Experimental Determination, Modeling, and Simulation of Nonlinear Thermal Effects in Bipolar Transistors under Static Conditions: A Critical Review and Update
Abstract
:1. Introduction
2. Thermal Resistance Dependence on Nonlinear Thermal Effects
3. Review of Experimental RTH Extraction Techniques Accounting for Nonlinear Thermal Effects
3.1. Bovolon et al.
3.2. Yeats
3.3. Marsh
3.4. Paasschens et al.
3.5. Menozzi et al.
3.6. Berkner and Balanethiram et al.
3.7. Huszka et al.
4. Numerical Simulation
4.1. Devices under Test
4.2. Results
5. RTH Formulations Available in Compact Transistor Models and Update
5.1. VBIC
5.2. Mextram 504
5.3. AgilentHBT (AHBT)
5.4. HiCUM
5.5. Proposed RTH Modeling Approaches
6. Conclusions
- 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.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
βF | common-emitter forward current gain |
BVCBO | open-emitter breakdown voltage [V] |
BVCEO | open-base breakdown voltage [V] |
∆Tj = Tj − TB | junction temperature rise over backside temperature [K] |
fMAX | maximum oscillation frequency [Hz] |
fT | unity-gain cut-off frequency (or transition frequency) [Hz] |
IB | base current [A] |
IC | collector current [A] |
IE | emitter current [A] |
k | thermal conductivity [W/µmK] |
PD | dissipated power [W] |
RB | parasitic base series resistance [Ω] |
RE | parasitic emitter series resistance [Ω] |
RTH or RTH(TB,PD) | self-heating thermal resistance [K/W] |
RTH00 = RTH(TB = T0,PD→0 W) | self-heating thermal resistance at TB = T0 and a very low PD (ideally for PD→0 W) |
RTH0P = RTH(TB = T0,PD) | self-heating thermal resistance at TB = T0 and an arbitrary PD |
RTHB0 = RTH(TB,PD→0 W) | self-heating thermal resistance at a generic TB and a very low PD (ideally for PD→0 W) |
TB | backside (or baseplate, or ambient) temperature [K] |
Tj | average temperature of the base-emitter junction (junction temperature) at arbitrary TB and PD [K] |
Tj00 | junction temperature at TB = T0 and a very low PD [K] |
Tj0P | junction temperature at TB = T0 and an arbitrary PD [K] |
TjB0 | junction temperature at an arbitrary TB and a very low PD [K] |
T0 = 300 K | reference temperature [K] |
VAF | forward Early voltage [V] |
VBE | externally applied base-emitter voltage [V] |
VBEj = VBE − RB·IB − RE·IE | internal (or junction) base-emitter voltage [V] |
VCB | externally applied collector-base voltage [V] |
VCE | externally applied collector-emitter voltage [V] |
Abbreviations
AlGaAs | ternary alloy formed by introducing a given mole fraction of aluminum (Al) in the place of Ga in the GaAs lattice |
BJT | bipolar junction transistor |
Cu | copper |
ET | electrothermal |
FEM | finite-element method |
GaAs | gallium arsenide |
GaN | gallium nitride |
HBT | heterojunction bipolar transistor |
InGaAs | ternary alloy created by introducing an assigned mole fraction of indium (In) in the place of Ga in the GaAs lattice |
InGaP | ternary alloy formed by replacing a given mole fraction of Ga with indium (In) in the GaP lattice |
InP | indium phosphide |
NDR | negative differential resistance |
NTC | negative temperature coefficient |
PDR | positive differential resistance |
PTC | positive temperature coefficient |
Si | silicon |
SiC | silicon carbide, with 4H-SiC and 6H-SiC polytypes |
SiGe | silicon-germanium: binary alloy formed by substituting an assigned mole fraction of Si with germanium (Ge) in the Si lattice |
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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] |
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]. |
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 |
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 |
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d’Alessandro, V.; Catalano, A.P.; Scognamillo, C.; Müller, M.; Schröter, M.; Zampardi, P.J.; Codecasa, L. Experimental Determination, Modeling, and Simulation of Nonlinear Thermal Effects in Bipolar Transistors under Static Conditions: A Critical Review and Update. Energies 2022, 15, 5457. https://doi.org/10.3390/en15155457
d’Alessandro V, Catalano AP, Scognamillo C, Müller M, Schröter M, Zampardi PJ, Codecasa L. Experimental Determination, Modeling, and Simulation of Nonlinear Thermal Effects in Bipolar Transistors under Static Conditions: A Critical Review and Update. Energies. 2022; 15(15):5457. https://doi.org/10.3390/en15155457
Chicago/Turabian Styled’Alessandro, Vincenzo, Antonio Pio Catalano, Ciro Scognamillo, Markus Müller, Michael Schröter, Peter J. Zampardi, and Lorenzo Codecasa. 2022. "Experimental Determination, Modeling, and Simulation of Nonlinear Thermal Effects in Bipolar Transistors under Static Conditions: A Critical Review and Update" Energies 15, no. 15: 5457. https://doi.org/10.3390/en15155457
APA Styled’Alessandro, V., Catalano, A. P., Scognamillo, C., Müller, M., Schröter, M., Zampardi, P. J., & Codecasa, L. (2022). Experimental Determination, Modeling, and Simulation of Nonlinear Thermal Effects in Bipolar Transistors under Static Conditions: A Critical Review and Update. Energies, 15(15), 5457. https://doi.org/10.3390/en15155457