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630 IEEE TR ASSAC TIONS o s ELECTROS DEVICES, VOL. ED-13, x os . 8/9, AUGUST/SEPTEMBER, 1966
Some New Aspects of Thermal Instability of the
Current Distribution in Power Transistors
F. BERGMANN AND D. GERSTNER
Abstract-An importantcharacteristic of secondbreakdown in
p-n junction s is the current onstriction to a small region. This may
becaused by a thermal eedbackmechanism, as discussed by
Scarle tt and Shockley, and by Bergmann and Gerstner.
A brief review of this theory is given, illustrated by experimental
results of a simple model arrangement consisting of three thermally
coupled transis tors. hessentialarameters influencing the
therma l stability of the cur ren t distribution are device geometry,
power density, and tempera ture d ependence of current.
I t is widely known that second breakdown occurs at high voltages
at a much lower power level than at low voltages. To allow a more
detailed discussion of this effect in view of thermal stability, we
determined experimentally the temperature coefficient of transistor
current for various Si planar transistors as a function of current,
voltage, and junction t emperature. The experimental procedure is
describ ed and the re sults are discussed.
Theexperimentalvalues of the emperature coefficient rangefrom 0.08 to 0.01 1 / "C . Thevalues forhigh currentsaremuch
lower than predicted by the theory of Ebers and Moll. It th us can
easily be understood why, in the case of high current, and low volt-
age, the thermaI stability of the current distribution is much better
than in the case of low current and high voltage.
INTRODUCTION
HE PHENOMENOK of second breakdown in
transistors and diodes is associated with a current
constriction to a small area [1]-/7]. In many cases
thismay becausedbya thermal feedbackmechanism
according to a theory proposed by Shockley and Scarlett
181, [9], and independently by Bergmann and Gerstner lo]in 1963. If the thermal eedback is overcritical, he current
distribution becomes unst'able,and small parts of the
transistor bear almost all of the current. This instability
can occur ndependently of possible diffusion defects of
the ransistor.Although he ransistor is loaded below
the theoreticalmaximum power dissipation,calculated
from the thermal resistance, local overheating (hot spots)
may occur. When a critical temperature is reached at a
hot spot, an intrinsic zone is formed which short circuits
the space charge region of the p-n junction [ l ], 1121. This
results in a typical voltage reduction over the transistor.
I n some cases evenmolten zones have been observed
[13], [14]. With reverse-bias conditions, in addiOion to t'hethermal effects, electrical fields have o be taken nto
account,but nevertheless the hermal nstability seems
to be an important featureof second breakdown.
The purposeof this paperis to point out the importance
of the essential parameters influencing the thermal sta-
The authors are with Telefunken Aktiengesellschaft, Heilbronn,Manuscript received December 7 , 1965.
Germany.
bility of the current distribution. These parameters aredevicegeometry, power density, nd emperature de-
pendence of current. Someexperimental esults on he
temperature coefficient of currentre presentedor
typical Si planar transistor configurations. These results
allow an interpreta tion of the fact that the thermal sta-
bility of the current distribution s much better in the
case of high current and low voltage than it s in the case
of low current and high voltage.
MODELDEMONSTRATIONF THERMALNSTABILITY
A large area HF power transistor can be regarded as a
parallel connection of many small area transistors. Actu-
ally power transistors are constructed in this way, e.g.,when annterdigitatedtructure (comb structure) is
used. The question is: how will thecurrentdistribute
itself amongst the single transistors?
Figure 1 shows a imple model arrangement [ lo ] to
measure the current istributionnhreeransistors
working inparallel. The hree ransistorswithin he
dashed ine epresentherea arge area ransistor.The
single collector currents J,,, J o , J,, of the transistor
parts are measured. The sum of these currents and the
common collector voltage VCEetermine the power load-
ing of the whole transist,or onfiguration. This power
loading is stabilized to a nearly constant value by means
of the emither resistor R, which is common to the tran-sistors of the model.
Figure 2 shows the measured currents J,,,J,,, J c sas a function of t, ime after application of power. At the
beginning the currents J c , , J c 2 , Jo areapproximately
equal o 10 mA. Thecurrent n 'ransistor 1 increases
steadily,whereas thecurrent n ransistor 2 decreases.
The current n ransistor3 a t first increases a small
amount, and then also decreases. Finally after 5 minutes,
transistor 1 carries 96 percent of the total current, which
implies also almost all of the supplied power,while the
remaining'ransistors arry less than 3 percent ach.
Throughout, hesum of thecurrents remainsapproxi-
mately onstant J , w 30 mA. Thus, fromonly ex-ternalvoltage VCR ndcurrent J , observations of our
power transistor model, the transistor seems to operate
well.
Under the special conditions of this model, transistor
1mas not dest'royed because the whole system was driven
far below its maximum allowableower dissipation.
Bu t under real, operational conditions of a power tran-
sistor when loaded near the theoretical maximum power
dissipation, alculated rom the hermal resistance, i t
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1966 BERGMANN AND GERSTNER:HERMALNSTABILITY 63
1. 2. 3. I J C ,
Fig. 1. Arrangement for demonstration of currentdistribution nthree thermally coupled transistors.
A
0 2.5 5.0min
Time t
Fig. 2. Unstable currentdistribution among three ransistors nparallel or tw o different initia l conditions. At he right-handside, transistor 3 had been preheated.
may be tha t one part of the transistor would have burned
out.
It is important to note that this overloading of tran-
sistor 1 in our model arrangement is not predetermined
by adifferentelectricalcharacteristic of the transistor,
which would, orexample,correspond to aweakpoint
in the diffusion of the power transistor. This is demon-stratedat he ight-handside of Fig. 2. Here we had
preheated ransistor 3 a short whilebefore application
of the power. Now transistor 3ncreases its urrent
steadi ly and finally carries approximately 97 percent of
the otalcurrent , while transistors 1 and 2 retain he
remaining current.Thus,he urrentnstabilitys a
fundamental one, and can not be prevented by carefully
avoidingweakpoints n the ransis tor electricalstruc-
ture only.
This behaviorcanbeexplainedbya thermal feed-
backmechanism, which ismade possible by hevery
rapid increase of the collector current J , (approximately
equal to the emitter current J ,) with a rise in tempera-
ture.For abroad emperature range, this emperature
dependence may be pproximatedby an exponential
relation.
In our 3-transistor model the three transistors receive
the samevoltage V,, because theyare connected in
parallel. So a transistor tha t has a temperature slightly
higher than the others will bear more c,urrent than the
others.Thismeans he ransistor will dissipatemore
power, and if the emperature unbalance s not leveled
out either by hermal coupling t o the heat sink, or by
thermal coupling with ts neighbors, it will increase it s
temperature more than heothers,This n urn will
cause a further increase of the current, and so on. It is
not mportant how large the emperature difference or
the other differences between the various transistors are
at the beginning. When the feedback mechanism is over-
critical, the currentdistribution will beunstable,and
current crowding will occur.
STABILITYARAMETERS
From a quali tative discussion of our 3-transistor model,
three parameters can be predicted th at will influence the
thermalstability of currentdistribution n our model
or in a real large-area transistor.
A. Device Geometry.
Device geometry determines whether or not tempera-
ture differences between the various active parts of the
power transistor will be leveled out. If thehermal
couplingetween thehreeransistorsnd/orheir
thermal coupling to heheat sink sverystrong, the
temperature differences will be mall ven when thetransistorsdonotdissipate hesame power. Suppose,
as a limiting case, the whole transistor is concentrated at
one mathematical point; then no temperature differences
within this point would be possible.
B. Temperature Coe$icient of the Transistor Current.
A current nstability of the described type is only
possible when there is an increase of transistor current
with a rising temperature. Instability will be more serious
when the tempera ture coefficient of the transistor current
is high.
C . Power Level.
For a certain transistor system, thermal stability will
be better at a low rather han a t ahigh power level.
Suppose in our 3-transistor model, there is a temperature
unbalance tha t results in a 5 percent increase of current
for one transistor. At a power level of, say, 20 mW, this
means additional power of only 1 mW,whereas a t 200
mW this means 10 mW additional power.
For geometricallyimple transistortructuresnd
simple boundary conditions, the conditions for stable or
unstable current distribution can be calculated by solving
the differentialequation of heat conduction [lo], [ls].For a power transistor with a simple rectangular shape
with ength a , thickness h, the result of thestability
analysis is shown inFig. 3. At he op surface,a dis-
tributed heat source (e.g., a very fine interdigitated tran-
sistor structure),and at he undersurface, a flatand
perfect heat drain, have been assumed.
Again, here a re the three parameters that we already
know from the qualitative discussion. Using the stability
chart, (Fig. 3), it is possible to point o ut the relative im-
portance of theseparameters. Devicegeometry for this
transistor tructure s described by a,%, the ratio of
length o hickness of the ransistor crystal. This indi-
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632 IEEE TRANSACTIONS ON ELECTRON DEVICES AUGUST/SEPTEMBER
cates th at in a large area ransistor, hermal coupling
depends on the greatestdistance between the active parts
of the ransistorand he thickness of the chip. The
temperaturedependence of t'he ransistorcurrent at th e
certain operating point is characterized by the tempera-
ture coefficient B, which is defined by
B = L % lJ C dT V B E - C O a S t .
This emperature coefficient depends on the semicon-ductormaterial bandgap), he operatingpoint of the
transistor (especially current density), and he junction
temperature. AT is the emperature difference between
the ransistor junctions and he heat sink. AT may be
calculated as he product of theheatproduced in the
collector unction, and he hermal resistancebetween
the transistor unctions and the heat ink. AT is a measure
fo r the power density at which the transistor is operated.
In the stability chartwe find three regions.
1 . Stable mgion, the lowpower region. In th is region,
stableoperation of transistors is possible, regardless of
their geometry (large or small area).
2. Conditionally stableegion. In this region, stableoperation is possible if the total power is limited; that is,
if the rans istor is stabilizedagainst thermal runaway.
As canbe seen from the stabilitychart, tabilization
against thermal runaway a t a certain value of B -AT can
be achieved only for suitably low values of a/h; that is,
only for transistors of sufficiently small area.
3. Unstable egion. In his region, thermal coupling
between the active part s of the transistor is oo weak.
Thus the current distributionn the transistor is unstable
and current crowding will occur. This type of instability
cannot be avoided by limiting the total dissipation to a
constant value,as, for example, by an emitter resistor.
Whencomparing he concept of thermal nstabilitywith experimental results on the phenomenon on second
breakdown, one finds tha t the thermal concept fits very
well the main features observed in forward-bias second
breakdown. A complete understanding of second break-
down, including reverse-bias second breakdown, has not
ye t been achieved 1161. But it is hoped that by proper
refinement of t he existing heories, further peculiarities
will be understood. An example is the emperature de-
pendence of triggering energy or of the delay time [5], [16].
It seems possible that this temperature dependence can
be explained by taking into account thevariation of heat
conductance with temperature, which is quite important,
for example, in silicon. Another well-known feature is
the depen.dence of forward-bias second breakdown be-havior on operating point (Fig. 4).As can be seen from
the lowest curve, which is for the dc case, the tendency
of the ransistor obedishrbedby unstable current
distribution ismuchhigher at highvoltages and low
currents than it s a t low voltages and high currents. For
example, n the 10 volt ange, this ransistormaybe
10distributed heat source
55
. ., , . . . . . . :. ;
b < a
.B.AT-Fig. 3. Stability chart for a power transistor of rectangular shape.
I I50 100 2
Col lectorvol tage V ---b
0
Fig. 4. Safe operating range of a power transistor.
loaded to approximately 100 watts, whereas at 100 volts
the power allowable is only 20 watts.
How can hisbe explained? We believe that t is a
consequence of thecurrentdependence of temperature
coefficient onransistor urrent.This arameter asalready been mentioned when we were discussing the
stabilitychart.Remember, for example, a reduction of
the temperature coefficient by, say, a factor of two will
allow theemperature difference, and, therefore, also
the power dissipation, o ncreaseby thesame actor
withoutchanging hestabilitybehavior of thecurrent
distribution. Indeed a current dependence of this temper-
ature coefficient can be predicted by t'he simple theory of
the junction ransistor, orexampleby he heory of
Ebers and Moll 1171. By this heory, one would expect
for reasonable current densities, a value of B = O . lO /OC ,
which means that the transistor current is increased by
10 percent when the junction temperature is increased by
1°C. Concerning the current dependence f B, this theorypredicts tha t B decreases by 2 . 3 / T ; hat is, about 0.007/"C
for a ten times increase in current density [8].Obviously
this heoreticalcurrentdependence of the emperature
coefficient B is too weak and would not explain the large
differences of current stability which are observed experi-
mentally.
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1966 BERGMANN AND GERSTNER: THERMAL INSTABILITY 633
MEASUREMENTF TEMPERATURECOEFFICIENT
Since the transistor heory s based on many simpli-
fyingassumptions which are generally not fulfilled, we
havemeasured the emperature coefficients fo r silicon
planar transistors and their dependence on current level
and junction temperature.
In order to avoid instability of current distribution a t
too low power levels, we used small area transistors. Bu t
these transistors were similar to typical large area tran-
sistors when junction depth, base width, diffusion layer
sheet resistance, and so on, are considered.
On the right-hand side of Fig. 5, a sketch of our meas-
uring ircuit is given. The ransistors were put n a
thermostat, and for fixed collector currents J , the base
emitter voltage VB ,was measured with a high precision
digital voltmeter as a function of junction temperature
Ti.t should be mentioned th at the voltage drop over
the base resistance, which has a stabilizing influence, has
been subtracted from the measured values of VBE. hu s
the following considerations are also valid for open base
conditions, which are very mportant n practice. Toobtain he right value of Ti, e firstmeasured V B E t
several collector voltages for fixed collector current as a
function of case temperature T,. This measurement gave
aset of parallelcurves which could be extrapolated t o
zero dissipated power using the relation'
T i T,, +~- which has been derived in [ lS].cz - T c ,-- -C E l 1
v C E 2
In this way we obtained the set of curves shown on the
left-hand side of Fig. 5 for a 2 N 1613 transistor.
In order t o determine the emperature coefficient B,
this set of curves can be replotted in the manner shownin Fig. 6. This s a log plot of collector current J , against
junction temperature T i or constant base emitter volt-
age V B E . By graphical differentiation of these curves, th e
experimental values of the temperature coefficient B are
obtained. Throughout the whole curve set, it can be seen
that the slope is much less steep at high currents than itis at low currents, and that the lope is not very tempera-
ture dependent.
In Fig. 7, B has been plotted for two emperatures
(50 and 100°C) as a function of transistor current. For
comparison, two heoretical curves-the dot ted ones-
are given. These show the weak current and temperature
dependencementionedabove. In contrast o he theo-retical curves, the experimental values of B tend t o much
lower values when the current density is increased. Foran example, when J , is increased from 10 t o 100 mA, B
'or the determination of the thermal resistance between th e transis-1 This measurement procedure is actually a simple static method
;or junctions an d the case [18]. We noticed that the thermal resis-jances given by the dataheets were not accurateenough to calculate;he junction temperature from case temperature and power dissipa-ion, especially a t high power levels.
Tj -ig. 5. Base emitter voltage V B Evs. junction temperature Tifor various collector currents J c for a silicon planar ransistor2 N 1613.
mA
t lou7
1
".I , -0 50 - 100 150 200 "C
Tj -Fig. 6 . Collector current J c vs. unction temperature T i fo r a
silicon planar ransistor 2 N 1613. Parameter 1s base emittervoltage VBE.
-- heoretlca
0.00 !0.1 1 10 100mA
Jc
Fig. 7. Temperature coefficient B vs. collector current JC for asilicon planar transistor 2 N 1613.
decreases from 0.045 t o 0.015 according t o the 100°Ccurve;
tha t is, by about a factor of three.
Our experiments show that at high current densities,
the temperaturedependence of current is muchower tha n
predicted by theory . Thus itan be easily understood why,
in a large-area transistor a t acertain power level, the
stabilityagainst second breakdown is much better for
high current than for low current operation.
CONCLUSIONS
In large-area transistors, an instability of current dis-
tribution can occur due to a thermal feedback mecha-
nism. This type of instability which may initiate second
breakdownhas been demonstratedexperimentally na
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634 IEEE TRANSACTIONS O K ELECTRON D E V I C E S AUGUST/SEPTEMBER
Fig. 8. Second breakdown proof power transistor.The large-area
resktor.device is divided into eight cells, each having a stabilizing emitter
simple model arrangement.There rehree ssential
parameters influencing the stabil ity of the current distri-
bution: deviceeometry, temperature dependence of
current, and power level.
In order to improvestabilityagainst second break-
down initiated by this thermal mechanism, th e following
precautions may be taken.
1) Device geometry should be designed so as to pro-
vide strong thermal coupling.
2) The temperature coefficient of current should be re-
duced or the single activeparts he ransistor s
composed of .
3) The power level should be reduced to a value which
assures stable current distribution.
The hird possibility does not seem to bevery atis-
factory, since it does not cure the inst’ability.I t is merely
a withdrawal from the dangerous high power region and
must be paid for by inconveniently high capacitance in
the ransistorandbyaddit ional yield problems when
producing such overdimensioned devices. But there is a
very promising possibility for he construction of a second
breakdown proof power transistor by combining the first
tm7o stability design concepts [19]. The large-area power
transistor is divided into single cells which are small
enough t o assurestabilitywithineach cell. Then hese
stable cells are combined to a large area configuration with
stabilizing emitter resistors for each cell. Of course, the
internal B within these cells is not influenced, but th at is
not necessary because the single parts are designed in an
undercritical size. Figure 8 shows, as an example, a power
transistor which is divided into eight small-area devices,
eachhaving a stabilizing emitter resistor. The resistors
in this example are evaporated KiCr resistors.
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