Chapter 5

FUNDAMENTAL LIMITATIONS ON POWER OUTPUT FROM SOLID-STATE MICROWAVE DEVICES

INTRODUCTION

In this chapter, we shall derive some fundamental limits on the output power which can be expected from solid-state microwave sources (oscillators). These limits will be derived with specific reference to the geometry of Gunn and IMPATT devices (DeLoach, Jr., 1967; Johnson, 1965; Scharfetter, 1971). It should be realized that very similar expressions can also be derived for other devices, to be discussed in later chapters, and we will make brief mention of such limitations further on, as well. The two fundamental physical effects which limit the amount of DC power a device can dissipate are (1) there is generally a maximum operating temperature beyond which failure modes of different types will occur; thus, we must be concerned with how effectively heat can be transported out of the active region of the device, so that the temperature rise can be limited, and (2) avalanche break-down limits both the maximum voltage and the maximum current of the device. Finally, the power conversion efficiency will determine the fraction of the maximum DC power which will be converted to RF power.

THE THERMAL LIMIT

Solid state sources are quite small in size, and consequently will be op­erating at high current densities and dissipated power densities. The devices are typically mounted on a heatsink made of a material with good thermal conductivity, such as copper or diamond, but there is invariably some distance between the location at which most of the heat is dissipated, and the heatsink. This path, which the heat dissipated must follow, becomes the "bottle-neck" which determines how much power that can be dissipated. A typical geometry is illustrated in Figure 5.1., where it should be noted that the vertical (height) dimension of the semiconductor "puck" or "mesa" has been exaggerated rel­ative to the horizontal (width) dimension, for clarity. Although the power dissipation may be distributed along the height dimension, we assume for sim­plicity that it can be localized to a particular height inside the semiconductor device.

We must next review the basic equations which govern transport of heat in a solid (see e.g. Kittel (1976), Swan et al. (1967) and Scharfetter (1971». These are easiest to remember if we note the equivalence with the case of

S. Yngvesson, Microwave Semiconductor Devices© Kluwer Academic Publishers 1991

128

T

Active, RegiOn '-=

Microwave Semiconductor Devices

Ribbon

Contact

Semiconductor Device

Discrete Heat

Source

Fig. 5.1.a

Heat Reservoir

J=======~. ________ I '.:m'm ".,," 14.5 ~mAu t

Diamond

Copper

Fig. 5.1.h

Chapter 5

0.5

.. 1.0

g 1.5 ~ a 2 .. Q

2.5

3

3.5

o

Solder

Seml-Inflnn. copper heat.lnk

1.5 3 Radius (mils)

Fig. 5.1.c

129

4.5 6

Figure 5.1. (a) Simplified geometry of a two-terminal semiconductor de­vice placed on a heat sink. (b) Actual geometry of an IMPATT diode placed on a diamond heat sink, with a thin intermediate layer of gold. (c) Heat Jluz curves in the heat sink of an IMPATT device. The curve. define .ur­faces which contain the fraction of the total heat Jluz noted. Parts (b) and (c) have been adapted from HOLWAY, Jr., L.H., and ADLERSTEIN, M.G. (1977). "Approzimate Formula. for the Thermal Resistance of IMPATT Diodes Compared with Computer Calculations," IEEE Tran6. Electron De­vice., ED-24, 156, @1977 IEEE.

electrical conduction, thus the equivalent expressions are noted in parallel. Corresponding to Ohm's law, we have the equation for heat flow:

(5.1)

Here,

aT = temperature difference, (OC or OK)

P = heat flow (watts)

R = heat resistance (OC/watt)

The geometry is assumed to be as shown in the sketch below (Figure 5.2):

130 Microwave Semiconductor Device,

r I ..

-:-1-------- -----

I

I I I

Area I A I

I Heat ~ Flow

Figure 5.2. Simplified geometry of a two-terminal device, used to e,timate the thermal re,istance in the tezt.

For this (let's say circularly cylindrical) geometry, the heat resistance can be found in analogy with that of the electrical resistance for the same geometry, i.e.

R = _d_ COmRE R = _d_ (= dP) A· K.S A· (1' A

(5.2)

Here we have introduced K.s, which is the thermal conductivity of the semiconductor with units of W fcm and 0 C or oK. If we have a device which consists of several sections made from different materials along the height di­mension, we can find the total thermal resistance by adding the individual resistances in series as in the electrical case.

In passing the plane where the heat sink begins, we have a different case, since the cross-sectional area of the heat sink typically is much larger than that of the device. While in the semiconductor we can reasonably assume that the heat flow is uniform over the cross-sectional area, the heat flow to the heat sink will clearly spread, which will increase the amount of heat which can be transported. This type of flow is also familiar from the electrical case, and the resistance for this case is termed the "spreading" resistance. If we assume that the heat-sink is essentially infinitely thick, then the spreading resistance is given by :

(5.3)

A third situation occurs if the cross-sectional diameter is increased but the thickness can not be assumed to be infinite, as illustrated in Figure 5.3.

A convenient approximation to use in this case is to assume that the heat­flow is still approximately uniform, and spreading through a conical volume where the angle of the cone is 8. The effective radius at the lower surface of the material with larger. radius thus is

7'N+1 = rN + dN tan8j (5.4)

Chapter 5 131

(a)

(b)

45°

(e)

Figure 5.3. Rlustration o/the t18e o/the conical approzimation/or calculating the spreading resistance in different cases. (a) A GaAs device placed on a heat sink comp08ed 0/ two layers. (b) A case in which the radius 0/ the heat sink is large. (c) A case in which the radius of the heat sink is smaller than the cross·sectional radius 01 the cone at a point in8ide the bottom layer. Adapted from HOLWAY, Jr., L.H. and ADLERSTEIN, M.G. (1977), see caption lor Fi9ure 5.1, @1977 IEEE.

The heat resistance becomes

(5.5)

If one uses a value of () = 45°, (5.5) will give a heat resistance which matches that of an accurate calculation quite well, provided that all materials have reasonably good thermal conductivity.

Current Crowding

Because realistically the temperature must be highest in the center of the device, current crowding toward the rim will result in an IMPATT device, since the ionization coefficient ct decreases with increased temperature. This

132 Microwave Semiconductor Device8

effect will be more serious in lower frequency diodes, and is also less serious in double-drift devices [Masse' et aI., 1985].

Quadri-Mesas

One can improve the heat transfer per unit total area by using separate adjacent "mesas" with the same total area as a single mesa (see Figure 5.4). If the spreading resistance dominates, using a "quadri-mesa" configuration will decrease the thermal resistance by a factor of two if there are no inter-mesa heating effects. If we take the latter into account, the thermal resistance will be increased by a factor (1 + I) where I is given by:

1= ~ (2+ ~) ~: (5.6)

a = mesa separation; 1'0 = mesa radius;

Therlnal Time-Constants

For pulsed devices, we must find the thermal time-constants of the de­vice/heatsink combination. The thermal "circuit" becomes equivalent to an electrical circuit which has resistive and capacitive elements, and the time­constants are the familiar RC time-constants (Masse' et aI. (1985». For a cylindrical semiconductor, with height ds , we find

d2 tD =...§...

4as

Here, as is the thermal diffusivity given by

KS as= --

pCp

(5.7)

(5.8)

where p is the density, and Cp is the heat capacity per unit mass. The units for a are cm2 /sec. Time-constants for the semiconductor device are fairly short (tens of nanoseconds) since the heat capacity is not very large. The value of a for GaAs is 0.121 cm2 /sec. A corresponding expression for the heatsink is

(5.9)

The value of a for copper is 1.09 cm2 /sec. Some other thermal materials constants are given in Table 5.1. * Typical heatsink thermal timeconstants are in the tens of microseconds range. For typical pulselengths of 100 nsec to 1 microsec., the heat sink will not show substantial pulse-to-pulse fluctuations in temperature, but will slowly approach a temperature which is given by the average power dissipated. The device temperature will fluctuate rapidly,

* See p. 141

Chapter 5

BEAM LEADS

(b)

1-----020~ t- 008~

_ ••. 02" __

~I'---- 040.-----1"'1

133

jJ.mTHiCKj

Figure 5.4. Scanning Electron Microscope pictures of Millimeter Wave IM­PATT diode,. (a) Single Mesa (b) Quadrimesa (c) Geometry of the diode" in (a) and (b). Adapted from ADLERSTEIN, M.G., McCLYMONDS, J. W., and MASSE', D. (1982). "Gallium Arsenide IMPATT Diodes at 20 GHz, II IEEE Intern. Microw. Symp. Dig., p. 143, @1982 IEEE.

134

HEAT RESERVOIR

I-- GoAs ---I

Microwave Semiconductor Device,

PULSE SOURCE

\--GOAS --1

HEAT SINK

R.

'11---1 Rz Rs

II

C, c'l cs

1 c.

-=-

Figure 5.5. Lumped parameter circuit model of the thermal path, in a GaA, IMPATT diode and its heat ,ink. The parameter value, are:

d 4 Ra = --- Ca = 2dApCp;

A/I;QIIA. 11'

.&. == 1 C4 = 1I'8/2rks /I;HS J Tp ;

1I'rHSN.HS 2 DaHs

Tp = pulse length;

D = duty factor;

Cl , C2 , R2 are analogous to Ra and Ca; Note: No head conduction occur. to the left of Ct. Adapted from MASSE', D., ADLERSTEIN, M.G., and HOLWAY, Jr., L.H. {1985}. "Millimeter Wave GaA. IMPATT Diode,," in Infrared and Millimeter Wave., K.J. Button, Ed., Academic Pre .. , 1-1, 291, with permiuion.

however, and will reach a peak value of

(5.10)

Both time-constants will vary essentially inversely proportional to P (for the device time-constant case, the length scales as 1//, and for the heatsink, the radius scales the same way if we keep the capacitive reactance constant, see the discussion later on in this section). Thus the ratio of the time-constants is expected to be more or less independent of frequency. The time constants above can also be expressed in terms of equivalent circuit elements. These are summarized in Figure 5.5.

Chapter 5 135

The Thermal Limit Expression

We are now ready to pull together the above information into a general expression for the thermal limit. At first sight, it may seem that in order to increase the power capability of a device, we only need to increase the area, which will decrease the thermal resistance to the required value. There is a limit to this method, however, which is set by the lowest impedance level for which a microwave circuit can be designed. The typical device has a capacitive reactance, as we have seen, and in the actual oscillator realization, we must resonate this capacitance with an inductive reactance from the circuit. The condition is:

2..=wL we (5.11)

Details of the circuit aspects of oscillators will be discussed in Chapter 6. The lowest reactance levels which can be realized in practise are of the order of a few ohms or tenths of ohms, more or less independent of frequency. We therefore must impose the constraint of a minimum realizable reactance level. This can be phrased in terms of the device capacitance, i.e.

1 fA Xc = -- = constant j C = -j

27rIC ds (5.12)

Re-arranging this we can obtain

ds -- = 27r£Xc A·I (5.13)

On the other hand, the maximum dissipated power, which is roughly the total DC power, if the efficiency is not very high, is given from our previous discussion by

l:J.T ds PDc=-j R=--

R A· K.S (5.14)

where l:J.T must be limited to some value in the range 200-300°C.

If we insert (5.13) in (5.14), and multiply the DC power by the frequency, we thus find

P I __ l:J.....,T::-x_K..:;:S_ DC X =

f X Xc x 2r (5.15)

In this expression, all quantities on the right-hand side are constants for a given material, or constants such as l:J.T and Xc. Thus (5.15) can be sum­marized by saying that the maximum dissipated DC power is inversely pro­portional to the frequency, if the thermal limitation applies. The RF power generated will also follow this law if the power conversion efficiency is con­stant. Since the efficiency does not increase with increased frequency, in any case, we must conclude that the maximum RF power output must always fall as the frequency goes up. In the next section, we will find that the "electronic"

136 Microwave Semiconductor Device6

limitation predicts an even faster fall-off, which will eventually prevail as the frequency is increased further.

THE ELECTRONIC LIMIT

Maximum Voltage

The voltage is limited by the occurence of avalanche break-down in the drift region. The maximum possible voltage can be approximately calculated from the condition that the electric field is equal to the maximum field for break-down in the entire device, i.e.

Vm = Em X dj d = drift - region width (5.16)

Maximum Current

As we discussed earlier, the moving concentration of charge modifies the electric field distribution, see Chapter 3 and Figure 3.17. If we want to restrict the maximum electric field to be less than Em, we found in (3.46) that

JO,max = Ef Em = Ev,Em /2d (3.45a)

also using (3.10) for an IMPATT device (the current limit for a Gunn-device will be twice as large).

The maximum current is obtained by multiplying by the cross-sectional area, A,

(5.17)

Multiplying (5.16) by (5.17), we have for the DC power:

P - V. 1. _ EE~v,A m- mm- 2 (5.18)

As in the previous section, we use the constraint on the reactance

d d ~ Xc = --- => EA = --- = ---j

21rf:Af 21rfXc 41rf2Xe (5.19)

If (5.19) is used in (5.18), we find the final expression for the electronic limitation:

(5.20)

Thus, the output power due to this limit is proportional to 1/ p. Whereas the relevant constants for the thermal limit are thermal quantities and XC! the

Chapter 5 137

Ul

I: 4( :J:

II: W :J: 0 Q.

I-:J Q. I-:J 0

CW OUTPUT POWER - TWO-TERMINAL DEVICES 10. r----r--~~~~~~~--~~~,,~~~~~~_rTTTC

10'

100

10-'

saAa IMPATT

SeAa SUNN

.......................... InP SUNN

..... -., ...... \

'. . ' . •

........

•

• BARITT

•

" "\

\ \ \ \ \ \ \S1 IMPATT

\ \ \ \ \ \ \ \ \

10~L---~--~~~WLLU----WL--L-~LL~LL----L-~~-L~WWU

100 10' 10· 10'

FREQUENCY - SHZ

Figure 5.6. Typical mazimum CW power output of two-terminal 06cillator device6, plotted ver6U6 frequency. The main 60urce6 from which the data have been compiled are: SZE (1981); ADLERSTEIN and CHU (198-1) @198-1 IEEE; SHIH and KUNO (1989) @1989 Horizon Hou6e-Microwave, Inc.

138 Microwave Semiconductor Devicel

GeAs IMPATT

20

51 IMPATT 15

10

SeAs SUNN 5

O~~~~~ __ ~WW~W--4-L~~ 100 10' 10. 10·

FREGUENCY. SHZ

Figure 5.7. Typical mazimum efficiency of CW two-terminal lemiconduc­tor olcillators, plotted al a function of frequency. The main 60urce, from which the data have been compiled are: SZE {1981}; ADLERSTEIN and CHU {198-1}, @198-1 IEEE; SHIH and KUNO (1989).

10·

~

ri w ~ 0 Q.

.... ~ 10' Q. .... ~ 0

c W III .J ~ Q.

10· 10·

FREGUENCY. SHZ

Figure 5.8. Typical mazimum pulsed output power for IMPATT and Gunn o,cillatorl, versul frequency. The main source, from which the data have been compiled are: SZE (1981); ADLERSTEIN and CHU (198-1), @198-1 IEEE; SHIH and KUNO (1989).

Chapter 5 139

constants for the electronic limit are the well-known Em and v" as well as X •. The numerical constant 811' varies from author to author, depending on the exact assumptions made.

Again, if we want to find the RF power, we multiply the DC power by the efficiency.

MEASURED DATA FOR RF POWER

Figure 5.6 shows measured output power versus frequency for Gunn and IMPATT devices, derived from several sources. Generally, two regions with the frequency dependencies derived above are observed.

It is fairly clear that the maximum output power is primarily determined by the thermal limit. In order to achieve the very highest powers, diamond heatsinks are used, especially for IMP ATT devices. Our discussion of the increased relative size ofthe avalanche region (com pared with the drift region), in millimeter wave IMP ATTs, also is in agreement with the faster fall-off of PRF

for IMPATT devices close to f = 100 GHz, although the analysis performed in this chapter undoubtedly is over-simplified. The lower power of Gunn devices is in agreement with the fact that they do not reach avalanche break-down. Another factor is the lower efficiency of Gunn devices - typical values are given in Figure 5.7, and are compared with those for IMPATT devices. BARITT devices have a much smaller power output due to maximum current densities of only about 100A/cm2 , and lower efficiency than IMPATTs (-5%). RTDs, finally, have too low output power to be included in Figure 5.6. The reader may refer to Figure 4.16, however, for typical output power levels of these devices.

Since the limit on power output is typically due to limited heat dissipation capability, we expect that pulsed devices may be able to operate with consid­erably higher RF power, while constraining the average DC power to a value which leads to an acceptable average operating term perature. This conclusion is confirmed in Figure 5.8, which shows pulsed output power versus frequency for several devices.

Problems, Chapter 5

1. Assume a double-drift IMPATT-diode, as shown in Figure 5.9 below. For simplicity, the saturation velocities, v" are assumed to be the same for both holes and electrons. Compare the maximum power output of this diode, with that of a single-drift diode at the same frequency and with the same efficiency and circuit reactance, X •.

a) Use the electronic limitation for power output

140 Microwave Semiconductor Device~

HOLES ELECfRONS

--------w --- ---w ---

A V ALANCHE ZONE

Figure 5.9. Geometry of a double-drift IMPATT device, for problem 1.

w w

--- AVALANCHE ZONE

CUHEATSINK

Figure 5.10. Geometry of a double-drift IMPATT device on a Cu heat ~ink, for problem 2.

b) Use the thermal limitation, and assume that the thermal spreading resistance dominates.

2. A GaAa double-drift IMPATT diode is mounted on a copper heat-sink as shown (Figure 5.10). Choose w for operation at 40 GHz (the saturation velocity is 6 x 108 em/sec). The break-down voltage is 27 volts. Determine the area so that Xc = Ion results.

a) Find the maximum current at which the diode can be operated if the junction temperature must not exceed 250°C, and the heat-sink is at 20°C. Use thermal data from Table 5.1. Assume that all heat production is localized to the avalanche zone.

b) If the efficiency of the osciliator is 10%, how much RF power will be produced?

c) Also find how these values change if you assume an intermediate layer of 0.1 mm thick gold between the GaAa and the heat-sink. The gold layer has a diameter much larger than the diode.

Chapter 5

Table 5.1 Average Thermal Parameters with Junction at 250°C

n(W/cmOC) pcp(J Icm3 °C)

a(cm2/sec)

GaAs Au Cu Diamond Solder 0.25 2.92 3.8 9.0· 2.5 2.06 2.51 3.5 1.24 2.5

0.121 1.16 1.09 7.26 1.0

141

• The thermal conductivity of diamond can range from 9 to 20 and is also tem­perature dependent.

REFERENCES

ADLERSTEIN, M.G., and CHU, S.L.G. (1984). "GaAa IMPATT Diodes for 60 GHz," IEEE Electron Device Lett., EDL-5, 97.

KITTEL, C. (1976). "Introduction to Solid State Physics," 5th Edition, John Wiley, New York.

KRAMER, N.B. (1981). "Sources of Millimeter-Wave Radiation: Traveling­Wave Tube and Solid-State Sources," in Infrared and Millimeter Waves, K.J. Button, Ed., Academic Press, Orlando, FL., Vol. 4, p. 151.

DeLOACH, B.C., Jr. (1967). "Recent Advances in Solid State Microwave Generators," in Advances in Microwave., L. Young, editor, Academic Press, New York.

JOHNSON, E.O. (1965). "Physical Limitations on Frequency and Power Pa­rameters for Transistors," RCA Rev. 26, 163.

MASSE', D., ADLERSTEIN, M.G., and HOLWAY, L.H., Jr. (1985). "Mil­limeter-Wave GaAB IMP ATT Diodes", in Infrared and Millimeter Wave., K.J. Button, Ed., Vol. 14, Ch. 5, 291.

SCHARFETTER, D.L. (1971). "Power-Impedance-Frequency Limitations of IMPATT Oscillators from a Scaling Approximation," IEEE Tran •. Electron Device., ED-18, 537.

SHIH, Y.C., and KUNO, H.J. (1989). "Solid-State Sources from 1 to 100 GHz," in Microwave Journal - 1989 State of the Art Reference, Supplement to Microw. J., Sept. 1989, p. 145.

SWAN, C.B., MISAWA, T., and MARINACCIO, L. (1967). "Composite Avalanche Diode Structures for Increased Power Capability," IEEE Tran8. Electron Device., ED-18, 536.

SZE, S.M. (1981). "Physics of Semiconductor Devices," Second Edition, John Wiley & Sons, New York.