T

f
T
School of Communication and
Information Technology
Royal Institute of Technology
Email: mwhus@kth.se
School of Communication and
Information Technology
Royal Institute of Technology
Email: arusu@kth.se
Abstract —This paper models the temperature dependence of f
T in 4H-SiC bipolar devices. The proposed model describes variation of the constituent parameters of f
T as a function of temperature. The model assumes complete ionization of dopants in 4H-SiC. However, this assumption hampers the model’s utility at temperatures below 300 ◦ C. The model was simulated at temperatures between 300 ◦ C and 700 ◦ C and a drop in f
T was observed. However, measurements are required to prove the correctness of the model or lack thereof.
TABLE I
D OPANT C ONCENTRATIONS OF THE S I C BJT
Region
Emitter
Base
Intrinsic Collector
Buried Collector
Concentration ( cm −
3
)
2 X 10
19
1 X 10
18
5 X 10
16
1 X 10
19
I. I NTRODUCTION
The commercial availability of high quality wafers for wide bandgap semiconductors (WBS), like silicon carbide
(SiC) and gallium nitride (GaN) has brought a revolution in high temperature electronics. Such electronics find myriad of applications in industries like automotive, deep well drilling, space exploration and turbine engine technology. WBS devices can be used to extract and propagate relevant information from extremely hot ambient to regions of temperature levels where ordinary electronics, with better processing capability, can work. Parameters of interest e.g, pressure levels in a furnace or seismic activity on some hot planetary surface can be extracted using sensors [1][2]. The extracted information is further amplified [3], digitized [4] and communicated to their destination. Common communication channels like copper wires or optical fibers are either unreliable or outright impossible to operate at very high temperature. Thus, necessitating the use of high temperature RF transceivers for wireless communication.
This paper presents an analytical model to characterize the bipolar devices required to build such high temperature RF transceivers. In particular, the developed model establishes a relationship between temperature and unity current gain bandwidth ( f
T
) of bipolar junction transistors (BJTs) in 4H-
SiC technology. Modeling the temperature dependence of f
T enables quick estimation of a device performance at a given temperature. It also provides useful insight into the behavior of device parameters as a function of temperature, thereby, enabling the possibility to improve a particular design aspect in subsequent generation of devices.
Mobility of charge carriers ( µ ) is a vital parameter in physics based models of bipolar devices. This electric field, temperature and dopant concentration dependent quantity has a strong impact on f
T
. Mobility influences f
T through resistivity
TABLE II
D IMENSIONS OF THE S I C BJT
Parameter
Emitter length
Emitter width
Emitter thickness
Base thickness
Intrinsic collector thickness
Buried collector thickness
Distance between the center of buried collector and collector contact
Abbreviation l e w e t e t b t ic t bc l x
Value ( µm )
70
24
0.75
0.3
0.5
1
15
(=1/qN µ ) and diffusion constant (= V
T
µ ), where N and V
T are dopant concentration and thermal voltage, respectively.
Numerous temperature and dopant concentration dependent low field mobility equations have been reported in the past.
A model describing f
T for 3C/6H heterojunction bipolar transistors (HBTs) [5] uses the experimental results from [6] to develop an empirical relationship for µ . Another model to predict DC and AC behavior of 3C-SiC BJTs [7] uses experimental results from Rosengreen [8] to describe µ . A recent work on 4H-SiC BJTs [9] presents a comparison between
Arora mobility model [10] and Balachandran mobility model
[11]. The latter of the two models demonstrated relatively better agreement with the measurement results. Consequently, mobility equations in [11] are chosen to model the temperature dependence of f
T in 4H-SiC BJTs as they are the most promising and relevant choice.
Furthermore, dopants in SiC have high ionization energies.
They completely ionize at temperatures around 300 ◦ C [9]. The effect of incomplete ionization of dopants is not incorporated into the proposed model because of an intrinsic ambiguity.

This ambiguity is associated with modeling incomplete ionization of nitrogen, a common donor dopant in 4H-SiC. Nitrogen in 4H-SiC can occupy two nonequivalent sites located in the cubic or hexagonal C-lattice, which implies the presence of two donor levels with different ionization energies. However, the model that accounts for incomplete ionization [12] assumes dopants to have a single energy level. The choice of one ionization energy level over the other can lead to different results [9] and hence the ambiguity. To avoid this situation, the proposed model assumes complete ionization of dopants in
SiC and consequently, should be used for temperatures at and above 300 ◦ C. A vertical npn 4H-SiC BJT, with the physical parameters presented in Table I and Table II, is used for the simulation of f
T
.
The paper is organized as follows. Section II describes the temperature dependence of individual delays in the transit time equation. In Section III, simulation results obtained from the model of f
T
Section IV.
are presented. Finally, conclusions are drawn in
II. T RANSIT T IME
The model of f
T is based on a well known set of transit time equations [13]. For negligible parasitic capacitance and emitter bulk resistance, these equations are given by:
τ ec
= τ e
+ τ b
+ τ c
+ τ d
= r e
C j,be
+ r e
C d,be
+ ( r e
+ r c
) C j,bc
+ x d
2 v sat
(1)
(2) where τ ec is the total transit delay from emitter to collector,
τ e is the base-emitter charging delay, τ b is the base transit delay, τ c is the base-collector charging delay and τ d is the base-collector depletion delay.
r e is the small signal input resistance of a BJT in common base configuration and r c is the collector resistance.
C j,be
, C d,be and C j,bc are the base-emitter junction, the base-emitter diffusion and the basecollector junction capacitance, respectively.
x d is the width of base-collector junction and v sat is the saturation velocity of
SiC at room temperature with a value equal to 2.10
7 cm/s.
Finally, f
T in terms of transit time is given by: f
T
=
1
2 πτ ec
(3) f
T varies with temperature because of the temperature dependence of the parameters given in (2). This section will describe the behavior of each of these parameters as a function of temperature.
A. Base-Emitter Charging Delay
Base-emitter charging delay is equal to the product of r e and C j,be
. The former parameter is a function of temperature through the common-emitter current gain ( β ), the ideality factor ( n ) and the thermal voltage. It is given by: r e
= nV
βI
T b
= nV
I c
T where I b and I c are the base and collector currents, respectively. The value of β for 4H-SiC BJT increases monotonically with temperature above 300 ◦ C [9][11]. However, to keep the model simple, I c
= βI b was assumed to be a constant.
Moreover, variation of n with temperature is rather weak in the active region with a value usually very close to one. Hence, n was also assumed to be a constant with a value equal to one. Both these assumptions left V
T as the only temperature dependent quantity in r e
.
The second parameter of T e is base-emitter junction capacitance. Base-emitter junction capacitance is given by:
C j,be
=
A be ǫ
(5)
W where W is the base-emitter junction width, A be
= w e
X l e is the base-emitter junction area and ǫ is the permittivity of SiC.
For an abrupt junction, W is given by: where N −
A
W and
=
N n i s
+
D
2 ǫ q
N −
A
N −
A
+ N +
D
N
+
D
( V bi −
V a
) (6) and for non-degenerate semiconductor, built-in potential ( V bi
) is given by:
V bi
= V
T ln
N −
A
N +
D n 2 i
(7) are the concentrations of ionized acceptor and donor dopants in base and emitter regions, respectively.
Moreover, V a is the applied reverse-biased voltage and n i is the intrinsic carrier concentration. Since complete ionization of
+ dopants was assumed, so N −
A
= N
A and N
D
= N
D
, where N
A and N
D are the dopant concentrations in the base and emitter, respectively. Under these assumptions, W becomes function of temperature indirectly through V
T and n i
; which, in turn, makes C j,be temperature dependent. Now, the temperature dependence of n i is given by [14]:
= p
N c
N v exp −
E g
/ 2 κ
B
T
(8) where T is the temperature in Kelvin and κ
B constant.
N c
, N v and E
G is the Boltzmann are the effective electron density of states, the effective hole density of states and the bandgap of the semiconductor, respectively. They are physical properties of semiconductor crystal and also temperature dependent. The formulas for N c
, N v and E
G as a function of temperature are given by [14] :
(4)
N c
=2
2 πm ∗ n
κ
B
T h 2
2 πm ∗ p
κ
B
T
N v
=2
E g
= E g
(0)
− h 2
αT 2
β + T
3 / 2
3 / 2
=
=
N
N c v
(300)
(300)
T
300
T
300
1 .
5
1 .
5
(9)
(10)
(11) where h is Planck’s constant, m ∗ n and m ∗ p are the effective masses of electrons and holes in a crystal lattice, respectively.
E g
(0) is the bandgap energy at 0K and N c
(300) , N v
(300) , α

TABLE III
C ONSTANTS FOR CALCULATING n i
AS A FUNCTION OF TEMPERATURE
120
115
E g
(0)
α
β
N c
(300)
N v
(300)
3.265 eV
6.5X1
0 −
4
1300
1.69X1
0 19 cm −
3
2.4X1
0 19 cm − 3
110
105
100
95
300 400 500 600
Temperature (
°
C)
700
250
200
150
I c
= 5mA
I c
= 10mA
I c
= 20mA
Fig. 2. Simulation results for τ b
100
50
0
300
B. Base Transit Time
C
400 500 600
Temperature (
°
C ) d,be
= t 2 b
2 D n
.
1 r e
700
Fig. 1. Simulation results for τ e at various biasing currents and β are material specific constants. For SiC, the values of these constants are given in Table III [15]. The temperature dependence of V
T
= κ
B
T /q is rather straightforward.
Finally, the temperature dependence of T e was obtained by simulating the product of (4) and (5) from 573 K (300 ◦ C) to
1000 K (727 ◦ C) at βI b
V. Values of I c
= I c
= 5, 10, 20 mA and V a
= 2.3
were chosen arbitrarily. However, the specific value of V a was chosen to be 2.3V to prevent C j,be from acquiring an imaginary value within the simulated temperature range. The situation with imaginary C j,be because V bi can potentially arise decreases with temperature, therefore, making
( V bi −
V a
) term in (6) negative. In this particular case, V bi decreased to a minimum value of 2.4 V at 1000K calculated using (7) . Hence, to keep (6) real, V a was chosen to be 0.1 V smaller than the minimum built-in potential. The simulation results are shown in Figure 1. Base-emitter charging delay increases monotonically with temperature for a given bias current. Moreover, it decreases with increasing bias current at a given temperature.
Base transit time is equal to the product of C d,be and r e
Under the assumptions that: there is a negligible recombination
.
of minority charge carriers in the base region, the base is uniformly doped, the base-collector junction is devoid of minority carriers, the emitter injection efficiency is close to one and the base-emitter voltage variations are much slower than the minority carrier lifetime of electrons in the base; the
C d,be can be approximated by [16]:
(12) where t b is the thickness and D n is the diffusion constant of electrons, respectively, in the quasi-neutral base region.
The depletion approximation in base-collector junction, for a nonzero collector current, implies infinite velocity of carriers in base-collector depletion region. However, electrons can not go faster than their saturation value, in turn, giving rise to an additional component of charge in the base region. This charge component can be accounted for, by modifying (12) as follows:
C d,be
= t 2 b
2 D n
+ t b v sat
.
1 r e
(13)
Under aforementioned assumptions, τ b is equal to the bracketed term in (13) and becomes independent of the bias current.
In Table I, it can be seen that the concentration of dopants in the base is orders of magnitude larger than the concentration of dopants in the intrinsic collector. Hence, variation of the collector-base voltage ( V cb
) will have insignificant influence on t b as long as it is not increased beyond a point where the basecollector depletion region extends into the buried collector. On the other hand, V cb should be sufficiently large to speedup the charge carriers in base-collector depletion region to their saturated value. The temperature dependence of v sat taken into consideration since available data on v sat was not of 4H-
SiC showed little variation with temperature [17]. Moreover, it was also assumed that the collector current levels are below the onset of KirK effect [18]. Under these conditions, τ b became function of temperature through D n
. The diffusion constant was simulated indirectly using Einstein equation ( D n
= V
T
µ n
), where µ n is the mobilty of electrons in the base and is calculated using Balachandran mobility model. Finally, τ b was simulated using bracketed term of (13). The simulation result is shown in Figure 2.
C. Base-Collector Depletion Delay
The time it takes for charge carriers to cross the basecollector depletion region is nonzero because of their finite velocity. For nonuniform velocity profile in base-collector depletion region, this delay is given by:
τ d
= x d
2 v sat
(14)

1.11
1.09
1.07
1.05
300 400 500 600
Temperature (
°
C)
700
Fig. 3. Simulation results for τ d at V cb
=8 V
13
12
11
10
9
300 400 500 600
Temperature (
°
C)
700
Fig. 4. Simulation results for τ c at I c
=20 mA and V cb
=8 V where x d is the width of base-collector depletion region and is given by (6), with N
A and N
D representing base and intrinsic collector dopant concentrations, respectively. To simulate this delay, a collector-base voltage equal to 8 V was chosen.
Its the maximum V cb that would ensure that base-collector depletion region would not extend into buried collector within the simulated temperature range, thereby, conforming with a similar assumption in Section II-B. Moreover, electric filed corresponding to V cb
= 8 V would have a minimum value of
160 kV/cm when x d would approach t ic
. This value is ample enough to produce velocity saturation.
Finally, (14) was used to simulate the base-collector depletion delay and the simulation result is shown in Figure 3. This delay is very small as compared to other constituent delays of transit time equation, especially for large device sizes; and it is presented here only for completion.
D. Base-collector charging delay
Base-collector charging delay is equal to the product of
( r c
+ r e
) and C j,bc
. The value of r c for a vertical BJT device is a sum of three components [19]: 1) vertical intrinsic collector resistance ( r i
), 2) lateral buried collector resistance ( r b
) and
3) collector contact resistance ( r n
).
r i was ignored because of its insignificant contribution to the total collector resistance.
Consequently, the total collector resistance for equidistant collector contacts on n sides is given by: r c
=
1 n
( r b
+ r n
) with r b and r n given by:
(15) r b
= ρ c l x t bc l e
(16) r n
=
ρ n
L
T
Z
(17) where ρ c is the resistivity of buried collector, ρ n is the specific contact resistance, L
T is the transfer length, and Z is the length of the collector contact.
The value of r c varies with temperature because its constituent components are temperature dependent.
r b is a function of temperature through ρ c
=1/(qN bc
µ n
), where µ n is the mobility of electrons in the buried collector region. Mobility
1.4
1.3
1.2
1.1
1
0.9
0.8
300 400 500 600
Temperature (
°
C)
700
Fig. 5. Simulated f
T at I c
=20 mA equations from Balachandran model were used to simulate µ n
.
Under the assumption of complete ionization of dopants, r b increases with temperature
The contact resistance, on the other hand, usually decreases with increase in temperature since the excited electrons at high temperature can cross the metal-semiconductor interface much more readily. The values of ρ n and L
T for a Nickel contact on n-type 4H-SiC were measured at various temperatures and dopant concentrations in [20] and they exhibited a decreasing trend with temperature. For donor dopant concentration of
1.10
19 cm −
3 and sheet thickness of 1 µ m, L
T decreases from 2.5
µ m at 25 ◦ C to 1.5
µ m at 500 ◦ C and ρ n varies from 7 Ω cm 2 at 25 ◦ C to 2.5
Ω cm 2 at 500 ◦ C. Rather than using fitting formulas for L
T and ρ n
, they were assumed to have a constant value equal to the average of their respective aforementioned values spread. This assumption would make r n a pseudo temperature independent quantity. Z was assumed to be equal to the emitter length.
The other temperature dependent parameter of τ c is C j,bc
. It was calculated using (5) with N
A and N
D equal to base dopant and intrinsic collector dopant concentrations, respectively.
Base-collector depletion area was assumed to be equal to baseemitter depletion area. The result of base-collector junction charging time for one sided collector contact at I c
= 20 mA and V a
= -8 V is shown in Figure 4. It shows an increasing trend with temperature owing to increase in r c and C j,bc
.

III. R ESULTS
All the constituent delays of transit time equation in (1) were modeled and simulated in Section II. Finally, (3) is used to simulate f
T and the simulation result is shown in Figure 5.
The decrease in f
T with temperature is dominated by τ b since all other delays are relatively small.
The assumption of negligible parasitic capacitance can potentially cause discrepancy between the simulated and the measured f
T
. For modeling f
T
, important parasitic capacitances in BJTs include extrinsic base-collector, fringing and interconnect capacitance. These capacitances are cumbersome to model and it is a common practice to neglect them.
However, the inclusion of parasitic capacitance is predicted to result in downward shift of f
T curve in Figure 5.
IV. C
ONCLUSION
This paper presented a temperature based model of f
T for
4H-SiC BJTs. Formulas for the constituent parameters of f
T were explained and simulated as a function of temperature.
Various assumptions were made to simplify the model so that it can facilitate a quick estimation of f
T
. Above temperatures at which dopants in 4H-SiC are completely ionized, the model is believed to predict f
T with a reasonable accuracy.
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A CKNOWLEDGMENT
The author would like to thank Knut and Alice Wallenberg foundation for funding this project.
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