1

Chapter 4

Performance Analysis

4.1 Introduction

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.

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

2

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.

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.

4.2 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.

This causes the discrete impurity level to split and form an impurity band, as shown in figure 1. 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 It can be seen that the

overall effect of the high dopant concentration is to reduce the bandgap from Ego to Ege.

3

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.

Fig-4.1 Effect of High doping on energy bands.

For ease of modelling, these effects are rolled into a single parameter called the apparent bandgap

narrowing or the dopinginduced bandgap narrowing in the emitter ∆Ege, which is defined by

thefollowing equation:

������ � ���2 � ���2 ��� ���

Where ∆Ege = Ego − Ege, nie is the intrinsic carrier concentration in the emitter, and nio is the

intrinsic carrier concentration for lightly doped silicon. As the name implies, the apparent bandgap

narrowing is not the same as the bandgap narrowing obtained from optical measurements, because

it includes the effects of Fermi-Dirac statistics.

4

4.3 Performance Parameter

Summarized below are some the key performance parameters of the Heterojunction Bipolar

Transistor.

4.3.1 Current Gain

The current gain of the HBT for an abrupt emitter-base junction was given previously

by (2.14). It is clear from the above that for ∆Ev > kBT, the current gain is large and an exponential

function of ∆Ev. This factor is sufficiently large in many cases so that the base doping can be made

higher than the emitter doping while achieving a reasonable current gain. � � �������������� ��� �∆�� �

Analysis of Current Gain –

A simple way of modelling bandgap narrowing in the emitter is through an effective doping

concentration in the emitter ( )Edeff NN due to Bandgap Narrowing[9].

∆−==

kT

EN

n

nNN ge

deie

iodedeff exp

2

2

[2]

For heavily doped, n-type silicon, the model developed by del Alamo [10] gives a reasonably

accurate description of the apparent bandgap narrowing. In this model, the apparent bandgap

narrowing in the emitter geE∆ is described by the following empirical equation

( )

meVcmN

E dege 17

3

107ln7.18

×=∆

−

[3]

The effects of bandgap narrowing in the base can be simply modeled using an effective doping

concentration in the base ( )Baeff NN due to Bandgap Narrowing is given as[11]

kT

EN

n

nNN gb

abib

ioabaeff

∆−== exp

2

2

[4]

The structure under consideration is npn SiGe HBT. In a SiGe HBT the ratio of the base and

collector currents is given by [7]

5

∆=Tk

E

NWD

NWD

I

I

B

v

BBpE

EEnB

B

C exp [1]

where vE∆ is the valence band discontinuity at the emitter-base heterojunction. For Si-SiGe

heterojunctions, the consensus from the literature is that gv EE ∆=∆ so 0≈∆ cE [8]. 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.

So putting the value of Effective doping effective

( ){ }kTEE gegbBGN /exp0 ∆−∆= ββ [5]

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 xxGeSi −1 (b) due

to Ge. The Ge dependent energy bandgap of the SiGe is given by ( ) meVxGeEg 700=∆ where x

is the Ge content in mole fraction. The bandgap reduction due to Ge content has been incorporated

by using the above function,

( ) ( ){ }kTEx geBGN /700exp ∆−= ββ [6]

where, 0β is the gain calculated considering Boltzman Statistics. geE∆ is constant as Emitter

doping is constant.

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.

6

4.3.2 Cutoff Frequency

Usually the gain of the device is independent of the frequency at low frequencies. However, at

high frequencies the current gain decreases due to capacitances within the device and transit time

effects. The frequency at which the transistor incremental gain (βac) drops to unity is called the

cutoff frequency (fT ). For HBTs, the base transit time is significantly reduced by the reduction in

base width possible due to the heavy doping in the base, which reduces the overall τec of the HBT.

Also, the emitter delay time τe is reduced since Cje is proportional to N1/2 DE and the NDE is reduced.

4.3.3 Maximum Frequency of Oscillation In SiGe HBTs, the base resistance of a HBT is reduced due to the higher base doping and the fT is

improved due to the thinner base. Together, these improve the maximum frequency of oscillation

(fmax).

���� � � � 8"#$%&'

Analysis of Cutoff frequency

Cutoff frequency is given by,

fT = ()*+ where τ = τe+ τb+ τc .

τe 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.

τe = �, -. (Cje+Cjc)

where Cje and Cjc denote the junction capacitances for the base-emitter and base collector

junctions, respectively, I denotes the collector currentand n denotes the ideality factor of the

device.

τc ,The collector charging time is given by,

τc= (RE + RC )CjC

7

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.

τb 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, (Wb2)/2Dn where Wb is base width

and Dn is diffusivity. Dn is a linear function of mobility and mobility is given by,

nSiGe(x)= (1+3x) nSi(x)

Hence, while increasing the Germanium fraction mobility is increased, consequently base transit

time is decreased.

In this study we have taken CONMOB, FLDMOB ie Concentration Dependent and Field

Dependent mobility [14]

μ01E3 � μ04 5 ((67 89:1;3<=>?@ AB;?>@C

DB;?>@

(VSATN and BETAN are constants.)

So when we consider the bandgap Narrowing due to high doping effect high electric field can

cause decrement in mobility, hence base transit time is decreased after high Ge fraction( as Ge

fraction is responsible for band gap narrowing) by following analysis.

∆Eg – effective band gap reduction in the base due to the presence of Ge (∆EgGe) and due to

heavy doping effects (∆EgDOP)

∆Eg(x) = ∆EgGe(x) + ∆EgDOP(x)

∆EgGe(x) The band gap narrowing due to the presence of Ge is assumed to have a linear

dependence on Ge concentration ∆Eg,Ge = 700x(meV) and ∆EgDOP(x) given by equation 3.

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.

8

Conductance-

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 [13],

µph � 400 G 29� G4737�), thereby reducing the base resistance.

Fig.-4.3 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

9

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 (0.25) emitter resistance decreases rapidly as shown in graph.

Fig.4 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.

10

Capacitive Effect-

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.

11

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.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

12

E1�3 � 1.155 N 0.43� G 0.206�) for �>0.25 Eg1�3 � 2.010 N 1.47� for 0.85< � P0.25

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 ∆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 ∆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).

4.4 Conclusion

It is to conclude that there is trade off between current gain and cutoff frequency (unity gain

bandwidth). So for high speed application (LAN and Mobile Communication), where Current

Gain is not the major concerned, low Germanium fraction (<0.25) in SiGe Base region must be

used. At this fraction Band gap narrowing has lower effect on Cutoff frequency and current gain

also. Beyond this fraction, Junction Capacitive effects significantly come into existence which

causes a drastically decrement in Cutoff frequency. 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. Thus we conclude that

compensating the bandgap narrowing effect due to high doping can be compensated by adequate

Germanium fractionin SiGe base region. So Femi Dirac statistics must be considered for Gain

and Cutoff frequency estimation.

13

Chapter 5

Simulation Software and Technique

5.1 Introduction

Computer Aided Design(CAD) are software packages that are powerful tools to economically

simulate semiconductor devices and obtain information which cannot be experimentally

observed. This information can help to understand the device physics and characteristics and the

factors limiting device performance. In addition, CAD simulation also provides a means to

predict device behavior that can reduce device prototyping and development costs.

Semiconductor devices can be modeled in two ways; one is to determine the terminal electrical

properties of a device based on a fit to empirical results and the other is to study the carrier

transport processes taking place within the device. The method used for this work is a physical

model, which can be used to predict both terminal characteristics and examine the transport

phenomena within the device. Physical device models are based on a description of carrier

transport physics, so that they calculate electron and hole concentrations within the device, their

spatial variation, and their response to material properties and biasing. They can be used to

characterize the DC and AC operation of the device and to provide detailed insight into the

physical aspects of device operation, provided they include all the important device physics. An

14

important advantage of this type of model is that it can not only be used to predict the terminal

current-voltage characteristics of the devices for comparison with device measurements, but also

to investigate the origins of the observed behavior.

5.2 SILVACO-TCAD Tool

Silvaco International produced the device modeling and simulation software utilized in this

work. Silvaco’s ATLASTM is a versatile and modular program designed for one, two, and three-

dimensional device simulation. BLAZETM and GIGATM, ATLASTM sub-modules (Figure 8),

perform specialized functions required for advanced materials, heterojunctions, and temperature-

dependent conditions. To control, modify, and display the modeling and simulation, the Virtual

Wafer Fabrication (VFW) Interactive Tools, namely DECKBUILDTM and TONYPLOTTM, were

utilized (Figure 4.1).

Unlike some other modeling software, Silvaco uses physics-based simulation rather than

empirical modeling. In truth, empirical modeling produces reliable formulas that will match

existing data but physics-based simulation predicts device performance based upon physical

structure and bias conditions. To perform the modeling, the Silvaco software graphically

represents a device on a two-dimensional grid with designated electronic meshing parameters. At

every mesh intersection, the program simulates carrier transport by means of differential

equations derived from Maxwell’s laws. To achieve accuracy, the program incorporates the

appropriate physics via numerical procedures.

To accurately model the III-V semiconductors, ATLAS must employ the BLAZE

program extension to modify calculations that involve energy bands at heterojunctions. The

heterojunctions require changes in calculating current densities, thermionic emissions, velocity

saturation, and recombination-generation.

15

ATLAS attempts to find solutions to carrier parameters such as current through

electrodes, carrier concentrations, and electric fields throughout the device. ATLAS sets up the

equations with an initial guess for parameter values then iterates through parameters to resolve

discrepancies. ATLAS will alternatively use a decoupled (Gummel) approach or a coupled

(Newton) approach to achieve an acceptable correspondence of values. When convergence on

acceptable values does not occur, the program automatically reduces the iteration step size.

ATLAS generates the initial guess for parameter values by solving a zero-bias condition based

on doping profiles in the device.

5.2.1 ATLAS/BLAZE

The software employed in this study is a commercial simulator called ATLAS produced by

Silvaco and configured to run on LINUX Operating System. This software package is a

comprehensive and flexible tool for simulating the electronic performance of arbitrary two

dimensional device structures, which can be composed of semiconductors, insulators, conductors

and terminals. The simulation can be conducted to get not only DC and small signal AC analysis

of the device terminal behavior, but also analysis of internal device parameter distributions, such

as the conduction and valence band energies, electrostatic potential, electric fields, electron and

hole concentrations and other current components. It also includes the interface physics models,

which are provided for all topology combinations, such as heterojunctions, metal contacts, and

surfaces. These capabilities enable accurate simulation of parasitics, leakage currents, trapped

charges and contacts. ATLAS predicts the electrical characteristics of the device by solving the

key semiconductor equations including the Poisson’s equations, the current continuity equations

for electron and holes and the current density equations that describe the physics of the device

operation. The simulator consists of a device simulation sub framework and a modular set of

application oriented tools like BLAZE, which is a 2-D device simulator for III-V material

devices and devices with position dependent band structure (heterojunctions), as well as a

Graphic User Interface called TONYPLOT for graphic presentation.

16

5.2.2 ATHENA

It also has a two dimensional fabrication process simulator called ATHENA. The ATHENA Two-

Dimensional Process Simulation Framework is a comprehensive software tool for modeling semiconductor

fabrication processes. ATHENA provides facilities to perform efficient simulation analysis that substitutes for

costly real world experimentation. ATHENA combines high temperature process modeling such as impurity

diffusion and oxidation, topography simulation, and lithography simulation in a single, easy to use framework.

Simulation of a device for its characteristics involves the following methodology. The physical

structure of the device, along with material parameters in the different regions and the doping

profiles are initially specified by the user. A mesh is defined for the initial structure which

generates the nodes at which the device equations are solved. In order to avoid overloading the

computer, the mesh is defined very dense only near the junctions where quantities change

rapidly. Physical models are incorporated that specify the dependence of one parameter on the

other, such as carrier mobility on doping levels.

Appropriate physics-based models are then selected by the user including recombination models

(Auger, ConSRH), mobility models(Klaassen, Fldmob, Conmob), and bandgap narrowing

models (BGN). Then, the bias conditions are specified for performing the AC, DC and transient

simulation, if any. Appropriate numerical techniques for solving can also be specified.

17

Fig5.1 Flow Diagram of Simulation.

5.3 Models and Material

The default material parameters can be overridden using the MATERIAL, IMPACT, MODEL,

and MOBILITY statements.In this project following Models are used-

5.3.1 Parallel Electric Field-Dependent Mobility (FLDMOB)

As carriers are accelerated in an electric field, their velocity will begin to saturate at high enough

electric fields. This effect has to be accounted for by a reduction of the effective mobility since

18

the magnitude of the drift velocity is the product of the mobility and the electric field component

in the direction of the current flow. The model used in

ATLAS to represent this reduction is FLDMOB. The following Caughey and Thomas(Equation

1-2) expression is used to implement a field dependent mobility that provides a smooth

transition between low field and high field behaviour.

μQ1E3 � μQ4 5 ((67 89:1;3<=>?R AB;?>RC

DB;?>R [1]

μ01E3 � μ04 5 ((67 89:1;3<=>?@ AB;?>@C

DB;?>@ [2]

where E the parallel electric field and µno and µpo are the low field electron and hole mobilities,

respectively. The model parameters BETAN and BETAP are curvature parameters that are set

equal to unity for these simulations in the absence of adequate data for the Ge dependence in

SiGe alloys.

5.3.2 Concentration Dependent Mobility (CONMOB)

The model used here is based on empirical data for the doping dependent low-field mobilities of

electrons and holes in silicon at T= 300K. The mobility of holes and electrons is found to

decrease with increase in doping concentration (Equation 3) due to ionized impurity scattering.

The dependence of the hole mobility on doping can be given by the following equation-

μ�1S3 � μTUQ G V:( 6 � RRWXY�Z [3]

where µmin is the carrier mobility at its minimum value and N is the doping level.

19

5.3.3 Bandgap Narrowing (BGN)

In the presence of heavy doping, i.e. greater than 1018, the pn product in silicon becomes doping

dependent. As the doping increases, a decrease in ∆Eg occurs, where the conduction band is

lowered by approximately the same amount as the valence band is raised. These bandgap

narrowing effects are enabled by specifying the BGN parameter of the MODELS statement. The

Ge dependence of the band gap narrowing has been analysed in this project.

5.3.4 Fermi-Dirac Statistics (FERMI)

Electrons in thermal equilibrium at temperature TL within a semiconductor lattice obey Fermi-

Dirac statistics. That is, f(E) is the probability that an available electron state with energy E is

occupied by an electron. The expression for this relation is given by

f1E3 � ((6\]0�;^;_`?a � [4]

where EF is a spatially independent reference energy known as the Fermi level and k is

the Boltzmann’s constant. By utilizing the Fermi-Dirac statistics instead of Boltzmann

statistics, more accurate simulation results are obtained.

In the limit that - EF >> kTL Equation 3-25 can be approximated as

�13 � exp 7e_f egha A [5]

The use of Boltzmann statistics instead of Fermi-Dirac statistics makes subsequent calculations

much simpler. The use of Boltzmann statistics is normally justified in semiconductor device

theory, but Fermi-Dirac statistics are necessary to account for certain properties of very highly

doped (degenerate) materials.

20

The remainder of this section outlines derivations and results for the simpler case of Boltzmann

statistics which are the default in ATLAS. Users can specify that ATLAS is to use Fermi-Dirac

statistics by specifying the parameter FERMIDIRAC on the MODEL statement.

5.3 Simulation Technique

The simulation method described in this chapter is the basis employed in a commercial device

simulator, ATLAS, by Silvaco International [3]. It is a fully numerical model, which is based on

using partial differential equations that describe the important device physics and include the

different regions of a device in one unified manner. The semiconductor devices are governed by

the basic semiconductor equations, such as Poisson’s equation and the carrier continuity

equations, and are described as follows.

5.3.1 Poisson’s Equation i1j ik3 � Nl

where φ is the electrostatic potential, + is the local permitivity, and ρ is the local charge density,

which can be a function of position. The reference potential can be defined in various ways for

ATLAS. For the simulator used in this work, φ is always the intrinsic

fermi level.

5.3.2 Carrier Continuity Equation

The carrier continuity equations for electrons and holes are

m�mn � (- i o� G p� N &�

m�mn � (- i o� G p� N &�

21

where n and p are the electron and hole concentrations, Jn and Jp are the electron and hole current

densities, Gn(Rn) and Gp(Rp) are the generation (recombination) rates for the electrons and holes,

and q is the magnitude of the charge on the electron. All of these (except q) are, in general, a

function of position. The electron and hole current densities (Jn and Jp) are calculated based on

the drift-diffusion model and are described as follows

Jn � qnμnE G qDn∇n Jp � qnμpE G qDp∇p

where µn and µp are the electron and hole mobilities, Dn and Dp are the electron and hole diffusion

constants, Eis the local electric field, and ∇ is the three dimensional spatial gradient

The simulation principle incorporated in the ATLAS model is to numerically solve the above

Poisson’s equation and the carrier continuity equation self-consistently to get electron and hole

concentrations and electrostatic potential (ψ) at all points within the device structure. In general,

the doping level as well as the material parameters in the above equation vary with location in

the device structure. The solution is obtained by solving the above described differential

equations subject to boundary conditions imposed by the device’s contacts and biasing.

Other models incorporated in the simulation include ones for the carrier generationrecombination

mechanism, mobility models that depend on the doping level, band gap narrowing models, and

density of states for different materials. Also included are velocity saturation effects at high

electric fields.

22

Chapter 6

Conclusion

23

Appendices (A)

SILVACO Source Code

go athena

# SiGe HBT simulation

# Establish initial grid and substrate material

line x location=0.0 spacing=0.08

line x location=0.5 spacing=0.05

line x location=0.7 spacing=0.05

line x location=1.2 spacing=0.08

line x location=2.2 spacing=0.18

#

line y location=0.0 spacing=0.01

line y location=0.1 spacing=0.02

line y location=0.5 spacing=0.05

line y location=0.8 spacing=0.15

#

init silicon c.phos=2e16

structure outf=tmpr1.str

init inf=tmpr1.str flip.y

implant phos energy=60 dose=3e15

diffuse time=5 temp=1000

struct outf=temp

init inf=temp flip.y

24

# Deposit Silicon germanium with composition fraction 0.2 for base

deposit sige thick=.1 divis=12 ydy=0.05 dy=0.02 c.frac=0.2 c.boron=1e15

implant boron energy=10 dose=1.0e13

# Deposit silicon for the emitter #implanatation

deposit silicon thick=0.2 divis=10 ydy=0.08 dy=0.04 c.phos=1.e15

implant boron energy=12 dose=3e14

diffuse time=0.5 temp=920

# Mask and implant the emitter

deposit photo thick=.5 divis=5

etch photo left p1.x=0.5

#implant phos energy=40 dose=6e15

implant phos energy=38 dose=6e15

diffuse time=5 temp=920

strip

# Deposit and pattern the contact metal

deposit aluminum thick=0.05 div=1

etch aluminum start x=0.5 y=10.

etch cont x=0.5 y=-10.

etch cont x=1.2 y=-10.

etch done x=1.2 y=10.

# Define the electrodes

electrode name=emitter x=0.0

electrode name=base x=2.0

electrode name=collector backside

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# Define impurity characteristics in each material

impurity i.boron acceptor sige

impurity i.phos donor sige

structure outfile=hbtay03_0.str

go atlas

# Material parameter and model specification

material material=Si taun0=1e-7 taup0=1e-7

material material=SiGe taun0=1.e-8 taup0=1.e-8

model bgn consrh auger fldmob conmob fermi print temp 300

output con.band val.band

solve init

#

# Calculate Gummel plot and AC parameters versus Vbe (Vce) at 1 MHz

solve prev

# 1 - emitter 2 - base 3 - collector

log outf=hbtay03_1.log

solve v2=0.01 v3=0.01 ac freq=10 direct

solve v2=0.025 v3=0.025 vstep=0.025 electr=23 nstep=2 ac freq=1e6 direct

solve v2=0.1 v3=0.1 vstep=0.1 electr=23 nstep=5 ac freq=1e6 direct

solve v2=0.65 v3=0.65 vstep=0.05 electr=23 nstep=6 ac freq=1e6 direct

save outf=hbtay03_2.str

solve v2=0.975 v3=0.975 vstep=0.025 electr=23 nstep=3 ac freq=1e6

#

# Frequency domain AC analysis up to 100 GHz

#

log outf=hbtay03_3.log s.param gains inport=base outport=collector width=50

load inf=hbtay03_2.str master.in

solve previous ac freq=1 direct

solve ac freq=10 fstep=10 mult.f nfstep=8 direct

solve ac freq=2e9 direct

solve ac freq=5e9 direct

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solve ac freq=1e10 direct

solve ac freq=2e10 fstep=2e10 nfstep=5 direct

# Extraction of parameters

extract init inf="hbtay03_1.log"

# Maximum cutoff frequency

extract name="Ft_max" max(g."collector""base"/(6.28*c."base""base"))

#

# Base bias at maximum cutoff frequency

Extract name="Vbe@Ft_max" x.val from curve (v."base", g."collector""base"/(6.28*c."base""base" )) where y.val=$"Ft_max"

# Input (base) capacitance at maximum cutoff frequency

extract name="Cbb@Ft_max)" y.val from curve (v."base", abs(c."base""base" )) where x.val=$"Vbe@Ft_max"

#

# Transconductance at maximum cutoff frequency

extract name="Gm@Ft_max)" y.val from curve (v."base", abs(g."collector""base" )) where x.val=$"Vbe@Ft_max"

# Gummel plot

tonyplot hbtay03_1.log -set hbtay03_1_log.set

# AC current gain versus frequency

tonyplot hbtay03_3.log -set hbtay03_4_log.set

# S12 & S21 polar coordinates

tonyplot hbtay03_3.log -set hbteay03_4_s12.set

#

# S11& S22 Smith chart

tonyplot hbtay03_3.log -set hbtay03_4_s11.set

#

quit

Appendices (B)

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References

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[19] P. Ashburn, C. Bull, K.H. Nicholas and G.R. Booker, ‘Effects of dislocations in silicon transistors with implanted bases’, Solid State Electronics, 20, 731 (1977).

[20] J. del Alamo, S. Swirhun and R.M. Swanson, ‘Simultaneous measurement of hole lifetime, hole mobility, and bandgap narrowing in heavily doped n-type silicon’, IEDM Technical Digest, 290 (1985).

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[23] Silvaco International Atlas Manual, version 1998.

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