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An Ecient Thermal Model for Multinger SiGe HBTs
Under Real Operating Condition
Nidhin K, Shubham Pande, Shon Yadav, Suresh Balanethiram, Deleep R
Nair, Sebastien Fregonese, Thomas Zimmer, Anjan Chakravorty
To cite this version:
Nidhin K, Shubham Pande, Shon Yadav, Suresh Balanethiram, Deleep R Nair, et al.. An Ecient
Thermal Model for Multinger SiGe HBTs Under Real Operating Condition. IEEE Transactions
on Electron Devices, Institute of Electrical and Electronics Engineers, 2020, 67 (11), pp.5069-5075.
�10.1109/TED.2020.3021626�. �hal-03015948�
1
An Efficient Thermal Model for Multifinger SiGe
HBTs under Real Operating Condition
Nidhin K, Shubham Pande, Shon Yadav, Suresh Balanethiram, Deleep R Nair, Sebastien Fregonese, Thomas
Zimmer, Senior Member, IEEE Anjan Chakravorty, Member, IEEE
Abstract—In this work, we present a simple analytical model
for electrothermal heating in multifinger bipolar transistors
under realistic operating condition where all fingers are heating
simultaneously. The proposed model intuitively incorporates the
effect of thermal coupling among the neighboring fingers in the
framework of self-heating bringing down the overall model com-
plexity. Compared to the traditional thermal modeling approach
for an n-finger transistor where the number of circuit nodes
increases as n
2
, our model requires only n-number of nodes.
The proposed model is scalable for any number of fingers and
with different emitter geometries. The model is validated with 3D
thermal simulations and measured data from STMicroelectronics
B4T technology. The Verilog-A implemented model simulates
40% faster than the conventional model in a transient simulation
of a five-finger transistor.
Index Terms—SiGe HBTs, multifinger transistor, electrother-
mal effect, thermal modeling, self-heating, thermal resistance.
I. INTRODUCTION
S
ILICON germanium heterojunction bipolar transistors
(SiGe HBTs) are popularly used as power amplifiers in
the RF front end modules. Power amplifiers are often expected
to have large emitter area in order to allow large amount of
currents. However, large emitter widths lead to a higher base
resistance resulting in a lower maximum oscillation frequency
(f
max
) of the device. Therefore, it has been a common practice
to partition a large emitter into smaller fingers each having
small enough emitter width (W
E
) leading to a multi-finger
transistor. Although in such a structure, each emitter finger
is electrically isolated by shallow trenches (ST), they are
thermally coupled through the common Silicon substrate.
Self heating is a serious problem in modern bipolar transis-
tors where lateral dimensions are significantly scaled down and
additional trench isolations are used. In case of multi-finger
transistors, additional thermal coupling from nearby fingers
further increases the device temperature. Conventionally, this
additional increment in temperature is captured by considering
Nidhin K, S. Pande, D. R. Nair, A. Chakravorty are with the Depart-
ment of Electrical Engineering, IIT Madras, Chennai 600036 India. email:
anjan@ee.iitm.ac.in.
S. Balanethiram is with the Department of Electronics and Communica-
tion Engineering, IIIT Tiruchirappali, Trichy 620015. email: sureshbalanethi-
ram@gmail.com
S. Yadav is with Globalfoundries, Bangalore.
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.
This work was supported in part by the EU under Project Taranto, in part
by ISRO project ELE/17-18/176/ISRO/ANJA and in part by Department of
Science and Technology, India, under Project EMR/2016/004726.The authors
would like to thank STMicroelectronics for providing the B9MW wafer.
thermal coupling effects from other nearby fingers. For the
calculation of the overall temperature at a finger, thermal
effect of each finger is considered at a time and finally all
effects are added up assuming the validity of superposition.
The temperature dependence of thermal conductivity makes
the temperature-power relationship nonlinear. This makes the
straightforward application of superposition to include both
the self-heating and thermal coupling effects in calculating
the overall finger temperature questionable. The real operating
condition within a power amplifier circuit exciting all the
fingers in a multi-finger transistor together instead of exciting
one finger at a time was elaborated in [1]. A state-of-the-
art static thermal model to cater the self-heating as well as
thermal coupling effects in an n-finger transistor requires n
2
number of nodes as reported in [2]. Besides, modeling thermal
coupling using voltage controlled-voltage-sources (VCVS) in
series with the self-heating resistances degrades the speed
performance of a thermal network as illustrated in [3].
In this work, we present an intuitive thermal model to
predict the overall temperature at each finger in a multifinger
transistor when all the fingers are excited together. Unlike the
conventional methods, our model uses no VCVS as the thermal
coupling effects are considered within the framework of self-
heating. Eventually, the use of superposition can be avoided
altogether resulting in just ‘n’ number of nodes for an n-finger
transistor. The proposed model also considers the temperature-
dependent thermal conductivity of the semiconductor material
and does not need to use any superposition theorem to calcu-
late the overall device temperature at any finger. The paper is
organized as follows. Section II presents the elaborate model
formulation with a hint towards model implementation. Sec-
tion III presents a detailed model validation against 3D TCAD
simulation and experimental data. The speed performance of
the proposed model is also compared with the conventional
thermal model. Finally, we present our conclusions in section
IV.
II. MODEL FORMULATION
Electrothermal effect in HBTs causes heat generation at
the base-collector junction. Most of the generated heat flows
down towards the substrate contact because of the high thermal
resistance offered by the upward path due to the interlayer
dielectric of the back-end-of-line (BEOL) structures as re-
ported in [1], [4], [5]. Therefore, the upward heat-flow is
neglected in the initial analysis and is added later by adding an
effective BEOL thermal resistance in parallel. In the front-end-
of-line (FEOL) portion, the base and emitter regions are also
Published in: IEEE Transactions on Electron Devices (Volume: 67 , Issue: 11 , Nov. 2020)
2
T
j
T
amb
W
E
θ θ
x
z
y
z = 0
z = H
Fig. 1. Cross sectional view of a single finger bipolar transistor structure with
no trench isolation showing heat source, heat sink, isothermal lines and the
imaginary boundaries.
neglected because of their negligible thickness. This allows us
to model the emitter finger as a rectangular heat source on a
semi-infinite substrate as shown in Fig 1. Note that the effect
of base and emitter region can always be clubbed with the
BEOL thermal resistance. For modeling purpose, it is assumed
that the bottom of the substrate is maintained at an ambient
temperature (T
amb
) and the substrate extends to infinity in the
lateral directions. In case of multifinger transistor, multiple
heat sources are to be considered simultaneously under real
operating condition. The modern application circuits such as
power amplifiers are equipped with temperature insensitive
bias techniques to ensure a near constant operating current
[6]–[9]. Adopting such biasing techniques in a multifinger
transistor yields similar amount of collector current (I
C
)
through all the fingers at a given collector-emitter voltage
(V
CE
). This leads to an identical amount of power dissipation
(P
diss
) in each finger (since P
diss
≈ I
C
× V
CE
). Accordingly
the modeling framework presented in this work assumed that
identical amount of P
diss
is generated at each finger.
Fig. 1 depicts the heat spread from a single heat source in
a semi-infinite environment. Vertical position (z) dependent
temperature variation, T (z), inside the system from the heat
source (at a temperature T
j
) to the heat sink (at T
amb
) is
indicated by the isothermal contours. In order to obtain a
simplified model for T (z) in such a system, a single heat-
spreading angle (θ) under conical approximation is used
defining the imaginary thermal boundary [10]. If P
diss
is the
power dissipated by the heat source, using average thermal
conductivity formulation [11], the temperature variation along
the z-direction (T (z)) can be written as
T (z) =
T
1−α
ref
+ P
diss
f
G
(z)
1 − α
β
1/(1−α)
(1)
where T
ref
= T
amb
at z = H (signifying a heat-sink at the
substrate contact). α, β are the parameters of temperature
dependent thermal conductivity in Silicon, κ
si
(T ) = βT
−α
[12]. f
G
(z) signifies the position dependent geometry factor
of the heat spread [13]. First, T (z) is evaluated at the heat sink
(z = H) with T
ref
= T
amb
in (1). Eventually, for calculating
T (z− ∆z), T (z) from the previous calculation is taken as T
ref
in (1). This is repeated to obtain the temperature variation for
all z in the system from heat sink to the heat source.
The above formulation can be generalized for an asymmetric
heat spread of a planar heat source with W as width in the
x-direction and L as length in the y-direction. In that case,
T
j1
T
j2
T
amb
s
θ θ
x
z
y
(a)
T
j1
T
j2
T
j3
T
amb
θ
θ
1
θ
1
θ
x
z
y
(b)
Fig. 2. Cross sectional view of (a) two-finger and (b) three-finger transistor
structure with no trench isolation showing heat source, heat sink, isothermal
lines and the imaginary boundaries.
the position-dependent cross-sectional area of the heat spread
can be written as [14]
A(x, y, z) = (W + (H − z)Θ
x
).(L + (H − z)Θ
y
) (2)
with
Θ
x
= tan(θ
x,l
) + tan(θ
x,r
) and Θ
y
= tan(θ
y,l
) + tan(θ
y,r
)
(3)
where H is the total height of the heat spread. Here θ
x,l
(θ
y,l
)
and θ
x,r
(θ
y,r
) are the heat spreading angles on the left and
right sides along W
E
(L
E
) respectively. Resultant geometry
factor can be obtained as
f
G
(z) =
Z
z
0
dz
0
A(x, y, z
0
)
=
ln
h
W (L+zΘ
x
)
L(W +zΘ
y
)
i
W Θ
y
− LΘ
x
. (4)
This equation presents a generic formulation and can be
used to estimate the geometry factor of any asymmetric heat
spread. Note that (4) need not be used in case of vertical heat
spread, where a simple ratio between the height and the cross
sectional area yields f
G
(z). Accordingly, the depth-dependent
temperatures (including that at the heat source, z = 0) can be
obtained using (1). Note that our aim is to find out the finger
temperature i.e., at z = 0 and as such there is no need to
compute any other T (z). However, this calculation is exercised
in order to showcase the model’s capability to capture the
underlying physics and accordingly gain a confidence on the
proposed model.
Estimation of the temperatures becomes difficult for a
system with two or more heat sources. Consider a case in
which two heat sources (having same area) each dissipating
a power of P
diss
are kept close to each other (with centre-
to-centre spacing s in between them) along the x-direction as
shown in Fig. 2(a). Similar to the single-heat-source structure,
total heat spread of the system can be shown by dashed lines
with spreading angle θ (which may be equal to 45
◦
) [10], [15].
For such a system, one can obtain the rise in temperature by
3
considering this overall heat spread and total power dissipation
of 2P
diss
. Alternatively, one can also obtain temperature rise at
one finger by defining individual heat spread. As the structure
is symmetric, one can intuitively say that T
1
(z) = T
2
(z).
Hence, the total heat spread in the x-z plane must be divided
into identical halves as shown by the dashed line in Fig. 2(a).
The vertical thermal boundary or zero spreading angle from
the z-point where the spread of the two heat sources intersect
essentially ensures equal heat flow volume in this case. This
divided heat-flow volume with power dissipation P
diss
yields
the same finger temperatures. In case of a system with three
heat sources as shown in Fig. 2(b), the thermal boundaries
of heat source in the middle are governed by two adjacent
heat sources. On the other hand, for the first and third heat
sources, only one thermal boundary depends on the adjacent
(middle) heat source and the other boundary is defined by a
heat spreading line of θ. Depending on the spreading angles
for individual heat sources, this framework can allow a higher
temperature rise at the second heat source compared to the
other two. However the first and third heat sources have an
equal rise in temperature (T
1
= T
3
) due to identical boundary
conditions. This is also evident from the identical isothermal
lines (obtained from TCAD simulation) under the first and
third heat sources as shown in Fig. 2(b). Under the second heat
source, these lines appear more crowded indicating a higher
temperature than the rest.
In order to accurately predict the temperature at each finger,
an effective heat spreading angle (θ
1
) has to be defined
between the adjacent heat sources as shown by the dashed lines
in Fig. 2(b). Prior to the intersection of the heat sources, the
spreading angles of all the heat sources were same θ. If θ
1
is
estimated accurately, separate geometry factors corresponding
to all heat sources can be obtained. Since all the fingers are
dissipating the same amount of power and since the heat flux
is continuous, the same amount of heat flux lines must be
crossing the x-y plane at the depth of intersection of the flux
lines from different fingers. Therefore, estimation of θ
1
can be
guided by a principle based on the symmetric total spreading
as elaborated below. In case of two-finger device, the first heat
spread in Fig. 2(a) is characterized by θ on one side and 0
◦
on
other. A similar approach follows for the second heat spread
as well. Therefore, both the spreads have effective spreading
angle of θ + 0
◦
. In case of three-finger device (Fig. 2(b)), heat
spread for the first finger has an angle of θ in one side and
−θ
1
in the other. Similarly for the second finger, the angle is
θ
1
in both the sides. Equating the effective total angles of the
two fingers results into θ − θ
1
= 2 × θ
1
yielding θ
1
= θ/3.
Therefore, the essential principle is to find out the total heat
spreading angle associated with each finger and to equate them
in order to obtain the effective heat spreading between the
adjacent heat sources. Extending this technique to a system
with n number of fingers, one can obtain a single equation
for heat spreading angles as
θ
i
=
θ
n
(n − 2i) i = 1, 2, 3, ...
n
2
for even n,
i = 1, 2, 3, ...
n − 1
2
for odd n.
(5)
Here θ is the angle of outer heat spreading for the corner
fingers. Note that the constant spreading angle framework does
not always ensure θ = 45
◦
as have been reported in several
cases [15]–[17]. While testing against TCAD simulation (as
presented in the next section) we have found that in most of
the geometries θ = 46
◦
yields excellent accuracy. Once the
heat spreading angles are obtained, they are used to compute
the cross-sectional area of each heat spread and the geometry
factors corresponding to each heating finger using (2) to (4)
and subsequently the temperature at each finger using (1).
In practice, each transistor finger is to be modeled using
separate electrical model where a thermal sub-circuit is avail-
able in order to capture the self-heating effect. Conventionally
the thermal coupling effect is captured by using a more com-
plicated thermal network that uses voltage-controlled voltage
sources [2]. In the present work, since we have computed the
geometry factor (f
G
) for each heating finger, the corresponding
thermal resistance is easily obtained and can be used within
the already existing self-heating network. This way one can get
rid of the thermal coupling network altogether and reduce the
overall node-count of the thermal network from n
2
(required
for conventional model) to just n for an n-finger transistor.
The resulting improvement in the simulation speed will be
discussed in the next section.
III. MODEL EVALUATION
A. Comparison with 3D TCAD simulation
First, we test our proposed model against 3D TCAD ther-
mal simulation results of multifinger SiGe HBT structures
having no trench isolation. Using Synopsys Sentaurus [18],
we simulated a single finger structure and four multifinger
structures with two, three, four and five fingers, respectively,
having an emitter area (A
E
) of 0.2×5 µm
2
for each finger.
Following the modeling framework, each heat source of area
0.2×5 µm
2
corresponding to each transistor finger is located
at the top of the Silicon substrate. A constant power of
30 mW is dissipated at each heating finger. This simulation
scenario is illustrated through a cross-sectional view (x-z
plane) of a 5-finger structure in Fig. 3. Each finger is isolated
from the neighboring finger with a centre-to-centre spacing
of s = 2.5 µm. The total simulated area of the substrate
is 300×300 µm
2
with a Silicon thickness of 100 µm. In the
TCAD simulation, values for the temperature coefficients of
thermal conductivity for Silicon, α = 1.263 and β = 2099
are used. For all the TCAD simulations, the bottom surface
is maintained at ambient temperature (T
amb
=300 K) and the
heat source injects a constant uniform heat flux into the
silicon substrate. All other surfaces, including the part of the
top surface not covered by the heat source, are considered
adiabatic.
Fig. 4(a) compares our model results for the z-dependent
temperature variation along the middle of the heat source
(T (z)) with TCAD simulation data corresponding to a single
finger structure and corner (first) fingers in case of multifinger
structures, without any trench isolation. Similarly, Fig. 4(b)
compares our model results for the T (z) variation with TCAD
simulation data corresponding to the internal fingers in multi-
finger structures without any trench isolation. Note that for
4
T
j2
T
j3
T
j4
T
j5
T
j1
θ
θ
1
θ
2
θ
2
θ
1
θ
z [µm]
x [µm]
Fig. 3. The imaginary boundary used in the model overlaid on the TCAD
simulated temperature profile in the x − z plane of a 5-finger device without
any trenches. θ
1
and θ
2
can be calculated using (5). For θ = 46
◦
, the resulting
values are θ
1
= 27.6
◦
and θ
2
= 9.2
◦
.
10
−4
10
−3
10
−2
10
−1
10
0
10
1
10
2
300
320
340
360
380
400
z [µm]
T (z) [K]
1-Finger
2-Finger
3-Finger
4-Finger
5-Finger
(a)
10
−4
10
−3
10
−2
10
−1
10
0
10
1
10
2
300
320
340
360
380
400
420
z [µm]
T (z) [K]
3-Finger pos 2
4-Finger pos 2
5-Finger pos 2
5-Finger pos 3
(b)
Fig. 4. T (z) variation at the middle of (a) corner (first) fingers (b) inner
fingers in multifinger structures without any trench isolation: a comparison
between TCAD simulation (symbols) and proposed model (lines).
optimum model fitting, we have used θ = 46
◦
for all the
structures. Accordingly the spreading angles of internal fingers
are calculated using (5). Excellent model agreement with the
TCAD simulated T (z) data for all the structures builds up our
confidence in the calculation of all the heat spreading angles
corresponding to each finger and subsequently the imaginary
boundaries. This motivates us to extend our modeling frame-
work for shallow trench isolated (STI) isolated multifinger
transistor structures as detailed below.
In case of multifinger structure with STI, each finger is
housed within a STI with a constant finger-edge to trench-
edge distance of 0.14 µm along both x- and y-directions.
Cross sectional view of an ST-isolated 5-finger HBT structure
simulated in TCAD is shown in Fig. 5. Corresponding to
STMicroelectronics B4T technology, the width (and height)
of STI is chosen as 0.45 µm (and 0.36 µm) [19]. The material
chosen for the trench has very low thermal conductivity (SiO
2
with κ = 0.014 W/cm-K) compared to that of the substrate
material (Silicon). The dimensions of the substrate, emitter
finger(s) and finger spacing remain identical with the previous
study carried out for no-trench devices. A constant power of
30 mW is dissipated at each finger for the TCAD simulation
as well as for the model. In this case, the heat flow volume of
each finger is divided into three regions as shown in Fig. 5.
The symmetric trapezoidal volume within STI, followed by
T
j2
T
j3
T
j4
T
j5
T
j1
θ
θ
1
θ
2
θ
2
θ
1
θ
STI STI
z [µm]
x [µm]
Fig. 5. The imaginary boundary used in the model overlaid on the TCAD
simulated temperature profile in the x − z plane of a 5-finger device with
STI. Inset shows the region near STI.
10
−4
10
−3
10
−2
10
−1
10
0
10
1
10
2
300
320
340
360
380
400
420
440
z [µm]
T (z) [K]
2-Finger
3-Finger
4-Finger
5-Finger
(a)
10
−4
10
−3
10
−2
10
−1
10
0
10
1
10
2
300
350
400
450
z [µm]
T (z) [K]
3-Finger pos 2
4-Finger pos 2
5-Finger pos 2
5-Finger pos 3
(b)
Fig. 6. T (z) variation at the middle of (a) corner fingers (b) inner fingers
in multifinger structures with STI: a comparison between TCAD simulation
(symbols) and proposed model (lines).
the cuboidal volume within STI, and the rest below STI. Note
that in order to evaluate the temperature profile in the section
below STI, one can apply the proposed model from the bottom
of STI to the heat sink in the same fashion as explained in the
no-trench case. Temperature profile within STI can be obtained
following the technique generally applied for calculating the
self-heating temperature since no effect of thermal coupling
from other fingers need to be considered within this heat flow
volume. A simple depth/area ratio is used to calculate f
G
(z)
inside the cuboidal section and (4) is used for the remaining
sections.
Fig. 6(a) compares our modeling results for the T (z)
variation with the 3D TCAD simulation data corresponding to
the corner fingers in case of multifinger structures with STI.
Thermal boundaries corresponding to the internal fingers from
the bottom of the STI are estimated using (5). Eventually using
θ and θ
i
in (4) and then using (4) in (1) the overall temperature
profile is obtained. In Fig. 6(b), we compare our modeling
results for T (z) variation with the TCAD simulation data
corresponding to the internal fingers in multifinger structures
with STI. Excellent model agreement is observed both in
Fig. 6(a) and (b). Although our model assumes the STI to
be a prefect thermal insulator, TCAD simulation considers a
slight heat leakage through the STI due to the non-zero κ of
SiO
2
. However, model correlation with TCAD data justifies
our approximation of the perfect insulator used for SiO
2
. This