have their usual meaning. The well-known boundary conditions of the system (1) and (2) are: ueb uð0Þ ¼ exp 1; ð3Þ VT J ðwÞ qnðW Þvs ðW Þ;
ou p ¼ J: ox kT ln n2ie
ð8Þ
In addition, using an approximation on/ot op/ot in combination with (6), we can also transform (2) as:
2. The PiN diode model
ð4Þ
where ueb is the diode external biasing, and vs(W) and n(W) are the minority carrier effective velocity and concentration, respectively, taken at the QNR’s end (x = W). Also, VT(=kT/q) denotes the thermal voltage. The total carrier recombination R appearing in Eq. (2) represents the sum of Shockley–Read–Hall RSRH and Auger recombination RA as [16]: R ¼ RSRH þ RA ¼
and J ¼ J n ;
and the approximate expression for spatially varied built-in electric field K(x) in QNR region [17]: kT d pðxÞ ln 2 KðxÞ ; ð7Þ q dx nie ðxÞ
−
14
pn n2ie ; p 0 n0
pn n2ie þ ðC n n þ C p pÞðpn n2ie Þ; sp0 ðn þ nie Þ þ sn0 ðp þ nie Þ ð5Þ
where sn0 and sp0 are the electron and holes doping-dependant minority carrier lifetimes at low injection level, nie is the effective intrinsic carrier concentration incorporating band-gap-narrowing effects and Cn and Cp are Auger constants [16].
oJ n2 ou ¼ qR þ q ie ox p þ n ot
ð9Þ
The Eqs. (8) and (9) are analogues to well-known telegraphers’ equations usually appearing in transmission line theory in the form: ou 0 0 o ¼ R þL J; ð10Þ ox ot oJ o ¼ G0 þ C 0 u; ð11Þ ox ot They describe the equivalent non-linear inhomogeneous lossy transmission line (TL) with spatial and voltage dependant line parameters R 0 , L 0 , G 0 , C 0 . They are easily defined from (8) and (9) as: R0 ¼
p ; kT ln n2ie
and 0
G ¼
qn2ie
C0 ¼ q
n2ie ; L0 ¼ 0 pþn
1 Cnn þ Cpp þ ; sp0 ðn þ nie Þ þ sn0 ðp þ nie Þ
ð12Þ
ð13Þ
Consequently, u(0) becomes the input biasing voltage of the equivalent TL with loading impedance Zl derived from (4) as: Zl ¼
uðW Þ pðW Þ : J ðW Þ qn2ie vs ðW Þ
ð14Þ
The expressions (10)–(14) implies that an arbitrarily doped QNR can be described by the equivalent lossy TL represented with a cascade of finite number (N) of RC segments of the length w(=W/N). Fig. 2 shows an example of QNR segmentation method in case of N = 3. Each kth segment (k = 1:N) is taken as uniformly doped with doping concentration Nk = NA((k 1/2) Æ w) corresponding to the segment position. The non-linear spatially and voltage dependent lumped elements Rk, Gk, Ck and Zl of a single kth segment are expressed with analytical formulas (A6), (A7) derived in the Appendix A.
N. Jankovic et al. / Solid-State Electronics 51 (2007) 719–725
Fig. 2. Illustration of arbitrarily doped QNR represented by the equivalent TL with three RC segments.
2.2. The circuit model Fig. 3 shows the circuit implementation of PiN power diode model. A core model is shown in Fig. 3a. It consists of a voltage variable junction capacitor C(ueb), a variable resistor Rd (ueb), and the current-controlled current source id. The capacitance model has the same form as in the conventional SPICE diode model [18]. The current-controlled current source id in Fig. 3a mirrors the current flowing trough the voltage source f(ueb) of the auxiliary sub-circuits shown in Fig. 3b and c. In this work, the QNRs of P+/P emitter and the N+/N base are represented with N = 2 and N = 4 RC segments, respectively. In spite of choosing a relatively small N, the good modeling accuracy can be achieved as will be shown in Section 5. Note that the model accuracy can be improved by using TLs with more RC segments (higher N) [14,20], but at the expense of prolonged simulation time and increased sub-circuit complexity. The conductivity modulation effect [19] is included in the model by defining the voltage dependant non-linear resistor Rd(ueb) as: Rd ðueb Þ ¼ Rd0 þ RMF expðukl =NFÞ for ueb > 0;
ð15Þ
721
where Rd0 is the unmodulated diode resistance, while the second term represents the conductivity modulated base resistance with RMF and NF as fitting parameters. The variable ukl is the voltage of the last node of the Nbase TL (the node between the third and the fourth RC segment of the base in Fig. 3b), which exactly follows the dynamics of N base charging when the diode is switched into a forward-bias condition. The efficiency of (15) is demonstrated in Fig. 4 showing the forward recovery waveforms obtained with the new diode model. In these simulations, the diode was supplied with ideal current ramp having ia,max = 1.1 A as shown in the inset of Fig. 4. Three cases of ramp rise times have been considered: 0.2 A/lm, 0.1 A/lm and 0.07 A/lm. Results shown in Fig. 4 confirm that the new diode model including (15) is able to predict the forward voltage overshoots and, moreover, the experimentally confirmed effect [4–6] of lowering the voltage peaks with extending the rise time of current pulse. The effect of moving boundary condition [19] is also included in the new diode model trough the dependence of N base QNR width Wb on the inverse biasing voltage ueb (<0). Namely, in the model, the width w of all RC-segments modeling the transport trough N base layer is taken in (A7) as: w¼
pffiffiffiffiffiffiffiffiffi
W ðjueb jÞ W b ¼ 1 EARLY jueb j N N
if ueb < 0; ð16Þ
where EARLY is another model parameter. The expression (16) is based on the depletion approximation theory [16] assuming that the N+/N junction (Fig. 1) is super-abrupt. The base lifetime s10 parameter appearing in (A5) is highly important for accurate modeling of the power PiN diodes. During the PiN diode manufacturing, a special fabrication step for adjusting s10 in the N base layer is usually performed [19] in order to obtain a device faster response. Consequently, a new lifetime parameter s is introduced as: s ¼ TAU s10
ð17Þ
8
Anode Voltage (V)
7
1.1A
0.2 A/μs
6 5
i a (t)
0.1A/μs
4
0
0.07A/μs
3 2 1 0
0
2
4
6
8
10
Time (μs) Fig. 3. (a) The PiN diode circuit model; (b) the auxiliary TL sub-circuit (c) the basic RC segment.
Fig. 4. Forward recovery waveforms of PiN power diode supplied by an ideal current ram (shown in the inset) simulated with the new model for different pulse rise times.
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1.5
0
0
0.6
-100
0.3 -150 0.0 TAU=1 TAU=0.1 TAU=10 EARLY=0.45
-0.3 -0.6 5.00
5.01
-200
5.03
5.04
5.05
5.06
-1000
The model Silvaco R=1Ω
5.0
5.1
50
1.2
1.2
0
1.0
-50
0.8
Anode current (A)
-150
0.3
-200
0.0
-250
-0.3 -0.6
EARLY=0.45 EARLY=0.045 EARLY=0.9 TAU=10
-0.9 5.00
5.01
5.02
-300
Anode voltage (V)
-100
-350 -400
5.03
5.04
5.05
Anode current (A)
1.5
0.6
5.2
5.3
5.4
5.5
5.6
Time (μs)
Time (μs)
0.9
L=100μH
L=10μH
-500
-1500 4.9
-250
5.02
Anode voltage (A)
L=1μH
-50
0.9
Anode voltage (V)
Anode current (A)
1.2
The model Silvaco R=1 Ω L=100μH
0.6 0.4 L=10μH
0.2 0.0
L=1μH
-0.2 -0.4 -0.6 4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
Time (μs)
5.06
Time (μs) Fig. 5. Reverse recovery voltage and current waveforms of PiN diode simulated for various (a) TAU and (b) EARLY parameters.
It replaces s10 of (A7) only in those RC-cells representing the N base layer. The use of TAU parameter allows a better model calibration by separate fine tuning of the minority hole lifetime in the N base layer.
1.2 slope ~ 1/Vt
-1
Anode current (A)
1.0 0.8
Anode current (A)
10
0.6
-3
10
-5
10
SILVACO The model
-7
10
T=300K
-9
10 0.25
0.50
0.75
1.00
1.25
Fig. 7. The reverse recovery waveforms of (a) the anode voltages and (b) the anode currents simulated for different values for L and for constant R = 1 X in the test circuit shown in the inset.
The sensitivity of simulated diode reverse recovery waveforms on TAU and EARLY parameters is illustrated in Fig. 5. In these simulations, the test circuit shown in the inset of Fig. 7a has been analyzed with L = 1 lH and R = 1 X . Results shown in Fig. 5a predict correctly that the reverse recovery current peak decreases with decreasing TAU (e.g., imposing a shorter lifetime in the N base layer), whereas the voltage peak is weakly sensitive on the same parameter. In contrast, the results in Fig. 5b show that the variation of EARLY parameter influences both voltage and current peaks in a complex manner. The results presented in Fig. 5 confirm the potentials of TAU and EARLY parameters for successful calibration of the new diode model.
1.50
Anode voltage (V)
0.4
3. Electro-thermal model
T=398K 0.2 0.0 0.0
The model Silvaco
0.5
T=300K 1.0
1.5
2.0
2.5
Anode voltage (V) Fig. 6. Simulated forward DC characteristics of PiN power diode operating at different temperatures. The figure in inset shows a log-scale of anode current where the transition from low-to-high injection operation is clearly observed at around 0.6 V.
The electro-thermal PiN diode model is based on temperature dependences of C1–C9 parameters calculated trough (A7), (A9) and (A10), assuming that the impurity atoms are fully ionized for T > 120 K, hence ND 5 f(T). In addition, the increase of s10 in low-doped N base layer [21] followed by simultaneous decrease of l0n,p [22] with temperature increase are included with semi-empirical expressions of s and Rd as:
N. Jankovic et al. / Solid-State Electronics 51 (2007) 719–725
ð18Þ ð19Þ
where b and NRES are the fitting parameters, and T0 is the referent temperature (K). 4. Parameter extraction The electro-thermal power PiN diode model described in this paper requires seven parameters Rd0, NF, RMF, NRES, TAU, b and EARLY, since the constancies C1– C9 can be calculated from (A7) using a known diode doping profile and geometry. In addition, a standard SPICE diode capacitance model [18] is employed in this work with well-known parameters CJ0, VJ, and M. They are extracted in this work from diode C–V characteristics obtained form AC simulations with ATLAS. The (TAU, b) and (Rd0, NRES) parameters are obtained from tuning the model with simulated DC forward characteristics at high and low current regions, respectively. The base resistance parameters NF and RMF are determined from simulated forward recovery waveforms. The EARLY parameter is extracted from the resistive and/or inductive turn-off diode characteristics obtained with the mixed-mode numerical simulations. For final calibration, however, TAU and EARLY have to be fine tuned simultaneously (for the reason illustrated in Fig. 4), which eventually yield to a minimal error of modeling both the DC and the transient diode characteristics. The extracted parameter values with their notation and meanings are listed in Table 1 shown in the Appendix B. 5. Modeling results and numerical simulations The electrical characteristics of 25 lm2 area power PiN diode described by its one-dimensional net doping profile in Fig. 1 are simulated using the test circuit shown in the inset of Fig. 7a. First, the DC characteristics are obtained for L = 0 and R = 1 X at different temperatures using the mixed-mode numerical device simulator ATLAS [15]. They are used to extract TAU, b, Rd0, NF, NF, RMF, and NRES parameters. Note that C1–C9 parameters are precalculated from (A7) using the physical constants given in the Appendix A. Fig. 6 shows a good match between the model and ATLAS results of simulated DC characteristics obtained for both high and low level injection regions. The later is better observed when plotting the anode current in log-scale as shown in the inset of Fig. 6. It shows that the new model yields a smooth transition from low to high injection operation depicted by the change of curve grading from 1 to 0.5 at around 0.6 V. It stems from the all-injection level TL modeling approach presented that solves virtually a coupled minority-carrier drift–diffusion and continuity equations in the diode QNR regions [14,20].
Following the DC calibration, the diode reverse recovery waveforms are then compared with ATLAS simulations as shown in Figs. 7 and 8. An excellent match between TL model and ATLAS results is obtained for different L and R values set in the test circuit. The extended electro-thermal model including temperature effects is also validated by simulating instantaneous power dissipation P(t, T) = ia(t, T) Æ ua(t, T) developed during the diode reverse recovery. The test is repeated for two different temperatures, T = 300 K and T = 398 K. An excellent agreement is obtained between the new model and ATLAS results as shown in Fig. 9. Finally, it is worth to point out that the emitter recombination effect is inherently included in the diode model trough the equivalent TL modeling of the emitter QNR. For example, the one would obtain a substantially different reverse recovery waveforms (not shown in Fig. 7 and 8.) departing from ATLAS results if the two RC segments of P+/P emitter region (see Fig. 3b) are omitted in the diode model. It confirms that the minority carrier’s distribution in the P+/P emitter QNR has substantial influence on the diode transient behavior, which has also been emphasized in Refs. [24–26].
50 0
Anode voltage (A)
b T sðT Þ ¼ TAU s10 ; T0 NRES T Rd0 ðT Þ ¼ Rd0 ; T0
R=0.01Ω
-50 -100
R=1Ω
-150 -200 -250
R=0.1Ω
L=1μH
-300
The model Silvaco
-350 4.90
4.95
5.00
5.05
5.10
5.15
5.20
Time (μs) 6
The model Silvaco L=1μH
R=0.01Ω
5
Anode current (A)
723
4
R=0.1Ω
3 2 1
R=1Ω
0 -1 -2 4.90
4.95
5.00
5.05
5.10
5.15
5.20
Time (μs) Fig. 8. The reverse recovery waveforms of (a) the anode voltages and (b) the anode currents simulated for different values for R and for constant L = 1 lH.
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N. Jankovic et al. / Solid-State Electronics 51 (2007) 719–725
where p(u, x), n(u, x) and NA(x) are the spatially and voltage dependent hole and electron densities and the doping concentration, respectively. Combining (A1) with (6) yields: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# " N A ðxÞ 4uðxÞn2ie ðxÞ 1 þ 1 þ nðu; xÞ ¼ ; ðA2Þ 2 N 2A ðxÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# " N A ðxÞ 4uðxÞn2ie ðxÞ 1þ 1þ pðu; xÞ ¼ : ðA3Þ 2 N 2A ðxÞ
70 The model Silvaco
Power dissipation (W)
60
R=1Ω L=1μH
50 40 T=300K
30 20
T=398K
10 0 4.90 4.95 5.00 5.05 5.10 5.15 5.20 5.25 5.30
Time (μs) Fig. 9. The instantaneous power dissipation P(t, T) = ia(t, T) Æ ua(t, T) developed during the diode reverse recovery transient time simulated at different temperatures (R = 1X, L = 1 lH).
6. Conclusions Novel power PiN diode model based on the generalised all-injection level minority carrier drift–diffusion theory is presented in this paper. The equivalent lossy transmission lines describing the carriers’ transport trough arbitrarily doped emitter and base quasi-neutral regions are defined. The effect of the moving boundary condition and the emitter recombination effect are included in the model. The extended electro-thermal diode model including the temperature as parameter is also described and implemented in circuit simulator PSPICE. By tuning a small set of model parameters, a very good agreement between modelling results and numerical simulation results of the realistic power PiN diode is obtained for different switching conditions and temperatures. Apart from thermal equilibrium simulations, it is also doable to employ this new diode model using a standard approach with auxiliary thermal analog sub-circuit [27] to predict the device’s dynamic self-heating. However, due to the complex temperature dependences of the model parameters C1–C9 expressed by (A7)–(A10), the present form of electro-thermal diode model has exhibited the substantial convergence problems during self-heating simulations with SPICE. Hence, in order to obtain the convergence-free self-heating simulations, the authors have been developing an improved version of this electro-thermal diode model with optimized expressions for C1–C9 temperature dependences, which will be discussed elsewhere.
where l0 is the doping-dependant low-field mobility, K is the effective electric field and vsat is the carrier drift saturation velocity. The doping-dependant carrier lifetime is expressed as [23]: s10 s0n;p ¼ ; ðA5Þ 1 þ NN AR where s10 = 5 · 107 s and NR = 5 · 1016 cm3. Finally, by replacing p(u,x), n(u,x) in (12)–(14) and (A8) in (A4), one can derive the analytical expressions for Ck, Gk, Rk, and Zl as: C2 C k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ C 1 u2k Gk ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ C 5 ð1 þ 1 þ C 1u2k Þ C 3 ð1 þ C 1 u2k Þ þ C 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ C 6 ð1 þ 1 þ C 1 u2k Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rk ¼ C 7 ð1 þ 1 þ C 1 u2k Þ Z l ¼ C 9 ð1 þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 u2k1 u2kþ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ C 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ C 1 u2k ð1 þ 1 þ C 1 u2k Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ C 1 u2nþ1 Þ
ðA6Þ where u2k1, u2k+1 and u2k are the input, the output and the middle node voltage, respectively, of the kth RC segment. The associated constancies C1–C9 are expressed as: C1 ¼
4n2ie ; N 2k
C4 ¼
2s10 ; qnie w
C2 ¼
qn2ie w ; Nk
C5 ¼
C3 ¼
s10 N k ; qn2ie w
qn2ie C n N k w ; 2
qn2ie C p N k w N kw ; C7 ¼ ; 2 2qV T n2ie l0 2 l0 C 1 V T N k¼N : ; C9 ¼ C8 ¼ vsat 2w 2qn2ie vs ðW Þ
C6 ¼
Appendix A The derivations of analytical expressions for Rk, Gk, Ck and Zl are described in what follows. One can start with quasi-neutrality approximation [17]: pðu; xÞ N A ðxÞ þ nðu; xÞ
The high-field carrier mobilities are calculated by following formula [22]: l0n;p ln;p ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA4Þ
ffi; l0n;0p K 2 1 þ vsat
ðA1Þ
ðA7Þ
Note that the effective electric field Kk appearing in each kth segment is approximately derived from (7) and (A3) as:
N. Jankovic et al. / Solid-State Electronics 51 (2007) 719–725 Table 1 The PiN power diode model parameters Parameter
Description
Value 2
Diode surface area (cm ) Unmodulated total resistance (X) Grading factor for conductivity modulated base resistance Zero-bias conductivity modulated base resistance (X) Exponent for diode resistance temperature dependence Adjustment factor for carrier lifetime in lowdoped base region Exponent for carrier lifetime temperature dependence in low-doped base region Pre-factor for moving boundary condition effect Zero-bias junction capacitance (F/cm2) Junction potential (V) Grading coefficient for junction capacitance
AREA Rd0 NF RMF NRES TAU b EARLY CJ0 VJ M
Kk
25 · 108 107 2.5 3 · 105 0.95 0.1 8 0.05 2 · 107 1.5 0.5
kT 1 C1 u2k1 u2kþ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; q ð1 þ 1 þ C 1 uÞ 2 1 þ C 1 u w ðA8Þ
and it appears in the mobility Eq. (A4). The temperature and doping dependence of parameters l0n,p and n2ie are calculated from published semi-empirical expressions [22–24] as: l0n ¼ 88 T 0:57 n þ
7:4 108 T 2:33 n ; 0:146 1 þ N D =ð1:26 1017 T 2:4 n Þ 0:88 T n
l0p ¼ 54:3 T n0:57 1:36 108 T 2:23 n 0:146 1 þ N D =ð2:35 1017 T 2:4 n Þ 0:88 T n qV g0 qDV g0 n2ie ¼ C n T 3 exp exp ; kT kT 8 9 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 < N = N A A þ ln þ C c1 ; DV g0 ¼ V ref ln : N ref ; N ref þ
ðA9Þ
ðA10Þ
where Tn = T/300, vsat = 1.1 · 107 cm/s, Cn = 10.38 · 1032 cm6 K3, Vg0 = 1.206 V, Vref = 0.009 V, Nref = 1.3 · 1017 cm3, Cc1 = 0.5. Appendix B See Table 1. References [1] Massmoudi N, M’bairi D, Allard B, Morel H. On the validity of the standard SPICE model of the diode for simulation in power electronics. IEEE Trans Ind Electron 2001;48:864–7. [2] Vogler T, Schroder D. A new and accurate circuit-modeling approach for the power-diode. In: Proceedings of power electronics specialists conference, vol. 2; 1992. p. 870–6. [3] Strollo AGM. A new SPICE subcircuit model of power p-i-n diode. IEEE Trans Power Electron 1994;9:553–9.
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