## 1. Introduction:

The past few old ages have seen important advancement in the development of SiGe heterojunction bipolar transistor ( HBT ) engineering. Today, the usage of SiGe-base HBTs is going progressively popular in radio and high-velocity digital communications [ 1 ] .

The most important stuff parametric quantity to be specified in the simulation of SiGe HBTs is the bandgap contracting induced by incorporation of a Ge fraction in the base. In add-on to the Ge-induced bandgap narrowing, the high doping in the base induces extra bandgap narrowing, similar to that observed in Si. Although several bandgap contracting theoretical account and its affect have been proposed for Si [ 2 ] – [ 4 ] , the new effects and niceties of operation of SiGe HBT are still being uncovered and as transistor grading progresss with different application marks steadily increasing, the comprehensive intervention of its working is still desired.

While planing the SiGe HBT, doping is considered a critical issue as it affects bandgap contracting. In lightly doped semiconducting materials the dopant atoms are sufficiently widely spaced in the semiconducting material lattice that the moving ridge maps associated with the dopant atoms ‘ negatrons do non overlap. The energy degrees of the dopant atoms are hence distinct. Furthermore, it is sensible to presume that the widely spaced dopant atoms have no consequence on the perfect cyclicity of the semiconducting material lattice, and therefore the borders of the conductivity and valency sets are aggressively defined. In to a great extent doped semiconducting materials the dopant atoms are close plenty together that the moving ridge maps of their associated negatrons overlap.

In add-on, the big concentration of dopant atoms disrupts the perfect cyclicity of the Si lattice, giving rise to a set tail alternatively of a aggressively defined set border. At high doping concentrations, the Fermi degree approaches the set border and can even travel above the set border. In these fortunes, the Boltzmann statistics used in are inaccurate and it is necessary to utilize Fermi-Dirac statistics to cipher the place of the Fermi degree. To pattern heavy doping effects in the emitter of a bipolar transistor, it is necessary to unite the effects of bandgap narrowing and Fermi- Dirac statistics. For easiness of modeling, these effects are rolled into a individual parametric quantity called the evident bandgap narrowing or the dopinginduced bandgap narrowing.

The energy set spread of the SiGe metal lies between the set spread of Silicon ( 1.1eV ) and Germanium ( 0.66eV ) . The bandgap is further decreased by the compressive strain in the metal bed when grown on a Silicon substrate [ aˆ¦ ] . In labored SiGe grown on a Si substrate, most of the bandgap decrease consequences in a valency set discontinuity ( about 75meV for each 10 % of Ge ) [ aˆ¦ ] . For SiGe HBTs, the Silicon stuff forms the wider bandgap stuff while the SiGe metal is the narrow set spread stuff. In other words, the conductivity set and valency set borders of the labored beds of SiGe in the base prevarication within the set borders of the underlying Si in the aggregator and the overlying Si in the emitter, a circumstance which favors this stuff combination and the usage of bandgap technology to construct faster Si bipolar transistors. A large advantage of holding 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 possible energy barrier in the conductivity set at the emitter-base junction is lowered leting more negatrons to be injected into the base and thereby leads to an addition the aggregator current. Further, hole back injection is besides reduced by the big valency set discontinuity cut downing the base current. Overall, this addition in aggregator current and decrease in the hole back injection dramatically improves the current addition.

This high field increases the velocity and addition of the device by helping the conveyance of negatrons across the base, therefore, diminishing the base theodolite clip and bettering the conveyance efficiency. The strain introduced during the growing of SiGe on the individual crystal Silicon fortuitously besides contributes to high negatron mobilities in the base thereby besides increasing the velocity of the device. The mobility is increased due to the fact that the negatrons occupy the conductivity set vales for which the effectual mass is lower by the add-on of Ge. The high base doping besides consequences in an betterment in the Early Voltage of the SiGe HBT. But there is non any important alteration in Early Voltage due to low base doping as in this paper low base doping is considered.

Although important sum of work on Fermi Dirac Analysis to gauge the bandgap narrowing has been done but its consequence on some other parametric quantities like addition, cutoff frequence etc. needs some more account. A public presentation comparing of Fermi Dirac Statistics with Boltzman attack is reviewed in this paper with the aid of natural philosophies based theoretical account and fake consequences. These consequences are obtained utilizing 2D SILVACO device simulator known for its hallmark in the industry. An effort has been done in this paper to analyze the impact of Ge fraction ( in SiGe base ) on bandgap contracting sing the of import design parametric quantities and issues like conductance, current addition, cutoff frequence, maximal oscillation frequence and junction electrical capacities.

## 2. Model:

The construction under consideration is npn SiGe HBT. In a SiGe HBT the ratio of the base and aggregator currents is given by [ aˆ¦ ]

[ 1 ]

where is the valency set discontinuity at the emitter-base heterojunction. For Si-SiGe heterojunctions, the consensus from the literature is that so [ aˆ¦ ] . ATLAS has a parametric quantity called ALIGN that allows the user to integrate the per centum of the energy spread difference at a heterojunction to the conductivity set.

A simple manner of patterning bandgap narrowing in the emitter is through an effectual doping concentration in the emitter due to Bandgap Narrowing [ aˆ¦ ] .

[ 2 ]

For to a great extent doped, n-type Si, the theoretical account developed by del Alamo [ 1-2 ] gives a moderately accurate description of the evident bandgap contracting. In this theoretical account, the evident bandgap narrowing in the emitter is described by the undermentioned empirical equation [ aˆ¦ ]

[ 3 ]

The effects of bandgap narrowing in the base can be merely modeled utilizing an effectual doping concentration in the base due to Bandgap Narrowing is given as [ aˆ¦ ]

[ 4 ]

So seting the value of Effective doping effectual

[ 5 ]

This equation clearly indicates that bandgap narrowing has the consequence of cut downing the effectual doping concentration in the emitter, and therefore besides the addition of the bipolar transistor. The addition can be manipulated if the doping concentration in the emitter Nde is replaced by the effectual doping concentration Ndeff.

In basal part there are two beginning of Bandgap Narrowing ( a ) due to the strained ( B ) due to Ge. The Ge dependent energy bandgap of the SiGe is given by [ aˆ¦ ]

[ 6 ]

where ten is the Ge content in mole fraction. The bandgap decrease due to Ge content has been incorporated by utilizing the above equation ( 5 ) and ( 6 )

[ 7 ]

where, is the addition calculated sing Boltzman Statistics. is changeless as Emitter doping is changeless.

## 3. Consequences and Discussion:

Sing the Fermi Dirac and Boltzman Statistics and consequences obtained on Atlas we came to reason that Bandgap Narrowing has great affect on extremely doped SiGe HBT ( 1*1020 cm3 ) . The most of import parametric quantity, the current addition, is reduced from 180 to 145. There is a corresponding decrease in Collector current from 3.12 mas to 1.25 ma. Hence Fermi Dirac Stats is indispensable for accurate mold of extremely doped SiGe HBT.

So increasing the Ge fraction we can counterbalance the mistake happening 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. Mistake generated due to Bandgap norrowing is suppressed at a peculiar Ge fraction.

This graph clearly shows the decrease of mistake by increasing the Ge concentration as discussed above.

## Cutoff Frequency and Parasitic Effect-

The SiGe bed with a changeless Germanium mole fraction consequences in a Valence set discontinuity at both the emitter base and the aggregator base junctions. The Valence set discontinuity prevents back injection of holes from the base to the emitter.

The public presentation of the SiGe HBT greatly depends on the Ge profile in the base. In the instance of changeless Ge profile, similar scaling is assumed on the emitter side every bit good the device with changeless Ge profile is much greater than that with ranked Ge profile.

The ground for the improved public presentation with addition in Ge fraction can be attributed to an increased hole mobility and improved current addition. The hole mobility in the SiGe increases with increasing Ge content as given by,

thereby cut downing the base opposition.

Fig.-2 Dark colour line shows the fluctuation of Conductance of Emitter base junction in ( Siemens/mm ) .It clearly shows that upto some extent it increases ( correspondingly base opposition lessenings ) but for higher Ge fraction in SiGe it decreases. This analysis is for sing Fermi Dirac Statistics. Light colour line does non demo the existent fluctuation in conductance as it considers the Boltazman Statistics which does non number Bandgap Narrowing.

It shows the decrease in Conductance which signifies the lessening of mobility for high Ge fraction. It shows the Base opposition is increasing and therefore Cutoff frequence is diminishing. Mobility of holes besides depends on Doping and Electric field ( CONMOB, FLDMOB Internet Explorer Concentration Dependent and field dependent mobility ) in our Simulator ATLAS. So increasing Ge fraction by a certain bound ( .25 ) emitter opposition decreases quickly as shown in graph.

## Formula used to cipher Cutoff frequency-

In our simulator, ATLAS peak cutoff frequence is derived from,

## Ft= g. ” aggregator ” ” base ” / ( 2*3.14*c. ” base ” ” base ” )

## g. ” aggregator ” ” base ” = Transconductance ( Siemens/micron )

## c. ” base ” ” base ” = Input Base Capacitance ( Farad/micron )

But as a consequence, cutoff frequence is increased so the maximal frequence of oscillation additions with Ge content. In instance the of a changeless Ge profile, the current addition and therefore cutoff frequence are improved due to the presence of the emitter-base heterojunction at the valency set, which significantly reduces back injection of holes from the base into the emitter. The valency set discontinuity additions with Ge fraction at the emitter base junction, thereby bettering the public presentation of the device bettering the current addition. It besides reduces the base theodolite clip, which gives a larger cutoff frequence. The slump at higher current denseness ( thereby cut downing cutoff frequence ) is much steeper when the Ge mole fraction is higher ( & gt ; 24 % as from Fig.-2 ) at the aggregator junction due to the formation of the parasitic barrier. The supplanting of the heterojunction off from the p-n junction consequences in a parasitic barrier at the base-collector junction that degrades the public presentation of the SiGe.

Fig.3 It shows the formation of Junction Capacitance at Peak Cutoff Frequency. Reasoning all the theories on the formation of Parasitic Barrier this graph justify that there is an disconnected alteration in electrical capacity 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 densenesss, hole accretion at the basal terminal of the collector-base junction induces electron stack up at the collector terminal of the base-collector junction. This leads to the formation of a parasitic field which acts as a possible barrier to the negatron flow in the conductivity set as shown This barrier increases the recombination in the base and produces a impregnation inclination in the aggregator current, both of which degrade the current addition.

Fig.3 Graph shows that there is a decrease in Transconductance ( used in expression ) with higher Ge fraction.Which is the chief cause to diminish the Peak Cutoff frequence.

This barrier besides increases the base theodolite clip and accordingly degrades the cutoff frequence ( foot ) and maximal frequence of oscillation ( fmax ) . It has been found that parasitic barriers at both junctions are besides formed due to nonalinement of the p-n junction and heterojunction.

Fig.4 This graph justify the above discussed theory on parasitic consequence on Cutoff frequence. Using a bit by bit reduced bandgap in the quasi-neutral part an speed uping electric field may be introduced for diminishing the base theodolite clip. However, a big electric field in the base can be counter-productive as a consequence of mobility debasement in the high-field parts. In this survey it has shown that Ge fraction about 24 % can do decrease in Cutoff frequence

The usage of SiGe in the base of Si/SiGe/Si NPN HBT causes the presence of a heterojunction at the collector-base junction every bit good as the emitter-base junction. Since, the energy spread difference between the two stuffs is chiefly in the valency set, the conductivity set difference is about negligible. This is desirable since the presence of a I”Ec at the collector-base heterojunction forms a possible energy barrier hindering electron injection from the base into the aggregator. For SiGe HBTs, the valency set discontinuity due to the heterojunction at the base-collector junction prevents holes from sloping into the aggregator from the base, at the oncoming of basepushout. Since the I”Ev barrier prevents holes from traveling into the aggregator, a net positive charge accumulates at the collector terminal of the base, a corresponding negative charge signifiers nearby in the aggregator and a parasitic barrier signifiers in the conductivity set at the collector-base junction. The presence of this parasitic barrier limits the aggregator current, causes the base current to increase and the current addition of the device drops drastically. Further, the base theodolite clip additions which decreases the cutoff frequence ( foot ) and the maximal frequence of oscillation ( fmax ) .

Mathematical Interpretation of Change in Cutoff frequency- Cutoff frequence is given by,

where

I„e is the emitter bear downing clip and is defined as the clip required to alter the base potency by bear downing up the device electrical capacities through the differential base-emitter junction opposition

where Cje and Cjc denote the junction electrical capacities for the base-emitter and base aggregator junctions, severally, and n denotes the ideality factor of the device.

I„b is the basal theodolite clip and is defined as the clip required to dispatch the extra minority bearers in the base through the aggregator current. It is given as, when we consider the Bandgap Narrowing,

Vs the negatron impregnation speed decreases well as the Ge concentration is increased. This decrease in V is given by the additive map

Where

where ni0 – intrinsic bearer concentration in undoped Si, k – the Boltzmann invariable, T – temperature in Kelvin, a?†Eg – effectual set spread decrease in the base due to the presence of Ge ( a?†EgGe ) and due to heavy doping effects ( a?†EgDOP )

a?†Eg ( x ) = a?†EgGe ( x ) + a?†EgDOP ( x )

a?†EgGe ( ten ) The set spread contracting due to the presence of Ge is assumed to hold a additive dependance on Ge concentration I”Eg, Ge = 700x ( meV ) and a?†EgDOP ( x ) is

Bandgap Narrowing due to Heavy doping consequence )

Mobility can be expressed as

In this survey we have taken CONMOB, FLDMOB Internet Explorer Concentration Dependent and field dependent mobility

VSATN and BETAN are invariables.

Sing all the footings ( Internet Explorer. Mobility, Effective doping, Electron Saturation Velocity ) in base theodolite clip depends on Bandgap Narrowing which has high consequence on high Ge concentration. Further if we consider the Bandgap narrowing there is drastic alteration in Base theodolite Time which accordingly consequence Cutoff frequence.

where RE is the emitter opposition and RC is the aggregator opposition. The value of this charging clip depends greatly on the parasitic emitter opposition RE and aggregator opposition RC.

## Result-

Percentage decrease in Peak Cutoff frequence can be analysed by the undermentioned graph.

Comparing this figure to Fig.1 ( Gain Difference Curve ) we came to reason that there is trade off between current addition and cutoff frequence ( unity addition bandwidth ) . But there is crisp decrease in extremum cutoff frequence curve ( Fig.5 ) at high Ge fraction. Besides in Fig.-1 addition Difference mistake lessening 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 addition but pay off of cutoff frequence occurs. So for high velocity application ( LAN and Mobile Communication ) , where Current Addition is non the major concerned, low doping must be used. At this fraction Band spread narrowing has lower consequence on Cutoff frequence and current addition besides. Sing our analysis Boltzman Statistics ( which we by and large use ) , does non give an accurate appraisal of Device parametric quantities. Mistake caused by this estimate is really high at low fraction of Ge. So Femi Dirac statistics must be considered for Gain and Cutoff frequence appraisal.