624 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 42, NO. 4, APRL 1995

Application of the TLM Method to Transient Thermal Simulation of Microwave Power Transistors

Paul W. Webb and

Abstract-The transmission line matrix (TLM) explicit method of numerical simulation has been used to model the transient thermal properties of various microwave heterojunction bipolar transistor (HBT’s) power structures, used in a pulsed mode. Control of the time step during the simulation is of paramount importance and the paper outlines some of the problems en- countered using time step control methods currently published and describes an improved algorithm. This improved time step control method has been implemented in a general purpose 3D TLM transient thermal simulator. Some simulation results are described for a variety HBT transistor structures with very different thermal time constants.

I. INTRODUCTION HE application of the transmission line matrix method T (TLM) to solving electromagnetic scattering problems

was first described by Johns and Beurle [l] in 1971 and its utility in diffusion modeling by Johns [2] in 1977. The TLM method may also be used to describe drift, diffusion and re- combination mechanisms of charge carriers in semiconductors and this was first described by Saleh [3] in 1991. The TLM method is well established as a technique for solving diffusion problems and there are numerous engineering examples to be found in the literature, for example, [4]-[6].

Transmission line matrix modeling is a time domain nu- merical technique that involves the use of circuit analogues to model fields or other physical phenomena. A lumped element circuit model renders the formulation of a problem that is discrete in space, and a combination of some of the circuit components, inductance and capacitance, of the modeling circuit render the problem discrete in time. The details of the TLM diffusion modeling are described in [1]-[5]. The technique uses an explicit time stepping algorithm and this makes the TLM method very suitable for solving non linear problems where the material parameters change in the course of the simulation. The variation of thermal conductivity of a semiconductor with temperature is an important example.

In reality, the general procedure for describing a geometry and studying thermal diffusion within it, is very similar to that encountered using the finite difference method, indeed the thermal resistance matrix (mesh) involved in both formulations could be identical. In the finite difference method the time step used in the algorithm to describe the diffusion of heat is limited, as it becomes numerically unstable above a certain

Manuscript received received June 1, 1994; revised November 7, 1994. The

The authors are with the School of Electronic and Electrical Engineering,

IEEE Log Number 9409050.

review of this paper was arranged by Associate Editor J. Xu.

The University of Birmingham, Edgbaston, Birmingham B15 2TT U.K.

Ian A. D. Russell

value. The maximum time step is related to the smallest thermal time constant to be found within the structure, being the thermal capacity of that “node” divided by the sum of the thermal conductances which connect that node to its surrounding “nodes,” see for example [7]. In TLM diffusion modeling the time step is numerically unconditionally stable and this is one of its main advantages, but also one of its problems. It is the algorithm for increasing the time step during the simulation of the diffusion process which is so important in improving the utility of the TLM technique.

The authors are interested in the transient thermal simulation of high power density microwave devices both MESFETs and Gallium Arsenide heterojunction transistors and have used the TLM method to write a general purpose three dimensional simulator to study these structures. For the sorts problems outlined in this article the time stepping routines which have been described in the literature do not work at all well, especially where the simulation is for time periods which approach steady state. This article describes a method of time step control which sought to achieve three objectives namely,

1) It sets an initial time step consistent with the geometry

2) It changes the time step continuously in a way consistent

3) The time step will not increase beyond some reasonable

These objectives have been achieved in a three dimensional TLM general purpose thermal simulator written by the authors. The results described here were obtained using this software and it is this simulator which will be referred to in this article. Where possible the accuracy of this simulator has been checked against various finite difference simulators.

and material properties of the structure.

with some specified accuracy.

limit.

11. TLM BASICS In order to describe the time step problem it is necessary

to outline some of the basic principles and background of TLM diffusion modeling. The nature of the problem will be considered in one dimension but the principle may be easily extended to two or three dimensions. Fig. 1 shows the circuit representation of a one dimensional TLM node of length Ax used for diffusion modeling. The parameters of the circuit R, L and C are the unit length values of resistance, inductance and capacitance. The equations which describe the circuit are,

(1) -- SV Ax = I(R$) + ( L $ ) g Sa 2

0018-9383/95$04.00 0 1995 IEEE

WEBB AND RUSSELL APPLICATION OF THE TLM METHOD TO TRANSIENT THERMAL SIMULATION OF MICROWAVE POWER TRANSISTORS 625

: Fig. 1. Electrical Equivalent Circuit of a one dimensional TLM node.

and SI SV -AX = -CAX-. S X St

If these equqtions are simplified and rearranged then after some differentiation and algebraic manipulation the relation- ship between V, St and 6x is given by (3) the telegrapher’s equation.

S2V SV S2V - = RC- + LC-. 6x2 St St2 (3)

This equation should be compared with (4), which is the differential equation used to describe heat flow in solids where p, c and IC are respectively the material density, specific heat and thermal conductivity.

(4) S2T pc ST 6x2 IC S t .

It seems that the circuit of Fig. 1 will model the diffusion equation provided the last term in (3) is made small compared to the penultimate term. The voltage models temperature, current heat flow and thermal resistance and capacitance are modeled by their electrical equivalents.

In a TLM node the inductance and capacitance become transmission lines and the iterative solution commences with injecting heat into the model as delta voltage impulses using current generators. These pulses travel and scatter and connect through the TLM matrix along the transmission lines. The delay in all transmission lines are made identical and the local voltage impulse population provides the solution to the diffusion equation at any half time step interval, as the temperature distribution may be calculated both at the node centres and at the edges of the nodes.

- - - _ _

111. STEP CHANGE

The algorithm for achieving a time step change in a rectan- gular parallelepiped TLM matrix has already been published by Webb 1992 [8] and a method of controlling the time step in a two dimensional square TLM matrix was first published by Pulko in 1990 [9]. Some observations on the time step control method proposed by Pulko, and its relationship to accuracy and TLM node size, may be found in reference [lo].

A. The Initial lime Step

If the TLM method is be useful for diffusion simulation, then a minimum time step would seem to be that associated with a value given by an equivalent finite difference solution (Note (9) in the next section.). In the TLM simulator the time

step can be an input parameter or alternatively a suitable time step can be calculated. A node in the power dissipating region is automatically chosen to monitor time step change and if necessary, its characteristics are used to determine an initial time step. The time constant of the node is found, which is the same as the stable maximum for the finite difference method, and the time step is set to one twentieth of this value. Normally the nodes in the dissipating region will be the smallest requiring the shortest time constants, if this is not the case some problems in time step control can arise and this matter is discussed later. In the simulator the time step is limited to values greater than the initial value.

B. lime Step Control

In one of the time step control methods proposed by Pulko [9] the ratio of the factors in (3) was used to monitor and control the time step. A factor ‘m’ is defined according to (5) , the postulate being that the smaller the value of ‘m’ then the TLM equations will more closely represent the thermal diffusion equation.

A difference equation may be formulated to calculate ‘m’ from alternate consecutive values of V . For example if the time step is constant then ( 5 ) may be written as (6) where the subscripts refer to the appropriate consecutive values of V (Or in this case temperature T). The characteristic impedance Z = fl = St/C in the TLM modeling equations,

St [V, - av,-1 + V,-Z] m = - RC [V, -Vn-l

In the method outlined in reference [9], the time step is constant for a sufficient number of iterations so that ‘m’ may be calculated, then the time step is changed appropriately by 10% up or down. Various properties of the algorithm are demonstrated in reference [ l l ] . The aim in the simulator is to change the time step in a quantitative way at every iteration, and generally, in the sort of problems of interest to the authors, one expects the time step to gradually increase as the temperature rises ( or falls ) to a steady state. If the time step is continuously changing then (5) may be written as (7) noting the equivalence of V and T. Some liberties have been used in writing down this equation, St is an average time step. Three consecutive temperature values are considered with temperature changes ST1 then ST2 corresponding to time steps 6tl and 6t2.

Equation 7 may be rearranged giving (8).

or, approximating Stl + St

St STi/Stl RC ST2/St2

m = - [ ~ - 11. (9)

626 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 42, NO. 4, APRIL 1995

We notice from (9) that if St, the time step was constant and the diffusion process was developing towards a steady state, ST1 would always be larger than ST2. This is consistent with the observed fact that ‘m’ decreases during a simulation with constant time step [lo] as the ratio (STl/ST2) will become smaller during a realistic simulation. Notice also that RC is the time constant of the node and if this value is used for the initial time step a small value of ‘m’ will result, as the ratio of the temperature steps will be near to unity.

The temperature rise (or fall) for a practical simulation can be modeled as a series of exponential terms each with its own characteristic time constant. In (5) the ratio of the first and second derivative terms would be constant for each exponential term and equal in magnitude, for each term, to the reciprocal of its characteristic time constant. In the initial phase of a simulation with a step input of power to the system the smaller time constants would dominate the temperature change implying a short time step. The time step would then increase as the simulation proceeds because the effective time constant would increase. For a simulation producing a temperature rise or fall which could be modeled with a single time constant the time step should be constant according to (5) and so also would the bracketed term in (9).

The problem which arises using this method of time step control is illustrated in Fig. 2, a simulation using an alternative time step control algorithm, for a device to be described in a later section, (Device ‘A’ with constant material parameters and input power for the 100 micron chip.) where the tem- perature rise and the corresponding ‘m’ value are plotted. The value of ‘m’ rises well above values which would be considered appropriate using the ratio of the factors shown in (5). For the graph of Fig. 2, if ‘m’ was chosen to be 0.01 the time step would become constant at a simulation time of under seconds and this would be of little practical value. In Fig. 2 the initial values of ‘m’ show some erratic behavior compared to similar curves to be found in the literature. In this simulation there are many current generators representing a single power source and it takes a number of time steps before the effect of them all become established at the ‘test node’, and a physically realistic thermal distribution has been established. Calculation of the ‘m’ value using (6) can be a further source of problems as there is considerable variance in consecutive calculated values which become increasingly large as the simulation proceeds. The problem is not entirely numerical but due to the intrinsic temperature fluctuations observed in the TLM method [9].

Another way of viewing the relationship of (3) to the diffusion equation is to consider making the last term small in absolute terms, that is, LC(S2V/6t2) 4 0. Differencing this term in a similar way to that previously described would yield (10).

rearranging this equation and using the fact that the propaga- tion velocity on the matrix for the delta pulses is given by v = l/m = Sx/St and making ST an average change in

4 ) / 1 150

Log 10 ‘m’ Factor

0

2 ,

4 .

Centigrade

6 30 -12 .10 8 6 4

Time Secondr Log10

Fig. 2. time.

The variation of the error factor ‘m’ (after Pulko) with simulation

temperature over the time period gives (1 1).

n = - ST [- STl/Stl - 11. Sx2 ST2/St2

It is worth noting that the value of ST must fall as a simulation tends towards the steady state, making the whole term progressively smaller, and for a constant time step the bracketed term would always be greater than zero. The value of ‘n’ could only be zero if the time step was decreasing. The fact that the bracketed term in (1 1) cannot be zero for an increasing time step and its appearance in the value for ‘m’ in (9) perhaps indicates why the time step method based on (5) is unsatisfactory in this application. The numerator in (5) will always be finite for a constant or increasing time step and could only become zero for a decreasing time step.

A control of the time step may be implemented by moni- toring the value of bracketed term in (1 1) and keeping it at a small value ( E ) as shown in (12). If the algorithm for changing the time step is successful one can neglect 6T and 6x2 in (1 l), as the latter is constant and the former will be of decreasing value. The Sx2 term will manifest itself in the choice of initial time step for the simulation as it will be related to the time constant of the test node.

A possible predictive relationship between a current time step and the next time step is given by (13).

= StlF. (13)

The current time step should be multiplied by a factor which could be greater or less than unity and would depend on the difference between a constant (1 + E ) and the ratio of the (STISt) terms. The multiplying factor F for the time step from one iteration to the next would be given in (14).

In the simulator the values of the ratios of the derivatives (ST/&) needed to calculate the bracketed term are stored over

WEBB AND RUSSELL APPLICATION OF THE TLM METHOD TO TRANSIENT THERMAL SIMULATION OF MICROWAVE POWER TRANSISTORS 621

two iterations, and averaged. At the end of each iteration the time step is multiplied by the factor F which is calculated from (14). The time step will be changed consistent with the criterion shown in (1 1) where the value of ‘n’ will be small and related to the value of (1 + E ) . The calculation accuracy of the derivative terms will deteriorate as the simulation reaches a steady state and this problem is considered in the next section.

An altemative formulation may be obtained from (13) by solving it for the ratio of consecutive time steps. The relationship between the consecutive time steps is then given by (15). In this case only the temperature steps ST need be monitored. Both equations give identical simulation results when implemented.

C. The Maximum Time Step

The question of what algorithm should be used to limit the maximum time step in the simulation process is of practical importance as the value of 6T becomes zero in the steady state leading to computational problems. It so happens that the time step change algorithm described here in fact produces a multiplying factor for the next step equal to unity as the steady state is approached but some method of inhibiting its use is still required because it is not possible to numerically calculate the ratio (STl/Stl)/(ST2/St2) or (STl/ST2) as the temperature becomes constant. The following algorithm worked best on all the test structures. Three factors are monitored.

1) The ratio of the rate of increase of temperature divided

2) The ratio of the power leaving the heat sink to the input

3) The multiplying factor for the time step.

As steady state is reached the factor in ‘1’ falls rapidly and that in ‘3’ becomes near to unity.

The following algorithm was used to fix the maximum time step.

Provided the value in ‘1’ is below a set point (typically lop4) when the time step multiplying factor becomes equal to or less than unity, the time step is fixed at the current value. The time step will also be fixed at the maximum recorded value if the power leaving the device reaches 99.8% of that input.

The ratio in ‘1’ involving the rate of increase in temperature is sufficient to distinguish when the steady state is being reached as the multiplying factor can take erratic values at the commencement of the simulation. The ratio of the power leaving through the heat sink, to that input, is also used to warn of any error in the simulation. If the time step increases too rapidly or becomes too long, then the temperature always overshoots the steady state value and this can give rise to a condition where the power leaving the device is greater than that being input. Also the fact that the temperature starts to decrease having reached a maximum is also a good indication that the time step has increased too rapidly (i.e. ‘E ’ was too large). The maximum time step which is reached using the procedure outlined can often be increased further without

by the total rise (or fall) in temperature.

power.

.-

producing any unexpected problems with the simulation but there does not seem to be any simple algorithm which can be applied to ultimately limit the time step. It would not seem to be an important issue as the algorithm described works adequately up to a point where the maximum temperature has been reached. The simulator has been run on various machines in single and double precision, the worst scenario being an IBM3090 in single precision. The algorithm to control the time step worked as described but the maximum time step was limited to a smaller value with decreasing computational precision.

IV. GaAs MICROWAVE TRANSISTOR TEST STRUCTURES

Fig. 3 shows some GaAs/GaAlAs heterojunction power transistor test structures which have been fabricated as part of a test programme although device ‘A’ has not been fabricated with the air bridge. In structure ‘A’ the substrate is heavily doped and the collector contact is at the heat sink whereas in structure ‘B’ the collector contact is through the top surface via a heavily doped surface layer. Device ‘A’ represents a section through a structure having a large number of identical parallel emitter fingers. Device ‘B’ is a single emitter finger mounted centrally on a large chip. These structures will have the very different thermal time constants necessary to test the time step control. The time constant for device ‘A’ can be reduced by decreasing the chip thickness, as indicated in the diagram of Fig. 3. The emitter, base and collector contacts are indicated on the diagrams. In both cases one half of the structure is simulated there being planes of symmetry as indicated in Fig. 3 and the surface metallisation patterns are included in the 3D simulation. The simulator takes into account the variation of the thermal conductivity of gallium arsenide with temperature according to the datas given by Brice [13] and the GaAlAs regions are assumed to have the same thermal properties as GaAs. (As with GaAs, the thermal properties of GaAlAs are thought to be very dependent on doping and also on the molecular composition. The thermal conductivity is reported to be poorer than GaAs.) The gold has a constant thermal conductivity, specific heat and density of 320 W d K , 130 J/Kg . K and 19300 Kg/m3 respectively. Device ‘A’ was also simulated with a bridge which spans the base contact, the object being to increase the thermal capacity in the region of the dissipation and to decrease the thermal resistance. In this case the base would need to be a mesa or implanted as indicated in Fig. 3. Practical measurements on a structure which uses a heat spreading thermal shunt of this type have been reported by Bayraktaroglu [14] 1993. The use of a heat shunt can alleviate the problems of high temperature operation and current channelling within the transistor. The power and power densities used in the power dissipating regions are given in Table I.

V. SOME SIMULATION RESULTS

Fig. 4 shows a simulation for structure ‘A’ where the gallium arsenide substrate thickness is 30 microns. It shows the temperature rise with time, up to steady state, and the multiplication factor for the time step. In this case the value

628 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 42, NO. 4, APRIL 1995

Structure A Structure B

l?

1.759 0.4 & 0.44

% v

A

150 I :

I I /

I I : I

I

. . j-24- . _ _ ;...i... .. . ........ . . ;. .... 1 . .. . .. ..... 3 1 , .

I 1 : i /

i ! : i : i

i !

I ; j . .

I I w / j c i i

Heat Sink

Planes of Symmetry - _ _ __ - - Bridge - Heat Sources Gold

All dimensions in microns (Not to scale)

Fig. 3. Some typical Heterojunction Bipolar Transistor Structures, used to demonstrate the time step control algorithms. The diagram is not to scale. In structure ‘A’ Emitter width = 100 microns, chip dimension normal to the plane shown = 272 microns. In structure ‘B’ Emitter width = 30 microns, chip dimension normal to the plane shown = 450 microns.

TABLE I POWER DENSITIES USED IN THE SIMULATIONS

Pow input

WfD/tiUga

Powa density

structure ‘A’

0.347 0.275 0.07

I 1- c’

0.98 M -12 -10 8 6 4

nma seconds Log10

Fig. 4. microns. (a). Time step multiplier. (b). Temperature degrees centigrade.

Simulation for the structure ‘A’ with a substrate thickness of 30

of ‘E’ was 0.012. The multiplication factor tends to unity as the steady state is reached and is fixed at this value.

The time step is constant for the first 7 iterations so that sufficient data is available to make an adequate prediction for

the following time step. As in the other simulations the steady state temperature is high as the devices are intended for pulsed operation.

A number of additional simulations were undertaken on the structure ‘A’ for a substrate thickness of 30 microns and the results are shown in Fig. 5. The curve ‘a. . .a’ indicates a simulation using the time step routine already outlined. The curve ‘e . . . e’ is a similar simulation except that the value of ‘E ’ was increased to 0.024, a figure which is clearly too large. Curves ‘ d . . . d ’ and ‘ b . . . b ’ were obtained using a constant time step and therefore required many iterations. In the latter case the initial time step was too large leading to some inaccuracy at the start of the simulation. It would be totally unrealistic to simulate the whole characteristic using the time constant of the test node. These results indicate that the time step control routine works satisfactorily. Except where indicated, a value of ‘E’ equal to 0.012 was used in the remainder of the simulations and the starting temperature was 30 degrees Centigrade. Repeating a simulation using different values of ‘E’ does of course give different results. If the simulation shown in Fig. 5 is run for two values of E , 0.006 and 0.0 12, the curves would be indistinguishable when overlaid on the diagram but examination of the numerical output would reveal small differences of about 0.02 degrees maximum at some point in the simulation and naturally the smaller value of ‘E’ results in a larger number of iterations.

With reference to the simulation of ‘a . . .a’ in Fig. 5, Table II indicates the approximate number of time steps which would be required under the condition stated up to the point where the power leaving the structure reached 99.9% of the input.

The total simulation time (Not CPU time.) to reach the steady state for the simulation was about seconds, in some simulations the simulation times approach lo-’ seconds

WEBB AND RUSSELL APPLICATION OF THE TLM METHOD TO TRANSIENT THERMAL SIMULATION OF MICROWAVE POWER TRANSISTORS

5

0 .e

x F

a * C

f

-9 0 I

.10

200 1.010

1”

1 .m6

1.m

1.002

180

160

140

1 20

I W

80

80

40

--@ (I

Temperature Centigrade

1.0

c / d b

1 F

a c t 0

a

9 1

10

11

20 I

0.m

0.98

0.W

0.92

-11 -10 -9 -8 -7 -6 -5 4 -3 Time Seconds Log 10

Fig. 5. Simulation of the structure ‘A’ with a substrate thickness of 30 microns. The curves compare the result shown if Fig. 4, with those using a constant time step over various sections of the simulation. The letters mark the start and finish times of the various curves. The oscillatory curve is obtained using a value of ‘E ’ = 0.024.

TABLE Il ESTIMATED NWER OF TIME STEPS REQUIRED

and under these circumstances only the time step control method described gives realistic computation times.

A simulation of the ‘A’ structure using a 100 microns substrate is shown in Fig. 6. The temperature rise, time step, time step multiplying factor and power reaching the substrate are all shown. Notice that the multiplying factor for the time step is related to the slope of the rise in temperature, the shape of this curve is similar for all reasonable values of ‘ E ’ . Results for the structure ‘B’ are shown in Fig. 7 and as expected the temperature rise is much more gradual because of the smaller power input relative to the size of the gallium arsenide chip. In this case the time step algorithm indicated a decrease in time step at the start of the simulation, but the time step is not decreased in the simulator as the initial time step is taken as a minimum value. If the minimum time step criterion is removed then the time step initially decreases and then starts to increase. The number of time steps for the ‘A’ and ‘B’ structures were 2280 and 2783 respectively with ‘E’ = 0 .012 up to the time that the power entering the heat sink was 99.9% of the input power. In the simulation of structure ‘B’ using ‘E’ = 0 .02 the number of time steps is reduced to 1537 but there were some small oscillations in the temperature as the steady state was reached amounting to 0.003% of the steady state temperature.

The simulation results for the structure ‘A’ with a 100 microns chip thickness and the bridge in place are shown in Fig. 8. Curve ‘a’ is without the bridge, curve ‘b’ has the bridge made of gallium arsenide, curve ‘b’ the bridge material is gold and for curve ‘c’ the bridge is constructed unrealistically, at the moment anyway, of diamond. In the case of the bridge being

-11 .10 -9 8 .l d 4 4 -3 -2 Time Seconds Log10

Fig. 6. Simulation for the structure ‘A’ with a substrate thickness of 100 microns, showing (a) temperature, (b) time step, (c) multiplying factor for the time step and (d) power leaving the heat sink. Scale &loo%.

-11 . i o .9 a .7 .E .s 4 J .z Time Seconds Log10

Fig. 7. Simulation of structure ‘B’, showing (a) temperature, (b) time step (log scale), (c) multiplying factor for the time step and (d) power leaving the heat sink scale &loo%.

constructed of gold or diamond there were nodes within the bridge on a thermally conducting path down to the heat sink which had thermal time constants less than the test node. This problem is to be expected using a single node to control the time step. In the simulation with the gold bridge the value of ‘E’ was changed to 0.004 resulting in 6242 iterations using the same measurement criterion as before, compared with 2500 for structure with the gallium arsenide bridge. It would be feasible to search through the range of node time constants in the structure and modify the value of ‘E ’ accordingly. The method described presupposes that the shortest time constant lies in the dissipating region which is often, but not necessarily, the case.

The effect of the gold bridge on the thermal distribu- tion during the transient and reduction in the final thermal resistance is significant. One of the problems with HBT structures is the fact that the collector region, whether it takes the form of the ‘A’ structure, or if it is a diffusion or implantation which is contacted from the surface, it is a fairly heavily doped volume which is directly under the power dissipating region. This doping would reduce the thermal conductivity by a factor of about 0.7 compared to intrinsic gallium arsenide giving a corresponding increase in the total thermal resistance. In the HBT structure there could be some design compromise between collector electrical resistance and

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IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 42, NO. 4, APRIL 1995

Temperature Centigrade

- ___._ G M 22 . , . 3 . , . I . ,

0 4 8 12 16 20 24

Scales in Microns Temperature Degrees Centigrade

Fig. 9. Temperature profile through the ‘B’ structure at a time of 5 microseconds into the simulation.

device thermal resistance. The bridge structure would help to reduce these effects, but at the expense of increased emitter to base capacitance. Figs. 9 and 10 show thermal contour maps through the central region of the ‘A’ structure without and with the gold bridge after a time interval of 5 microseconds from the start of the simulation. Inspection of the thermal distributions at varying time intervals can be most useful in determining the effect of geometrical changes on the thermal behavior. As a final demonstration of the time step control algorithm, Fig. 11 shows the temperature variation, time step and percentage of the input power leaving the heat sink for the structure ‘A’ with a 100 micron chip thickness subjected to a 20 microsecond power pulse with a 25% duty cycle. Notice that the average power leaving the heat sink is 25% of the input power, and there is an appropriate phase lag between switching the power input and the flow of heat from the heat sink. One suspects that even in small device structures, time varying mechanical stress caused by thermal cycling, could be responsible for eventual device failure.

- _____ G M

22 0 4 8 12 18 20 24

Scales in Microns Temperature Degrees Centigrade

Fig. 10. a time of 5 microseconds into the simulation.

Temperature profile through the ‘B’ structure with a gold bridge at

am, I 3 5 50%

3

25%

P 0 w e r

0%

0 1 2 3 4 6 6 1 4 8 Time Secondl r10

Fig. 11. Temperature (a), time step (b), and power as a percentage of the input power leaving the heat sink (c), for structure ‘A’ for a 20 microsecond repetitive power pulse having a 25% duty cycle.

To give some measure of the computer time taken for the simulations the simulator was transferred to a 5OMHz 486 personal computer and structure ‘A’ was simulated. The number of nodes in the simulation was about 7000 and the simulation time was for 0.5.10-4 seconds, as in Fig. 4. The number of iterations taken was 2082 and the computation time was 55 minutes.

VI. CONCLUSIONS

Perhaps the two most important features of the TLM method for simulating diffusion processes are the ease with which non linear parameters may be modeled and the fact that the algorithm is unconditionally stable. The utility of TLM diffusion simulation compared with other numerical methods depends on the development of improved time step control. For example, the algorithms used in explicit finite difference methods are very simple and as in the TLM method, it is easy to implement non linear effects. Compared with finite element simulators it is certainly more straight forward to implement non linear effects using the TLM method. In this article a method of controlling the time step in a TLM simulator,

I 1

WEBB AND RUSSELL: APPLICATION OF THE TLM METHOD TO TRANSIENT THERMAL SIMULATION OF MICROWAVE POWER TRANSISTORS 631

designed for studying thermal transients in electronic power structures, has been demonstrated and the results obtained are far superior to those which have been achieved using the methods currently described in the literature. The development of improved time step control is important as it reduces the computation time very considerably and allows the user to strike a balance between simulation accuracy and computation time.

Some simulation results have been presented for typical microwave power structures, particularly those based on het- erojunction bipolar technologies, and this type of simulation is obviously important in the development of pulsed and continuous wave applications. The simulator not only gives important information about the heat flow during transients but also gives the steady state thermal distribution. The bridge structure which was the subject of one of the simulations seems to be a geometry worthy of further investigation.

ACKNOWLEDGMENT

The work reported in this article has resulted from research supported by the Defence Research Agency, Malvern, UK. I. A. D. Russell acknowledges their financial support.

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121 P. B. Johns, “A simple explicit and unconditionally stable numerical routine for the solution of the diffusion equation,” Int. J. Numer. Meth. Eng., vol. 11 pp. 1307-1328, 1977.

[ 3 ] M. Y. AI-Zeben, A. H. M. Saleh, and M. A. AI-Omar, “TLM modeling of diffusion, drift, and recombination of charge carriers in semiconduc- tors,” Int. J. Numer. Mod.: Elect. Network, Devices and Fields, vol. 5 , pp. 219-225, 1992.

[4] X. Cui, P. W. Webb, and G. Gao, “Use of the 3-dimensional TLM method in the thermal simulation and design of semiconductor devices,” IEEE Trans. Electron Devices, vol. 39, no. 6, June 1992.

[5] D. deCogan and S. A. John, “The calculation of temperature distri- bution in punch through structures during pulsed operation using the transmission line modeling method,” J. Phys. D: Appl. Phys., vol. 15, pp. 1979-1990, 1982.

[6] S. H. Pulko and C. P. Phizacklea, “A thermal model for cyclic glass pressing using TLM modeling,” in Proc. Instn. Mech. Engrs., Pr. B: J . Eng. Manut, vol. 205, pp. 187-194, 1991.

[7] J. P. Holman, Heat Transfer, 5th ed.. New York: McGraw-Hill, p. 145, 1981.

[8] P. W. Webb and X. Cui, “Implementation of time step changes in TLM diffusion modeling,” Int. J. Numer. Mod.: Elect. Network, Devices, vol.

[9] S . H. Pulko, A. Mallik, and P. B. Johns, “Automatic time stepping in TLM routines for modeling thermal diffusion processes,” Int. J. Numer. Mod.: Elect. Network, Devices and Fields, vol. 3, pp. 127-136, 1990.

[IO] X. Gui, P. W. Webb, and D. deCogan, “An error parameter in TLM diffusion modeling,” Int. J. Numer. Mod.: Elect. Network, Devices, vol.

5 , pp. 251-257, 1992.

5 , pp. 129-137, i992. 111 X. Cui and P. W. Webb, “A comparative study of two TLM networks

for the modeling of diffusion processes,” Int. J. Numer. Mod.: Elect. Network, Devices, vol. 6, pp. 161-164, 1993.

121 P. W. Webb, “Transmission line matrix modeling applied to thermal diffusion problems,” in IEE Electronics Division Tutorial Colloquium on Transmission Line Matrix Modelling, Professional Groups E15 and C6, Digest No: 1991/157, Oct. 18, 1991.

131 J. C. Brice, “Properties of gallium arsenide,” EMIS Data Reviews Ser., no. 2, London, 1986.

[14] B. Bayraktaroglu, J. Barrette, L. Kehias, C. I. Huang, R. Fitch, R. Neidhard, and R. Scherer, “Very high power density CW operation of GaAdAlGaAs microwave heterojunction bipolar transistors,” IEEE Electron Device Lett., vol. 14, no. 10, Oct. 1993.

Paul Webb received the B.Sc. degree in electrical engineering in 1959 and Ph.D. degree in 1963, both from the University of Birmingham, UK.

He was a lecturer at the University of Notting- ham, UK, until 1971, and is now a Lecturer at the University of Birmingham. His current interests include thermal imaging techniques to measure tran- . . . . ,

electronic devices and circuits, and thermal mod- eling using numerical techniques. He is manager of the Wolfson Thermal Imaging Laboratory at the

University of Birmingham, and has undertaken thermal design and measure- ment studies for many industrial organizations and research laboratories.

Ian Russell received the B.Sc. (Joint Hons.) degree in electronic engineering and computer science in 1984 from the University of Birmingham, UK. He is currently a research fellow at the University of Birmingham. His current research interests include thermal and electrical modeling and design of elec- tronic devices using numerical methods.