- Email: [email protected]
Transient temperature response of an avalanche diode
Solid State
Electronics
TRANSIENT
Pergamon
Printed
Press 1970. Vol. 13, pp. 799406.
TEMPERATURE
RESPONSE
in Great Britain
OF AN AVALANCHE
DIODE G. GIBBONS Sperry Rand Research (Received
Center,
Sudbury,
Massachusetts,
30 June 1969; in rwised_form
8 August
U.S.A.
1969)
Abstract-The thermal impedance is calculated for an avalanche diode under pulse bias conditions. The diode is represented by a circular heat source on the surface of an infinite solid cylinder and the calculation is made for the application of a single pulse and for periodic pulses. The temperature response is characterized by a time constant t, = IF/a, where R is the radius of the diode and TVis the diffusivity of the heat sink. The results can be used to derive peak and average power limits and energy per pulse limits for high-efficiency avalanche diode oscillators operating under pulsed conditions. A simple, non-destructive technique is described for measuring the temperature of a junction when pulsed into avalanche breakdown. R&sum&L’impedance thermique est calculee pour une diode a avalanche sous des conditions de polarisations d’impulsions. La diode est representee par une source de chaleur circulaire sur la surface d’un cylindre infiniment solide et les calculs sont faits pour l’application d’une impulsion unique et d’impulsions periodiques. La reponse de temperature est caracterisee par une constante de temps tc R2/or, oh R est le rayon de la diode et ccla diffusion du recepteur de chaleur. Les resultats peuvent &tre employ& pour d&river les limites de c&e et moyennes des puissances et energie par impulsion des oscillateurs B diodes a avalanche ayant de t&s grands rendements operant sous conditions d’impulsions. On decrit une technique non-destructive simple pour mesurer la temperature d’une jonction impulsee a la rupture d’avalanche. Zusammenfassung-Die thermische Impedanz einer im Pulsbetrieb laufenden Lawinendiode wird berechnet. Die Diode wird dargestellt durch eine kreisfiirmige Wiirmequelle an der Oberflache eines unendlich ausgedehnten festen Zylinders. Die Rechnung wird durchgefiihrt fiir einzelne und fiir periodische Impulse. Das Temperaturverhalten wird charakterisiert durch eine Zeitkonstante 1, = R’/a, wo R der Diodenradius und G(die WBrmediffusionskonstante sind. Das Ergebnis kann dazu dienen die Grenze der Maximalleistung, der mittleren Leistung sowie der Energie je Impuls abzuleiten fiir Lawinendiodenoszillatoren mit hohem Wirkungsgrad. Es wird eine einfache, zerstorungsfreie Messmethode beschrieben, zur Bestimmung der Temperatur im p-n-Ubergang bei pulsfijrmiger Aussteuerung in den Lawinendurchbruchsbereich.
INTRODUCTION
the device is the product of the efficiency, the input power density, and the junction area. Therefore, in assessing the power limitations of these oscillators, at least three factors have to be considered; the minimum realizable impedance level in a practical microwave circuit (which sets an upper limit for the junction area), any current density limitation intrinsic to the microwave power generating process and the degradation of performance resulting from internal heating of the junction. Since the power can, at least in principle, be increased without limit by making the device
RECENTLY discovered modes of oscillation in avalanche diodes have led to the development of microwave generators with d.c.-to-r.f. conversion efficiencies on the orderof 50 per cent.(lV4) These devices operate efficiently at power densities well in excess of the optimum power density required for normal transit time oscillations. They are therefore well suited for application as highpulsed-power microwave sources where highefficiency and high-input power are required simultaneously. The microwave output power from 799
G.
800
GIBBOSS
bigger, the ultimate limitation is set by the first factor, minimum impedance level. This has been discussed by DELOACH(~) and will not be considered here. We will consider the limits imposed by the internal heating of the junction when the input power is applied in the form of periodic pulses. The maximum power that can be dissipated is determined by the maximum safe operating temperature of the junction (about 300°C for Si) and the thermal impedance of the structure which is a measure of how efficiently the heat is dissipated. Under pulsed bias we can define a transient thermal impedance 8, as the junction temperature reached at the end of a heating cycle when a peak pulse power of one watt is applied. 8, depends on the pulse length f, and the duty cycle (tJta), where f, is the period of successive pulses. A simple, nondestructive method for measuring the junction temperature directly from the voltage increase between the start and end of the pulse is described. This technique is used to measure experimentally the transient thermal impedance of a diode. Reasonably good agreement with the calculated value is obtained over a range of three orders of magnitude variation in pulse length. Plots of maximum peak power vs. pulse length are given for 100 p, 250 p and 625 p diameter diodes. DIODE
MODEL
In a typical diode structure, shown in Fig. 1, the heat generated at the junction flows through a region of heavily doped semiconductor and a
3ntac: (Au)
FIG. 1. Physical
structure of typical diode.
silicon
lrvalanche
metallic contact, usually gold, before reaching the copper heat sink. \Ve shall assume that all of the heat is generated in a plane at the junction and shall restrict our considerations to the case where heat is removed through one end of the device only. This structure, although not optimum for high-power devices, is the one most commonly
used to date in experimental models. The flow of heat away from the junction into the surrounding mass is governed by the laws of diffusion. The heat penetrates into the semiconductor and metal contact regions and then spreads out into the copper half space. Associated with each section of the structure is a thermal impedance determined by the thermal and geometrical properties of the section. In the steady-state case, where power is applied continuously, the total thermal impedance of the diode is obtained by adding the contributions from each section. When the input power is applied in a pulse that is shorter than the time required for the heat to diffuse through the structure and establish a steady temperature distribution, the thermal impedance is determined by the distance penetrated bv the heat during the heating cycle. As a result, transient thermal impedance may not be added for sections in series since any- section will contribute only if the heat reaches it. A method for calculating the thermal impedance of a multisection device has been presented by DIEBOLD and LwI-.‘~’ In this paper, we will simplifv our problem byconsidering only the spreading resistance and neglect completely the semiconductor and metal between the junction plane and the regions surface of the heat sink. Thus the model assumed for the diode is a circular heat source on the surface of an infinite solid cylinder. This model differs from those of MORTENSON’~) and DIEBOI.D(~) who have also considered the transient heating of semiconductor devices. ;\‘IORTESSO#) has calculated the transient temperature response of a uniform bar of semiconductor suspended between two posts at constant temperature. This analysis applies only to the case of one-dimensional rectangular flow of heat. In addition, DIEBOLD’*’ has considered the radial flow of heat into an infinite solid from a hemispherical source imbedded in the surface of the solid. This approximation is frequently made for the calculation of the spreading resistance of a small circular source. However, considerable error may result’“’ because of the importance of the region near the origin. Neglecting the parasitic series resistance, the current I flowing through a back biased junction is related to the voltage drop V across the junction by the cspression V = VBo+AVB+R,,I
(1)
TRANSIENT
TEMPERATURE
RESPONSE
where V,, is the room temperature value of the breakdown voltage, AV, is the incremental change in breakdown voltage caused by internal heating and R,, is the space charge resistance. If a constant voltage is applied the current decreases during the pulse as the diode heats up and the applied power is given by va vv,, v ---Air, P=K-- SC R,, R,,
OF AN AVALANCHE
801
DIODE
spatially uniform, time-varying heat flux F(t) located at the surface of a semi-infinite copper cylinder. TRANSIENT
THERMAL
IMPEDANCE
In a cylindrical coordinate system possessing circular symmetry, the temperature distribution is given by ii2T 1 aT ii2T 18T
i.e.,
-F(t)/K P = constant
When a constant current is applied the voltage increases during the pulse as the diode heats up and the applied power is given by P = V,,I+
R,,Z2+AVBI
I.e., P = constant
+IAVa.
Therefore, an accurate calculation of the transient temperature response must be made by an iterative scheme where the input flux is continuously adjusted according to the junction temperature. We will consider only the first approximation and assume that the input flux density is constant when the pulse is on. This is obviously the case of most practical interest. Further, we shall assume that the input flux is uniform over the entire area of the junction. Actually, for constant voltage bias, the flux is larger around the outer edge of the diode.(lO) Since the diode is hottest at the center the current will adjust to make the power density smaller there. For the steady-state case, this effect has been shown to produce about a 10 per cent reduction in the thermal impedance.‘lO) The model of the diode is illustrated in Fig. 2. The diode is assumed to be a heat source radius R with a Uniform
O
-cAVa. SC
flux density
F(t)
FIG. 2. Model assumed for the purpose of calculation of thermal impedance.
(3)
r>R,Z=O
where CL,K are the thermal diffusivity and conductivity of copper. T is the temperature rise above room temperature hereinafter referred to simply as temperature. Solutions to this problem have been obtained (11012)for cases where heat flux is applied in a single pulse or in periodic pulses. Both are discussed below. Single pulse Since the maximum temperature is reached at the end of the pulse we need only consider the application of a step function in power, i.e. F(t) = 0
t
F(t) = F.
t > 0I
(4)
then for any time t, > 0 the temperature calculated is that obtained by application of a single pulse of length t,. The temperature after the end of the applied pulse (during the cooling cycle) can be calculated by superposing two such step functions, namely, where
F(t) = F,(t) +Fa(t)
F,(t) = 0,
t < 0 : r;;(t) =
F,(t) = 0,
t < tl:F2(t)
Fo,
= -F,,
t > 0 t > t, I ’
The maximum temperature occurs at the center of the diode and for the single pulse of applied power is given by solution of (2-4), yielding’ll)
802
G.
GIBBONS
where R is the radius of the diode. A natural time constant for the diode is the quantity f, = (R2/a): when the pulse length is much smaller than this time constant (fi < tJ, (5) becomes T = 2F,(~.t,)~‘~/y’z-K
(6)
and the temperature is proportional to the square root of the pulse length independent of the diode size: i.e. the temperature is the same as though the entire surface of the heat sink were exposed to the flux density F,. It is easily shown that for long pulses (tl $ R”,h), the temperature given in (5)
spreading
resistance
decreases
with
the radius
1 O,, = p. rKR Periodic pulses The case of most practical interest is the application of periodic input power pulses of length t, and period t,. For this case F(t) is given by F = F,
nt,<
t < nt,+t,
F = 0
nt, i-t,
I1 = 0, 1, 2 . . . (8)
< t < (n+ l)tz
~__._ FO ‘ih +l, +---+ ‘. r.‘
AFpl,ed fli;x dens’y
:‘;
mole @se
_
130~
-1
FULSE LENGTH
FIG.
reduces
to the steady-state
3.
Thermal impedance of diode for various lengths (single pulse applied).
value (7)
T2$.
8, has From (5) the transient thermal impedance been calculated and the results are shown in Fig. 3 for various pulse lengths and junction radii. For pulses equal in length to the diode time constant (indicated by the arrows in Fig. 3), the thermal impedance has reached 75 per cent of the steady-state value. For pulses considerably shorter than the time constant the thermal impedance decreases with the area of the junction 112 0,
=
2
““1 (
in contrast
?r
( seconds)
The maximum temperature occurring at the end of a heating cycle and at the center of the diode is given by”‘)
RF, h
0
?‘=y
7
-+(rt,)1’2
to the steady-state
case in which
R ------+1
R
1 A
-e-RZi4crt1)
(
erfc
2&/2(,t,)l’”
(d,)l’”
(9) 1
1s; S ,* x
the
R
2
2 *(l
where
1
77KR”
F,tla1i2 f---x
’ 2
e-dz2{e-_(l-d)ZZ_e-~2
__ 1
pulse
c 0
-
} (
1 - ~os(~
x2(1 -emf2)
dx
TRANSIENT
TEMPERATURE
RESPONSE
and d is the duty cycle (tJta>. When the period between pulses t, is much longer than the diode time constant t,, the contribution to (9) from the integral I is negligible and (9) then reduces to
R +(at,)l/’
R -_ erfc 2(at,)1/2 I
(10)
which is identical to the single-pulse response given in (5). The temperature, as given in (9), is composed of two parts: one is an average temperature based on the steady application of the average flux density F,(t,/t,) and therefore given by
(11) For high duty cycles the temperature response is not fast enough to follow the rapidly changing power and the junction assumes a constant average value given by (11). The other part of the solution represents a periodic temperature variation which reduces to the single-pulse response (5) when the period between pulses is much greater than the time constant of the structure (t2 9 tc). Thus, for low duty cycles the junction temperature follows the applied power waveform more exactly. From (9) the transient thermal impedance has been calculated for a 100 p radius diode for various pulse
OF
.4N
AVALANCHE
DIODE
803
lengths and duty cycles. The results are shown in Fig. 4. For high duty factors the thermal impedance approaches the average value given by where Ocwis the continuous, steady-state (+a)& value. For low duty factors the thermal impedance is given by the single-pulse result and is independent of the duty cycle; the effect of the duty cycle becomes evident only when the period between pulses ta is equal to the diode time constant t,, as shown by the arrows in Fig. 4. EXPERIMENTAL RESULT In this section, a simple, nondestructive technique is described for measuring the temperature rise in the active region of an avalanche diode under pulsed bias. The voltage drop across the junction is used as a measure of the temperature rise produced by the internal heating. The diode is mounted in a non-oscillating circuit and biased with a low duty cycle constant current pulse. Under these conditions the voltage increases during the pulse and the voltage difference between the start and end of the pulse is proportional to the temperature rise at the junction. The technique can readily be extended to high duty cycles by measuring the voltage increase at the start of the pulse as the duty cycle is increased. Here we shall consider only the single pulse (low duty cycle) case. The voltage increase is given by
where T is the junction temperature and T, is room temperature. The avalanche breakdown voltage of abrupt junctions increases linearly with temperature(13) J’,(T)
WTY
FACTOR
FIG. 4. Thermal impedance of 100 p radius diode for various pulse lengths and duty cycles (periodic pulses
applied).
= vn(To)+p(T-
To)vs(T,).
(13)
Measurements of V,(T) have been made from 300 to 500°K on abrupt junction Si diodes with breakdown voltages V,(T,) from 30 V to 120 V. For all diodes measured the linear variation given by (13) was found to be a good fit to the experimental data. The values of Bv,(T,) are plotted vs. vn(Ta) in Fig. 5. An Si abrupt junction diode with radius of 80 p and breakdown voltage of 120 V was used. The variation of the space charge resistance with temperature was measured. (ARJAT) was = 0.1 n/C. Thus, the spacech arge resistance
804
G.
GIBBONS
I
FIG. 5. Temperature dependance of breakdown voltage for abrupt-junction silicon diodes vs. room-temperature value. term
in (12) can be neglected
provided
and the voltage increase between the start and the end of the pulse is then given by AV = hT/Wa(T,).
(14)
The bias current was chosen to be 100 mA to satisfy the above condition and the voltage rise for various pulse lengths was measured. The temperature rise at the junction is calculated using
(14). From these results the thermal impedance can be calculated approximately using the average value for the peak input power. The nonconstant power during the pulse leads to some error when this technique is applied to cases of large tempcrature rise. However, in the case considered here (when the average pulse power is used) the masimum error is _+7 per cent. Figure 6 shows the results of the thermal impedance measurements for pulse lengths from 1 psec to 1 msec. Also shown is the transient thermal impedance calculated using (5) for a 80 p radius diode. Although this particular diode did not conform well to the model of Fig. 2, since the junction was 2 ,LLfrom the surface and the Au contact was approximately 1.5 p thick, the agreement between the measured and calculated values of 0 is reasonably good. DISCUSSION
The transient thermal impedance results, presented above, can be used to calculate the power and energy limits for pulsed avalanche diode oscillators. The discovery of the high-efficiency has greatly (anomalous) mode of operation increased the interest in avalanche diodes as practical, high-power, pulsed microwave generators. In addition to the high efficiency, this mode has the advantage of operating at much higher power densities than the normal transit-time mode. Figure 7 shows efficiency vs. input power density for an abrupt junction Si I!VIPATT diode
FIG. 6. AIeasured and calculated thermal impedance for silicon diode under single-pulse bias.
TRANSIENT
TEMPERATURE
RESPONSE
OF
AN
AVALANCHE
DIODE
805
FIG. 8. Peak power vs. pulse length for 100, 250, 625 p dia. Si avalanche diodes.
FIG.
7. d.c.-to-r.f. conversion efficiency for both and high-efficiency modes as a function of
IMPATT
input power density.
operating in the normal transit-time mode at 6 GHz. The efficiency peaks at an input power density of 7 x 104W/cm2 are in good agreement with HOEFFLINGER’S(~~’ theory. The same theory predicts that at 1.5 GHz the optimum power density for the transit-time mode is -3 x lo4 W/cm2. For comparison in Fig. 7 is shown efficiency vs. input power density on a ‘punch-through’(3) p+-n-n+ Si diode operating in the high-efficiency mode at 1.5 GHz. There is essentially no degradation in efficiency as the input power density is increased from 6 x lo4 W/cm2 to 3 x lo5 W/cm2 which is about an order of magnitude higher than the optimum power density in the transit-time mode at the same frequency. Thus, at a given frequency and with a given size diode we expect the high-efficiency mode to yield about fifty times the power that can be obtained from the transit-time mode. The data shown in Fig. 7 were obtained with 1 psec pulses with a repetition rate of 1000 pulses/set. The diodes were approximately 100 p in diameter. For Si diodes, the maximum safe temperature rise at the junction is about 250°C. Using this value and the singlepulse thermal impedance results, the maximum peak power vs. pulse length has been plotted in
Fig. 8 for 100, 250 and 625 p dia. Si diodes. Assuming conversion efficiencies of 50 per cent (power generated equal to power dissipated) these results represent the maximum peak microwave output power attainable from Si avalanche diode sources having the structure of Fig. 1. It should be emphasized that, because of the simplifications in the diode model, these results represent upper limits. In practice, the output power is limited also by diode size and by the maximum usable power density. As the device is made bigger it becomes more difficult to get a uniform avalanche junction with low leakage current. Also, nonuniform distribution of current in large area devices becomes a serious problem and reduces the total effective area of the junction.“” The diode area is also limited by the impedance level that can be realized in a practical microwave circuit. CONCLUSION
The principal conclusions of this paper are summarized as follows. The temperature response of an avalanche diode under pulsed bias is characterized by a thermal time constant t,, given by
t, = R2ix where R is the junction radius fusivity of the heat sink. When pulses is much longer than this thermal response of the diode a single pulse and the thermal
and M is the difthe period between time constant, the is the same as for impedance for this
806
G.
GIBBONS
case is given in (5) and Fig. 3. When the period between pulses is short (high duty cycle) the thermal impedance approaches a value equal to the product of the duty cycle and the steady state, cw thermal impedance. For any value of duty cycle and pulse length the thermal impedance is given by (9) and in Fig. 4 (for a 100 p radius diode). Using the voltage drop across a reverse biased junction, as a measure of the operating temperature, the thermal impedance of an Si avalanche diode has been measured as a function of pulse length. These results can be used to develop the power-energy limits for pulsed avalanche diode oscillators. The maximum peak power vs. pulse length has been calculated for 100, 250 and 625 p dia. diodes.
Acklzowledgment-The author is indebted to L. W. CURRIER for technical assistance in the fabrication and testing of the diodes.
REFERENCES 1. H. J. PR.~GER, K. K. N. CHANG, and S. WEISBROU, Proc. IEEE%, 586 (1967). 2. R. L. JOHNSTON, D. L. SCH~RFETTER and I>. J. BARTELINK,Proc. IEEE 56, 1611 (1968). 3. IZ’I. I. GRACE and G. GIBBONS, Electron. Lett. 4, 564 (1968). 4. C. P. SNAPP and B. HOEFFLINGER, Int. Electron Devices Conf., Washington, D.C. (1968). 5. B. C. DELOACH, Advances in Microwaves (Ed. LEO YOUNG). Academic Press, New York (1967). 6. E. J. DIEBOLD and IV. LUFT, Trans. AIEE Part I, Commun. Electron. 719 (1961). 7. K. E. MORTENSON, Proc. I.R.E. 45, 504(1957). 8. E. J. DIEBOLD, Trans. .4IEE Part I, Com~wrr. Electron. 76, 593 (1957). 9. H. S. CARSLAW and J. C. JAEGER, Comiuction of IIeat in Solids. D. 217. Clarendon Press. Oxford (1959). 10. G. GIBBoNsand +. MIsaw.4, Solid-‘St. Elect&. ii, 1007 (1968). 11. H. S. CARSLAW and J. C. JAEGER,Cor2ductiofr of lieat in Solids, p. 264. Clarendon Press, Oxford (1959). 12. J. C. JAEGER,Aust. Quart. Math. 11,132 (1953). 13. K. G. MCKAY, Phys. Rev. 94,877 (1954). 14. B. HOEFFUNGER, Microzcave J. 12, 101 (1969).
Electronics
TRANSIENT
Pergamon
Printed
Press 1970. Vol. 13, pp. 799406.
TEMPERATURE
RESPONSE
in Great Britain
OF AN AVALANCHE
DIODE G. GIBBONS Sperry Rand Research (Received
Center,
Sudbury,
Massachusetts,
30 June 1969; in rwised_form
8 August
U.S.A.
1969)
Abstract-The thermal impedance is calculated for an avalanche diode under pulse bias conditions. The diode is represented by a circular heat source on the surface of an infinite solid cylinder and the calculation is made for the application of a single pulse and for periodic pulses. The temperature response is characterized by a time constant t, = IF/a, where R is the radius of the diode and TVis the diffusivity of the heat sink. The results can be used to derive peak and average power limits and energy per pulse limits for high-efficiency avalanche diode oscillators operating under pulsed conditions. A simple, non-destructive technique is described for measuring the temperature of a junction when pulsed into avalanche breakdown. R&sum&L’impedance thermique est calculee pour une diode a avalanche sous des conditions de polarisations d’impulsions. La diode est representee par une source de chaleur circulaire sur la surface d’un cylindre infiniment solide et les calculs sont faits pour l’application d’une impulsion unique et d’impulsions periodiques. La reponse de temperature est caracterisee par une constante de temps tc R2/or, oh R est le rayon de la diode et ccla diffusion du recepteur de chaleur. Les resultats peuvent &tre employ& pour d&river les limites de c&e et moyennes des puissances et energie par impulsion des oscillateurs B diodes a avalanche ayant de t&s grands rendements operant sous conditions d’impulsions. On decrit une technique non-destructive simple pour mesurer la temperature d’une jonction impulsee a la rupture d’avalanche. Zusammenfassung-Die thermische Impedanz einer im Pulsbetrieb laufenden Lawinendiode wird berechnet. Die Diode wird dargestellt durch eine kreisfiirmige Wiirmequelle an der Oberflache eines unendlich ausgedehnten festen Zylinders. Die Rechnung wird durchgefiihrt fiir einzelne und fiir periodische Impulse. Das Temperaturverhalten wird charakterisiert durch eine Zeitkonstante 1, = R’/a, wo R der Diodenradius und G(die WBrmediffusionskonstante sind. Das Ergebnis kann dazu dienen die Grenze der Maximalleistung, der mittleren Leistung sowie der Energie je Impuls abzuleiten fiir Lawinendiodenoszillatoren mit hohem Wirkungsgrad. Es wird eine einfache, zerstorungsfreie Messmethode beschrieben, zur Bestimmung der Temperatur im p-n-Ubergang bei pulsfijrmiger Aussteuerung in den Lawinendurchbruchsbereich.
INTRODUCTION
the device is the product of the efficiency, the input power density, and the junction area. Therefore, in assessing the power limitations of these oscillators, at least three factors have to be considered; the minimum realizable impedance level in a practical microwave circuit (which sets an upper limit for the junction area), any current density limitation intrinsic to the microwave power generating process and the degradation of performance resulting from internal heating of the junction. Since the power can, at least in principle, be increased without limit by making the device
RECENTLY discovered modes of oscillation in avalanche diodes have led to the development of microwave generators with d.c.-to-r.f. conversion efficiencies on the orderof 50 per cent.(lV4) These devices operate efficiently at power densities well in excess of the optimum power density required for normal transit time oscillations. They are therefore well suited for application as highpulsed-power microwave sources where highefficiency and high-input power are required simultaneously. The microwave output power from 799
G.
800
GIBBOSS
bigger, the ultimate limitation is set by the first factor, minimum impedance level. This has been discussed by DELOACH(~) and will not be considered here. We will consider the limits imposed by the internal heating of the junction when the input power is applied in the form of periodic pulses. The maximum power that can be dissipated is determined by the maximum safe operating temperature of the junction (about 300°C for Si) and the thermal impedance of the structure which is a measure of how efficiently the heat is dissipated. Under pulsed bias we can define a transient thermal impedance 8, as the junction temperature reached at the end of a heating cycle when a peak pulse power of one watt is applied. 8, depends on the pulse length f, and the duty cycle (tJta), where f, is the period of successive pulses. A simple, nondestructive method for measuring the junction temperature directly from the voltage increase between the start and end of the pulse is described. This technique is used to measure experimentally the transient thermal impedance of a diode. Reasonably good agreement with the calculated value is obtained over a range of three orders of magnitude variation in pulse length. Plots of maximum peak power vs. pulse length are given for 100 p, 250 p and 625 p diameter diodes. DIODE
MODEL
In a typical diode structure, shown in Fig. 1, the heat generated at the junction flows through a region of heavily doped semiconductor and a
3ntac: (Au)
FIG. 1. Physical
structure of typical diode.
silicon
lrvalanche
metallic contact, usually gold, before reaching the copper heat sink. \Ve shall assume that all of the heat is generated in a plane at the junction and shall restrict our considerations to the case where heat is removed through one end of the device only. This structure, although not optimum for high-power devices, is the one most commonly
used to date in experimental models. The flow of heat away from the junction into the surrounding mass is governed by the laws of diffusion. The heat penetrates into the semiconductor and metal contact regions and then spreads out into the copper half space. Associated with each section of the structure is a thermal impedance determined by the thermal and geometrical properties of the section. In the steady-state case, where power is applied continuously, the total thermal impedance of the diode is obtained by adding the contributions from each section. When the input power is applied in a pulse that is shorter than the time required for the heat to diffuse through the structure and establish a steady temperature distribution, the thermal impedance is determined by the distance penetrated bv the heat during the heating cycle. As a result, transient thermal impedance may not be added for sections in series since any- section will contribute only if the heat reaches it. A method for calculating the thermal impedance of a multisection device has been presented by DIEBOLD and LwI-.‘~’ In this paper, we will simplifv our problem byconsidering only the spreading resistance and neglect completely the semiconductor and metal between the junction plane and the regions surface of the heat sink. Thus the model assumed for the diode is a circular heat source on the surface of an infinite solid cylinder. This model differs from those of MORTENSON’~) and DIEBOI.D(~) who have also considered the transient heating of semiconductor devices. ;\‘IORTESSO#) has calculated the transient temperature response of a uniform bar of semiconductor suspended between two posts at constant temperature. This analysis applies only to the case of one-dimensional rectangular flow of heat. In addition, DIEBOLD’*’ has considered the radial flow of heat into an infinite solid from a hemispherical source imbedded in the surface of the solid. This approximation is frequently made for the calculation of the spreading resistance of a small circular source. However, considerable error may result’“’ because of the importance of the region near the origin. Neglecting the parasitic series resistance, the current I flowing through a back biased junction is related to the voltage drop V across the junction by the cspression V = VBo+AVB+R,,I
(1)
TRANSIENT
TEMPERATURE
RESPONSE
where V,, is the room temperature value of the breakdown voltage, AV, is the incremental change in breakdown voltage caused by internal heating and R,, is the space charge resistance. If a constant voltage is applied the current decreases during the pulse as the diode heats up and the applied power is given by va vv,, v ---Air, P=K-- SC R,, R,,
OF AN AVALANCHE
801
DIODE
spatially uniform, time-varying heat flux F(t) located at the surface of a semi-infinite copper cylinder. TRANSIENT
THERMAL
IMPEDANCE
In a cylindrical coordinate system possessing circular symmetry, the temperature distribution is given by ii2T 1 aT ii2T 18T
i.e.,
-F(t)/K P = constant
When a constant current is applied the voltage increases during the pulse as the diode heats up and the applied power is given by P = V,,I+
R,,Z2+AVBI
I.e., P = constant
+IAVa.
Therefore, an accurate calculation of the transient temperature response must be made by an iterative scheme where the input flux is continuously adjusted according to the junction temperature. We will consider only the first approximation and assume that the input flux density is constant when the pulse is on. This is obviously the case of most practical interest. Further, we shall assume that the input flux is uniform over the entire area of the junction. Actually, for constant voltage bias, the flux is larger around the outer edge of the diode.(lO) Since the diode is hottest at the center the current will adjust to make the power density smaller there. For the steady-state case, this effect has been shown to produce about a 10 per cent reduction in the thermal impedance.‘lO) The model of the diode is illustrated in Fig. 2. The diode is assumed to be a heat source radius R with a Uniform
O
-cAVa. SC
flux density
F(t)
FIG. 2. Model assumed for the purpose of calculation of thermal impedance.
(3)
r>R,Z=O
where CL,K are the thermal diffusivity and conductivity of copper. T is the temperature rise above room temperature hereinafter referred to simply as temperature. Solutions to this problem have been obtained (11012)for cases where heat flux is applied in a single pulse or in periodic pulses. Both are discussed below. Single pulse Since the maximum temperature is reached at the end of the pulse we need only consider the application of a step function in power, i.e. F(t) = 0
t
F(t) = F.
t > 0I
(4)
then for any time t, > 0 the temperature calculated is that obtained by application of a single pulse of length t,. The temperature after the end of the applied pulse (during the cooling cycle) can be calculated by superposing two such step functions, namely, where
F(t) = F,(t) +Fa(t)
F,(t) = 0,
t < 0 : r;;(t) =
F,(t) = 0,
t < tl:F2(t)
Fo,
= -F,,
t > 0 t > t, I ’
The maximum temperature occurs at the center of the diode and for the single pulse of applied power is given by solution of (2-4), yielding’ll)
802
G.
GIBBONS
where R is the radius of the diode. A natural time constant for the diode is the quantity f, = (R2/a): when the pulse length is much smaller than this time constant (fi < tJ, (5) becomes T = 2F,(~.t,)~‘~/y’z-K
(6)
and the temperature is proportional to the square root of the pulse length independent of the diode size: i.e. the temperature is the same as though the entire surface of the heat sink were exposed to the flux density F,. It is easily shown that for long pulses (tl $ R”,h), the temperature given in (5)
spreading
resistance
decreases
with
the radius
1 O,, = p. rKR Periodic pulses The case of most practical interest is the application of periodic input power pulses of length t, and period t,. For this case F(t) is given by F = F,
nt,<
t < nt,+t,
F = 0
nt, i-t,
I1 = 0, 1, 2 . . . (8)
< t < (n+ l)tz
~__._ FO ‘ih +l, +---+ ‘. r.‘
AFpl,ed fli;x dens’y
:‘;
mole @se
_
130~
-1
FULSE LENGTH
FIG.
reduces
to the steady-state
3.
Thermal impedance of diode for various lengths (single pulse applied).
value (7)
T2$.
8, has From (5) the transient thermal impedance been calculated and the results are shown in Fig. 3 for various pulse lengths and junction radii. For pulses equal in length to the diode time constant (indicated by the arrows in Fig. 3), the thermal impedance has reached 75 per cent of the steady-state value. For pulses considerably shorter than the time constant the thermal impedance decreases with the area of the junction 112 0,
=
2
““1 (
in contrast
?r
( seconds)
The maximum temperature occurring at the end of a heating cycle and at the center of the diode is given by”‘)
RF, h
0
?‘=y
7
-+(rt,)1’2
to the steady-state
case in which
R ------+1
R
1 A
-e-RZi4crt1)
(
erfc
2&/2(,t,)l’”
(d,)l’”
(9) 1
1s; S ,* x
the
R
2
2 *(l
where
1
77KR”
F,tla1i2 f---x
’ 2
e-dz2{e-_(l-d)ZZ_e-~2
__ 1
pulse
c 0
-
} (
1 - ~os(~
x2(1 -emf2)
dx
TRANSIENT
TEMPERATURE
RESPONSE
and d is the duty cycle (tJta>. When the period between pulses t, is much longer than the diode time constant t,, the contribution to (9) from the integral I is negligible and (9) then reduces to
R +(at,)l/’
R -_ erfc 2(at,)1/2 I
(10)
which is identical to the single-pulse response given in (5). The temperature, as given in (9), is composed of two parts: one is an average temperature based on the steady application of the average flux density F,(t,/t,) and therefore given by
(11) For high duty cycles the temperature response is not fast enough to follow the rapidly changing power and the junction assumes a constant average value given by (11). The other part of the solution represents a periodic temperature variation which reduces to the single-pulse response (5) when the period between pulses is much greater than the time constant of the structure (t2 9 tc). Thus, for low duty cycles the junction temperature follows the applied power waveform more exactly. From (9) the transient thermal impedance has been calculated for a 100 p radius diode for various pulse
OF
.4N
AVALANCHE
DIODE
803
lengths and duty cycles. The results are shown in Fig. 4. For high duty factors the thermal impedance approaches the average value given by where Ocwis the continuous, steady-state (+a)& value. For low duty factors the thermal impedance is given by the single-pulse result and is independent of the duty cycle; the effect of the duty cycle becomes evident only when the period between pulses ta is equal to the diode time constant t,, as shown by the arrows in Fig. 4. EXPERIMENTAL RESULT In this section, a simple, nondestructive technique is described for measuring the temperature rise in the active region of an avalanche diode under pulsed bias. The voltage drop across the junction is used as a measure of the temperature rise produced by the internal heating. The diode is mounted in a non-oscillating circuit and biased with a low duty cycle constant current pulse. Under these conditions the voltage increases during the pulse and the voltage difference between the start and end of the pulse is proportional to the temperature rise at the junction. The technique can readily be extended to high duty cycles by measuring the voltage increase at the start of the pulse as the duty cycle is increased. Here we shall consider only the single pulse (low duty cycle) case. The voltage increase is given by
where T is the junction temperature and T, is room temperature. The avalanche breakdown voltage of abrupt junctions increases linearly with temperature(13) J’,(T)
WTY
FACTOR
FIG. 4. Thermal impedance of 100 p radius diode for various pulse lengths and duty cycles (periodic pulses
applied).
= vn(To)+p(T-
To)vs(T,).
(13)
Measurements of V,(T) have been made from 300 to 500°K on abrupt junction Si diodes with breakdown voltages V,(T,) from 30 V to 120 V. For all diodes measured the linear variation given by (13) was found to be a good fit to the experimental data. The values of Bv,(T,) are plotted vs. vn(Ta) in Fig. 5. An Si abrupt junction diode with radius of 80 p and breakdown voltage of 120 V was used. The variation of the space charge resistance with temperature was measured. (ARJAT) was = 0.1 n/C. Thus, the spacech arge resistance
804
G.
GIBBONS
I
FIG. 5. Temperature dependance of breakdown voltage for abrupt-junction silicon diodes vs. room-temperature value. term
in (12) can be neglected
provided
and the voltage increase between the start and the end of the pulse is then given by AV = hT/Wa(T,).
(14)
The bias current was chosen to be 100 mA to satisfy the above condition and the voltage rise for various pulse lengths was measured. The temperature rise at the junction is calculated using
(14). From these results the thermal impedance can be calculated approximately using the average value for the peak input power. The nonconstant power during the pulse leads to some error when this technique is applied to cases of large tempcrature rise. However, in the case considered here (when the average pulse power is used) the masimum error is _+7 per cent. Figure 6 shows the results of the thermal impedance measurements for pulse lengths from 1 psec to 1 msec. Also shown is the transient thermal impedance calculated using (5) for a 80 p radius diode. Although this particular diode did not conform well to the model of Fig. 2, since the junction was 2 ,LLfrom the surface and the Au contact was approximately 1.5 p thick, the agreement between the measured and calculated values of 0 is reasonably good. DISCUSSION
The transient thermal impedance results, presented above, can be used to calculate the power and energy limits for pulsed avalanche diode oscillators. The discovery of the high-efficiency has greatly (anomalous) mode of operation increased the interest in avalanche diodes as practical, high-power, pulsed microwave generators. In addition to the high efficiency, this mode has the advantage of operating at much higher power densities than the normal transit-time mode. Figure 7 shows efficiency vs. input power density for an abrupt junction Si I!VIPATT diode
FIG. 6. AIeasured and calculated thermal impedance for silicon diode under single-pulse bias.
TRANSIENT
TEMPERATURE
RESPONSE
OF
AN
AVALANCHE
DIODE
805
FIG. 8. Peak power vs. pulse length for 100, 250, 625 p dia. Si avalanche diodes.
FIG.
7. d.c.-to-r.f. conversion efficiency for both and high-efficiency modes as a function of
IMPATT
input power density.
operating in the normal transit-time mode at 6 GHz. The efficiency peaks at an input power density of 7 x 104W/cm2 are in good agreement with HOEFFLINGER’S(~~’ theory. The same theory predicts that at 1.5 GHz the optimum power density for the transit-time mode is -3 x lo4 W/cm2. For comparison in Fig. 7 is shown efficiency vs. input power density on a ‘punch-through’(3) p+-n-n+ Si diode operating in the high-efficiency mode at 1.5 GHz. There is essentially no degradation in efficiency as the input power density is increased from 6 x lo4 W/cm2 to 3 x lo5 W/cm2 which is about an order of magnitude higher than the optimum power density in the transit-time mode at the same frequency. Thus, at a given frequency and with a given size diode we expect the high-efficiency mode to yield about fifty times the power that can be obtained from the transit-time mode. The data shown in Fig. 7 were obtained with 1 psec pulses with a repetition rate of 1000 pulses/set. The diodes were approximately 100 p in diameter. For Si diodes, the maximum safe temperature rise at the junction is about 250°C. Using this value and the singlepulse thermal impedance results, the maximum peak power vs. pulse length has been plotted in
Fig. 8 for 100, 250 and 625 p dia. Si diodes. Assuming conversion efficiencies of 50 per cent (power generated equal to power dissipated) these results represent the maximum peak microwave output power attainable from Si avalanche diode sources having the structure of Fig. 1. It should be emphasized that, because of the simplifications in the diode model, these results represent upper limits. In practice, the output power is limited also by diode size and by the maximum usable power density. As the device is made bigger it becomes more difficult to get a uniform avalanche junction with low leakage current. Also, nonuniform distribution of current in large area devices becomes a serious problem and reduces the total effective area of the junction.“” The diode area is also limited by the impedance level that can be realized in a practical microwave circuit. CONCLUSION
The principal conclusions of this paper are summarized as follows. The temperature response of an avalanche diode under pulsed bias is characterized by a thermal time constant t,, given by
t, = R2ix where R is the junction radius fusivity of the heat sink. When pulses is much longer than this thermal response of the diode a single pulse and the thermal
and M is the difthe period between time constant, the is the same as for impedance for this
806
G.
GIBBONS
case is given in (5) and Fig. 3. When the period between pulses is short (high duty cycle) the thermal impedance approaches a value equal to the product of the duty cycle and the steady state, cw thermal impedance. For any value of duty cycle and pulse length the thermal impedance is given by (9) and in Fig. 4 (for a 100 p radius diode). Using the voltage drop across a reverse biased junction, as a measure of the operating temperature, the thermal impedance of an Si avalanche diode has been measured as a function of pulse length. These results can be used to develop the power-energy limits for pulsed avalanche diode oscillators. The maximum peak power vs. pulse length has been calculated for 100, 250 and 625 p dia. diodes.
Acklzowledgment-The author is indebted to L. W. CURRIER for technical assistance in the fabrication and testing of the diodes.
REFERENCES 1. H. J. PR.~GER, K. K. N. CHANG, and S. WEISBROU, Proc. IEEE%, 586 (1967). 2. R. L. JOHNSTON, D. L. SCH~RFETTER and I>. J. BARTELINK,Proc. IEEE 56, 1611 (1968). 3. IZ’I. I. GRACE and G. GIBBONS, Electron. Lett. 4, 564 (1968). 4. C. P. SNAPP and B. HOEFFLINGER, Int. Electron Devices Conf., Washington, D.C. (1968). 5. B. C. DELOACH, Advances in Microwaves (Ed. LEO YOUNG). Academic Press, New York (1967). 6. E. J. DIEBOLD and IV. LUFT, Trans. AIEE Part I, Commun. Electron. 719 (1961). 7. K. E. MORTENSON, Proc. I.R.E. 45, 504(1957). 8. E. J. DIEBOLD, Trans. .4IEE Part I, Com~wrr. Electron. 76, 593 (1957). 9. H. S. CARSLAW and J. C. JAEGER, Comiuction of IIeat in Solids. D. 217. Clarendon Press. Oxford (1959). 10. G. GIBBoNsand +. MIsaw.4, Solid-‘St. Elect&. ii, 1007 (1968). 11. H. S. CARSLAW and J. C. JAEGER,Cor2ductiofr of lieat in Solids, p. 264. Clarendon Press, Oxford (1959). 12. J. C. JAEGER,Aust. Quart. Math. 11,132 (1953). 13. K. G. MCKAY, Phys. Rev. 94,877 (1954). 14. B. HOEFFUNGER, Microzcave J. 12, 101 (1969).