Analysis of Electrothermal Effects in Devices and Arrays in InGaP/GaAs HBT Technology
Abstract
:1. Introduction
- More details of the individual transistor model and its subcircuit implementation are provided.
- Differently from [15], where the arrays were assumed to lie on an unthinned GaAs substrate (as typical for known-good-die identification), here they are considered in a realistic phone-board environment, i.e., the substrate is thinned and attached on a laminate, the bottom of which is at TB = 358 K.
- Similar to [21], in this work the linear power-temperature feedback is described by invoking FANTASTIC [22,23], which is fed with the COMSOL geometry/mesh accompanied with additional information (on position/shape of heat sources, boundary conditions, and thermal conductivities), and rapidly extracts an ETN based on the RTH matrix without performing simulations. Contrary to conventional ETNs (like the one used in [15]), the FANTASTIC network allows reconstructing the overall temperature field for selected bias conditions in a post-processing step.
- In [15], the Kirchhoff’s transformation was applied by assuming that all materials share the same nonlinear thermal behavior as GaAs. Unfortunately, this was found to lead to a perceptible overestimation of ET effects. Here, more realistic results are achieved by carrying out a suitable preliminary calibration procedure, similar to that made in [21].
2. Devices and Arrays
- an In0.5Ga0.5As cap to reduce the contact resistance with the gold (Au)-based emitter metallization;
- a grading InxGa1-xAs layer (with x spanning from 0.5 to 0) used to ensure a good lattice continuity with the underneath layer;
- a GaAs layer acting as a set-back for an easier manufacturing process;
- an n-doped In0.49Ga0.51P emitter layer, the bottom surface of which corresponds to the metallurgical base-emitter junction.
3. Simulation Approach
3.1. General Description
- Each four-finger unit cell is represented with one SPICE-compatible subcircuit (Section 3.3) implementing a simple analytical transistor model (Section 3.2). This assumption relies on the following considerations: (i) the two metals uniformly distribute the temperature over the base-emitter junction; (ii) the electron currents emerging from the four closely-spaced individual emitters are expected to spread and give rise to only one heat source. The subcircuit uses (i) a basic/standard bipolar transistor at reference (and unchangeable) temperature T0 as a core component, and (ii) linear and nonlinear controlled sources to account for the variation of the temperature-sensitive parameters during the simulation run, as well as for other specific mechanisms. According to the TEOL, the temperature rise ΔTj = Tj − T0 averaged over the base-emitter junction (which mainly influences the ET device behavior) is actually a voltage, while the dissipated power PD is treated as a current. In addition to the standard transistor terminals (emitter, base, and collector), the unit-cell subcircuit is also equipped with an input node carrying the “voltage” ΔTj and with an output node offering the “current” PD.
- The power-temperature feedback is described with a SPICE-compatible thermal feedback block (TFB), the construction of which is carried out in a pre-processing stage. The TFB contains an ETN including the matrix of self-heating (SH) RTHs of the unit cells and mutual RTHs among them. The inputs of the ETN are the powers PD dissipated by the cells (represented with currents), and the outputs are their temperature rises ΔTjlin = Tjlin − T0 for the test devices or ΔTjlinB = Tjlin − TB for the arrays (all emulated with voltages) under linear thermal conditions.
- The ETN is automatically determined through the following procedure. First, an accurate 3-D geometry/mesh of the domain is built in the COMSOL environment using an in-house routine; then, the geometry/mesh, along with additional information concerning position/shape of heat sources, boundary conditions, and thermal conductivities, is fed to FANTASTIC, which extracts the ETN in a really short time without the need of user’s intervention/expertise or onerous FEM simulations (Section 3.5). Generally, the whole process is very fast and error-free. The adoption of FANTASTIC is an improvement over our prior contribution [15], where (i) the RTH matrix was calculated by performing N purely-thermal static COMSOL simulations by activating only one heat source at a time, and (ii) the simple ETN adopted did not allow a post-processing reconstruction of the whole temperature map in the domain.
- As mentioned above, the ETN only accounts for linear thermal conditions. However, nonlinear thermal effects can be significant when particularly high temperatures are reached. Such effects are taken into account by making use of the Kirchhoff’s transformation, which converts the linear temperature rises ΔTjlin (test devices) or ΔTjlinB (arrays) offered by the ETN into the nonlinear counterparts ΔTj = Tj − T0 and ΔTjB = Tj − TB, respectively. Contrary to [15], here the transformation was properly calibrated (Section 3.4) to improve the ET simulation accuracy.
- Besides the ETN, the TFB also includes N voltage-controlled voltage sources that apply the calibrated Kirchhoff’s transformation to the ETN linear outcomes; as a result, the nonlinear temperature rises ΔTj (test devices) and ΔTjB (arrays) are computed; only for the arrays, the increment TB − T0 is added to ΔTjB to get the N nonlinear ΔTj = Tj − T0 to be provided to the unit-cell subcircuits.
- The subcircuits are then connected to the TFB in the environment of a commercial circuit simulation tool. As a result, the whole domain is transformed into a purely-electrical macrocircuit, which inherently accounts for ET effects (Section 3.6): the temperature, and thus the temperature-sensitive parameters, are allowed to vary during the simulation run. The task of solving this macrocircuit is given to the powerful and robust engine of the circuit simulation tool, with very low computational effort and minimized occurrence of convergence issues compared to other numerical methods.
3.2. Bipolar Transistor Model
- ICnoAV [A] is the collector current in the absence of impact-ionization (II), or avalanche, effects;
- IAV [A] is the collector current component only induced by avalanche;
- VCB [V] is the collector-base voltage;
- VAF [V] is the forward Early voltage;
- M (≥ 1) is the dimensionless VCB-dependent avalanche multiplication factor;
- AE [µm2] is the emitter area;
- JS0 [A/µm2] is the reverse saturation current density at the reference temperature T0 = 300 K;
- η is the ideality coefficient at T0;
- VT0 = 0.02586 V is the thermal voltage at T0;
- VBEj [V] is the internal (junction) base-emitter voltage, that is, VBEj = VBE − RB·IB − RE·IE, where VBE is the externally-accessible base-emitter voltage, IB, IE [A] are the base and emitter currents, and RB, RE [Ω] are the parasitic base and emitter resistances, respectively;
- the temperature rise ΔTj [K] is defined as Tj − T0, Tj being the temperature averaged over the base-emitter junction;
- ϕ [V/K] is the temperature coefficient of VBEj;
- BHI (≥1) is an IE-dependent dimensionless term introduced to describe the attenuation dictated by high-injection (HI) effects, i.e., the Kirk-induced gain roll-off.
3.3. SPICE Unit-Cell Subcircuit
3.4. Construction of the Geometry/Mesh in COMSOL and Calibration of the Kirchhoff’s Transformation
3.5. FANTASTIC
Algorithm 1: SCTM extraction |
Set V:=0 |
for each heat source n=1, …, N do |
1 Solve (21) for Θn |
2 Update matrix V by appending Θn |
3 Generate a SCTM projecting (12)–(14) onto V |
3.6. Construction of the Macrocircuit
3.7. Extension to the Dynamic Case
- The selected transistor model must be provided with a power (output) node, and the internal one- or two-pair thermal network has to be deactivated.
- All parameters of the model must be extracted from experimental data, which is a nontrivial task.
- As mentioned in Section 3.5, if FANTASTIC is also fed with the mass density and specific heat for all materials, it can be enabled to extract a DCTM of the domain and the associated ETN accounting for the dynamic heat propagation [21,22,23]. Such an ETN, together with the Kirchhoff’s transformation sources, will constitute the SPICE- and ADS-compatible TFB.
- Lastly, the macrocircuit has to be built in ADS by connecting the model instances among them and with the TFB.
4. Results and Discussion
4.1. Test Devices
4.2. Transistor Arrays
- The collapse onset in an IBTOT-constant ICTOT–VCE curve is associated to a uneven current/temperature distribution, wherein the right-column cells (in particular, the inner ones) bear almost all the current.
- The steeper ICTOT drop in the collapse region is induced by a bifurcation mechanism involving the symmetric cells belonging to the right column. In the base-ballasted case (with RBext = 400 Ω) this leads to three adjacent cells conducting all the current (either the top or the bottom ones, depending on small technological/layout discrepancies); a further VCE increase makes the current flow in two cells, and eventually in only one cell (the outer one). In the unballasted and emitter-ballasted case (with REext = 4 Ω), for VCE slightly higher than that entailing the bifurcation, the current flows in only one of the inner cells (#9 or #10). This leads to a very sharp and linear temperature increment vs. VCE of this cell (plainly illustrated for cell #9 in Figure 23b).
- Such a linear nature of the ΔTj9–VCE behavior can be straightforwardly explained as follows. Neglecting II effects, which is reasonable in both the unballasted and emitter-ballasted cases, ΔTj9 is approximately equal to RTH99(Tj9)·βF(Tj9)·IBTOT·VCE + 58 K (IB9 ≈ IBTOT), where RTH99 is the SH thermal resistance of cell #9, and 58 K is the difference between TB and T0; as VCE increases, there is a compensation between the NTC of βF and the increase in RTH99 with temperature due to nonlinear thermal effects.
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
CB | common base |
CE | common emitter |
DCTM | dynamic compact thermal model |
DoF | degree of freedom |
ET | electrothermal |
ETN | equivalent thermal network |
FANTASTIC | FAst Novel Thermal Analysis Simulation Tool for Integrated Circuits |
FEM | finite-element method |
GaAs | gallium arsenide |
HBT | heterojunction bipolar transistor |
HI | high injection |
II | impact ionization (avalanche) |
MOR | model-order reduction |
NDR | negative differential resistance |
NTC | negative temperature coefficient |
PA | power amplifier |
PTC | positive temperature coefficient |
RTH | thermal resistance [K/W] |
SCR | space-charge region |
SCTM | static compact thermal model |
SH | self-heating |
SOG | silicon-on-glass |
TEOL | thermal equivalent of the Ohm’s law |
TFB | thermal feedback block |
T0 | reference temperature: 300 K |
TB | temperature of the laminate bottom for the arrays: 358 K |
Tj | temperature averaged over the base-emitter junction under nonlinear thermal conditions |
Tjlin | temperature averaged over the base-emitter junction under linear thermal conditions |
ΔTj | temperature rise Tj-T0 |
ΔTjlin | temperature rise Tjlin-T0 |
ΔTjB | temperature rise Tj-TB |
ΔTjlinB | temperature rise Tjlin-TB |
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Parameter | Value |
---|---|
Common-emitter current gain at 300 K and medium current levels βF0 | 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 | 82 GHz |
Parameter | Value |
---|---|
AE | 164 µm2 |
JS0 | 3.5 × 10−26 A/µm2 |
η | 1.01 |
VAF | 1000 V |
βF0 | 135 |
ϕ0 | 5.4 mV/K |
ΔEG/k | 200 K−1 |
JHI | 0.35 mA/µm2 |
nHI | 1 |
BVCBO | 27 V |
nAV | 9 [32] |
RE | 1 Ω |
RB | 3.5 Ω |
Material | k(T0) (W/µmK) | k(TB) (W/µmK) | Temperature Dependence |
---|---|---|---|
Si3N4 | 18.5 × 10−6 [33] | 19.6 × 10−6 | (8), α = −0.33 [33] |
In0.5Ga0.5As | 0.048 × 10−4 [33] | 3.9 × 10−6 | (8), α = 1.175 [33] |
InxGa1-xAs (0 < x< 0.5) | 0.092 × 10−4 [33] average in the layer | 7.4 × 10−6 | (8), α = 1.212 [33] |
GaAs | 4.6 × 10−5 [33] | 3.69 × 10−5 | (8), α = 1.25 [33] |
ion-implanted GaAs | 0.046 × 10−5 [33] | 0.0369 × 10−5 | (8), α = 1.25 [33] |
In0.49Ga0.51P | 0.052 × 10−4 [33] | 0.041 × 10−4 | (8), α = 1.4 [33] |
Au | 3.18 × 10−4 [34,35] | 3.14 × 10−4 | (9), β = 6.98 × 10−8 W/μmK2 [34,35] |
Pt | 0.71 × 10−4 [34,35] | 0.71 × 10−4 | independent |
Ti | 0.22 × 10−4 [34,35] | 0.22 × 10−4 | independent |
Ni | 0.91 × 10−4 [35] | 0.863 × 10−4 | (9), β = 8.1 × 10−8 W/μmK2 [35] |
Ge | 0.6 × 10−4 [33] | 0.48 × 10−4 | (8), α = 1.25 [33] |
Cu | 3.98 × 10−4 [35] | 3.95 × 10−4 | (9), β = 5.83 × 10−8 W/μmK2 [35] |
Glue | 1 × 10−4 | 1 × 10−4 | independent |
Polybenzoxazole (PBO) | 0.0014 × 10−4 | 0.0014 × 10−4 | independent |
laminate dielectric | 0.0065 × 10−4 | 0.0065 × 10−4 | independent |
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d’Alessandro, V.; Catalano, A.P.; Scognamillo, C.; Codecasa, L.; Zampardi, P.J. Analysis of Electrothermal Effects in Devices and Arrays in InGaP/GaAs HBT Technology. Electronics 2021, 10, 757. https://doi.org/10.3390/electronics10060757
d’Alessandro V, Catalano AP, Scognamillo C, Codecasa L, Zampardi PJ. Analysis of Electrothermal Effects in Devices and Arrays in InGaP/GaAs HBT Technology. Electronics. 2021; 10(6):757. https://doi.org/10.3390/electronics10060757
Chicago/Turabian Styled’Alessandro, Vincenzo, Antonio Pio Catalano, Ciro Scognamillo, Lorenzo Codecasa, and Peter J. Zampardi. 2021. "Analysis of Electrothermal Effects in Devices and Arrays in InGaP/GaAs HBT Technology" Electronics 10, no. 6: 757. https://doi.org/10.3390/electronics10060757