# Modeling and Extraction of Parameters Based on Physical Effects in Bipolar Transistors

Authors: Agnes Nagy; Alicia Polanco; Manuel Alvarez
Journal: Active and Passive Electronic Components
 (2011)

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## Abstract

The rising complexity of electronic systems, the reduction of components size, and the increment of working frequencies demand every time more accurate and stable integrated circuits, which require more precise simulation programs during the design process. PSPICE, widely used to simulate the general behavior of integrated circuits, does not consider many of the physical effects that can be found in real devices. Compact models, HICUM and MEXTRAM, have been developed over recent decades, in order to eliminate this deficiency. This paper presents some of the physical aspects that have not been studied so far, such as the expression of base-emitter voltage, including the emitter emission coefficient effect (n), physical explanation and simulation procedure, as well as a new extraction method for the diffusion potential VDE(T), based on the forward biased base-emitter capacitance, showing excellent agreement between experimental and theoretical results.


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## Body

## 1. Introduction

Every day, products and processes for microsystems become more complex and lead to high degrees of accuracy and stability. The simulation of the design must be very precise to increase the stability and reproducibility of the devices. Between the elements of electronic circuits used in microsystems, IC temperature sensors and temperature-compensated voltage references are frequently found. Reliable design of these circuits, fabricated in advanced bipolar and BiCMOS technologies has become seriously affected by the deficiencies of SPICE Gummel-Poon model (SGPM) as discussed by Schroter et al. [1] and need more physical-based and accurate models.The base-emitter voltage, due to its extremely reproducible exponential characteristic, is the basic element in the design of I.C. temperature sensors and temperature-compensated voltage references. To achieve the required accuracy and stability in these circuits, an accurate description of the base-emitter voltage temperature dependenceVBE(T) is needed.The expressions ofVBE(T) reported in the literature have been obtained from collector current. The collector current of an NPN bipolar transistor working in the active region, at low current levels, neglecting emitter emission coefficient and direct and inverse Early effects, is given by Tsividis [2]:
(1)IC(T)=IS(T)[exp⁡⁡(qVBE(T)kT)-1],
where T is the absolute temperature, VBE the base-emitter voltage, q the electron charge, and k the Boltzmann constant (8.617×10-5 eV/K). The −1 term can be neglected when the injected current is much greater than the saturation current IS(T).The expression ofVBE(T)  obtained from (1) by Tsividis [2] is known as the “accurate expression” and is given by
(2)VBE(T)=VG(T)-VG(Tr)TTr+VBE(Tr)TTr-(η-m)kTqln⁡TTr,
where VBE(Tr) and VG(Tr) are the base emitter and bandgap voltages at the reference temperature Tr. The accuracy of expression (2) depends on the accuracy of the bandgap voltage model VG(T) used. There are various models of VG(T) reported in the literature as discussed elsewhere [2–6], obtained by different methods for intrinsic silicon. In 1982, Meijer [7], using linear approximation for bandgap voltage VG(T)=VG0-αT [5], obtained the following expression of VBE(T):
(3)VBE(T)=VG0[1-TTr]+TTrVBE(Tr)-(η-m)kTqln⁡TTr,
where VG0 is the extrapolated value of bandgap voltage at zero Kelvin and η a parameter related to the dependence of global mobility with temperature through β:μg∝T-β as discussed by Slotboom and Graaff [8], which results from scattering mechanisms caused by thermal vibrations and ionized impurities as discussed by Amador et al. [9],
(4)η=4-β.
The values of VG0 and η can be obtained from experimental measurements of VBE at three temperatures of the operating range. Experimental measurements of η and VG0 in diffused wafers and in different types of bipolar transistors as reported elsewhere [7, 10] showed a negative correlation between both parameters as reported in the literature.In order to improveVBE(T) description, direct and inverse early effects have been discussed elsewhere [11, 12]. Nevertheless, the emitter emission coefficient n(T) and its influence on VBE(T)  has not been  well studied up to now.


## 2. Modeling and Extraction of Parameters

The modeling and extraction of the parameters treated here are based on experimental measurements of forward biased base-emitter capacitance, using the measurement set-up shown in Figure1.Figure 1
Measurement setup.A Haake liquid bath thermostat guarantees temperature stability better than 0.05°C. The DC measurements ofVBE were performed with a 6 1/2 digits Agilent multimeter. Ce measurement error in the bridge due to parallel base-emitter diode conductance is negligible except near the maximum value of Ce, where it is only 2%. Therefore, no corrections are introduced in the experimental values of Ce.From experimental measurements of Ce realized with MAT01 standard bipolar transistors, an unexpected exponential increase under significant forward bias has been observed. This behavior is not considered by Ideal, Gummel-Poon nor the first MEXTRAM and HICUM Ce models as can be seen elsewhere [13–15]. It may be seen from Figure 2 that the three models fail to follow the exponential behavior of Ce and begin to increase at higher bias voltages. For this reason, it was necessary to develop a new forward biased emitter capacitance model of bipolar junction transistors, published by Nagy et al. [16], which includes a region with exponential behavior of the emitter capacitance Ce as forward bias increases. The physical explanation of the exponential increase of Ce was found in the theory for the transition region of the p-n junction as discussed by Shockley et al. [17]. Modern compact bipolar transistor models such as MEXTRAM and HICUM as discussed elsewhere [18, 19] take the neutral charge into account using a charge-transit time model.Figure 2
Experimental and simulated values of Ce at 29.3°C.Modeling and extraction of parameters treated here are based on the new forward bias base-emitter capacitance model as discussed by Nagy et al. [16]. A brief description is given as follows.


### 2.1. Brief Description of New Bipolar Transistor Base-Emitter Capacitance Model

The new model is built using two expressions linked with a smoothing function:  the three-parameter Ideal expression that predominates at low forward bias and the neutral capacitance expression that predominates as forward bias increases, as discussed by Nagy et al. [16]. The fit of the new Ce model to experimental results is notable. This model of Ce in function of base-emitter voltage and temperature is given by
(5)Ce(VBE,T)=f(VBE)CSc(T)+Cn(T),
where f(VBE) is a smoothing function defined for a voltage interval [VBE1,VBE2] given by
(6)f(VBE)=11+eλ(VBE-α),
where
(7)α=VBE1+VBE22,λ=ln⁡⁡99δ,δ=VBE2-VBE12,(i)
VBE2 is the base-emitter voltage VBEmax at maximum Ce,(ii)
VBE1 is the top value of base-emitter voltages that satisfy the space charge condition taking into account experimental measurements.In the Ideal or space charge capacitance model,(8)CSC(T)=Cje0(T)[1-VBE/VDE(T)]Z,
where Cje0(T) is the zero-bias capacitance.The temperature dependence of the zero-bias emitter capacitance is given by(9)Cje0(T)=Cje0(Tr)[VDE(Tr)VDE(T)]Z.
It is also known as reported by Schroter et al. [13] that the diffusion voltage VDE depends on temperature through
(10)VDE(T)=VG0[1-TTr]+VDE(Tr)TTr-3VTln⁡⁡TTr.
The expression of the neutral capacitance was obtained using Shockley’s theory on p-n junction transition region capacitance. For the case of a bipolar transistor, it can be written as shown by Nagy et al. [16] as
(11)Cn(VBE,T)=CS(T)[VDE(T)-VBEkT/q-1]exp⁡⁡(qVBEkT)
with
(12)CS(T)=CS(Tr)(TTr)2exp⁡⁡[qVG0k(1Tr-1T)],
where Tr is a reference temperature and VG0 is the zero Kelvin extrapolated bandgap voltage.Based on expression (11) and the condition for a maximum capacitance dCn/dVBE=0, an important relationship between VBEmax and VDE (that allows the experimental determination of VDE at any temperature) is found,
(13)VBEmax⁡(T)=VDE(T)-2kTq.
The simulation of experimental measured Ce values at 39.3°C was used to assess the temperature dependence of the new Ce model. The measured and simulated values of Ce with the new model at two temperatures are shown in Figure 3.Figure 3
Experimental and simulated values of Ce at 29.3°C and 39.3°C with the new model.The simulation parameters, extracted from the experimental measurements at 29.3°C, areCje0(Tr)=8.52 pF, z=0.329, VDE(Tr)=0.610 V, Cemax(Tr)=84 pF, Cs(Tr)=4.259×10-8 pF, and VBEmax(Tr)=0.558 V. The limits of the smoothing function interval at T=Tr=29.3°C are VBE1=0.400 V and VBE2=0.558 V while at T=39.3°C are VBE1=0.372 V and VBE2=0.535 V. The measured and predicted values using the new model are summarized in Table 1.Table 1
Values measured and predicted with the new model.
Measured values at 29.3°CPredicted values at 39.3°CMeasured values at 39.3°CVBEmax=0.558 VVBEmax=0.5346 VVBEmax=0.532 VCemax=84 pFCemax=81.54 pFCemax=75.2 pFVDE=0.610 VVDE=0.5885 VVDE=0.586 VCS=4.259×10-8 pFCS=19.47×10-8 pFCS=19.7×10-8 pFA very good 1% extrapolation accuracy is observed except for Cemax with almost 8% error that can be attributed to ±5% time and temperature drift of the capacitance bridge during measurements.


## 2.1. Brief Description of New Bipolar Transistor Base-Emitter Capacitance Model

The new model is built using two expressions linked with a smoothing function:  the three-parameter Ideal expression that predominates at low forward bias and the neutral capacitance expression that predominates as forward bias increases, as discussed by Nagy et al. [16]. The fit of the new Ce model to experimental results is notable. This model of Ce in function of base-emitter voltage and temperature is given by
(5)Ce(VBE,T)=f(VBE)CSc(T)+Cn(T),
where f(VBE) is a smoothing function defined for a voltage interval [VBE1,VBE2] given by
(6)f(VBE)=11+eλ(VBE-α),
where
(7)α=VBE1+VBE22,λ=ln⁡⁡99δ,δ=VBE2-VBE12,(i)
VBE2 is the base-emitter voltage VBEmax at maximum Ce,(ii)
VBE1 is the top value of base-emitter voltages that satisfy the space charge condition taking into account experimental measurements.In the Ideal or space charge capacitance model,(8)CSC(T)=Cje0(T)[1-VBE/VDE(T)]Z,
where Cje0(T) is the zero-bias capacitance.The temperature dependence of the zero-bias emitter capacitance is given by(9)Cje0(T)=Cje0(Tr)[VDE(Tr)VDE(T)]Z.
It is also known as reported by Schroter et al. [13] that the diffusion voltage VDE depends on temperature through
(10)VDE(T)=VG0[1-TTr]+VDE(Tr)TTr-3VTln⁡⁡TTr.
The expression of the neutral capacitance was obtained using Shockley’s theory on p-n junction transition region capacitance. For the case of a bipolar transistor, it can be written as shown by Nagy et al. [16] as
(11)Cn(VBE,T)=CS(T)[VDE(T)-VBEkT/q-1]exp⁡⁡(qVBEkT)
with
(12)CS(T)=CS(Tr)(TTr)2exp⁡⁡[qVG0k(1Tr-1T)],
where Tr is a reference temperature and VG0 is the zero Kelvin extrapolated bandgap voltage.Based on expression (11) and the condition for a maximum capacitance dCn/dVBE=0, an important relationship between VBEmax and VDE (that allows the experimental determination of VDE at any temperature) is found,
(13)VBEmax⁡(T)=VDE(T)-2kTq.
The simulation of experimental measured Ce values at 39.3°C was used to assess the temperature dependence of the new Ce model. The measured and simulated values of Ce with the new model at two temperatures are shown in Figure 3.Figure 3
Experimental and simulated values of Ce at 29.3°C and 39.3°C with the new model.The simulation parameters, extracted from the experimental measurements at 29.3°C, areCje0(Tr)=8.52 pF, z=0.329, VDE(Tr)=0.610 V, Cemax(Tr)=84 pF, Cs(Tr)=4.259×10-8 pF, and VBEmax(Tr)=0.558 V. The limits of the smoothing function interval at T=Tr=29.3°C are VBE1=0.400 V and VBE2=0.558 V while at T=39.3°C are VBE1=0.372 V and VBE2=0.535 V. The measured and predicted values using the new model are summarized in Table 1.Table 1
Values measured and predicted with the new model.
Measured values at 29.3°CPredicted values at 39.3°CMeasured values at 39.3°CVBEmax=0.558 VVBEmax=0.5346 VVBEmax=0.532 VCemax=84 pFCemax=81.54 pFCemax=75.2 pFVDE=0.610 VVDE=0.5885 VVDE=0.586 VCS=4.259×10-8 pFCS=19.47×10-8 pFCS=19.7×10-8 pFA very good 1% extrapolation accuracy is observed except for Cemax with almost 8% error that can be attributed to ±5% time and temperature drift of the capacitance bridge during measurements.


## 3. Effect of Emitter Emission Coefficient

The variation of emitter emission coefficient (n) with temperature has been measured and reported elsewhere [20, 21]. At very low temperature (230 K–300 K), an increase of the value of n was found. Until now, there is no satisfactory explanation for this phenomenon.In this paper, a physical approach ofn(T) behavior through of emitter-base capacitance Ce(VBE,T) is presented. Considering the effect of emission coefficient n(T), the collector current IC(T)  can be written as
(14)IC(T)=IS(T)[exp⁡(qVBE(T)n(T)kT)-1],
from which VBE(T) expression can be obtained as
(15)VBE(T)=n(T)[VG(T)-VG(Tr)TTr]+VBE(Tr)TTrn(T)n(Tr)-n(T)(η-m)kTqln⁡⁡TTr.
Equation (15) shows that VBE(T) is affected by both the value of emission coefficient and its variation with temperature. Three cases can be differentiated:(a)
n=1for all of the temperature range: in this case, (15) is reduced to the expression reported in the literature;(b)
n>1and constant for all of the temperature range: in this case n(T)=n(Tr), so only the linear term is not affected;(c)
n>1and varies withT: in this case, all of the terms are affected.This makes necessary to characterize the variation ofn as a function of bias and temperature.


### 3.1. Physical Approach to Emitter Emission Coefficient

Many authors  consider the emission coefficient, as in the SPICE Gummel-Poon model (SGP), a constant value close to the unity to explain the nonideality of the exponential dependence of collector currentIC on emitter-base voltage as discussed elsewhere [2, 10] (see (14)). Nevertheless, as was discussed by Gummel and Poon [14], in the original Gummel-Poon model n is not a parameter, and it is defined at given VCB and T as the reciprocal of the slope of the Gummel plot given at low level injection by
(16)n(T)=1+kTCeqQb0.
Extending the Gummel-Poon model as discussed by Nagy et al. [22] definition of n in temperature results in
(17)n(T)=1+kTCe(T)qQb0,
where Ce(T) is the emitter capacitance and Qb0 the zero-bias charge in the base, which can be considered constant. This expression allows a physical approach of its behavior with forward bias and temperature through Ce(T),  which has been widely studied elsewhere [13, 14, 23, 24].


### 3.2. Simulation of Emitter Emission Coefficient

A new simulation procedure was elaborated to characterize the behavior of the emission coefficient of a bipolar transistor (biased at constant collector current) with the temperature. This procedure uses the original Gummel-Poon’s expressions ofn including its temperature dependence (see (17)) and the new Ce model developed as a function of base-emitter voltage and temperature.A flow chart showing the simulation procedure used to obtainn(T) is presented in Figure 4, which includes experimental and simulated values of Ce at 29.3°C.Figure 4
Flow chart: simulation procedure to obtainn(T).


### 3.3. Simulation Results

The emission coefficient was calculated in the temperature range of 240 K–300 K using (17), where the value of Qb0  was considered constant and equal to 6.0 pF. The simulation parameters extracted from experimental Ce measurements at reference temperature (29.3°C) were the following: Cje0(Tr)=8.52 pF, VDEideal(Tr)=0.719 V, z=0.329, CS(Tr)=4.259×10-8 pF, and  VDEneutral(Tr)=0.61, Cemax(Tr)=84 pF, and VBEmax(Tr)=0.558 V. Bias conditions for simulation at Tr were given by VBE=0.5 V and VBE=0.46 V, both selected into the region with exponential behavior of the emitter capacitance Ce as forward bias increases.Simulation results of MAT01 standard bipolar transistor withη=3.54, VG0=1.185 V and vertical PNP transistors fabricated in 0.5 μm CMOS technology with η=5.1, VG0=1.12046 V at two bias levels are shown in Figures 5 and 6.Figure 5
Variation ofn(T) of standard bipolar MAT01 transistor with η=3.54, VG0=1.185 V.Figure 6
Variation ofn(T) for vertical PNP transistors in 0.5 μm CMOS technology with η=5.1, VG0=1.12046 V.The temperature sensitivity ofn is related to TCe(T) product, as can be seen in (17). It implies a dependence of emitter area confirmed through the experimental measurements of n(T) at low temperatures for different emitter areas as discussed by Wang and Meijer [20]. From the simulation results presented, it can be observed that the increase of n at low temperature also depends on bias conditions and technological parameters η and VG0. In the case of vertical pnp transistors fabricated with 0.5 μm CMOS technology, the simulated behaviour of n(T) shows a similar trend than measured data reported.


### 3.4. Assessment ofn(T) Effect on VBE(T)

The estimation of emitter emission coefficient effect was obtained using the linear approximation forVG(T) in (15) of VBE(T),  resulting in
(18)VBE(T)=n(T)VG0[1-TTr]+VBE(Tr)TTr[n(T)n(Tr)]-n(T)ηkTqln⁡⁡TTr,
where n(T)VG0=VG0′ and n(T)η=η′ represent the best fitting parameters for VBE(T), extracted from experimental measurements. Now (18) is modified as
(19)VBE(T)=VG0′[1-TTr]+VBE(Tr)TTr[n(T)n(Tr)]-η′kTqln⁡⁡TTr.
The error ΔVBE(T) introduced, neglecting the variation of emission coefficient with  temperature, was estimated for transistors fabricated with different technologies, using (19). The results are shown in Table 2, where the error ΔVBE(T), expressed  in equivalent temperature, was estimated considering a typical variation of  −2 mV/°C for base-emitter voltage.Table 2
Calculated values ofΔVBE and its equivalent temperature error.
Data atTr=300 K and T=240 KResultsTechnologyVBE(Tr)Vn(Tr)n(T)VG0′ Vη′ΔVBE(T) mVΔTeq°CNPN (Bipolar)0.461.074961.077641.120465.11.7110.8550.501.124591.157901.120465.110.405.2PNP (CMOS)0.461.074721.070031.185003.54−1.6060.8030.501.123441.128371.185003.541.7560.878These results assess the convenience of taking into account the variation of emission coefficient with temperature in the design of circuits based on base-emitter voltage, as I.C. temperature sensors and compensated bandgap voltage references.


## 3.1. Physical Approach to Emitter Emission Coefficient

Many authors  consider the emission coefficient, as in the SPICE Gummel-Poon model (SGP), a constant value close to the unity to explain the nonideality of the exponential dependence of collector currentIC on emitter-base voltage as discussed elsewhere [2, 10] (see (14)). Nevertheless, as was discussed by Gummel and Poon [14], in the original Gummel-Poon model n is not a parameter, and it is defined at given VCB and T as the reciprocal of the slope of the Gummel plot given at low level injection by
(16)n(T)=1+kTCeqQb0.
Extending the Gummel-Poon model as discussed by Nagy et al. [22] definition of n in temperature results in
(17)n(T)=1+kTCe(T)qQb0,
where Ce(T) is the emitter capacitance and Qb0 the zero-bias charge in the base, which can be considered constant. This expression allows a physical approach of its behavior with forward bias and temperature through Ce(T),  which has been widely studied elsewhere [13, 14, 23, 24].


## 3.2. Simulation of Emitter Emission Coefficient

A new simulation procedure was elaborated to characterize the behavior of the emission coefficient of a bipolar transistor (biased at constant collector current) with the temperature. This procedure uses the original Gummel-Poon’s expressions ofn including its temperature dependence (see (17)) and the new Ce model developed as a function of base-emitter voltage and temperature.A flow chart showing the simulation procedure used to obtainn(T) is presented in Figure 4, which includes experimental and simulated values of Ce at 29.3°C.Figure 4
Flow chart: simulation procedure to obtainn(T).


## 3.3. Simulation Results

The emission coefficient was calculated in the temperature range of 240 K–300 K using (17), where the value of Qb0  was considered constant and equal to 6.0 pF. The simulation parameters extracted from experimental Ce measurements at reference temperature (29.3°C) were the following: Cje0(Tr)=8.52 pF, VDEideal(Tr)=0.719 V, z=0.329, CS(Tr)=4.259×10-8 pF, and  VDEneutral(Tr)=0.61, Cemax(Tr)=84 pF, and VBEmax(Tr)=0.558 V. Bias conditions for simulation at Tr were given by VBE=0.5 V and VBE=0.46 V, both selected into the region with exponential behavior of the emitter capacitance Ce as forward bias increases.Simulation results of MAT01 standard bipolar transistor withη=3.54, VG0=1.185 V and vertical PNP transistors fabricated in 0.5 μm CMOS technology with η=5.1, VG0=1.12046 V at two bias levels are shown in Figures 5 and 6.Figure 5
Variation ofn(T) of standard bipolar MAT01 transistor with η=3.54, VG0=1.185 V.Figure 6
Variation ofn(T) for vertical PNP transistors in 0.5 μm CMOS technology with η=5.1, VG0=1.12046 V.The temperature sensitivity ofn is related to TCe(T) product, as can be seen in (17). It implies a dependence of emitter area confirmed through the experimental measurements of n(T) at low temperatures for different emitter areas as discussed by Wang and Meijer [20]. From the simulation results presented, it can be observed that the increase of n at low temperature also depends on bias conditions and technological parameters η and VG0. In the case of vertical pnp transistors fabricated with 0.5 μm CMOS technology, the simulated behaviour of n(T) shows a similar trend than measured data reported.


## 3.4. Assessment ofn(T) Effect on VBE(T)

The estimation of emitter emission coefficient effect was obtained using the linear approximation forVG(T) in (15) of VBE(T),  resulting in
(18)VBE(T)=n(T)VG0[1-TTr]+VBE(Tr)TTr[n(T)n(Tr)]-n(T)ηkTqln⁡⁡TTr,
where n(T)VG0=VG0′ and n(T)η=η′ represent the best fitting parameters for VBE(T), extracted from experimental measurements. Now (18) is modified as
(19)VBE(T)=VG0′[1-TTr]+VBE(Tr)TTr[n(T)n(Tr)]-η′kTqln⁡⁡TTr.
The error ΔVBE(T) introduced, neglecting the variation of emission coefficient with  temperature, was estimated for transistors fabricated with different technologies, using (19). The results are shown in Table 2, where the error ΔVBE(T), expressed  in equivalent temperature, was estimated considering a typical variation of  −2 mV/°C for base-emitter voltage.Table 2
Calculated values ofΔVBE and its equivalent temperature error.
Data atTr=300 K and T=240 KResultsTechnologyVBE(Tr)Vn(Tr)n(T)VG0′ Vη′ΔVBE(T) mVΔTeq°CNPN (Bipolar)0.461.074961.077641.120465.11.7110.8550.501.124591.157901.120465.110.405.2PNP (CMOS)0.461.074721.070031.185003.54−1.6060.8030.501.123441.128371.185003.541.7560.878These results assess the convenience of taking into account the variation of emission coefficient with temperature in the design of circuits based on base-emitter voltage, as I.C. temperature sensors and compensated bandgap voltage references.


## 4. Built-in Voltage

The difference of potential that exists across a pn junction in equilibrium, known as contact potential, diffusion potential, or “built-in potential,”VDE, is a fundamental parameter in the theoretical study and modeling of the pn junction and semiconductor  devices. This potential is present in  the theoretical expressions of space charge, in Ideal model of  space charge capacitance of the pn junction  in the emitter-base capacitance Poon-Gummel model, HICUM and Mextram compact models as discussed elsewhere [15, 23–26]. The determination of VDE’s value, being a parameter associated with the equilibrium, is only possible indirectly, from the measurement of the capacity of the corresponding junction.As was reported by Schroter [13], the diffusion voltage VDE depends on temperature through
(20)VDE(T)=VG0[1-TTr]+VDE(Tr)TTr-3VTln⁡⁡TTr,
where VDE(Tr)  is the diffusion potential at reference temperature Tr.


### 4.1. Measurement ofVDE

The determination ofVDE(T) values is based on (13) using experimental measured values of VBEmax(T), obtained with the measurement setup in Figure 1.For the comparison of the experimental and theoretical behavior of the diffusion potential with temperature, the theoretical values ofVDE(T) by means of expression (20) were calculated, taking VDE(Tr) as the experimental value of VDE(T) at the lowest temperature of measurement and the value of VG0=1.185 V brought by Meijer [27] for the transistor MAT01. The results are plotted in Figure 7.Figure 7
Variation of diffusion potential versus temperature.Results show an excellent agreement between theory and experiments with a maximum error lower than 2%. These results validate the method of measurement used for the determination of diffusion potential and assess (20) to describe the behavior of the diffusion potential with temperature.


## 4.1. Measurement ofVDE

The determination ofVDE(T) values is based on (13) using experimental measured values of VBEmax(T), obtained with the measurement setup in Figure 1.For the comparison of the experimental and theoretical behavior of the diffusion potential with temperature, the theoretical values ofVDE(T) by means of expression (20) were calculated, taking VDE(Tr) as the experimental value of VDE(T) at the lowest temperature of measurement and the value of VG0=1.185 V brought by Meijer [27] for the transistor MAT01. The results are plotted in Figure 7.Figure 7
Variation of diffusion potential versus temperature.Results show an excellent agreement between theory and experiments with a maximum error lower than 2%. These results validate the method of measurement used for the determination of diffusion potential and assess (20) to describe the behavior of the diffusion potential with temperature.


## 5. Conclusions

A physical point of view in the analysis of the approximations used  inVBE(T) expression allows to reach an improved description of base-emitter voltage temperature dependence, considering the effect of emitter emission coefficient n and its direct dependence on base-emitter capacitance Ce.The obtained results allow a physical approach ton(T) behavior with forward bias and temperature through Ce(T) and a procedure to simulate n(T) values, using a new Ce model.The estimation of errorΔVBE(T) introduced neglecting the emission coefficient shows that it is convenient to take into account the variation of emission coefficient with temperature in the design of circuits based on base-emitter voltage, as I.C. temperature sensors and compensated bandgap voltage references.A new method for measurement of diffusion potentialVDE(T) based on the theory of pn junction neutral capacitance and the measurement of emitter capacitance Ce versus VBE,  was developed. The measurements of VDE(T) show good concordance with theoretical behavior, confirming the validity of both, the method used and the theoretical equation reported in the literature.



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