The past few years have seen significant progress in the development of SiGe heterojunction bipolar transistor (HBT) technology. Today, the use of SiGe-base HBTs is becoming increasingly popular in wireless and high-speed digital communications .
The most significant material parameter to be specified in the simulation of SiGe HBTs is the bandgap narrowing induced by incorporation of a Ge fraction in the base. In addition to the Ge-induced bandgap narrowing, the high doping in the base induces additional bandgap narrowing, similar to that observed in silicon. Although several bandgap narrowing model and its affect have been proposed for silicon -, the new effects and nuances of operation of SiGe HBT are still being uncovered and as transistor scaling advances with different application targets steadily increasing, the comprehensive treatment of its working is still desired.
While designing the SiGe HBT, doping is considered a critical issue as it affects bandgap narrowing. In lightly doped semiconductors the dopant atoms are sufficiently widely spaced in the semiconductor lattice that the wave functions associated with the dopant atoms' electrons do not overlap. The energy levels of the dopant atoms are therefore discrete. Furthermore, it is reasonable to assume that the widely spaced dopant atoms have no effect on the perfect periodicity of the semiconductor lattice, and hence the edges of the conduction and valence bands are sharply defined. In heavily doped semiconductors the dopant atoms are close enough together that the wave functions of their associated electrons overlap.
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In addition, the large concentration of dopant atoms disrupts the perfect periodicity of the silicon lattice, giving rise to a band tail instead of a sharply defined band edge. At high doping concentrations, the Fermi level approaches the band edge and can even move above the band edge. In these circumstances, the Boltzmann statistics used in are inaccurate and it is necessary to use Fermi-Dirac statistics to calculate the position of the Fermi level. To model heavy doping effects in the emitter of a bipolar transistor, it is necessary to combine the effects of bandgap narrowing and Fermi- Dirac statistics. For ease of modelling, these effects are rolled into a single parameter called the apparent bandgap narrowing or the dopinginduced bandgap narrowing.
The energy band gap of the SiGe alloy lies between the band gap of Silicon (1.1eV) and Germanium(0.66eV). The bandgap is further decreased by the compressive strain in the alloy layer when grown on a Silicon substrate[â€¦]. In strained SiGe grown on a Si substrate, most of the bandgap reduction results in a valence band discontinuity (about 75meV for each 10% of Ge) [â€¦]. For SiGe HBTs, the Silicon material forms the wider bandgap material while the SiGe alloy is the narrow band gap material. In other words, the conduction band and valence band edges of the strained layers of SiGe in the base lie within the band edges of the underlying Si in the collector and the overlying Si in the emitter, a circumstance which favors this material combination and the use of bandgap engineering to build faster silicon bipolar transistors. A big advantage of having Germanium in the base of a SiGe HBT is that there is a formation of a heterojunction at the emitter-base junction of the transistor. Therefore, the potential energy barrier in the conduction band at the emitter-base junction is lowered allowing more electrons to be injected into the base and thereby leads to an increase the collector current. Further, hole back injection is also reduced by the large valence band discontinuity reducing the base current. Overall, this increase in collector current and reduction in the hole back injection dramatically improves the current gain.
This high field increases the speed and gain of the device by aiding the transport of electrons across the base, thus, decreasing the base transit time and improving the transport efficiency. The strain introduced during the growth of SiGe on the single crystal Silicon fortunately also contributes to high electron mobilities in the base thereby also increasing the speed of the device. The mobility is increased due to the fact that the electrons occupy the conduction band valleys for which the effective mass is lower by the addition of Ge. The high base doping also results in an improvement in the Early Voltage of the SiGe HBT. But there is not any significant change in Early Voltage due to low base doping as in this paper low base doping is considered.
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Although significant amount of work on Fermi Dirac Analysis to estimate the bandgap narrowing has been done but its effect on some other parameters like gain, cutoff frequency etc. needs some more explanation. A performance comparison of Fermi Dirac Statistics with Boltzman approach is reviewed in this paper with the help of physics based model and simulated results. These results are obtained using 2D SILVACO device simulator known for its authentication in the industry. An attempt has been done in this paper to study the impact of Ge fraction (in SiGe base) on bandgap narrowing considering the important design parameters and issues like conductance, current gain, cutoff frequency, maximum oscillation frequency and junction capacitances.
The structure under consideration is npn SiGe HBT. In a SiGe HBT the ratio of the base and collector currents is given by [â€¦]
where is the valence band discontinuity at the emitter-base heterojunction. For Si-SiGe heterojunctions, the consensus from the literature is that so [â€¦]. ATLAS has a parameter called ALIGN that allows the user to incorporate the percentage of the energy gap difference at a heterojunction to the conduction band.
A simple way of modelling bandgap narrowing in the emitter is through an effective doping concentration in the emitter due to Bandgap Narrowing[â€¦].
For heavily doped, n-type silicon, the model developed by del Alamo [1-2] gives a reasonably accurate description of the apparent bandgap narrowing. In this model, the apparent bandgap narrowing in the emitter is described by the following empirical equation[â€¦]
The effects of bandgap narrowing in the base can be simply modeled using an effective doping concentration in the base due to Bandgap Narrowing is given as[â€¦]
So putting the value of Effective doping effective
This equation clearly indicates that bandgap narrowing has the effect of reducing the effective doping concentration in the emitter, and hence also the gain of the bipolar transistor. The gain can be manipulated if the doping concentration in the emitter Nde is replaced by the effective doping concentration Ndeff.
In base region there are two source of Bandgap Narrowing (a) due to the strained (b) due to Ge. The Ge dependent energy bandgap of the SiGe is given by [â€¦]
where x is the Ge content in mole fraction. The bandgap reduction due to Ge content has been incorporated by using the above equation (5) and (6)
where, is the gain calculated considering Boltzman Statistics. is constant as Emitter doping is constant.
3. Results and Discussion:
Considering the Fermi Dirac and Boltzman Statistics and results obtained on Atlas we came to conclude that Bandgap Narrowing has great affect on highly doped SiGe HBT (1*1020 cm3). The most important parameter, the current gain, is reduced from 180 to 145. There is a corresponding reduction in Collector current from 3.12 mA to 1.25 mA. Hence Fermi Dirac Stats is essential for accurate modeling of highly doped SiGe HBT.
So increasing the Ge fraction we can compensate the error occurring by bandgap narrowing.
Fig.1 This graph clearly shows Difference between two stats which signifies the presence and absence of Bandgap Narrowing. It Increases as Ge Concentration increases. Error generated due to Bandgap norrowing is suppressed at a particular Ge fraction.
This graph clearly shows the decrement of error by increasing the Ge concentration as discussed above.
Cutoff Frequency and Parasitic Effect-
The SiGe layer with a constant Germanium mole fraction results in a Valence band discontinuity at both the emitter base and the collector base junctions. The Valence band discontinuity prevents back injection of holes from the base to the emitter.
The performance of the SiGe HBT greatly depends on the Ge profile in the base. In the case of constant Ge profile, similar grading is assumed on the emitter side as well the device with constant Ge profile is much greater than that with graded Ge profile.
The reason for the improved performance with increase in Ge fraction can be attributed to an increased hole mobility and improved current gain. The hole mobility in the SiGe increases with increasing Ge content as given by,
thereby reducing the base resistance.
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Fig.-2 Dark color line shows the variation of Conductance of Emitter base junction in (Siemens/mm).It clearly shows that upto some extent it increases (correspondingly base resistance decreases) but for higher Ge fraction in SiGe it decreases. This analysis is for considering Fermi Dirac Statistics. Light color line does not show the actual variation in conductance as it considers the Boltazman Statistics which does not count Bandgap Narrowing.
It shows the decrement in Conductance which signifies the decrease of mobility for high Ge fraction. It shows the Base resistance is increasing and hence Cutoff frequency is decreasing. Mobility of holes also depends on Doping and Electric field (CONMOB, FLDMOB ie Concentration Dependent and field dependent mobility) in our Simulator ATLAS. So increasing Ge fraction by a certain limit (.25) emitter resistance decreases rapidly as shown in graph.
Formula used to calculate Cutoff frequency-
In our simulator,ATLAS peak cutoff frequency is derived from ,
g."collector""base"= Transconductance (Siemens/micron)
c."base""base"= Input Base Capacitance (Farad/micron)
But as a result, cutoff frequency is increased so the maximum frequency of oscillation increases with Ge content. In case the of a constant Ge profile, the current gain and hence cutoff frequency are improved due to the presence of the emitter-base heterojunction at the valence band, which significantly reduces back injection of holes from the base into the emitter. The valence band discontinuity increases with Ge fraction at the emitter base junction, thereby improving the performance of the device improving the current gain. It also reduces the base transit time, which gives a larger cutoff frequency. The falloff at higher current density (thereby reducing cutoff frequency) is much steeper when the Ge mole fraction is higher (>24% as from Fig.-2) at the collector junction due to the formation of the parasitic barrier. The displacement of the heterojunction away from the p-n junction results in a parasitic barrier at the base-collector junction that degrades the performance of the SiGe.
Fig.3 It shows the formation of Junction Capacitance at Peak Cutoff Frequency. Concluding all the theories on the formation of Parasitic Barrier this graph justify that there is an abrupt change in capacitance at high value of Ge concentration
The formation of the Parasitic barrier at the emitter-base and base-collector junctions due to high current effects and base dopant outdiffusion at high current densities, hole accumulation at the base end of the collector-base junction induces electron pile up at the collector end of the base-collector junction. This leads to the formation of a parasitic field which acts as a potential barrier to the electron flow in the conduction band as shown This barrier increases the recombination in the base and produces a saturation tendency in the collector current, both of which degrade the current gain.
Fig.3 Graph shows that there is a decrement in Transconductance(used in formulae) with higher Ge fraction.Which is the main cause to decrease the Peak Cutoff frequency.
This barrier also increases the base transit time and consequently degrades the cutoff frequency (fT) and maximum frequency of oscillation (fmax). It has been found that parasitic barriers at both junctions are also formed due to nonalignment of the p-n junction and heterojunction.
Fig.4 This graph justify the above discussed theory on parasitic effect on Cutoff frequency. Using a gradually reduced bandgap in the quasi-neutral region an accelerating electric field may be introduced for decreasing the base transit time. However, a large electric field in the base can be counter-productive as a result of mobility degradation in the high-field regions. In this study it has shown that Ge fraction about 24% can causes decrement in Cutoff frequency
The use of SiGe in the base of Si/SiGe/Si NPN HBT causes the presence of a heterojunction at the collector-base junction as well as the emitter-base junction. Since, the energy gap difference between the two materials is primarily in the valence band, the conduction band difference is almost negligible. This is desirable since the presence of a Î”Ec at the collector-base heterojunction forms a potential energy barrier impeding electron injection from the base into the collector. For SiGe HBTs, the valence band discontinuity due to the heterojunction at the base-collector junction prevents holes from spilling into the collector from the base, at the onset of basepushout. Since the Î”Ev barrier prevents holes from moving into the collector, a net positive charge accumulates at the collector end of the base, a corresponding negative charge forms nearby in the collector and a parasitic barrier forms in the conduction band at the collector-base junction. The presence of this parasitic barrier limits the collector current, causes the base current to increase and the current gain of the device drops drastically. Further, the base transit time increases which decreases the cutoff frequency (ft) and the maximum frequency of oscillation (fmax).
Mathematical Interpretation of Change in Cutoff frequency- Cutoff frequency is given by,
Ï„e is the emitter charging time and is defined as the time required to change the base potential by charging up the device capacitances through the differential base-emitter junction resistance
where Cje and Cjc denote the junction capacitances for the base-emitter and base collector junctions, respectively, and n denotes the ideality factor of the device.
Ï„b is the base transit time and is defined as the time required to discharge the excess minority carriers in the base through the collector current. It is given as, when we consider the Bandgap Narrowing,
Vs the electron saturation velocity decreases substantially as the Ge concentration is increased. This reduction in vs is given by the linear function
where ni0 - intrinsic carrier concentration in undoped Si, k - the Boltzmann constant, T - temperature in Kelvin , âˆ†Eg - effective band gap reduction in the base due to the presence of Ge (âˆ†EgGe) and due to heavy doping effects (âˆ†EgDOP)
âˆ†Eg(x) = âˆ†EgGe(x) + âˆ†EgDOP(x)
âˆ†EgGe(x) The band gap narrowing due to the presence of Ge is assumed to have a linear dependence on Ge concentration Î”Eg,Ge = 700x(meV) and âˆ†EgDOP(x) is
Bandgap Narrowing due to Heavy doping effect)
Mobility can be expressed as
In this study we have taken CONMOB, FLDMOB ie Concentration Dependent and field dependent mobility
VSATN and BETAN are constants.
Considering all the terms (ie. Mobility, Effective doping, Electron Saturation Velocity) in base transit time depends on Bandgap Narrowing which has high effect on high Ge concentration. Further if we consider the Bandgap narrowing there is drastic change in Base transit Time which consequently effect Cutoff frequency.
where RE is the emitter resistance and RC is the collector resistance. The value of this charging time depends greatly on the parasitic emitter resistance RE and collector resistance RC.
Percentage reduction in Peak Cutoff frequency can be analysed by the following graph.
Comparing this figure to Fig.1 (Gain Difference Curve) we came to conclude that there is trade off between current gain and cutoff frequency (unity gain bandwidth). But there is sharp decrement in peak cutoff frequency curve (Fig.5) at high Ge fraction. Also in Fig.-1 gain Difference error decrease at high Ge fraction.So it is desirable that Ge fraction in SiGe Base is high (between 0.24 and 0.26).At higher Concentration of Ge we get High gain but pay off of cutoff frequency occurs. So for high speed application (LAN and Mobile Communication), where Current Gain is not the major concerned, low doping must be used. At this fraction Band gap narrowing has lower effect on Cutoff frequency and current gain also. Considering our analysis Boltzman Statistics (which we generally use), does not give an accurate estimation of Device parameters. Error caused by this approximation is very high at low fraction of Ge. So Femi Dirac statistics must be considered for Gain and Cutoff frequency estimation.