Study on Possible Double Peaks in Cutoff Frequency Characteristics of AIGaAs / GaAs HBTs by Energy Transport Simulation

By using an energy transport model, we simulate cutoff frequency f T versus collector current density I C characteristics of n p n − n AlGaAs/GaAs heterojunction bipolar transistors (HBTs) with various n − -collector thickness and n − -doping densities. It is found that the calculated f T characteristics show double peak behavior when the n − - layer is thick enough and the n − -doping is high enough to allow existence of neutral n − - region. The mechanism of the double peak behavior is discussed by studying energy band diagrams, electron-energy profiles and electron-velocity profiles. Particularly, we discuss the origin of the second peak (at higher I C ) which is not usually reported experimentally.


INTRODUCTION
Recently, A1GaAs/GaAs heterojunction bipolar transistors (HBTs) have received great interest for application to high-speed and high-frequency devices.Since non equilibrium carrier transport becomes important in the HBTs, carrier energy should be considered in the modeling of them.For this purpose, the Monte Carlo simulation [1,2] and so-called an energy transport model [3][4][5][6][7] have been applied to analyze the characteristics of A1GaAs/GaAs HBTs.
Cutoff frequencyfis one of the figure-of-merits of HBTs' high-frequency performance.Usually, the cutoff frequency of A1GaAs/GaAs HBTs increases with the collector current density Ic and begins to decrease at a certain Ic, showing a single peak in the experimental fT'-Ic character- istics.However, according to the simulation using a drift-diffusion model, the fT-Ic characteristics show a steep second peak in some cases [5,8].This is attributed to the fact that in the drift-diffusion approximation, electron mobility is given as a function of local electric field and the electron velocity versus electric field curve of GaAs shows a peak behavior.By the other simulation methods, the second peak has not been reported yet.
In this work, we have systematically and care- fully analyzed the fr characteristics of A1GaAs/ GaAs HBTs by using an energy transport model [9] in which electron mobility is determined by an electron energy (not by local electric field) and velocity overshoot can be treated.As a result, we have found that the second peak can arise (at rather high current levels) also when using this model.Therefore, we discuss here the physical reason why the double peak behavior in the characteristics arises.
2. PHYSICAL MODEL density and carrier energy.We have simulated the device characteristics as parameters of n--collector thickness Lcl and its doping density NCl.
Next we describe the derivation of electron transport equations.Assuming parabolic energy bands, three conservation equations, that is, a particle conservation equation, a momentum conservation equation and an energy conservation equation are obtained by taking moments of the Boltzmann equation [10].To make these equations tractable and applicable to the HBTs, the follow- ing assumptions are made [5].Firstly, we adopt relaxation time approximation for collision terms.
Secondly, we adopt an equivalent one-valley model.If we assume that electron drift energy is negligible small as compared to its thermal energy, the average electron energy w, can be written as where T. is electron temperature, Fv is upper- valley fraction and ALU is energy difference between upper and lower valleys.To treat HBTs, some factors are also considered.Carrier recom- bination is treated by SRH statistics, and an effective field acting on electrons which arises due to the position dependence of band structure is included.Then the electron transport equations are simplified as follows.
dz q U (2) Jn -qnvn q#n(Wn) n En "+ z n---- (3)   d {Jn } Wn--Wo dz --(wn + kTn) JnEn wn U-n.rw where U is the recombination rate, v is the average electron velocity, # is the electron mobility, En is the effective field acting on electrons, 'w is the energy relaxation time and w0 is the equilibrium value of w.In the electron transport equations, parameters that should be given are the electron mobility n, the energy relaxation time -w and the upper-valley fraction Fu.
For holes, we use drift-diffusion type equations.In addition to the transport equations, we include Poisson's equation, which completes the basic equations for device simulation.

Transport Parameters
Here we describe methods of giving transport parameters such as n, 7"w and Fu.To do this, usually, homogeneous bulk is assumed (d/dz=O).
Then, the following tWO equations are obtained from the previous electron transport equations.
V n --n E (5) Wn WO q "rw Vn E (6) where E is electric field.By using the Monte Carlo method, v,, w, and Fu are obtained as a function of electric field E. Thus, by using Eqs.( 5) and ( 6), the energy-dependent #, w and Fu can be obtained.
In this work, we must give the transport parameters as functions of A1 composition x, electron energy Wn and doping density N. To do this, we first evaluate #, w and Fv for x 0, 0.05, 0.1, 0.15, 0.2, 0.25 and 0.3 by a Monte Carlo method.Doping densities are also varied.Once these parameters are available as fit curves or tables for a given doping density, parameters for any x between 0 and 0.3 can be obtained (as a function of w) by a linear extraporation method.
That is, if x lies between X and x2, the parameter f is given by the following equation.
where fl and f2 are corresponding values of f for x X and x2, respectively.Next we show some examples of transport parameters estimated by a Monte Carlo method.FIGURE 2 Electron mobility #,,, energy relaxation timeand upper-valley fraction Fu versus electron energy COn curves as a parameter of x in AlxGal_x As, calculated by a Monte Carlo method.The energy difference between upper and lower valleys ALu and upper-valley effective mass mu are set to 0.284- 0.605 x (eV) and 0.23 m0, respectively.
parameter of A1 composition x.The doping density is 2 1018 cm-3.Here, we consider L-valley as upper valley and use L-valley's parameters for ALU and upper-valley effective mass.We also treat a case somewhat considering X-valley's contribution, but x and energy dependences of estimated parameters are essentially similar to those shown in Figure 2 [9].As described before, once the figures like Figure 2 are obtained for a given doping density, the transport parameters for any x and for any electron energy are obtained by the linear extraporation method.Of course, when the doping density becomes different, another figure is required.Furthermore, the parameters must be evaluated in every mesh point.However, this approach is simple in itself and can be easily implemented.

SIMULATED fr-Ic CHARACTERISTICS
We calculate fr-Ic characteristics of the A1GaAs/ GaAs HBT as parameters of the ncollector thickness Lc1 and its doping density NCl.Here fr is calculated from the following equation.
fr (1/27r)(OIc/OQn)vcE (8) where Q, is electron charges in the device and VCE is the collector-emitter voltage.VCE is set to 1.5 V in this study.
Figure 3 shows calculatedfr-Ic characteristics of the HBT as a parameterof n--layer doping density Nc1, where Lc1 is set to 0.5 Im. Figure 4 shows calculated fr-Ic characteristics of the HBT as a parameter ofn--layer thickness Lcl, where NCl is set to 5 x 1016 cm-3.As Ic increases,fr increases because the emitter charging time and the collector charging time are reduced.From these figures, we see that for lower Ncl (1016 cm -3 in Fig. 3) and for thinner Lc (0.1 gm and 0.2 gm in Fig. 4), fr characteristics show a single peak.These cases correspond to the situation that n--collector layer is almost or fully depleted already when the base- emitter voltage (VBE) is 0 V.In the other cases,two peaks are clearly seen in the fr characteristics.In these cases, neutral n--region exists at VBE 0 V. Up to the first peak, fr is higher for higher Nc1, as seen from Figure 3. Also, the value of Ic where fr begins to decrease is higher for higher Ncl.
These are because around the peak region, the transit time through n--collector depletion layer (which is thinner for higher Ncl) is dominant [5] and fr begins to decrease due to a high injection effect which leads to expanding the depletion layer  and increasing the collector transit time.As seen in Figure 4, the fr characteristics are essentially similar between the two cases with Lcl =0.5 and 0.7 txm.This is because the thickness of n-collector depletion layer is determined by Nc].As shown in Figure 3, when Ncl is higher, the value of Ic where fT takes the second peak is higher.We will discuss below why the fT characteristics show double peak behavior.
Figure 5 shows energy band diagrams as a parameter of Ic for the HBT where Ncl 5 1016 cm -3 and Lcl =0.5 lxm.Figures 6 and 7 show the corresponding electron-energy profiles and elec- tron-velocity profiles, respectively.In these figures Ic 10 4 A/cm 2, 3 x 10 4 A/cm 2, 8 10 4 A/cm 2, 1.2 x 10 5 A/cm 2 and 1.5x 10 5 A/cm 2 correspond to the regions before the first peak, around the first peak,  at the local minimum, at the second peak, and after the second peak respectively, in the fr-Ic characteristics.From these, we interpret the double peak behavior in the following way.
As is understood from Figure 5, the first peak arises because the depletion layer in the n-collector layer expands due to an high injection effect, and hence the collector transit time in- creases.The fall of Ic should last until the n-- collector layer becomes entirely depleted.Around Ic 8 x 10 4 A/cm 2, the n--layer is entirely depleted as seen from Figure 5 and at this current, fr characteristics show the local minimum.As can be seen from the band diagrams, the electric field at the n--collector layer near the base becomes weaker when Ic increases further.Hence, as shown in Figure 6, the electron energy in this region becomes lower.Consequently, the electron mobi- lity becomes high, leading to the higher electron velocity than the saturation velocity, for example, at Ic =l.2xlO 5 A/cm 2 as shown in Figure 7.
Therefore, the collector transit time becomes shorter temporarily and hence fr begins to increase again.In the end, however, fr falls because of the base-push-out effect (Kirk effect) which leads to a lower electron velocity around the base-collector interface and higher collector capacitance due to the injected electrons whose densities become higher than Nc1.Thus the second peak arises.As is evident from the above discussion, the value of Ic where fr shows a second peak becomes higher for higher Mil.
As described above, we have shown theoretically that double peak behavior can be seen in theft-Ic characteristics of A1GaAs/GaAs HBTs.Physical mechanism of this behavior has been explained.

CONCLUSION
By using an energy transport model, we have simulated fr-Ic characteristics of A1GaAs/GaAs HBTs with various n--collector thickness and n-doping densities.It is found that the calculated fr characteristics show the double peak feature when the n--layer is thick enough and the n--doping is high enough to allow existance of the neutral n-region.It is interpreted that the first peak arises because the depletion region in the n--layer begins to expand due to a high injection effect and the collector transit time increases.The fall offr lasts until the n--layer becomes entirely depleted.When the base voltage is raised further, the electric field in the n--layer near the base becomes lower, leading to the lower electron energy there.Then, the electron velocity in the n--layer becomes higher, resulting in shorter collector transit time.
Therefore, fr begins to increase again.Finally, fr decreases due to the base-push-out effect (Kirk effect), resulting in the second peak.We can say that the double peak behavior can be seen in the fr

Figure 2
Figure 2 shows electron mobility #, energy relaxa- tion time w and upper valley fraction Fu as a

FIGURE 4
FIGURE 4 Calculated fr-Ic curves of A1GaAs/GaAs HBTs as a parameter of n--collector thickness Lcl.The.n--collector doping density Ncl is 5 x 1016 cm-3.

FIGURE 3 FIGURE 5
FIGURE 3 Calculated fr-lc curves of A1GaAs/GaAs HBTs as a parameter of n--collector doping density Ncl.The n-collector thickness Lcl is 0.5 tm.