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Extraction of True Finger Temperature from Measured
Data in Multi-Finger Bipolar Transistors
Aakashdeep Gupta, K. Nidhin, Suresh Balanethiram, Rosario d’Esposito,
Sebastien Fregonese, Thomas Zimmer, Anjan Chakravorty
To cite this version:
Aakashdeep Gupta, K. Nidhin, Suresh Balanethiram, Rosario d’Esposito, Sebastien Fregonese, et al..
Extraction of True Finger Temperature from Measured Data in Multi-Finger Bipolar Transistors.
IEEE Transactions on Electron Devices, Institute of Electrical and Electronics Engineers, 2021, 68
(3), pp.1385-1388. �10.1109/TED.2021.3054602�. �hal-03273341�
1
Extraction of True Finger Temperature from
Measured Data in Multi-Finger Bipolar Transistors
Aakashdeep Gupta, K Nidhin, Suresh Balanethiram, Rosario D’Esposito, Sebastien Fregonese, Thomas
Zimmer, Senior Member, IEEE, Anjan Chakravorty, Member, IEEE
Abstract—In this brief, we propose a step-by-step strategy to
accurately estimate the finger temperature in a silicon based
multi-finger bipolar transistor structure from conventional mea-
surements. First we extract the nearly zero-power self-heating
resistances (R
th,ii
(T
a
)) and thermal coupling factors (c
ij
(T
a
)) at
a given ambient temperature. Now, by applying the superposition
principle on these variables at nearly zero-power, where the
linearity of the heat diffusion equation is preserved, we estimate
an effective thermal resistance (R
th,i
(T
a
)) and the corresponding
revised finger temperature T
i
(T
a
). Finally, the Kirchhoff’s trans-
formation on T
i
(T
a
) yields the true temperature at each finger
(T
i
(T
a
, P
d
)). The proposed extraction technique automatically
includes the effects of back-end-of-line metal layers and different
types of trenches present within the transistor structure. The
technique is first validated against 3D TCAD simulation results
of bipolar transistors with different emitter dimensions and then
applied on actual measured data obtained from state-of-the-art
multi-finger SiGe HBT from STMicroelectronics B5T technology.
It is observed that the superposition of raw measured data at
around 40 mW power underestimates the true finger temperature
by around 10%.
Index Terms—SiGe HBT, multi-finger transistor, measurement,
self-heating, thermal coupling, parameter extraction
I. INTRODUCTION
In order to increase the current handling with minimized
current crowding, multi-finger bipolar transistor structures are
preferred in power amplifier applications [1]. Eventually, a
large amount of power is dissipated in the base-collector junc-
tions of all the fingers. Since, the fingers are thermally coupled
through the common substrate (although electrically isolated
by shallow trenches), modeling of such transistors requires
accurate estimation of both self-heating in each finger and
thermal coupling among the fingers [2], [3], [4], [5]. In order
to develop a reliable thermal model, accurate characterization
of thermal resistances and junction temperatures for all the
fingers is of utmost importance. State-of-the-art measurement
techniques allow us to determine the self-heating thermal
resistance (R
th,ii
) and corresponding rise in junction temper-
ature (∆T
ii
) for each (i
th
) finger [6], [7]. On the other hand,
A. Gupta, K. Nidhin, A. Chakravorty are with the Department of Electrical
Engineering, IIT Madras, Chennai 600036 India. email: anjan@ee.iitm.ac.in.
S. Balanethiram is with the Department of Electronics and Communication
Engineering, IIIT Tiruchirappalli, Trichy 620015. email: sureshbalanethi-
ram@gmail.com
S. Fregonese, and T. Zimmer are with IMS Laboratory, University of Bor-
deaux, 33400 Talence, France. email: sebastien.fregonese@ims-bordeaux.fr,
thomas.zimmer@ims-bordeaux.fr.
R. D’Esposito is with Micron Semiconductor, 67051 Avezzano, Italy.
This work was supported in part by the EU under Project Taranto under
Grant 737454, in part by ISRO project ELE/17-18/176/ISRO/ANJA and in
part by DST, India, under Project EMR/2016/004726.
thermal coupling effects are determined from measurements
of coupling coefficients (c
ij
) based on heat-sense technique
[5], [8] between a given pair of fingers (i, j). This is done
by passing high current through one (j
th
) finger, raising its
temperature to ∆T
jj
, while sensing temperature (∆T
ij
) at the
other (i
th
) finger. The ratio ∆T
ij
/∆T
jj
yields c
ij
. From these
measurements, the total rise in junction temperature above the
ambient temperature (T
a
) for the i
th
finger, in an n-finger
system, is estimated as [3], [5],
∆T
i
= ∆T
ii
+
n
X
j=1,j6=i
∆T
ij
= P
d,i
R
th,ii
+
n
X
j=1,j6=i
c
ij
P
d,j
R
th,jj
(1)
where the first term indicates the rise in junction temperature
due to self-heating and the second term considers the effect
of thermal coupling from all other fingers. P
d,i
(P
d,j
) is the
electrical power dissipation in the i
th
(j
th
) finger.
Although the accuracy of the thermal characterization is im-
proved over the decades of research, the fundamental problem
lies in using superposition to obtain the overall finger temper-
ature from self-heating and thermal coupling temperatures as
done in (1). This is because any measurement automatically
includes the temperature dependence of thermal conductivity
of substrate Si (or any other material in the heat flow path).
Eventually, the resulting model based on the heat diffusion
equation will be non-linear [9], [10]. Therefore, one cannot use
superposition principle to obtain the total finger temperature
from two separate measurements. Although the work reported
in [11] attempted to obtain the c
ij
without ignoring the self-
heating in the sensing finger, the original problem of handling
the non-linearity with superposition could not be avoided.
To the best of the authors’ knowledge, a suitable technique
to accurately estimate the true finger temperature from the
measurements of self-heating and thermal coupling in a multi-
finger transistor system under real operating condition (when
all fingers are heating simultaneously) is not only missing
in the literature, but also in high demand in the modeling
community. This paper addresses this fundamental research
gap. In section II, we present the extraction methodology in
detail. Section III demonstrates a primary validation of our
technique using 3D TCAD simulation data. Note that TCAD
simulation cannot substitute the actual measurements required
to extract compact model parameters and subsequently to
develop the process design kit. Therefore, after validating
with TCAD simulation, we apply the proposed technique on
2
DT
ST
M8
M7
M6
M1-M5
x
y
z
Intrinsic Si
BEOL
Fin-1
E
C
C
B
B
C
C
B
B
Fin-2
Fin-3
substrate
Fig. 1. 3D cross-sectional view of a five-finger TCAD structure of
STMicroelectronics B55 process [12] containing shallow and deep
trenches along with the eight metal layers (M1 to M8) in BEOL. A
set of emitter (E), base (B), and collector (C) contacts are indicated
for the rightmost corner finger.
0 1 0 2 0 3 0 4 0 5 0
3 0 0
3 5 0
4 0 0
4 5 0
5 0 0
5 5 0
6 0 0
6 5 0
T
i
[ K ]
F i n - 1 , T C A D c a s e - B
F i n - 3 , T C A D c a s e - B
F i n - 1 , T C A D c a s e - A w i t h e q . ( 1 )
F i n - 3 , T C A D c a s e - A w i t h e q . ( 1 )
W i t h S T - D T
T
a
= 3 0 0 K
P
d
[ m W ]
W
E
= 0 . 2 µm , L
E
= 5 µm
Fig. 2. Dissipated power-dependent true temperature (T
i
) of 1
st
and
3
rd
finger in a five finger transistor. Circles correspond to case-B
TCAD simulation. Solid lines are the results obtained following (1)
from case-A TCAD simulations by exciting one finger at a time.
measured data of multi-finger transistor fabricated in state-of-
the-art B5T technology from STMicroelectronics and provide
a true finger temperature. We also demonstrate the underesti-
mation of finger temperature compared to the true value if (1)
is applied directly on the measured data. Finally, we conclude
in section IV.
II. EXTRACTION OF TRUE FINGER TEMPERATURE
We start with a set of TCAD simulated data of a five-finger
bipolar transistor whose dimensions correspond to the state-of-
the-art SiGe HBT from STMicroelectronics B55 technology
[12]. Fig. 1 shows the device structure where each finger
is electrically isolated by shallow-trenches (ST) and all the
five-fingers are additionally housed within deep-trenches (DT)
on all four sides. Note that for TCAD simulation we have
included back-end-of-line (BEOL) metal layers till M1 as
most of the BEOL thermal resistance is offered by M1
[13], [14]. Essentially two types (case-A and B) of TCAD
simulation are carried out on the same five-finger device
structure. Case-A refers to the situation when one finger is
0 1 0 2 0 3 0 4 0 5 0 6 0
2 . 0
2 . 4
2 . 8
3 . 2
3 . 6
4 . 0
0 1 0 2 0 3 0 4 0 5 0 6 0
0
5
1 0
1 5
2 0
2 5
W i t h S T - D T
R
t h , 1 1
( T
a
)
R
t h , 3 3
( T
a
)
R
t h , 2 2
( T
a
)
F i n - 1
F i n - 2
F i n - 3
T
a
= 3 0 0 K
C a s e - A
T C A D
F i t o v e r T C A D
W
E
= 0 . 2 µm , L
E
= 5 µm
W i t h S T - D T
T
a
= 3 0 0 K
C a s e - A
F i n - 1 h e a t i n g
T C A D
F i t o v e r T C A D
c
2 1
( T
a
)
c
3 1
( T
a
)
c
4 1
( T
a
)
c
5 1
( T
a
)
W
E
= 0 . 2 µm , L
E
= 5 µm
R
t h , i i
[ k K / W ]
P
d
[ m W ]
( a )
P
d
[ m W ]
( b )
c
i , 1
[ % ]
Fig. 3. Self-heating and thermal coupling results obtained from case-A
TCAD simulations: (a) P
d,i
-dependent self-heating thermal resistance
R
th,ii
of all three individually heating fingers (Fin-1, 2 and 3) and
(b) coupling coefficients obtained for case-A when only 1
st
finger is
heating (j = 1) and rest are sensing. Ambient variables R
th,ii
(T
a
)
and c
ij
(T
a
) are also indicated.
excited at a time (analogous with the heat-sense measurement
condition) and summing up the effects of self-heating and
thermal coupling using (1) for the overall finger temperature.
Case-B corresponds to a situation when all fingers are excited
simultaneously which emulates the real operating condition
of the device, yielding true finger temperature. Fig. 2 shows
the dissipated power-dependent temperatures of finger-1 and
finger-3 obtained from case-A and case-B simulations. One
can observe a significant difference between the true finger
temperature (case-B result) and the ones obtained following (1)
(case-A approach), particularly at large P
d
. Since, the temper-
ature dependence of thermal conductivity of Si is considered
in both the TCAD simulations, the original heat diffusion
equation becomes non-linear. Therefore, the application of (1)
results into underestimation of finger temperature as shown
in Fig. 2. In the following we present the methodology to
achieve the true finger temperature (results of case-B) from
the conventional heat-sense measurement (case-A) technique.
Fig. 3(a) presents case-A TCAD simulation results of dissi-
pated power (P
d,i
) dependent self-heating thermal resistance
(R
th,ii
) of the first three fingers (i = 1, 2, 3) at an ambient
temperature, T
a
= 300 K. Note that symmetry of the structure
ensures R
th,44
= R
th,22
and R
th,55
= R
th,11
. The solid lines
in Fig. 3(a) are polynomial fit capturing the non-linear P
d,i
-
dependent trend of TCAD data. Extrapolation of the solid line
at zero P
d,i
yields R
th,ii
(T
a
). Similarly, Fig. 3(b) contains
TCAD simulated P
d,1
-dependent coupling coefficients c
i,1
from which one can obtain c
i,1
(T
a
) by extrapolating each
solid line for P
d,1
→ 0. Other c
ij
(T
a
) can be obtained
from similar plots of P
d,j
-dependent c
ij
data. Note that at
P
d
→ 0, thermal conductivity of Si depends only on T
a
and the
corresponding differential equation for heat diffusion remains
linear. Therefore, one can apply the superposition principle
to obtain a total effective thermal resistance corresponding to
each finger as
R
th,i
(T
a
) = R
th,ii
(T
a
) +
n
X
j=1,j6=i
c
ij
(T
a
)R
th,jj
(T
a
). (2)
Subsequently, the rise in junction temperature ∆T
i
(T
a
)
can be estimated as ∆T
i
(T
a
) = R
th,i
(T
a
)P
d,i
. Finally,
3
0 1 0 2 0 3 0 4 0 5 0
3 0 0
3 5 0
4 0 0
4 5 0
5 0 0
5 5 0
6 0 0
6 5 0
7 0 0
0 1 0 2 0 3 0 4 0 5 0
3 0 0
3 5 0
4 0 0
4 5 0
5 0 0
, , F i n - 1 , 2 , 3 T C A D
F i n - 1 , f r o m e q . ( 3 )
F i n - 2 , f r o m e q . ( 3 )
F i n - 3 , f r o m e q . ( 3 )
, , F i n - 1 , 2 , 3 T C A D
F i n - 1 , f r o m e q . ( 3 )
F i n - 2 , f r o m e q . ( 3 )
F i n - 3 , f r o m e q . ( 3 )
C a s e - B
T
a
= 3 0 0 K
W i t h S T - D T
P
d
[ m W ]
T
i
[ K ]
α = 1 . 1 4
W
E
= 0 . 2 µm , L
E
= 5 µm
α = 1 . 1 7
W
E
= 0 . 2 µm , L
E
= 1 0 µm
W i t h S T - D T T
a
= 3 0 0 K
C a s e - B
( a )
( b )
P
d
[ m W ]
T
i
[ K ]
Fig. 4. A comparison of dissipated power-dependent temperatures T
i
, at
fingers 1, 2 and 3 obtained from the proposed extraction scheme (solid lines)
against case-B TCAD simulation (symbols) for devices with emitter area (A
E
)
of (a) A
E
= 0.2 × 5 µm
2
and (b) A
E
= 0.2 × 10 µm
2
.
the application of Kirchhoff’s transformation [15] on the
calculated ∆T
i
(T
a
) takes care of the temperature dependence
of material thermal conductivity and provides us with the
true finger temperature as
T
i
(T
a
, P
d
) = T
a
1 +
(1 − α) R
th,i
(T
a
)P
d,i
T
a
1
1−α
. (3)
Here the parameter α is originated from the temperature
dependent thermal conductivity of the heat flow medium which
is modeled as κ(T ) = κ(T
a
)(T/T
a
)
−α
. All the terms in the
r.h.s. of (3) are already known except (1 − α) which can be
obtained by non-linear parametric fitting of P
d,i
-dependent
R
th,ii
data (Fig. 3(a)) using the well-known relation [16]
R
th,ii
(T
a
, P
d
) =
T
a
P
d,i
"
1+
(1 − α) R
th,ii
(T
a
)P
d,i
T
a
1
1−α
−1
#
.
(4)
Thus the true finger temperature T
i
can be obtained as a
function of P
d,i
using (3) with extracted R
th,ii
(T
a
), c
ij
(T
a
)
and (1 − α).
III. VALIDATION OF PROPOSED TECHNIQUE AND
APPLICATION ON MEASURED DATA
Figs. 4 (a) and (b) compare the dissipated power-dependent
true temperatures obtained from the proposed extraction
method (solid lines) following (3) against case-B TCAD
simulations (circles) for first three fingers in two different five-
finger transistors with A
E
= 0.2 × 5 µm
2
and A
E
= 0.2 × 10
µm
2
, respectively. Note that the temperatures of 4
th
and
5
th
fingers are identical with those of 2
nd
and 1
st
fingers,
respectively. Highest temperature of the central finger is due
to high amount of coupling from the neighboring fingers. An
excellent agreement of the extracted temperatures with case-B
TCAD data demonstrates the accuracy of the overall extraction
methodology. The values of α extracted from our approach are
1.14 and 1.17 for A
E
= 0.2× 5 µm
2
and A
E
= 0.2× 10 µm
2
,
respectively. Note that the proposed technique automatically
takes into account the effects of trenches and BEOL metal
layers as the extraction deals directly with the characterised
data. Therefore, the extracted α does not necessarily corre-
spond to that of the Si material but also incorporates the effect
of trenches and BEOL.
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0
3 . 2
3 . 6
4 . 0
4 . 4
4 . 8
5 . 2
5 . 6
6 . 0
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0
0
2
4
6
8
1 0
1 2
1 4
1 6
1 8
2 0
2 2
F i n - 3
C a s e - A
F i n - 2
M e a s u r e m e n t
F i t o v e r m e a s .
W i t h S T - D T
F i n - 1
P
d
[ m W ]
R
t h , i i
[ k K / W ]
W
E
= 0 . 1 8 µm , L
E
= 5 µm
T
a
= 3 0 0 K
T
a
= 3 0 0 K
R
t h , 3 3
( T
a
)
M e a s u r e m e n t
F i t o v e r m e a s .
R
t h , 1 1
( T
a
)
c
5 1
( T
a
)
R
t h , 2 2
( T
a
)
c
4 1
( T
a
)
c
3 1
( T
a
)
c
2 1
( T
a
)
W i t h S T - D T
P
d
[ m W ]
c
i , 1
[ % ]
W
E
= 0 . 1 8 µm , L
E
= 5 µm
( a )
( b )
C a s e - A
F i n - 1 h e a t i n g
Fig. 5. Measured dissipated power-dependent (a) self-heating thermal resis-
tance of 1
st
, 2
nd
and 3
rd
finger, and (b) coupling coefficients obtained when
only 1
st
finger is heating. Extracted R
th,ii
(T
a
) and c
ij
(T
a
) are also marked.
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0
3 0 0
3 5 0
4 0 0
4 5 0
5 0 0
5 5 0
6 0 0
6 5 0
T
a
= 3 0 0 K
C a s e - B
α = 0 . 7 6
P
d
[ m W ]
T
i
[ K ]
W i t h S T - D T
F i n - 1 , f r o m e q . ( 3 )
F i n - 3 , f r o m e q . ( 3 )
F i n - 3 , f r o m e q . ( 1 )
W
E
= 0 . 1 8 µm , L
E
= 5 µm
Fig. 6. Extracted values of true finger temperatures (star-lines) for 1
st
and
3
rd
finger of a five finger device from B5T technology [17]. An underesti-
mated finger-3 temperature (crosses) obtained from a direct superposition of
measured self-heating and thermal coupling temperatures is also shown for
comparison.
Now we present the outcomes of our technique when
applied on the actual measured data. For this purpose we
chose a five-finger SiGe HBT transistor fabricated in B5T
technology from STMicroelectronics with six metal layers
at BEOL [17]. Each transistor finger is isolated from the
neighbouring fingers with shallow trenches and the whole
transistor structure is housed within deep trenches from all
four sides. The emitter fingers of the test structure (each
with A
E
= 0.18 × 5 µm
2
) are individually accessible with a
common collector while the bases are all connected to ground
as detailed in [8]. On-wafer measurements were carried out
on the test structure using a SUSS MicroTec probing station
equipped with a thermal chuck to obtain the self-heating
thermal resistances (R
th,ii
) and coupling coefficients (c
ij
)
following the techniques elaborated in [7] and [8], respectively.
Heating and sensing bias conditions of the test structures used
to extract coupling coefficients in our work are the same as in
[8]. Base-emitter voltage (V
BE
) of the heating finger is varied
from 0.4 to 0.98 V while a constant emitter current (I
E
) of
1 µA is forced at the sensing fingers. Collector-base voltage
(V
CB
) is kept constant at 0.5 V for all the fingers. Fig. 5(a)
shows the dissipated power-dependent R
th,ii
values obtained
from measurements (circles) for the first three fingers. Fig. 5(b)
4
shows the measured c
ij
values (circles) of the neighboring
fingers when finger-1 is heating. Subsequently, R
th,ii
(T
a
) and
c
ij
(T
a
) are extracted from the polynomial fit (solid lines) over
the measured data at P
d
= 0. Finally, using the extracted
values of R
th,ii
(T
a
), c
ij
(T
a
) and (1− α) we estimate the true
junction temperature (T
i
) following (3). Fig. 6 presents the
true temperatures of 1
st
and 3
rd
fingers obtained from mea-
surements following the proposed technique. The temperature
of the 3
rd
finger obtained from a direct superposition over the
measured data following (1) is also shown on the same plot for
comparison. It is observed that the conventional approach of
direct superposition underestimates the total junction temper-
ature (T
i
) by around 10% at around 40 mW power dissipation.
Such an underestimation will not only influence a self-heating
aware device design with insufficient data, but also the error
is not acceptable from the perspective of an accurate model
development.
IV. CONCLUSION
We presented an extraction technique to accurately estimate
the true finger temperature in a multi-finger transistor under
real operating condition when all the fingers are heating simul-
taneously. The proposed technique requires no additional mea-
surements than the conventional heat-sense based approach.
The methodology is first validated against TCAD simulations
and then applied on actual measured data. The effects of back-
end-of-line metal layers and different types of trenches present
within the transistor structure are automatically included in
this approach. However, any potential error that can originate
from measurement uncertainty is not ruled by the proposed
method. Therefore, one has to be careful while extracting
the R
th,ii
(T
a
) and c
ij
(T
a
) by means of non-linear fitting
and extrapolation towards zero dissipated power since the
measurement uncertainty tends to increase at lower dissipated
power. It is demonstrated that an application of superposition
of self-heating and thermal coupling directly on the measured
data underestimates the true finger temperature (under real
operating condition) by around 10% at around 40 mW power
dissipation.
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