However, if RE1 is insufficient, the first unstable point will not go to infinity, un-stable operation may occur.
4.4 The Effect of Finger Separation
The finger separation used in the previous example, a large value of 40 µm, is specially chosen to demonstrate all of the unstable points inside a readable simula-tion range. But this large separasimula-tion is not commonly used in real device. A reason-able value of finger separation is about 15 µm which includes, for example, 3 µm contact width for one emitter and one collector, 1.5 µm base contact width for two bases, 1 µm emitter ledge width for two sides of the emitter, and 2 µm base-collector contact spacing of two sides of the base pedestal. The corresponds coupling ther-mal resistances are Rth1 = 800 ◦C/W, Rth2 = 185.8 ◦C/W, Rth3 = 94.4 ◦C/W, Rth4 = 57.3◦C/W, and Rth5 = 38.0◦C/W computed by the effective thermal conduc-tivity kth,ef f = 0.4124 W/cm-◦C as discussed in Section 4.1. Other parameters use the same value as the previous sections.
We firstly use these parameters to simulate the case of RE = 3 Ω as shown in Fig. 4.8(a) and (b). It can be found that the highest current of the side fingers is about 15.0 mA which is higher than the 40 µm separation case. It means that the thermal stability is better as the finger separation becomes smaller. The temperature distribution curves can also reveal this feature. The temperature of the center finger for the 15 µm separation case is only higher before unstable point. After unstable point, the temper-ature increases slower than the 40 µm separation case that results in a more uniform temperature distribution. This thermal stability enhancement feature is more obvious in the next example.
In Fig. 4.8(a) and (b), we use the ideal emitter ballasting resistance RE2,ideal = 3.55 Ω or ∆RE2,ideal = 0.55 Ω to calculate the IC−VBE curves. The eigen-currents are ICλ1 = 7.4 mA, ICλ2 = 21.0 mA, and ICλ3 = 188.3 mA. It can be found that the first unstable
4.4 The Effect of Finger Separation
1.25 1.26 1.27 1.28 1.29 1.30 1.31 0.00
0.01 0.02 0.03 0.04 0.05
Current(A)
Base-Emitter Voltage Vbe (V) f1a
f2a
f3a
f1b
f2b
f3b
(a)
0.00 0.05 0.10 0.15 0.20
300 400 500 600 700 800
Temperature(K)
Total Current (A) f1a
f2a
f3a
f1b
f2b
f3b
(b)
Fig. 4.8. The simulation results of a 3f 3x40 s15 HBT with RE13 = RE2 = 3 Ω compared with the results of the 3f 3x40 s40 HBT in Fig. 4.1. (a) IC−VBE curves and (b) the finger temperature as a function of the total current. f2 is the center finger. f1 and f3 are the side fingers. The letter a indicates the results of a 3f 3x40 s40 HBT. The letter b indicates the results of a 3f 3x40 s15 HBT.
4.4 The Effect of Finger Separation
1.25 1.26 1.27 1.28 1.29 1.30 1.31 0.00
0.01 0.02 0.03 0.04 0.05
Current(A)
Base-Emitter Voltage Vbe (V) f1a
f2a
f3a
f1b
f2b
f3b
(a)
0.00 0.05 0.10 0.15 0.20
300 400 500 600 700 800
Temperature(K)
Total Current (A) f1a
f2a
f3a
f1b
f2b
f3b
(b)
Fig. 4.9. The simulation results of a 3f 3x40 s15 HBT with RE13 = 3 Ω and RE2 = RE2,ideal = 3.55 Ω compared with the results of the 3f 3x40 s40 HBT in Fig. 4.2. (a) IC−VBE curves and (b) the finger temperature as a function of the total current. P0 marks the bend-over point, P1 marks the first unstable point, and P2 marks the second unstable point. f2 is the center finger. f1 and f3 are the side fingers. The letter a indicates the results of a 3f 3x40 s40 HBT. The letter b indicates the results of a 3f 3x40 s15 HBT.
4.4 The Effect of Finger Separation
1.22 1.25 1.28 1.31 1.34 1.37 1.40 0.00
Base-Emitter Voltage Vbe (V) f1c
0.00 0.05 0.10 0.15 0.20
300
Fig. 4.10. The simulation results of a 3f 3x40 s15 HBT with the absolutely stable condition, RE13 = RE13,opt = 4.23 Ω and RE2 = RE2,ideal = 4.78 Ω, and with the no bend-over condition, RE13 = RE13,no-bend = 6.48 Ω and RE2 = RE2,ideal = 7.03 Ω compared with the results of Fig. 4.5. (a) IC−VBEcurves and (b) the finger temperature as a function of the total current. The temperature curves of these two conditions are identical. f2 is the center finger. f1 and f3 are the side fingers. The letter a and c indicates the results of a 3f 3x40 s40 HBT. The letter b and d indicate the results of a 3f 3x40 s15 HBT.
4.4 The Effect of Finger Separation
point is higher than the 40 µm separation case although the bend-over point is lower.
The reason can be explained by (4.11) and (4.12). Since the bend-over current level is inverse proportional to the summation of all effective thermal resistance, smaller finger separation implies larger coupling thermal resistance and results in larger summation of the effective thermal resistance and smaller bend-over current. But the unstable point is determined by the inverse of the difference of R∗th1 and R∗th3. Larger coupling thermal resistance will result in smaller difference and larger unstable current level. Shrinking finger separation actually improves the thermal stability. The small penalty we must pay is the higher finger temperature resulted from the higher total thermal resistance.
Fig. 4.10(a) and (b) show the absolutely stable condition, RE13,opt = 4.23 Ω and RE2,ideal = 4.78 Ω, and the no-bend-over condition, RE13,no-bend = 6.48 Ω and RE2,ideal = 7.03 Ω. As one can expect, the value of the emitter ballasting resistance used in the 15 µm separation case for the absolutely stable condition is smaller than the 40 µm separation case as a result of better thermal stability. On the contrary, the no-bend-over condition uses larger resistance in the 15 µm separation case. Besides, for the absolutely stable condition, smaller separation will result in more serious bend-over IC−VBE curves with smaller base-emitter voltage at the bend-over point. It is a result of higher finger temperature, as shown in Fig. 4.10(b). The higher the finger temperature is, the more bending the curves will have. For the no-bend-over condition, the IC−VBE curves are overlapped in both 15 µm and 40 µm separation cases. The temperature curves for both the absolutely stable and no-bend-over conditions are also overlapped that is the same situation as Fig. 4.5.
Fig. 4.11 is the collection of the simulated total currents in Fig. 4.1 for RE = 3 Ω and in Fig. 4.2 for RE13/RE2,ideal = 3/3.26 Ω conditions for the 40 µm separation case, and in Fig. 4.8 for RE = 3 Ω and in Fig. 4.9 for RE13/RE2,ideal = 3/3.55 Ω conditions for the 15 µm separation case. It shows that smaller finger separation will result in smaller bend-over voltage and higher unstable current. Fig. 4.12(a) and (b) shows the IC−VBE curves and the temperature distribution curves for a five-finger device with RE15 = 3 Ω
4.4 The Effect of Finger Separation
1.25 1.26 1.27 1.28 1.29 1.30 1.31 0.00
0.03 0.06 0.09 0.12 0.15
TotalCurrent(A)
Base-Emitter Voltage Vbe (V) A1
A2
B1
B2
Fig. 4.11. The simulation results of the total current versus VBE curves collected from Fig. 4.1-Fig. 4.2 for the 3f 3x40 s40 HBT and Fig. 4.8-Fig. 4.9 for the 3f 3x40 s15 HBT. A1 correspond to RE= 3 Ω for a 3f 3x40 s40 HBT. A2 correspond to RE13/RE2,ideal = 3/3.26 Ω for a 3f 3x40 s40 HBT. B1 correspond to RE = 3 Ω for a 3f 3x40 s15 HBT. B2 correspond to RE13/RE2,ideal = 3/3.55 Ω for a 3f 3x40 s15 HBT.
and RE2/3/4 = REideal = 3.37/3.45/3.37 Ω. In opposition to the complexity of Fig. 4.7, most unphysical solutions of the 15 µm separation case are lifted up outside of the real plane. The eigen-currents are ICλ1 = 6.4 mA, ICλ2 = 13.3 mA, ICλ3 = 35.7 mA, ICλ4 = −44.6 mA, and ICλ5 = −1.037 A. The negative eigen-currents mean that the corresponding unstable points do not exist physically. This device has only two unstable points, now, as shown in Fig. 4.12. The no-bend-over requirement is RE1,no-bend = 7.05 Ω and the absolutely stable requirement is RE1,opt = 4.94 Ω. The same as the three-finger device, the thermal stability is also improved although the first unstable point does not change much.
4.4 The Effect of Finger Separation
1.20 1.22 1.24 1.26 1.28
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
f1
f2
f3
f4
f5
Current(A)
Base-Emitter Voltage Vbe (V)
(a)
0.00 0.05 0.10 0.15 0.20
300 400 500 600 700 800
Temperature(K)
Total Current (A) f1a
f2a
f3a
f1b
f2b
f3b
(b)
Fig. 4.12. The simulation results of a 5f 3x40 s15 HBT with RE15 = 3 Ω and RE2/3/4 = REideal = 3.37/3.45/3.37 Ω. (a) IC−VBE curves and (b) the finger temperature as a function of the total current.
P1-P4 mark the four unstable points. f3 is the center finger, f1 and f5 are the outer fingers, and f2 and f4 are the side fingers beside the center finger.
4.5 Summary
4.5 Summary
Multiple-finger transistors with nonuniform distribution of ballasting resistance have been analyzed by using the simple model. Analytical formulas for the best ballasting re-sistance distribution for optimum thermal stability operation were derived. Comparing with the conventional method of using uniform ballasting resistance, the new schemes with optimized design could result in a significant increase in the device current un-der stable operation. With the ideal ballasting resistance distribution, it is possible to achieve absolutely stable operation, where the device never becomes unstable, by using the optimum ballasting resistance. This optimum value could be obtained by solving an eigenvalue equation. The second largest real positive eigenvalue of this eigenvalue equation was the optimum ballasting resistance. A design procedure for the thermally stable optimum design was developed. The effect of the finger separation is also dis-cussed. For a three-finger device, reducing finger separation will improve the thermal stability.
Chapter 5
Thermally Stable Optimum Design of Multifinger HBTs in Accurate Model
W
e had developed a design procedure to determine the values of the emit-ter ballasting resistance needed for thermal stable operation from a set of thermal-electrical feedback equations in simple model. The simple model gives us the basic concept of the thermally stable design by using the cancelation of positive and negative feedback. Based on this model, two ideas, the ideal ballasting resistance distri-bution and the absolutely stable condition, are derived and discussed. In this chapter, we extend our model to the accurate model and take the temperature dependence of thermal conductivity into account. A more physical model will give us a more practical result.By this manipulation, although the behavior of the IC−VBE curves are quit different from the simple model, the basic concept of the ideal ballasting resistance distribution and the absolutely stable condition are still similar and only need small modification.
We could use the same derivation procedure to derive the ballasting resistance equations for the accurate model. Two practical design procedures based on uniform current and uniform temperature consideration were obtained and described [8]. The results with the temperature dependent and constant thermal conductivity were also compared.
In the calculation presented below, we use the same 3 × 40 InGaP/GaAs HBT on a
5.1 The Accurate Model
substrate with dimension 1000 µm × 1000 µm and 100 µm thickness as mentioned in Chapter 4 to illustrate the design procedure. The parameters used in Chapter 4 com-bining with the parameters of the accurate model described in Chapter 3 and the tem-perature dependent thermal conductivity described in Chapter 2 will be used in this chapter. We relist these parameters here for convenience. They are IO = 6 × 10−25 A, Eg0∗ = 1.483 eV, α∗ = 4.88 × 10−4 eV/K, γ = 4.21, kth,ef f = 0.4124 W/cm-◦C, and b = 1.25, VC = 6 V. The calculated coupling thermal resistances are Rth1 = 800◦C/W, Rth2 = 66.9◦C/W, and Rth3 = 23.5◦C/W for the 40 µm finger separation device, and Rth1 = 800◦C/W, Rth2 = 185.8◦C/W, and Rth3 = 94.4◦C/W for the 15 µm finger separation device.
5.1 The Accurate Model
The difference between the simple model and the accurate model is that we use the general SPICE model current equation (3.9) instead of the linearized model current equation (3.15). The accurate coupled thermal-electrical feedback equation (3.36) had been discussed in Section 3.2. Using a three-identical-finger transistor as an example, the accurate coupled current-voltage equations with self-heating are
VBE = Eg0∗ − φ∗1T1+ RE13IC1
5.1 The Accurate Model
and the parameters had been defined previously. The ideality factor is set to unity for simplicity as before. IS0is obtained from (3.16) as
IS0 = IO µTA
300
¶−γ exp
· q
kTA(Eg0∗ − α∗TA)
¸
. (5.3)
For our device with IO = 6 × 10−25 A, IS0 = 0.494 A. From (5.1), a relation between RE13 and RE2 can be obtained as
RE2 = IC1
IC2RE13+ φ∗2T2− φ∗1T1 IC2
= IC1
IC2(RE13+ ∆R∗E2)
(5.4)
where ∆R∗E2is the effective emitter resistance distribution difference and the real emitter resistance distribution difference ∆RE2 defines as
∆RE2 = RE2 − RE13. (5.5)
This distribution is no longer a constant but is a function of the current and the tem-perature of all fingers because that the thermal-electrical feedback coefficient is not a constant now. Therefore, the procedure we developed in Chapter 4 is not suitable for the accurate model. However, from (5.4), we found that there are two degrees of freedom to design our thermally stable transistors under some conditions. They are the uniform cur-rent design and the uniform temperature design that will be discussed in the following sections.
The effect of temperature dependent thermal conductivity will affect the thermal sta-bility significantly. When the thermal conductivity is a function of temperature, the tem-perature of fingers will become a very nonlinear function of the dissipated power. This problem can be solved by using the Kirchoff transform through defining a linearized temperature U(x, y, z) as had mentioned in Section 2.2.1. The solutions of U(x, y, z)
5.1 The Accurate Model
are linear functions of currents. For the three-finger case,
U1 = TA+ VC(Rth1IC1 + Rth2IC2 + Rth3IC3) U2 = TA+ VC(Rth2IC1 + Rth1IC2 + Rth2IC3) U3 = TA+ VC(Rth3IC1 + Rth2IC2 + Rth1IC3).
(5.6)
Once the temperature dependence of the thermal conductivity is known, we can solve for the actual junction temperature T (x, y, z) using (2.11) and (5.6). Here, we use the temperature dependence expression of (2.1) and parameter values of Table 2.1. By using this relation, the actual junction temperature T (x, y, z) can be obtained as (3.28). For the three-finger case,
T1 = TA
·
1 −b − 1
TA VC(Rth1IC1 + Rth2IC2 + Rth3IC3)
¸−1
b−1
T2 = TA
·
1 −b − 1
TA VC(Rth2IC1 + Rth1IC2 + Rth2IC3)
¸−1
b−1
T3 = TA
·
1 −b − 1
TA VC(Rth3IC1 + Rth2IC2 + Rth1IC3)
¸−1
b−1 .
(5.7)
From now on, we have a very nonlinear relation between temperature and current. This relation will seriously affect the behavior of the thermal-electrical feedback equation, as we will see. Similar results of the constant thermal conductivity case can be obtained by simply setting b = 0 to the results of the temperature dependent thermal conductivity case. In this chapter, we will discuss both the constant thermal conductivity case and the temperature dependent thermal conductivity case.
It is interesting to point out that in the temperature dependent thermal conductivity case, the linearized temperature U has an upper limit. It is because that the junction temperature T increase faster than U and will go to infinity as U increases to a certain value. It happens when the second term in the bracket of (5.7) is equal to unity. And as a result, the currents of the fingers also have a theoretical upper limit and the hottest
5.1 The Accurate Model
finger, the center finger under normal operation, decides the limit. By setting all currents the same, i.e. IC1 = IC2 = IC3 = IC, the maximum current level ICmax is
ICmax = TA
(b − 1) VC(Rth1 + 2Rth2) (5.8) Substituting the parameters, we have ICmax = 171 mA for the 3f 3x40 s15 HBT and ICmax = 214 mA for the 3f 3x40 s40 HBT. For the N-finger case, the upper limit be-comes
where Rthij is the thermal resistance matrix element. Because of the existence of this upper limit, setting the current to infinity when deriving the absolutely stable condition in Section 4.2 is not valid. Even so, the basic idea of these equations is still correct. All the change we need to do, instead of setting the current to infinity, is setting the current to a specific current level as we will do in Section 5.2.
Following the similar steps in Section 4.2, we derive the eigenvalue equations of the ballasting resistance for the accurate model. Firstly, from (5.2), the derivatives of (5.2) with respect to VBE are
φ∗10 = −k
Then, the derivatives of (5.1) with respect to VBE are obtained as
1 = −φ1T10 +
5.1 The Accurate Model
where φiis defined in (3.21) and (3.23). We consider the temperature dependent thermal conductivity case here. The results of the constant thermal conductivity case can be easily obtained from the temperature dependent thermal conductivity case. Because of the temperature dependence of the thermal conductivity, the derivatives of the finger temperatures with respect to the base-emitter voltage become
T10 = VC¡
Combining (5.11) with (5.12) and using the relation of the emitter resistance (5.4) and the definition of the effective thermal resistance (4.4), Rth∗i = φVCRthi, the derivatives of the current-voltage equations with respect to VBE (5.1) become
φ is an arbitrary constant that we choose to be equal to 1 mV/◦C as before. We can multiply the second equation of (5.13) by r to eliminate the prefatory term of RE13 in