A Critical Review of Techniques for the Experimental Extraction of the Thermal Resistance of Bipolar Transistors from DC Measurements—Part I: Thermometer-Based Approaches

Abstract

This paper presents a critical and detailed overview of experimental techniques for the extraction of the thermal resistance of bipolar transistors from simple DC current/voltage measurements. More specifically, this study focuses on techniques based on a thermometer, i.e., the relation between the base-emitter voltage and the junction temperature. The theory behind the techniques is described with a unified and comprehensible nomenclature. Advantages, underlying approximations, and limitations of the methods are illustrated. The accuracy is assessed by emulating the DC measurements with PSPICE electrothermal simulations of a transistor model, applying the techniques to the simulated currents/voltages, and comparing the extracted thermal resistance data with the values obtained from the target formulation embedded in the transistor model. An InGaP/GaAs HBT and an Si/SiGe HBT for high-frequency applications are considered as case-studies.

1. Introduction

Electrothermal (ET) effects plague modern high-frequency bipolar transistors in multiple ways, as they can lead to distortion in the I–V curves, which modifies the DC bias and shrinks the safe operating area [1,2,3,4], degradation of the small-signal low-frequency behavior [5,6], reduction in cut-off frequency caused by the higher scattering rate [7], and even irreversible failure, likely to occur in multifinger devices due to thermally-induced current hogging [8,9,10,11]. This holds true regardless of the technology and is dictated by the high operating current (and power) densities and the high self-heating thermal resistances. In gallium arsenide (GaAs)-based heterojunction bipolar transistors (HBTs) like InGaP/GaAs and AlGaAs/GaAs, considered the dominant technology for handset power amplifier design, the high thermal resistances are a consequence of (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 interlevel dielectric films [9,10,12,13,14,15]. In silicon/silicon-germanium (Si/SiGe) HBTs for mm-wave and near-THz applications, namely, wireless and optical communication, medical equipment, and automotive radars, the increase in thermal resistances is due to technology strategies devised to boost the frequency performance, like (i) adoption of oxide-based shallow/deep trenches and reduction of the spacing between intrinsic transistor and trenches, which hinder the lateral heat propagation from the power dissipation region, and (ii) horizontal scaling of the emitter, which drives higher current (and power) density; such factors have contributed to push the thermal resistances of single-finger HBTs into the thousands of K/W [16,17,18,19,20,21,22].

In a bipolar transistor, the thermal resistance R_TH [K/W] is defined as the ratio between the temperature rise ΔT_j = T_j − T_B [K] divided by the dissipated power P_D [W], where T_j [K] is the temperature averaged over the base-emitter junction and T_B [K] is the backside temperature. This definition is reasonable since the electrical characteristics of the device markedly depend on T_j. Accurately assessing the thermal resistance from experimental data is of utmost importance in terms of thermal characterization, modeling/simulation for device/circuit design, as well as reliability estimation.

In general, there are two experimental approaches for assessing the temperature of a transistor: the direct one is based on the detection of temperature maps over the top surface of the exposed chip (through e.g., infrared imaging and liquid crystal methods), while the indirect one relies on the measurement of currents/voltages and only allows determining a single temperature value in a relevant device region. For downscaled bipolar transistors for high-frequency applications, indirect techniques are the common choice, as direct methods can only detect the temperature over the top surface (the base-emitter junction is not exposed) and suffer from limited space resolution. As a result, a single “average” T_j value is indirectly extracted, but this does not represent a problem for aggressively scaled devices, in which the base-emitter temperature is expected to be almost uniform. Within the realm of indirect techniques, those based on DC measurements are often preferred to low-frequency AC or pulsed transient methods, since they are easier to perform and require cheaper equipment. DC indirect techniques can be in turn subdivided into the following categories.

• Techniques using a thermometer, i.e., the relation between a temperature-sensitive electrical parameter (TSEP) and the temperature in a relevant device region [18,23,24,25,26,27,28,29,30,31]. The TSEP typically adopted in a bipolar transistor is the base-emitter voltage V_BE, as it varies with temperature more linearly than the common-emitter forward current gain β_F [24].
• Techniques exploiting intersection points [20,22,32,33,34].
• Techniques based on the measurement of the base current I_B [35,36,37,38,39].
• A technique relying on analytical assumptions that allows the full evaluation of nonlinear thermal effects [40].

A review of all the above methods has been recently published [41], which clarifies that none of them can be considered the absolute best, as the accuracy of each technique depends on specific circumstances (biasing conditions and technology under test).

This paper critically investigates and compares only thermometer-based DC indirect techniques, while follow-up papers will be dedicated to other approaches. The work is intended to extend and complete the analysis conducted in [41] by providing much more details on the theory on which the techniques are based, as well as on the reasons of extraction inaccuracy.

In Section 2, an extensive theoretical background is offered, which explains the temperature dependence of the collector current, gives the thermal resistance definition, and provides some details on nonlinear thermal effects. Section 3 presents the devices selected as case-studies and probes into the circuit-based simulation approach needed to analyze the accuracy of the techniques of interest. Section 4 describes the theory behind the techniques in a tutorial style with a unified and comprehensible nomenclature, clarifies advantages, limitations, and approximations, draws simple guidelines for their correct applications, shows and discusses the results. Conclusions are finally given in Section 5.

2. Theoretical Background

2.1. Temperature Dependence of the Collector Current

The collector current I_C of a bipolar transistor operated in forward active mode can be expressed as

$$I_C=M·I_{CT}$$

(1)

where M (≥1) is the dimensionless V_CB-dependent avalanche multiplication factor, V_CB [V] being the collector-base voltage, and I_CT [A] (≈I_E, emitter current) represents the minority transport current flowing across the quasi-neutral base region [42]. I_CT is given by

$$I_{CT}=\left(1+\frac{V_{CB}}{V_{AF}}\right)·\frac{1}{B_{HI}}·\frac{q·A_E·D_{nB}\left(T_j\right)·n_{iB}^2\left(T_j\right)}{W_B·N_B}·\exp \left(\frac{V_{BEj}}{\eta ·V_T}\right)$$

(2)

where

• V_AF [V] is the forward Early voltage;
• B_HI (≥1) is an I_C-dependent dimensionless term included to empirically describe the attenuation dictated by high-injection (high-current) effects leading to the gain roll-off;
• q [C] is the absolute value of the electron charge (or elementary charge);
• A_E [cm² or µm²] is the emitter area;
• T_j [K] is the average temperature over the base-emitter junction (also simply referred to as junction temperature), as mentioned in Section 1;
• D_nB [cm²/s] is the average electron diffusivity in the quasi-neutral base region;
• n_iB [cm⁻³] is the intrinsic carrier concentration in the base;
• W_B [cm or µm] is the quasi-neutral base width;
• N_B [cm⁻³] is the average base doping;
• V_BEj [V] is the “internal” (junction) base-emitter voltage, that is, V_BEj = V_BE − R_B·I_B − R_E·I_E, where V_BE [V] is the externally-accessible base-emitter voltage, I_B and I_E [A] are the base and emitter current, respectively, and R_B and R_E [Ω] are the parasitic base and emitter resistances, respectively;
• η is the dimensionless ideality coefficient;
• V_T = kT_j/q [V] is the thermal voltage at T_j, k = 8.617 × 10⁻⁵ eV/K being the Boltzmann constant.

Using the Einstein relation, D_nB can be expressed as

$$D_{nB}\left(T_j\right)=V_T·{\mu }_{nB}\left(T_j\right)=\frac{k{T}_j}{q}·{\mu }_{nB}\left(T_0\right)·{\left(\frac{T_j}{T_0}\right)}^{-m_B}$$

(3)

where T₀ = 300 K is the reference temperature, µ_nB [cm²/Vs] is the electron mobility, and m_B (>0) is a doping-dependent power factor. Moreover,

$$n_{iB}^2\left(T_j\right)=A·T_j^3·\exp \left[-\frac{E_G\left(T_j\right)-\Delta E_{GB}}{k{T}_j}\right]=A·T_j^3·\exp \left[-\frac{V_G\left(T_j\right)-\Delta V_{GB}}{k{T}_j/q}\right]$$

(4)

where A is a temperature-insensitive term, E_G [eV] is the bandgap of the base semiconductor, ΔE_GB [eV] is a potential bandgap narrowing (e.g., due to heavy doping or presence of a Ge mole fraction), and V_G and ΔV_GB are the voltage equivalents of E_G and ΔE_GB, respectively. Since for temperatures higher than 250 K V_G(T_j) can be reasonably approximated by

$$V_G\left(T_j\right)\approx V_{G0}-\chi ·T_j$$

(5)

where V_G0 = 1.21 V and χ = 2.85 × 10⁻⁴ V/K for Si [43], V_G0 = 1.57 V and χ = 4.85 × 10⁻⁴ V/K for GaAs, (4) becomes

$$n_{iB}^2\left(T_j\right)=A·\exp \left(\frac{\chi ·q}{k}\right)·T_j^3·\exp \left(-\frac{V_{G0}-\Delta V_{GB}}{k{T}_j/q}\right)=B·T_j^3·\left(-\frac{V_{G0}-\Delta V_{GB}}{k{T}_j/q}\right)$$

(6)

Using (2), (3) and (6), (1) turns into

$$\begin{array}{ll}{I}_C& =M\left(V_{CB}\right)·\left(1+\frac{V_{CB}}{V_{AF}}\right)·\frac{1}{B_{HI}\left(I_C\right)}·C·T_j^{4-m_B}·\exp \left(-\frac{V_{G0}-\Delta V_{GB}}{k{T}_j/q}\right)·\exp \left(\frac{V_{BEj}}{\eta ·k{T}_j/q}\right)\\ & \approx M\left(V_{CB}\right)·\left(1+\frac{V_{CB}}{V_{AF}}\right)·\frac{1}{B_{HI}\left(I_C\right)}·C·T_j^{4-m_B}·\exp \left[\frac{V_{BEj}-\left(V_{G0}-\Delta V_{GB}\right)}{\eta ·k{T}_j/q}\right]\end{array}$$

(7)

where C is a temperature- and bias-independent term.

From (7), it can be easily found that

$$V_{BEj}=V_{G0}-\Delta V_{GB}-\eta ·\frac{k{T}_j}{q}·\ln \frac{M\left(V_{CB}\right)·\left(1+\frac{V_{CB}}{V_{AF}}\right)·C·T_j^{4-m_B}}{I_C·B_{HI}\left(I_C\right)}$$

(8)

Let us now evaluate the (positive) temperature coefficient ϕ [V/K] given by

$$\varphi =-{\left.\frac{\partial V_{BEj}}{\partial T_j}\right|}_{I_C,V_{CB}}$$

(9)

which inherently assumes that T_j increases by varying T_B. From (8),

$$\begin{array}{ll}\varphi & =\eta ·\frac{k}{q}·\left(4-m_B\right)-\eta ·\frac{k}{q}·\ln \frac{I_C·B_{HI}\left(I_C\right)}{M\left(V_{CB}\right)·\left(1+\frac{V_{CB}}{V_{AF}}\right)·C·T_j^{4-m_B}}\\ & =\eta ·\frac{k}{q}·\left(4-m_B\right)+\eta ·\frac{k}{q}·\ln \frac{C·T_j^{4-m_B}}{A_E·J_{S0}}-\eta ·\frac{k}{q}·\ln \frac{I_C·B_{HI}\left(I_C\right)}{M\left(V_{CB}\right)·\left(1+\frac{V_{CB}}{V_{AF}}\right)·A_E·J_{S0}}\\ & \approx \eta ·\frac{k}{q}·\left(4-m_B\right)+\eta ·\frac{k}{q}·\ln \frac{C·T_0^{4-m_B}}{A_E·J_{S0}}-\eta ·\frac{k}{q}·\ln \frac{I_C·B_{HI}\left(I_C\right)}{M\left(V_{CB}\right)·\left(1+\frac{V_{CB}}{V_{AF}}\right)·A_E·J_{S0}}={\varphi }_0-\eta ·\frac{k}{q}·\ln \frac{I_{CT0}}{A_E·J_{S0}}\end{array}$$

(10)

where ${\varphi }_0=\eta ·\frac{k}{q}·\left[\left(4-m_B\right)+\ln \frac{C·T_0^{4-m_B}}{A_E·J_{S0}}\right]$ [V/K] is a bias- and almost-temperature-independent parameter (typically falling in the range 3 to 6 mV/K) [1,18], J_S0 [A/cm² or A/µm²] is the reverse saturation current density at the reference temperature T₀, and I_CT0 [A] is the collector current without avalanche, Early, and high-injection effects. Coefficient ϕ does not markedly depend on the transistor layout (i.e., on A_E).

Let us consider a practical case where the bipolar transistor is operated in a common-base configuration with an assigned I_E and assume that V_CB is kept low enough to avoid the avalanche effect (I_C = I_CT ≈ I_E) and that the device does not suffer from a significant Early effect. From (8) it is obtained that

$$V_{BEj}=V_{G0}-\Delta V_{GB}-\eta ·\frac{k{T}_j}{q}·\ln \frac{C·T_j^{4-m_B}}{I_E·B_{HI}\left(I_E\right)}$$

(11)

If the junction temperature T_j is not higher than 400 K, (11) can be reasonably approximated as

$$V_{BEj}\approx V_{G0}-\Delta V_{GB}-\eta ·\frac{k{T}_j}{q}·\ln \frac{C·T_0^{4-m_B}}{I_E·B_{HI}\left(I_E\right)}$$

(12)

which can be expressed as

$$V_{BEj}\approx V_{BEj}\left(T_0\right)-\varphi ·\left(T_j-T_0\right)$$

(13)

where V_BEj(T₀) = V_BEj(T_j = T₀). The temperature coefficient $\varphi =-{\left.\frac{\partial V_{BEj}}{\partial T_j}\right|}_{I_E}$, calculated from (11), becomes

$$\varphi ={\varphi }_0-\eta ·\frac{k}{q}·\ln \frac{I_E·B_{HI}\left(I_E\right)}{A_E·J_{S0}}$$

(14)

From (13), it is clear that

• by increasing the backside (or baseplate, or ambient) temperature T_B through a thermochuck at given values of I_E and V_CB, the junction temperature T_j increases, and V_BEj decreases almost linearly with T_j;
• by increasing V_CB at T_B = T₀ and at an assigned I_E, the dissipated power P_D [W] increases, the junction temperature T_j increases, and V_BEj decreases almost linearly with T_j.

Note that (13) can be expressed in terms of the externally-measurable V_BE as follows:

$$V_{BE}-R_B\left(T_j\right)·I_B\left(T_j\right)-R_E\left(T_j\right)·I_E\approx V_{BE}\left(T_0\right)-R_B\left(T_0\right)·I_B\left(T_0\right)-R_E\left(T_0\right)·I_E-\varphi ·\left(T_j-T_0\right)$$

(15)

By neglecting the I_B variation with T_j, and assuming temperature-insensitive parasitic resistances R_B and R_E [18], (15) can be rewritten as

$$V_{BE}\approx V_{BE}\left(T_0\right)-\varphi ·\left(T_j-T_0\right)$$

(16)

that is, the V_BE–T_j characteristic is only shifted upward with respect to the V_BEj−T_j counterpart, while exhibiting the same slope $\left(\varphi =-{\left.\frac{\partial V_{BEj}}{\partial T_j}\right|}_{I_E}\approx -{\left.\frac{\partial V_{BE}}{\partial T_j}\right|}_{I_E}\right)$. If I_E is selected not too high to neglect high-injection effects (B_HI ≈ 1), then (14) reduces to [1,10,15,16,18,19,31,44,45]

$$\varphi ={\varphi }_0-\eta ·\frac{k}{q}·\ln \frac{I_E}{A_E·J_{S0}}$$

(17)

2.2. Thermal Resistance

As mentioned in Section 1, the static thermal behavior of a semiconductor device is well 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 (simply denoted as heat source). By specifically referring to a bipolar transistor, R_TH is defined as

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

(18)

where ΔT_j is the junction temperature rise above backside and P_D is the dissipated power, given by

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

(19)

The thermal resistance 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 since for the same P_D the dissipated power density is higher, and therefore T_j is also higher. Similarly, the adoption of materials with low thermal conductivities hinders the heat flow, thus leading to an increase in R_TH.

In addition, it must be considered that the thermal conductivities k [W/µmK] of semiconductors and metals in a transistor decrease with temperature ([46] and references therein), 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. The device temperature in turn increases for two distinct physical mechanisms: (i) the increase in backside temperature T_B (nonlinear thermal effect due to 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), where the dependence on T_B and P_D implicitly comes from the k reduction with increasing temperature [47].

3. Simulation Approach

3.1. Devices under Test

Similar to [46], the analysis was conducted on two NPN HBT technologies.

The InGaP/GaAs NPN HBT is a mesa-isolated device manufactured by Qorvo with four 2 × 20.5 µm² emitter fingers (and thus the total emitter area 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. The key features of this device are reported in

Table 1. Key features of the InGaP/GaAs NPN HBT under test.

Parameter	Value
Common-emitter current gain βF at 300 K and medium current levels	150
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

. Further technological details are provided in [15].

The Si/SiGe NPN HBT was fabricated by Infineon Technologies AG in the framework of the European Project DOTFIVE. The device has only one base and one collector contact (BEC configuration), and belongs to the latest project technology stage, also denoted as set #3 in [18,44]. The drawn emitter area is equal to 0.2 × 2.8 µm², and the substrate is 185 µm thick. The figures of merit of this device are listed in

Table 2. Key features of the Si/SiGe NPN HBT under test.

Parameter	Value
Common-emitter current gain βF at 300 K and medium current levels	2200
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

.

3.2. Transistor Model

We chose to resort to an in-house analytical model to describe the DC operation of the bipolar transistor, as it is simple, accurate enough, and enables a low-effort parameter extraction procedure. This offers high flexibility throughout the whole investigation. The collector current I_C in forward active mode is expressed as [15,44,45]

$$I_C=M·I_{CT}=M·\left(1+\frac{V_{CB}}{V_{AF}}\right)·\frac{1}{B_{HI}}·A_E·J_{S0}·\exp \left[\frac{V_{BEj}+\varphi ·\left(T_j-T_0\right)}{\eta ·V_{T0}}\right]$$

(20)

where all terms have the same meaning as in Section 2.1, and coefficient ϕ is given by the logarithmic law (10). In this approach, the temperature dependence of I_C is taken into account with a V_BEj shift, while the reverse saturation current density J_S0 and the thermal voltage V_T0 = kT₀/q are kept at their T₀ values (e.g., [48]).

As far as the avalanche factor M is concerned, any model can in principle be adopted. For the InGaP/GaAs HBT under test, we chose the classic Miller formulation given by [49]

$$M=\frac{1}{1-{\left(\frac{V_{CB}}{B{V}_{CBO}}\right)}^{n_{AV}}}$$

(21)

where BV_CBO [V] is the open-emitter breakdown voltage and the dimensionless n_AV (>0) is a fitting power factor. For the Si/SiGe HBT, we selected the more complex model [50,51]

$$M=1+a_{AV}·\frac{V_{CB}/B{V}_{CBO}}{1-V_{CB}/B{V}_{CBO}}·\exp \left[-b_{AV}·{\left(\frac{V_{CB}}{B{V}_{CBO}}\right)}^{-c_{AV}}\right]$$

(22)

where a_AV, b_AV, and c_AV (all > 0) are dimensionless fitting parameters.

The high-injection (high-current) attenuation term is modeled as [15,44,45,52]

$$B_{HI}=1+{\left(\frac{I_C}{A_E·J_{HI}}\right)}^{n_{HI}}$$

(23)

where J_HI [A/cm² or A/µm²] and n_HI (>0) are fitting parameters.

The common-emitter forward current gain β_F is described as [15,18,44,45]

$${\mathsf{\beta }}_F={\mathsf{\beta }}_{F0}·\left(1+\frac{V_{CB}}{V_{AF}}\right)·\frac{1}{B_{HI}}·\exp \left[\frac{\Delta E_{GEB}}{k}·\left(\frac{1}{T_j}-\frac{1}{T_0}\right)\right]$$

(24)

where β_F0 is the gain at T₀, at medium current levels (i.e., before the high-injection-induced fall-off) and in the absence of Early effect, while ΔE_GEB = E_GE − E_GB [eV] is the difference between the bandgaps of emitter and base, which is positive in HBTs and entails a negative temperature coefficient of β_F, and negative in BJTs (due to the band-gap narrowing in the heavily doped emitter), where it leads to a positive temperature coefficient.

The base current I_B is given by

$$I_B=I_{BT}-I_{AV}$$

(25)

where I_BT [A] is the current of holes injected into the emitter and I_AV is the avalanche-induced current of holes exiting the base terminal (or equivalently of electrons entering the collector terminal). Considering that

$$I_{BT}=\frac{I_{CT}}{{\beta }_F}$$

(26)

and that I_AV can also be expressed as

$$I_{AV}=I_C-I_{CT}=\left(M-1\right)·I_{CT}$$

(27)

then (25) becomes [15,42,45]

$$I_B=\frac{I_{CT}}{{\mathsf{\beta }}_F}-\left(M-1\right)·I_{CT}=I_{CT}·\left(\frac{1+{\mathsf{\beta }}_F}{{\mathsf{\beta }}_F}-M\right)=I_C·\left(\frac{1}{{\mathsf{\alpha }}_F·M}-1\right)$$

(28)

α_F being the common-base forward current gain. The emitter current I_E is obviously given by I_C + I_B.

The static power-temperature feedback (i.e., the evaluation of T_j from P_D) including nonlinear thermal effects is accounted for according to the conclusions reached in [46], which can be summarized as follows. 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,

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

(29)

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 low-power thermal resistance R_THB0 at an arbitrary T_B in the range 250 to 450 K (nonlinear thermal effect due to the backside temperature) can be calculated as [22,26,47,53]

$$R_{THB0}=R_{TH00}·{\left(\frac{T_B}{T_0}\right)}^{\mathsf{\alpha }}$$

(30)

where α (>0) is a dimensionless fitting parameter. The further thermal resistance growth due to the increase in P_D (nonlinear self-heating effect) can be accounted for by invoking the Kirchhoff transformation as [5,22,26,47,53]

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

(31)

where α is the same parameter applied in (30). Using (30) in (31), the following R_TH expression is obtained:

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

(32)

In [46], R_TH was determined for the InGaP/GaAs HBT and the Si/SiGe HBT under test in reasonably wide ranges of T_B and P_D by extremely detailed nonlinear COMSOL [54] simulations, and it was observed that (32) allows obtaining a good agreement with COMSOL data if a “brute-force 2-D search” of parameters R_TH00 and α is performed, as suggested in [26]. Conveniently, it was found that the optimized R_TH00 is very close to that computed by COMSOL, regardless of the HBT technology.

3.3. Circuit-Based Electrothermal Simulation

The model detailed in Section 3.2 was implemented in the popular PSPICE circuit simulator [55] as a subcircuit, where the standard bipolar transistor instance is used as a core component at temperature T₀. Besides the collector, emitter, and base terminals, the subcircuit is equipped with an additional (input) thermal node and an additional (output) power node. The thermal node is fed with the temperature rise ΔT_j = T_j − T_B, while the power node provides the dissipated power P_D, internally computed according to (19). Apart from the standard transistor, the subcircuit is enhanced with linear/nonlinear controlled voltage/current sources to enable the variation of the temperature-sensitive parameters during the simulation run, as well as to account for avalanche, Early, and high-injection effects. Further details are given in [15].

Equation (32) is in turn implemented as follows (solution #1 in [46]). First, R_THB0 is evaluated in the pre-simulation stage from (30) at the assigned T_B, and the dissipated power P_D (a current in PSPICE) is forced to flow into R_THB0 (an electrical resistance). The temperature rise (a voltage drop) given by R_THB0·P_D is provided as input to a behavioral block (a nonlinear voltage-controlled voltage source) that calculates ΔT_j as

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

(33)

which is then forced to the thermal node of the subcircuit; T_j = T_B + ΔT_j influences the collector current I_C (20) and the common-emitter current gain β_F (24).

3.4. Validation Methodology

The parameters of the transistor model described in Section 3.2 were tuned to achieve a good agreement between the DC ET characteristics of the devices under test simulated with PSPICE and experimental data. Here the calibration procedure is omitted for the sake of brevity. In (30) and (32), R_TH00 = 460 K/W and α = 0.95 for the InGaP/GaAs HBT, and R_TH00 = 6855.8 K/W and α = 1.333 for the Si/SiGe HBT.

Then, DC ET simulations of the devices are performed in PSPICE to emulate the experimental current/voltage data needed for the application of the thermometer-based techniques (Section 4). The extracted R_TH results are then compared to the target formulation (32) embedded in the transistor model. This is equivalent to feeding the techniques with ideal (noiseless) measurements; as a consequence, any disagreement between the extracted data and (32) is only ascribable to the nature of the adopted extraction technique. The approach of using simulation data (also referred to as synthetic data) based on a known target model has been already applied in many previous works dealing with bipolar transistors [20,22,31,41,56,57,58].

It is worth noting that we are currently conducting a similar analysis also for other HBT technologies by means of Keysight ADS [59] simulations of advanced compact transistor models such as AgilentHBT [60] and HICUM [61] equipped with an R_TH given by the target (32). Until now, the findings are in line with those shown and discussed in Section 4.

4. Analysis of Thermometer-Based Experimental R_TH Extraction Techniques

In this section, the thermometer-based R_TH extraction techniques are discussed in chronological order by using a unified and comprehensible nomenclature. First, the analytical theory behind them is explained in detail; then, the techniques are applied to synthetic current/voltage data obtained with DC ET PSPICE simulations of the transistor models corresponding to the devices under test introduced in Section 3.1; finally, the extracted data are compared to the target (32).

The pioneering method of Waldrop et al. [23], which exploits β_F as a TSEP, is excluded from the review, since it is based on a complex and time-consuming procedure difficult to implement in an extraction code, as correctly noticed in [40,41]. Consequently, all the techniques presented in the following make use of V_BE as TSEP. All of them must be applied to a transistor operated in forward active mode at a backside temperature T_B ≥ T₀.

4.1. Dawson et al. [24]

In the historically-important paper of Dawson et al. [24], the R_TH extraction is made in a two-fold way, namely, using either β_F or V_BE as TSEPs. Hereafter only the approach based on V_BE is considered.

As a first step, the authors calibrate the thermometer on a device with a ground-signal-ground (GSG, with grounded emitter) configuration as follows. V_BE is measured as a function of T_B by keeping constant I_E and V_CE. To ensure a constant I_E, I_B is tuned at each T_B, which is not a simple task. Dawson et al. observe that V_BE shows a nearly linear decrease with T_B and determine the absolute value ϕ′ of the slope of the straight line described by

$$V_{BE}=V_{BE}\left(T_B=T_0\right)-{\varphi }^{\prime }·\left(T_B-T_0\right)$$

(34)

allowing the best fit at low/medium T_B values. Subsequently, they assume negligible self-heating, i.e., T_j ≈ T_B, ϕ′ ≈ ϕ, so that (34) can be approximated with (16).

By exploiting (18), (16) becomes

$$V_{BE}=V_{BE}\left(T_0\right)-\varphi ·\left(T_B+R_{TH}·P_D-T_0\right)=V_{BE}\left(T_0\right)-\varphi ·\left(T_B-T_0\right)-\varphi ·R_{TH}·P_D$$

(35)

The V_BE–T_B characteristics are then measured at different V_CE values (low enough to avoid avalanche) at the same I_E. This allows plotting the I_E-constant V_BE–V_CE curves at each T_B, and from simple elaboration also the corresponding V_BE–P_D curves, which exhibit a linearly decreasing behavior as well. By considering the T_B = T₀ case, (35) reduces to

$$V_{BE}=V_{BE}\left(T_0\right)-\varphi ·R_{TH}·P_D$$

(36)

Dawson et al. extract the (negative) slope

$$\nu =\frac{d{V}_{BE}}{d{P}_D}=-\varphi ·R_{TH}$$

(37)

whence R_TH is simply calculated as

$$R_{TH}=-\frac{\nu }{\varphi }=\frac{\left|\nu \right|}{\varphi }$$

(38)

Apart from the practical problem of continuously tuning I_B to hold I_E constant (as pointed out in [25]), which can be solved by fabricating an identical device with accessible emitter pad, this technique is based on a simple mathematical theory and seems to be reliable. However, there are various mechanisms that potentially affect the accuracy of the R_TH extraction (and can in principle also jeopardize improved variants of the technique). Such mechanisms are explained in detail in the following subsections.

4.1.1. Significant Self-Heating in the Thermometer Calibration

In (38), Dawson et al. do not use the coefficient ϕ describing the slope of the V_BEj–T_j behavior, but rather the absolute value ϕ′ of the slope of the straight line matching with the experimental I_E- and V_CE-constant V_BE–T_B characteristic, that is, they actually evaluate R_TH as

$$R_{TH}=\frac{\left|\nu \right|}{{\varphi }^{\prime }}$$

(39)

The approximation ϕ′ ≈ ϕ could in principle be inaccurate, since ϕ depends on I_E and thus the I_E used in the first measurement (thermometer calibration) must be the same applied in the second measurement (leading to the I_E-constant V_BE–P_D characteristic at T_B = T₀), where an appreciable self-heating should be ensured; consequently, the I_E used in the first measurement cannot be very low, and this unavoidably leads to non-negligible self-heating even for the smallest V_CE driving the transistor into forward active mode. This mechanism can be analytically described as follows [18]. As explained in Section 2.1, for an assigned I_E, by neglecting the variation of the voltage drops over the parasitic base and emitter resistances due to self-heating, if T_B is swept, V_BE linearly decreases with T_j according to (16), where the absolute value ϕ of the slope only depends on I_E for HBTs marginally impacted by the Early effect. Let us start from (35), derived by (16) and (18), and let us use (19) for the dissipated power P_D. It is obtained that

$$\begin{array}{ll}{V}_{BE}& =V_{BE}\left(T_0\right)-\varphi ·\left(T_B-T_0\right)-\varphi ·R_{TH}·\left(I_E·V_{BE}+I_C·V_{CB}\right)\\ & \approx V_{BE}\left(T_0\right)-\varphi ·\left(T_B-T_0\right)-\varphi ·R_{TH}·I_E·\left(V_{BE}+V_{CB}\right)\end{array}$$

(40)

whence

$$V_{BE}=\frac{V_{BE}\left(T_0\right)-\varphi ·R_{TH}·I_E·V_{CB}}{1+\varphi ·R_{TH}·I_E}-\frac{\varphi }{1+\varphi ·R_{TH}·I_E}·\left(T_B-T_0\right)=V_{BE}\left(T_B=T_0\right)-{\varphi }^{\prime }·\left(T_B-T_0\right)$$

(41)

It should be noted that (i) in the common-emitter I_E-constant V_BE–T_B measurement performed by Dawson et al., V_CB is almost unchanged (in other techniques, this measurement is executed under common-base conditions with constant V_CB); (ii) T_B = T₀, in the presence of self-heating, corresponds to T_j > T₀; consequently, V_BE(T_B = T₀) is lower than V_BE(T₀) = V_BE(T_j = T₀); (iii) as the most important finding inferred from (41), coefficient ϕ′ extracted by Dawson et al. is lower than the ϕ that should be actually adopted in (38), and this could in principle lead to an overestimation of R_TH. However, it must be remarked that this analysis only focuses on linear thermal effects; the influence of nonlinear thermal effects is discussed in Section 4.1.4.

4.1.2. Significant Temperature-Induced Variation of the Voltage Drop over the Base Resistance

If a high I_E is selected for the extraction, the variation in the base current I_B with temperature cannot be disregarded any longer (especially if R_B is high), and (16) no longer holds; instead, for temperature-insensitive parasitic resistances, (15) reduces to

$$V_{BE}-R_B·I_B\left(T_j\right)=V_{BE}\left(T_0\right)-R_B·I_B\left(T_0\right)-\varphi ·\left(T_j-T_0\right)$$

(42)

By applying (18)

$$V_{BE}=V_{BE}\left(T_0\right)+R_B·\left[I_B\left(T_j\right)-I_B\left(T_0\right)\right]-\varphi ·R_{TH}{P}_D-\varphi ·\left(T_B-T_0\right)$$

(43)

and then (19)

$$V_{BE}=V_{BE}\left(T_0\right)+R_B·\left[I_B\left(T_j\right)-I_B\left(T_0\right)\right]-\varphi ·R_{TH}·I_E·\left(V_{BE}+V_{CB}\right)-\varphi ·\left(T_B-T_0\right)$$

(44)

it is obtained that

$$V_{BE}=\frac{V_{BE}\left(T_0\right)-\varphi ·R_{TH}·I_E·V_{CB}}{1+\varphi ·R_{TH}·I_E}+\frac{R_B·\left[I_B\left(T_j\right)-I_B\left(T_0\right)\right]}{1+\varphi ·R_{TH}·I_E}-\frac{\varphi }{1+\varphi ·R_{TH}·I_E}·\left(T_B-T_0\right)$$

(45)

By increasing T_B for a constant I_E, I_C(T_j) decreases due to the negative temperature coefficient of β_F, and thus I_B(T_j) increases. When such effect is not negligible, the actual absolute value ϕ′ of the slope of the V_BE–T_B characteristic is even lower than $\frac{\varphi }{1+\varphi ·R_{TH}·I_E}$, and this could cause an overestimation of R_TH more marked than that dictated only by the linear self-heating [18].

On the other hand, let us focus on the second step aiming at the R_TH assessment, i.e., the extraction of the slope ν of the I_E-constant V_BE–P_D curve at T_B = T₀. In this case, (43) becomes

$$V_{BE}\approx V_{BE}\left(T_0\right)+R_B·\left[I_B\left(T_j\right)-I_B\left(T_0\right)\right]-\varphi ·R_{TH}{P}_D$$

(46)

By increasing P_D at a constant I_E, I_B(T_j) grows, and then the absolute value of the slope |ν| is lower than ϕ·R_TH, which could lead to an underestimation of R_TH [18]. In conclusion, applying a high I_E gives rise to two counteracting effects: a further reduction of the extracted ϕ′ with respect to ϕ that could lead to an overestimation in R_TH; a reduction of the slope |ν| with respect to ϕ·R_TH that could yield an underestimation in R_TH. In general, it is difficult to predict which effect dominates.

4.1.3. Significant Early Effect

If significant, the Early effect can be misinterpreted by the extraction technique as an additional overheating, thus leading to an overestimation of R_TH [19,31,56]. This can be analytically explained as follows. Let us neglect that ϕ is sensitive to the Early effect and increases with V_CB. Under biasing conditions at which avalanche and high-injection effects can be neglected, (20) reduces to

$$\begin{array}{cc}\hfill I_C& =\left(1+\frac{V_{CB}}{V_{AF}}\right)·A_E·J_{S0}·\exp \left[\frac{V_{BEj}+\varphi ·\left(T_j-T_0\right)}{\eta ·V_{T0}}\right]=\left(1+\frac{V_{CB}}{V_{AF}}\right)·A_E·J_{S0}·\exp \left[\frac{V_{BE}-R_B·I_B-R_E·I_E+\varphi ·\left(T_j-T_0\right)}{\eta ·V_{T0}}\right]\hfill \\ & \approx \left(1+\frac{V_{CB}}{V_{AF}}\right)·A_E·J_{S0}·\exp \left[\frac{V_{BE}-R_{EB}·I_C+\varphi ·\left(T_j-T_0\right)}{\eta ·V_{T0}}\right]\hfill \end{array}$$

(47)

where $R_{EB}=\frac{R_B+R_E}{{\beta }_F}+R_E$. Assuming T_B = T₀ and absence of self-heating, then T_j = T₀ and (47) becomes

$$I_C=\left(1+\frac{V_{CB}}{V_{AF}}\right)·A_E·J_{S0}·\exp \left[\frac{V_{BE}\left(T_0\right)-R_{EB}·I_C}{\eta ·V_{T0}}\right]$$

(48)

from which V_BE(T₀) can be determined as

$$V_{BE}\left(T_0\right)=R_{EB}·I_E+\eta ·V_{T0}·\ln \frac{I_E}{\left(1+\frac{V_{CB}}{V_{AF}}\right)·A_E·J_{S0}}$$

(49)

where use has been made of the approximation I_C ≈ I_E. By substituting (49) into (16),

$$V_{BE}=R_{EB}·I_E+\eta ·V_{T0}·\ln \frac{I_E}{\left(1+\frac{V_{CB}}{V_{AF}}\right)·A_E·J_{S0}}-\varphi ·\left(T_j-T_0\right)$$

(50)

and using (18)

$$V_{BE}=R_{EB}·I_E+\eta ·V_{T0}·\ln \frac{I_E}{\left(1+\frac{V_{CB}}{V_{AF}}\right)·A_E·J_{S0}}-\varphi ·R_{TH}·P_D-\varphi ·\left(T_B-T_0\right)$$

(51)

which at T_B = T₀ becomes

$$V_{BE}=R_{EB}·I_E+\eta ·V_{T0}·\ln \frac{I_E}{\left(1+\frac{V_{CB}}{V_{AF}}\right)·A_E·J_{S0}}-\varphi ·R_{TH}·P_D$$

(52)

It can be inferred that, due to the Early effect, the absolute value |ν| of the slope of the I_E-constant V_BE–P_D curve at T_B = T₀ is higher than that due to self-heating only, as V_CB increases and leads to a reduction in V_BE(T₀); this is misinterpreted by the techniques as an additional self-heating, so that R_TH could be overestimated.

It will be seen that the Early effect has a perceptible influence only on the extraction results corresponding to the Si/SiGe HBT under test, the V_AF of which is high, but lower than that of the InGaP/GaAs transistor.

4.1.4. Significant Nonlinear Thermal Effects

For the sake of simplicity, let us disregard the mechanisms explored in Section 4.1.2 and Section 4.1.3. As mentioned in Section 2.2, nonlinear thermal effects make the thermal resistance a monotonically-increasing function of T_B and P_D; as a consequence, (41) should be more correctly formulated as

$$V_{BE}=\frac{V_{BE}\left(T_0\right)-\varphi ·R_{TH}\left(T_B,P_D\right)·I_E·V_{CB}}{1+\varphi ·R_{TH}\left(T_B,P_D\right)·I_E}-\frac{\varphi }{1+\varphi ·R_{TH}\left(T_B,P_D\right)·I_E}·\left(T_B-T_0\right)$$

(53)

In the first measurement, P_D is almost constant, and R_TH(T_B,P_D) increases due to the T_B sweep. By approximating the V_BE–T_B data with a straight line, it is found that the absolute value ϕ′ of the slope of this line is higher than $\frac{\varphi }{1+\varphi ·R_{TH00}·I_E}$, which means that the error due to the presence of the nonlinear thermal effect associated to the T_B increase gives rise to an unquantifiable compensation of the error associated to the presence of the linear self-heating.

In the second measurement, T_B = T₀ and (36) becomes

$$V_{BE}=V_{BE}\left(T_0\right)-\varphi ·R_{TH}\left(T_0,P_D\right)·P_D$$

(54)

Here, V_BE reduces more than linearly with P_D, as R_TH(T₀,P_D) increases with P_D (nonlinear self-heating effect), which leads to an extracted |ν| higher than that dictated by the linear self-heating; this causes an overestimation of R_TH with respect to R_TH00, which is expected to be exacerbated as the P_D range wherein the extraction is carried out increases.

As a rule of thumb, it would be better to apply the lowest I_E giving rise to perceptible self-heating and a rather narrow P_D range in order to prevent a significant nonlinear self-heating effect during the second measurement. Unfortunately, the self-heating and nonlinear thermal effect due to the T_B increase during the thermometer calibration cannot be fully disregarded: the extracted ϕ′ can be lower than ϕ if the first effect prevails or higher than ϕ if the second dominates.

For the InGaP/GaAs HBT, I_E = 10 mA and V_CE = 2 V were chosen with the aim to keep the device in forward active mode, minimize the self-heating and the nonlinear thermal effect due to T_B in the first measurement, and (with reference solely to I_E = 10 mA) ensure a perceptible self-heating during the second measurement. The extracted ϕ′ is 1.175 mV/K (higher than ϕ = 1.141 mV/K, as the nonlinear thermal effect due to T_B prevails over the linear self-heating), while the extracted R_TH is 468.1 K/W by superiorly limiting the P_D range to 0.03 W, with an error of 1.76% with respect to R_TH00 = 460 K/W induced by the nonlinear self-heating effect in the second measurement.

For the Si/SiGe HBT, I_E = 1 mA and V_CE = 1 V were selected. The extracted ϕ′ is 0.856 mV/K (higher than ϕ = 0.829 mV/K for the same reason). The extracted R_TH is 7196.7 K/W by superiorly limiting the P_D range to 1.5 mW, with an error equal to 5% with respect to R_TH00 = 6855.8 K/W due to the Early and the nonlinear self-heating effect in the second measurement.

In conclusion, in both cases R_TH > R_TH00, since the inaccuracy in the second step dominates over the error leading to ϕ′ > ϕ in the first step, which paradoxically plays a beneficial compensation role.

4.2. Bovolon et al. [25]

Dawson et al. [24] observe that only two backside temperatures T_B and T_B + ΔT_B and two power dissipation levels P_D and P_D + ΔP_D are in principle needed to determine R_TH (“differential” variant of the approach). Coefficient ϕ can be indeed calculated as

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

(55)

and the (negative) slope ν as

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

(56)

so that (38) becomes

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

(57)

The technique proposed by Bovolon et al. [25] can be considered as an extension of the “differential” variant of the approach by Dawson et al., as it allows determining the influence of nonlinear thermal effects (i.e., the T_B and P_D dependences of R_TH). In the assumption of validity of (16), Bovolon et al. observe that the linear V_BE decrease with T_j also takes place locally, i.e., around a certain junction temperature T_j*:

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

(58)

By exploiting (18) and accounting for nonlinear thermal effects, (58) becomes

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

(59)

If (59) is applied to another backside temperature T_B + ΔT_B at the same dissipated power P_D,

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

(60)

Let us subtract (60) from (59) by assuming ΔT_B sufficiently low to neglect the related R_TH variation; this leads to

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

(61)

so that

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

(62)

If (59) is applied to another dissipated power P_D + ΔP_D,

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

(63)

Let us subtract (63) from (59) by assuming that ΔP_D is sufficiently small to disregard the related R_TH variation and the ϕ dependence on I_E; it is obtained that

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

(64)

Finally, by combining (62) and (64),

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

(65)

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, the technique requires the measurement of V_BE and I_C by sweeping V_CE under I_B-constant conditions at T_B and T_B + ΔT_B. The evaluation of R_TH(T_B,P_D) at the given T_B is carried out as follows. Once a dissipated power P_D is chosen, two points with close dissipated powers P_D and P_D + ΔP_D are selected on the V_BE–V_CE curve corresponding to T_B, and the related V_BE(T_B,P_D) and V_BE(T_B,P_D + ΔP_D) values are used for the calculation of the numerator of (65). Then, the point with dissipated power P_D has to be also identified on the V_BE–V_CE curve corresponding to T_B + ΔT_B, and the associated V_BE(T_B + ΔT_B,P_D) allows the calculation of the denominator of (65). This procedure is repeated for other P_D values. Then the whole process is applied to another T_B.

Due to its differential nature (R_TH is determined by two differences between V_BE values measured in two points only), this technique is expected to suffer from inaccuracy induced by the following reasons: (i) ΔT_B and ΔP_D should be chosen sufficiently small to ensure (61) and (64), respectively. However, if ΔT_B and ΔP_D are too small, the error associated to noisy data is emphasized when calculating R_TH with (65); (ii) the collector currents corresponding to P_D and P_D + ΔP_D should be quite close to safely neglect the ϕ dependence on I_E; (iii) since for assigned P_D and ΔP_D values the measurement will not exactly provide V_BE(T_B,P_D + ΔP_D) on the first characteristic and V_BE(T_B + ΔT_B,P_D) on the second, an interpolation between adjacent points might be needed.

For the InGaP/GaAs HBT, the extraction was performed by applying I_B = 0.7 mA, ΔP_D = 10 mW, ΔT_B = 10 K. For the Si/SiGe HBT, it was carried out by applying I_B = 2.5 µA, ΔP_D = 0.1 mW, ΔT_B = 10 K. Unfortunately, in the latter case, the low BV_CEO value (=1.6 V) leads to a quite narrow P_D range where the avalanche multiplication is negligible and thus the technique can be adopted. Figure 1 shows the comparison between the R_TH data extracted from DC ET simulations with the target ones evaluated from (32) for both technologies. It can be inferred that the technique of Bovolon et al. roughly predicts the increase in R_THB0 with T_B but fails to describe the R_TH increase with P_D at a given T_B (nonlinear self-heating effect) due to the issues mentioned earlier.

4.3. Yeats [26]

The technique conceived by Yeats [26] can also be reviewed as an improved version of that presented by Dawson et al. [24] in that it allows accounting for both nonlinear thermal effects. Electrical common-base measurements are performed on a transistor equipped with a directly-accessible emitted pad. First, some I_E-constant 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 various backside temperatures T_B. Through a quadratic polynomial with I_E-dependent coefficients, 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, described by means of another quadratic polynomial with I_E-dependent coefficients, and then inverted into T_j–V_BE. 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 (18).

It is worth noting that this technique directly uses the thermometer to evaluate R_TH, differently from other methods that are also based on the differential calculation ΔV_BE/ΔP_D [25] or the more robust extraction of the slope ν of the V_BE–P_D curve [24,27,29] (or equivalently the extraction of the slope γ of the V_BE–V_CB curve [18,31]) under I_E-constant conditions.

This technique is often applied to GaAs-based HBTs, especially for T_j estimates used in reliability.

For the InGaP/GaAs and the Si/SiGe HBTs, I_E = 20 mA and 2 mA were applied, respectively. Results are shown in Figure 2, where again the extracted data are compared to the target formulation (32). As can be seen, for the InGaP/GaAs HBT, the agreement is excellent. A slight R_TH overestimation is instead obtained for the Si/SiGe HBT while the increase induced by the nonlinear self-heating effect is well described. Such overestimation can be explained as follows. The temperature T_j is predicted with a very small overestimation; however, subtracting T_B and dividing by the small P_D dissipated in this scaled transistor to calculate R_TH = ΔT_j/P_D magnifies the error.

Although this technique works fairly well on synthetic data, it must be observed that it directly uses the thermometer to determine R_TH, and therefore it is very sensitive to the V_BE values, which can be noisy since are difficult to measure with a high degree of accuracy.

4.4. Pfost et al. [28]

Pfost et al. [28] perform measurements on a transistor in a GSG configuration. They first measure V_CE-constant I_C–V_BE characteristics at various T_B values, and then derive by interpolation the V_BE–T_B curves at various I_C values. The V_CE is chosen sufficiently low to safely neglect the avalanche effect, while the Early effect was assumed negligible. The authors then calibrate the thermometer by extracting the absolute value ϕ′ of the slope of the straight line ensuring the best fit with the V_BE–T_B curves, and observe that this coefficient depends on the technology, is almost insensitive to the transistor layout, and decreases with I_C. Then they assume that there is negligible self-heating and thus ϕ′ ≈ ϕ.

Subsequently, Pfost et al. consider that in the absence of avalanche, Early, and high-injection effects, the I_C formulation (20) reduces to

$$I_C=A_E·J_{S0}·\exp \left[\frac{V_{BE}-R_{EB}·I_C+\varphi ·\left(T_j-T_0\right)}{\eta ·V_{T0}}\right]$$

(66)

At low V_BE (low I_C), self-heating and resistive effects can be neglected, and (66) turns into

$$I_C\approx A_E·J_{S0}·\exp \left(\frac{V_{BE}}{\eta ·V_{T0}}\right)$$

(67)

Parameters J_S0 and η can be extracted by comparing the experimental V_CE-constant I_C–V_BE curve at T_B = T₀ with (67) at low V_BE (low I_C). From (66), it can be obtained that

$$V_{BE}=R_{EB}·I_C-\varphi ·\left(T_j-T_0\right)+\eta ·V_{T0}·\ln \frac{I_C}{A_E·J_{S0}}$$

(68)

while the V_BE determined by extrapolating the low-current behavior (referred to as V_BELC) can be derived from (67) as

$$V_{BELC}=\eta ·V_{T0}·\ln \frac{I_C}{A_E·J_{S0}}$$

(69)

The deviation of the measured V_BE from V_BELC can be obtained by subtracting (69) from (68):

$$\Delta V_{BE}=V_{BE}-V_{BELC}=R_{EB}·I_C-\varphi ·\left(T_j-T_0\right)$$

(70)

Using (18) with T_B = T₀, (70) can be rewritten as

$$\Delta V_{BE}=R_{EB}·I_C-\varphi ·R_{TH}·P_D$$

(71)

where the I_C dependence of coefficient ϕ has been determined in the first step of the procedure.

Pfost et al. measure the I_C–V_BE curves at T_B = T₀ applying various V_CE values. Then they select a quite high current I_C (which should in principle be low enough to avoid high-injection effects) and obtain the experimental ΔV_BE vs. P_D behavior (P_D being approximatively given by I_C·V_CE) at I_C. By extrapolating the behavior to P_D→0 W, R_EB·I_C can be assessed, from which (I_C being assigned) R_EB is also determined. Then, they consider that (71) can be rearranged as

$$R_{TH}=\frac{R_{EB}·I_C-\Delta V_{BE}}{\varphi \left(I_C\right)·P_D}$$

(72)

As ϕ is known for the assigned I_C, and R_EB has been estimated, R_TH(T₀,P_D) can be obtained by using the given P_D and the measured ΔV_BE values in (72).

For the InGaP/GaAs HBT, the extracted ϕ′ at I_C = 8 mA was 1.174 mV/K (higher than ϕ = 1.16 mV/K since the nonlinear thermal effect due to the T_B increase prevails over the linear self-heating). Concerning the further step for the R_TH assessment, it must be remarked that the measurement of the T_B = T₀ V_CE-constant I_C–V_BE curves by sweeping V_BE is critical, especially at high V_CE, as the marked self-heating may give rise to an irreversibly destructive thermal runaway (or equivalently to a flyback followed by a negative differential resistance branch by sweeping I_C [1,3]). In this technology, ΔV_BE given by (71) is negative for each V_CE (each P_D at I_C = 8 mA) since the self-heating prevails over the resistive effect. The R_EB value is assessed with very good accuracy. The extracted R_TH(T₀,P_D) vs. P_D is compared to the target (32) in Figure 3a; in the narrow P_D range analyzable in this approach, the R_TH increase dictated by the nonlinear self-heating effect is not accurately described, which can be attributed to the sensitivity of the method to the V_BE values to be identified on the V_CE-constant T_B = T₀ I_C–V_BE curves at a chosen I_C. Anyway, the error corresponding to a given P_D is not so high.

For the Si/SiGe HBT, I_C = 3 mA was chosen to apply the procedure. A lower I_C would lead to problems in identifying the V_BE values on the V_CE-constant I_C–V_BE curves. Unfortunately, at this I_C the high-injection and nonlinear self-heating effects play a role. Coefficient ϕ′ was extracted to be 0.696 mV/K (higher than ϕ = 0.675 mV/K since the nonlinear thermal effect due to T_B dominates). In this technology, ΔV_BE given by (71) is instead positive for each V_CE due to the prevailing resistive effect. The R_EB value is not properly extracted, as the high-injection effects, not included in (66) and thus (72), are misinterpreted as an additional resistive contribution. Due to the overestimated aggregate voltage drop R_EB·I_C, the extracted R_TH(T₀,P_D) is higher than the target counterpart given by (32), as shown in Figure 3b.

4.5. Rieh et al. [7,27]

Rieh et al. [7,27] perform common-base measurements on a transistor with a directly-accessible emitter pad. First, I_E and V_CB are assigned, T_B is swept in a range of practical interest, and V_BE is measured, so that the I_E- and V_CB-constant V_BE–T_B curve is available. Then the transistor is biased with the same I_E, T_B = T₀ is applied, V_CB is increased, and V_BE, I_B are measured; this allows obtaining the I_E-constant V_BE–V_CB curve, and, calculating P_D as I_E·V_BE + (I_E − I_B)·V_CB, also the corresponding V_BE–P_D characteristic. The two datasets can then be combined; more specifically, V_BE is eliminated to obtain T_B as a function of P_D.

From a mathematical point of view, the approach is articulated as follows. It was found that in the absence of avalanche, assuming temperature-insensitive parasitic resistances, and neglecting the drop $R_B·\left[I_B\left(T_j\right)-I_B\left(T_0\right)\right]$, (16) holds. From (16), the V_CB- and I_E-constant V_BE–T_B curve can be modeled by (53), which inherently accounts for self-heating and the nonlinear thermal effect due to T_B. By neglecting self-heating, (53) reduces to

$$V_{BE}\approx V_{BE}\left(T_B=T_0\right)-\varphi ·\left(T_B-T_0\right)$$

(73)

In the second measurement, by disregarding the nonlinear self-heating effect, (36) is obtained. By combining (36) and (73)

$$V_{BE}\left(T_B=T_0\right)-\varphi ·\left(T_B-T_0\right)=V_{BE}\left(T_0\right)-\varphi ·R_{TH}·P_D$$

(74)

whence the experimental T_B–P_D curve is described by

$$T_B=T_0-\frac{V_{BE}\left(T_0\right)-V_{BE}\left(T_B=T_0\right)}{\varphi }+R_{TH}·P_D$$

(75)

On the basis of (75), R_TH is obtained as the slope of the straight line fitting this curve, as shown in Figure 4.

Subsequently, Rieh et al. use the following procedure to obtain the T_j–P_D behavior starting from the available T_B–P_D one. First, they note that the above straight line intercepts the y-axis (P_D→0 W) at a T_B value < T₀, referred to as T_B(P_D = 0 W); then, they consider that the desired T_j–P_D curve would intercept the y-axis at T_j = T₀; hence, they simply add T₀ − T_B(P_D = 0 W) to the T_B–P_D curve to achieve the T_j–P_D counterpart.

Critical points of the technique are listed below:

• To eliminate V_BE, an interpolation process leading to the same V_BE values (i) in the I_E- and V_CB-constant V_BE–T_B data and (ii) in the I_E-constant V_BE–P_D data at T_B = T₀ is needed.
• As noted by Vanhoucke et al. [29] and also by Rieh and his co-workers in [7], (73) is not correct due to the self-heating (and the T_B-induced nonlinear thermal effect) occurring during the measurement of the I_E- and V_CB-constant V_BE–T_B curve.
• The presence of the Early and of the nonlinear self-heating effect during the second measurement might lead to an R_TH overestimation.

For the InGaP/GaAs HBT, I_E = 10 mA and V_CB = 0.5 V were applied during the first measurement, and the same I_E value was adopted for the second measurement. Coefficient ϕ′, extracted over a T_B span from 300 to 400 K, turned out to be 1.164 mV/K (higher than ϕ = 1.141 mV/K as the nonlinear thermal effect due to T_B prevails over the linear self-heating). Despite this, R_TH was extracted to be 468 K/W by limiting the P_D range to 0.03 W (with an error of 1.74% compared to R_TH00) and 486.3 K/W by limiting the P_D range to 0.082 W (with an error of 5.7%) due to the nonlinear self-heating effect during the second measurement.

For the Si/SiGe HBT, I_E = 1 mA and V_CB = 0.2 V were applied during the first measurement. Coefficient ϕ′ was extracted to be 0.849 mV/K (higher than ϕ = 0.829 mV/K). The extracted R_TH was 7241 K/W by superiorly limiting the P_D range to 1.55 mW (with an error of 5.62% with respect to R_TH00) due to the Early and the nonlinear self-heating effect in the second measurement. This conclusion was supported by a further analysis performed by removing ad hoc the Early effect from the Si/SiGe HBT model and limiting the R_TH extraction to P_D = 1.3 mW; using the resulting synthetic data, the extracted R_TH was 6940 K/W (with an error of 1.23% with respect to R_TH00).

4.6. Vanhoucke et al. [29]

Vanhoucke et al. [29] recognize that during the first measurement the transistor can be affected by self-heating; as a consequence, they assume that the I_E- and V_CB-constant V_BE–T_B can be well approximated by a straight line described by (41), while the I_E-constant V_BE–P_D curve at T_B = T₀ can be modeled using (36). By eliminating V_BE from (41) and (36), it is obtained that

$$T_B=T_0+\left(1+\varphi ·R_{TH}·I_E\right)·\left[-\frac{V_{BE}\left(T_0\right)-V_{BE}\left(T_B=T_0\right)}{\varphi }+R_{TH}·P_D\right]$$

(76)

where V_BE(T₀) = V_BE(T_j = T₀). From (76), it can be easily inferred that R_TH is not given by the slope of the experimental T_B–P_D characteristic, as described by Rieh and his co-authors [7,27]. Vanhoucke et al. suggest improving the R_TH estimation as follows. From the knowledge of ${\varphi }^{\prime }=\frac{\varphi }{1+\varphi ·R_{TH}·I_E}$ and ν = −ϕ·R_TH (negative slope of the straight line ensuring the best fit with the V_BE–P_D curve), ϕ can be eliminated to obtain

$${\varphi }^{\prime }=\frac{-\frac{\nu }{R_{TH}}}{1-\nu ·I_E}=\frac{\left|\nu \right|}{R_{TH}·\left(1+\left|\nu \right|·I_E\right)}$$

(77)

whence R_TH is calculated as

$$R_{TH}=\frac{-\nu }{{\varphi }^{\prime }·\left(1-\nu ·I_E\right)}=\frac{\left|\nu \right|}{{\varphi }^{\prime }·\left(1+\left|\nu \right|·I_E\right)}$$

(78)

For the InGaP/GaAs HBT, again I_E = 10 mA and V_CB = 0.5 V were applied during the first measurement, and the same I_E value was applied during the second measurement. Clearly, the extracted ϕ′ value does not change compared to Rieh et al. (=1.165 mV/K), while the R_TH values slightly reduce: 465.5 and 483.6 K/W by limiting the P_D range to 0.03 and 0.09 W, respectively.

For the Si/SiGe HBT, again I_E = 1 mA and V_CB = 0.2 V were applied during the first measurement. Coefficient ϕ′ was equal to 0.849 mV/K. The extracted R_TH was 7196 K/W by limiting the P_D range to 1.55 mW.

It must be noted that in practice the slight accuracy improvement obtained with this technique is due to an unintentional increase of the error associated to the first measurement, where ϕ′ is assumed to be lower than ϕ (while being actually higher), so that the inaccuracy corresponding to the second measurement is better compensated.

4.7. University of Bordeaux

A thermometer-based technique relying on DC measurements was conceived and applied at University of Bordeaux, as reported in some PhD theses (e.g., [30]). Measurements are performed on a bipolar transistor in a GSG configuration. The base current I_B is kept constant, V_CE is swept, and V_BE, I_C are measured at various T_B values. By evaluating the dissipated power P_D with (19), it is possible to plot the I_B-constant V_BE–P_D curves at all the applied T_B, and, choosing a specific V_BE, the I_B- and V_BE-constant T_B–P_D characteristic can be derived. Let us then consider that from (18)

$$T_B=T_j-R_{TH}·P_D$$

(79)

By arbitrarily assuming T_j constant along this characteristic, R_TH can be determined from the slope.

For the InGaP/GaAs HBT, I_B was chosen equal to 0.7 mA and V_BE equal to 1.3 V; lower V_BE values would have pushed the P_D range under analysis to higher values, thus aggravating the nonlinear self-heating effect. The extracted R_TH was found to be 488 K/W, higher than R_TH00 due to the slightly decreasing T_j along the curve and to the nonlinear self-heating effect.

For the Si/SiGe HBT, I_B was chosen equal to 20 µA and V_BE equal to 0.88 V; choosing a lower I_B does not allow applying the technique, as it is impossible to find a V_BE at which two avalanche-free points can be intercepted. The extracted R_TH was found to be 7278 K/W considering the T_B span from 300 to 340 K, and 7424 K/W considering the T_B span from 300 to 360 K, as dictated by the slight T_j decrease along the T_B–P_D curve and to the nonlinear self-heating effect.

4.8. d’Alessandro et al. [18,31]

The technique developed by d’Alessandro et al. [18] is articulated as follows. Given a bipolar transistor with accessible emitter operated in common-base configuration, the V_CB- and I_E-constant V_BE–T_B characteristic is measured at very low I_E. Under these conditions, the self-heating (and thus the nonlinear thermal effect due to T_B) can be safely neglected; consequently, (53) reduces to (73). Hence, coefficient ϕ can be extracted at the applied I_E. The measurement is then repeated at other, very low as well, I_E values. Hence, the experimental ϕ vs. I_E behavior is obtained, and parameter ϕ₀ of the logarithmic law (17) can be easily calibrated. In [18], it has been demonstrated that, for a given technology node, ϕ₀ is almost independent of the transistor layout, in accordance with the theory formulated in Section 2. The accuracy of the ϕ₀ calibration is verified through the following procedure. In the absence of avalanche, Early, high-injection, and resistive effects, (20) reduces to

$$I_C\approx A_E·J_{S0}·\exp \left[\frac{V_{BE}+\varphi ·\left(T_j-T_0\right)}{\eta ·V_{T0}}\right]$$

(80)

If the self-heating is negligible, (80) becomes:

$$I_C\approx A_E·J_{S0}·\exp \left[\frac{V_{BE}+\varphi ·\left(T_B-T_0\right)}{\eta ·V_{T0}}\right]$$

(81)

Here, ϕ is a function of I_E according to (17). Considering that I_E ≈ I_C, (17) can be rewritten as

$$\varphi ={\varphi }_0-\eta ·\frac{k}{q}·\ln \frac{I_C}{A_E·J_{S0}}$$

(82)

Substituting (82) into (81), it is obtained that

$$\frac{I_C}{A_E·J_{S0}}=\exp \left[\frac{V_{BE}+{\varphi }_0·\left(T_B-T_0\right)-\eta ·\frac{k}{q}·\left(T_B-T_0\right)·\ln \frac{I_C}{A_E·J_{S0}}}{\eta ·V_{T0}}\right]$$

(83)

By applying the logarithm to both sides of (83),

$$\eta ·V_{T0}·\ln \frac{I_C}{A_E·J_{S0}}=V_{BE}+{\varphi }_0·\left(T_B-T_0\right)-\eta ·\frac{k}{q}·\left(T_B-T_0\right)·\ln \frac{I_C}{A_E·J_{S0}}$$

(84)

whence

$$I_C=A_E·J_{S0}·\exp \left[\frac{V_{BE}+{\varphi }_0·\left(T_B-T_0\right)}{\eta ·V_{T0}+\eta ·\frac{k}{q}·\left(T_B-T_0\right)}\right]=A_E·J_{S0}·\exp \left[\frac{V_{BE}+{\varphi }_0·\left(T_B-T_0\right)}{\eta ·\frac{k{T}_B}{q}}\right]$$

(85)

If the experimental V_CE-constant (with low V_CE) I_C–V_BE curves measured at various T_B are favorably described by (85) with optimized ϕ₀ at low V_BE (low I_C), the accuracy of the calibration is verified [18,19].

Once ϕ₀ is calibrated, (17) can be exploited to determine ϕ for higher I_E values, thus avoiding all the mechanisms leading to the extraction of a ϕ′ different from ϕ; improved accuracy at particularly high I_E values can be obtained using (14), which however requires a preliminary calibration of parameters J_HI and n_HI. To summarize, this technique solves all the issues associated to the thermometer calibration, as it allows an accurate evaluation of the ϕ value corresponding to the I_E to be used in the second measurement.

As usual, the second measurement is performed at T_B = T₀ by forcing an I_E value high enough to entail perceptible self-heating, and V_CB is increased; the I_E value and the V_CB range should not lead to significant nonlinear self-heating effect to allow the extraction of an R_TH close to R_TH00. In this case, (36) is assumed to be valid. From simple elaboration of the experimental data, V_BE is obtained as a function of P_D, the V_BE–P_D behavior is almost linear, with a slope ν = −ϕ·R_TH, and R_TH is calculated with (38).

As an equivalent alternative, the measured V_BE–V_CB data can be directly used without further elaboration. Let us consider that (36) can be recast as

$$V_{BE}\approx V_{BE}\left(T_0\right)-\varphi ·R_{TH}·I_E·\left(V_{BE}+V_{CB}\right)$$

(86)

from which

$$V_{BE}=\frac{V_{BE}\left(T_0\right)}{1+\varphi ·R_{TH}·I_E}-\frac{\varphi ·R_{TH}·I_E}{1+\varphi ·R_{TH}·I_E}·V_{CB}$$

(87)

where ϕ at the selected I_E is computed with (17) or (14). As can be seen, the I_E-constant V_BE–V_CB behavior is also expected to be linear. By extracting the (negative) slope γ of the straight line ensuring the best matching with experimental data, R_TH can be calculated as

$$R_{TH}=\frac{\left|\gamma \right|}{\varphi ·I_E·\left(1-\left|\gamma \right|\right)}$$

(88)

For the InGaP/GaAs HBT, the extracted R_TH was equal to 476.5 K/W at I_E = 10 mA by superiorly limiting V_CB to 2.3 V (with an error of 3.59% compared to R_TH00 due to the nonlinear self-heating effect in the second measurement). It can be inferred that the error is paradoxically higher than that corresponding to less elaborate techniques since in this case coefficient ϕ is well evaluated, and the compensation of errors does not take place.

For the Si/SiGe HBT, the extracted R_TH was equal to 7412.6 K/W at I_E = 1 mA by limiting V_CB to 0.75 V (with an error of 8.12% with respect to R_TH00), which is due to the Early and the nonlinear self-heating effect, as well as to the absence of compensation of errors. Removing ad hoc the Early effect from the HBT model, and repeating the extraction on the new synthetic data, R_TH = 7104.7 K/W by limiting the simulation to V_CB = 0.5 V (with an error of 3.63%).

Improved variants of the technique by d’Alessandro et al. were developed to purify the R_TH extraction from the Early effect [31], which is expected to play a relevant role in PNP HBTs and in Si bipolar junction transistors (BJTs), regardless of their application. For the sake of brevity, here we describe only the first approach, which requires the preliminary determination of V_AF from common-emitter I_C–V_CE measurements; another strategy presented in [31,56] and inspired by an early paper of Sparkes [62] is not based on the knowledge of V_AF.

Making use of (19), (52) can be rewritten as (I_C ≈ I_E):

$$V_{BE}\approx R_{EB}·I_E+\eta ·V_{T0}·\ln \frac{I_E}{\left(1+\frac{V_{CB}}{V_{AF}}\right)·A_E·J_{S0}}-\varphi ·R_{TH}·I_E·\left(V_{BE}+V_{CB}\right)$$

(89)

from which

$$V_{BE}=\frac{1}{1+\varphi ·R_{TH}·I_E}·\left[R_{EB}·I_E-\varphi ·R_{TH}·I_E·V_{CB}+\eta ·V_{T0}·\ln \frac{I_E}{\left(1+\frac{V_{CB}}{V_{AF}}\right)·A_E·J_{S0}}\right]$$

(90)

Hence, the I_E-constant V_BE–V_CB behavior at T_B = T₀ is still nearly linear, with an absolute value |γ| of the slope higher than that in the absence of the Early effect, and approximately given by

$$\left|\gamma \right|\approx \frac{\varphi ·R_{TH}·I_E}{1+\varphi ·R_{TH}·I_E}+\frac{1}{1+\varphi ·R_{TH}·I_E}·\frac{\eta ·V_{T0}}{V_{AF}}$$

(91)

Once |γ| is extracted, R_TH is easily determined as

$$R_{TH}=\frac{\left|\gamma \right|-\frac{\eta ·V_{T0}}{V_{AF}}}{\left(1-\left|\gamma \right|\right)·\varphi ·I_E}$$

(92)

Exploiting this approach for the Si/SiGe HBT (where V_AF = 110 V), the extracted R_TH was equal to 7113 K/W.

4.9. Summary of the Main Findings

Some techniques like those developed by Dawson et al. [24], Rieh et al. [7,27], and Vanhoucke et al. [29] are based on a first measurement aimed at calibrating the thermometer, which, if the emitter pad is accessible, is performed by assigning I_E and V_CB, measuring V_BE as a function of T_B, and extracting the absolute value ϕ′ of the slope of the straight line ensuring the best fit with experimental data. Then, the R_TH assessment is carried out by measuring V_BE as a function of P_D (or equivalently as a function of V_CB) by keeping I_E constant at the same value as in the first measurement and T_B = T₀, extracting the (negative) slope ν of the straight line providing the best alignment with the V_BE–P_D characteristic (or equivalently the slope γ of the straight line matching with the V_BE–V_CB characteristic), and elaborating this slope with the temperature coefficient ϕ of the “internal” base-emitter voltage V_BEj.

Unfortunately, the techniques of Dawson et al. [24] and Rieh et al. [7,27] are based on the assumption that the extracted ϕ′ coincides with ϕ. Since the chosen I_E is expected to trigger a perceptible self-heating in the second measurement, self-heating will also take place in the first measurement, even though V_CB is low. In the ideal absence of nonlinear thermal effects, the linear self-heating would lead to ϕ′ < ϕ. Instead, the increase in R_TH induced by the T_B sweep implies that ϕ′ > ϕ; moreover, the higher is the T_B range, the higher ϕ′. Using ϕ′ in combination with |ν| (or |γ|) for the final thermal resistance evaluation could in principle give rise to an underestimation of R_TH.

In the second measurement, two mechanisms deserve attention. First, the Early effect can be misinterpreted as an additional overheating and thus could lead to an overestimation of R_TH. Second, the nonlinear self-heating effect due to the increase in P_D makes the thermal resistance grow along the V_BE–P_D (or V_BE–V_CB) curve. Consequently, although ϕ′ is higher than ϕ, the extracted R_TH is always higher than R_TH00. This overestimation is exacerbated when the extraction of ν (or γ) is performed over a larger P_D (or V_CB) range; hence, if the aim is to assess R_TH00, the maximum P_D (or V_CB) should be selected as small as possible.

Paradoxically, the adoption of ϕ′ > ϕ in the elaboration for the R_TH assessment mitigates the above overestimation, that is, the accuracy improves due to a compensation of errors.

Vanhoucke et al. [29] disregard the nonlinear thermal effect due to T_B in the first measurement, and thus assume that ϕ′ < ϕ due to the linear self-heating, which would in principle lead to an R_TH overestimation. Owing to this, they conceive an improved formula to evaluate R_TH from |ν| and ϕ′ that accounts for the self-heating in the first measurement, and extract an R_TH slightly lower than that obtained by Dawson et al. [24] and Rieh et al. [7,27]. However, in practice ϕ′ > ϕ, so the technique of Vanhoucke et al. [29] unintentionally exacerbates the error associated to the first measurement, and thus allows obtaining a slightly higher accuracy in the R_TH evaluation due to a more marked compensation of errors.

The technique of d’Alessandro et al. [18] allows solving the self-heating issue related to the first measurement by determining an accurate ϕ model as a function of the applied emitter current I_E. Unfortunately, in the basic approach, the resulting error is higher than that obtained with other techniques, as the discrepancy between R_TH and R_TH00 due to the Early and the nonlinear self-heating effect is no longer compensated by the error made in the thermometer calibration. Improved variants of the technique allow purifying the result from the impact of the Early effect.

Among the techniques aimed at determining the impact of nonlinear thermal effects on R_TH, namely, those developed by Bovolon et al. [25] and Yeats [26] for the whole R_TH(T_B,P_D) behavior, and Pfost et al. [28] for the R_TH(T₀,P_D) behavior, the one of Yeats has been found to be very accurate when applied to simulated data. However, in practice this technique is very sensitive to the measured V_BE values, as it is based on the direct use of the thermometer to find the junction temperature T_j and then the R_TH at the assigned I_E. Consequently, noisy V_BE data are expected to perceptibly affect the R_TH extraction despite the polynomial fitting. The method by Pfost et al. is cumbersome to apply to experimental data, as it is based on the detection of points at the same I_C on various I_C–V_BE characteristics at T_B = T₀ and different V_CE values; as this detection must be carried out at high I_C, the extraction accuracy can be affected by high-injection effects, and in some critical cases the HBT could be even destroyed by the occurrence of a thermal runaway. All these considerations are summarized in

Table 3. Main features of the thermometer-based R_TH extraction techniques.

Technique	Advantages, Approximations, and Limitations
Dawson et al. [24] and Rieh et al. [7,27]	The technique developed by Rieh et al. can be considered a variant of the classical approach of Dawson et al., the latter being more complex as it is applied to a device with grounded emitter, and the first being simpler as it is applied to a device with accessible emitter, for which it is possible to force an assigned emitter current. In both techniques, coefficient ϕ′ extracted in the first measurement is higher than the desired ϕ due to the nonlinear thermal effect induced by the TB increase on the RTH. Conveniently, the resulting error (which would give rise to an RTH underestimation) is compensated by another error in the second measurement dictated by the Early and nonlinear self-heating effects (such an error tends to overestimate RTH). By keeping IE sufficiently low in both measurements and choosing a limited PD range in the second measurement, these techniques allow extracting RTH00 with a fairly good accuracy for the InGaP/GaAs and Si/SiGe HBT devices under test, which do not exhibit a significant Early effect. However, it must be remarked that this accuracy originates from a compensation of errors; hence, further analyses should be performed to establish if a similar compensation takes place also in other HBT technologies devised for RF applications.
Bovolon et al. [25]	This technique can be reviewed as an extension of the differential variant of the approach of Dawson et al. conceived to extract RTH as a function of TB and PD. Unfortunately, due to its differential nature and to the underlying assumptions, this technique suffers from a marked inaccuracy in describing the impact of the nonlinear self-heating effect, regardless of the HBT technology.
Yeats [26]	This technique represents an extension of the approaches of Dawson et al. and Rieh et al. aimed to extract RTH vs. TB and PD. The method allows determining accurate results when applied to simulated data (corresponding to ideal noiseless measurements). However, as the approach is based on the direct use of the thermometer to evaluate the junction temperature Tj, noisy VBE data coming from real measurements are expected to jeopardize the extraction accuracy.
Pfost et al. [28]	This approach is developed to extract the RTH dependence on PD at TB = T0; differently from other techniques, the method operates on IC–VBE characteristics and is quite critical, as it is based on the detection of points at the same IC on characteristics measured at different VCE values. This detection is indeed possible when IC (VBE) is high, but in severe cases increasing VBE can lead to thermal runaway, and in milder cases the extraction accuracy can be affected by high-injection effects.
Vanhoucke et al. [29]	This technique is conceived to improve the approaches of Dawson et al. and Rieh et al. by mitigating the error due to self-heating in the first measurement. However, Vanhoucke et al. improperly assume that only linear self-heating takes place, while the prevailing mechanism is the RTH increase due to the nonlinear thermal effect induced by the TB sweep. Hence, this technique unintentionally exacerbates the error associated to the first measurement, and paradoxically improves the compensation of errors, thus leading to a slightly higher accuracy in the RTH00 extraction. Again, as the accuracy derives from a compensation of errors, it is difficult to predict what might happen by applying this method to other HBT technologies.
University of Bordeaux	This technique is based on common-emitter measurements performed at various TB values by sweeping VCE and keeping IB constant. By elaborating the results and selecting a VBE value, the IB- and VBE-constant TB–PD curve is obtained, and the RTH is determined from its slope. Unfortunately, the IB and VBE values are not simple to choose, which makes the method quite difficult to apply. For the HBT technologies under test, the extracted RTH is higher than RTH00 due to an underlying approximation and to the nonlinear self-heating effect.
d’Alessandro et al. [18,31]	The technique in [18] aims to improve the accuracy in the extraction of coefficient ϕ with respect to the approaches of Dawson et al. and Rieh et al., the price to pay being an increased elaboration effort. Consequently, in this case the compensation of errors does not take place, and this technique overestimates RTH with respect to RTH00 due to the nonlinear self-heating effect and the Early effect in the second measurement. The inaccuracy dictated by the nonlinear self-heating effect can be alleviated by limiting the PD range in which the RTH extraction is performed. The extended versions in [31] allow purifying the extraction from the Early effect, which is misinterpreted as additional self-heating. As a result, differently from all other techniques, the approaches in [31] can be adopted not only to NPN HBTs, but also to PNP HBTs and Si BJTs, where the Early effect plays a more relevant role.

and sketched in Figure 5, which illustrates the main features of the investigated techniques and the correlations among them as a function of the year of publication.

5. Conclusions

In this paper, a critical review of DC indirect techniques for the experimental extraction of the thermal resistance of bipolar transistors from straightforward current/voltage measurements has been presented. The accuracy verification has been performed by (i) simulating the DC characteristics of the devices under test through PSPICE electrothermal simulation of a simple, yet accurate enough, transistor model, (ii) applying the techniques to the current/voltage data, and (iii) comparing the extracted thermal resistance to the target one implemented in the transistor model. The impact of both nonlinear thermal effects has been explained in detail. An InGaP/GaAs HBT and a Si/SiGe HBT for high-frequency applications have been selected as case-studies. Results obtained by performing Keysight ADS simulations of compact transistor models such as AgilentHBT and HICUM also for other HBT technologies are in line with those shown and discussed in this paper.