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NOTE A 3D THERMAL MODEL FOR ELECTROSTATIC DISCHARGE THERMAL RUNAWAY IN SEMICONDUCTOR DEVICES (Received 3 June 1993; in revisedform 22 September 1994)

MOSFET as shown in Fig. 1[8]. Since the failure mechanism for damage of the transistors under ESD stress is the second breakdown at the drain junction, the main parameters of the transistor related to second breakdown are the vertical depth of the current flow in the depletion region at the drain junction, the lateral depth of the depletion region. and the width of the drain junction. The heat generation is P(f)= V(r)l(r) through the volume of the heat source. The temperature function T(.r, y, z, 1) in the vicinity of the volume under consideration is given by the solution to the heat flow equation[5,9]. In this analysis, it is assumed that the heat is generated uniformly throughout the volume and the hottest point is at the center of the heat source (x =r = z = 0). The solution to this heat equation can be written as an integral over the time and the volume of the heat source yielding a general temperature distribution function as follows:

NOTATION 4

b, c heat source dimensions

G

D f(t) K PO P(l) Pf

if TUI TO

T(r) Vol

V(l) P

specific heat capacity thermal diffusivity (= K/pC,) current thermal conductivity peak power power failure power time failure time melting temperature ambient temperature temperature function volume of heat source (Vol = abc) voltage density discharge time constant

T(0, t)=To+&exp

P

1. INTRODUCFION

Electrostatic discharge (ESD) is a serious reliability problem in VLSI technology, often resulting in thermal runaway. Thermal runaway is modeled by a localized heat source, where the instantaneous power, voltage and current are related by P(t) = V(t)I(r). In the case of a human body model (HBM) ESD pulse, Speakman[l] applied the Wunsch and Bell (W-B) model[2] for constant pulses to exponential ESD pulses. The discharging current profile is expressed by the exponential function with decay time constant. Pierce et a/.[31 developed a waveform conversion model and obtained the threshold current using numerical iterations. Dwyer et a/.[41 extended Pierce’s work and obtained the failure threshold using the Dawson integral and the time interval of the pulse which causes the damage. Recently, Choi and DeMassa[S] derived a closed-form thermal model and the failure threshold from the one-dimensional heat flow equation. The closed-form model is only valid for large junction area and relatively small juction depletion width. Dwyer et a/.[61 developed a 3D thermal model for a constant pulse based on a heat source in the form of a rectangular box. The method is based on Green’s function formalism and the Duhamel formula, which relates constant and varing power pulses. Franklin et a/.[71 have applied the model successfully to thermal breakdown in GaAs MESFETs caused by electrical overstress (EOS). In this paper, a solution for an ESD input to semiconductor devices is developed from the heat flow equation with the heat source assumed to be a rectangular box. Using an error function aproximation, a simplified 3D closed-form thermal model is obtained, thus eliminating 3D numerical calculations. 2.

THERMAL

MODELING

d.u’ dr’ d:‘.

(1)

where p(t-u)=poexp

(f-u)

( > -7

for the human body model ESD pulse and Y’ is integrated from -a/2 to a/2, y’ from -b/2 to b/2 and :’ from -c/2 to c/2. The temperature at the center of the heat source is obtained by setting x =y = z = 0, yielding:

T(0, t)= To+

x erf($===krf(&)

da.

(2)

It is convenient to define diffusion times related to a, b and c as follows:

These represent approximate times for thermal equilibrium to be established in the .Y,~3and -_directions, respectively. Since c

Channeling current causes joule heating of the heat source. The heat source is modeled as a rectangular box for a

-1 939

.\-

(4a)

X&/1.9.

(4b)

Note

940

(lo)

I .9&(4

exp( -21/3~) - &

exp( - 21,137))I

oxide

If c is very small, as is the case for a reverse biased p-n junction, (10) reduces to the ID thermal model[5]. If c has finite length, the 3D closed-form model must be used. The ESD power/time to failure relationship becomes:

/

c

p-sub

pr=

heat dissipation region

Fig. 1. The approximated heat dissipation region for a reverse-biased NMOST. The shaded area is associated with the depletion region around the reverse-biased junction[5].

(T,-

TOW, Vol

exptrt-/r)

exp(t/35) - J;r exp(fJ3r)) + r exp $ ( (J-l)

1.9$&/Z

(11) Region Ill

The approximation of eqns (4a) and (4b) is excellent except in the range of 0.6 c x < I .82 and in this range acceptable. We obtain, for example, the following equations: erf(-$)r0.95

J

if r>0.9rC

(5a)

if f gO.92,.

(5b)

Region I

In the time region tb< r C r., thermal equilibrium exists in both they and z directions at which point the corresponding functions (!r and c) enter the linear region. The temperature distribution function in this time interval is given by:

ihexp(-fl7) W, 1)= TO+ pc P vo,

r(exp(:)’

l)+0.95z&,

i.2

du

(12)

+ ,.9J;;(JXexp($)-help($))

During the time interval 0 < r GO.91,all error functions are approximately equal to one, i.e.

iyo, ~)=T~+~r[l-exp(-rl~)l.

(6)

Since HBM ESD is a short pulse, for small integrand term is approximated as follows:

u/7

the

P

If failure occurs in this time interval, the pulse power/time to failure relationship becomes:

exp z 0

J

u

du=log u+;+T+T+T+.

(7) In this region, the failure power is very large for small failure time. Region II

In the time region (I,, tb), the error function in the z-direction is not one and enters the linear region in the equation. During this time interval, the temperature distribution function is given by:

z1ogu+;.

(13)

The temperature distribution function is then: p0exp -i ( ) T(0, r)= TO+ PC vo, -+xp($

X

+

I)

1.9&(&exp($)-J,exp(=$))

(14)

+o.95’&(log;+~) x[O.95fi[[$)dz4+[exp(F)da].

(8)

The ESD power/time to failure relationship for thermal breakdown in this time domain becomes: pr=

The integral terms in eqn (8) are simplified in order to obtain a closed-form temperature function[5] as follows:

Urn- To)pC, Vol exp : 0

x

I

z(exp(:)-

(9)

I)+ *.9J(Jbexp(&)

-j;;,xp($))+0.9SZ&X(iog~+~)

and

(15)

Note Region IV

Finally, for times greater than t., all of the error functions vary linearly. The temperature distribution function then becomes: p0exp -5 ( ) PC Vo, 7-(0, rl=To+ P

1

[kp(;)- I)

Time

[ns]

Fig. 3. The input power vs failure time plot. +l.,J;;(Jiexp($)-Jiexp($))

x

+o.9s&(log~+~).

2(exp( -!+o.5_r-05)

I

+0.953&z

++($ exp(&)-,:;lexp(&)

I_

In this case, if the melting temperature T, is reached, the ESD power/time to failure relationship is given by: pr= (T, - To)pCpVol . exp 9

0

power vs time relationship should display all four time dependencies. The quantities C, and K are taken to be temparature independent and are calculated at the temperature of interest. This simplifies the calculations and has been shown not to have a significant effect on the temperature rise vs time relationship. One example can be seen in Fig. 2 which shows the increase in temperature T(r) - TO with time in a Si p-n diode with heat source dimensions of 0.3 x0.6x 100 pm subject to an ESD input with peak power of 10 W. The diffusion time constants are t,=0.55 ns, rb=2.2 ns and r&=61.4 ps. Typical HBD ESD pulses have a time constant of about 150 ns or less. Thus, Fig. 2 shows only time ranges less than 200 ns and thus, region IV is not shown in Fig. 2. The input power to failure time relationship is shown in Fig. 3. The lowest input power indicates the maximum temperature in Fig. 2. The dashed curve increases slightly after the maximum temperature due to decreasing temperature. But the allowable input power should be the same as the lowest input power.

r(exp(:)-l)+1.9&(&exp($) 3. CONCLUSIONS -i;;exp($))+0.9SJGb(log~+~)

*(exp( -$+O’-r?) t 0.953Jr,rbt, I

++(&exp($)-J;Iexp(;)

1 (17)

Hence, there are four separate time domains for all device shapes. The analysis presented herein shows that the

In conclusion, a closed-form solution of the heat flow equation in three dimensions is obtained using approximations to the error function. The three-dimensional solution of the heat flow equation is then applied to predict the temperature rise of semiconductor devices. Also, the input power to failure time relationship is obtained and is useful for designing thermal protection devices. The closed-form model provides accuracy as well as computation time savings over the three-dimensional numerical model calculations. This closed-form thermal model is also useful for thermal device design. Department of Electronics Engineering National Fisheries University of Pusan Pusan Korea

5 2 ‘L S 3 E d E c

800

Center for Solid-State Electronics Research Arizona State Universit? Tetnpe AZ 85287 U.S.A.

800 400 200

H. H. Choi

T. A. DeMassa

REFERENCES

Of00 Time

Fig. 2. Temperature

[ns]

rise vs time plot for a= 100 pm, b=0.3 pm and c=0.6pm.

1. T. S. Speakman, Proc. In!. Rel. Phgs Symp.. p. 60 (1974). 2. D. C. Wunsch and R. R. Bell, IEEE Trans. Nucl. Sri. 15, 244 (1968).

3. D. G. Pierce, W. Shiley, B. D. Mulcahy, K. E. Wagner and M. Wunder, EoS/ESD Symp. 10, 137 (1988).

Note

942

4. V. M. Dwyer, A. J. Franklin and D. S. Campbell, IEEE Trans. Electron Deoices ED-37, 2381 (1990). 5. H. H. Choi and T. A. DeMassa, Solid-St. Electron. 36,

I51 I (1993). 6. V. M. Dwyer, Solid-St.

A. J. Franklin

Electron.

and

33, 553 (1990).

D. S. Campbell.

7. A. J. Franklin, Solid-St.

Electron.

V. M. Dwyer

and

D. S. Campbell.

33, 1055 (1990).

8. A. Amerasekera. L. Roozendaal. J. Bruines and F. Kuper, IEEE Trans. Elec/rons Devices ED-38 (1991). 9. H. S. Carslaw and J. C. Jaeger, Conduction of’ Heat in Solids. Clarendon, Oxford (1959).