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27
Heat Sink Temperature Profile of Ring
Geometry DDR IMPATT Diode
Aritra Acharyya and J. P. Banerjee
Abstract— In this paper analytically the Laplace Equation of heat transfer is solved with proper boundary conditions to obtain
Temperature Distribution inside the semi-infinite heat sink for ring structured DDR IMPATT source. This type of temperature distribution analysis is very much important to design the heat sinks for IMPATT sources for proper and safe operation of them in
continuous wave (CW) mode during steady state. Typical heat sinks are designed for a millimeter-wave Si-IMPATT (ring diode)
source using diamond and copper as heat sink materials. The analytical solutions of the Laplace Equation with appropriate
boundary conditions which provide the temperature distributions inside the heat sinks are presented in graphical forms. Results
are provided in the form of necessary graphs and tables.
Index Terms— Analytical Solution, IMPATT Diode, Laplace Equation, Ring Structure, Temperature Distribution.
—————————— ——————————
1 INTRODUCTION
T
HE recent advances in IMPATT device technology
has made it possible to use this device in various
communication systems operating in millimeter wave
and sub-millimeter wave bands of frequency ranges
which provides advantages such as increased resolution
and use of low voltage power supplies. The availability of
several atmospheric window frequencies in the mm-wave
frequency range (30-300 GHz) has further increased the
usefulness of IMPATT diodes which can now produce
appreciable amounts of millimeter wave power at very
high frequencies of operation. Last four decades, several
workers have been exploring the possibility of highpower generation either from a single IMPATT diode or
from the several diodes by using power combining techniques. It is well known that the efficiency of the IMPATT
diode is relatively low [5-15% for CW operation], a large
fraction of the DC power is dissipated as heat in the highfield region. The temperature of the junction rises above
ambient, and in many cases output power of oscillator is
limited by the rate at which heat can be extracted from
the device. As junction temperature increases, the reverse
saturation current rises exponentially and eventually
leads to thermal runway phenomenon resulting the burning
out of the device. Since in contrast to the avalanche current, the reverse saturation current does not require a
large voltage to sustain it, the voltage begins to decrease
when the junction gets hot enough for the reverse current
to constitute a significant fraction of the total current. A
thermally induced DC negative resistance is produced,
causing the current to concentrate in the hottest part of
the diode. As well as leading to the eventual burn-out of
the junction, the increased saturation current at elevated
temperatures produces a degradation in the oscillator
performance at power levels below the burn-out value.
The increased reverse saturation current produces a faster
build-up of the avalanche current and degrades the negative resistance of the device. Thus, in general, the oscillator efficiency will begin to decrease at power levels just
below the burn-out power. As the band-gap of the semiconductor becomes larger the reverse saturation current
associated with it becomes smaller and consequently the
burn-out temperature of the junction rises. Thus GeIMPATTs (Eg = 0.7 eV) are lower power devices than either
Si (Eg = 1.12 eV) or GaAs (Eg = 1.43 eV) devices.
Thus properly designed Diode and Heat sink is required for proper operation of the IMPATT Oscillator.
Temperature distribution inside the heat sink must be
known to obtain the proper thermal design. Internal temperature distribution inside the heat sink depends on the
shape and dimensions of the diode and heat sink and also
on the temperature of the diode heat sink interface. In this
paper, ring structured IMPATT diode is considered instead of solid mesa diode [3]. Ring structure, which is
obtained by removing the central part of the solid circular
diode structure, has the advantage of smaller thermal
impedance than the solid circular diode (i.e. ordinary mesa diode). In this paper authors have find out 2-D Temperature Distribution inside the semi-infinite heat sink for
ring structure of IMPATT diode [Fig. 1] by using analytical approach to solve Laplace Equation in cylindrical coordinate system with proper boundary conditions. Here
the same approach is followed by the authors as in [1].
The rest of the paper is organized as follows. Section 2
provides the detailed solution procedure of the Laplace
Equation with appropriate boundary conditions to find
out the temperature distribution inside the heat sink dur————————————————
ing steady state operation (CW) of the IMPATT oscillator.
• Aritra Acharyya is with the Institute of Radio Physics and Electronics,
In Section 3 different necessary results in the form of
University of Calcutta, Kolkata, India.
• J. P. Banerjee is with Institute of Radio Physics and Electronics, University graphical plots are presented. Finally the paper conof Calcutta, Kolkata, India.
cludes in Section 4.
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2 SOLUTION OF THE LAPLACE EQUATION
Actual Structure of the Ring Diode [1][2] [Double Drift
Region (DDR) IMPATT] is shown in Fig. 1. During the
steady state (CW-mode) operation actually the heat is
generated at the p-n junction. This heat must be transferred to the ambience with a constant rate through heat
sink to avoid the device burn out. Since the thickness of
the diode is very small, it can be assumed that the junction temperature is equal to the diode heat sink interface
temperature during steady state.
Fig. 2. Figure illustrating the Ring diode Semi-infinite heat sink
interface.
Here heat is supplied over the circular area r1 ≤ r ≤ r2 in
the plane z = 0 [Fig. 2]. Since the diode thikness is very
small, again it will be valid approximation that
temperature distribution at the junction of the diode is
almost same as at the diode heat sink interface
[Temperature distribution at the interface is taken as
uniform here]. For Ring diode, the steady state heat sink
temperature rise from ambient is found using the
Fig. 1. Ring Structure of DDR IMPATT Diode.
following assumptions:
1) 1) The heat sink is semi-infinite, with a constant tharmal
Now the temperature distribution inside the semiconductivity k. It’s temperature approaches that of the
infinite heat sink will be found out. The temperature
ambient at a great distance from the origin. The ambient
distribution within a homogeneous solid cylinder of
temperature is taken to be zero for convenience.
isotropic media must satisfy Laplace Equation. Laplace
2) 2) The diode acts as a constant thermal flux source with a
equation in cylindrical co-ordirnate [4][5][6],
flux density. in practice, a substantial variation could, of
∇ 2T (r , z ,θ ) = 0
(1)
course, take place, owing to in-homogeneity in material
Due to symmetry, temperature is only the function of two and temperature variation in the transverse direction.
spatial variables r, z only,
3) 3) The region z ≥ 0 with flux f over the circular area r1 ≤ r ≤
r2 and zero elsewhere. At the plane of the diode-heat sink
∂ 2T (r , z ) 1 ∂T (r , z ) ∂ 2T (r , z )
+
+
=0
(2)
2
2
interface
(z = 0).
r ∂r
∂r
∂z
Equation (2) is satisfied by [6][7],
e
−λ z
J 0 (λr )
(3)
∂T
∂z
=−
z =0
∫e
−λ z
r1 ≤ r ≤ r2
=0
For any λ, thus,
∞
f
k
J 0 (λr ) f (λ )dλ
(4)
0
Equation (4) will be solution of the problem, if f (λ ) can
be chosen to satisfy the prescribed conditions in the plane
z = 0. The two most interesting cases follow from the well
known integrals involving the Bessel functions [7],
∞
dλ
−1  R0 
r > R0
∫ J 0 (λr ) sin(λR0 ) λ = sin  r 
0
π
=
r ≤ R0
(5)
2
r > R0
Using (4), (6) & (7) we get, the temperature distribution
measured from the ambient, is (z ≥ 0),
T(r, z) =
f  ∞ −λz
dλ ∞ −λz
dλ 
r2 ∫ e J0 (λr)J1 (λr2 ) − r1 ∫ e J0 (λr)J1 (λr1 ) 
k  0
λ 
λ
0
1
2 R0
r = R0
=
1
R0
r < R0
(8)
A solution of (8) at the plane z = 0 is given by,
2
1 1
1 1
r 2 
f 
r2  r
T (r ,0 ) = r2 F  ,− ;1; 2  − 1 F  , ;2; 12 
2 2
k 
r 
r2  2 r  2 2


for
r1 ≤ r ≤ r2
F (a , b; c; z ) = 1 +
0
=
(7)
and
z =0
(9)
The hyper-geometric functions are evaluated using the
series,
∞
∫ J 0 (λr )J 1 (λR0 )dλ = 0
elsewhere
(6)
ab z a (a + 1)b(b + 1) z 2
+
+ .....
c 1!
c(c + 1)
2!
(10)
The improper integrals of (8) are evaluated by numerical
method. Detail of the evaluation procedure of the integrals is given in APPENDIX A.
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3 RESULTS AND DISCUSSIONS
During the steady state (CW) operation of the IMPATT
oscillator the heat generated at the p-n junction of the
diode must be transferred to the ambience with a very
high rate. To achieve this purpose heat sink made of some
special materials having very high thermal conductivities
[e.g. copper or diamond] has to be attached bellow the
diode. Heat sink dimensions must be chosen such that at
the end surface of the heat sink temperature falls to ambient temperature during steady state. In TABLE 1 the
dimensions of the heat sinks are given for both copper
and diamond as heat sink materials.
TABLE 1
COPPER AND DIAMOND HEAT SINK DIMENSIONS
HEAT SINK MATERIAL
RADIUS,
RH (μm)
THICKNESS,
LH (μm)
COPPER (k = 396 Watt/moC)
900
1000
400
1000
DIAMOND (k = 1200
Watt/moC)
The evaluation procedure of (8) is available from APPENDIX A. Using this (8) authors have plotted Temperature Distribution inside the Copper and Diamond Semi infinite Heat Sinks as a function of r, where z is used as parameter as well as a function of z, where r is used as parameter. Here diode junction temperature [which is assumed as equal to diode-heat sink interface temperature]
is typically [for Silicon IMPATT diode (Ring Structure)]
taken as 530 K. For Silicon IMPATTs the burn out temperature is 575 K, so here junction temperature is much bellow the burn out temperature of the device. Here r1 and r2
values are 80 μm and 100 μm respectively.
steady state operation heat is uniformly generated precisely at the junction. Naturally the temperature at the
ring shaped junction surface becomes maximum. So, it
can be concluded that here the shape of the heat source is
just like a two dimensional circular ring (not like two dimensional circular disk which is associated with ordinary
mesa diode [1]). The heat generated at the circular ring
shaped junction is conducted below towards the heat sink
through p+, Cr and Au-layers. Since the total thickness of
the p+, Cr and Au-layers is very small, that is why it is a
valid approximation that the temperature is distributed at
the bottom surface of the diode (i.e. at the bottom of AuElectroplated layer as shown in Fig.1.) just like as it is at
the junction. So, heat source at the upper surface of the
heat sink is approximately circular ring shaped (it’s dimensions are approximately equal to the dimensions of
the junction; i.e. r1 and r2) and temperature (approximately equal to junction temperature) is uniformly distributed
throughout the source (Fig. 2.) as we have previously assumed that temperature is uniformly distributed
throughout the junction. Due to this kind of heat source
geometry naturally the temperature distribution at the
upper surfaces of the heat sinks possess two peaks having
the temperature almost equal to the junction temperature.
These temperature peaks are diminished as one goes inside the heat sinks from the upper surface and finally at
the bottom surface of the heat sinks the temperature distributions become almost flat. Effectively the temperatures at every point of the bottom surfaces of the heat
sinks decreased just below the ambient temperature.
Fig. 3. Temperature Distribution inside the Semi-infinite (a) Copper
and (b) Diamond Heat Sink.
Fig. 4. 3-D Plot of Temperature Distribution inside the Copper Heat
Sink.
Fig. 3. (a) and (b) can be explained from the actual
structure of the Ring Diode shown in Fig. 2. The axial portion of the diode is not solid, like ordinary mesa diodes
[1]. From the upper surface of the diode, almost to the
end of the p-epitaxial layer the base material (here Silicon)
is etched out to obtain the desired circular ring shape near
the junction region. Now during the continuous wave
Three dimensional plots of temperature distribution
inside the copper and diamond heat sinks are shown in
Fig. 4 and Fig. 5. From these figures temperature of any
point either on the surface or inside the heat sinks can be
obtained. It can be observed from the temperature distribution plots that at the lower surfaces and side walls of
the heat sinks, temperatures are lowered to almost to the
ambient temperature [i.e. 300 K].
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 r  1   r  1 
J 0  ln   J 1  2 ln  
 z  x    z  x   dx
I= ∫
1
0.000001
ln 
x
0.999999
 r  1   r  1 
J 0  ln   J 1  1 ln  
 z  x    z  x   dx
− ∫
1
0.000001
ln 
x
0.999999
(13)
Now Numerical technique can be used to find out the
integrations of above (13). Here Simpson’s 1/3 – Rule is
used. According to this Rule,
Fig. 5. 3-D Plot of Temperature Distribution inside the Diamond
Heat Sink.
xn
h
∫ y.dx = 3 [ y0 + 4( y1 + y3 + y5 + .......... + yn−1 )
x0
+2( y 2 + y4 + y6 + ........... + y n− 2 ) + y n
(14)
Where, xn = x0 + nh, n = 999998, xn = 0.999999, x0 =
0.000001 and h = 0.000001.
4 CONCLUSIONS
In this paper analytical approach to solve the Laplace Equation to obtain the temperature distribution inside the
semi-infinite heat sink during steady state (CW) operation
of ring structured IMPATT diode oscillator is presented.
Temperature distributions for typical copper and diamond heat sinks are also plotted graphically to visualize
the actual fact. This approach is very much useful to design heat sinks for CW IMPATT (ring geometry) oscillators. It can be easily observed that as the thermal conductivity of diamond is almost three times larger than that of
copper near 500 K, i.e. why required diamond heat sink is
much smaller in size than copper heat sink for equivalent
operation.
 r  1   r  1 
J 0  ln   J 1  2 ln  
z  x   z  x 
Integrants, y 1 = 
1
ln 
 x
 r  1   r  1 
J 0  ln   J 1  1 ln  
z  x   z  x 
y2 = 
1
ln 
x
and
(15)
(16)
Evaluation of the Integrations has been carried out by
using MATLAB software.
ACKNOWLEDGMENT
APPENDIX A
The improper integrals of (8) can be evaluated by following way using Numerical Method [Simpson’s 1/3 - Rule].
∞
dλ ∞ − λz
dλ
I = ∫ e −λz J 0 (λr )J 1 (λr2 )
− ∫ e J 0 (λr )J 1 (λr1 )
(11)
λ
λ 0
0

1
1  1 
λ = − ln(x ) = ln 
z
z  x 

Put, e −λz = x
⇒ e −λ z dλ = −
λ
x
0
1
The authors wish to thank Miss Moumita Mukherjee [Research Associate, Centre of Millimeter wave Semiconductor Devices and Systems, Institute of Radio Physics And
Electronics, University College of Science And Technology (University of Calcutta)] for her valuable suggestions
and immense help.
dx
z
REFERENCES
[1]
∞
0
After changing variable (11) becomes,
 r  1   r  1 
 r  1   r  1 
J  ln J1 1 ln 
J  ln J1 2 ln 
1 0
1 0
x
z
x
z




 
dx−  z  x   z  x dx
I =∫ 
∫
1
 
 1
0
0
ln 
ln 
 x
 x
[2]
(12)
Here one can see that the integrants are undefined at x = 1
& x = 0. To avoid these discontinuities it can be assumed
that the lower limit as x = 0.000001 & upper limit as x =
0.999999 with maintaining sufficient accuracy.
[3]
Aritra Acharyya, Baisakhi Pal, J.P. Banerjee, “Temperature Distribution inside Semi-Infinite Heat Sinks for IMPATT Sources”,
International Journal of Engineering Science and Technology”,
vol 2, issue-10, pp. 5142-5149, October-2010.
Aritra Acharyya, Baisakhi Pal, J.P. Banerjee, “Comparison Between 2-D Temperature Distribution Analysis Inside a Semiinfinite Copper Heat Sink for Mesa and Ring Structure of SiIMPATT Diodes using Analytical Method and Finite Difference
Method”, International Journal of Electronic Engineering Research, vol. 2, no. 4, pp. 553-567, 2010.
Baisakhi Pal, Aritra Acharyya, Arijit Das, J.P Banerjee, “Temperature Distribution in a Mesa Structure of Si-IMPATT Diode
on a semi-infinite copper heat-sink”, Proceedings of National
Conference on MDCCT 2010, Burdwan, pp. 58-59, March 2010.
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JOURNAL OF TELECOMMUNICATIONS, VOLUME 6, ISSUE 1, DECEMBER 2010
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[4]
[5]
[6]
[7]
D. P. Kennedy, “Spreading resistance in cylindrical
semiconductor devices”, J. Appl. Phys., vol. 31, pp. 1490-1497,
1960.
G. Gibbons, T. Misawa, “Temperature and current distribution
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Holman J. P.: Heat Transfer, Tata McGraw Hill [Ninth Edition].
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Oxford: Clarendon.
Aritra Acharyya received his M.Tech. Degree
from Institute of Radio Physics and Electronics,
University of Calcutta, Kolkata, W.B., India.
Earlier he obtained his B.E. Degree from Bengal
Engineering and Science University, Shibpur,
Howrah, W.B., India.He worked on Studies on
Series Resistance of Millimeter Wave SiIMPATT Diodes for his Master’s thesis, with
emphasis on various physical effects like effect
of carrier diffusion and tunneling current on
series resistance of IMPATT devices. His research interest is Microwave
Semiconductor Devices more specifically IMPATT Devices.
Professor (Dr.) J.P. Banerjee obtained B.Sc.
(Hons.) and M.Sc. in Physics and Ph.D. in Radio
Physics and Electronics from University of
Calcutta.
He worked as a senior scientist of a Department of Electronics project in Institute of Radio
Physics and electronics, C.U. during 1986-1989.
He joined the Department of Electronic Science,
C.U. in 1989 as a reader. He has been working
as a professor in the Institute of Radio Physics
and Electronics, C.U. since 1998.
He is the recipient of Indian National Science Academy Award of a visiting fellowship and Griffith Memorial Prize in Science of the Calcutta
University in 1986. He is the principle co-author of more than 100 research
papers in International Journals in the fields of Semiconductor Science and
Technology, Microwave and Millimeter wave avalanche transit time devices and avalanche Photo Detectors. He has successfully carried out a number
of research projects of Government of India on IMPATT diodes. A collaborative research work in the field of computer analysis, fabrication and
characterization of V-Band silicon double low high low IMPATTs was
successfully carried out by Dr. Banerjee for the first time with Dr. J.F. Luy
the eminent German Scientist of Daimler Benz research centre. He is a
Fellow of Institute of Electronics and Communication Engineers (IETE), a
life member of society of EMI and EMC and Semiconductor Society, India.
He is an expert committee member of All India Council of Technical Education and served as a referee for various technical journals.