14

In this era of the communication revolution, progress of humankind is faster than ever. The whole credit goes to the invention, modification and application of various semiconductor devices that are being used in various ways. Starting from the bipolar device, today's semiconductor market consists of unipolar devices like MOSFET (metal-oxide-semiconductor field-effect transistor) or HEMT (High electron mobility transistor). The basic bipolar device had experienced vast improvements and HBT (heterojunction bipolar device) was formed.

These heterostucture devices are typically consisting of SiGe or III-V materials. The improvement in SiGe HBT leads to a new technology in 1992 called BiCMOS (bipolar complementary metal oxide semiconductor) [1] which quickly adjusted in various system applications. The SiGe ICs are representing a growth rate of 42% per year [2], a remarkable figure by any economic standard. The SiGe technology is used vastly in cellular application but optical networking, storage drives and radar system implementation are also in cards.

a) SiGe HBT used in high frequency operation

To achieve higher current gain and cutoff frequency, the basic structure of bipolar transistor has to be altered. The improvement, following the band gap engineering, consists of a combination of Si (wide band gap emitter) and Ge (narrow band gap base) to form the SiGe HBT. The first results of SiGe HBT were presented by IBM in the IEDM conference in 1987 [3]. Since then, the continuous modification to obtain high operating frequency has become an important research field and binds thousands of researchers of that field. In IEDM 2010, IHP, one of the DOTFIVE [4] members, presented maximum oscillation frequency of 500GHz, which is till date, considered the highest for SiGe devices [5]. In  PhD work

As the operating frequency approaches near the cut off frequency of transistor, the quasi static approach does not provide the accurate phase shift of dynamic parameters. The effect is called Non Quasi Static (NQS) effect. Also the compact bipolar transistor models become often inaccurate at higher frequency. The main aim of this work is to investigate the NQS effect from the physical point of view and also investigate the compact modeling approach of this effect. The whole work is divided into five different parts.

1. First of all, literature concerning the NQS effect is studied. Starting from the basic transistor equations dynamic parameters are calculated and compared with the previously implemented models.

2. Next, a popular physics based compact model (HICUM) [9] is analyzed. The circuit implementation of NQS phenomena gives a clear understanding of compact modeling approach.

3. Measurement results are modeled at high frequency using HICUM model and the NQS circuit implementation is verified. Also, a scalable approach is investigated. The necessity of an improved model is shown and a new model is verified with these same measurements.

4. Physical device simulation (TCAD) is performed in a modern high speed device up to very high frequency (500GHz) to deeply investigate the NQS behavior. Then simulated results are then modeled using HICUM.

5.

A new concept device with a SiGe spike inside emitter is fabricated and simulation has been performed in that device to show NQS behavior. After, the simulation results are modeled using HICUM model.

Chapter 1

[NQS THEORY]

Introduction

Steady state transistor operations were always analyzed using basic drift diffusion equations which explained junction transistor behavior fairly well. As time progressed more complicated effects were observed and introduction of new approach became prominent. The problem arises while investigating the dynamic behavior of transistors like the AC or fast switching operation. For a transient operation certain time delay was observed for both switch on and switch off cases [1]. As an example, Figure 1.1 highlights the variation of electron concentration profile inside the quasi neutral base region in transient operation Scholars were engaged to introduce new concept that would help modeling transistor at high frequency. At that time Winkel applied a time delay inside the current gain and recalculated conductance parameters [2]. This was the starting point of including time delays inside steady state calculations which afterwards we call NQS calculation. At the beginning of 90's many researchers calculated NQS related parameters following different approaches. The famous charge partitioning concept was introduced [3] [4] and reported in agreement with simulations. Afterwards Rinaldi [6] introduced a general closed form solution for all injection level and his work allowed achieving all the previously described models from a closed form solution with the help of minor approximations. In this part first we will consider the charge control approach and then recalculate NQS related parameters using basic device equations.

Classical charge control model and extension for high frequency application

To examine a junction transistor in more details we need to calculate the terminal currents at three nodes (Emitter, Base, and Collector) of a transistor. Terminal currents obviously depend on the excess electron concentration at the narrow p type base region. The calculation is very similar to that used to calculate the narrow base diode currents [7] The charge control approach is useful for analyzing the transistor terminal currents, particularly in ac applications. Following this approach we can separate arbitrary excess electron distribution in the base into the normal and inverted distribution ( Here EN I , CN I represent the normal and EI I , CI I represent the inverted components of currents at emitter and collector terminals respectively. The currents for the normal and inverted modes in terms of storage charges can be constructed and reads as follows,

E EN EI I I I   1.1 C CN CI I I I  
These equations (1.1) are the basic form of the Ebers-Moll equations [8]. Expressing these currents as a function of storage charge, we have,

Δn E Δn C 0 W B Δn E 0 W B I EN I CN Δn C 0 W B I EI I CI (a) (b) (c) , b b b EN CN tN n tN Q Q Q I I       1.2 , ( )
I I I EI CI tR tR n Q Q Q I I        
Where b Q and I Q are the storage charges in the normal and inverted modes, With the help of equations (1.1) and including the effect of time dependence of storage charge we can rewrite the equations (1.2) as,

(

)

b I E b tN n tR dQ Q IQ dt        1.3 11 ( ) b I C I tN tR n Q dQ I Q dt        b b I I B nn Q dQ Q dQ I dt dt     
These equations can be expressed in the general form,

i i B i Q dQ I dt     1.4 i i C oi Q dQ I dt     ( , , ) i i BE CB C Q f V V I  Here i Q represents the total charge at a region i.
For a low level injection and reversely biased base-collector junction, we can neglect the time dependent contribution of

c i and equations set (1.4) becomes b b B b Q dQ I dt   1.5 b C ob Q I   0 (exp 1) BE b T V QQ V   b  , ob
 are two constants having the dimension of time.

Changing the time domain into the frequency domain we get two sets of ac parameters. In the common-emitter configuration the current gain becomes,

0 / 1 1 / b ob b j j             1.6
Where

1 b     and 0 / b ob    
Transconductance and input admittance are also calculated as,

c fe m T I y g V  1.7 0 1 ( ) fe ie m T y y g j       

With transition frequency 1

T ob

  

These small signal expressions (Equations 1.6) display a number of well known properties correctly. The behavior of ()

 is constant at low frequencies with a cutoff at   and falling at the rate of 20dB/dec. But the main drawback is that excess phase in  is not accounted in this calculation. At that very point a correction term inside the charge control approach was introduced by Winkel [2]. A close similarity between the differential equations that govern the current flow in transistor and the equations that describe the flow of electric signal in transmission line was observed. When the carrier lifetime and electric field in the transistor are constant, the analogy will be a uniform transmission line. We can approximate the network as a summation of two networks, one with minimum phase type and another that only changes the phase of the signal. In the time domain a suitable approximation of the second network would be a constant delay.

Replacing (1.5) we will get a new set of equations

( ) ( ) () b b b b Q t dQ t it dt    1.8 1 ( ) () b c ob Qt it     2 0 () ( ) (exp 1) be b T vt Q t Q v    
And the small signal parameters changed accordingly [2]

0 1 0 exp( ) 1 / T j j          1.9 1 2 exp{ ( )} fe m y g j      2 0 1 ( ) exp( ) ie m T y g j j      
Equations 1.9 represent the first attempt to include time varying phase factor inside calculations. These equations prove their significance in vast domain as one of the modern compact models HICUM [9] basically applies it to model NQS effect.

Calculation of current components in a drift transistor in small signal operation

In quasi-static approach, basic current components are calculated only using the time independent part of the drift diffusion equation [7]. For the small signal analysis where the time dependence of minority carrier becomes predominant, it is obvious to consider time variation of the excess carrier concentration.

In the following calculation an npn transistor is considered with exponential acceptor doping profile. Carrier mobility is assumed to be constant inside the base for all injection level and an average value of electron diffusion coefficient has been applied for the calculation i.e. heavy doping effect has not been considered. The classical drift diffusion equation in the quasi neutral base region for the electron and hole charge reads [10] ( , ) ( , ) ( , ) Here, all the symbols represent their usual meaning. We consider negligible recombination inside the base region and omitting recombination term from the basic continuity equation, we get, ( , ) ( , ) 1

N j x t n x t t q x     1.11
Differentiating (1.10) with respect to the space variable x and putting in (1.11) gives a second order partial differential equation in ( , )

n x t . 2 2 ( , ) ( , ) ( , ) 0 nB T nB n x t n x t n x t V x x t             1.12
Next, assuming a small signal perturbation is introduced over the DC bias condition; all the electrical parameters can be constructed as a summation of a DC and a small signal AC component. Where,

B T W A V   1 x B x u W  2 4 1 2 fl j A     2 B nB T W V    1.16
This equation (1.15) is admittedly rather complicated and the electron distribution described is not immediately apparent. The factor () nx is the sinusoidal varying reference and the expression describes the phase and the amplitude at any point with respect to the base width.

Equation (1.15) is a general solution for the excess carrier concentration and it is always modified depending upon the injection condition. Low and high injection are treated separately and presented below.

 Low Injection (LI) level

For LI we approximate (N<<N A , P  N A ) and the built in electric field can be calculated by setting ( , ) 0 P j x t  . (Equation 1.10)

1 T A AB dN V N dx W     1.17
Here we have assumed a drift transistor with an exponential base doping, 0 exp( )

AA B x NN W   1.18
Comparing equation (1.16) with (1.17) we find A=η for low injection level.

It is customary to express Ω (equation 1.16) in terms of the base transit time. For a drift diffusion transistor, base transit time can be calculated easily

2 1 2 B B nB T W V    considering diffusion transistor [7] 2 2 1 exp( ) () B B nB T W V        
for drift transistor in low level of injection [11] 2 1 . 4

B B nB T W V   
for transistor in high level of injection [11] 1. [START_REF] Hamel | An accurate charge control approach for modeling excess phase shift in the base region of bipolar transistors,‖ Electron Devices[END_REF] Taking equation (1.19) into account fl  can be expressed as a function of

B  , 4 1 2 1 exp( ) B fl j        1.20
Next figure shows the magnitude of normalized carrier concentration variation over the base width. To obtain these plots, some device parameter values are assumed such as 20 The next step is to obtain the small signal current components which can be obtained from the current density, 00 0

n e E nB xx x T n I i qA D n Vx            1.21 BB B n c E nB x W x W xW T n I i qA D n Vx            
Using equation (1.15) a set of small signal terminal currents can be achieved for low injection level. For a transistor operating in common base configuration, the current gain ( ) is defined as the ratio of the collector current to the emitter current. Small signal current gain can be easily formulated using equation 1.22 Again coming back to the current calculation, we observe, at the limit of 0 The ratio of frequency dependent and independent collector currents is

  , equation (1.22) changes to, /2 0 0 ( / 2) .sinh / 2 c nB E B e i n qD A W       1.24 0 0 coth 1 22 nB E e B n qD A i W         
  _ sinh () /2 0 sinh / 2 fl fl c c norm c i i i         1.25

Equation (

1.25) is also noted in [12] The next figure shows the variation of normalized small signal current over frequency for different drift factors. The current component is the same as in [6]. Here we have assumed that no recombination is present inside the base region, i.e. 0 C dI dx  inside the base. As the base width changes with the external bias variation terminal currents depend only on the external bias points.

DC excess carrier concentration is achieved simply by solving second order differential equation in () Nx and is given by,

  0 ( ) 1 1 x u N N x e e         1.27
Where static excess carrier concentration at the edge of base emitter junction is given by, Up to now, we have calculated the excess carrier concentration for Small signal as well as for DC bias variation. At the base emitter junction, the relation between small signal and DC carrier concentrations can be arrived following a few steps. Considering the law of injection at the edge of the base emitter junction, we can express carrier concentration on the emitter side as,

2 0 0 0 ( ) exp( ) BEtot i T V N n P n V  1.28 2 00

exp(

)

BE i T V N P n V 
From equations (1.28) assuming that T vV we get directly the relation between excess carrier concentration at emitter side for LI condition. This ratio of transfer currents can be readily compared with the normalized current formulation (equation

1.25).
The only difference is the factor

T v V
which represents the relation between the small signal and quasi static excess carrier concentration. In the limit of 0   small signal current is directly proportional to the static forward current and the proportionality constant depends upon the magnitude of small signal perturbation.

0 0 ( 1) c c nB E T B T N vv i I qD A V W e V        1.32
Terminal currents are calculated and compared in different condition like the DC and small signal application. It is noted that the magnitude of applied small signal voltage plays a significant role for small signal current. The ratio of small signal base charge and DC charge is calculated and the magnitude and phase variation is shown in Figure 1.7 The above analysis provides some basic small signal equations in LI domain. These equations will be used after in admittance parameters calculation.

 High Injection (HI) level

In the previous part, small signal analysis in the LI condition is formulated. Next, the same analysis will be carried out in High Injection (HI) condition. The Distributive effect of parasitic resistance and capacitance are neglected (i.e ac crowding effect) [13] while considering HI level. Though it is an important factor which contributes significantly to the total delay time, it is assumed to be -extrinsic‖ part of the device and hence it is overlooked inside this calculation. At HI level carrier mobility is reduced due to electron-hole scattering [7] which reduces diffusion coefficient. This effect is also neglected to provide a simple calculation.

In HI, We approximate (N P>>N A ) and the equation ( The doubling of the electron diffusion coefficient is known as Webster effect.

Following the same procedure as for the LI level, a second order differential equation for the excess carrier concentration is constructed. Common emitter current gain (  ) at HI is calculated using

1     
and plotted to compare, In the above analysis excess minority carrier concentration and terminal currents for small signal has been calculated considering both injection levels. This analysis provide frequency dependent current components which will be used in advanced calculations.

Y parameter calculation for Low injection and High injection level

The Ac characteristics of a transistor can be described in terms of two voltages and two currents. We can express input and output currents as a function of input and output voltages and define this proportionality constant as admittance (y).

In the condition of dynamic short of the collector, the y parameters are defined as [10] 11 These equations of small signal Y parameters are compared with [6] and found to be the same. The CB y parameters can be approximated in terms of polynomials of complex variable [6] [14] where different time constants are taken into account,

         
With suitable approximation to the general equations (1.46) all the previously described models in the literature can be obtained [6].

Analysis of different NQS models

Within the last decade, much interest had been devoted, to develop small signal AC and large signal transient models for bipolar transistors. The quasi static capacitance based models such as Gummel-Poon model [15] became more and more inadequate in high frequencies or under fast transient condition. A widely used method was the partitioned charge based (PCB) modeling concept which basically separated the quasi neutral base charge into emitter and collector terminals. The most popular PCB models are named after Klose and Wieder [16] and Hummel and Selvakumara [4]. Both these models deal with LI condition and calculate differently α11, the Charge Partitioning Factor (CPF). An analytical solution of LI minority carrier transport equations in the Laplace domain was given by Wu and Lindholm [17] which includes heavy doping effect and also extended upto arbitrary level of injection. The model of Seitchik [5] It is a bit difficult task to compare all the previously established models because all the models use different time constants to establish admittance equation [START_REF] Rinaldi | Reply to comments on "Modeling of small-signal minority-carrier transport in bipolar devices at arbitrary injection levels[END_REF].

In this section different models will be analyzed and compared with the general equation that has been established previously (Equation 1.46).

 Quasi Static (QS) model:

It is the simplest model following charge control concept and does not include the base delay time inside calculation. The basic Y parameter calculation is very simple (equations 1.6). It can be reproduced from the general equation (1.46) [6] while considering only the first order term in the numerator and neglecting all the frequency dependent terms in the denominator. The characteristic time can be identified as the differential forward base transit time τ f .

 Charge partitioning model:

The essence of charge partitioning concept relied on the fact that for a narrow diode operation only the two third of the stored charge was provided when the diode turned on applying a step voltage [10]. This idea was extended to the transistor and the ratio of the stored charge flowing to the emitter terminal to that flowing to the collector terminal was calculated to be 2:1 [1].

In the classical charge control approach the total stored charge at the quasi neutral base region was obtained while integrating the excess carrier concentration over the whole base width (W B ). If however the same integration is down in two steps i.e. first integrating up to an arbitrary distance x and then up to the base width, the reclaimable charge shows a pure partition over transistor terminals. Diffusion capacitances are then calculated considering different partitioning functions [3] [16] [4] Taking only the first order term from the general expression (equation 1.46), one obtain [6]

1 11 1 1 11 1 1 (1 ( )) (1 ) 1 b N f f N D f f D i y g g i g i i                1.49 21 1 21 1 1 (1 ) (1 ) 1 b f f D f f D y g g i g i i             
The quantities 11

 and 21

 are defined as the charge partitioning factors as, In this model the basic drift diffusion equations were solved applying Laplace transform method considering LI level and heavy doping effects. The model can be applied inside the quasi neutral base as well as in the emitter part of the transistor [17] Taking Laplace transform to the both sides, basic drift diffusion equations can be written,

11 { ( , )} { ( , )} () n n d s L n x t L j x t x q dx      1.50 { ( , )} { ( , )} (ln ) { ( , )} n n A dd L j x t qD L n x t N L n x t dx dx

   

A set of iterative analytical equations can be constructed using the basic equations.

1 01 1 0 1 0 { ' ( , )} 1 { '( , )} ( ) 1 ( ) ( ) x n n L j x t L n x t n x dx q D x n x      1.51 2 1 1 1 1 0 { ' ( , )} { ' (0, )} ( , ). ( ). { '( , )} x n n n L j x t L j t q x s D x L n x t dx   
The analytical series solution coming from equations (1.51) is

0 2 1 2 11 1 { '( , )} ( ) 1 { ' (0, )}. ( , ). ( ,
)
n i i ii L n x t n x L j t A x s B x s q           1.52
Changing the integration limit to W B instead of x and truncating the matrix elements, y parameters were constructed which can be achieved from the general equation (1.46) considering the first order terms only.

1 11 1 1 1 b N f D i yg i      1.53 21 1 1 1 b f D yg i    Seitchik model:
Seitchik model is basically developed from the Thomas Moll model. This model [5] includes three parameters to describe the ac response of the quasi neutral base region. This small signal model can be achieved when including the second order term in the expression of y parameter

(equation 1.45) [6], 11 2 1 f e f i yg i     1.54 1 21 2 exp( ) 1 e f i yg i     
Where, the exponential term accounts for the excess phase shift. Comparing the Taylor series expansion of 21 e y and keeping in mind the charge partitioning model, one gets [6] 1 2

1 21 Df         1.55
This relation is also noted by Hamel [START_REF] Hamel | An accurate charge control approach for modeling excess phase shift in the base region of bipolar transistors,‖ Electron Devices[END_REF] who approximated,

1 2 1 /2 D     1.56
From this analysis it is clear that the time constants in the Seitchik model is not only function of the first order term, but the second order term also.

Small signal gain is modelled by,

1 0 0 exp( ) 1 f i i        1.57  Winkel model:
This model is analysed in the previous section in details. Equation 1.9 presents the set of necessary small signal parameters. To compare it with the general formulation (equation

e f f y g i i     1.58 21 1 2 exp( ( )) e f y g i      1 0 0 exp( ) 1 f i i       
Comparing Winkel's model with Seitchik model, it is obvious that both the model uses similar kind of formulation for the current gain and admittance parameters. The accuracy, Phase of Y 21 is almost the same for these two models. It also established that Seitchik and

Winkel's model are the best available formulation for high frequency operation. In practice, there is a possibility of covering NQS behaviour due to the parasitic resistances and capacitances in modern devices. However, for a practical application, simpler models like Wu-Lindholm and Hummel also provide considerable accuracy in NQS modelling.

Transit time calculation for small signal analysis

When we consider transistors at high frequency operation, there are some parameters that effect high speed transistor operations. Some of these frequency limitations are junction capacitances, the charging time associated with the capacitance and the transit time of carrier.

The total transit time can be written as a summation of transit time coming from the different parts of the structure [7] Phase(Y 21

)(deg)

Frequency(Hz) f E BED B BCD C            1.59
Total transit time is mainly due to the base transit time and it is well known for a drift diffusion transistor (equations 1.19).

In the following section we will try to develop analytic expression for BCD  (BC junction transit time) and E  (emitter transit time) for the small signal application.

Base collector junction transit time

To achieve high speed performance it is essential to downscale vertical dimensions to reduce the overall delay [START_REF] Van Der Ziel1 | High-frequency response of microwave transistors[END_REF]. As in the modern HBT transistor deals with graded base profile, the built in electric field reduces the base transit time significantly. As a consequence of this process, the signal delay in the BC junction has become an important factor for the total delay and hence included inside the model [START_REF] Rinaldi | Modelling of collector signal delay effects in bipolar transistors[END_REF] Base collector junction is considered as high field region where carrier travels at a constant velocity which is the saturation velocity inside this region. We assume that the excess minority carrier concentration at the base collector junction boundary is zero and no recombination takes place inside the junction. For a junction of thickness bc x and carrier saturation velocity sat v , the displacement current inside this junction is assumed to be [22]

sat cc bc qv q i x   1.60
Assuming that the transfer current encounters an exponential delay when travelling base collector junction, we calculate the current by multiplying the transfer current ( () c ix) over the current function ( cc i ) and integrating it over time. 0 ( ) ( ). .

jt cc cc i i x i x e dt q       1.61
Integration gives the result, sin( ) ( ) ( ). ( ). .

Emitter transit time

In modern technology, heavily doped polysilicon is used inside the emitter [13]. This predominantly increases the emitter transit time ( E  ) and sometimes comparable to the base transit time ( B  ) [24]. As a consequence partitioned charge based models predict wrong results when applied to the emitter region as all the models assume negligible recombination inside the calculation. Our approach is to investigate E  for small signal operation considering a simple structure and applying almost the same procedure that has been presented while calculating the base dynamics for a drift transistor.

Small signal analysis has been carried out inside the quasi neutral emitter region considering constant doping profile at the emitter. Next Rewriting drift-diffusion equations for a n type emitter region and considering recombination inside the continuity equation, we obtain,

P pE pE p j q n qD x     1.63 N nE nE n j q p qD x     1 P p j pp q x t       
Where p  stands for the hole recombination lifetime insider the emitter. Where, the terms are presented as, 1 2

(1 )

EE p pp WW j LL      1.68 1 x E x u W  2 0 exp( ) i BE D T T n V v p N V V  After differentiating ()
px, excess hole current density (equation 1.67) at both ends of the quasi neutral emitter becomes, 0 0 coth( )

E p p p x p W j qD p L L   1.69 0 csc ( ) E E p p p xW p W j qD p L h L  
As the hole diffusion length (L p ) inside the emitter is much greater than the emitter length (W E ) we can approximate

1 1 2 2 2 2 ' E E E E E p p p p p W W j W W j W L L D D L                            The quantity ' E p W L
can also be represented as

1 1 2 2 ' 0 E E pp W jj W LD          
The total small signal charge is obtained by integrating equation (1.67) which is given by, 0 tanh( ) 2

E ac E p p W Q qA p L L  1.70
Emitter transit time for small signal operation is obtained dividing diffusion capacitance (C D ) by the transconductance (g m ).

Base emitter diffusion capacitance for the small signal operation is defined as,

2 ' tanh( ) exp( ) 2 ac E i E BE Dp D T p T dQ qA n WV CL dv N V L V  1.71
Transconductance is also calculated from the current density, coth( ) exp( )

Ep En E i p xx E BE m D T p p T Aj Aj qA n D WV g v v N V L L V       1.72
And the emitter transit time for the small signal operation is given by,

2 tanh( ) tanh( ) 2 p D E E e m p p p L C W W g D L L    1.73
Similarly the calculated transit time due to the DC bias variation reads, In this section, an attempt has been made to investigate the frequency dependence of emitter transit time when the system is exited with a small signal perturbation. Although, the calculation is fairly simple, the small signal emitter transit time (equation 1.73) shows frequency dependence. A more detailed investigation in this subject is necessary to conclude the effect of emitter transit time in the total time delay.

Conclusion

In this chapter, the physical approach towards the high frequency dynamic parameter calculation is presented. Both the charge control concept and small signal calculations are examined in this part. Dynamic parameters (transconductance and input admittance) are calculated from the small signal analysis and compared with the popular models present in the literature.

For all the calculation of bias dependent time delay, it has been assumed that the delay time is proportional to the total transit time. The contribution in total transit time of a modern bipolar transistor comes from different transistor parts. In this chapter, base collector (BC) junction transit time and neutral emitter transit time are calculated for small signal operation and said to be an important part of the total time delay. Consequently, the results obtained from the BC junction delay will be used after to suggest a new phase shift network in the HICUM model.

Introduction

Communication and information system frequently undergoes rapid and sophisticated changes. Nowadays, Heterojunction Bipolar Transistor (HBT) and High Electron Mobility

Transistor (HEMT) serve mostly as the building blocks of the communication system [1]. The Maintaining the TCAD flow the initial step was to virtually fabricate the transistor structure by applying step by step process simulation. Generation and orientation of suitable mesh is a valuable point of consideration at that step. Once the process simulation is completed the device is simulated using the known physical models to calibrate the structure i.e. to obtain the similar result as from measurements. In this chapter we shall mainly focus on the various physical models necessary for device simulation [3].

Process simulation

The main goal of process simulation is to model a virtual device with geometry and properties identical to the real structure. Sentaurus TCAD process [4]

m          2.2
In device simulation, charge density is the summation of total ionized and mobile charge and total trap charge, ()

D A trap q p n N N       2.3
So the Poisson equation becomes,

()

D A trap m q p n N N          2.4
Including the total recombination, charge continuity equation reads, ( , ) .

N net n x t J q qR t      2.5 ( , ) . P net p x t J q qR t     
Where net R is the total electron hole recombination rate and N J , P J are electron and hole current density respectively.

 Drift diffusion model

The drift diffusion model can be derived from the Boltzmann transport equation [9] by applying the basic principle of irreversible thermodynamics [10]. This model uses Poisson and continuity equation in addition to the current density caused by the drift field of immobile ionized charge.

n N n n n J nqµ E nqµ      2.6 p P p p p J pqµ E pqµ     
Where n µ and p µ are electron and hole nobilities, and n  , p  are electron and hole quasi

Fermi potentials. Quasi Fermi potential helps to calculate electron and hole concentration inside an intrinsic or a compound semiconductor. Electron and hole concentration can be written as, [2] exp( )

n i T nn V    2.7
exp( )

p i T pn V   
Here, i n is the effective intrinsic density.  , the potential which is above the quasi Fermi level for n type and below the quasi Fermi level for the p type semiconductor. This potential is calculated from the reference potential which is set equal to Fermi potential for intrinsic semiconductor like Si [10] Combining equation 2.6 and 2.7 we arrive to the basic drift diffusion equations [2] ntot N nn

J q nE qD n     2.8 ptot P pp J q pE qD p    
Where ntot E the total electric field is represented by,  

  ln ntot Ti E V n      2.9   ln ptot Ti E V n      Equations 2.
  N n n n J nqµ P T      
1.5 ln td N n C n n n n n J qµ n E KT n Kn T nKT m f          2.11   1.5 ln td P p V p p p p n J qµ p E KT p Kp T pKT m f        
The first term takes into account the contribution due to the spatial variations of electrostatic potential, electron affinity, and the band gap [7]. The three remaining terms in take into account the contribution due to the gradient of concentration, the carrier temperature gradients, and the spatial variation of the effective masses n m and p m .

Band structure

Band structure plays the most important role in device simulation [10]. Realistic band structure is rather complex and hence are fully applied only inside the Monte Carlo simulation. For the general use the effective mass for the electrons and holes and effective density of states in intrinsic and compound semiconductors are mainly analyzed.

 Intrinsic density

For an intrinsic semiconductor the intrinsic carrier density can be represented as a function of band gap energy [7] ( ) ( ) ( ) exp( ) 2 ,0 ,0 (0)

g i C V E n T N T N T KT   2.12
g g g E E E   2.15
The effective band gap is constructed by reducing the amount of energy reduced by band gap narrowing.

, ( ) ( ) ( )

g eff g gbn E T E T E T  2.16

 Bandgap narrowing

Band gap narrowing (BGN) is one of the most crucial heavy doping effects observed in modern semiconductors. Increased doping of the semiconductor affects the periodic property of the crystal lattice and hence band widening, band tailing and electron electron interaction occur [10]. Band gap narrowing has a great impact on transistor behavior as it changes the bandgap difference between the emitter and base. For a highly doped base region, BGN increases the emitter injection efficiency and reduces turn on voltage. In HBTs BGN effects the alignment at the base emitter heterointerface which influences the tunneling and thermionic emission current across the junction [10] [7] Sentaurus Device provides different bandgap models namely BennettWilson, delAlamo,OldSlotboom, and Slotboom (the same model with different default parameters),JainRoulston and TableBGN [7].

The model of Slotboom [11] is the most popular BGN model in Silicon [10]. This model reads, 2 ln ln 0.5

tot tot g ref ref ref NN EE NN                         2.17
The shift of the conduction band and valence band edges are calculated as [10]

C Cg g E EE E          2.18 V g C E E E     
The Slotboom model is based on experimental measurements of

 Effective carrier mass and Effective density of states

Considering parabolic conduction and valence band, density of states (DOS) can be calculated from the calculation of the effective mass [10]. The effective DOS masses scaled by electron mass represent relative carrier mass. For the direct band gap semiconductor  value gives the value of the relative mass. For indirect band gap semiconductor like Si, Ge, the transverse and the longitudinal mass from the X valley (L for Ge) are taken into account [10].

In Sentaurus device lattice temperature dependent electron mass is described by [7]  

1 2 2 3 3 6 n t l m m m m m  2.19
Where the temperature dependence for Si is described by [12] 0 (0) () () [START_REF] Van Der Ziel1 | High-frequency response of microwave transistors[END_REF] Parameters are defined inside the parameter file for the device. DOS for the conduction band follows, Similarly, the relative carrier mass for the holes are represented by [7]

g t g E mT a m E T  2.
           

2.22

The effective DOS for V N is represented by,

Carrier mobility

Electron and hole mobility are the basic carrier transport parameter that influence the electrical behavior. It is supposed to be constant for undoped materials and decay in compound semiconductor due to high electric filed or the interaction between lattice vibration, impurity or surface. Different mobility (bulk and surface) combine following the Mathiessen's rule [7].

 Philips unified mobility model

The Philips unified mobility model proposed by Klaassen [13] provides unified description of the majority and minority carrier bulk mobilities. In addition to describing the temperature dependence of the mobility, the model includes the effect of electron-hole scattering, screening of ionized impurities by charge carriers, and clustering of impurities. Because the carrier mobility is given as an analytical function of the donor, acceptor, electron and hole concentrations, and the temperature, this model is well suited for simulation proposes, and excellent agreement is obtained with published experimental data on Si [10].

In the Philips unified mobility model carrier mobility is considered to be composed of two parts, first part representes the phonon scattering and another part accounts for all other bulk scattering mechanisms.

The partial mobility of each carrier is given by, ,

i b i L i DAeh µ µ µ  , , 1 1 1

2.25

The index i takes the value e for electron and h for holes. First part in the equation 2.25 is due to interaction with lattice vibration which is represented by, ,max , 1 300

i i iL T µ µK      

2.26

The second contribution has the form, , ,

1

i i ref i sc i N i c i DAeh i sc eff i sc i sc eff N N np µµ µ N N N                       

2.27

Effective donor or acceptor concentrations are calculated by introducing some clustering function and then approximated by some analytical functions in Sentaurus device [7].

 High field mobility models

At high electric field, carrier velocity becomes no longer proportional to the electric field as it saturates to the saturation velocity. In Sentaurus device high field saturation model is composed of actual mobility model, velocity saturation model and drivind force model. In the Canali [14] model carrier mobility is represented by, 1

( 1)

1 low low hf sat aµ µF a µ F s a v              2.28
Where, low µ denotes the low field mobility, hf F is the deriving field and sat v represents the carrier saturation velocity. In Sentaurus device energy dependent energy relaxation time is applied as a table function and it can be accessible from the parameter file.

Energy and momentum relaxation

Generation and recombination

Carrier generation and recombination process is a very important phenomenon particularly for bipolar devices. For a typical device total carrier lifetime is assumed to be composed of contribution from the different recombination processes by an inverse law,

1 1 1 1 1 tot SRH AUG DIR SURF         

2.29

This includes doping dependent Shockley-Read-Hall (SRH), Auger recombination, direct recombination and surface SRH recombination.

 Shockley-Read-Hall (SRH) recombination

Electron transition between the bands passes through a new energy state created by the impurity of lattice. The impurity state can absorb the difference between the momentums of two carriers and hence recombine. For indirect band gap semiconductors like Si, SRH recombination is dominant [10].

In Sentaurus device the net SRH recombination is defined as,

SRH i eff net pn np n n n p p R       2.30 2 , 11 ( ) ( )
With, 1,

exp( )

trap i eff E nn KT  2.31 1,

exp( )

trap i eff E pn KT   trap E
is the difference between the defect level and intrinsic level. For Si devices the default value of trap E is set to zero. Minority carrier lifetime ( n  ) are modeled as a product of doping dependent, field dependent and temperature dependent factor.

 SRH doping dependence

Doping dependence is modeled using the Scharfetter relation inside Sentaurus device, max min min ,0 ,0

1 dop AD ref NN N            2.32
Such dependence arises from the experimental data [15]. These parameters are accessible from the parameter file for the Scharfetter relation inside the simulator.

 SRH temperature dependence

To date there is no correspondence of temperature dependence of carrier lifetime [7]. This appears due to a different understanding of recombination lifetime as [16] net n R   

2.33

Where, n  stands for excess carrier concentration. For a power device it is calculated that lifetime increases with increasing temperature [16]. This is modeled in Sentaurus device by power law or by exponential law, 0 () 300

T T K      

 Auger recombination

Auger recombination is very important for the high doping regions inside the device. The band to band Auger recombination is modeled as,

   2 , A n p i eff net C n C p np n R   

2.35

Where n C and p C are the Auger parameters. Comparing different direct band gap III-V semiconductor materials it is established that Auger co efficient increases with decreasing bandgap [7].

Temperature dependence of Auger coefficient is modeled by simple power law (for Si) or by polynomial expansion.

,300 () 300

AUG AUG K T T K CC       2.36 2 12 , 300
() 300 300

AUG AUG K TT T K K KK CC                

Conclusion

The main aim of this chapter is to understand the physics behind TCAD device simulation. It In the next part, the main physical equations inside the HICUM model are discussed with some details.

HICUM model equations  Transfer current

The 1D GICCR result for transfer current is described in HICUM [3] as,

10 ' ' '' [exp( ) exp( )] BC BE TC pT T T CV V II Q V V    3.1
Where, 10 C is a model parameter which is calculated from the Generalized Integral Charge Control Relation (GICCR) relation.

 

2 2 10 E T nB i C qA V µ n  3.2
In HICUM model, total base charge ( pT Q ) is considered as an important internal variable. Total base charge ( pT Q ) is composed of zero bias hole charge ( 0 p Q ), weighted sum of depletion charges and weighted sum of forward and reverse minority charges as,

, 0 , , p T p jei jEi jci jCi f T r T Q Q h Q h Q Q Q      3.3
Here, 0 p Q is the zero bias hole charge inside base. Depletion charges due to the BE junction and BC junction are jEi Q and jCi Q which are modeled using two HICUM parameters ( jei h , jci h ). Other two parts represent the forward and reverse component respectively. The forward component of base charge ( , fT Q ) can be summed as,

,0 f T f fE Ef Bf fC Cf Q Q h Q Q h Q        3.4
Hare, Bhrec I is the recombination current due to the BC barrier effect [5] at high current densities.

0 f Q is the low current charge component, Ef Q  Bf Q  Cf Q  are
B E B E jBEi BEiS REiS Bhrec BEi T REi T V V i i i I m V m V                                  3.5

Saturation currents

This quasi static assumption works well also in high frequencies. Similarly the quasi-static base current injected across the emitter periphery is given by,

* ' * ' exp 1 exp 1 B E B E jBEp BEpS REpS BEp T REp T VV i i i m V m V                                     3.6
In hard saturation current component at the base collector junction becomes prominent which is modeled as,

 Depletion capacitance

In HICUM depletion charge and capacitance are assumed to follow the classical formulation, 0

(1

) j j j Z D C C VV   3.9 0 ' V jj Q C dV  
Here, D V is the built in potential and 0 j C is the zero bias capacitance.

D

VV  is a pole of equation 3.9 which is not suitable for the compact model formulation. Moreover simple theoretical expression does not predict the behavior of depletion capacitance at high reverse bias (punch through). HICUM capacitance parameters are discussed below,

Base-emitter junction

With the classical model parameters DEi V Zei and an additional parameter jEi a which is defined as the ratio of maximum capacitance to the zero bias capacitance, the expression for the base emitter junction capacitance reads, 0 0 ' ' ' '

.1 ( 1)

jEi j j jEi jEi jEi Zei j DEi B E B E C dv dv C a C V V dv dv       

3.10

Where the auxiliary voltage j v uses a hyperbolic smoothing expression, 1

Ei Z f DEi jEi V V a   3.11 2 2 fj j f T x x a v V V   '' f B E T VV x V   (1 )
The corresponding charge equation becomes,

  (1 ) 0 0 ' ' 11 1 Ei Z jEi DEi j jEi jEi jEi B E j Ei DEi C V v Q a C V v ZV               3.12

Internal base collector junction

The BC junction is usually operated by reverse bias. If the internal voltage (vcb) becomes greater than the effective punch through voltage the epitaxial collector becomes fully depleted. The effective punch through voltage is given by

DCi PTCi jPCi V V V  

3.13

For large forward bias the parameter

DCi V becomes, ) 1 ( 1 Ci z jCi DCi fCi a V V   

3.14

Large reverse bias starts from the voltage,

T PTCi r V V V 4 1 . 0  

3.15

The total base collector depletion capacitance is modeled as a summation of three bias dependent components, large reverse bias, medium bias and high forward bias. ).

At medium bias, 0 , , , ,, (1 ) (1 )

BCx jCx BCpar

C C C 

3.18

The total capacitance The factor BCpar f is geometry and technology specific parameter. The capacitance splitting is described inside HICUM as,

(1 )

BCx BCx BCx BCpar BCx BC BCx C C C f C f C      3.20

 Transit time

The extraction of transit time parameters is a crucial point for precise modeling of bipolar transistors using the HICUM model. The transit time formulation used in HICUM model [3] is much more complex and more precise than that used within the Gummel-Poon model [6],

which requires an accurate and rigorous method of parameter extraction. The transit time is defined from the transition frequency f T (the frequency for which the gain becomes unity for small signal current) by the relation,

1 ( ) ( ) 2 T F je jc CX E jc TC V C C R R C fI        3.21
Where je C and jc C are the total base emitter and base collector junction capacitance, E R and CX R are emitter series resistance and extrinsic collector resistance. In HICUM the expression of total transit time ( f  ) is associated with charges in each region of transistor.

f E BED B BCD C            3.22
Where, the different parts represent the different regions inside the transistor.

The current and voltage dependence of the total transit time is modeled inside HICUM by two components.

' ' 0 ' ' ' ' ( , ) ( ) ( , ) f C E Tf f B C f C E Tf v i v v i     

3.23

Where, 0 f  represents the low current transit time and f   is the increase of transit time at high current densities.

Low current density

The BC voltage dependence of 0 f  is described by,

0 0 0 1 ( 1) ( 1)
f h Bfvl C c           3.24
The first term 0  represents the total transit time at V BC =0V. The second term models the net change caused by the Early effect with the help of normalized internal BC depletion capacitance C . The third term takes into account the carrier jam at BC space charge region at low V CE . The HICUM transit time parameters for low current density are 0h   and Bfvl  .

Medium and high current densities

At high current density a sudden rise of transit time occurs due to the Kirk effect. In HICUM the onset of high current effect is characterized by critical current (

Ck I ). 2 2 0 lim 1 1 2 1 ickpt ceff CK ci cef x x a v I r v V          

3.25

With x connects the high and low electric field inside the collector. ( , ) ( , )

fh Tf C E Ef Tf C E i V i V       
i Q Q Q i w g           

• Breakdown

In HICUM weak avalanche effect at the base collector junction is modeled by avalanche current,

' ' 1/ ' ' ( ) exp( ) ( ) Ci AVL AVL Tf AVL DCi B C Z jCi DCi B C q i i f V v C V v = - - - 3.34 AVL f , AVL q
Are the model parameters.

• Self heating HICUM uses a thermal network (Figure 3.1) to model the rise of junction temperature due to the self heating. The sub circuit requires a thermal resistance R TH and a thermal capacitance C TH as model parameters and a current source to dissipate power of the device. Node voltage corresponds to the junction temperature. This thermal circuit is solved together with the transistor model.

NQS implementation inside HICUM

For fast transient or at high frequency operations, the quasi static approach becomes inadequate and an excess phase shift is necessarily included inside compact models. In some BJT models NQS phase is included inside gm which indeed predicts inconsistency between ac and transient modeling. The bias dependent delay should only introduce an excess phase shift without affecting the magnitude of electrical parameters. In a popular bipolar model VBIC [7] excess phase is introduced by an excess phase network. In HICUM the approach is similar and the excess phase network introduces an phase shift in transfer current as well as in minority charge.

The approach of Winkel [8] is implemented inside the HICUM model. In this approach, time delay is assumed to be proportional of the total transit time. As the total transit time is a function of V BE and V CE , delay time also becomes bias dependent. The starting equation with the transfer current and excess charge is presented below, However these relations are valid for lower frequency range. Beside, they are derived using the linear case of small signal analysis.

In HICUM model time delay is introduced in the transfer current and also in the excess charge. That is why in the HICUM model NQS circuit, we find two independent excess phase

( * * ( 1)) ( 2)

tzf xf it D dd I Q V xf T V xf V xf dt dt      3.38 2 ( 1) ( ) ( 2)
( * / 3* ( 2)) ( 2)
xf it D dd V xf Q V xf T V xf V xf dt dt      And, ( 2)
txf I V xf 

3.39

With the help of equations 3.38 and 3.39, a second order differential equation in time domain can be constructed, The NQS charge is represented by the next equation,

1 dei deix D Q Q sT  

3.42

Last of all, HICUM L2V24 NQS modeling code in VerilogA is presented for the transfer current and excess charge.

HICUM modeling in BiCMOS7RF

BiCMOS7RF, a SiGe-C technology which is widely used inside oscillator for wireless communications, is developed by ST microelectronics. In the next figure, the roadmap of ST process technologies is shown indicating the BiCMOS7RF layout [9]. The transistor used for modeling contains emitter length of 12.8µm and emitter width of 0.4µm.

Extraction strategy using HICUM model

Single transistor parameter extraction was performed following the extraction procedure mentioned below [10]. The main intention of this parameter extraction is to verify the NQS related parameters when the transistor works beyond its cut off frequency. The parameter set obtained by single transistor parameter extraction can't assume to be accurate. In this first step extraction temperature parameters were ignored. Also the external resistances are very difficult to extract from a single geometry. However, global optimization ensures the best possible fit for measurements.

Modeling results

 Gummel plot:

Static measurement is performed setting V BC =0V and varying V BE from 0.5V to 0.98V which contains both the low and high injection regions. The modeling results are presented below which represent modeling of Gummel plot and Beta. Both these are modeled properly as shown in the Figure 3.9.

Static parameters

 Output characteristics

Output graphs are presented in Figure 3.10 showing I C as a function of V CE considering different V BE s. The maximum base emitter voltage is chosen beyond peak f T at high current region. Base collector breakdown is not reached in the measurement. Also the effect of self heating is not predominant in output characteristics. Previous modeling results show an acceptable result for both the Y and h parameters at high frequency. It is also pointed that at very high frequencies, the NQS modeling is not always perfect while considering the imaginary part of Y 21 or phase of Y 11 parameters.

Though the measurements were performed beyond the transistors maximum cut off frequency (60GHz) there was no significant effect observed in y parameters except that there is a sudden decrease of y 11 magnitude at higher frequency (Figure 3.13).

Modeling with HICUM model

Scaling of NQS related parameters with HICUM model

In The nature of the parameters as shown in Figure 3.21 is not totally followed when the transistors with higher emitter widths are considered. The difference is clearly observed in the high current density region, where the alqf parameter undergoes a sharp fall from the average value.

Chapter 3 [NQS IMPLEMENTATION INSIDE COMPACT MODEL]

108

To compare the extracted parameters as a function of real emitter area, only two bias points below f T are considered (V BE =0.76V, 0.8V) because the high injection parameters are not consistent for large devices. The real emitter area is calculated after subtracting the spacer width of 55nm at both sides. While the alit parameter is almost constant at 1, alqf suddenly decrease at high emitter area transistors at V BE = 0.8V. Last of all, variation of these parameters along emitter length is shown considering two bias points. From the previous analysis it is clear that the NQS parameters (alit, alqf) are very close to have constant values over emitter length but alqf decreases rapidly over real emitter area It is, therefore, possible to fix the parameters value (alit=1, alqf=0.5) inside the model code to have an overall fit over dynamic parameters without losing the accuracy of NQS modeling.

HICUM model with an extended phase network

It is shown in the previous part that at high frequency, the phase of h 21 cannot be modeled properly with the existing HICUM model. As the phase network comes from a physical understanding of NQS behavior, the improvement should also contain physical basis. In this work, the effect of base collector (BC) junction delay, which was not previously included inside this model, is investigated. A new excess phase circuit is proposed including the BC junction delay. The modeling results show improvement in h 21 phase at higher frequencies.

Model calculation

In a modern HBT transistor, with graded base profile, the built-in electric field reduces the base transit time significantly. As a consequence of this process, the signal delay due to the BC junction has become an important contributing factor in the total delay. Base-collector junction (BC) is considered as high field region where carrier travels at a constant velocity which is the saturation velocity inside that region. Assuming that the excess minority carrier concentration at the base collector junction boundary is zero and no recombination takes place inside the junction, it has been shown in the chapter 1 (equation 1.61 and equation 1.62) that the BC junction introduces a delay which is proportional to the collector transit time. Quoting equation 1.62 of chapter 1 that assumes a small signal transfer current, entering the BC junction, the resultant current leaving the junction is described by,

.

Implementation inside HICUM

To include the new formulation inside HICUM L2V24, equation 3.35 is restructured to give a simple form,

)

f Ci Ci f j altc II j alit      3.47
altc is a new parameter representing the BC junction delay. Again, changing the domain by putting sj   in equation 3.47 and with a suitable approximation [12], we get the final form of transfer current which will be implemented inside HICUM via VerilogA code In NQS equivalent networks, 1Ω resistors are commonly used at the output node. Doing so, the current through the resistor is equal to the voltage over its nodes. When writing the circuit node equations, the 1Ω multiplication factor is omitted resulting in a mix-up of current and 112 voltages but the aim of the NQS network is only the mimic of a mathematical relationship. In the improved excess phase circuit, output current of the first branch is taken as an input of the second branch (Figure 3.25). The circuit equations are formulated below.

( * * ( 1))

( 1)

tzf lit f d I T V xf V xf dt   3.49 ( 1) ( * * ( 2)) ( 2
)
ltc f d V xf T V xf V xf dt  
And the transfer function [12] which gives the final branch current (Equation

3.48) is, 2* ( 2) ( 1) txf I V xf V xf  3.50

Modeling result with the extended model

Conclusion

In this chapter, compact modeling implementation of NQS effect is discussed while showing some modeling results. Two types of transistors with different technologies are chosen for modeling. The first technology is BiCMOS7RF, which provides a cut off frequency of 60GHz. In this transistor, a step by step modeling and HICUM specific parameter extraction was performed before proceeding to the NQS modeling. High frequency modeling was performed and the modeling results show a considerable fitting of dynamic parameters up to high frequency. It is also shown from the input admittance modeling that there is a scope to further improve the excess phase network.

In the next step, transistor with a very modern technology is considered to perform the NQS modeling. There a scalable model card is used and NQS modeling is performed in a vast range of transistors with different emitter lengths and widths. Modeling of high frequency phase of small signal current gain, input admittance and transconductance are presented. From the modeling results it is clear that the existing HICUM excess phase network does not provide an exact fitting in transistors having higher emitter dimension. To overcome the inconsistency, an improved network is proposed which accomplish a time delay due to the base collector junction. The implementation of the new model is done by VerilogA code in the HICUM model. After that, the modeling results compare these two models and clearly establish that the new model works better than the previous one.

Introduction

In this chapter, physical device simulation results are presented for SiGe HBT devices. Here, two kinds of device are considered-a conventional high speed HBT and a new type of HBT device with a SiGe spike incorporated inside emitter. The working device is first calibrated with measurements and then DC and AC simulations are performed with the calibrated parameter file. As all the simulations are performed in a calibrated structure, the simulation results can be treated as measured data with the real device. The simulated data are then exported to ICCAP to perform compact modeling. For compact modeling purpose, HICUM model is chosen due to its ability to model high injection operation properly. A systematic modeling was performed and the results are shown in this chapter.

In the device simulation, very high frequency simulation (500GHz) was performed and simulated data were modeled using HICUM model. The aim is to verify the HICUM excess phase circuit at very high frequencies. The modeling results of high frequency phase shift of dynamic parameters, shown in this chapter, provide a better understanding of the NQS phenomena. The next table (

Simulation of IMEC structure

The device used for physical simulation is incorporated in 130 nm SiGe BiCMOS process technology by IMEC [1]. This advances HBT has a high cut off frequency (fTpeak= 240GHz) and it is widely used for high speed application [2]. Keeping in mind the symmetry of

Calibration

To use the device as a substitute for the real transistor, first the device has to be calibrated with measurements. TCAD Sentaurus [3] software tools have been used to perform the 2D device simulations using hydrodynamic model (HD) equations with a set of suitable physical models. The physics of semiconductor device is discussed in chapter 2. In this simulation, the carrier temperature equation for the dominant carriers is solved together with the electrostatic Poisson equation and the carrier continuity equations. The carrier mobility's have been calculated using Philips unified mobility model, the high field saturation was calculated 122

Above figures represent the calibration of the HBT device. These plots confirm that simulated curves follow the measurement quite well. The structure is said to be calibrated and can be used to perform high frequency simulations.

DC and AC simulation modeling with HICUM

Device simulation results are exported to ICCAP to perform HICUM [4] modeling. Both DC and AC simulations results are exported and modeled with HICUM L2.24 model. As previously described that the device structure is calibrated with measurement, all these device simulation results can be considered as measured data. Parameter extraction is performed while following the standard extraction procedure which is described in chapter 3. Some of these extraction results are presented below.

NQS modeling

To observe NQS behavior, high frequency simulation is carried out in the IMEC device structure. This modern device structure admits the simulation to go much beyond the cut off In the forward active mode a sinusoidal perturbation of 10mV is superimposed to the positive static bias, such that the total base emitter voltage becomes v be =V BE +10mVsin (ωt). Keeping the base collector junction voltage fixed (V BC =0V), for different bias points (V BE =0.7V, 0.8V, 0.9V, 0.95V) and for different frequency values (f=1KHz, 1MHz, 1GHz, 10GHz, 50GHz, 75GHz, 100GHz, 150GHz, 200GHz) the same device is simulated using the Hydrodynamic equations and using calibrated parameter file. Steps of this process are outlined below.

From the device simulation results, both the base and emitter excess charge are extracted (by integrating the excess carrier concentration over the vertical dimension of the device) and plotted against time. Excess base charge is plotted against different frequencies considering different bias points (V BE =0.7V, 0.8V, 0.9V, 0.95V). The results are presented below and discussed after.

 At low frequency (1GHz) and moderate injection (0.8V), the variation of excess minority carrier follows the variation of the applied external signal.

 At high frequency (150GHz) and moderate injection (0.8V), a delay is present between the variation of the excess minority carrier and the applied external signal.

 At high frequency (150GHz) and high bias (0.95V), the delay between the excess minority carrier density and the external signal is further enhanced. The charge is constant for a wide range of frequency but at high frequencies (>100GHz) it starts to decrease. The frequency dependence of base charge is not considered inside compact model.

It has been shown that the excess base charge encounters a delay at higher frequencies (Figure Base charge delay increases as the biasing voltage increase. Also, the phase shift (ωτ) increase as the frequency increases.

From the above analysis, it can be concluded that the base excess charge encounters a delay at higher frequencies. The delay increases as the bias increases and it also depends upon frequency. In the compact model formulation, phase network of excess charge is considered  At low frequency (1GHz) at moderate injection (0.8V), the variation of excess minority carrier follows the variation of the applied external signal.

 At high frequency (200GHz) and moderate injection (0.8V), a delay is present between the variation of the excess minority carrier and the applied external signal.

 At high frequency (200GHz) and high bias (0.95V), the delay between the excess minority carrier density and the external signal is further enhanced. The magnitude of excess carrier concentration at emitter side does decay over frequency as it is shown in Figure 4.14 for different bias points. As expected the phase delay at low frequency is negligible and increases as frequency increase. This delayed integrated excess carrier (i.e. minority charge) certainly affects the dynamic behavior at high frequency as well as at high bias. The bias variation is shown in

Introduction

The advancement of SiGe HBT over the past decade has enormously increased due to the capability of providing higher cut-off frequency in high-speed application. To achieve higher f T , transistor dimension is drastically reduced as well as the doping profile is scaled in the neutral base and collector region. The inclusion of Ge inside the base increases the current gain but reduces the open base breakdown voltage (BV CEO ). The only way to increase the breakdown voltage without affecting the RF performance is to increase the base current by the band gap engineering in the base or emitter region. One way to increase the base current is the inclusion of a SiGe spike inside the neutral [5][6]. It is reported to increase the base current without affecting the f T and thus also increase the breakdown voltage. In the next section a SiGe spike mono emitter will be considered and simulation results will be presented for both DC and AC operations.

In the next figure (Figure 4.17 The constructed structure is used to perform electrical device simulation. For device simulation, a parameter file is used which had been calibrated for the simple emitter structure.

Also, the physical used for the device simulation are the same as used before. The inclusion of SiGe spike inside the mono emitter locally increases the recombination. The total recombination is a summation of SRH recombination, Auger recombination and recombination at the Si SiGe interface. To calibrate the total recombination in the emitter region, a slight modification of the hole recombination lifetime is made. Total recombination rate is then extracted for both mono emitter and SiGe spike mono emitter and is shown in From the basic device simulation results, it is observed that the structure works properly. The next step is to use this structure to perform all the electrical simulations. These simulation results will help to verify that inclusion of a SiGe spike only increases the base current but do not affect the ft characteristics. Also, the effect of this excess charge will be investigated.

Comparison with mono-emitter

Gummel plot: As discussed before, addition of SiGe spike inside emitter region locally increases the recombination and hence it is expected to observe an increase in base current without affecting the collector current. Cut off frequency: Though the effect in the base current is evident in the Gummel plot, there is no difference in the dynamic characteristics of the device namely the transit frequency is the same for both the devices. Transit frequency is calculated from the magnitude of h 21 parameter at a frequency where the slope of h 21 shows a 20dB/dec roll off. This behavior is expected from the theory and also reported in [5]. The next figure (Figure 4.24) compares the cut off frequency for both the devices. Current gain: As the magnitude of base current in SiGe spike mono emitter is more than the simple mono emitter, current gain is reduced in the spiked emitter. In the dynamic performance, magnitude of h 21 presents the current gain. It is compared for the devices with and without SiGe spike inside the emitter. Magnitude of h 21 is calculated from AC simulation while keeping V BC fixed at 0V and for different V BE = 0.85V and 0.9V. From the figure (Figure 4.25) it is clearly evident that at higher frequency (>10GHz), there is no difference in the h 21 magnitude for both the transistors. So, the cut off frequency, which is calculated by multiplying the magnitude of h 21 with frequency, is the same for spiked and non spiked emitters.

Current gain can also be calculated for different frequencies from the transient analysis and can be compared with the value calculated from the ac simulation. A small signal voltage (10mV) is applied on the DC bias point for the transient simulation. The simulation is performed with a vast range of frequencies and the difference between the magnitudes of maximum and minimum of Ic and Ib currents at a frequency are divided to obtain the small signal current gain (Equation 4.1).

(max) (min) (max) (min) Ic Ic Ib Ib    

4.1

The next figure shows the small signal current gain for a SiGe spike transistor.

Modeling with HICUM

The SiGe spike mono emitter structure is simulated with the calibrated parameter file. Device simulation results with the SiGe spike mono emitter are exported into ICCAP. The simulated results are accepted as measured data and model with HICUM model [7]. To properly extract all parameters, a step by step procedure is applied which was discussed before (Chapter 3). In the next part, some of the modeling results are discussed. From the figures (4.29 and 4.30), it can be concluded that an additional parameter is necessary to model the base current increase at medium injection. TCAD device simulation reminds that the sudden increase of base current is basically due to the SiGe spike inside emitter.

Gummel plot

Transit frequency:

Dynamic behavior of this transistor is shown in Figure 4.31 where a reasonable fitting is observed in modeling. As the cut off frequency is not affected by the Ge spike, the modeling with HICUM provides a good result.

NQS parameters:

After getting required parameters for forward current and transit time and axis resistance, high frequency modeling is done with that parameter set. For the NQS modeling, first h 21 magnitude and phase is observed after optimizing the NQS parameters (alit and alqf). Next figure shows magnitude and phase of h 21 parameter in SiGe spike mono emitter. This controlled current source can be easily implemented parallel to IjBEi inside HICUM model circuit. 148

Results

Similarly, for the y 11 parameter, at low frequency, the phase shift is more for the SiGe spike transistor and modeled satisfactorily by the extended model.

Conclusion

In this chapter, two different types of devices are considered for TCAD device simulation.

The first one is a modern IMEC device with a high cut off frequency dedicated for high frequency operation. To verify NQS effect, both AC and small signal transient operations are With all the analysis, it is shown that high frequency characteristics of HBT has become an important topic for modeling engineers. The available compact models provide a global fitting of dynamic parameters but to apply in high frequency circuits, exact fitting is necessary. The difficulty is to implement complex models inside the model code and convergence issues are also there. Nevertheless, this work provides NQS analysis in both physical and compact modeling point of view. It can be treated as a starting point of high frequency analysis for modern day HBT devices.

4. 4 .GLOSSARYpalit

 4 Conclusion ............................................................................................................................ 149 General conclusion .............................................................................................................................. 151 Small signal e hole density at the base emitter junction v Small signal voltage BE V Base emitter DC voltage  Common base (CB) current gain  Common emitter (CE) current gain time constant at the BC barrier for high forward injection (default is v2.1 compatible) Ratio of maximum to zero-bias value of internal B-Exponent factor for current dependence of neutral emitter storage time hcs  Saturation time constant at high current densities Factor for additional delay time of transfer current alqf Factor for additional delay time of minority charge General introduction [GENERAL INTRODUCTION]

  Figure I: SEM cross section of a new HBT device from DOTFIVE project.

Figure II :

 II Figure II: Application of RF and analog/mixed signal technology [6].

Figure

  Figure IV: SiGe:C HBT technology BiCMOS/200mm [7]

Figure 1 . 1 :

 11 Figure 1.1 : Transient phenomena in base region. (a) switch off (b) switch on[1]

Figure 1 . 2 )Figure 1 . 2 :

 1212 Figure 1.2 : excess electron distribution in neutral base in normal and inverted mode of operation. (a) Approximation with emitter and collector junction forward biased. (b) Emitter and collector currents in normal mode. (c) Emitter and collector currents due to inverted mode of operation.



  is the minority carrier lifetime in the base region.

  is considered between 1GHz to 500GHz. The drift factor ( ) is a positive number and attended integer values upto 8. It is notable that at high drift field ( 3   ), the transistor becomes a purely drift transistor.

Figure 1 . 3 :

 13 Figure 1.3 : (a) Normalised excess carrier as a function of normalised distance for different frequencies (1GHz, 10GHz, 100GHz, 200GHz, 300GHz, 500GHz). The plot shows a clear frequency dependence at high frequency. (b) Normalized excess carrier as a function of normalized distance at a constant frequency for different drift factors.

4 Figure 1 . 4 :

 414 Figure 1.4 : Magnitude and phase variation of common emitter current gain for different drift factors.

Frequency

Figure 1 . 5 :

 15 Figure 1.5 : Magnitude and phase variation of normalized collector current at different drift factor.

Equation 1 .Figure 1 Figure 1 . 6 :

 1116 Figure 1.6 : comparison between small signal and quasi static excess carrier concentrations at a constant drift factor. Quasi static part is shown in blue circle and small signal part in line.

Figure 1 . 6

 16 Figure 1.6 suggests that increasing frequency shifts the magnitude of excess carrier inside the quasi neutral base region. Difference in carrier concentration means additional small signal charge inside base region.

  With the help of equations (1.28 and 1.25) a relation between the small signal excess carrier concentration and forward transfer current can be established, relation between small signal transfer current ( () c i  ) and static transfer current (

Figure 1 . 7 :

 17 Figure 1.7 : Magnitude and phase variation of normalized charge at LI level.

Figure 1 . 7

 17 Figure 1.7 typically shows the phase variation due to the charge accumulation at higher frequencies.

Figure 1 . 9 :

 19 Figure 1.9 : Comparison between small signal current gain  in LI and HI condition.

FrequencyFigure 1 . 10 :Figure 1 . 11 : 1 . 11 .

 110111111 Figure 1.10 : Calculated magnitude and phase variation of small signal common emitter Y parameters at Low Injection condition. Parameter set is the same as before.

Chapter 1 [Figure 1 . 12 :

 1112 Figure 1.12 : phase of Y 21 with different models-Winkel (in symbol), Seitchik (in line). Three different curves correspond to different values of fitting parameter alit (alit= 0.1, 0.3, 0.5) which relates delay time to the total transit time (τ f ).

Figure 1 . 13 :

 113 Figure 1.13 : Magnitude and phase variation of BC junction delay function for different collector transit times ( c  =0.25ps, 0.5ps, 0.75ps, 1ps).

Figure 1 . 14 :

 114 shows the diagrammatic representation of the transistor under consideration.

Figure 1 . 14 :

 114 Figure 1.14 : Small signal excess carrier concentration inside base and emitter.

52

  Assuming an ac perturbation over the DC bias variation all the electrical parameters can be expressed a summation of a DC and a time varying part. Expressing total minority carrier concentration as, Assuming the emitter doping profile is constant, equations 1.63 can be combined for low level of injection condition (n  N D ). The resulting second order equation for small signal bias 65) is a second order homogeneous differential equation which can be solved easily with the help of boundary conditions i.e. at x=0,

  (1.73) and (1.74) reduces to the wellknown equation for emitter transit time,

  advancement of downscaling of Si heterostructure devices have improved the circuit performance and also reduced the cost per function. Another intelligent way that reduced the time and cost of technology development was the vast use of Technology Computer Aided Design (TCAD). TCAD is used in the fabrication of process design, semiconductor device design, technology characterization of circuit design and also in the computer integrated manufacturing [2] [1]. It is not only useful to predict the device behavior or the product cost, but also significantly reduces the product cycle time. Next Figure 2.1 points the relationship between TCAD, process development and circuit implementation [1].

Figure 2 . 1 :

 21 Figure 2.1: product cycle for technology development [1]

P

  are the absolute thermoelectric power. Hydrodynamic modelThe downscaling of modern semiconductor devices assisted to achieve high speed transistors. Meanwhile non local effects which are neglected in the drift diffusion model are prominent in these sophisticated transistors. Both the internal and external device characteristics are found to be incorrect as it cannot predict the velocity overshoot in III-V semiconductors[7]. In hydrodynamic transport model carrier temperature is considered different from the lattice temperature. In general hydrodynamic model contains Poisson and continuity equation and energy conservation equation for electron, hole and lattice.

N 13 

 13 represent the density of states at conduction and valence band respectively. For a doped semiconductor, intrinsic carrier density changes due to doping dependent bandgap narrowing. It can be represented by, Bandgap energy Various models define the temperature dependence of bandgap energy in semiconductor. For the basic semiconductor material temperature dependence of band gap is calculated as, energy at 0K. As Sentaurus device uses different bandgap models, (0) g E is assumed to be a summation of common term and a variable term (equation 2.15)



  for p-n-p transistor with different base doping concentrations and a 1D model for the collector current.

  Energy relaxation time models are used in the relaxation time terms in Hydrodynamic equations. A constant energy relaxation times or a quadratic dependence on the electron temperature are usually assumed. Using non constant term for energy relaxation inside mobility models provides better modeling characteristics at high field region (i.e. velocity overshoot). It also improves the bias dependent and small signal simulation[10]  Relaxation time model There are mainly two procedures to model the energy relaxation time. The direct method assumes an average of electron energy and uses MC simulation result. From the interpolation of MC results for different electric fields, relaxation time is obtained directly as a function of electron temperature.

  reduces the recombination lifetime by field enhancement factors. This is due to the Trap Assisted Tunneling (TAT) which is an important mechanism for the indirect band gap semiconductors. There are two types of field enhancement available in Sentaurus device simulator, Hurkx TAT[17] and Shenk TAT model.

  is fairly well established that in modern days, TCAD tools are an integral part of semiconductor device industry. Starting from the process simulation, there exist numerous tools for device simulation or parameter extraction. In this chapter, the different physical models used in TCAD device simulations are analyzed. This analysis helps to build a physical understanding behind different electrical behaviors of real time measurements. Again, for the calibration process, the understanding of different physical models is important. In calibration, only a few parameters are tuned to achieve the desired output results and will be shown in the last chapter.

3. 1

 1 Introduction: HICUM model overview HICUM is a semi physical compact bipolar transistor model for arbitrary transistor configuration [1]. The name HICUM was derived from HIgh CUrrent Model, indicating the operating region at high current densities, a very important region for high speed transistor applications. The model is based on an extended and Generalized Integral Charge Control Relation (GICCR) and GICCR concept is applied consistently without inadequate simplifications in HICUM. Next figure (Figure 3.1) shows a schematic cross section of SiGe bipolar transistor fabricated with non selective epitaxial growth process and circuit diagram of HICUML2 model representation of the different parts of the transistor [2].

Figure 3 . 1 :

 31 Figure 3.1: Upper figure: Schematic deep trench isolated SiGe HBT. Lower figure: large signal circuit diagram (HICUM L2) with the self heating network [2]. The internal transistor is shown with different color.

  minority charge at neutral emitter, base and collector regions. The weighting factors like fE h fC h depend upon the grading and bandgap of various transistor regions in HBT. The next figure (Figure 3.2)shows the different charge component inside the transistor.

Figure 3 . 2 :

 32 Figure 3.2: HICUM formulation of different parts of base charge inside HBT in both forward and reverse operation [4].

Equation 3 .

 3 16 contains classical capacitance part on the first term and also the smoothing function at the end.At high forward bias and high reverse bias the capacitance equation looks like,

  The external base collector capacitance consists of a bias dependent base collector external depletion capacitance ( jCx C ) and a bias dependent external parasitic capacitance ( BCpar C

  density the increase of the transit time is represented by,



  Base resistanceIn HICUM internal base resistance and the external base resistance are treated separately. The internal base resistance depends on the operating point, temperature and the transistor operation condition. The dc internal base resistance is modeled by,

I

  the quasi-static values and t 2 and t m are corresponding additional delay times. In the first order approximation t 2 and t m are associated with the total transit time t and α m are charge partitioning factors used as NQS model parameters in the HICUM model.

Figure 3 . 3

 33 Figure 3.3 shows the rough assumption of α parameters extracted as a function of collector current density from 1D device simulation.

Figure 3 . 3 :

 33 Figure 3.3: HICUM time delay parameters as a function of current density [1]

Chapter 3 [

 3 NQS IMPLEMENTATION INSIDE COMPACT MODEL]90 networks-one for the transfer current and another for the excess charge. In the next part, NQS networks in the HICUM model are analyzed separately.Bias independent phase shift in transfer current is implemented in compact model via Weil-McNamee formulation as the equivalent circuit represents. This approach is suitable for all kind of circuit analysis like ac or transient.

Figure 3 . 4 :The implementation of this network (Figure 3 . 4 )Figure 3 . 5 :

 343435 Figure 3.4: Weil-McNamee implementation of excess phase circuit.

Chapter 3 [ 3

 33 And mapping ddt to s, the corresponding expression in frequency domain for itxf becomes, gyrator equivalent network implements the same equation as the previous modelimplements. NQS IMPLEMENTATION INSIDE COMPACT MODEL] 92The excess charge phase shift network is similar to the transfer current network (Figure3.5)in HICUM L2V23 model. The only different is that the input current ( tzf I ) is replaced by the excess charge ( dei Q ). In the new version of HICUM model (L2V24), a simple RC network is implemented which is presented in figure

Figure 3 . 6 :

 36 Figure 3.6: excess charge phase network in HICUM L2V24

  ) <+ ddt(Qxf1); I(br_bxf2) <+ Ixf2; I(br_cxf2) <+ ddt(Qxf2); ) <+ Ixf; I(br_cxf) <+ ddt(Qxf);

Figure 3 . 7 :

 37 Figure 3.7: Different Si process technologies developed by ST microelectronics [9].

Figure 3 . 8 :

 38 Figure 3.8: single transistor parameter extraction methodology[10]

Figure 3 . 9 :

 39 Figure 3.9: (a) Ic and Ib (measured and simulated) as a function of V BE (Gummel plot) at V CB =0V (b) Beta Vs V BE plot for the same configuration.

Figure 3 . 10 :Figure 3 .

 3103 Figure 3.10: (a) Ic-V CE for different V BE = 0.75V, 0.8V, 0.85V, 0.9V, 0.95 V, (b) Ib V CE for the same configuration.

Figure 3 . 11 :Figure 3 . 12 :From the previous figure 3 . 12 ,Figure 3 . 13 :

 311312312313 Figure 3.11: f T vs V BE and f T vs Ic characteristics for different V CE .

Figure 3 . 14 :

 314 Figure 3.14: ST BiCMOS9MW layout. Left figure shows the TEM picture and on the right hand side the device structure [11].

Figure 3 . 15 :Figure 3 . 16 :

 315316 Figure 3.15: Modeled phase h 21 for different emitter lengths (L E = 0.6, 3, 5, 10 µm) at an emitter width W E = 0.27µm at V CE = 1.2V and V BE =0.76V, 0.8V, 0.84V, 0.88V, 0.92V. All these transistors are having one base one emitter and one collector contact respectively. Emitter lengths are (a) L E = 0.6µm, (b) L E = 3 µm, (c) L E = 5 µm, (d) L E = 10 µm.

Figure 3 . 17 :

 317 Figure 3.17: (a, b) Magnitude and Phase of Y 11 parameter for a single transistor at V CE = 1.2V and V BE =0.76V, 0.8V, 0.84V, 0.88V, 0.92V. (c, d) Magnitude and Phase of Y 21 parameter for a single transistor at V CE = 1.2V and V BE =0.76V, 0.8V, 0.84V, 0.88V, 0.92V.

Figure 3 . 18 :Figure 3 . 19 :Figure 3 . 20 :

 318319320 Figure 3.18: Phase h 21 for different emitter widths (W E = 0.27µm, 0.54 µm 0.84 µm, 1.08µm) at an emitter length L E = 10µm at V CE =0.8, 1.2V and V BE =0.76, 0.8, 0.84, 0.88, 0.92V. Emitter widths are (a) W E = 0.27µm, (b) W E = 0.54µm, (c) W E = 0.84µm, (d) W E = 1.08µm respectively.

Figure 3 . 19 , 3 . 20 )Figure 3 . 21 :

 319320321 Figure 3.21: HICUM NQS related parameters as a function of collector current density in a transistor where L E = 3µm, W E = 0.27µm at V CE = 1.2V and V BE =0.76V, 0.8V, 0.84V, 0.88V, 0.92V.

Figure 3 . 22 :

 322 Figure 3.22: HICUM NQS parameters in two different transistors (a) L E = 10µm, W E = 0.27µm and (b) L E = 10µm, WE= 0.54µm at V CE = 1.2V and V BE =0.76V, 0.8V, 0.84V, 0.88V.

Figure 3 . 23 :

 323 Figure 3.23: alit and alqf as function of emitter area at V CE = 1.2V and V BE =0.76V, 0.8V.

Figure 3 . 24 :

 324 Figure 3.24: alit and alqf as function of emitter length at V CE = 1.2V and V BE =0.76V, 0.8V.

Chapter 3 [g 1 Equation 3 .Equation 3 .

 3133 NQS IMPLEMENTATION INSIDE COMPACT MODEL] 110Where, c  is the collector transit time which depends upon the BC junction width and on the carrier saturation velocity in the junction and x is the length dimension in the device. The magnitude of, equals to, is nearly unity even at higher frequencies. So the small signal transfer current only encounters an exponential delay without changing the current magnitude while passing the base collector junction Recalling Winkel's formulation of common emitter (CE) transconductance parameter, represents the low frequency forward conductance and * includes the bias dependent time delay. Including the BC junction delay inside the admittance parameter calculation of Winkel [8], Equation 3.44 becomes, 45 can be simplified after expanding the BC junction time delay function in Taylor's series and considering up to the second term, 46 shows the final form of transconductance where c  represents the collector transit time. The first part of this equation is the same as before (Equation 3.44) and the multiplicative constant represents an additional part due to the BC junction delay.

Figure 3 . 25 :

 325 Figure 3.25: Modified excess phase network for transfer current.

Chapter 3 [

 3 NQS IMPLEMENTATION INSIDE COMPACT MODEL]

Figure 3 .

 3 Figure 3.26: h 21 : phase with extended HICUM with L E =10µm, W E =0.27µm at a constant V CE =1.2V and V BE =0.8V, 0.84V, 0.88V.

Figure 4 . 1 ) 1 .Figure 4 . 1 :

 41141 Figure 4.1: (a) IMEC HBT device showing the vertical cut at the middle of the base (inset: IMEC 130nm process TEM picture.) (b) The doping profiles in the emitter, base and collector regions.

Figure 4 . 2

 42 Figure 4.2 shows measured and simulated collector (Ic) and base (Ib) currents at V CB =0V (Gummel plot). To adjust the collector current at high current region some lumped resistances (R E , R C ) have been added. Self heating is included by adjusting the thermal resistance. For the base current, simulation is performed with surface and volume recombination models. SRH recombination parameters are slightly optimized to fit the measurement at low and medium injection.Figure 4.2 indicates that the simulated collector current (Ic) shows an optimistic

Figure 4 . 2 Figure 4 . 2 : 4 . 3 .Figure 4 . 3 :Figure 4 . 4 :

 4242434344 Figure 4.2: Gummel plot comparison between measurement and TCAD simulation at V CB =0V

Figure 4 . 5

 45 Figure 4.5 shows the forward Gummel plot where the symbol represents the TCAD device simulated result and line represents HICUM model simulation. The modeled curve shows an exact fit at low, medium and high injection regions.

Figure 4 . 5 :

 45 Figure 4.5: Measured and simulated ic, ib as a function of V be (Forward Gummel plot).

123Figure 4 . 6 :

 46 Figure 4.6: Measurement and simulation of (a) f T _V BE and (b) f T _ic curve at different V CE .

Figure 4 . 7 :

 47 Figure 4.7: Magnitude and Phase of h 21 at different V BE (0.75, 0.79, 0.83, 0.87, 0.91, 0.95V) and at V CE = 0.5, 1V

Figure 4 . 8 : 4 . 2 . 4 .

 48424 Figure 4.8: (a, b) Magnitude and phase of Y 11 at different V BE (0.75, 0.79, 0.83, 0.87, 0.91, 0.95V) and at V CE = 0.5, 1V. (c,d) Magnitude and phase of Y 21 at different V BE (0.75, 0.79, 0.83, 0.87, 0.91, 0.95V) and at V CE = 0.5, 1V.

Figure 4 . 9 :

 49 Figure 4.9: Variation of excess minority carrier (e) density (red line) inside neutral base over time. Blue graph corresponds to the applied sinusoidal bias variation over time. For (a) V BE =0.8V, f=1 GHz, (b) V BE =0.8V, f=150GHz (c) V BE =0.95V, f=150GHz

Figure 4 . 10 :

 410 Figure 4.10: Excess carrier (e) concentration (/cm -2 ) over frequency in the neutral base for V BE = 0.8V, 0.9V, 0.95V. It shows a gradual decay as frequency increases.

Figure 4 .

 4 Figure 4.10 shows the frequency dependence of base charge at different base emitter voltage.

4. 9 )Figure 4 . 11 :Figure 4 . 12 :

 9411412 Figure 4.11: Fitting of excess carrier concentration over time. The fitting function used is of the form, y=y0+a*sin (pi*(x-xc)/W). The parameter xc gives the time delay.

  phase shift of dynamic parameters. The device simulation results hence prove the significance of this type of networks.In the next part small signal transient results are shown at the neutral emitter side and the excess charge is calculated. The excess emitter charge also shows a comparable time delay for the device. The results for medium and high injection are shown below.

Figure 4 . 13 :

 413 Figure 4.13: Variation of excess minority carrier density (red line) over time. Blue graph corresponds to the applied sinusoidal bias variation over time. For (a) V BE =0.8V, f=1 GHz, (b) V BE =0.8V, f=200GHz (c) V BE =0.95V, f=200GHz

Figure 4 . 14 :Figure 4 .

 4144 Figure 4.14: Excess carrier concentration (cm -2 ) over frequency for V BE = 0.8V, 0.9V, 0.95V. It shows a gradual decay as frequency the increase.

Figure 4

 4 Figure 4.16.

Figure 4 . 16 :

 416 Figure 4.16: variation of excess phase over bias point.

  ) a diagram is presented to show the position of the SiGe spike inside mono emitter transistors[5].

Figure 4 . 17 :

 417 Figure 4.17: schematic cross section of different emitters and the minority carrier profile inside these emitter (a) emitter without SiGe spike, (b) emitter with SiGe spike.

Figure 4 . 18 :134Figure 4 . 19 :Figure 4 . 20 :

 418419420 Figure 4.18: Device structure of SiGe spike mono emitter and emitter without spike. Device structure is showing the device dimension and emitter, internal base and collector region. Arrow indicates the position of the vertical cut.

Figure 4 . 21 .Figure 4 . 21 :Figure 4 Figure 4 . 22 :

 4214214422 Figure 4.21 . At the position of the spike, there is a certain hill due to sudden increase of recombination.

Figure 4 .

 4 23 represents Gummel plot showing base and collector currents for both SiGe spike emitter and mono emitter transistors. The simulation region covers both the low and high injection region.

Figure 4 .Figure 4 . 23 :

 4423 Figure 4.23: Gummel plot showing Ic and Ib for both SiGe spike and without spike emitter configuration.

Figure 4 . 24 :

 424 Figure 4.24: Ft Ic plot for two different transistors with and without SiGe spike inside emitter for different V CE .

Figure 4 .

 4 Figure 4.25: magnitude of h 21 calculated from AC simulation keeping V BC =0V.

Figure 4 . 26 :

 426 Figure 4.26: Comparison between current gain calculated from AC and transient simulation in a SiGe spike mono emitter transistor keeping V BC =0V and V BE = 0.85V and 0.90V.

Figure 4 .Chapter 4 [Figure 4 Figure 4 . 28 :From Figure 4 .

 4444284 Figure 4.27: Comparison between device simulation results of h 21 Phase with and without SiGe spike inside emitter. High frequency (200GHz) simulation is performed with a constant V CE = 0.5V and with different V BE = 0.75V, 0.79V, 0.83V, 0.87V. Difference is clear at low frequency.

Figure 4 .

 4 Figure 4.29: Forward Gummel plot at V CB =0V with measurement and simulation.

Figure 4 .Figure 4 .

 44 Figure 4.29 shows Gummel plot for the SiGe spike device. Modeling of transfer current and forward base current is done with the HICUM L2.24 model. Although, the collector current is fitted well the results show that at the medium current range the base current is not fitted properly with the HICUM L2.24 model. In the HICUM model, base current is modeled as a summation of two diode currents and a recombination current. At the low injection range, parameters ireis and mrei model the base current. Over that region ibeis and mbei are the other two parameters to model base current and at high injection a recombination current term (tbhrec) due to the Ge drop in BC sunction in SiGe HBTs, helps modeling the base current.the modeling inaccuracy can be more prominent in the next graph, where ib/exp(V BE /V T ) is plotted,

Figure 4 .

 4 Figure 4.31: (a) fT vs V BE and (b) fT vs ic plot at different V CE . Measurement is in circle and modeling result in line.

Figure 4 . 32 :Figure 4 . 33 : 4 . 3 . 4 .

 432433434 Figure 4.32: Magnitude and Phase variation of h 21 parameter at V BE =0.75V, 0.79V, 0.83V, 0.87V, 0.91V at V CE =1V

Figure 4 .Figure 4 . 34 :

 4434 Figure 4.34 presents the base current modeled with the conventional HICUM L2V24 model and also with the modified model. As discussed above the sudden increase of the base current is modeled perfectly with the help of additional recombination current.

Figure 4 . 35 :

 435 Figure 4.35: Phase of h 21 parameter modeled with HICUM and extended HICUM up to high frequency (200GHz) at a constant V CE = 0.5V and with different V BE = 0.75V, 0.79V, 0.83V, 0.87V. Symbols represent the TCAD simulation, dash line HICUM L2V24 model and solid line extended model.

Chapter 4 [

 4 SIMULATION RESULTS]

Figure 4 . 36 :

 436 Figure 4.36: Phase of y 11 parameter modeled with HICUM and extended HICUM upto high frequency (500GHz) at a constant V CE = 0.5V and with different V BE = 0.75V, 0.79V, 0.83V, 0.87V.

  considered after calibrating the basic structure. High frequency Device simulation results are modeled with HICUM model. From the transient simulation results, time delays corresponding to the excess charge are calculated and it proves the importance of NQS excess phase circuit. Although, the frequency dependence of accumulated charge is not included inside compact models. Small signal simulations also indicate that the neutral emitter region, which is not so well accounted inside models, introduces a delay quite similar to the neutral base region. It provides a scope for further investigation in this subject. From the dynamic parameters simulation part, high frequency device simulation results are shown for the device.The modeling results with the excess phase network provide a considerable fitting up to very high frequency which in turns proves the significance of the HICUM excess phase circuit.The second part of this chapter considers a new kind of device which includes a SiGe spike inside emitter. The simulation results show a base current increase compared to the conventional transistor where there is no effect in the transition frequency. The reason behind the current increase is the sudden increase of carrier recombination inside emitter region.Effect of this excess charge is investigated and it is shown that the phase of dynamic parameters such as current gain or input admittance is also affected at low frequency.Modeling of device simulated data is performed with the HICUM model. The modeling results show that there requires an additional current source to model the base current at moderate injection. A recombination time constant, introduced as a parameter inside the HICUM model, relates the excess emitter charge. After that the modeling results show perfect modeling of base current and also the low frequency phase of dynamic parameters with the extended model.Nowadays, when the SiGe HBT has proved their capability to become the building blocks for RF applications, accurate compact modeling of SiGe devices at high frequency has become a crucial step for circuit engineers. Today a modern HBT device can gain a very high (around 500GHz) cut off frequency but the existing models can predict the transistor behavior only upto 60-80GHz range. At that point, the so called Non Quasi Static (NQS) effect is considered very important and requires careful investigation.In this whole work, one sole question is investigated, what NQS effect is and how one can model this effect. Starting from the beginning, semiconductor physics is analyzed for small signal operation. The whole physical process based on the solution of the transport and the continuity equation in the most important part of a bipolar device-the quasi static base region.The aim is to calculate the admittance parameters for low and high injection condition. Then, there are various approximated models available in the literature. In this work, calculations are made by hand to reach the previously published results. All the previously described models are discussed and compared with each other. This calculation helps to understand the physical mechanism behind the NQS effect and gives correct formulation to implement inside compact model. By now, calculation is restricted up to the base region, which is considered as the most important part of the bipolar device. Though, the most part of the total transit time comes from the base region, modern semiconductor shows a significant contribution coming from the emitter and base collector junction. Again, in the NQS analysis we are mainly interested on the excess phase shift of dynamic parameters. So, we conclude that the emitter or base collector junction transit time influence the phase behavior without much influencing the magnitude of dynamic parameters. In chapter 1 base collector junction and emitter region are separately investigated. This investigation helps to improve the compact modeling approach.TACD simulation is an integral part for the semiconductor device research. It not only helps to understand the basic physical models properly but also it can provide useful information to improve the device characteristics. To study the NQS behavior, we need to go up to a very high frequency (more than the cut off). Our most efficient measurement test bench can only provide electrical measurement up to 110GHz, which is far below the transition frequency of modern HBT transistors. At that point, TCAD simulation tool is used to simulate a calibrated HBT structure beyond its cut off frequency. Dynamic parameters are extracted and these simulated data then eventually treated as measurements at very high frequency. After that, a step by step parameter extraction is performed with a compact bipolar model, HICUM. All the necessary parameters are extracted before proceeding to the high frequency modeling. The modeling results show that the existing HICUM model provide a good fitting of dynamic parameters.There is another part of the work, to see how the excess phase network in HICUM model works in real measurements. The transistor used here is a modern SiGe HBT with a cut off frequency of 240 GHz. The modeling results show that HICUM provides an overall fitting to the high frequency current gain or the admittance parameters but it can be improved. Then we try to apply the physical understanding of high frequency behavior and proposed a new circuit. In this new excess phase circuit, phase shift due to the BC junction is taken into account. With the improved circuit, more accurate modeling is achieved at very high frequencies.Last of all, a new kind of device is considered. This HBT device is having a SiGe spike inside the neutral emitter region. The spike acts as a virtual contact and increases the base current significantly but does not affect the AC behavior of the transistor. This device is constructed with a modification of previously used HBT device. The TCAD device simulation results show an improvement in base current which lead to a high collector to emitter breakdown voltage. While comparing SiGe spike emitter HBT with the conventional one, input admittance and current gain show some ambiguity at low frequency. This is due to the excess charge formed inside the neutral emitter region. Similarly, in the HICUM modeling of SiGe spike transistor, we observe that another recombination term is necessary to model the excess base current. Including an ideal current source in the HICUM model circuit, modeling results show significant improvement in base current and also in the low frequency admittance parameters.

factors x/W B Low Injection High Injection Mag(Normalised carrier conc) Chapter 1 [NQS THEORY] 38

	Terminal currents for HI condition are constructed thereafter
	e i		2 qA D E nB	0 B n W	coth  fH	fH	1.38
	2 d n x 2 () dx 2 c i qA D 2 nB j nx ( ) 0 D   0 sinh fH E nB B fH n W   	1.36
	Considering the same boundary conditions used for LI condition i.e. at x=0, ñ=ñ 0 and at In the LI condition, small signal current gain was calculated with the help of terminal current
	equations. Similarly for HI condition common base current gain is calculated from equation x=W B , ñ=0, solution of equation 1.36 becomes, 1.38,
	0 n x n ()  1 cosh  	sinh( sinh( ) fH x fH u   	)	1.37 1.39
						fH
	Where		fH	D jW 2 B 	2	2	j 	B
								nB



At LI level excess carrier concentration depends vastly upon the base drift factor ( ). At HI level, as there is no influence of the drift field, carrier concentration is merely a straight line for a vast range of frequencies as shown in Figure

1

.8 Figure 1.8 : Magnitude of normalized carrier concentration at different injection level. Different curves in LI represent different drift

Table 1 .1: Comparison

 1 uses Thomas-Moll equation and exponential base doping to calculate different time constants at LI level of operation. Rinaldi[6] [14] clearly compared all the models by developing a closed form solution of transport equations in arbitrary level of injection. The basic differences between the theoretical NQS models are shown in Table1.1. between different NQS models available in the literature.

	Model Name	Model Type	Doping	Injection	Comments
	QS model [6]	Charge Control	Constant	LI	Initial model not
		(CC)			including time delay.
	Winkel [2]	CC	Arbitrary	LI	Introduces two delay
					times
	Klose Wieder [16] TICCR (PCB)	Exp	LI	Calculated CPF
	Hamel [4]	GTCCR (PCB)	Exp	LI	Calculate CPF α11
	Wu [17]	CC	Arbitrary	LI	Calculated	base
					transit time and delay
					times.
	Rinaldi [6] [14]	Small signal	Exp	LI+HI	Compare all the
					models to provide a
					close form solution.

2.3.1 Transport equation  Poisson and continuity equation

  Device simulation is important part in TCAD simulation because it gives insights into complex phenomena that cannot be obtained separately through experimentation. Many of these phenomena involved in semiconductor can be described by partial differential equations (PDEs): as a result, much of the TCAD software is based on the solution of PDEs[2] [6]. To solve the PDEs approximate methods like finite element or the finite difference methods are used by the simulator.The main governing models of semiconductor devices are discussed below, Poisson equation is one of the three governing equations in the semiconductor device physics[7]. In mathematics Poisson equation, a partial differential equation with vast application in electrostatics, theoretical physics and mechanical engineering, is named after the French mathematician Poisson[8]. Poisson equations relates the electric field and charge density  as, Electric field vector can be represented as gradient of potential function. As vector calculus predicts that curl of E is zero, Poisson equation can be written as a function of electrostatic

			Input		Process Steps	Output
			Ambient
	potential, .( )	2	Temperature Pressure Time Rate Dose Energy		 Diffusion  Oxidation  Ion Implantation  Deposition  Etching  Masking	Structure Mesh Layer thickness Doping profile Function depth
			Thickness	. . .	 Epitaxy  Stress	. . .
	. Assuming m m m E   				2.1

is an advanced, complete, and highly flexible multi-dimensional process simulator for developing and optimizing silicon and compound semiconductor process technologies. It offers unique predictive capabilities for modern silicon and non-silicon technologies, which is calibrated to a wide range of the latest experimental data using proven calibration methodology. It is powerful numerical algorithms simulate process steps like implantation, diffusion and dopant activation, etching, deposition, oxidation, and epitaxial growth in different semiconductor material with efficient meshing for robust and stable calculation. The fabrication of integrated circuit devices requires a series of processing steps called a process flow. A process flow is simulated by issuing a sequence of commands that corresponds to the individual process steps. In addition to a several control commands allows selecting physical models and parameters, grid strategies, and graphical output preferences. Also process conditions like the ambient chemical composition, temperature, pressure, etc. during individual fabrication steps can be controlled. The final output is a 2D or 3D device structure which can be used for device simulation. The major processing steps involved in the manufacturing of integrated circuit devices are shown in Figure

2

.2 . Each of these steps containing numerous possible variations in process controls, and may take several weeks to complete

[5]

Figure

2

.2: Major process steps involved in the manufacturing of integrated circuit devices

[2]

2.3 Device simulation (physical model)

E is the electric field in a medium where the permittivity is m  ,

  8 represent current magnitude as well as the direction which indicates a steady flow of carrier in the direction of concentration gradient. Overall, drift diffusion model is a simple model which assumes full impurity ionization and non degenerate statistics. It cannot encounter the non local effects since the electron gas is assumed to be in thermal equilibrium with the lattice temperature[7]. Drift diffusion model can be extended to thermodynamic

model just by including lattice heat flow equation (temperature gradient) inside the formulation. Assuming a lattice is in thermal equilibrium with the carrier at temperature T, the thermodynamic model reads,

[7]

Table 3 .

 3

1: VerilogA code for excess phase circuit of transfer current and excess charge in HICUM L2 V24 model.

Table 4 .

 4 1) indicates the basic schematic of the overall process.

	STEP	Input	output
	Measurement	Real transistor on wafer	Electrical measurement data
	Device Simulation	Device structure	Device simulated data
		Device structure	Electrical measurement
	Calibration		≈
			Device simulation
		Device simulation	 Model parameter file
	Compact model		 NQS modeling result

Table 4 .

 4

	1
	2
	3
	1: A step by step procedure to obtain NQS related compact modeling parameters from
	device simulation results.

. SiGe spike mono-emitter

  In this part, simulation and modeling results for an npn SiGe spike mono emitter are shown for both DC and high frequency operation. First, the results obtained for a SiGe spike emitter is compared with the conventional HBT. Then parameter extraction and compact modeling with the help of HICUM is performed in the new device. The modeling approach introduces a new recombination time constant inside the existing HICUML2.24 model. Modeling results indicate a good agreement not only in DC characteristics but also in dynamic behavior.

						1.40
	4.3							ωτ_0.7
								ωτ_0.8
								ωτ_0.9
								ωτ_0.95
						ωτ
						1.00E+09		1.00E+10	1.00E+11 Frequency(Hz)
		1.5
			ωτ_50GHz
		1.3	ωτ_100GHz ωτ_150GHz
		1.1	ωτ_200GHz
	ωτ	0.9
		0.7
		0.5
		0.3
		0.65	0.7	0.75	0.8	0.85	0.9	0.95	1
					V BE (V)

00'

LIST OF PUBLICATIONS International Conferences

Chapter 2

[TCAD SIMULATION AND DEVICE PHYSICS]

Chapter 3

[NQS IMPLEMENTATION INSIDE COMPACT MODEL]

General conclusion [GENERAL CONCLUSION]

Abstract

Modern high speed (RF) transistors encounter certain delay while operated at high frequency or under fast transient condition. This effect is named as Non Quasi Static (NQS) effect. In the current work, NQS effect is analyzed in a concise manner so that it can be readily implemented in a compact model using the VerilogA description language. The basic physics behind this effect is investigated in small signal domain and the results are compared with the published work. In popular bipolar model HICUM lateral and vertical NQS are examined separately and uses the same model for both transient and AC operation which requires an additional minimum phase type sub circuit. Compact modeling with HICUM model is performed in both measurement and device simulated data. At last, an improved excess phase circuit is proposed to model the NQS effect.

Key words

Computer Aided Design (CAD), Bipolar transistor, SiGe HBT, Compact modeling, HICUM model, Parameters extraction.

Résumé

Les transistors rapides actuels présentent un retard lorsqu'ils fonctionnent à très hautes fréquences ou en régime transitoire rapide. Cet effet est appelé effet non quasi-statique (NQS). Dans cette thèse, l'effet NQS est analysé de manière concise de façon à être directement implanté dans les modèles de composant pour les bibliothèques de circuit en utilisant le langage standard VerilogA. Les mécanismes physiques à la base de l'effet NQS sont évalués dans le domaine de fonctionnement petit signal et les résultats sont comparés aux travaux déjà publiés. S'agissant du modèle standard bipolaire HICUM, les effets NQS latéraux et verticaux sont examinés séparément à partir du même modèle, en régime de fonctionnement transitoire et fréquentiel grâce à un sous-circuit dédié au calcul de la phase du signal. A partir de ce sous-circuit, la modélisation compacte avec HICUM est comparée aux données issues de mesures et issues de simulation amont. Enfin, un nouveau sous-circuit calculant l'excès de phase est proposé pour prendre mieux en compte les effets non quasistatiques dans les transistors bipolaires.

Mots-clefs

Conception assistée par ordinateur (CAO), Transistor bipolaire, TBH SiGe, Modélisation compacte, Modèle HICUM, Extraction des paramètres.