Solid-State Electronics Vol. 36, No. 2, pp. 125-132, 1993 0038-I 101/93 $6.00 + 0.00

Printed in Great Britain. All rights reserved Copyright fQ 1993 Pergamon Press Ltd

THEORETICAL CALCULATIONS OF TEMPERATURE AND CURRENT PROFILES IN MULTI-FINGER

HETEROJUNCTION BIPOLAR TRANSISTORS

WILLIAM LIU and BURHAN BAYRAKTAR~GLU?

Central Research Laboratories, Texas Instruments, P.O. Box 655936, M/S 134, Dallas, TX 75265, U.S.A.

(Received 4 August 1992)

Abstract-A theoretical analysis is presented to determine both the temperature profile and current distribution among the fingers for a multi-finger heterojunction bipolar transistor designed for microwave power applications. This analysis allows for arbitrary transistor geometries and non-uniform emitter ballast resistors in the fingers. The effects of varying ballast resistances, emitter width, emitter length, and spacing between the fingers will be discussed. _

1. INTRODUCTION

Microwave power heterojunction bipolar transistors (HBTs) are designed to deliver large amounts of power at high frequencies. To increase HBT capability in handling large voltages and currents, designers gener- ally employ multi-finger structures so that the total active device area increases[l,2]. To date, 120-finger AIGaAs/GaAs heterojunction bipolar transistors (HBTs) having a total emitter length of 3600 pm have demonstrated an C.W. output power of 12.5 W and a power added efficiency of 3 1% at X-band[ 11.

When multi-finger HBTs are biased under high power operating conditions, the base-emitter junc- tions of the fingers heat up to different temperatures, with the temperature being higher for the central fingers than the outer fingers[3]. Because the base- emitter bias required to turn on a given current decreases with increasing junction temperature, the central fingers conduct a larger amount of current than others[4]. Without proper transistor layout designs, the ratio of the current flowing through the central fingers to the current flowing through the outer fingers can be large. This non-uniform current distribution reduces the maximum power output achievable compared to the operating condition where all the fingers share roughly an equal amount of current. In this study, a theoretical analysis is presented to determine both the temperature profile and the current distribution for multi-finger HBTs. This analysis is applicable to HBTs with arbitrary transistor geometries and non-uniform emitter ballast resistors in the fingers, thus enabling one to design multi-finger transistor layouts which give rise to a relatively uniform current distribution across the fingers. Calculations of the temperature profile and

tPresent address: North Carolina State University, College of Engineering, Box 7903 Raleigh, NC 27695-7903, U.S.A.

the current distribution are performed for various practical transistor layout as a function of the total device collector current. The effects of varying ballast resistances, emitter width, emitter length, and spacing between fingers are discussed.

The theoretical approach of this study to find the temperature profile and current distribution differs from various published approaches[3,5,6]. The ap- proaches in Refs [5] and [6], which apply to transis- tors in general rather than specifically to bipolar transistors, requires an a priori known current distri- bution (and thus the input power distribution) to calculate the temperature profile. However, since the collector current flowing through a finger depends critically on the local finger temperature, the current distribution in a multi-finger bipolar transistor is not known prior to the determination of the temperature profile. Another published approach resolves this difficulty by including the current distribution, in addition to the temperature profile, as unknown and sought for a separate set of equations governing the current distribution[3]. While this is a significant improvement to the accurate determination of the temperature profile and current distribution, each of two sets of equations governing the temperature pro- file and the current distribution contains coefficients which are variables of the other set of equations. Simultaneously solving these two sets of coupling equations is thus difficult. In contrast, the approach presented here removes the coupling between the two sets of equations by first expressing explicitly the tem- perature profile as a function of the current distri- bution. The two sets of equations are then reduced to a new set of equations whose coefficients are indepen- dent of both the temperature and the current. From this new set of equations, the current distribution is first found deterministically with a known computer algorithm. The solution of the temperature profile is subsequently determined. The advantages of the pre- sent approach will be elaborated upon later.

125

126 WILLIAM LIU and BURHAN BAYRAKTAROGLU

2. THEORETICAL ANALYSIS

The theoretical approach in this study applies to arbitrary transistor layouts, such as the one schemat- ically shown in Fig. 1. Each finger of the transistor is subdivided into several elements to account for the possible temperature variation within an individual finger. The X and Y coordinates of the center of 4th element are denoted as X, and Y,, respectively. We also allow the ballast resistors, which are placed at the emitter lead of each element, to vary in value from finger to finger. The value of the ballast resistor in the Ath element is denoted as R,,.

The collector current flowing through the Rth element, lc,, as a function of base-emitter junction temperature of the Ith element, T,, can be expressed as[4]:

where 1% is the collector saturation current; 4 is the negative of the rate of the change of base-emitter voltage per temperature change for constant emitter current and it is experimentally determined for an AlGaAs-GaAs junction to be -2.06mV/“C; T, is the ambient temperature, taken to be 300 K in this analysis; kT,/q is the room temperature thermal voltage, equal to 0.0258 V; Vbej”, is the base-emitter junction voltage in the Rth element. Vbelnr is equal to the applied base-emitter bias, Vbe, subtracted by the voltage drop across the emitter ballast resistor in the r&th element:

Vsbej,, = Vb, - Ret . Ic, (2)

Note that the applied base-emitter bias is the same for all elements, whose base and emitter leads are connected in parallel.

The steady state temperature at any point of the wafer can be determined from the heat flow equation[7]:

d2T d2T a2T Q+,+,Z,=o,

aY

and appropriate boundary conditions of the second kind[3,6]:

i3T

ax r=0.L=03 dT

av y=o,w = 0,

K f? th r~- = -P(X,Y),

uz lz=o

TI,=, = TA,

where Kth is the thermal conductivity in GaAs, and p (x, y) is a distribution function of input power den- sity on the chip’s top surface. p(x, v) is zero except at the fingers where the device power is dissipated.

An analytical solution to eqn (3) and the boundary conditions has be determined[3,6,8]. In this investi- gation, the solution is re-arranged and explicitly written as a function of the elemental currents flowing through the N elements: I,, , . . , IcN, where N is the total number of elements[3]. The re-arranged equation is given as follows. (However, one notes that the temperature solution expressed in Ref. [3] missed a factor of 4 in the third term, In both Refs [6] and [8] which determined the temperature solutions for slightly different problems, this factor of 4 was shown to exist. This factor of 4 is included in this analysis.)

Tk =

x Fm, cos(l, . A’,) 1

. Ic, + + ‘UJ

6, COS(K 1 . ‘c,

x F,,,, G,, cod&n xi NW& . Y, 1 1

. 4,

+kw+i I,,+T,, th,-I

(8)

w

I

t

.

W

Fig. 1. A schematic transistor layout. There are three emitter fingers, all subdivided into elements of area w x P. The portion of the wafer where the device is located has dimension W x L x I.

where F,,,j and G, are given as,

F~~=sin,.(,,3_,,,.(,4), (9)

G,=sinp.(Y,+T)-sinp,,(Y,-y). (10)

In the above equations, I,, p”, ymn are Eigen values associated with the Eigen solutions when one applies the technique of separation of variables to solve eqn (3). L, IV, e, W, and t are the geometrical dimensions of the transistor layout (Fig. 1), and V, is the applied emitter-collector bias. [More generally, eqns (8)-(10) can be used to find the temperature at any point of the wafer having coordinates X, and Y, by dropping the subscript n.]

Equations (1) and (8) represent a total of 2N equations with 2N unknowns of I,, , . . , IcN, and T I,“‘, TN. Since these 2N unknowns appear in both eqns (1) and (8), a conceptually simple approach to solve these two sets of coupling equations is a self- consistent iteration approach. It first assumes that the total collector current is equally divided in all the elements. A preliminary temperature profile is evalu- ated from eqn (8) with this assumption, and then, this preliminary temperature profile is used to re-calculate a current distribution from eqn (1). Subsequently, the updated current distribution is used to recalculate the temperature profile. This process is repeated until a self-consistent solution exists simultaneously for both the temperature profile and the current distri- bution. However, such a self-consistent solution is often difficult to obtain within a reasonable accuracy, especially when the device is assumed to dissipate a huge amount of power.

In contrast, in the present approach, we take advantage of the fact that T, can be explicitly expressed as a function of the elemental currents, as shown in eqn (8). Therefore, eqns (1) and (8) can be combined and reduced to a set of N nonlinear equations with N unknowns of I,, , . . , I,,. This new set of equations is given as follows and the derivation of these equations will be described shortly:

N

‘CT = c ‘c,, (11.1) I=,

(11.2)

Current profiles in multi-finger transistors 127

x [cos(l, . X‘)COS( & . Y;)

- cos(l, . X,)cos( p” . Y,)]

+-&{S,z.Re,-6,.Re,}. (12) A

6, is 1 when i = j, and is 0 otherwise. Note that dv are independent of both the temperature profile and current distribution. These coefficients are readily determined once a transistor layout is specified.

Equations (11.1) to (11 .N) are a set of N nonlinear equations with N unknowns: Z,, , . . . , IcN. Equation (11 .l) simply ‘states that the given total collector current, IcT, is the sum of the collector currents of the individual elements. Equation (11.2) is obtained by first substituting both eqns (2) and (8) into eqn (1) for both elements 1 and 2, and then taking the difference of the two resultant equations. This relationship between the currents of elements 1 and 2 is similarly obtained between element 1 and other elements, as shown in eqns (11.3)-( 11 .N). In general, an arbitrary set of nonlinear equations can not be solved system- atically[9]. However, the equations to be solved here are fairly regular, all having very similar forms. Therefore, the solutions of Ic,, . . . , IcN are easily obtained by a Newton-Raphson algorithm[lO]. Note that the solution is found in a deterministic manner by this algorithm. Once the individual element cur- rents are determined, the temperature at each element can be determined from eqn. (8).

3. RESULTS AND DISCUSSION

I,, = I,, . exp (1 l.N)

where d,/ is expressed as,

Calculations for the temperature profile and cur- rent distribution are performed for a practical transis- tor layout design as shown in Fig. 2. The transistor has 5 fingers, each having an area of 2 x 20pm’. For convenience, it is assumed that the temperature in a given finger is uniform; hence, each finger of the transistor is taken to consist of only one element. This assumption is valid when the die containing the transistor has a considerably larger dimension than that of the transistor. Note that the emitter fingers have rectangular shape, even though the pre- sent analytical approach allows arbitrary transistor geometries.

x F,Jcos(l, . Xi) - cos(l, . X,)]

44 2 vce .S tanhh. 4 f--‘-t pc;: kT,LW.w Kthn=,

x G, [cos( P, Y; I- CM A . Y, )I

Figure 3 illustrates the calculated current distri- bution among the fingers for two extreme cases: one with zero emitter ballast resistance, and another with uniform emitter ballast resistance of R, = 20 R per finger (corresponding to an equivalent specific con- tact resistance of 8 x 10e6R. cm*). As shown in Fig. 3, for the device with R, = 0, the total collector current flows only through the central finger even when I, is still low (0.01 A). This crowding of current _.

128 WILLIAM LIU and BURHAN BAYRAKTAR~CLU

Fig. 2. A practical transistor layout for a HBT with five fingers. Each emitter finger has area = 2 x 20 fim2.

an

flowing in just the central finger can be explained as follows. As shown in eqn (l), when the junction temperature of a finger rises, the current flowing through that particular finger increases exponentially. Since the central finger is naturally hotter than other fingers, the current flowing in the central finger increases at a much larger proportion compared to those in the rest of the fingers. Eventually, when the base-emitter junction of the central finger rises to a certain temperature which is much larger than the temperatures in other fingers, such proportion becomes so large that the central finger conducts essentially the entire current. This non-uniform cur- rent conduction is undesirable because the entire device power is dissipated in one finger, rather than being distributed among all the fingers. In contrast,

in the device with R, = 20 R/finger, Fig. 3 shows that all fingers conduct appreciable amounts of current, even when ZCT reaches 0.04 A. This result agrees with the implication of eqns (1) and (2), where R, is seen to provide negative feedback to the base-emitter junction voltage, stabilizing the current conduction among the fingers. The importance of using emitter ballast resistors to prevent non-uniform current con- duction has also been observed experimentally[ 11,121.

Figures 4 and 5 illustrate the calculated profiles across the fingers (along the Y-axis) for the devices with and without the ballast resistors of 20 R/finger, respectively. The devices have the same transistor layout shown in Fig. 2. As shown in Fig. 4, the junction temperatures among the fingers in the device with the ballast resistors are relatively the same at low

10-4 i

Re =20 LY’nger

II t t

Re=O12

0 ’ I I I I I I 1 I

12345 1 2 3 4 5

Finger Location

Fig. 3. The calculated current distribution among the fingers for the HBT shown in Fig. 2. The total collector current is varied from 0.005 to 0.04 A. Two values of uniform emitter ballast resistance per finger are used: R, = 0 and R, = 20 a. V,, = 10 V. The curves of (a), (b), (c), and (d) correspond to IcT = 0.04,

0.02, 0.01, and 0.005A. respectively.

Current profiles in multi-finger transistors 129

160 180 200 220 240 260 280 300

Distance (pm)

Fig. 4. The temperature profile across the fingers for the HBT shown in Fig. 2 whose R, = 20 R/finger. v,= 1ov.

Zcr, and are increasingly different as I,, increases. However, even though the temperature difference between the central finger and the outermost finger reaches 40°C when Z,, = 0.04 A, all of the fingers still conduct appreciable amount of current as was demonstrated in Fig. 3. In contrast, Fig. 5 shows that for devices without the ballast resistors, the junction temperature of the central finger rises sharply com- pared to those of other fingers (> 100°C difference), even when the current levels are still low (0.01 A). This large temperature difference reflects the phenom- enon observed in Fig. 3 that the central finger con- ducts almost the entire Zcr, rendering the central finger to be the only location where the entire device power is dissipated. In practical devices, however, the

junction temperature is unlikely to reach a tempera- ture of 840 K as predicted for ZcT = 0.04 A. Some breakdown mechanisms would likely occur prior to such a temperature rise and destroy the device. Nonetheless, a comparison of Figs 4 and 5 clearly demonstrates the importantance of ballast resistors in allowing devices to handle higher power dissipation.

A new device layout is used to compare the effects of varying the emitter width. This new transistor layout is exactly the same as the layout shown in Fig. 2, except that the emitter widths of this new device are 3, 2, 1.5, 2 and 3 pm for the 5 fingers. The distance between two centers of adjacent fingers remains to be 15 pm. This device is to be compared to the device with R, = 20 Q/finger discussed above. Because the

900

800

700

600

300 160 180 200 220 240 260 280 300

Distance (pm)

Fig. 5. The temperature profile across the fingers for the HBT shown in Fig. 2 whose R, = 0. V,, = IO V

130 WILLIAM Lru and BLJRHAN BAYRAKTAR~CLU

440

G

k 420 $ 400

% E 380 c

360

340

320

300 160 180 200 220 240 260 280 300

Distance (pm)

Fig. 6. The temperature profile across the fingers for a HBT slightly modified from that shown in Fig. 2. The emitter widths of the fingers are 3, 2, 1 S, 2, and 3 pm, instead of 2 pm for all the five fingers. The distance (along Y-axis) between the centers of any adjacent fingers is still 15 pm. R, = 13.3 R, 20 R, 26.7 Q,

20 R, and 13.3 Q for the fingers. V, = 10 V.

emitter contact resistivity is independent of device geometry, the ballast emitter resistance increases with decreasing emitter area. With the same emitter con- tact used to result in the uniform R, = 20 R/finger for the device whose fingers have equal emitter widths (Fig. 2), the new device with varying emitter widths then has R, = 13.3, 20, 26.7, 20, and 13.3 R across its 5 fingers. These non-uniform ballast resistors can potentially result in a more uniform current distribution than the current distribution attainable in the device with uniform R, = 20 R/finger. That is, since the junction temperature naturally tends to be highest in the central finger (Fig. 3), the tendency of

junction current being largest in the central finger is offset with a larger designed value of ballast resistor there[3].

Figure 6 shows the calculated temperature profile of this device with varying emitter width. As shown, the temperature becomes dramatically higher in the outermost fingers than in the central finger as I,, increases. The observation that the temperature profile has sharply peaked regions in the outermost fingers rather than being a flatter profile suggests that most of the device current flows through the outer- most fingers. Hence, Fig. 6 shows that the value of the non-uniform ballast resistors of the present device are

1000

g, 700

E 600

300 160 180 200 220 240 260 280 300

Distance (pm)

Fig. 7. The temperature profile across the fingers for a HBT slightly modified from that shown in Fig. 2. The emitter length varies from 15, 20, 25, to 40 pm V, = 10 V and IcT = 0.05 A.

Current profiles in multi-finger transistors

650

s 600 L

$ 550 a t 8 500

c 450

400 Spacing = 50 pm

350

300 100 150 200 250 300 350 400 450 500

Distance (pm) Fig. 8. The temperature profile across the fingers for a HBT slightly modified from that shown in Fig. 2. The distance (along Y-axis) between the centers of any adjacent fingers varies from 5, 15, to 15pm.

v,= 1ov.

131

not optimized for uniform current conduction. Furthermore, the maximum junction temperature is -460 K when ZcT = 0.03 A. This temperature is higher than that of the device with uniform ballast resistor at the same current level (N 430 K, Fig. 4). Therefore, this comparison demonstrates that while non-uniform ballast resistors would conceptually maintain a uniform temperature profile, their resist- ance values need to be carefully optimized to actually achieve a desired uniform temperature profile. If these resistance values were chosen improperly so that the central emitter ballast resistance was too high, the outermost fingers would then conduct the entire current.

Figure 7 illustrates the effects of varying the emitter length. The device layout is the same as Fig. 2, except that the emitter length varies from 15, 20, 25, to 40 pm. The assumed ZcT and V,, are 0.05 A and 10 V, respectively, and the device has uniform ballast resis- tors of 20R in each of the fingers. As the emitter length decreases, the current density increases for a given Z,, . Therefore, the base-emitter temperature increases as emitter length decreases. It is observed from Fig. 7 that with the junction temperature in- creases rapidly as the length is decreased. When the length decreases to 15 pm, the power density dissi- pated in the fingers becomes too high and the central finger conducts the entire Z,, . Consequently, a sharp peak in the junction temperature results.

Figure 8 illustrates the calculated temperature profile as the spacing between adjacent fingers is varied. The transistor layout is the same as Fig. 2, except the distance between two adjacent fingers (15 pm as indicated in Fig. 2) is varied from 5, 15, to 50pm, while the distance between the outermost fingers to the edges of the die remains to be 200 pm.

This calculation assumes a uniform ballast resistance R, = 20 R/finger, and ZcT and V, are 0.05 A and 10 V, respectively. Figure 8 demonstrates the effectiveness of increasing the spacing to reduce the junction temperatures of the fingers for a given operating condition. For example, the junction temperature of the central finger decreases from 650 to 550 K, and 450 K, when the spacing increases from 5 to 15 pm, and 50 pm, respectively. In addition, Fig. 8 demonstrates that at a spacing of 5 pm, the spacing becomes so narrow that the temperature profile does not show distinctive five peaks as observed in cases with wider spacings.

4. SUMMARY

A theoretical analysis is developed to calculate the both temperature profile and current distribution for multi-finger power transistors having arbitrary transistor layouts. The effects of varying the ballast resistance, emitter width, emitter length, and spacing between fingers are discussed for a practical transistor layout. It is shown that uniform emitter ballast resistors are critical to maintain a fairly flat tem- perature profile and uniform current distribution. Furthermore, when non-uniform ballast resistors are used, their values need to be designed properly to achieve the desired temperature profiles. Lastly, it is shown that increasing emitter length and the spacing between the fingers reduces the difference between the temperatures at the central fingers and outermost fingers.

Acknowledgements-The authors would like to thank S. K. Fan, T. Henderson, D. Hill, M. A. Khatibzadeh. and J. Wilson for useful discussions.

132 WILLIAM LIU and BURHAN BAYRAKTAROGLU

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