Summary

model, such as Dawson’s and Bovolon’s, because of b = 0, the ratio of these two deriva-tives gives the constant thermal resistance. Equation (2.30) is more general to be able to include the temperature dependent effect of thermal resistance.

Now, we use Bovolon’s method and (2.30) to measure the thermal resistance of the same 2×20 InGaP/GaAs HBT device in Section 2.3.1. The measured data we need is the same I_C−V_C curves in Fig. 2.7 as Marsh’s method. And then, we convert these curves to current gain versus dissipated power β−P curves as shown in Fig. 2.10. By using the data of Fig. 2.10, we can calculate these two derivatives we need. The derivatives are taken by averaging the slopes of two adjacent data points. The slopes are calculated by the forward difference, i.e. slope = (y_i+1− y_i)/(x_i+1− x_i). The first and last points of each data sequence will vanish after differentiating. Careful measurement is needed when applying Bovolon’s method because of the use of derivatives by which any small fluctuation will be amplified and cause large error. The measured R_th and the fitted lines by using (2.30) are shown in Fig. 2.11. The measured Rth for each temperature almost lay on the same level as (2.30) expected. It is a good proof that Bovolon’s method is theoretically correct and better than Marsh’ method. The fitted parameters Rth0 = 1100^◦C/W and b = 1.15 are obtained.

2.4 Summary

The values of the thermal conductivity and its temperature dependence of GaAs and InGaP in literature had been reviewed. The values used in this work are given in Table 2.1. The effect of temperature dependent thermal conductivity on the heat flow equation and the thermal resistance was also explained. Theoretical calculation of the thermal resistance was shown and the properties of the thermal resistance dependent on device geometry were discussed. A small, long and narrow finger is better for the sake of reducing thermal resistance. Thinning the substrate is not an effective way to reduce thermal resistance. Tow methods of DC measurement were investigated and had been

2.4 Summary

0.06 0.07 0.08 0.09 0.10 0.11 0.12

20

Fig. 2.10. Measured curves of current gain β versus dissipated power of a 2 × 20 InGaP/GaAs HBT for IB = 1 mA at heat plate temperature changing from 300 K to 450 K with 30 K step. The specified dissipated power increases from 100 mW to 112 mW with 1 mW step to extract the thermal resistance data of Fig. 2.11.

0.100 0.102 0.104 0.106 0.108 0.110 0.112 1200

Fig. 2.11. Extracted thermal resistances for different dissipated power and heat plate temperature by using Bovolon’s method. Symbols are measured data and lines are fitted lines. The fitting equation is (2.10). The fitted parameters and variances of parameters are listed in the figure. “Chiˆ2/DoF” is the reduced chi-square. “Rˆ2” is the R-square or the square of correlation coefficient.

2.4 Summary

applied to a 2 × 20 InGaP/GaAs HBT. It was found that Marsh’s method is only suitable in low power operation and Bovolon’s method is preferred.

Chapter 3

The Coupled Current-Voltage Equations

W

hen the temperature dependence of the current-voltage equation is taken into account, the device current will be affected by the device temperature and in turn the device temperature will be changed by the device current. This interaction between current and temperature is called self-heating because the device temperature usually becomes higher than that without this effect. The self-heating effect will make the current-voltage equation nonlinear and a method to solve current and temperature si-multaneously is needed. The current-voltage equations we need to solve will be defined first in this chapter.

For a multi-finger transistor, the equation that describes the current-voltage behavior of the whole device is not just to multiply one finger equation to the total number of fin-gers. This simple scaling is only correct when the device is operated under low power condition. As the power increases, the temperature of each finger is not the same any-more. Usually, the fingers at the center of transistor will be hotter than the fingers near the edge. Because that each finger temperature will be affected by the dissipated power of other fingers, the current-voltage behavior of each finger will also be affected by the other fingers and the current of each finger will not be the same. It is the phenomenon

3.1 The Current-Voltage Equation with Self-Heating

of the thermal coupling effect. When this coupling effect happens, a single expression cannot describe the current-voltage behavior of the whole device. We need a set of sep-arated coupled current-voltage expressions for each finger and the current of each finger is a function of other finger currents. In this chapter, the coupled I_C−V_BE equations will be derived and the method to solve them will also be discussed. The equations and the solving method will be used in the following chapters to design a thermally stable multi-finger transistor.

3.1 The Current-Voltage Equation with Self-Heating

The classical Ebers-Moll expression of collector current versus emitter-base voltage equation, I_C−V_BE equation, is [19]

I_C = I_S

· exp

µqV_BEj kT

¶

− 1

¸

(3.1)

where I_S is the temperature dependent reverse saturation current and V_BEj is the base-emitter junction voltage given by

VBEj = VBE − (IC+ IB)(RE0+ REb) − IB(RB+ RBb)

= VBE − ICRE (3.2)

where R_E0 and R_B are the emitter and base resistance, R_Eb and R_Bb are the ballasting resistors for emitter ballasting and for base ballasting, and R_E is the total resistance seen into the emitter and given by

RE = µ

1 + 1 β

¶

(RE0+ REb) + 1

β(RB+ RBb) (3.3)

where β is the current gain. In most case, R_E can be treated as a constant because that RE0, RB, REb, and RBb are almost temperature independent and the temperature

3.1 The Current-Voltage Equation with Self-Heating

dependence of β is small, especially true for InGaP/GaAs HBT. The ballasting resistors will be discussed in the following chapters.

There are two expressions of I_S depending on its origin of physics. For drift-diffusion limited devices, I_S is

I_S = qAD_nN_CN_V WBNB

exp

³

− q kTE_g

´

(3.4)

where A is the device area, Dn is the diffusion constant, NC is the conduction band density of states, N_V is the valence band density of states, W_B is the base width, N_Bis the base doping, and E_g is the band-gap voltage of the base layer material. We known NC and NV have the temperature dependence ∼ T^1.5. If we substitute the Einstein rela-tion D_n = (kT /q)µ_n, where µ_nis the mobility, into (3.4) and assume the temperature dependence of mobility as µ_n ∼ T^−x where x > 0, the temperature dependent reverse saturation current IS of (3.4) can be obtained as

I_S ∼ T^4−xexp h

− q

kTE_g(T ) i

. (3.5)

For thermionic emission limited devices, I_S is [30]

I_S = AA^∗T²exph q

kT(−E_g− V_p− ∆E_C) i

. (3.6)

where A^∗ is the effective Richardson constant, V_p is the energy difference between the valence band edge and the hole quasi-Fermi level E_{f p} of the base layer, i.e. E_V − E_{f p}, and ∆ECis the conduction band discontinuity at base-emitter junction. Here, we neglect the voltage drop of the base depletion region because that the base doping is very high in modern HBT devices. It implies that the base depletion region is negligible small and the voltage drop on it is also small. In order to eliminate the variable Vp in (3.6), we substitute the relation N_B = p = N_V exp(qV_p/kT ) into (3.6). Although this relation is derived from the Boltzmann statistics which is valid for non-degenerate semiconductor,

3.1 The Current-Voltage Equation with Self-Heating

it still could give us a qualitative analysis of the temperature dependence of I_S. After substituting this relation, (3.6) becomes

I_S = AA^∗T²N_V

If we assume the effect of ∆E_C can be written as

exp

the temperature dependent reverse saturation current I_S of (3.6) can be obtained as

I_S ∼ T^3.5+yexph

− q

kTE_g(T )i

(3.7)

where y > 0 because of the fact that ∆E_Cis almost temperature independent and results in as the temperature increasing the exponential term will also increasing. As the results of (3.4) and (3.6), we know that no matter the dominated effect of device current is drift-diffusion limited or thermionic emission limited, I_S has a general form as

I_S = I_S0

where IS0 is the temperature independent reverse saturation current or the reverse satu-ration current at 300 K, and the exponent γ is a constant. From (3.5) and (3.7), we know that if γ > 4, the current is thermionic emission dominated and if γ < 3.5, the current is drift-diffusion dominated. In the range 3.5 < γ < 4, these two mechanisms are both possible.

In this work, because we only concern about the normal operation of devices, i.e.

the base-emitter junction forward biased, the second term in the square brackets of (3.1),

−1, can be neglected safely. Combining (3.2) and (3.8), we have the general expression

3.1 The Current-Voltage Equation with Self-Heating

Fig. 3.1. The fit of the Gummel data of a 2 × 20 InGaP/GaAs HBT to (3.11) which measured at heat plate temperature changing from 300 K to 450 K with 30 K step. The fit range is limited to the low power data which is selected from VBE≥ 1.0 V to IC≤ 2 mA. Small symbols are measured data, large symbols are the measured data used in fit, and the line is the fit curve of (3.17). The fitted parameters and variances of parameters are listed in the figure. “Chiˆ2/DoF” is the reduced chi-square. “Rˆ2” is the R-square or the square of correlation coefficient.

of the I_C−V_BE equation as

and take the logarithm of both sides of (3.9) as

VBE = kT

Fig. 3.1 shows the fit of the varied temperature Gummel data, which are the I_C−V_BE curves measured with collector and base shorted, of a 2 × 20 InGaP/GaAs HBT. The

3.1 The Current-Voltage Equation with Self-Heating

where n is the ideal factor which is close to one, n = 1.022 for our device, and will set to unity in the following discussions for simplicity. We use the equation (3.17), which will be discussed later, for E_g(T_A) and take the band-gap narrowing effect into account.

We use Harmon’s [31] equation for GaAs to compute the band-gap narrowing value of the base material, i.e.

∆E_g = 2.55 × 10⁻⁸N_B¹³ V (3.12)

where N_B is the base doping level. For our devices, N_B = 4 × 10¹⁹ cm⁻³ and the corresponding ∆E_g = 87.2 mV. The extracted γ is about 4.21 which implies that our device is most possibly thermionic emission dominated. It is a common property of HBTs. But the value of fitted γ is still close to the largest possible value of the drift-diffusion dominated case. And we cannot exclude this possibility completely. It may be because that the conduction band discontinuity ∆ECbetween InGaP and GaAs is small, electrons could pass through the base-emitter junction with little blocking.

From the definition of the thermal-electrical feedback coefficient φ and (3.10), we can obtain an expression as [32]

φ = −∂V_BE

The negative sign in the definition is to make φ positive because that V_BE decreases as T increases. We can see that the thermal-electrical feedback coefficient φ = φ(I_C, T ) is actually a function of collector current and temperature even though it is usually treated as a constant because of the slow variations of ln I_Cand ln T . Although (3.9) and (3.10) are the more accurate equations, we often want a simplified formula in practice to look inside the physics of the thermal feedback effect. We use the Taylor series to expanse

3.1 The Current-Voltage Equation with Self-Heating feedback I_C−V_BE equation [2].

I_C = I_Oexp

We will call (3.15) the simplified I_C−V_BE equation. In this equation, it is very clear that the contribution of the self-heating effect on collector current is opposite to the effect from the emitter resistance. The emitter resistance acts as a negative feedback element but the self-heating effect acts as a positive feedback element. In order to achieve a thermal stable transistor, we can utilize the negative feedback effect of emitter resistance to compensate the positive feedback effect of self-heating. This is the original ideal of this work.

The last parameter we have mentioned but not discussed in IC−VBE equation is the band-gap voltage E_g which is an important factor to affect the thermal behavior of devices. The temperature dependence of band-gap voltage E_gcan be represented by the well-known Varshni’s empirical formula [33].

E_g(T ) = E_g0− αT²

T + T_β (3.17)

3.1 The Current-Voltage Equation with Self-Heating

200 300 400 500 600 700

1.24 1.28 1.32 1.36 1.40 1.44 1.48

Linear Regression for Eg:

Y = A + B * X

Parameter Value Error

A 1.56986 7.13956E-4

B -4.88055E-4 1.44759E-6

R SD N P

-9.99807E-1 1.30343E-3 46 <0.0001

BandgapEnergy(eV)

Temperature (K) Eg

Linear Fit of Eg

Fig. 3.2. The linear fit of (3.18) in the temperature range from 250 K to 700 K. The thick line is the curve of (3.17), and the thin line is the fitted line (3.20). The fitting equation and the fitted parameters are listed in the figure. “R” is the correlation coefficient. “SD” is the standard deviation of the fit.

For GaAs, that is

E_g,GaAs(T ) = 1.519 − 5.405 × 10⁻⁴ T²

T + 204 V. (3.18)

Sometimes, it is more convenient to represent E_g as a linear function of temperature.

We use the linear fit of (3.18) in the temperature range from 250 K to 700 K as the linear expression of Eg, i.e.

E_g,GaAs(T ) = 1.570 − 4.88 × 10⁻⁴T V. (3.19)

Fig 3.2 is a comparison between (3.18) and (3.19) and shows that two curves fit very closely. The maximum deviation in the fitted temperature range is about 3.3 mV and in the temperature range from 290 K to 670 K is less than 1.5 mV. The validity of (3.19) in this temperature range is confirmed. We still need to take the band-gap narrowing effect into account to obtain our final expression of the band-gap voltage by subtract

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