Implementation and Validation of a Physics-based Circuit Model for
IGCT with Full Temperature Dependencies
X. Wang, A. Caiafa, J.L. Hudgins, and E. Santi
P.R. Palmer
Department of Electrical Engineering
Department of Engineering
University of South Carolina
Columbia, SC 29208, USA
Wangx@engr.sc.edu
Abstract -- This paper presents a physics-based Fourier solution
IGCT model for circuit simulation with full temperature
dependencies. Besides the external electrical characteristics, the
model can also provide internal physical and electrical
information of the device, such as the junction temperature, and
the charge distribution. The model is shown to give good
agreement with experimental waveforms and accurately predicts
the device behavior under changing temperatures.
I.
INTRODUCTION
The Integrated Gate Commutated Thyristor (IGCT) is an
advanced GTO-based semiconductor device with its gate drive
unit coaxially integrated with a Gate Commutated Thyristor,
resulting in an extremely low gate-cathode parasitic
inductance ranging from 5nH to 15 nH [1]. Since it was
commercially introduced, the IGCT has rapidly gained
acceptance in major areas of high power electronics, such as
propulsion inverters for mass transit and locomotives, high
power industrial drives for steel and paper mills, and utility
power conditioning, including static VAR compensation and
flexible AC transmission [2]. Due to the rapidly increasing
application of IGCT in high power applications, an IGCT
model for circuit simulation is needed.
However, a
bibliography search shows only one IGCT model reported in
[3], in which all partial derivatives were approximated by
finite differences in order to solve the ambipolar diffusion
equation. That model is implemented by Saber MAST and
does not give detailed description of the temperature
dependent behavior of IGCT. For the circuit simulation
purpose, models with full thermal response and implemented
by widely used circuit simulator such as PSpice are desired.
This paper presents a new physics-based circuit model for
IGCT with full temperature dependent features. This IGCT
model was developed by employing Fourier-based solution for
ambipolar diffusion equation, and was implemented in circuit
simulator PSpice™. Besides the external electrical
characteristics, the model can also provide information on
physical quantities inside the device, such as the junction
temperature, hole and electron currents at different junctions
inside the device, and the charge dynamic distribution in the
buffer layer and base regions. To validate the model, a
chopper circuit with an inductive load was built to test the
IGCT switching behavior. The switching characteristics of
IGCTs at ambient temperatures ranging from –40 to 50 °C
University of Cambridge
Trumpington Street
Cambridge CB2 1PZ, UK
were tested. Good agreement has been obtained between the
simulation results and experimental data.
II.
MODEL DEVELOPMENT
Fig.1 shows the IGCT semiconductor structure, charge
profile and current components inside the device. The dashed
lines represent the charge profile and depletion boundary
while junction J2 is reverse biased. The proposed physicsbased IGCT model employs the Fourier expansion to solve the
ambipolar diffusion equation for the N-base region, uses a
charge control approach to describe the charge behavior in the
P-base region, applies quasi-static equations for the buffer
layer, and considers the N+ emitter as the minority sink. All
these regions interact through their current components and
charge densities at junction boundaries.
G
K
A
+
N buffer
IA
In1
Ip1
P
PH0
-
N base
N
IG
Px1
In2
Ip2
Idis
Px2
(Px2)
x1
J1
(x2)
Ip3
nb1
PHW (Px1)
J0
+
P base
IK
In3
nb2
(nb1)
(nb2)
x2
b2
J2
b1
J3
Fig.1 The IGCT semiconductor structure, charge profile and current
components
A. Modeling the N-base region
During the IGCT operation process, the thick and lowdoped N-base can be divided into two regions — the
undepleted region (storage charge zone) and the space charge
layer. The undepleted region contains the storage charge, and
the space charge layer sustains most of the voltage drop
applied to the device. The total width of the two regions is
equal to WN (N-base width) and each region can vary from
zero to WN depending on the applied external voltage.
1) The undepleted N-base region
Under the assumption of one-dimensionality and high-level
injection, the charge dynamics in the undepleted N-base
region can be described by the well-known ambipolar
diffusion equation:
D
p ( x , t ) ∂p ( x , t )
∂ 2 p( x, t)
=
+
τ
∂t
∂x 2
(1)
where p(x, t) is the excess carrier concentration, τ the carrier
lifetime, and D the ambipolar diffusion coefficient.
Many attempts have been made to solve this equation for
device modeling purposes, such as algorithmic solutions [4],
Hefner’s solution [5], and Fourier-based solutions [6] [7]. An
algorithmic solution is practical but cumbersome in the
context of use with a circuit simulation while Hefner’s
solution introduced mathematical oversimplifications to the
diffusion equation itself. The Fourier-based solution solves
the ambipolar diffusion equation quite exactly by an electrical
circuit analog, without extra-assumptions concerning the drive
or the boundary conditions. Diode [8] [9] and IGBT [7] [10]
[11] models have been successfully created using this Fourierbased solution and have been validated. The Fourier solution
modeling technique provides a good trade-off between
simulation accuracy and speed.
Based on the Fourier solution modeling technique, the
excess carrier concentration p(x,t) can be represented as a
discrete cosine Fourier expansion:
∞
 kπ ( x − x1 )
p( x, t ) = P0 (t ) + ∑ Pn (t ) cos

n =1
 x 2 − x1 
Vk
dV
+ Ck k + I k (k = 0, 1, 2, 3, 4, 5…)
Rk
dt
(3)
(6)
C0 = x2 − x1
(7)
∞
 dx
n dx 2 
I 0 = ∑ Pn (t ) 1 − (− 1)
dt 
 dt
n =1
(8)
For k ≠ 0:
  kπ
2
Rk =
D
x 2 − x 1   x 2 − x 1

Ck =
Ik =
2

1
 + 
τ


−1
(9)
x 2 − x1
2
(10)
Pk (t ) d ( x 2 − x1 ) ∞ n 2 Pk (t )  dx1
dx 
+∑ 2
− ( −1) k + n 2 
2 
4
dt
dt 
n =1 n − k  dt
V2
I even
V2n
R0
R2
R2n
C0
C2
C2n
I2
I0
 ∂p ( x , t )
= D
 ∂x
(11)
Reven
I2n
∂p ( x , t ) 
x2 −
x1 
∂x

V1
V3
V2n+1
R1
R3
R2n+1
C1
C3
C2n+1
I1
I3
I2n+1
x2
+
Rodd
∂p ( x , t ) 
x1 
∂x

Fig.3 Equivalent RC circuit describing the Fourier coefficients
Ik
Fig.2 The RC cell used to calculate Pk (k=0,1,2,3…)
The force current Iforce is function of the boundary conditions:
D  I n2 − I n1 I p2 −I p1  (for k = 0, 2, 4…)
−


2qA Dn
Dp 
τ
x2 − x1
 ∂p ( x , t )
I odd = − D 
 ∂x
Rk
Ck
I force = I even =
R0 =
V0
Vk
Iforce
D  I n 2 + I n1 I p 2 + I p1  (for k = 1, 3, 5…) (5)
−


2qA  Dn
Dp 
The values of capacitor and resistor are determined by the
carrier lifetime, undepleted base width and the harmonic
number k, and the parallel current sources, Ik (k=0,1,2,3…),
represent the moving boundary of the depletion region.
For k = 0:
(2)
By doing some mathematic derivation it is found that the
Fourier coefficients P0(t) and Pn(t) can be obtained by solving
a series of ordinary differential equations, which can be
represented by the analogue electric equation shown in (3),
and can be modeled by equivalent RC cell as shown in Fig.2.
I force =
I force = Iodd = −
(4)
Therefore, the ambipolar diffusion equation can be
represented in the form of two RC networks, shown in Fig.3,
which correspond to the even and odd values of k. The
voltages across the successive cells represent the Fourier
series coefficients P0(t) and Pn(t). The implementation of the
entire Fourier series expansion would require an infinite
number of RC-cells in the series string. Practically, less than
ten terms are required to achieve accuracy within the limits of
precision related to process parameters of the IGCT.
Therefore, after a small but suitable number of terms (RCcells) have been used the remaining resistive components of
the series are added together to form the equivalent
resistances, Reven and Rodd. The value of Reven and Rodd ranges
from 10-4 to 10-5 s/cm.
2) The space charge layer
The space charge layer behaves as either a quasi-ohmic
region, if the junction J2 is forward-biased, or a depletion
region, supporting most of the applied terminal voltage, if the
junction J2 is reverse-biased. The space-charge behavior can
be described by the Poisson’s equation. Assuming a constant
space charge density qNeff, the relation between the voltage
and the depletion width is:
(12)
2ε ×Vsc
2ε ×Vsc
x2 − x1 = Wn −
qNeff
= Wn −

q nB + I A
qAVSat 

Here, the space-charge includes the ionized impurities and the
density of free carriers. The second term is most important if
the device operates in an inductive-load circuit, in which the
device conducts high currents even at high bias voltage. The
voltage across the space charge layer, Vsc, is calculated by:
(13)
Vsc = − K × Px 2 a & Vsc⊆[0,Vbreak]
Where, K is a very big positive constant; Px2a is the carrier
concentration at x2 calculated using the Fourier expansion
formula (2). It can be seen that, if Px2a is positive or zero,
junction J2 is forward biased and Vsc is equal to zero; on the
other hand, if Px2a is negative, J2 is reverse biased and Vsc can
have any value smaller than the junction breakdown voltageVbreak.
B. Modeling the P-base region
Due to the comparatively narrow width and high doping of
the P-base, the charge control approach can be applied to
model the charge behavior in the p-base region. Consider the
built-in NPN transistor on the cathode side of the IGCT in
Fig.4. Since the total base region charge QB is approximately
linear, it can be broken into two parts QF and QR. QF is the
total injected charge from the N+ emitter to the base, and QR is
the total injected charge from the N- collector into the base.
N-
Idis
Ip2
In2
J3
IG
G
J2
Ip3
In3
P
QB
QR
b2
QF
K
N+
b1
Fig.4 Charge and boundary current components for the base of the cathode
NPN transistor
Applying the continuity equation to the P-base region, the
charge transient can be described by the charge control
relation as:
dQB
Q
(14)
= I p2 + I G + I dis − I p3 − B
dt
τ BHL
Where, τBHL is the high-level lifetime in the P-base region; Idis
is the displacement current due to the changing depletion
width at J2, and can be obtained by:
dV J 2
dV J 2
1
(15)
I =C
= εA
dis
J2
W N − ( x 2 − x1 ) dt
dt
The hole current at J3 can be obtained by considering the N+
emitter as a hole sink:
(16)
I p3 = qAhn n 2b1
The parameter hn is the recombination coefficient and nb1 is
the charge concentration at the boundary of J3. Since the QF
and QR are assumed to be linearly distributed, the charge
concentrations at the boundaries are related to the forwardand reverse-injection charges by the following equations:
2Q F
(17)
n b1 =
qAW P
1
(18)
Q R = qAWP nb 2
2
Wp is the P-base width. The charge concentration at the
boundary of J2 can be obtained by the Boltzmann’s relation:
n2
 qV 
(19)
nb2 = i exp J 2 
PB
 kT 
where PB is the background doping of P-base region.
Combining (15) to (17) and considering that QB is the sum of
QF and QR, the charge concentration nb1 can be determined.
Substituting the value of nb1 into (14), Ip3 can be obtained.
The collector electron current In2 can be expressed by the
forward and reverse charge transportation described in
equation (20), in which the τF and τR are the normal and
reverse base transit times.
Q
Q
(20)
I n2 = F − R
τF τR
The hole current Ip2 can easily be determined as:
(21)
I p 2 = I A − I n 2 − I dis
Therefore, the above equations link together the P-base
region to the N-base region, gate region and cathode.
C. Modeling the buffer layer
Since the buffer layer is highly doped and very thin (few
µm), low-level injection and quasi-static assumptions can be
applied. The continuity equation of holes in the buffer layer
can then be expressed by:
d 2 p( x) p( x) p(x)
(22)
=
=
Dτ BF L2pH
dt2
Where τBF is the minority lifetime in the buffer layer, and LpH
is the diffusion length of holes. The general solution for
equation (22) is:
p (x ) = C 1 e
x
L pH
+ C2e
−
x
L pH
(23)
J1
J0
P+
-
N base
N buffer layer
PH0
Px1
p(x)
PHW
IA
Ip0
Ip1
In0
In1
x1
x
H
WH
0
Fig.5 Buffer layer carrier distribution and variables definition
If the hole concentrations at the two edges of the buffer layer
are PH0 and PHW, respectively (as shown in Fig.5), the solution
of this equation is:

W − x 

 (24)
1
 + PHW sinh x 
p( x ) =
PH 0 sinh H



W 
 LpH 
 LpH 
sinh H  
L 
 pH 
where, WH is the width of buffer layer. Under low-level
injection, the minority hole current is mainly diffusion current.
Thus, the hole currents at the buffer layer boundary of J0 can
be obtained from the derivative of p(x) (equation (24)):

 (25)
qADp
 WH 
dp( x)
I p0 = qADp
x =0
dt
=
On the other hand, the hole concentration at the boundary of
J1, - PWH, can be obtained by applying the Boltzmann’s
relation to J1 under the assumption of high-level injection in
the N-base (e.g. px1 >> nB).
p ( p + n B ) p x21
(31)
PHW = x1 x1
≈
nH
nH
The hole current at the boundary of J1 can be easily obtained
by taking the derivative of equation (24):

 W  (32)
qADp
dp(x)
I p1 = qADp
PH0 − PHW cosh H 
x=W =
dt
W  
 LpH 
LpH sinh H  
L 
 pH 
Therefore, equations (29) to (32) link the buffer layer to the
N-base and to the anode.
 − PHW 
PH 0 cosh
LpH 
 WH  



LpH sinh
L 
 pH 
D. The total voltage across the device
The total voltage across the device consists of the voltage
drop in the undepleted N-base region, the voltage supported
by the space charge layer, and the voltages across the four
junctions. Here, the Ohmic voltage drops in the P-base and
emitter regions are ignored due to the high background doping
concentrations.
The voltage drop across the undepleted N-base region
(charge storage zone) is:
I x2
(33)
Vbn (t ) = ∫ ρ ( x, t )dx
A x1
Applying the Boltzmann’s relation to J0 with the quasiequilibrium assumption, results in equation (26), which relates
the hole and electron concentrations, with NH as the
background doping concentration of the buffer layer.
VJ 0
(26)
N H PH 0 = n i2 e Vt
In addition, under low-level injection, the electron current at J0
can be expressed by (27), where Jsne is the emitter electron
saturation current density.
I n 0 = AJ sne e
VJ 0
Vt
(27)
Combining (26) and (27), the electron current can be
represented by:
I n 0 = AJ sne
N H PH 0
n i2
(28)
The sum of electron and hole currents at J0 and J1 should be
equal to the anode current:
I A = I p0 + I n0 = I p1 + I n1
(29)
Combining equations (25), (28) and (29), the hole
concentration at J0 can be obtained:
−1
 J sne AN H


 (30)
qAD P PHW
qAD P
PH 0 = 

ni2
+
 × I A +

L pH tanh( W H / L pH ) 
L pH sinh(W H / L pH ) 

Fig.6 Discretized carrier profile for simulation of charge storage region
voltage drop
A typical effective way to realize the integration calculation
in simulation is calculating the sum of discretized values. As
shown in Fig.6, the charge profile is assumed to be linear
between two adjacent points. The charge concentrations at
every node along x1 to x2, i.e. p(x1,t), p(m1,t),… p(x2,t), can be
calculated by the cosine Fourier expansion formula (2).
Considering the trade-off between simulation speed and
accuracy, seven sampling points uniformly distributed along
the charge storage zone are typically used. Therefore, the total
voltage drop across the undepleted N-base region can be
calculated by:
V bn
  (µ n + µ p ) p (m 1 , t ) + n B µ n
 ln 
(
)   (µ n + µ p )p ( x 1 , t ) + n B µ n
(t ) = I × x 2 − x 1  
6 Aq (µ n + µ p ) 
p (m 1 , t ) − p ( x 1 , t )






+ ...
 (µ n + µ p ) p ( x 2 , t ) + n B µ n
ln 
 (µ + µ ) p (m 5 , t ) + n µ
n
p
B
n

+
p ( x 2 , t ) − p (m 5 , t )
 
 
 





(34)
Parameter
Intrinsic Carrier
Concentration
On the other hand, the voltage drops across the junctions
can be obtained by Boltznmann’s relation:
P n 
(35)
VJ 0 = Vt ln H 02 H 
 ni 
n 
(36)
V J 1 = Vt ln  B 
 p x1 
p
V J 2 = 2V t ln  x 2
 ni



for
p x 2 ≥ ni
Table 1. Temperature Dependence of the IGCT Model Parameters [12] [13]
Electron
Mobility
Hole Mobility
Lifetime
E. The thermal sub-model
Assuming double-sided cooling, the junction temperature of
the device is estimated by the coupled thermal equivalent
circuit shown in Fig.7. The power input source (current
source) represents the heat dissipation in the device and can be
expressed by equation (40). The R and C values are based on
data sheet parameters or on package physical parameters and
constants (e.g. thermal conductivity and heat capacity of the
materials). All the temperature dependent parameters for the
IGCT model are listed in Table 1. Therefore, the heat
dissipation of the device can be instantaneously fed back to
adjust the temperature dependent parameters and affect the
model behavior, as shown in Fig.8.
(40)
Pin = I AV AK + I G × VGK
µ
III.
p
= 495 × (T
300
(T
)1 . 5
τ =τ
Recombination
coefficient
(37)
 n b1 (n b1 + n p )
(38)
V J 3 = Vt ln 

ni2


where high-level injection is assumed in the N-base region to
obtain (36) and (37). In addition, the value of px2 is limited to
be equal to or higher than ni, since equation (37) is only used
to calculate the forward voltage drop of J2. If px2 is less than
ni, then J2 is reverse biased, and the voltage across the J2 is
represented by the voltage drop across the space charge layer.
Therefore, the total voltage drop across the device can be
obtained by:
(39)
VD = Vbn + V J 0 + V J 1 + Vsc − V J 2 + V J 3
where Vbn can be calculated using (34) and Vsc is expressed by
(13).
Temperature Dependence Equation
− 7.02×103
ni = 3.87×1016T1.5 exp(
)
T
µ n = 1360 × (T 300 )− 2 . 42
300
300
h n = J spo (T 300 )
0 .5
)− 2 . 2
−2
q −1 n i e
14000 (
1
1
− )
300 T
MODEL IMPLEMENTATION in PSpice™
The proposed IGCT model is implemented using the widely
used circuit simulator– PSpice. Every mathematical equation
describing the IGCT model is represented by one or more
circuits in PSpice. For instance, behavioral model voltage
controlled voltage sources (E) and voltage controlled current
sources (G) are frequently used to realize the functions of
calculating changing parameters. Differentiation is normally
performed by measuring the current through a unity-valued
capacitor to which a voltage equal to the quantity to be
differentiated is applied.
Pdiss = I AV AK + I GVGK
A
IA
EJ0
Ebn
EJ1
Tj
Thermal model
EJ3
-EJ2
Esc
IK
IG
p x1
IA
PH0
G
px1
Ieven
Ip1
In1
Buffer
layer
K
Ip2
Fourier
RC
Iodd
±
d ( x 2 − x1 )
dt
In2
network
Idis
In2
IA
:
:
∑ Ln(..)
px2
px2
Idis
Ip2
pm1
nb1
P-base
IG
(x2-x1)
f
Vsc
Ip3
Ip2
nb1
Space
charge
layer
Ip3
qAh n n 2b1
Regulate temperature dependent parameters
Fig.8 The block diagram describing the IGCT model implementation in
PSpice
Fig.7 Coupled thermal equivalent circuit used to calculate the junction
temperature
The block diagram describing the IGCT model
implementation in PSpice is shown in Fig.8. The IGCT model,
MODEL VALIDATION
A. Experimental circuit and measured results
As part of the validation process, experimental
characterization of the snubberless switching behavior of a
4500V/340A IGCT was carried out. Fig.9 shows the testing
circuit. The Lload represents the inductive load, and Lcl the
stray inductance. The IGCT is placed in an environmental
chamber, so that the ambient temperature can be controlled.
The IGCT switching characteristics were tested at ambient
temperature from –40 to 50 °C, with different conduction
currents and clamping voltages. Fig.10 and Fig.11 are the
measured waveforms with conduction current of 300 A and
clamping voltage of 2500 V.
D0
Lc1
LLoad
VDC
3000
2500
-40 C
0C
25 C
50 C
2000
1500
Temperature increases
1000
500
0
-500
32
33
B. Some simulation results
Corresponding simulation was performed using the
proposed IGCT model. The simulation circuit is similar to the
experimental one, except that the gate drive is simply
represented by a piecewise linear voltage source in series with
a resistor and an inductor. Figures 12 to 14 present some of
the simulation results. The anode and gate currents during
turn-off in Fig.12 indicate that the IGCT model has a unity
turn-off gain, an important feature of the IGCT. Fig.13 shows
that the junction temperature is approaching dynamic
equilibrium after several switching cycles. This result proves
that the thermal sub-model presents a proper thermal response
for the semiconductor device. Fig.14 shows the evolution of
the charge concentration profile during turn-off. It shows that
the storage charge is swept out during turn-off process, and the
storage time is less than 0.9 µs. This storage duration is close
to the experimental data reported in [15].
3500
350
Va
Control Signal
250
Ia (A), Ig(A) & Vg(V)
20V
35
Fig.11 Experimental anode voltages during turn-off at temperatures from –40
to 50 °C
IGCT
C
34
Time (us)
2500
Ia
1500
150
500
50
-5 0 4 0
50
60
70
Ig
80
90
1 0 0-5 0 0
Vg
-1 5 0 0
-1 5 0
Fig. 9 Circuit for testing IGCT switching
Va (V)
IV.
3500
Voltage (Volts)
including its thermal sub-model, is wrapped in one sub-circuit,
and can be called by any external circuit. The IGCT model
consists of five sub-models described previously. They are:
the buffer layer sub-model, the Fourier RC networks
representing the undepleted N-base region, the space charge
layer sub-model, the P-base region sub-model, and the thermal
sub-model. The thermal sub-model links with other submodels through the junction temperature and the other four
sub-models are linked together by their boundary charge
densities and current components. All the sub-models are
linked together with the external circuits and can be solved by
the solver.
-2 5 0
-2 5 0 0
T im e (u s)
Fig.12 Anode and gate currents and voltages
350
320
300
200
Junction Temperature (Kelvin)
Current (A)
250
150
100
Temperature
increases
50
316
314
312
310
308
306
304
302
0
-50
Tj
318
-40C
0C
25C
50C
32
33
34
35
36
Time( us)
Fig.10 Experimental anode currents during turn-off at temperatures from –40
to 50 °C
300
0
100
200
300
Time (us)
400
500
600
Fig.13 Junction temperature corresponding to several switching cycles
V.
3.5E+16
t=0
t=0.7us
t=0.9us
t=1.3us
t=3us
t=5us
t=8us
Charge Density (cm-3)
3.0E+16
2.5E+16
2.0E+16
1.5E+16
1.0E+16
5.0E+15
0.0E+00
Buffer
Layer
0
50
100
150
200 250
N- base region
300
350
400
450
CONCLUSION
This paper presents a newly developed physics-based IGCT
model based on a Fourier-series solution modeling technique.
This IGCT model consists of five sub-models describing the
buffer layer, storage zone of the N-base region, the space
charge layer, P-base region and a thermal sub-model. All submodels were explained and the PSpice implementation was
discussed.
The proposed physics-based circuit model for IGCT works
properly in the PSpice simulator, and is shown to give good
agreement with experimental waveforms and correctly models
the device behavior at various temperatures.
P base
ACKNOWLEDGMENT
Fig.14 Charge distributions in the base regions during turn-off
C. Comparison between experimental and simulation results
Fig.15 and Fig.16 are the comparisons of experimental and
simulation waveforms during turn-off with conduction current
of 300 A, clamping voltage of 2500 V, and under ambient
temperature of 25°C and 50°C. Good agreement has been
obtained. Similar comparison results have been obtained at
other temperatures with different conduction currents and
clamping voltages. Due to the space limit, they are not
presented here.
400
3000
Va
350
2500
250
2000
Ia_exp_25C
Ia_sim_25C
Va_exp_25C
Va_sim_25C
Ia
200
150
1500
1000
100
Voltage (V)
Current (A)
300
500
50
0
0
-50 31
33
35
37
39
Time (us)
41 -500
Fig.15 The simulated and measured anode current and voltage during turn-off
@25°C
400
3500
350
3000
Va
300
Current (A)
200
Ia_Exp_50C
Ia_sim_50C
Va_exp_50C
Va_sim_50C
Ia
150
2000
1500
1000
100
500
50
0
0
-50
Voltage (V)
2500
250
32
34
36
38
40
42
-500
Time(us)
Fig.16 The simulated and measured anode current and voltage during turn-off
@50°C
This work was supported by the U.S. Office of Naval
Research under contract numbers of N00014-02-1-0623 and
N00014-03-1-0434.
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