Experimental Determination, Modeling, and Simulation of Nonlinear Thermal Effects in Bipolar Transistors under Static Conditions: A Critical Review and Update

Abstract

This paper presents a comprehensive overview of nonlinear thermal effects in bipolar transistors under static conditions. The influence of these effects on the thermal resistance is theoretically explained and analytically modeled using the single-semiconductor assumption. A detailed review of experimental techniques to extract the thermal resistance as a function of backside temperature and/or dissipated power from DC measurements is provided; advantages, underlying approximations, and limitations of all methods are clarified, and guidelines for their correct application are given. Accurate FEM thermal simulations of an InGaP/GaAs and a Si/SiGe HBT are performed to verify the accuracy of the single-semiconductor theory. The thermal resistance formulations employed in the most popular compact bipolar transistor models for circuit simulators are investigated, and it is found that they do not properly describe nonlinear thermal effects. Alternative implementations of the more accurate single-semiconductor theory are then proposed for the future versions of the compact models.

1. Introduction

Electrothermal (ET) effects are an important issue in modern high-frequency bipolar transistors in any technology, as they adversely impact the device behavior in multiple ways, namely, distortion in the I–V curves, which shifts the DC bias and shrinks the safe operating area [1,2,3,4], degradation of the low-frequency behavior [5], and even irreversible failure, likely to occur in multifinger devices due to thermally-induced current hogging [6,7].

Gallium arsenide (GaAs)-based heterojunction bipolar transistors (HBTs) like InGaP/GaAs and AlGaAs/GaAs, commonly considered the dominant technology for handset power amplifier design, are also plagued by strong ET effects due to the high operating power densities and high thermal resistances, the latter induced by (i) the low thermal conductivity of the GaAs substrate (one third of that of silicon), (ii) the lateral heat confinement due to mesa isolation, and (iii) the choice of interlevel dielectric films [4,6,8], ref. [9] and references therein, [10].

In silicon/silicon-germanium (Si/SiGe) HBTs for mm-wave and near-THz applications (wireless and optical communication, medical equipment, and automotive radars), ET effects are aggravated by the technology strategies employed to boost the frequency performance, i.e., (i) adoption of shallow and deep trenches filled with oxide or polysilicon coated with oxide, which suffer from low thermal conductivity and hamper the lateral propagation of the heat emerging from the power dissipation region; horizontal scaling (ii) of the emitter, which drives higher current (and power) density, and (iii) of the spacing between the intrinsic transistor and the surrounding trenches, which further inhibits the lateral heat flow. All these factors concur to increase the thermal resistances of single-finger transistors, which have been pushed into the thousands of K/W and beyond [11,12,13,14,15,16,17].

As the static thermal behavior of any device is well represented by its thermal resistance, accurately extracting this parameter from easy-to-perform measurements and describing it with simple formulations for compact transistor models is of utmost importance for many applications, e.g., simulation of the impact of ET effects on circuit performance, determination of optimum values of ballast resistors, and reliability estimations.

Unfortunately, the extraction and modeling tasks are complicated by the nonlinear nature of the heat conduction problem arising from the reduction of thermal conductivity of materials with increasing temperature (nonlinear thermal effects), which make the thermal resistance a monotonically growing function of the backside temperature and dissipated power. Effort has been dedicated to partially/fully account for such effects (i) in techniques for the experimental extraction of the thermal resistance, and (ii) in thermal resistance formulations conceived for compact transistor models. However, detailed discussions on advantages and limitations of the extraction techniques, as well as on the accuracy of the formulations, are still missing in literature.

This paper is intended to offer an extensive and critical overview of nonlinear thermal effects in modern high-frequency bipolar transistors from a multi-fold point of view. In Section 2, the separate impact on the thermal resistance of the backside temperature and dissipated power is theoretically explained and analytically described by resorting to the simple assumption of a single-semiconductor device. In Section 3, a detailed review of the available experimental techniques to extract the thermal resistance as a function of backside temperature and/or dissipated power is offered. In particular, the analysis focuses on indirect methods based on straightforward and cheap measurements of voltages/currents under DC conditions. The techniques are presented in a tutorial style with a unified nomenclature; advantages, limitations, and approximations are clarified, and simple guidelines for their correct application are provided. In Section 4, 3-D nonlinear thermal simulations of an InGaP/GaAs and a Si/SiGe HBT, chosen as case studies, are performed to obtain a reliable dependence of the thermal resistance as a function of backside temperature and dissipated power for both technologies. The resulting R_TH data are used to verify the accuracy of the single-semiconductor theory. In Section 5, the thermal resistance formulations embedded in compact transistor models available in circuit simulators are examined. It is observed that nonlinear thermal effects are either not considered or improperly accounted for, which affects the ET simulation results for medium/high temperatures. To tackle this issue, more accurate, yet numerically efficient and stable (not prone to convergence problems) implementations of the single-semiconductor theory are proposed for the future releases of compact transistor models. Conclusions are finally given in Section 6.

2. Thermal Resistance Dependence on Nonlinear Thermal Effects

The static thermal behavior of an electron device is effectively described by the self-heating thermal resistance R_TH [K/W], which represents an indicator of the inability of the component to remove heat from the power dissipation region (heat source). By specifically referring to a bipolar transistor, R_TH is defined as

$$R_{TH}=\frac{T_j-T_B}{P_D}=\frac{\Delta T_j}{P_D}$$

(1)

where T_j is the temperature averaged over the base-emitter junction [K] (also called junction temperature), T_B is the backside (or baseplate, or ambient) temperature [K] that can be assigned through a thermochuck/heater, and P_D [W] is the dissipated power, given by

$$P_D=I_B\cdot V_{BE}+I_C\cdot V_{CE}=I_E\cdot V_{BE}+I_C\cdot V_{CB}$$

(2)

In (2), all terms have their customary meaning, namely, I_B, I_C, I_E are the base, collector, and emitter currents, respectively, while V_BE, V_CE, V_CB are the externally applied base-emitter, collector-emitter, and collector-base voltages, respectively. The thermal resistance is also referred to as (i) junction temperature rise above T_B normalized to P_D or (ii) static thermal response to power dissipation.

R_TH depends on (i) device and heat source geometry, (ii) thermal conductivities of the materials crossed by the heat emerging from the source, and (iii) boundary conditions. A transistor with a horizontally- and/or vertically-scaled heat source suffers from a higher R_TH (e.g., [13,18,19]), as for the same P_D the dissipated power density is higher, and therefore T_j is higher. In a similar fashion, the adoption of materials with low thermal conductivities hinders the heat flow, thus leading to an increase in R_TH.

Additionally, it must be considered that the thermal conductivities k [W/µmK] of semiconductors and metals in a transistor decrease with temperature, thereby lowering the heat transfer efficiency. The thermally induced k degradation introduces a nonlinearity in the heat conduction equation, and the resulting effects are referred to as nonlinear thermal effects. It is well known that in a practically relevant temperature range the k of many semiconductors of interest reduces with a power law

$$k\left(T\right)=k\left(T_0\right)\cdot {\left(\frac{T}{T_0}\right)}^{-\mathsf{\alpha }}$$

(3)

where the reference temperature T₀ is equal to 300 K and α > 0, while for the metals the decrease is well described by the following linear model:

$$k\left(T\right)=k\left(T_0\right)-\mathsf{\beta }\cdot \left(T-T_0\right)$$

(4)

where β > 0. Commonly accepted values for k(T₀), α, β corresponding to the most relevant semiconductors and metals are reported in

Table 1. Quoted values of parameters in (3), (4) for the most important semiconductors and metals in electron devices.

Material	k(T0) [W/µmK]	α	β [K−1]
Si	1.422 × 10−4 [20] 1.45 × 10−4 [21] 1.48–1.54 × 10−4 [22] and references therein 1.56 × 10−4 [23]	1.25 [24] 1.3 [25], also reported in [26] 1.33 from elaboration of data in [20], and [27] from elaboration of data in [21] 1.4 [28] from elaboration of data in [23] 1.65 [22], although 1.3 seems better (see Ref. [2] in [29])	-
GaAs	0.44 × 10−4 [21] 0.37–0.46 × 10−4 [22] and references therein 0.55 × 10−4 [30]	1.25 [22] and references therein 1.27 [28] from elaboration of data in [30]	-
GaN	1.25–1.5 × 10−4 [22] and references therein	0.43 [22] and references therein	-
InP	0.68 × 10−4 [22] and references therein, [31] 0.696 × 10−4 [32]	1.4 [22] and references therein 1.48 [31], also reported in [28]
4H-SiC	3.7 × 10−4 [33,34]	1.29 [35]	-
6H-SiC	3.2–4.9 × 10−4 [22] and references therein 3.87 × 10−4 for the conductivity normal to the c axis; a value 30% lower for the conductivity parallel to the c axis [36] 4.9 × 10−4 [33,34,37]	1.29 [35] 1.49 for the conductivity normal to the c axis [36]	-
Al	2.39 × 10−4 from elaboration of data in [38]	-	2.1 × 10−8 from elaboration of data in [38]
Cu	3.97 × 10−4 from elaboration of data in [38]	-	5.2 × 10−8 from elaboration of data in [38]

.

The device temperature increases—and the thermal conductivities decrease—for two distinct physical mechanisms: (i) the raise in backside temperature T_B (nonlinear thermal effect due to the backside temperature) and (ii) the increase in dissipated power P_D (nonlinear self-heating effect). Consequently, R_TH is a monotonically growing function of both T_B and P_D and should be more properly formulated as R_TH(T_B,P_D), which inherently accounts for nonlinear thermal effects. This means that the observed dependence on T_B and P_D does not come from the explicit presence of these quantities on the RHS of (1), which is the basic definition of thermal resistance, but implicitly comes from the k reduction with increasing temperature [28].

Hereafter, R_TH00 will conventionally denote the thermal resistance of the bipolar transistor at T_B = T₀ and very low P_D (ideally for P_D → 0 W, i.e., in the absence of the nonlinear self-heating effect), that is,

$$R_{TH00}=R_{TH}\left(T_B=T_0,P_D\to 0\right)$$

(5)

In simple terms, R_TH00 represents the thermal resistance of the transistor if the thermal conductivities of all materials are equal to their k(T₀) value.

The following analysis is aimed at providing an illustrative overview of the R_TH dependence on T_B and P_D. Let us consider an ideal device homogeneously composed of a specific semiconductor, the thermal conductivity of which obeys (3) (single-semiconductor assumption), and let us first assume that the backside temperature T_B is equal to T₀. The typical approach used to include the impact of the nonlinear self-heating effect on the junction temperature T_j (and on R_TH) is the Kirchhoff transformation [39,40,41]. Let us define T_j00 as the junction temperature given by

$$T_{j00}=T_0+R_{TH00}\cdot P_D$$

(6)

where P_D is assumed to be very low. In this case, the temperature T_j0P (>T_j00) accounting for the nonlinear effect due to P_D can be calculated as [24,27,42,43,44]

$$T_{j0P}=T_0\cdot {\left[1-\left(\mathsf{\alpha }-1\right)\cdot \frac{T_{j00}-T_0}{T_0}\right]}^{\frac{-1}{\mathsf{\alpha }-1}}=T_0\cdot {\left[1-\left(\mathsf{\alpha }-1\right)\cdot \frac{R_{TH00}\cdot P_D}{T_0}\right]}^{\frac{-1}{\mathsf{\alpha }-1}}$$

(7)

It must be remarked that (7) also holds in the more general case of a multilayered structure where the thermal conductivities k(T₀) of the layers are different but share the same power dependence on temperature (3), that is, the same value for the power factor α [26]. The thermal resistance is obtained as

$$R_{TH0P}=R_{TH}\left(T_B=T_0,P_D\right)=\frac{T_{j0P}-T_0}{P_D}=\frac{T_0}{P_D}\cdot \left\{{\left[1-\left(\mathsf{\alpha }-1\right)\cdot \frac{R_{TH00}\cdot P_D}{T_0}\right]}^{\frac{-1}{\mathsf{\alpha }-1}}-1\right\}$$

(8)

and is obviously higher than the T_B = T₀ zero-power thermal resistance R_TH00.

Now let us consider an arbitrary backside temperature T_B ≥ T₀. The R_TH increase due to nonlinear thermal effects concurrently induced by T_B and P_D can be described as follows. Since for this ideal single-semiconductor device the thermal resistance is inversely proportional to the thermal conductivity [13,17,18,19,28,29,39,45,46], then the R_TH growth only due to T_B > T₀ (for P_D → 0 W, i.e., without the nonlinear self-heating effect) can be described by the power law [17,28,29,45]

$$R_{THB0}=R_{TH}\left(T_B,P_D\to 0\right)=R_{TH00}\cdot {\left(\frac{T_B}{T_0}\right)}^{\mathsf{\alpha }}$$

(9)

It is noteworthy that also for real devices R_THB0 is higher than R_TH00 since, as mentioned before, the thermal conductivities of semiconductors and metals at T_B are lower than the reference values at T₀. Let us designate as T_jB0 the junction temperature given by

$$T_{jB0}=T_B+R_{THB0}\cdot P_D$$

(10)

for very low P_D. The Kirchhoff transformation can again be used to account for the nonlinear self-heating effect [17,28,29,43,45]

$$T_j=T_B\cdot {\left[1-\left(\mathsf{\alpha }-1\right)\cdot \frac{T_{jB0}-T_B}{T_B}\right]}^{\frac{-1}{\mathsf{\alpha }-1}}=T_B\cdot {\left[1-\left(\mathsf{\alpha }-1\right)\cdot \frac{R_{THB0}\cdot P_D}{T_B}\right]}^{\frac{-1}{\mathsf{\alpha }-1}}$$

(11)

and the thermal resistance can be calculated as

$$R_{TH}\left(T_B,P_D\right)=\frac{T_j-T_B}{P_D}=\frac{T_B}{P_D}\cdot \left\{{\left[1-\left(\mathsf{\alpha }-1\right)\cdot \frac{R_{THB0}\cdot P_D}{T_B}\right]}^{\frac{-1}{\mathsf{\alpha }-1}}-1\right\}$$

(12)

which can also be written as

$$R_{TH}\left(T_B,P_D\right)=\frac{T_B}{P_D}\cdot \left\{{\left[1-\left(\mathsf{\alpha }-1\right)\cdot \frac{R_{TH00}\cdot P_D}{T_B\cdot {\left(\frac{T_0}{T_B}\right)}^{\mathsf{\alpha }}}\right]}^{\frac{-1}{\mathsf{\alpha }-1}}-1\right\}$$

(13)

where use has been made of (9).

As an illustrative example, we evaluated the R_TH behavior as a function of T_B and P_D with (13) assuming that the device is homogeneously composed of GaAs (α = 1.25), and that its T_B = T₀ zero-power thermal resistance R_TH00 is equal to 1000 K/W. Figure 1 shows R_TH(T_B,P_D) vs. P_D and vs. T_j at different T_B values. It is important to note that, by keeping T_j constant while increasing T_B (which can be obtained by reducing P_D), R_TH increases. This contrasts with the common—and wrong—belief that R_TH only depends on T_j, and can be easily explained by considering that, for higher T_B values, all materials encountered by the heat flowing through the device will suffer from a higher temperature (falling in the range T_B to T_j), and thus from a lower thermal conductivity. Hence, accounting for nonlinear thermal effects with an R_TH that only depends on T_j in a compact bipolar transistor model is an unfortunate choice, as the underlying physics of the problem is incorrectly represented.

Starting from the single-semiconductor theory, Walkey et al. [28], and subsequently Huszka et al. [17], resort to mathematical approximations to get an R_TH(T_B,P_D) formulation simpler than (13). First, the power law (3) is linearized as

$$k\left(T_B\right)\approx \frac{k\left(T_0\right)}{1+\frac{\mathsf{\alpha }}{T_0}\cdot \left(T_B-T_0\right)}$$

(14)

whence the zero-power thermal resistance at T_B can be expressed as

$$R_{THB0}\approx R_{TH00}\cdot \left[1+\frac{\mathsf{\alpha }}{T_0}\cdot \left(T_B-T_0\right)\right]=R_{TH00}\cdot \left[1+{\zeta }_B\cdot \left(T_B-T_0\right)\right]$$

(15)

which shows a linear increase of R_THB0 with T_B, also assumed in [47]. The R_TH(T_B,P_D) formulation given by (12), which also includes the nonlinear self-heating effect, is then expanded in a Taylor series that retains only the first two terms, thus leading to

$$R_{TH}\left(T_B,P_D\right)\approx R_{THB0}\cdot \left(1+\mathsf{\alpha }\cdot \frac{R_{THB0}}{2\cdot T_B}\cdot P_D\right)=R_{THB0}\cdot \left[1+{\zeta }_P\left(T_B\right)\cdot P_D\right]$$

(16)

By substituting (15) into (16),

$$R_{TH}\left(T_B,P_D\right)=R_{TH00}\cdot \left[1+{\zeta }_B\cdot \left(T_B-T_0\right)\right]\cdot \left[1+{\zeta }_P\left(T_B\right)\cdot P_D\right]$$

(17)

where the α-sensitive parameters ζ_B and ζ_P(T_B) account for the separate dependence on T_B and P_D, respectively. A linear dependence of R_TH0P on P_D was also proposed in [2,48].

Walkey et al. also suggest a fast procedure to determine ζ_B and ζ_P(T_B) from a few experimentally extracted R_TH values. In particular, ζ_B can be evaluated from the zero-power thermal resistances R_TH00 and R_THB10 = R_TH(T_B1,P_D → 0) as

$${\zeta }_B=\frac{1}{T_{B1}-T_0}\cdot \left(\frac{R_{THB10}}{R_{TH00}}-1\right)$$

(18)

while ζ_P(T_B) can be calculated for each T_B from two thermal resistances R_TH(T_B,P_D1) and R_TH(T_B,P_D2) as

$${\zeta }_P\left(T_B\right)=\frac{\frac{R_{TH}\left(T_B,P_{D2}\right)}{R_{TH}\left(T_B,P_{D1}\right)}-1}{P_{D2}-\frac{R_{TH}\left(T_B,P_{D2}\right)}{R_{TH}\left(T_B,P_{D1}\right)}\cdot P_{D1}}$$

(19)

3. Review of Experimental R_TH Extraction Techniques Accounting for Nonlinear Thermal Effects

The literature is populated by many techniques for extracting the thermal resistance of bipolar transistors from simple DC measurements (a complete review being given in [49]), which can be subdivided into (i) approaches using the relation between an easily-measurable electrical parameter (like V_BE) and T_j as a thermometer, and (ii) approaches that do not use the thermometer (e.g., those based on intersection points). This section aims to provide a systematic and comprehensive overview of the methods explicitly accounting for the influence of nonlinear thermal effects, namely, those determining the R_TH dependence on T_B (for T_B ≥ T₀) and/or P_D, or on T_j. The techniques are explained in detail in chronological order using a unified nomenclature; advantages/drawbacks are clarified for each of them, and guidelines for their correct use are provided.

3.1. Bovolon et al.

The technique proposed by Bovolon et al. [50] is “differential” and can be considered as an extension of the approach published by Dawson et al. [51].

Let us focus on the “internal” (junction) base-emitter voltage V_BEj, given by

$$V_{BEj}=V_{BE}-R_B\cdot I_B-R_E\cdot I_E=V_{BE}-\frac{R_B}{{\mathsf{\beta }}_F}\cdot I_C-R_E\cdot I_E\approx V_{BE}-\left(\frac{R_B}{{\mathsf{\beta }}_F}+R_E\right)\cdot I_C=V_{BE}-R_{EB}\cdot I_C$$

(20)

where R_B and R_E are the parasitic base and emitter series resistances, respectively, and β_F is the common-emitter current gain. From simple theoretical considerations, it can be demonstrated that in the absence of Early, high-injection (Kirk), and avalanche effects, under forward mode V_BEj decreases almost linearly with the junction temperature T_j at a given I_C ≈ I_E [13]

$$V_{BEj}\left(T_j\right)=V_{BEj}\left(T_0\right)-\mathsf{\varphi }\cdot \left(T_j-T_0\right)$$

(21)

By making use of (20), in the assumption of a temperature-insensitive R_EB, (21) becomes

$$V_{BE}\left(T_j\right)=R_{EB}\cdot I_C+V_{BEj}\left(T_0\right)-\mathsf{\varphi }\cdot \left(T_j-T_0\right)=V_{BE}\left(T_0\right)-\mathsf{\varphi }\cdot \left(T_j-T_0\right)$$

(22)

which states that the linear behavior also holds for the externally applied V_BE. From (22), it is clear that the linear behavior also takes place locally, i.e., around a certain junction temperature T_j*:

$$V_{BE}\left(T_j\right)=V_{BE}\left(T_j*\right)-\mathsf{\varphi }\cdot \left(T_j-T_j*\right)$$

(23)

Exploiting the definition of thermal resistance (1), (23) turns into

$$V_{BE}\left(T_B,P_D\right)=V_{BE}\left(T_j*\right)-\mathsf{\varphi }\cdot \left[T_B+R_{TH}\left(T_B,P_D\right)\cdot P_D-T_j*\right]$$

(24)

If (24) is applied to another backside temperature T_B + ΔT_B at the same dissipated power P_D,

$$V_{BE}\left(T_B+\Delta T_B,P_D\right)=V_{BE}\left(T_j*\right)-\mathsf{\varphi }\cdot \left[T_B+\Delta T_B+R_{TH}\left(T_B+\Delta T_B,P_D\right)\cdot P_D-T_j*\right]$$

(25)

By subtracting (25) from (24),

$$V_{BE}\left(T_B,P_D\right)-V_{BE}\left(T_B+\Delta T_B,P_D\right)=\mathsf{\varphi }\cdot \Delta T_B+\mathsf{\varphi }\cdot \left[R_{TH}\left(T_B+\Delta T_B,P_D\right)-R_{TH}\left(T_B,P_D\right)\right]\cdot P_D$$

(26)

If ΔT_B is chosen sufficiently low, (26) reduces to

$$V_{BE}\left(T_B,P_D\right)-V_{BE}\left(T_B+\Delta T_B,P_D\right)\approx \mathsf{\varphi }\cdot \Delta T_B$$

(27)

from which coefficient ϕ describing the temperature dependence of V_BE (the thermometer) can be determined as

$$\mathsf{\varphi }=\frac{V_{BE}\left(T_B,P_D\right)-V_{BE}\left(T_B+\Delta T_B,P_D\right)}{\Delta T_B}$$

(28)

Subsequently, by applying (24) to another dissipated power P_D + ΔP_D at the same backside temperature T_B,

$$V_{BE}\left(T_B,P_D+\Delta P_D\right)=V_{BE}\left(T_j*\right)-\mathsf{\varphi }\cdot \left[T_B+R_{TH}\left(T_B,P_D+\Delta P_D\right)\cdot \left(P_D+\Delta P_D\right)-T_j*\right]$$

(29)

By subtracting (29) from (24),

$$V_{BE}\left(T_B,P_D\right)-V_{BE}\left(T_B,P_D+\Delta P_D\right)=\mathsf{\varphi }\cdot \left[R_{TH}\left(T_B,P_D+\Delta P_D\right)\cdot \left(P_D+\Delta P_D\right)-R_{TH}\left(T_B,P_D\right)\cdot P_D\right]$$

(30)

If ΔP_D is chosen sufficiently small, (30) reduces to

$$V_{BE}\left(T_B,P_D\right)-V_{BE}\left(T_B,P_D+\Delta P_D\right)\approx \mathsf{\varphi }\cdot R_{TH}\left(T_B,P_D\right)\cdot \Delta P_D$$

(31)

and finally, combining (31) and (28),

$$R_{TH}\left(T_B,P_D\right)=\frac{V_{BE}\left(T_B,P_D\right)-V_{BE}\left(T_B,P_D+\Delta P_D\right)}{\mathsf{\varphi }\cdot \Delta P_D}=\frac{\frac{V_{BE}\left(T_B,P_D\right)-V_{BE}\left(T_B,P_D+\Delta P_D\right)}{\Delta P_D}}{\frac{V_{BE}\left(T_B,P_D\right)-V_{BE}\left(T_B+\Delta T_B,P_D\right)}{\Delta T_B}}$$

(32)

By repeating the extraction for different values of backside temperature T_B and dissipated power P_D, R_TH can be determined as a function of T_B and P_D without imposing analytical assumptions on both dependences.

For each (T_B,P_D) couple, the approach requires the measurement of two I_B-constant I_C−V_CE characteristics and of the corresponding V_BE−V_CE curves at T_B and T_B + ΔT_B. On the first I_C−V_CE characteristic, two points with close dissipated powers P_D and P_D + ΔP_D are selected, and the related V_BE(T_B,P_D), V_BE(T_B,P_D + ΔP_D) values are used for the calculation of the numerator of (32). Then, the point with dissipated power P_D has to be identified on the second characteristic, and the associated V_BE(T_B + ΔT_B,P_D) allows the calculation of the denominator of (32); if this point has not been exactly measured, an interpolation between the adjacent points is required.

The technique must be cautiously applied for multiple reasons: (i) ΔT_B should be chosen sufficiently small to ensure (27); (ii) ΔP_D should be chosen sufficiently low to lead to (31); (iii) moreover, the collector currents corresponding to P_D and P_D + ΔP_D should be quite similar, as coefficient ϕ is dependent on I_C [1,3,6,8,11,13,52,53]; (iv) finally, it must be remarked that both the ϕ (28) and R_TH (32) calculations are determined by a difference between V_BE values measured in two points only, which can in principle jeopardize the accuracy of the results. Due to all the above aspects, the practical application of the technique can be judged as nontrivial.

3.2. Yeats

The technique conceived by Yeats [45] can also be reviewed as an improved version of that presented by Dawson et al. [51]. First, some V_BE–V_CE (V_CE being given by the sum of the forced V_CB and the measured V_BE) characteristics are measured at a given I_E and various backside temperatures T_B. Through a quadratic polynomial, the V_BE values corresponding to V_CE = 0 V (P_D = 0 W and thus T_j = T_B) are extrapolated, so that the V_BE–T_j thermometer is defined at the assigned I_E and then inverted (turned into T_j–V_BE) and described by means of a quadratic polynomial with I_E-dependent coefficients. Hence, any V_BE is associated to the corresponding T_j value, and the experimental I_E-constant T_j–P_D curves at various T_B are straightforwardly determined. Consequently, the experimental R_TH(T_B,P_D) against P_D is known for each T_B from (1) without imposing assumptions on these dependences. This technique is often applied to GaAs-based HBTs, especially for T_j estimates used in reliability.

Yeats proposes to adopt the single-semiconductor theory shown in Section 2 to analytically describe the measured R_TH(T_B,P_D) data; hence, he uses (13), where (9) models the influence of T_B, and the Kirchhoff transformation is adopted to include the sensitivity to P_D. An optimization routine (referred to as “brute-force 2-D search”) is exploited, which allows achieving R_TH00 and α that favor the best aggregate match between all the experimental R_TH data and (13). Here Yeats remarks that, as the correlated parameters R_TH00 and α are simultaneously optimized, R_TH00 might differ from the extracted T_B = T₀ zero-power thermal resistance, although the calibrated (R_TH00,α) couple well describes the measured data; conveniently, the numerical analysis reported in Section 4 shows that the optimized R_TH00 is very close to the simulated value over wide T_B and P_D ranges, regardless of the HBT technology.

Although the technique seems to be simple and effective, there are two questionable aspects to point out: the first is the extrapolation procedure needed to calibrate the V_BE–T_j thermometer, the accuracy of which should be further investigated, and the second is the direct use of the thermometer to find the T_j, and thus the R_TH(T_B,P_D), associated to a specific V_BE, which makes the R_TH results too sensitive to the precision of the V_BE measurements (other thermometer-based techniques do not directly use the thermometer to evaluate R_TH, but, like Bovolon et al. [50], exploit another step requiring the “differential” calculation ΔV_BE/ΔP_D or even the derivative dV_BE/dP_D; this alleviates the impact of inaccuracies on the V_BE measurements).

3.3. Marsh

In contrast to [45,50], which require a calibrated thermometer, the technique conceived by Marsh [47] belongs to a category of indirect approaches (the word “direct” in the title could mislead) based on intersections between curves, and allows a relatively simple extraction of R_THB0 as a function of T_B. Marsh proposes to measure three I_C–V_CE characteristics by forcing the same I_B at three backside temperatures T_B1 < T_B2 < T_B3. In HBTs, a negative-differential-resistance (NDR) region arises (I_C reduces with V_CE) under forward mode due to the negative temperature coefficient (NTC) of the current gain β_F, in turn induced by the wider bandgap of the emitter compared to that of the base. In the NDR, three points can be identified, which correspond to the same I_C, but are associated to different V_CE (V_CE1 > V_CE2 > V_CE3), and thus different dissipated powers P_D (P_D1 > P_D2 > P_D3). As I_C and I_B are the same at these points, β_F is the same, and, since β_F only depends on T_j, T_j is also the same. Consequently, using (1), the following equations hold in the points:

$$\begin{array}{l}{T}_j=T_{B1}+R_{TH}\left(T_{B1},P_{D1}\right)\cdot P_{D1}\\ T_j=T_{B2}+R_{TH}\left(T_{B2},P_{D2}\right)\cdot P_{D2}\\ T_j=T_{B3}+R_{TH}\left(T_{B3},P_{D3}\right)\cdot P_{D3}\end{array}$$

(33)

Marsh assumes that R_TH is a linearly increasing function of T_B, while neglecting its P_D dependence, that is,

$$R_{TH}\left(T_B,P_D\right)\approx R_{THB0}=A_{Marsh}+B_{Marsh}\cdot T_B$$

(34)

By substituting (34) into (33), a three-equation system with three unknowns (A_Marsh, B_Marsh, T_j) is obtained

$$\begin{array}{l}{T}_j=T_{B1}+\left(A_{Marsh}+B_{Marsh}\cdot T_{B1}\right)\cdot P_{D1}\\ T_j=T_{B2}+\left(A_{Marsh}+B_{Marsh}\cdot T_{B2}\right)\cdot P_{D2}\\ T_j=T_{B3}+\left(A_{Marsh}+B_{Marsh}\cdot T_{B3}\right)\cdot P_{D3}\end{array}$$

(35)

from which it is possible to determine the junction temperature T_j at the selected points, as well as R_TH as a function of T_B as given by (34). Such a linear relation can also be written as

$$R_{THB0}=A_{Marsh}+B_{Marsh}\cdot T_0+B_{Marsh}\cdot \left(T_B-T_0\right)=R_{TH00}+B_{Marsh}\cdot \left(T_B-T_0\right)$$

(36)

It is worth noting that in principle this technique can also be applied to Si bipolar junction transistors (BJTs), which show a positive-differential-resistance (PDR) behavior in the I_B-constant I_C–V_CE curves due to the positive temperature coefficient (PTC) of β_F, in turn induced by the narrower bandgap of the highly-doped emitter compared to that of the base; however, in this case the R_TH extraction would be affected by the Early effect, which is not accounted for in the approach and can be misinterpreted as an additional self-heating.

From the numerical investigation reported in Section 4, it will be found that the R_THB0 vs. T_B behavior is almost linear in the T_B span 300 to 450 K. However, attention must be paid during the practical application of this technique, as P_D1, P_D2, P_D3 must be chosen sufficiently low to prevent a significant nonlinear self-heating effect.

3.4. Paasschens et al.

Paasschens and his co-authors [29] invoke the single-semiconductor theory shown in Section 2, and, like [17,28,45], consider (9) for the dependence of the zero-power thermal resistance R_THB0 on T_B, and the Kirchhoff-based (13) for the complete R_TH(T_B,P_D) formulation. The authors only perform low-power measurements of R_THB0 at various T_B values with the technique proposed in [54], which can be categorized as a “differential” approach not based on a thermometer. Once the thermal resistance R_TH00 is extracted, parameter α is optimized by comparison between (9) and experimental data, and then also used in (13). In other words, Paasschens et al. do not measure the impact of the nonlinear self-heating effect on R_TH, which is instead done in [45,50].

The technique presented in [54] can be summarized as follows. The transistor is in a common-emitter configuration with the backside at temperature T_B, and the forced V_BE and V_CE are chosen to keep P_D low.

Let us derive the R_TH expression used in the procedure. The collector current I_C is a function of the independent variables V_BE, V_CE, T_j, that is,

$$I_C=I_C\left(V_{BE},V_{CE},T_j\right)$$

(37)

If V_BE is kept constant, the variation of I_C due to small variations of V_CE and T_j (the latter dictated by variations of V_CE or T_B) is given by

$$d{I}_C=\frac{\partial I_C}{\partial V_{CE}}\cdot d{V}_{CE}+\frac{\partial I_C}{\partial T_j}\cdot d{T}_j$$

(38)

which, neglecting the Early effect, reduces to

$$d{I}_C\approx \frac{\partial I_C}{\partial T_j}\cdot d{T}_j$$

(39)

If dT_j is due to dT_B, (39) becomes

$${d{I}_C|}_{V_{CE}}\approx \frac{\partial I_C}{\partial T_j}\cdot {d{T}_j|}_{V_{CE}}$$

(40)

while, if it is due to dV_CE,

$${d{I}_C|}_{T_B}\approx \frac{\partial I_C}{\partial T_j}\cdot {d{T}_j|}_{T_B}$$

(41)

The variation of the junction temperature due to a small variation of T_B is

$${d{T}_j|}_{V_{CE}}=d{T}_B+R_{THB0}\cdot V_{CE}\cdot {d{I}_C|}_{V_{CE}}=d{T}_B+R_{THB0}\cdot V_{CE}\cdot \frac{\partial I_C}{\partial T_j}\cdot {d{T}_j|}_{V_{CE}}$$

(42)

where use has been made of (40). From (42), it is obtained that

$${d{T}_j|}_{V_{CE}}=\frac{d{T}_B}{1-R_{THB0}\cdot V_{CE}\cdot \frac{\partial I_C}{\partial T_j}}$$

(43)

The variation of the junction temperature due to a small variation of V_CE is (dT_B = 0 K)

$${d{T}_j|}_{T_B}=R_{THB0}\cdot \left(I_C\cdot d{V}_{CE}+{d{I}_C|}_{T_B}\cdot V_{CE}\right)=R_{THB0}\cdot \left(I_C\cdot d{V}_{CE}+\frac{\partial I_C}{\partial T_j}\cdot {d{T}_j|}_{T_B}\cdot V_{CE}\right)$$

(44)

by exploiting (41). From (44),

$${d{T}_j|}_{T_B}=\frac{R_{THB0}\cdot I_C\cdot d{V}_{CE}}{1-R_{THB0}\cdot V_{CE}\cdot \frac{\partial I_C}{\partial T_j}}$$

(45)

The base current I_B is a function of the independent variables V_BE, T_j, that is,

$$I_B=I_B\left(V_{BE},T_j\right)$$

(46)

By considering that V_BE is constant,

$$d{I}_B=\frac{\partial I_B}{\partial T_j}\cdot d{T}_j$$

(47)

If dT_j is obtained through a variation dT_B, (47) becomes

$${d{I}_B|}_{V_{CE}}=\frac{\partial I_B}{\partial T_j}\cdot {d{T}_j|}_{V_{CE}}$$

(48)

while, if it is induced by dV_CE,

$${d{I}_B|}_{T_B}=\frac{\partial I_B}{\partial T_j}\cdot {d{T}_j|}_{T_B}$$

(49)

Dividing (48) by (49) and making use of (43) and (45)

$$\frac{{d{I}_B|}_{V_{CE}}}{{d{I}_B|}_{T_B}}=\frac{{d{T}_j|}_{V_{CE}}}{{d{T}_j|}_{T_B}}=\frac{d{T}_B}{R_{THB0}\cdot I_C\cdot d{V}_{CE}}$$

(50)

from which

$$R_{THB0}=\frac{1}{I_C}\cdot \frac{{\frac{d{I}_B}{d{V}_{CE}}|}_{T_B}}{{\frac{d{I}_B}{d{T}_B}|}_{V_{CE}}}=\frac{1}{I_C}\cdot \frac{\frac{I_B\left(V_{BE},V_{CE}+d{V}_{CE},T_B\right)-I_B\left(V_{BE},V_{CE},T_B\right)}{d{V}_{CE}}}{\frac{I_B\left(V_{BE},V_{CE},T_B+d{T}_B\right)-I_B\left(V_{BE},V_{CE},T_B\right)}{d{T}_B}}$$

(51)

The technique can be easily extended to account for the Early effect, here omitted only to simplify the derivation; it suffices to multiply the denominator by 1 + V_CE/V_AF, V_AF being the forward Early voltage.

The terms on the RHS can be obtained by measuring three I_B–V_BE characteristics at (V_CE,T_B), (V_CE + dV_CE,T_B), and (V_CE,T_B + dT_B), and choosing a V_BE value that, concurrently with V_CE, ensures a low P_D but an appreciable self-heating. By repeating the process at various T_B, the experimental behavior of R_THB0 vs. T_B is obtained; by comparing it with (9), if R_TH00 has been measured, α is calibrated and then also used in (13).

3.5. Menozzi et al.

The technique developed by Menozzi et al. [48] allows the experimental extraction of R_TH as a function of T_B and P_D based on a couple of assumptions. The method requires the common-emitter measurements of various (seven can be enough [48]) I_C–V_CE characteristics at an assigned base current I_B for various T_B values, the lowest of which is referred to as T_B0. Let us denote as V_CE0 the lowest V_CE applied under forward mode for all curves and as I_C00 the collector current at V_CE0 and T_B = T_B0. Lastly, let us call P_D0(T_B) the dissipated powers corresponding to the V_CE0 values for all characteristics. The first assumption of the method is that, for any assigned T_B, R_TH(T_B,P_D) is a linearly increasing function of P_D, that is,

$$R_{TH}\left(T_B,P_D\right)=R_{TH}\left(T_B,P_{D0}\left(T_B\right)\right)+A_{Menozzi}\left(T_B\right)\cdot \left(P_D-P_{D0}\left(T_B\right)\right)$$

(52)

where the (positive) slope A_Menozzi is a function of the applied T_B. The second assumption is that the collector current I_C linearly decreases with T_j (no matter how T_j is increased) for an assigned I_B, which is only true for HBTs due to the NTC of β_F. This can be expressed as

$$I_C\left(T_j\right)=I_{C00}\cdot \left[1-\mathsf{\kappa }\cdot \left(T_j-T_{j00Menozzi}\right)\right]$$

(53)

T_j00Menozzi being the junction temperature corresponding to V_CE0 and T_B0, i.e.,

$$T_{j00Menozzi}=T_{B0}+R_{TH}\left(T_{B0},P_{D00}\right)\cdot P_{D00}\approx T_{B0}+R_{TH}\left(T_{B0},P_{D00}\right)\cdot V_{CE0}\cdot I_{C00}$$

(54)

where P_D0(T_B0) has been denoted as P_D00. Let us subtract (54) from (1)

$$\begin{array}{cc}{T}_j-T_{j00Menozzi}& =T_B-T_{B0}+R_{TH}\left(T_B,P_D\right)\cdot P_D-R_{TH}\left(T_{B0},P_{D00}\right)\cdot P_{D00}=\\ & =T_B-T_{B0}+\left[R_{TH}\left(T_B,P_{D0}\left(T_B\right)\right)+A_{Menozzi}\left(T_B\right)\cdot \left(P_D-P_{D0}\left(T_B\right)\right)\right]\cdot P_D-R_{TH}\left(T_{B0},P_{D00}\right)\cdot P_{D00}\end{array}$$

(55)

where use has been made of (52). By substituting the RHS of (55) into (53),

$$I_C\left(T_j\right)=a_2\left(T_B\right)\cdot P_D^2+a_1\left(T_B\right)\cdot P_D+a_0\left(T_B\right)$$

(56)

with

$$\begin{array}{cc}{a}_0\left(T_B\right)& =I_{C00}\cdot \left\{1-\mathsf{\kappa }\cdot \left[T_B-T_{B0}-R_{TH}\left(T_{B0},P_{D00}\right)\cdot P_{D00}\right]\right\}=\\ & =-\mathsf{\kappa }\cdot I_{C00}\cdot T_B+I_{C00}+\mathsf{\kappa }\cdot I_{C00}\cdot T_{B0}+\mathsf{\kappa }\cdot I_{C00}\cdot R_{TH}\left(T_{B0},P_{D00}\right)\cdot P_{D00}=\\ & =B_{Menozzi}\cdot T_B+C_{Menozzi}\end{array}$$

(57)

$$a_1\left(T_B\right)=-\mathsf{\kappa }\cdot I_{C00}\cdot \left[R_{TH}\left(T_B,P_{D0}\left(T_B\right)\right)-A_{Menozzi}\left(T_B\right)\cdot P_{D0}\left(T_B\right)\right]$$

(58)

$$a_2\left(T_B\right)=-\mathsf{\kappa }\cdot I_{C00}\cdot A_{Menozzi}\left(T_B\right)$$

(59)

Hence, I_C(T_j) is expressed as a function of T_B and P_D in any point of the measured curves.

As a next step, the I_C–P_D curves associated to the I_C–V_CE ones are determined by a simple elaboration of the data. Then, the values of coefficients a₀, a₁, a₂ favoring the best agreement between the experimental I_C–P_D curves and (56) can be found for each T_B with a 2nd-order polynomial fit, and the three a₀–T_B, a₁–T_B, a₂–T_B characteristics are available. By comparing the a₀–T_B curve and (57), B_Menozzi and C_Menozzi can be calibrated. Hence, κ is evaluated as

$$\kappa =-\frac{B_{Menozzi}}{I_{C00}}$$

(60)

and the thermal resistance R_TH(T_B0,P_D00) as

$$R_{TH}\left(T_{B0},P_{D00}\right)=\frac{\frac{C_{Menozzi}}{I_{C00}}-\mathsf{\kappa }\cdot T_{B0}-1}{\mathsf{\kappa }\cdot P_{D00}}$$

(61)

From (59), A_Menozzi(T_B) is obtained for each T_B as

$$A_{Menozzi}\left(T_B\right)=-\frac{a_2\left(T_B\right)}{\mathsf{\kappa }\cdot I_{C00}}$$

(62)

Consequently, from (58), R_TH(T_B,P_D0(T_B)) can be determined as

$$R_{TH}\left(T_B,P_{D0}\left(T_B\right)\right)=A_{Menozzi}\left(T_B\right)\cdot P_{D0}\left(T_B\right)-\frac{a_1\left(T_B\right)}{\mathsf{\kappa }\cdot I_{C00}}$$

(63)

Once A_Menozzi(T_B) and R_TH(T_B,P_D0(T_B)) are known, the linear R_TH vs. P_D increase for each T_B as given by (52) is achieved.

Although this technique is suited for HBTs, it can in principle also be applied to Si BJTs by assuming a linear increase of I_C with T_j in (53). However, in the presence of significant Early effect, which is not accounted for, the extracted thermal resistance will be overestimated.

3.6. Berkner and Balanethiram et al.

The technique of Berkner [55], subsequently also presented by Balanethiram et al. [15], belongs to the category of approaches based on intersection(s) between curves and does not rely on a thermometer. Under forward active mode, at V_BE not too high to avoid Kirk and resistive effects, if the Early effect is negligible and V_CE (V_CB) is sufficiently low to avoid avalanche, the collector current is given by

$$I_C\left(V_{BE},T_j\right)=I_S\left(T_j\right)\cdot \exp \left(\frac{V_{BE}}{V_T}\right)$$

(64)

The technique requires the measurement of two V_CE-constant I_C–V_BE characteristics, the first obtained by assigning (V_CE1,T_B1), and the second fixing (V_CE2,T_B2), where V_CE1 < V_CE2 and T_B1 > T_B2. For low V_BE values (low self-heating), the junction temperature T_j1 along the (V_CE1,T_B1) curve will be higher than T_j2 associated to the (V_CE2,T_B2) counterpart, as T_B1 > T_B2 dominates with respect to P_D1 < P_D2, that is, invoking (1),

$$T_{j1}>T_{j2}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\iff \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{T}_{B1}+R_{TH}(T_{B1},P_{D1})\cdot P_{D1}>T_{B2}+R_{TH}(T_{B2},P_{D2})\cdot P_{D2}$$

(65)

On the other hand, by increasing V_BE, P_D2 grows faster than P_D1, thus leading to

$$T_{j1}<T_{j2}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\iff \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{T}_{B1}+R_{TH}(T_{B1},P_{D1})\cdot P_{D1}<T_{B2}+R_{TH}(T_{B2},P_{D2})\cdot P_{D2}$$

(66)

As a result, there will be a value of V_BE where T_j1 = T_j2, that is,

$$T_{B1}+R_{TH}(T_{B1},P_{D1})\cdot P_{D1}=T_{B2}+R_{TH}(T_{B2},P_{D2})\cdot P_{D2}$$

(67)

This V_BE value can be easily identified as the one at which the two characteristics intersect, as in the assumption of validity of (64), T_j1 = T_j2 also implies I_C1 = I_C2 and therefore V_BE1 = V_BE2. Here Berkner and Balanethiram et al. assume that R_TH(T_B1,P_D1) = R_TH(T_B2,P_D2) = R_TH, and evaluate R_TH from (67) as

$$R_{TH}=\frac{T_{B1}-T_{B2}}{P_{D2}-P_{D1}}$$

(68)

and the corresponding T_j = T_j1 = T_j2 from either the LHS or RHS of (67). By repeating the procedure for different (V_CE,T_B) couples, they determine the R_TH vs. T_j behavior.

Although this technique is quite easy to apply, it is affected by some drawbacks. First, the authors seem to claim that R_TH is only dependent on T_j; however, the theory presented in Section 2 (Figure 1b) clearly demonstrates that R_TH is not univocally determined by T_j, but separately depends on T_B and P_D. For this reason, we think that the equality R_TH(T_B1,P_D1) = R_TH(T_B2,P_D2) and the extracted R_TH–T_j behavior are questionable. In addition, the intersection point must be taken where the Kirk effect does not play a role to prevent I_C from being V_CB dependent. Lastly, the method cannot be applied to Si BJTs, where the collector current would depend on V_CE also due to the Early effect.

3.7. Huszka et al.

Huszka and his co-workers [17] present an improved variant of the technique from Berkner [55] to account for the separate dependence of R_TH on T_B and P_D. They recast (67) as

$$\frac{T_{B1}-T_{B2}}{P_{D2}-P_{D1}}=\frac{R_{TH}(T_{B2},P_{D2})\cdot P_{D2}}{P_{D2}-P_{D1}}-\frac{R_{TH}(T_{B1},P_{D1})\cdot P_{D1}}{P_{D2}-P_{D1}}$$

(69)

where the LHS is the original Berkner’s thermal resistance given by (68). Then they propose to use (13) or the linearized (17) for R_TH(T_B1,P_D1) and R_TH(T_B2,P_D2), so that (69) becomes an equation with two unknowns, i.e., R_TH00 and α. The authors find many intersection points by varying V_CE and T_B; by using (69) for each point, an equation system with unknowns R_TH00 and α is obtained. Subsequently, α (given as an input) is varied in a reasonably wide range, and parameter R_TH00 is optimized for each α (“single-variable optimization”); the (input) α and the calibrated R_TH00 ensuring the minimum sum of the squared differences between LHS and RHS of all the system are finally selected.

Differently from [45,50], and similar to [29,47,48], here the analytical R_TH dependences on T_B and P_D are imposed by the technique. Conveniently, in Section 4 a numerical analysis will allow demonstrating that (13) or (17) with calibrated parameters favor a good agreement with realistic R_TH(T_B,P_D) data, regardless of the technology.

The need to find many intersection points between I_C–V_BE characteristics at different V_CE and T_B values in a well-defined range (see Section 3.6) seems to be a shortcoming of this approach.

Lastly, it is important to remark that also other techniques, not specifically conceived to extract the influence of nonlinear thermal effects on R_TH, can in principle be exploited to determine the R_THB0 vs. T_B behavior at low P_D. An example is given by the thermometer-based method proposed by d’Alessandro et al. [13] and its enriched variant accounting for the Early effect [14,56], which are slightly less straightforward, yet more robust, than “differential” methods or approaches based on the direct adoption of the V_BE–T_j thermometer to determine T_j from the measured V_BE. The technique is articulated as follows: (1) the slope ϕ(I_E) of the V_BE–T_B curve is extracted at various I_E sufficiently small to avoid self-heating (so that T_j ≈ T_B); (2) an analytical model for the ϕ dependence on I_E (valid also for I_E values entailing self-heating) is determined and verified; (3) R_TH is calculated from ϕ(I_E) and the slope of an I_E-constant V_BE–V_CB characteristic (or equivalently from the slope of the corresponding V_BE–P_D curve) at dissipated power sufficiently high to favor self-heating, but low enough to avoid the nonlinear self-heating effect. Step (3) can be then repeated at various backside temperatures T_B. Unfortunately, it is not possible to evaluate the influence of the nonlinear self-heating effect on R_TH, as P_D is varied in an assigned range during a single extraction.

The main features of the presented techniques are summarized in

Table 2. Main features of the experimental R_TH extraction techniques explicitly accounting for nonlinear thermal effects.

Technique	RTH Extraction Procedure	Advantages, Approximations, and Limitations
Bovolon et al. [50]	RTH is experimentally extracted as a function of TB and PD with an extended version of the approach in [51]. For each (TB,PD) couple, two IB-constant IC–VCE (and the related VBE–VCE) characteristics at TB and TB + ΔTB have to be measured.	The technique allows extracting RTH(TB,PD) without any analytical assumption on the dependence of RTH on TB and PD. However, the (TB,PD), (TB,PD + ΔPD), (TB + ΔTB,PD) couples needed for the evaluation of ϕ and RTH(TB,PD) must be cautiously selected. Moreover, the approach is “differential”, being based on differences between two VBE values, and thus it may suffer from a relatively high inaccuracy.
Yeats [45]	RTH is experimentally extracted as a function of TB and PD with an improved version of the approach in [51]. It requires the measurement of some IE-constant VBE–VCE characteristics at various TB values.	The technique allows extracting RTH(TB,PD) without any assumption on the dependence of RTH on TB and PD. Questionable points are the accuracy of the extrapolation step needed to calibrate the VBE–Tj thermometer, and the direct use of the thermometer to evaluate Tj for each VBE, which makes the extracted Tj (and the related RTH) too sensitive to the precision in the VBE measurements.
Marsh [47]	RTHB0 is experimentally extracted as a function of TB at low power. The technique requires the measurement of three IB-constant IC–VCE characteristics at different TB values.	A linear increase of RTHB0 on TB is assumed. The PD dependence of RTH is not considered.
Paasschens et al. [29]	RTHB0 is experimentally extracted as a function of TB at low power by resorting to the approach in [54], which for each TB requires the measurement of three IB–VBE characteristics.	The technique does not extract the PD dependence of RTH, and uses only low-power data to optimize parameter α to be applied in (13). In addition, the “differential” approach in [54] may suffer from a significant inaccuracy if not carefully applied.
Menozzi et al. [48]	RTH is experimentally extracted as a function of TB and PD. The technique requires the measurements of some IB-constant IC–VCE characteristics by varying TB.	A linear behavior of RTH(TB,PD) on PD with a TB-sensitive slope is assumed. The dependence of IC (βF) on Tj under IB-constant conditions is assumed to be linear, while being exponential. Lastly, the technique cannot be applied to Si BJTs with significant Early effect.
Berkner [55] and Balanethiram et al. [15]	RTH is experimentally extracted as a function of Tj by identifying the intersection point between two IC–VBE curves at different couples (VCE1,TB1) and (VCE2,TB2) with VCE1 < VCE2 and TB1 > TB2.	The technique is intrinsically affected by the assumption that RTH is only a function of Tj, while it can assume different values for a given Tj, depending on TB. The intersection point should be taken in a region where the Kirk effect does not occur. The method cannot be applied to Si BJTs with significant Early effect.
Huszka [17]	It is an improved variant of Berkner’s method [55]. It imposes (13) or the linearized (17) to describe the separate dependence of RTH on TB and PD. It requires the measurements of several IC–VBE curves at various VCE and TB values.	Reliable values for parameters RTH00 and α in (13) or (17) can be achieved by applying an optimization procedure to many intersection points determined as proposed by Berkner [55].

.

4. Numerical Simulation

In this Section, we resort to extremely detailed COMSOL [57] 3-D static finite-element method (FEM) thermal simulations of an InGaP/GaAs NPN HBT and a Si/SiGe NPN HBT chosen as well-representative case studies. For both devices, the thermal resistances R_TH were evaluated over reasonably wide ranges of T_B and P_D by activating the temperature dependence of the thermal conductivities. This simulation study is intended to emulate the results of experimental extraction techniques like [45,50], which do not impose any analytical dependence of R_TH on T_B or P_D. The data were used to test the accuracy of the single-semiconductor formulations introduced earlier to describe R_TH(T_B,P_D).

4.1. Devices under Test

The InGaP/GaAs NPN HBT is a mesa-isolated device manufactured by Qorvo, the key features of which are reported in

Table 3. Key features of the InGaP/GaAs NPN HBT under test.

Parameter	Value
Common-emitter current gain βF at 300 K and medium current levels	135
Open-emitter breakdown voltage BVCBO	27 V
Open-base breakdown voltage BVCEO	17 V
Peak cut-off frequency fT for VCE = 3 V	40 GHz
Collector current density JC at peak fT for VCE = 3 V	0.2 mA/µm2
Maximum oscillation frequency fMAX for VCE = 3 V	82 GHz

. The transistor cell consists of four emitter fingers, each with 2 × 20.5 µm² area (the total emitter area then amounts to 164 µm²). The GaAs substrate is 620-µm thick and equipped with 65 × 65 µm² pads in a ground-signal-ground configuration for bare-die experimental characterization through RF probes. Further technological details are provided in [10].

The Si/SiGe NPN HBT was fabricated by Infineon Technologies AG in the framework of the European project DOTFIVE. The device, whose figures of merit are listed in

Table 4. Key features of the Si/SiGe NPN HBT under test.

Parameter	Value
Common-emitter current gain βF at 300 K and medium current levels	1500
Open-emitter breakdown voltage BVCBO	5.5 V
Open-base breakdown voltage BVCEO	1.6 V
Peak cut-off frequency fT for VCB = 0.5 V	240 GHz
Collector current density JC at peak fT for VCB = 0.5 V	10 mA/µm2
Maximum oscillation frequency fMAX for VCB = 0.5 V	380 GHz

, has only one base and one collector contact (BEC configuration), and belongs to the latest project technology stage, also denoted as set #3 in [13,14]. The drawn emitter area is equal to 0.2 × 2.8 µm², and the substrate is 185-µm thick.

The 3-D geometries and the meshes of the devices under test in the COMSOL environment are displayed in Figure 2 and Figure 3. The unique accuracy in emulating the real structure of the transistors was obtained by virtue of an in-house routine relying on the MATLAB-COMSOL Livelink [9]. In both technologies, the heat source was considered to geometrically coincide with the base-collector space-charge region. An isothermal boundary condition at temperature T_B was applied to the substrate backside, while the lateral sides and the top surface were assumed adiabatic. The mesh of the InGaP/GaAs HBT presents 2.8 × 10⁶ tetrahedra and 3.6 × 10⁶ degrees of freedom, while that of the Si/SiGe HBT involves 1.7 × 10⁶ tetrahedra and 3.9 × 10⁶ degrees of freedom. The thermal conductivities of the ternary alloys In_xGa_1−xP and In_xGa_1−xAs (for the emitter of the InGaP/GaAs HBT), as well as of the binary alloy Si_1−xGe_x (for the base of the Si/SiGe HBT), x being the mole fraction of In (in the GaP or GaAs lattice) and Ge (in the Si lattice), were calculated using formulas available in [22]. For the Si/SiGe HBT, two additional conductivity degradation effects were included, namely, one dictated by the high phonon-impurity scattering in the highly doped regions, and another induced by the phonon scattering with lateral boundaries in narrow layers, like emitter tungsten contact, Si emitter, SiGe base, and Si volume embraced by the shallow trench [14]. The CPU time required for each nonlinear static simulation was about 5–10 min using a PC equipped with an AMD Ryzen 5 3600 and 16 GB of RAM.

4.2. Results

By making use of the aforementioned simulation strategy, the T_B = T₀ zero-power thermal resistances R_TH00 were found to be 457.6 K/W for the InGaP/GaAs HBT, and 6829 K/W for the Si/SiGe HBT, values very close to the experimental counterparts determined with the methods in [51] for the InGaP/GaAs HBT and [13,14] for the Si/SiGe HBT.

The simulated R_TH vs. P_D curves for various T_B values are shown in Figure 4.

A first investigation was aimed at testing the accuracy of the R_THB0 vs. T_B formulations, namely, the power (9) and linear (15) laws with parameters α and ζ_B optimized by means of a simple code performing a least squares fitting. Figure 5 shows the results. For the InGaP/GaAs HBT, (9) with α = 0.99 and (15) with ζ_B = 3.3 × 10⁻³ K⁻¹ almost provide the same accuracy, while for the Si/SiGe HBT, (9) with α = 1.2924 is slightly better than (15) with ζ_B = 4.53 × 10⁻³ K⁻¹.

Unfortunately, we found that the α value calibrated at zero power cannot be used to accurately account for the nonlinear self-heating effect through (13), as proposed in [29]. Figure 4 witnesses that α = 0.99 leads to an overestimation of R_TH at medium/high dissipated power for the InGaP/GaAs HBT, while α = 1.2924 gives rise to an underestimation for the Si/SiGe HBT.

Then we tested the modeling approach suggested by Yeats [45] that requires a “brute-force 2-D search” of the R_TH00 and α values ensuring the best agreement between all COMSOL R_TH data and (13). In this case, R_TH00 is not the COMSOL (or equivalently the experimental) T_B = T₀ zero-power thermal resistance, but becomes a parameter to calibrate. For the InGaP/GaAs HBT, the optimization led to R_TH00 = 459.4 K/W (with a 0.39% discrepancy with respect to the COMSOL value) and α = 0.951, while for the Si/SiGe HBT, R_TH00 = 6855.8 K/W (0.39%) and α = 1.333. Figure 6 shows that the resulting agreement between the COMSOL thermal resistances and (13) is very good for both technologies. This is an important result, as it demonstrates that the R_TH dependence on T_B and P_D of a real bipolar transistor, independently of the technology, can be accurately described with the simple single-semiconductor theory by calibrating two parameters only, namely, R_TH00 and α; in addition, R_TH00 does not lose its physical meaning since the optimized value is very close to the reference (COMSOL) one.

We also examined the accuracy of the linearized formulation (16). Adopting (16) with (9) for R_THB0 and the R_TH00 and α values calibrated with the “brute-force 2-D search” leads to a perceptible underestimation of COMSOL data at medium/high P_D (not shown in the figures). Instead, (16) reaches a level of accuracy comparable with (13) only using (9) for R_THB0 with the COMSOL (or equivalently the experimental) R_TH00 and α calibrated on low-power data, and optimizing coefficient ζ_P (which can be used as a model parameter in compact transistor models) at each T_B. In our analysis, we did not calibrate α in (9) on low-power data, as the R_THB0 values corresponding to each T_B were all available from COMSOL simulations. It was found that the optimized ζ_P slightly decreases for the InGaP/GaAs HBT (from 0.8 W⁻¹ at T_B = T₀ = 300 K to 0.733 W⁻¹ at T_B = 450 K) and marginally increases for the Si/SiGe HBT (from 20.09 W⁻¹ to 23.16 W⁻¹). The resulting R_TH vs. P_D curves are also shown in Figure 6.

5. R_TH Formulations Available in Compact Transistor Models and Update

In compact models of bipolar transistors accounting for self-heating, namely, VBIC [58], Mextram 504 [59], Agilent HBT (AHBT) [60], and HICUM [61,62], all available in Keysight ADS [63], the power-temperature feedback is implemented through an equivalent thermal network, i.e., an electrical circuit where currents, voltages, resistances, and capacitances represent dissipated powers, temperatures, thermal resistances, and thermal capacitances, respectively. More specifically, in VBIC, Mextram 504, and HICUM, the default network is the one depicted in Figure 7a, while in AHBT it is the one shown in Figure 7b. Conveniently, all these models are equipped with an externally accessible thermal node that allows the implementation of a thermal network in any form.

5.1. VBIC

In VBIC, R_TH is assumed to be a constant parameter, that is, nonlinear thermal effects are not captured. This is problematic when simulating ET effects at medium/high junction temperatures.

5.2. Mextram 504

Mextram 504 implements the impact of backside (or ambient) temperature on R_TH according to

$$R_{TH}=RTH\cdot {\left(\frac{273.15+TEMP}{273.15+TREF}\right)}^{ATH}$$

(70)

where TREF (default: 25 °C) is the reference temperature at which parameters are extracted, TEMP (default: 25 °C) is the ambient temperature defined at the device instance, RTH (default: 300 K/W) and ATH (default: 0, although a list of suitable values for the most important materials is provided in [59]) are model parameters. Unfortunately, although the Mextram 504 team was well aware of the influence of the nonlinear self-heating effect on R_TH [29], this is not taken into account in the present release, as also correctly stated in [17]. Such a choice was probably made to ensure unconditional simulation stability at the price of a reduced accuracy at high P_D. By using the nomenclature introduced in Section 2, and considering that 273.15 + TREF ≈ T₀, 273.15 + TEMP = T_B, and that RTH and ATH coincide with R_TH00 and α, (9) is obtained.

5.3. AgilentHBT (AHBT)

AgilentHBT (AHBT) implements the following R_TH model (Figure 7b):

$$R_{TH}=R_{TH1}+R_{TH2}=RTH1\cdot {\left(\frac{273.15+Tdev}{273.15+Tnom}\right)}^{XTH1}+RTH2\cdot {\left(\frac{273.15+Tdev}{273.15+Tnom}\right)}^{XTH2}$$

(71)

where Tnom (default: 25 °C) is the nominal temperature at which parameters are extracted, RTH1 (default: 1000 K/W), XTH1 (default: 0), RTH2 (default: 0 K/W), XTH2 (default: 0) are model parameters, and Tdev [°C] is the device temperature, given by

$$Tdev=Temp+deltaT$$

(72)

In (72), Temp (default: 25 °C) is the ambient temperature defined at the device instance, and deltaT (ΔT_j in Figure 7b) is dynamically calculated as

$$deltaT=\left(R_{TH1}+R_{TH2}\right)\cdot P_D=R_{TH}\cdot P_D$$

(73)

Let us now use the nomenclature introduced in Section 2. By considering that 273.15 + Tnom ≈ T₀, 273.15 + Temp = T_B, and Tdev = T_j, (71) can be rewritten as

$$R_{TH}=RTH1\cdot {\left(\frac{T_B+\Delta T_j}{T_0}\right)}^{XTH1}+RTH2\cdot {\left(\frac{T_B+\Delta T_j}{T_0}\right)}^{XTH2}$$

(74)

which by default reduces to the two-parameter formulation

$$R_{TH}=RTH1\cdot {\left(\frac{T_B+\Delta T_j}{T_0}\right)}^{XTH1}$$

(75)

5.4. HiCUM

In the 2.4 release of HICUM/L2, the thermal resistance is evaluated as [17,62]

$$R_{TH}=rth\cdot \left[1+alrth\cdot \left(Temp+\Delta T_j-Tnom\right)\right]\cdot {\left(\frac{273.15+Temp+\Delta T_j}{273.15+Tnom}\right)}^{zetarth}$$

(76)

where Tnom (default: 27 °C) is the reference temperature at which parameters are determined, Temp (default: 27 °C) is the ambient temperature defined at the device instance, rth (default: 0 K/W), alrth (default: 0 K⁻¹), zetarth (default: 0) are model parameters, and ΔT_j is dynamically computed as

$$\Delta T_j=R_{TH}\cdot P_D$$

(77)

Let us now adopt the nomenclature reported in Section 2. By considering that 273.15 + Tnom = T₀ and 273.15 + Temp = T_B, (76) can be rewritten as

$$R_{TH}=rth\cdot \left[1+alrth\cdot \left(T_B+\Delta T_j-T_0\right)\right]\cdot {\left(\frac{T_B+\Delta T_j}{T_0}\right)}^{zetarth}$$

(78)

5.5. Proposed R_TH Modeling Approaches

Starting from the findings achieved in Section 4, we propose to resort to (13) with optimized R_TH00 and α in compact bipolar transistor models. However, implementing R_TH as a nonlinear T_B- and P_D-dependent resistor according to (13) would lead to a markedly unphysical behavior under dynamic conditions, where R_TH will be directly affected by (even fast) P_D variations in time.

A first solution (denoted as #1) involves the use of (i) a constant resistor R_THB0 in the single-pole network depicted in Figure 7a, the value of which can be preliminarily calculated with (9), or (ii) a nonlinear resistor R_THB0 given by (9), and a behavioral block (e.g., a nonlinear voltage-controlled voltage source) converting R_THB0∙P_D into ΔT_j, with T_j given by (11). Besides T₀, two model parameters are needed: R_THB0 and α in case (i), and R_TH00 and α in case (ii). This approach was successfully used on simple SPICE-compatible InGaP/GaAs and Si/SiGe HBT models in [10] and [64], respectively.

A second solution (referred to as #2) is obtained by replacing P_D with ΔT_j in (12) using (1), which leads to

$$1=\frac{T_B}{\Delta T_j}\cdot \left\{{\left[1-\left(\mathsf{\alpha }-1\right)\cdot \frac{R_{THB0}}{R_{TH}\left(T_B,P_D\right)}\cdot \frac{\Delta T_j}{T_B}\right]}^{\frac{-1}{\mathsf{\alpha }-1}}-1\right\}$$

(79)

whence

$$R_{TH}\left(T_B,\Delta T_j\right)=R_{THB0}\cdot \frac{\mathsf{\alpha }-1}{1-{\left(\frac{T_B+\Delta T_j}{T_B}\right)}^{-\left(\mathsf{\alpha }-1\right)}}\cdot \frac{\Delta T_j}{T_B}$$

(80)

Then, in the network of Figure 7a, R_TH must be implemented with a nonlinear resistor given by (80), where R_THB0 is again either considered a model parameter to be calculated in the pre-processing stage at the assigned T_B with (9), or computed in the compact model through (9). Option #2 reduces the number of nodes in the compact model compared with #1.

A third solution (designated as #3) is to exploit a nonlinear current source controlled by ΔT_j to force through the thermal resistance branch a power given by [29]

$$P_D=\frac{\Delta T_j}{R_{TH}\left(T_B,\Delta T_j\right)}=\frac{T_B}{R_{THB0}}\cdot \frac{1-{\left(\frac{T_B+\Delta T_j}{T_B}\right)}^{-\left(\mathsf{\alpha }-1\right)}}{\mathsf{\alpha }-1}$$

(81)

which can be easily obtained from (80). For R_THB0, the considerations expressed before still hold.

Solutions #1, #2, #3 provide the same ΔT_j under static conditions. Conversely, approach #1 behaves differently from #2, #3 under dynamic conditions: by considering the network in Figure 7a and assuming the same thermal capacitance C_TH, #1 leads to a time constant shorter than that of #2 and #3, i.e., it gives rise to a faster response to a power step. Unfortunately, all approaches are expected to be inaccurate in describing the dynamic thermal behavior, mainly due to the simplicity of the single-pole model. A thermal network correctly accounting for nonlinear dynamic thermal effects has been presented in [65]; however, it is far more complex than the single-pole network used by default in VBIC, AHBT, HICUM and can only be derived from a detailed 3-D thermal model of the device under test.

A simplified, yet slightly less accurate, option (called #4) uses the linearized dependence on P_D given by (16). Here, again P_D must be substituted with ΔT_j, which leads to the following R_TH formulation:

$$R_{TH}\left(T_B,\Delta T_j\right)=\frac{R_{THB0}}{2}\cdot \left(1+\sqrt{1+2\frac{\mathsf{\alpha }}{T_B}\cdot \Delta T_j}\right)$$

(82)

or equivalently

$$R_{TH}\left(T_B,\Delta T_j\right)=\frac{R_{THB0}}{2}\cdot \left(1+\sqrt{1+4\frac{{\zeta }_P\left(T_B\right)}{R_{THB0}}\cdot \Delta T_j}\right)$$

(83)

in turn representing the linearized version of (80). In this case, R_TH in the thermal network of Figure 7a must be implemented as a nonlinear resistor given by (82). Owing to the findings obtained before, using (82) with R_TH00 and α optimized through the “brute-force 2-D search” underestimates R_TH at medium/high ΔT_j; a fairly good level of accuracy can instead be achieved by using (83) with ζ_P as a model parameter optimized at each T_B, or equivalently (82) with α optimized at each T_B.

It is worth noting that, apart from T₀, all the above solutions make use of two parameters only, namely, R_THB0 and α, if R_THB0 is calculated in the preprocessing stage, or R_TH00 and α, if R_THB0 is expressed with (9) in the compact model. In the latter case, the computational burden can be alleviated by exploiting the linearized (15), which does not significantly reduce the accuracy.

Unfortunately, the R_TH formulations (75) and (78) employed in AHBT and HICUM, respectively, do not allow obtaining a favorable match with the accurate single-semiconductor theory. By setting RTH1 in (75) and R_TH in (78) equal to the R_TH00 value simulated by COMSOL, the R_TH vs. ΔT_j behavior determined by (75) and (78) strongly disagree with that corresponding to (80), independently of the choice of parameters XTH1 in (75) or alrth and zetarth in (78). In conclusion, the available thermal networks embedded in compact transistor models either do not consider or improperly account for nonlinear thermal effects. Hence, one of the implementations of the single-semiconductor theory (#1 to #4) should be suggested for their future releases.

Lastly, it is important to remark that all the results obtained with the simulation and modeling analyses performed in Section 4 and Section 5 can be safely extended down to T_B = 250 K since the adopted laws for the thermal conductivities of all materials are still approximately valid in the range 250 to 300 K.

6. Conclusions

This paper is intended to offer a comprehensive overview of nonlinear thermal effects in bipolar transistors from a manifold perspective. In particular, the work includes: (i) a theoretical discussion on the underlying physics and an analytical model for the influence of such effects on the thermal resistance, rigorously valid only for an ideal, very simple device homogeneously composed of one semiconductor (single-semiconductor assumption); (ii) a critical review of the available experimental techniques to identify the impact of nonlinear effects on the thermal resistance; (iii) a detailed 3-D numerical simulation campaign of state-of-the-art HBTs in InGaP/GaAs and Si/SiGe technologies, the results of which are exploited to verify the accuracy of the single-semiconductor theory; (iv) a description of the thermal resistance formulations embedded in the most relevant compact bipolar transistor models; (v) some proposals for the implementation of the single-semiconductor expressions in future releases of the compact models.

The main results can be summarized as follows.

• The theoretical analysis has allowed emphasizing a sometimes-overlooked aspect, i.e., that the thermal resistance suffers from two distinct nonlinear thermal effects associated with the backside temperature and dissipated power, and cannot be considered solely a function of the junction temperature.
• Only the techniques conceived by Bovolon et al. and Yeats allow the extraction of thermal resistance as a function of the backside temperature and dissipated power without imposing any analytical formulation for these dependences. However, the technique of Bovolon et al. is “differential” and thus prone to errors if not cautiously applied, while that of Yeats is based on a delicate and questionable thermometer calibration procedure.
• The detailed 3-D simulation campaign, performed to obtain realistic thermal resistance data as a function of backside temperature and dissipated power, has allowed demonstrating that the nonlinear thermal effects in a real transistor can be accurately described with the single-semiconductor approach if a proper parameter optimization is carried out.
• Conversely, the thermal resistance formulations used in the latest releases of compact bipolar transistor models for circuit simulators either do not include nonlinear thermal effects or improperly account for them. Some proposals for improving the implementation of such effects have been made, all inspired by the theory of the single-semiconductor device.