RWR-509-FG-91274-FG Nat. Lab. Unclassified Report Nr. 013/91 F.J. Groeneveldt
Automated Design Optimization Optimization of (Heterojunction)Bipolar Transistors
Concerns Coach Period of work
: HTS Stage Report
: G.J.L.Ouwerling : February 1991 - July 1991
0Philips Electronics 1991 All rights ure reserved. Reproduction in whole or in part is prohibited without the written consent of the copyright owner.
AUTOMATED DESIGN OPTIMIZATION
OPTIMIZATION OF (HETEROJUNCTION) BIPOLAR TRANSISTORS
In opdracht van : Het Philips' Natuurkundig Laboratorium F. J. Groeneveldt PhilipsNatuurkundigLaboratorium
Afdeling: Advanced Semiconductor Materials Science Bedrijfmentor: Dr. Ir. G. J. L. Ouwerling
Eindhoven, juli 1991
U R 013/91
At this moment it is possible to calculate the electrical characteristics of a transistor with an arbitrary doping profile with for instance the program Hetrap (heterojunction bipolar transistor analyse program). A next step for computer aided transistor design is a tool to construct the dope profile which belongs to the electrical characteristics that are desired. This is called inverse modelling. This report contains the first experiments to asses the possibilities of automatic transistor design, by coupling an optimization driver (Profile) with the one-dimensional (heterojunction) transistor simulator. Before optimizing the design of a (heterojunction) bipolar transistor the operation of a (heterojunction) transistor is summarily described. The optimization driver that is used utilizes an optimization algorithm that minimizes the least squares errors between the specified and simulated data. The optimization criteria that are defined are the current gain, the cutoff frequency, the base resistance and the avoidance of collector breakdown. First some experiments have performed for a bipolar transistor, where the parameters that could varied by the optimization driver were the Gummelnumber (Nb), the base width (Wb), the collector width (Wc) and the collector dope (Nc). Afterwards, some experiments with a heterojunction bipolar transistor are done, for a device with a Si,-,Ge, base. The parameters of the heterojunction transistor are the bipolar transistor parameters in addition with two parameters (Rxge,Wxg), that determine the composition profile of the transistor base. Although break throughs in the design of bipolar transistors can not be reported, we think that the first experiments show that automatic design optimization is certainly feasible. Further work can be sought in the areas of optimization parameters refinements and in the choice of more suitable optimization algorithm.
NAT.LAB.Unclassified Report 013/91
F.J.Groeneveldt
3
U R 013/91
CONTENTS 3
CHAPTER 1
................. INTRODUCTION . . . . . . . . . . . . . . .
CHAPTER 2
TRANSISTOR PHYSICS
............
8
5 2.1
INTRODUCTION
........... . . . .
8
.... . . . .
8
..........
9
........
11
.............
12
ABSTRACT
5 2.1.2
SEMICONDUCTOR MATERIALS
5 2.1.2
ELECTRONS A N D HOLES
5 2.2
CARRIER TRANSPORT PHENOMENA
5 2.2.1
CARRIER DRIFT
5 2.2.2
CARRIER DIFFUSION
6
. . . . . . . . . . . 12
..........
5 2.2.2.1
EINSTEIN RELATION
5 2.2.2.2
CURRENT DENSITY EQUATIONS . . . . . . 14
....... ...............
14
5 2.3
GENERATION AND RECOMBINATION
15
5 2.4
P-N JUNCTION
16
5 2.4.1
THERMAL EQUILIBRIUM CONDITION
5 2.4.2
DEPLETION REGION
5 2.4.3
IDEAL CHARACTERISTICS
5 2.5
. . . . . 17
....... .....
18
.........
20
..... . . . . .
22
.........
22
..............
24
. . .....
25
.........
27
..........
31
........
32
............
37
...............
37
THE BIPOLAR TRANSISTOR
5 2.5.1
THE TRANSISTOR ACTION
5 2.5.2
CURRENT GAIN
5 2.5.3
IDEAL TRANSISTOR CURRENTS
5 2.5.4
STATIC CHARACTERISTICS
5 2.5.5
THE CUTOFF FREQUENCY
5 2.6
HETEROJUNCTION TRANSISTORS
CHAPTER 3
INVERSE SIMULATION
5 3.1
INTRODUCTION
4
U R 0131’91
5 3.2 5 3.4 5 3.4.1 5 3.4.2
THE PRINCIPLE OF AND MOTIVATION FOR INVERSE MODELLING ....
......
38
THE OPTIMIZATION METHOD . . . . . . . . . . 40 AN INTRODUCTION TO NONLINEAR OPTIMIZATION METHODS . . . .
......
41
THE LEVENBERG-MARQUARDT AND MDLS METHODS .......
......
42
..........
42
5 3.4.3
CONVERGENCE CRITERIA
5 3.4.4
GLOBAL PARAMETER SCALING AND PARAMETER CONSTRAINTS . . .
......
43
STATISTICAL ANALYSIS OF THE PARAMETER SOLUTION . . . . .
......
43
5 3.4.5 5 3.5
THE MDLS OPTIMIZATION DRIVER IN PROFILE . . 45
CHAPTER 4
(HETEROJUNCTION) BIPOLAR TRANSISTOR OPTIMIZATION PROCEDURE . . . . . . . . . .
46
5 4.1
DEFINITION OF THE OPTIMIZATION CRITERIA . . 46
5 4.2
DEFINITION OF THE CHANGEABLE PARAMETERS
5 4.2.1 5 4.2.2
DEFINITION OF THE BIPOLAR TRANSISTOR PARAMETERS . .
..
50
.......
50
DEFINITION OF THE HETEROJUNCTION TRANSISTOR PARAMETERS . . . . .
....
51
5 4.3
SOFTWARE IMPLEMENTATION OF THE INVERSE MODELLING SCHEME . . . . . . . . . . . . . 52
5 4.4
THE (HETEROJUNCTION) BIPOLAR TRANSISTOR OPTIMIZATION PROGRAM . . . . . . . . . . . 55
CHAPTER 5
OPTIMIZATION EXAMPLES
5 5.1
BIPOLAR TRANSISTOR OPTIMIZATION . . . . . . 64
5 5.2
HETEROJUNTION BIPOLAR TRANSISTOR OPTIMIZATION . . . . . . . . . .
CHAPTER 6
. . . . . . . . . .
.....
64
68
CONCLUSIONS . . . . . . . . . . . . . . . . 71 REFERENCES
. . . . . . . . . . . . . . . .
5
74
U R 013/91
CHAPTER1 INTRODUCTION
The performance of a bipolar transistor is determined by the thickness, doping profile, and various other physical properties of the emitter, base and collector regions. In heterojunction transistors, a further factor is the material composition. The optimal design of such a transistor, given various technological constraints, is a complicatedmatter. Amethod for automated design optimization may be a valuable tool for the device engineer. At this moment it is possible to calculate the electrical characteristics of a transistor with an arbitrary doping profile with the program HETRAP (Heterojunction Transistor Analyses Program). This program solves the semiconductor equations in one space dimension. This is called forward modelling. In the case of forward modelling, the external device characteristics are calculated from the material properties using a description and electrical boundary conditions. A next step for computer aided transistor design is a tool to construct the dope profile which belongs to the electrical characteristics we want. This is called inverse modelling. In the case of inverse modelling physical parameters describing the material properties are constructed from externally measured or desired device characteristics, using the same device model that is used for the forward simulations. The possibilities for automated design optimization can be studied by coupling an optimization driver with the one-dimensional (heterojunction) transistor simulator. We have done experiments to assess the possibilities of automatic transistor design optimization. Here, emphasis was given to the definition of optimization criteria. We started with the two most important optimization criteria of a (heterojunction) bipolar transistor namely the current gain and the cuttoff frequency. Further important optimization criteria that we researched were the intrinsic base resistance and the avoidence of collector breakdown. An other point that is of relevance for the automated design optimization is to choose the correct parametrization of the optimization driver, these must choose precisely and should be dependent of the optimization criteria. After some experiments we found the most important parameters for the optimization program. We have used the software tool PROFILE for the automated 6
optimization because this program is especially designed for inverse modelling. The program includes an optimization driver, that uses an optimization algorithm, which minimizes the difference between measured or desired and simulated data (the error sum) by adapting a parameter vector. To make the relative contribution of all measurements to the error sum equal a weighting factor can be defined. Weighting also allows for the different kinds of optimization criteria in one error sum. First the transistor physics are summarily described, afterwards the subject inverse modelling is explained. When the inverse model was explained the (heterojunction) bipolar transistor analyse program was implemented in the inverse scheme. With the made (heterojuction) bipolar optimization program are some experiments done, some first results with a limited number of design parameters are presented.
7
CHAPTER 2
TRANSISTOR PHYSICS
Before we can optimize ,,,e design of a (heterojunction) bipolar transistor we must understand the operation of the transistor. In this chapter the physics of the bipolar transistor are described. Special attention is given to those properties of the bipolar transistor that are important for the definition of optimization criteria. The information provided in this chapter was summarized from the more extensive desciptions of references [1,2,3,4].
5 2.1
5 2.1.1
INTRODUCTION
SEMICONDUCTOR MATERIALS
Solid-state materials can be grouped into three classes, insulators, semiconductors, and conductors. Insulators such as fused quartz and glass have very low conductivities, in the order of to S/cm. Conductors such as aluminum and silver have high conductivities, typically from lo4 to lo8 S/cm. Semiconductors have conductivities between those of insulators and those of conductors. The conductivity of a semiconductor is generally sensitive to temperature, illumination, magnetic field, and minute amounts of impurity atoms. This sensitivity in conductivity makes the semiconductor one of the most important materials for electronic applications.
1
EMPTY CONDUCTiON B A N D CONDUCTION BAND 0..
F
PARTIALLY FILLED CONDUCTION BAND
VALENCE BAND
Fig.1. Schematic energy band representation of (a) an insulator, (b) a semiconductor and (c) conductors
a
5 2.1.2
ELECTRONS AND HOLES.
At finite temperatures in semiconductor materials there will be a number of electrons excited from the valence band into the conduction band. For covalently bounded semiconductors like Si or Ge this means that some of the covalent bonds between neighboring atoms formed bij valence electrons are broken up, leaving a hole in the valence band and delivering a free electron to the conduction band. When the electrons and holes are in internal equilibrium at a temperature T their distribution functions f,(E) and f,(E), giving the occupation probability of an energy level E, are described by the Fermi-Dirac distribution function or by the approximation through the Boltzman distribution for low occupation probabilities: l+exp-
(-
exp
kT
~krm)
The Boltzmann approximations are only valid when the energy level E is several kT above or below the quasi-Fermi levels E,, or E,, for electrons or holes, respectively. These quasi-Fermi levels describe a situation in which both the electrons and holes are in internal thermal equilibrium in their bands but not in equilibrium with respect to one another. Such a description of a nonequilibrium state is appropriate when the relaxation time for electrons and holes to reach internal equilibrium with the lattice is short compared with the electronhole recombination time. In overall thermal equilibrium we of course have only one Fermi level I E,=E,,=E FP' The total number of electrons in the conduction band is obtained by integration of the density of states multiplied by the distribution function f, over the entire conduction band. As the function f,(E) is an exponentially decreasing function of energy above the Fermi level, the only important contributions to this integral are from the region up to several kT above the bottom of the conduction band. With the appropriate density of states for this region we can describe the electron concentration n in the form: 9
The approximate form is the Boltzmann approximation, valid when E,, is more than several kT below the conduction-band edge E,. The factor N, is called the effective density of states of the conduction band. In a similar way the hole concentration p is described:
with the factor N,, the effective density of states for the valence band. In thermal equilibrium (E,,=EFp=EF)the pn product is found to be independent of the Fermi-level position when the Boltzmann approximation is valid:
pn
-
-
Ne N, exp (-E,/kT) ni2
The intrinsic carrier concentration ni is the number of electrons, equal to the number of holes for a pure semiconductor; each electron in the conduction band orginates from the valence band leaving a hole behind. Doping a semiconductor with impurities can give extra electrons or holes but in thermal equilibrium the produkt pn remains constant at a constant temperature. For an intrinsic semiconductor in equilibrium the Fermi-level position is very close to the middle of the energy gap, as can be calculated with the equation: (6)
In nonequilibrium situations the pn product can be varied over many orders of magnitude by making the quasi-Fermi levels different. In such a case
pn
-
ni2 exp
EFn-EFp
kT
This can be realized, for instance, in the transition region of a p-n junction by the application of an external voltage across the junction, where the values of E,, and EFp are mainly determined by the voltage supplied to the n-region and p-region.
10
5 2.2 CARRIER TRANSPORT PHENOMENA 5 2.2.1
CARRIER DRIFT
Consider an n-type semiconductor sample with uniform donor concentration in thermal equilibrium. The conduction electrons in the semiconductor conduction band are essentially free particles. Under thermal equilibrium, the average thermal energy of a conduction electron can be obtained from the theorem for equipartition of energy: 1/2 kT units of energy per degree of freedom. The electrons in a semiconductor have three degrees of freedom; they can move about in three-dimensional space. Therefore, the average kinetic energy of the electrons is given by 1 m, 2
3 kT 2
2
Vth
where m, is the effective mass of electrons and vthis the average thermal velocity. When a small electric field E is applied to the semiconductor sample, each electron will experience a force -qE from the field and will be accelerated along the field during the time between collisions. Therfore, an additional velocity component will be superimposed upon the thermal motion of electrons. This additional component is called the drift velocity. We can obtain the velocity speed v, by equating the momentum (force*time) applied to an electron during the free flight between collisions to the momentum gained by the electron in the same period. The momentum applied to an electron is given by -qET, (7, is called the mean free time which is the average time between collisions) and the momentum gained is m,v,. We have -q E 7,
- mn vn
(9)
or
The last equation states that the electron drift velocity is proportional to the applied energy. The proportionally factor depends on the mean free time and the effective mass. The proportionally factor is called the electron mobility p, in units of cm2/Vs, or 11
Thus
Mobility is an important parameter for carrier transport because it describes how strongly the motion of an electron is influenced by an applied electric field. A similar expression can be written for holes in the valence band: vp
-
crp
E
where vp is the hole drift velocity and pp is the hole mobility. The negative sign is removed because the holes drift in the same direction as the electric field. The transport of carriers under the influence of an applied electric field produces a current called the drift current. The current density J, flowing in a sample can be found by summing the product of charge (-9) on each electron times the electron's velocity over all electrons per unit volume n:
where I, is the electron current which flows through an area A. A simular argument applies to the holes. By taking the charge of the hole to be positive, we have
The total current flowing in a semiconductor sample due to the applied field E can be written as the sum of the electron and hole current components:
5 2.2.2
CARRIER DIFFUSION
In the preceding section we considered the drift current. Another important current component can exist if there is a spatial variation of carrier concentration in the semiconductor material, that is, the carriers tend to move from a region of high 12
concentration to a region of low concentration. This current component is called diffusion current. To understand the diffusion process we assume an electron density that varies in the xdirection, as shown in Figure 2. The semiconductor is at uniform temperature, so that the average thermal energy of electrons does not vary with x, only the density n(x) varies. We shall consider the number of electrons crossing the plane at x=O per unit time and per unit area. Because of finite temperature, the electrons have random thermal motions with a thermal velocity vthand a mean free . The electrons at x= -1, one mean free path away pad 1 ( l=vthtc) on the left side, have equal chances of moving left or right and in a mean free time tc one half of them will move across the plane x=o.
CURRENT
ELECTRONS
n(o I
I
-I
o
I
DISTANCE I
Fig.2. Eiectron concentration versus distance; 1 is the mean free path. The directions of electron and current flows are indicated by arrows.
The net rate of carrier flow from left to right is
=
where Dn vth 1 is called the diffusivity. Because each electron carries a charge -q, the carrier flow gives rise to a current
13
The diffusion current is proportional to the spatial derivative of the electron density. Diffusion current results from the random thermal motion of carriers in a concentration gradient. 2020201 EINSTEIN RELATION We can write
and using the relationship l=v,,t, yields
Therefore, D,
- (5) p n
The last equation(21) is known as the Einstein relation. It relates the two important constants (diffusivity and mobility) that characterize carrier transport by diffusion and by drift in a semiconductor. The Einstein relation also applies between D, and p,.
§ 2020202 CURRENT DENSITY EUUATIONS When an electric field is present in addition to a concentration gradient, current will flow according to:
and
When the Boltmann approximation can be used, namely when E,, sufficiently far below E, then:
E,
-
Ei+kT In-n ni
and Em = Ei-kT ln-P ni
14
is
The total current density at any point is the sum of the drift and diffusion components can we describe with the above equations, the einstein relation, $=-Ei/q and E=-d$/dx.
where E is the electric field in the x direction. A similar expression can be obtained for the hole current:
We use the negative sum in the equation of J, because, for a positive hole gradient, the holes will diffuse in the negative xdirection. The total conduction current density is given by
Si 2.3 GENERATION AND RECOMBINATION Whenever the thermal-equilibrium condition is disturbed (pn c> ni2), processes exist to restore the system to equilibrium ( pn=ni2). In the case of the injection of excess carriers, the mechanism that restores equilibrium is recombination of the injected minority carriers with the majority carriers. Electrons and holes can be generated in a semiconductor in excess of their thermal-equilibrium concentrations by the incidence of energetic particles (ions,electrons,photons) with an energy above the band gap, by a forward biased p-n junction in the semiconductor. There are two main generation/recombination mechanism : a : Direct recombination. Consider a semiconductor in equilibrium. The continuous thermal vibration of lattice atoms causes some bonds between neighboring atoms to be broken. When a bond is broken, an electron-hole pair is generated. In terms of the band diagram, the thermal energy enables a valence electron to make an upward transition to the conduction band leaving a hole in the conduction band. This process is called carrier generation (G). When an electron makes a transition downwards from the conduction band to the valence band, an electron-hole pair is annihilated.
15
This reverse process is cailed recombination described by:
(R), Which can
where no and po represent the electron and hole densities in an semiconductor at thermpl equilibrium and rd the generation/recombination velocity. b: Shockley-Read-Hall (SRH). An electron makes an indirect transition from the conduction band to the valence band via localized energy states in the forbidden energy gap (so called traps with energy level Et).
Where N, is the concentration of centers in the semiconductor, the quantity o describes the effectiveness of the center to capture an electron and is measure of how close the electron has to come to the center to be captured and is the thermal velocity of the carriers. RecombinationVth /generation processes under illumination is shown in Figure.3.
Fig.3. Reconbination/generation process under i ttunination.
5 2.4 P-N JUNCTION In the preceding sections we have considered ,he carrier concentrations and transport phenomena in homogeneous semiconductor material. In the next paragraphs we describe the p-n junction.
16
5 2.4.1
THERMAL EOUILIBRIUM CONDITION
In Figure 4a we see two regions of p- and n-type semiconductor materials that are uniformly doped and physically separated before the junction is formed. A p-n junction is formed when these two regions are joined. Holes from the p-side diffuse into the n-side, and electrons from the nside diffuse into the p-side. As holes continue to leave the pside, some of the negative acceptor ions near the junction are left uncompensated, since the acceptors are fixed in the semiconductor lattice while the holes are mobile. --&
Fig.4. (a) Uniformly doped p-type and n-type semiconductors before junction is formed. (b) The electric field in the depletion region and the energy band diagram of a p-n junction in thermal equilibriun.
Similary, some of the positive donor ions near the junction are left uncompensated as the electrons leave the n-side. Consequently, a negative space charge forms the p-side of the junction and a positive charge forms the n-side. This space charge region or depletion region creates an electric field that is directed from the positive charge to the negative charge, as indicated in the upper illustration of figure 4b. The electric field is in the direction opposite to the diffusion current for each type of charge carrier. The lower illustration of figure 4b shows that the hole diffusion currents flows from the left to the right, while the hole drift current due to the electric field flows from right to left. The electron diffusion current also flows from left to right, while the electron drift current flows in the opposite direction. At thermal equilibrium the net current flow across the junction is zero. The unique space charge distribution and the electrostatic potential 9 are given by Poisson's equation:
17
For a p-type neutral region, we assume N,=O and p>>n. The electrostatic potentional of the p-type neutral region, designated as )I in Fig.4b can calculated with:
JrP =
1 --(Ei-EF) 4
kT = --In4
NA
ni
Similary, we obtain the electrostatic potentional of the n-type neutral region:
J'n
1 kT ND = --(Ei-,!?,) = -1n-
4
4
ni
The total electrostatic potential difference between the p-side and the n-side neutral regions at thermal equilibrium is called the built-in potential Vbi: Vbi = $ -I# n
kT = -1n4
p
NAND
n!
(33)
Moving from a neutral region towards the junction, we encounter a narrow transition region, shown in Fig.4~. Here the space charge of impurity ions is partially compensated by the mobile carriers. Beyound the transition region we enter the completely depleted region where the mobile carriers densities are almost zero. This is called the depletion region. For typical p-n junctions in silicon, the width of each transition region is small compared to the width of the depletion region. Therefore, we can neglect the transition region and represent the depletion by the rectangular distribution shown in Fig.5d. This is called the abrupt depletion approximation.
5 204.2
DEPLETION REGION
In this section we consider an abrupt junction, as shown in the lower illustration Fig.5. The overall space charge neutrality of the semiconductor requires thet the total negative space charge per unit area in the p-side must precisely equal the positive space charge per unit area in the n-side: NA xP
= ND xn
18
(34)
The total potential variation overthe depletion region, the builtin potential Vbi, can be calculated by:
Combining the last two equations, the total width of the depletion region can calculated with:
NEUTRAL pIC)
’
CHARGE OENSIPI OUE TO UNCOMPENSATED IMPURITY IONS
I
l
id)
REGION
Fig.5. (a) A p-n junction uith aprupt doping changes at the mettallurgical junction. (b) Energy band diagram of an abrupt juiction at thermal equilibriun. (c) Space charge distribution. (d) Rectangular approximation of the space charge distribution.
When the impurity concentration of the n-side of an abrupt junction is much higher than that of the p-side, the depletion layer width of the n-side is much smaller than the p-side and the expression for W can be simplified to:
19
5 2 - 4 - 3 IDEAL CHARACTERISTICS We shall now derive the ideal current characteristics based on a abrupt depletion layer. At thermal equilibrium, the majorority carrier density is essentially equal to the doping concentration. Hence, nno and nP are the equilibrium electron densities in the nand p-sides. Similary, , p and , p are the equilibrium hole densities in the n- and p-sides. The expression for the built-in potential can be written as
where the mass action law pPnno=ni2has been used. Rearranging the equation gives
n,,
9
-
npoeqvbf/kT and ppo pnoeQVb,/kT
(39)
When a forward bias is applied, the electrostatic potential difference is reduced to Vbi V,; but when a reversed bias is applied, the electostatic potential difference is reduced to Vbi + V,. Thus, equation 39 is modified to
-
eq(vb,-v)/kT
Nn
(40)
p
where nn and np are the nonequilibrium electron densities at the boundaries of the depletion region in the n- and p-sides, with V positive for bias and negative for reverse bias. For low-injection condition the injected minority carrier density is much smaller than the majority carrier density; therefore nn N nno. Substituting this condition and equation 39 into equation 4 0 , the following equation yields the electron density at the boundary of the depletion region on the p-side (x=-xp)and for the n-side
(x=x,) :
-
np-npo npoi eqVIkT-1) and p,-pno
-
pno(eqVlkT-i)
(41)
Figure 6 show the carrier concentration in a p-n junction under foxward-bias and reverse-bias conditions. Under idealized assumptions, no current is generated within the depletion region; all currents come from the neutral regions. In the neutral n-region there is no electric field, thus the staedystate continuity equation reduces to
20
-19 in
ia1
I
'1.7
Ibl
Fig.6. Deptetion region and carrier distrikition. (a) Forward bias. (b) Reverse bias.
The solution of equation 42 with the boundary equation mentioned gives before and p,,(x=a~)=p~ p,-pno
= p no (e q V i H - 1 ) e - (x-xn)
/Lp
(43)
where Lp, which is equal to JDprp, is the diffusion length of holes (minority carriers) in the n-region. At x=x,:
Similary, we obtain for the neutral region
where 4, is the diffusion length of electrons. The electron and hole currents are shown in figure 7. The total current is constant throughout the device and is the sum of eauation 44 and equation 45 and is called the ideal diode equation:
21
1
(0)
í b)
Fig.7. Electron end hole currents. (a) Forward bies. (b) Reverse bias.
5 2.5 THE BIPOLAR TRANSISTOR Bipolar devices are semiconductor devices in which both electrons and holes participate in the production process. In the following section we consider the transistor action and some physics phenomena.
5 2.5.1
THE TRANSISTOR ACTION
Figure la shows the idealized p-n-p transistor in thermal equilibrium, that is where all three leads are connected together or all are grounded.
N~-N;
X
C I
II
,
II I
I
v
X
I
/
i--
Fig.1. (a) A p-n-ptransistor uith all leads grounded. (b) Doping profile of a transistor uith abrupt impurity distributions. (c) Electric-fieldprofile. (d) Energy band diagram at thermal equi libriun
22
Figure lb shows the impurity densities in the three doped regions, where the emitter is more heavily doped than the collector, while the base dope is less than the emitter doping, but greater than the collector doping. Figure IC shows the corresponding electric-field profiles in the two depletion regions and figure Id shows the energy band diagram. At thermal-equilibrium there is no net current flow. Figure 2 illustrates the corresponding situations when the transistor in figure 1 is biased in the active mode. Figure 2 is a schematic of the transistor connected as an amplifier with the common-base configuration (the base lead is common to the input and output circuits). The emitter-base junction is forward-biased, holes are injected from the p+ emitter into the base, and electrons are injected from the n base into the emitter. Under the ideal-diode condition there is no generation-recombination current in the depletion region; these current components constitute the total emitter current.
EMITTER
BASE
COLLECTOR
N9-N;
X
r
I
I I I I
I I I I I
I
Fig.2. (a) The transistor under the active mode of operation. (b) doping profiles and the depletion regions under biasing conditions. (c) electric-field-profile.(d) Energy band diagram.
The collector-base junction is reverse-biased, and a small reverse saturation current will flow across the junction. If the base width is sufficiently narrow, the holes injected from the emitter can diffuse through the base to reach the base-collector depletion edge and then float up into the collector. This transport mechanism gives rise to the terminology of emitter, which emits or inject 23
carriers, and of collector, which collects these injected carriers from a nearby junction. If most of the injected holes can reach the collector without recombining with electrons in the base region then the collector current will be very close to the emitter current. Therefore, carriers injected from a nearby emitter junction can result in a large current flow in a reverse-biased collector junction.
§ 2.5.2
CURRENT GAIN.
Figure 3 shows the various current components in an ideal p-n-p transistor biased in the active mode. The holes injected from the emitter constitute the current IEP,which is the largest current component in a well-designed transistor. Most of the injected holes will reach the collector junction and give rise to the current Iep. There are three base current components, which are labeled IE", Is,, and I,,. I,, corresponds to the current arising from electrons being injected from the base to the emitter. However, a large value of I,, is not desirable, it can be minimized by using heavier emitter doping or a heterojunction. This will be shown in a later section.
EMITTER(P*,
HOLE CURRENT bMHOLE FLOW
BASE in)
COLLECTOR (PI
---
+ELECTRON
CURRENT
ELECTRON FLOW
Fig.3. Various current conponents in a p-n-p transistor w d e r the active mode of operation. The electron flou is in the o m s i t e direction to the electron current.
I,, corresponds to electrons that must be supplied by the base to replace electrons recombined with the injected holes (I,,= IEp-Icp). I,, corresponds to thermally generated electrons that are near the collector-base junction edge and drift from the collector to the base. As indicated in the figure, the direction of the electron 24
current is opposite to the direction of the electron flow. We can now express the thermal current in terms of the various current components :
I,
-
-
I,
IEP + I ,
I,
I,
-
IE - I, Im
+ +
I,
(471
(IEp - I@) -I,
An important parameter in the characterization of bipolair transistors is the common-base current gain a,,. This quantity is defined as
After substitution follows:
The first term on the right-handed side is called the emitter efficiency y, which measures the injected hole current compared to the total emitter current:
Y -
The second term is called the base transport factor aT, which is the ratio of the hole current reaching the collector to the hole current injected from the emitter:
Therefore, ao=yaT. For a well-designed transistor, both y and aT approach unity, and a,, is very close to 1.
5 2.5.3
IDEAL TRANSISTOR CURRENTS
In active mode the minority carrier distribution in the neutral base region can be described by:
25
where D, and rp are the diffusion constant and the lifetime of minority carriers. The boundary conditions f o r t h e active mode are:
P,(O)
-
(531
9v,/m
and
Figure 4 shown the minority carrier distribution in the base for different values of W/Lp. If W/Lp cfi-fik>
+
I (@-@k> 2
TH(pk>
-
(4)
with @ the gradient according to gj aXmax, the algorithm cannot find further improvement although effectively the method of steepest descent is used. The absolute value of the normalized and weighted errorsum.
42
- The relative change in the error.
If this is small, the algorithm has probably found the bottom of the valley in parameter space. The largest absolute change in the gradient the largest relative change in a parameter value If one of these criteria is reached the iterations are stopped.
-
5 3.4.4
GLOBAL PARAMETER SCALING AND PARAMETER CONSTRAINTS
To avoid the assignment of values to the parameters that in reality cannot be obtained, and possibly also to restrict the optimization process to a region of the parameter space where only one minimum is presented, it is necessary to constrain the parameters between bounds. On the other hand, to avoid roundoff error in the solution of the linear system, it is desirable to have the parameters approximately in the same value range through some form of global scaling. Both requirements are fulfilled by using Box's [12] triogonometric parameter transformation, in which a new, internal set of parameters p" is defined by
where pi is constrained within [pimx,pimx] and pi is unconstrained, but may be reduced to [O,a/2].
5 3.4.5
STATISTICAL ANALYSIS OF THE PARAMETER SOLUTION
The reduction of the error sum to a stable value is a necessary but not sufficient condition for a parameter extraction or 'inverse modelling' procedure to be succesful. In the first place, more than one minimum may be present in the parameter space: in the second place, a combination of one or more parameters may be redundant because the error sum is not or only weakly sensitive to changes in this combination of the parameters. If this is the case, small errors in locating the minimum may cause large variations in the values of the parameters. Evidently, in the absence of a priori knowledge the matter of multiple minima cannot be resolved rigorously unless by scanning the entire relevant parameter space. Physical insight is usually sufficient to restrict the solution to one relevant minimum, provided that the model parameters have a clear physical interpretation. On the other hand, it is possible to analyse the solution for 43
parameter sensitivity and redundancy by using the information that is contained in the Jacobian and Hessian matrices. Here, we limit ourselves to the investigation the curvature of the error function in the neighbourhood of the minimum with an eigenvalue analysis of the Hessian matrix. In this analysis it is assumed that in the minimum p s t h e gradient @(fis> is zero and that hence the errorsum can can be approximated by the quadratic function L
this approximation, contours of constant value of the error sum are desribed by ellipsoids around p' = fis in parameter space. Of these ellipsoids, the directions of the principal axes are given by the eigenvectors of H. Moreover, it can be shown that the lengths of the principal axes are inversely proportional to the square root of the corresponding eigenvalues, or
where xi and xj are two principal axes parallel to the eigenvectors vi and vj that were determined from the eigensystem H G - yvfor the eigenvalues y i andyj . Hence, the least sensitive direction in parameter space is given by the eigenvector corresponding to the smallest eigenvalue of H. This analysis can also be done for a singular matrix H, yielding one or more eigenvalues of zero If the ratio between largest and smallest eigenvalues ymax/ymin is larger than approximately lo3-lo4, then the direction along the eigenvector corresponding to ymin is illdetermined and one or two of the parameters that are dominant in the eigenvector should be set to a fixed value in the extraction procedure. An example of the use of the eigenvalue analysis,
44
Si 3.5 THE MDLS OPTIMIZATION DRIVER IN PROFILE. The MDLS optimization driver is implemented with the following Profile commands: LEVMAR CONSTRAIN SETLM SETEXT CALLEXT SCANSPACE TPLED
The optimization driver itself Constrain parameters in a physical interval Set operational parameters of the MDLS procedure Set file names and model call of the external model interface Call the external model once Do a scan of the errorsum in the parameter space Generate a parametrized input file for a forward model
45
CHAPTER 4
THE (HETEROJUNCTION) BIPOLAR TRANSISTOR OPTIMIZATION PROCEDURE
5 4.1 DEFINITION OF THE OPTIMIZATION CRITERIA
Before starting to optimize an (heterojunction) bipolar transistor profile it is important to define the optimization criteria. To define the right optimization criteria, it is neccesary to understand the physics of a (heterojunction) bipolar transistor. We start the optimization with the current gain (H,,) and the cutoff frequency (F,) as optimization criteria. We want a current gain of 100 and a cutoff frequency that is as high as possible. To make it possible to optimize the cutoff frequency to the highest value as possible we need to optimize the inverse of the cutoff frequency to zero. Otherwise this criterium does not fit in the framework of a least squares optimization alogorithm. These optimization criteria must be put in a format to be used by the optimization driver. With the weight function, which we introduced in chapter 3 as wj, we can influence the relative importance of each optimization criterium. To verify the adjustment of the importance of the optimization criteria, we calculate the error sum of each criterium separate and write them to a file. We try to start the optimization procedure with an error sum value of 1 for each criterium. The current qain criterium: The weight function for the current gain H,, we defined by (1)
With the term KHFE we can fix the weight for the H,, and thus the importance of H,, in the optimization. The importance of the H,, which is used in the optimization, is expressed by the error sum of the squared differences between desired and calculated data values, which is defined by N
1
target-H
2
model) 2
FE i 1-1
46
were wH, is the weight function of the current gain HFE,HFEtarget is the desired value of the current gain (in our optimizations 100) and H,, the by Hetrap calculated value of the current gain, and N is the number of applied base-emitter voltages. The cutoff freauencv criterium: The cutoff frequency F, of a transistor should be as high as possible. To realize this in the optimization program we defined GTtarget
-1
-.
0
(3)
FT
where Fltargetis the desired value, specified as zero to get the highest cutoff frequency. The weight function for the cutoff frequency F, is defined by wFT
-
KFT
(4)
With the factor KFT we can adjust the importance of the cutoff frequency F, in the optimization. The importance of the F, is calculated by the errorsum of the squared differences between desired and calculated data values
is the inverse value of the F, that is calculate by the where G, program Hetrap. After a trial optimization we found that we could not adjust the importance of the cutoff frequency with the weight function KF, . The cause that we found for this problem was that the value of the calculated cutoff frequency for low and high base-emitter voltages was almost zero. The top value of the F, was almost 5000 GHz. The first solution to solve this could be obtained by increase the wF, by 5000. A better solution proved to be the use of the cutoff frequency values which belong to the startprofile to define the weightfunction. The weightfunction is then defined by wFT
-
KFT + F+tart
47
(6)
where F,start represents the values that belong to the startprofile is the term to adjust the importance of the cutoff and KF, frequency in the optimization. The importance of the cutoff frequency F, which is used in the optimization can controlled by
--
€FT
N
(KFTi+F+tart) (GTio-GTimodel)
1
N
(7)
i-i
where GTmodel is:
where F, is the calculated cutoff frequency calculated by the Forward model Hetrap. After optimization with the up till here described formules, we found that the errorsum of the cutoff frequency could be adjusted precisely. The weight values should also be written in a format to be used by the optimization driver. After a succesful optimization of a bipolar transistor dope profile with the up till here mentioned optimization criteria we choose some additional criteria. The base resistance criterium : Because Philips uses the transistor mostly for analogue applications transistors are needed with a low base resistance to keep the noise low. This is the reason we optimize also the base resistance (rsheet). To get a low base resistance the target value of the base resistance is optimized to zero. The program Hetrap uses the term rsheet to define the base resistance. The weight function to influence the importance of the base resistance in an optimization procedure is defined by 1 Wisheet'
(9)
Krsheet
where Krsheet is the term to influence the importance of the optimization criterium rsheet. The importance is calculate by the errorsum of the squared differences between the desired and calculated base resistance, rsheet.
48
where rsheetdel is the base resistance calculated by Hetrap. The avalanche breakdown criterium : The last optimization criterium we defined concerns the avalanche breakdown. As , V increases to the value of the breakdown voltage, the collector current starts to increase rapidly. This increase is caused by the avalanche breakdown of the collector-base junction. To prevent the transistor against breakdown we need a optimization criterium. To optimize this criterium we used the in the transistor generated avalanche current which is calculated by the program Hetrap. The avalanche current is in Hetrap called aval. Avalanche breakdown may not occur below a certain voltage .,V To realise this, the target value of the avalanche current (aval) is optimized to zero in a second Forward model call. The weight function of the avalanche current is defined by
The factor J,start represents the values of the collector current Jc that belongs by the startprofile, with Kavalthe importance of the optimization criterium aval can changed in the wanted value. As described above, when , V increases to the value BVbc, the collector current starts to increase rapidly. This increase is due to the avalanche breakdown of the collector-base junction. With the weight function can defined the importance of the optimization criteria aval. Because the collector current increased when the breakdown voltage is reached, we used the collector current that belongs to the startprofile in the weight function. Sothat when the collector current increase the importance of the optimization criterium increase also. The contribution is expressed by the error sum, which is defined by
where Kava,is the term to define the importance of the optimization criteria aval in the optimization and M is the number of applied base-collector voltages . , V To optimize the avalanche current and the other up here mention optimization criteria it is neccesary to call Hetrap twice. This is needed because to simulate the breakdown voltage of a transistor 49
the voltage ,V must be varied. To calculate the other optimization criteria characteristics the voltage ,V of the transistor must be varied. The total error sum of the optimized criteria is calculated by
Ztotal(fl
- If?,,
+
€F,
+
Ersheet+ [aval
(13)
where H,, is the current gain, F, the cutoff frequency, rsheet the base resistance and aval the avalanche current. The desired targets and calculated weights that belong to the applied voltages are stored in a file called target.new to be used by the optimization driver.
5 4.2 DEFINITION OF THE CHANGEABLE PARAMETERS
To optimize the start dope profile to a dope profile which belongs to the electrical data that satisfies the optimization criteria, the parameters that can changed the startprofile by the optimizer must be defined. 5 4.2.1 DEFINITION OF THE BIPOLAR TRANSISTOR PARAMETERS
The parameters to optimize the current sain : To improve the current gain (HFE)of a bipolar transistor we can change the following dope profile parameters; the emitter dope (Ne) the base dope(Nb), the Gummelnumber (Gb), the base width (Wb). The parameters to optimize the cutoff freauencv : To increase the cutoff frequency (F,) the transistor should have a very narrow base thickness (Wb), a narrow collector region (Wc), and should be operated at a high-current level. The parameters to optimize the avalanche breakdown : As described up here to increase the cutoff frequency the transistor should have a very narrow collector thickness (Wc). As the collector width decreases, there is a corresponding decrease in the breakdown voltage. Therefore, compromises must be made for high frequency and voltage operation. This tells us that the changeable parameter of the optimization criterium aval can defined by the collector width (Wc) and also the collector dope (Nc). 50
The parameters to optimize the base resistance : To reduce the base resistance (rsheet), which is important for analogue applications, the base dope should be increased. To optimize rsheet we must define the base dope (Nb) and the base width (Wb) as the changable parameters. The parameters that we actually defined in our optimization procedures are shown in figure 1.
Fig.1. The changable parameters that the optimizer can use.
.
The parameters that can changed by the optimizer to optimize the dope profile of an bipolar transistor are the base width (Wb), the base Gummelnumber (Gb), the collector width (Wc) and the collector dope (Nc). With these parameters we express the new dope profile. So the base dope (Nb) is calculated from the base Gummelnumber (Gb) divided by the base width (Wb). The parameters that formed the dope profile are calculated in the Profile program extmod.pro.
5 4.2.2
DEFINITION OF THE HETEROJUNCTION TRANSISTOR PARAMETERS
The heterojunction bipolar transistor parameters are the parameters usedto optimize a conventional silicon transistor in addition with 51
some parameters that describe the material composition of the epitaxial base material, in our case silicon-germanium. The germanium composition profile which is used in the optimization program is shown in figure 2. The changeable parameters that we defined are Rwg, the distance between M 3 and M4, and Rxge, parameter to calculate the place in the base to put the middlepoint of the distance Rwg. These parameters are relative compared with the base width (Wb). The parameter Mat is defined on 1, which means that the base contains 20 % germanium, and is not changeable. With the changeable parameters that can be varied by the optimizer the other parameters can be calculated which deccribed the dope profile as described above in g4.2.1.
T I I
I
I
Fig.2. The germanium cornposition used in the optimization program.
The choices of changeable parameters that we made where also influenced by the possibilities of doping and composition profile definition present by standard in Hetrap. We did not modify Hetrap to accept other definitions.
5 4.3 SOFTWARE IMPLEMENTATION OF THE INVERSE MODELLING SCHEME The optimization program consists of the programs HETRAP and PROFILE. These two programs implemented we in the inverse modelling scheme, described in chapter 3 . Figure 1 shows the communication 52
between the first Profile instance, that is the optimization driver and the second Profile instance, that serves as a data conversion between the optimizer and the (heterojunction) bipolar transistor analyse program (Hetrap). The block \target.datal contains the data that is wanted, the optimization criteria. We write a program called \convtarg.pro' that change the file 'target.datal to a file \target.newl in a format to be used by the optimization driver. The first profile Constance, the optimization driver, can call the external model, Hetrap. In the first call the startprofile is given to Hetrap, which calculate the physical and electrical characteristics that belongs to this startprofile. The calculated electrical values are put in a file called \type.elecl. The file 'type.elecl must convert to a file in a format to be used by the optimization driver. To optimize the dope profile of an bipolar transistor, the values of the Gummelnumber Gb, the basewidth, the collectorwidth , the collectordope can varied by the optimizer. To optimize the dope profile of a heterojunction bipolar transistor the before mentioned values can varied but also the material composition of the Germanium can varied by the optimizer. The optimizion algorithme change the parameters to minimize the errorsum. The new calculated parameters are indicated as changeable parameter values in the template file. The template editor uses a template file, that is a previously prepare input file for the external program Hetrap, as a starting point. In this file, trap.tp1 in figure 2, the values of the parameters to be optimized must be replaced by a string of the form ' ?parameternamellformattl?I where format is a format statement in the style of the C-language. The format specification is optional. The parametername is an identifier that corresponds to a Profile scalar variabele as declared in the Profile Program extmod.pro. The template editor command then produces a copy of the template file, in which this string is replaced by the actual parameter as it was specified by the optimization driver in the parameter file 'paramS.datl. The external model program Hetrap calculates the new electrical data that belongs to the new optimized dope profile. The new calculated electrical data is written to the outputfile type.elec. This file is convert by the Profile program convelec.pro to the file called 'nlmout.datI in a format to be used by the optimization driver.
53
INVERSE MODELUNG SCHEME a-n 1
I IL U I
I L L
I
I
I
MODB C'ALL
-I-
1
I
I I I
i
I I I I I I I
i
-
T
I
1
=EEC
-*-
f
1
I I I I
Figure 1: Data comnunucation between the optimization driver and the external model. The optimizer is a first Profile instance, running the program optim.pro. The external model is formed by a second Profile instance, running the program extmod.pro. This second Profile instance calls the template editor tpled to prepare a parametrized inputfile for the (heterojunction)bipolar transistor analyse program, Hetrap.
54
NP N
-DOPE DESWPTIONWITTER
N
XI
XZ
1.00E19 1.00E16 0.00E16
0.00E-4 0.00E-4
O.lOE-4 0.20E-4 0.20E-I
0.20E-4
G 3.80E-6
GAUSS /ERROR /I/
1.00E-9 1.00E-9
1 i
1
BASE
?Nb1'35.4E0? ?Xb1'%5.4En? ?Xb2"t5.4EW? 1.00E-9 0.00E17 2.60E-5 4.28C-5 5.5OE-6 1 0.00E16 1.00E-1 0.00E-4 i.00E-9 1
COLtECTOR
?NCl.tS.IE'? l.OOE19
n)szRuL
-
?Xcl"t5.4E"? ?Xc2"t5.4En? 1.00E-9 ?XC3'*5.4E"? ?Xcla%5.4E"? 6.52E-6
/o/
1
1 1
?riat"%5.4E"? ?ìá2"15.4E"? ?M3"t5.4E"? ?X4"15.4E"? ?@!5"%5.4E"? Lineat material profile
KESE DEFINITIONHINIMUM STEPSIZE: 0.10E-9 MPXiM3n STEPSIZE: 1.00E-6 TOTAL LZNGET : 7tot1"%5.4Em?
N[RQmOF
-
INTZRVALS WTTE FIXED STEPSIZES: 2 XMIN XMIU: -ER OF STEPS ?X1'35.4E"? ?XZ*%S.IE"? 30 60 ?X3"%5.4E"? 1X4'25.4E-7
BIAS CONDITIONSVE9-UIN : -0.700
VEB-MAX : 5.300 VEE-STEP : 0.000 VCB-START: 0.000 YCB-STEP : 1.000 BASE CONTACT -TRUE FORIURD B U S E D -TRüE
-OOTPOTN[RQER DEFINITIONRITERNAL
OF OvTpüTS: 1 B U S CONDITION m C R S : 30
NüMTSZCAL TAB=
m
3 -FALSE i TAU FILE -FALSE
---CA EPSILON: l.E-5 U QUANTITIES-P E ï S I CTEMPERATORE: 2.9402 STANDARD MODEL PARAMETERS 'FALSE
-
--
BASE POINT SELECTIONDEFAULT BASE POINT BASE POINT POSITION
FALSE ?b.mp"15.4E"?
Fig.2. Prepared inputfile 'trap.tpl' for the external program Hetrep. The parameters between question-marks are the changeable parameters
The optimization driver optimizes the parameters until the maximium number of certain iterations is reached or a parameter reached a constrain before the maximum number of iterations is reached or the optimized curve fits, with given tolerance, on the target data curve. 5 4 3 THE (HETEROJUNCTION) BIPOLAR TRANSISTOR OPTIMIZATION PROGRAM
The described program is an optimization of a heterojunction bipolar transistor with the optimization criteria, the current gain (hfe), the cutoff frequency (ft) and the avalanche breakdown voltage. The changable parameters are the Gummelnumber(Gb), the basewidth(Wb) , the collectorwidth(Wc) , the number to calculate the place of the Germanium composition (Rxge), the quantity of the germanium that is used(Mat) and the dope of the collector(Nc1). 55
The program consists of two head programs. These programs used some programs to regulate the datatransport between the two head programs. The in this section described Profile programs are: optim.pro extmod.pro convtarg.pro
run by the first Profile instance (optimizer) run by the second Profile instance (forward model) prepares target data and weight for optimization; called in optim.pro convelec.pro prepares calculated data for optimization; called in extmod.pro errorcalc.pro calculates the errorsums of the optimization criteria to control the importance of the defined optimization criteria
................................................................... *** 1st Profile program (OPTIM.PRO) *** ................................................................... type veb target result ft $ var real Gb Xbl Xb2 Wb Xcl Wc Rxge Rwg Ncl $ Xbl=0.20E-4 Wb=O. 10E-4 Xcl=Xb2 Gb=5E12 Wcz0.58E-4 Rxge=O .5 Rwg=O. 5 Ncl=5E16 setext call profile extmod.pro > extmod.run get voltage.data $ callext veb result Gb Wb Wc Rxge Rwg Ncl $ get filel.elec $ put start. ft ft $ get file2. elec $ put start.jc jc $ exec convtarg.pro constrain Wb 0.01E-4 0.30E-4 constrain constrain constrain constrain
Gb 1E12 1E13 Ncl 1E16 5E17 Wc 0.1E-4 1.OE-4 Rxge 0.0 1.0
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20)
(21) (22) (23) (24)
constrain Rwg 0.0 1.0 56
setlm deltapr 0.02 setlm deltapa 1E-3 setlm talk 2 setlm itermax 20 levmar pro target veb result Gb Wb Wc Rxge Rwg Ncl $ quit
Procrram OPTIM.PR0 : Profile program OPTIM.PR0 Lines (i)-(2) declarations of variables; (3)-(9) initial values of parameters; (11) definition of external model (second Profile call) ; (12) reading base-emitter wanted voltages; (13) call the external program once to calculate the startprofile; (14) reading the startvalues which belongs to veb; (15) write the startvalues of the cutoff frequency to define the weight-function; (16) reading the startvalues which belongs to the base-collector voltages; (17) write the startvalues of the collector current to define the weightfunction of the avalanche breakdown (aval); (18) call the program convtarg.pro, this program reads the target data, calculates the weights and puts them in a format to be used by the optimizer; (19)-(24) constraining parameters within reasonable values; (25) relative perturbance of scaled parameter; (26) absolute perturbance of scaled parameter; (27) print information for each parameter update; (28) maximum number of parameter vector updates.(29) call the optimizer command Levmar.
................................................................. *** *** 2st Profile program (EXTMOD.PR0) ................................................................. var real Nbl Xbl Xb2 Wb Gb Ncl Wc Xcl Xc2 Xc3 Xc4 Wg Xg $ (1) var real M 2 M 3 M4 M5 tot1 X1 X2 X3 X4 Rxge Rwg basep delta $ (2) exec paramS.dat (3) Xbl=0.2E-4 (4) Xb2=Xbl+Wb (5) Xcl=Xb2 (6) Nbl=Gb/Wb (7) XC2=Xcl+Wc+(0.2E-4) (8) xc4=xc2 (9) Xc3=Xc4-(0.05E-4) (10) (11) totl=Xc4 basep=Xbl+(Wb*0.4) (12) Wg=Rwg*Wb (13) 57
Xg=Xbl+(Rxge*Wb) M2=Xbl M3=Xg- (O. 5*Wg) delta=O.OOl*Wb M3=(M3*pos(M3-(M2+delta)))+ ((M2+delta)*(l-pos(M3-(M2-delta)))) M5=Xb2 M4=Xg+ (O. 5*Wg) M4=(M4*pos( (M5-delta)-M4) )+ ((M5-delta)*(l-pos((M5-delta)-M4))) Xl=M2 X2=M3 X3=M4 X4=M5 tpled trapl.tp1 filel.trap1 Nbl Xbl Xb2 Xcl Xc2 Xc3 Xc4 Ncl M2 M3 M4 M5 totl X1 X2 X3 X4 basep $ tpled trap2.tpl file2.trapl Nbl Xbl Xb2 Xcl Xc2 Xc3 Xc4 Ncl M2 M3 M4 M5 totl X1 X2 X3 X4 basep $ % hetrap typel.lis % hetrap typel.lis exec convelec.pro exec errorcalc.pro quit
Proqram EXTMOD.PR0 : Profile program that envelopes the (heterojunction) bipolar transistor simulator. Lines (1)-(2) variable declarations; (3) reading the parameter values from the parameter file paramS.dat; (4)-(20) calculate the parameters of the new dope profile; (21)-(27) call the template editor tpled to put the new parameter values in de Hetrap input file; (28) Run the Hetrap program with a 'shell escape' % to calculate the outputdata that belongs to the desired base-emitter voltages; (29) Run the Hetrap program with a 'shell escape' % to calculate the outputdata that belongs to the desired base-collector voltages; (30) call the program convelec.pro, this program read and write the Hetrap output to be used by the optimizer; (31) call the program errorcalc.pro, this program calculates the errorsum of each optimization criterium separately.
58
................................................................... *** PROGRAM CONVTARG.PR0 *** ................................................................... type target voltage whfe wft waval $ var int nveb nvbc ntot $
(1) (2)
var real Khfe Kft Kava1 $ get targetl.data $ put dat.veb veb $ put dat.hfe hfe $ put dat.ft ft $ get targetl.data $ put dat.vcb veb $ put dat.ava1 aval $ clear get dat.veb $ insert 1 ndata get dat.veb $ nveb=ndata ndata=2*nveb ntveb=ndata put data.veb veb $ get dat.vbc voltage $ nvbc=ndata insert 1 ntveb get data.veb $ ndata=ntveb+nvbc put data.voltage voltage $ clear get dat.ft $ nveb=ndata hfe=ft put save hfe $ clear get save $ insert 1 nveb get dat.hfe $ ndata=2*nveb put data.hfe hfe $ get dat.ava1 $ hf e=aval insert 1 ntveb get data.hfe $ ndata=ntveb+nvbc
(3)
(4) (5) (6) (7) (8) (9)
(10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) 59
target=hfe put data.target target $ clear get start.ft $ Kft=6O O 0 wft=ft+Kft whfe=wft insert 1 nveb get dat.hfe $ Khfe=65 whfe=l/ (Khfe) ndata=2*nveb put data.hfe whfe $ get start.jc $ Kaval=sqrt (10) whfe=Kaval/jc whfe=whfe*sqrt (41/7 ) insert 1 ntveb get data.hfe $ ndata=ntveb+nvbc weight=whfe put data.weight weight $ clear get data.voltage $ get data.target $ get data.weight $ put target.new voltage target weight $ rm dat* Proqram CONVTARG.PR0 : Profile program that reads and writes the
target data and defines the weights and puts them in a file target.new in a format to be used by the optimizer. Lines (i)- (3) variable declarations; (4) (10) read and write the data to be used by the optimizer; (11)-(18) read and write the needed veb voltages in a format to be used by the optimizer; (19) (24) read and write the needed vcb voltages and the needed veb voltages in a format to be used by the optimizer; (25)-(42) read and write the target data in a format to be used by the optimizer; (43)-(63) read, calculate and write the weight in a format to be used by the optimizer; (64)-(68) read the voltages, the targets and the weights and put them in one file (target.new) in a format to be used by the optimizer.
-
-
60
................................................................... *** PROGRAM CONVELEC. PRO *** ................................................................... type target $ var int nveb ntveb nvbc naval nmaval nhulp yhulp ntimes $ var real Kft $ get file1.elec $ put dat.hfe hfe $ put dat.ft ft $ get file2.elec $ nvbc=ndata naval=7 var real y1 ylml dy $ extract aval ndata y1 nhulp=ndata-1 extract aval nhulp ylml dy=yl-ylml ntimes= (naval-ndata)+1 nhulp=ndata+l yhulp=yl+dy do ntimes assign aval nhulp yhulp $ yhulp=yhulp+dy nhulp=nhulp+l od ndata=7 put dat.ava1 aval $ clear get dat.ft $ nveb=ndata Kft=5E3 ft=l/ (ft+Kft) hfe=ft insert 1 nveb get dat.hfe $ ndata=2*ndata ntveb=ndata put dat.hfe hfe $ get dat.ava1 $ hfe=aval nvbc=ndata insert 1 ntveb get dat.hfe $ 61
(1) (2) (3) (4) (5)
(6) (7) (9) (10) (11) (12) (13) (14)
(15) (16) (17) (18)
(19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41)
ndata=ntveb+nvbc target=hfe rm dat.* put nlmout.dat target $
: program that converts the outputfile type.elec of the external model program Hetrap to a file nlmout.dat to be used by the optimizer. Lines (1)- (3) variable declarations; (4) (6) read the Hetrap outputfile typel.elec and write the optimization data each in a special file; (6)-(26) read the Hetrap outputfile type2.elec, the aval is the data we need. We want no breakdown below vcb=6V, the program normaly calculates the aval from O to 6V by steps of 1V. If there is avalanche it is possible that the outputfile type2 .elec contains less than 7 lines. Because the optimization driver needs 7 lines, we extrapolate, if the outputfile contains less than 7 lines; (27)-(45) read and write the optimization data in a format to be used by the optimizer. Proqram
CONVELEC.PR0
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................................................................... *** PROGRAM ERRORCALC. PRO *** ................................................................... type sumhfe sumft sumaval model hfetarg fttarg avaltarg $ type hfemod ftmod avalmod whfe wft waval $ var int nvb nvcb ntot $ var real tot1 tot2 tot3 $ get target.new $ get nlmout.dat model $ put datal target model weight I 1 41 put data2 target model weight I 42 82 put data3 target model weight 1 83 89 clear get datal $ tmpl=sqr((target-model)*weight)
(1) (2) (3)
(4) (5) (6)
(7) (8)
(9) (10)
(11) (12)
sum tmpl sumhfe totl=sumhfe/41 get data2 $
(13)
tmpl=sqr((target-model)*weight)
(16) (17) (18)
(14) (15)
sum tmpl sumft tot2=sumft/41 get data3 $
(19)
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nvcb=ndata tmpl=sqr((target-model)*weight) ntot=nvb-nvcb insert i ntot sum tmpl sumaval tot3=sumaval/41 remove 1 40 get errorsum.dat $ ndata=ndata+l assign sumhfe ndata tot1 $ assign sumft ndata tot2 $ assign sumaval ndata tot3 $ put errorsum.dat sumhfe sumft sumaval $ rm data* Proqram ERRORCALC.PR0 :Profile program that calculates and writes the errorsum of each optimization criteriumto a file errorsum.dat. Lines (1) (4) variable declarations; (5) (9) After each new calculated Hetrap outputfile the newly calculated model data that the optimizer used is writen to a file. This is necessary to calculate the errorsum of each optimization criterium; (10)-(14) read the target value, the model value and the weight of the optimization criterium hfe to calculate the errorsum; (15) (18) calculate the errorsum of the optimization criterium ft; (19)-(26) calculate the errorsum o f t h e optimization criterium ava1;(27)-(33) write the calculated errorsums to the file errosum.dat.
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The up to here described programs can very easily changed to an optimization program with other optimization criteria and changable parameters. Profile commands can be found in the Profile manual written by Ouwerling [13].
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CHAPTER 5
OPTIMIZATION EXAMPLES
5 5.1 BIPOLAR TRANSISTOR OPTIMIZATION
Startina dope Drofile Each optimization we start with the dope profile shown in Figure 1. The criteria that we optimize have been explained in the previous chapter. To compare the optimized dope profile and the electrical characteristics with the electrical characteristics that belong to the start profile the characteristics of the current gain HFE, the cutoff frequency F,, the base resistance rsheet and the avalanche current aval that belongs to this start profile are shown.In Figure 2 the current gain H,, and the Fig.1. The start profile (x in pn, dope in ~ r n . ~ ) cutoff frequency F, are shown Figure 3 shows the base resistance rsheet.
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Fig.3. The base resistance rsheet in n/û
Fig.2. The current gain HFE and the cuttof frequency F, (in HHz)
In this paragraph the changeable parameters that could varied by the optimizer are the Gummelnumber of the base (Gb), the base width (Wb), the collector width (Wc) and the collector dope (Nc). 64
The width of the lowly doped part of the emitter was also chosen as a changeable parameter but after a trial optimization we concluded that the optimizer makes the emitter width as narrow as possible. This can explained by the optimization criteria F,, the cutoff frequency is dependent of the collector width (Wc), the base width (Wb) and the emitter width (We). This is the reason we fixed the emitter parameter. ODtimization of the avalanche current :
current sain, cutoff freauencv and
the
The optimized dope profile is shown in figure 4. The changes of the parameters can be explained by the optimization criteria. To increase the cutoff frequency F, of the bipolar transistor the base thickness (Wb) and the collector region thickness (Wc) are made smaller. To improve the current gain H,,the optimizer increases the Gummelnumber (Gb). The base dope (Nb) which is dependent of Fig.4. The optimized dope profile, the optimized the Gummelnumber and the base criteria were the current gain HR, the width, so that to improve the cutoff frequency F, and the h - ( d ~ : \ c ~ ~ ~ ~ ~ current gain the base dope increased. -
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Fig.6. The avalanche current aval aid collector current Jc in A/cm"
Fig.5. The optimized current gain HFE and cutoff frequency F, characteristic
65
To prevent the transistor against a low breakdown voltage (aval) the collector dope is decreased. The current gain H,,and the cutoff charateristic F, are shown in figure 5. The highest cutofffrequency is 33 GHz. In figure 6 the avalanche current and the collector current is shown that belongs to the optimized dope profile. The collector current increase is because the current gain is increased. The avalanche current is much less than the collector current so that the avalanche breakdown voltage is not occured below the desired , V ( the desired , V is 6V ). ODtimization of the current aain, cutoff freauencv. the avalanche current and the base resistance with the same weiaht function : For analogue applications it is important to have transistors with a low base resistance. The next optimization which we did contains the previous optimization criteria in addition to the base resistance rsheet. By the first optimization, the importance of the weight function was adjusted for the current gain, the cutoff frequency and the base resistance on the same value (
