(0 NBS SPECIAL PUBLICATION 400-63

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards]

A FORTRAN Program

for Calculating the

Electrical Parameters

of Extrinsic Silicon
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RATIONAL BVMXAV

09 MAKPAUW

UBBAJtr

NOV 2 4 1980

Semiconductor Measurement Technology:

A FORTRAN Program for

Calculating the Electrical

Parameters of Extrinsic Silicon

R. D. Larrabee, W. R. Thurber, and

W. M. Bullis

Electron Devices Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Luther H. Hodges, Jr., Deputy Secretary

Jordan J. Baruch, Assistant Secretary for Productivity, Technology and Innovation

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued October 1980
Library of Congress Catalog Card Number: 80-600158

National Bureau of Standards Special Publication 400-63

Nat. Bur. Stand. (U.S.), Spec. Publ. 400-63, 54 pages(Oct. 1980) CODEN: XNBSAV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1980

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Price $3.75

(Add 25 percent for other than U.S. mailing)
Table of Contents

Page

Abstract 1

1 . Introduction 1

2. Description of Program Operation 3

2. 1 Options 5

2.1.1 DOUBLE ON 5 2.1.2 DOUBLE OFF 6

2.1.3 CONSTANTS 6

2.1.4 DOLOOP XX ARG(I) ARG(2) 7 2.1.5 STOP 8

2.2 Warning and Error Messages 8

2.3 Additional Information 9

3. Theory and Method of Computation 10

3.1 Location of the Fermi Level 10

3.2 Computation of Carrier Mobilities and Related Parameters . . 14

4. Annotated Program Listing 19

5. Illustrative Output Data Cards and Examples 36

5.1 Data Cards Used to Generate Output Listings 36

5.2 Illustration of a No-Option Output 37

5.3 Illustration of the DOUBLE ON Option Output 38

5.4 Illustration of the DOLOOP XX Option Output 41

5.5 Illustration of the CONSTANTS Option Output 45

6. Conclusions 46

Acknowledgments 46

References 46

List of Tables

Page

1. Symbols Used in Mobility Formulas 15

2. Formulas for Calculating Electron Mobility 16

3. Formulas for Calculating Hole Mobility 17

iii
Preface

The computer program described in this report evolved during the course of a number of projects in the Semiconductor Technology Program in the Electron

Devices Division at the National Bureau of Standards. This program serves to

focus NBS efforts to enhance the performance, interchangeability, and

reliability of discrete semiconductor devices and integrated circuits through

improvements in measurement technology for use in specifying materials and

devices in national and international commerce for use by industry in

controlling device fabrication processes. Its major thrusts are the

development of carefully evaluated and well-doctmiented test procedures and

associated technology and the dissemination of such information to the

electronics community.

iv
Semiaonductor Measurement Teahnology:

A FORTRAN Program for Calculating the Electrical Parameters

of Extrinsic Silicon

by

R. D. Larrabee, W. R. Thurber, and W. M. Bullis

National Bureau of Standards

Washington, DC 20234

ABSTRACT

Many electrical properties of silicon are strongly dependent upon the specific nature and density of the active

impurities present. Calculation of these electrical

properties hinges on the solution of the charge balance

equation to determine the position of the Fermi level for

each specific case of interest. A FORTRAN program is

presented that performs this determination and then calculates some of the often-used electrical parameters of

silicon as a function of temperature. Results obtained

from this program have proven useful in interpreting Hall

effect data, determining the degree of ionization of the

separate dopant states as a function of temperature, predicting the behavior of specimens when the dopant picture

is intentionally (or conceptually) changed, and understanding the variations in the relative roles of the different scattering mechanisms on carrier mobility as the

temperature is changed.

Key Words: Carrier density; computer program; electri- - cal properties of silicon; Hall effect; mobility; resistivity; silicon.

1 . INTRODUCTION

Trace impurities can act as donors or acceptors and thus contribute mobile

charge carriers that can have a profound effect on resistivity and other

electrical properties. Since carrier density and mobility are functions of

temperature as well as the density and type of impurities present, it is impractical to provide general tables listing all possible situations that

might occur in practice. Therefore, it becomes necessary to calculate the

desired electrical properties for each impurity situation and temperature of

interest.

The FORTRAN program presented in this report was written to accomplish this

objective for the case of simple donor and acceptor states in silicon (hereafter called dopant states). The density, activation energy, and degeneracy

of all active donors and acceptors are initially provided as inputs to the

program. These data are then used by the program to calculate the position

of the Fermi level by solving the detailed charge balance equation (i.e., the

sum of the positive charges in the valence band and ionized donors equals the sum of negative charges in the conduction band and ionized acceptors). After
the position of the Fermi level has been found, the program computes many of

the more often used electrical properties of silicon that depend on carrier

density or carrier scattering (e.g., resistivity, carrier mobility, and Hall

coefficient). The scattering mechanisms which are included in the calculation of the carrier mobility are lattice, ionized impurity, neutral impurity,

and electron-electron (or hole-hole) interactions. Both electron and hole

mobilities are calculated and an effective conductivity mobility is calculated as the sum of the electron and hole mobilities weighted by the density

of each. The effective conductivity mobility computed this way is the same

as the majority carrier conductivity mobility in extrinsic specimens. In intrinsic specimens, it corresponds to a hypothetical mobility for the total

carrier density (i.e., electron plus hole density). The expressions for calculating the resistivity and Hall coefficient include appropriately weighted

contributions from both charge carriers. In the absence of agreement as to

appropriate values, the Hall scattering factors (i.e., ratio of Hall to con- ductivity mobility) for both electrons and holes were taken to be unity.

In its present form, the program handles up to five independent impurity

states, two donors and three acceptors. The donor levels are assumed to be

neutral when occupied by an electron and to have a temperature-independent

activation energy referenced to the conduction band edge. The acceptor levels are assumed to be neutral when occupied by a hole and to have a temperature-independent activation energy referenced to the valence band

edge. Provision is made for the reduction in activation energy for both

donors and acceptors at the higher dopant densities. These assumptions are appropriate for impurities from column 3 and 5 of the periodic table (e.g.,

boron and phosphorus) and may, or may not, be appropriate for other impurities with larger activation energies or multiple energy levels.

The key algorithms in this program are the routine for calculating the position of the Fermi level and the routine for calculating the carrier mobilities. These algorithms are most accurate over the temperature interval from

100 to 500 K and for total doping densities below 10^^ cm~^. These

two algorithms have been written as subroutines in the present program so that they can be used as a starting point in the development of other programs that require these algorithms. Notice, however, that the mobility rou- tine requires knowledge of many of the parameters calculated by the Fermi

level routine, and thus it cannot be used without first executing the Fermi

level routine.

Variations of the present program have proven useful for interpreting Hall

data [1], predicting the electrical behavior when the dopant picture (e.g.,

compensation) is changed [2,3], determining the degree of ionization of the

active dopant state in extrinsic infrared detectors [4] , and understanding

the changing role of the various scattering mechanisms on mobility as the

temperature or dopant density changes.

The following section provides the information necessary to use the program.

Section 3 describes the various equations used in the program so that the interested user may verify that they are appropriate for his particular application. A detailed listing of the complete program is given in section 4.

This listing is heavily annotated so that the user can identify each equation

2
and follow each step of the program. Section 5 provides examples of typical

output formats.

2. DESCRIPTION OF PROGRAM OPERATION

The present program can be divided into four main parts as follows:

1. Read data cards and set up options.

2. Compute position of the Fermi level, occupancy of all dopant states,

and density of carriers in the conduction and valence bands.

3. Compute carrier mobilities and related parameters.

4. Print output listings and reinitialize for next run.

In its present form, the program can handle up to two donor states (called Dl

and D2) and up to three acceptor states (called Al, A2, and A3). For each

dopant state, the program expects to find a data card specifying:

1. the name of the dopant state (up to 12 alphanumeric characters),

2. the density of the dopant state in reciprocal cubic centimeters,

3. the activation energy of the dopant state at infinite dilution in

electron volts,* and

4. the degeneracy of the dopant state (an integex- > unity).

The format of these data cards is 2A6, E10.4, 2F10.4, corresponding to the

four items of input data - name, density, activation energy, and degeneracy,

respectively. The format for the name (2A6) assumes that the machine can

store 12 alphanumeric characters in two 6-character words. This may have to

be changed for machines with different alphanumeric storage formats. If a particular dopant state is absent, its data card must still be present to

inform the program of this fact. The data cards representing missing dopant

states use the name DUMMY and the rest of the card is left blank. The order

of the five data cards is:

1. Dl - first donor state.

2. D2 - second donor state.

3. Al - first acceptor state.

4. A2 - second acceptor state, and

5. A3 - third acceptor state.

It is recommended that this same order (i.e., Dl through A3) be the order of occurrence of the corresponding dopant states from the conduction band to the

valence band (if possible) to aid in remembering which symbol (i.e., Dl

through A3) represents each dopant state.

*For donor states, the activation energy is the magnitude of the energy

difference between the conduction band and the dopant state; for acceptor

states, it is the magnitude of the energy difference between the valence

band edge and the dopant state. Both of these quantities are assijmed to be

independent of temperature.

3
In the normal (no-option) output format, the program lists the following

quantities on the output page for each set of five input data cards supplied:

1 . the position of the Fermi level with respect to the valence band in

electron volts,

2. the density of electrons in the conduction band in reciprocal cubic

centimeters,

3. the density of holes in the valence band in reciprocal cubic

centimeters,

4. the effective conductivity mobility in square centimeters per

volt-second,

5. the Hall coefficient in cubic centimeters per coulomb,

6. the electrical resistivity in ohm-centimeters, and

7. the ionized density of dopant species Dl through A2 in reciprocal

cubic centimeters. (The DOUBLE ON option discussed below can be

used to output the ionized density of species A3.

)

These ten items are printed across a single line on the output listing in the

order given. Since these parameters are functions of temperature, the output

listing consists of a series of such lines, each preceded by its temperature

and the corresponding value of 1000/T. The resulting table is provided with

descriptive column headings and documentation of the appropriate input information from the data cards (see first sample output listing in sec. 5). The

program is capable of listing additional output information, as discussed in

the section on options to follow.

The above quantities are evaluated and listed as a function of temperature

over the range from 4.2 to 500 K. This temperature range is divided into

three regions, with break-points at 77 K (liquid nitrogen temperature) and

300 K (room temperature). The program will perform its evaluations at each

break-point (i.e., 77 and 300 K) , as well as at the extreme points (i.e., 4.2

and 500 K) . After completion of the first evaluation at 4.2 K, the program

jumps to some higher specified temperature and then starts to increase the

temperature with equal increments of 1 000/( temperature ) . The incremental

values of 1000 /(temperature) are specified independently in each of the three

temperature regions. As presently written, the temperature following 4.2 K

is 20 K (1000/T = 50), and then the temperature is increased in intervals of

1000/T = 2 until the first break-point at 77 K is reached. Within the interval from 77 K to 300 K, the temperature is increased in intervals of 1000 A =

1. Above 300 K, the temperature is increased in intervals of 1000/T = 0.5.

The FORTRAN parameter RUNNO keeps track of things as follows:

RUNNO = 1, first time through, T = 4.2 K,

RUNNO = 2, first region up to 77 K,

RUNNO = 3, second region from 77 to 300 K, and

RUNNO = 4, third region above 300 K.

Many of these details of the temperature sequence can easily be changed by

simply changing the appropriate defining statements in the program (e.g., the

first temperature of 4.2 K is defined by the FORTRAN statement numbered 170

in the program listing in sec. 4.).

4
2.1 OPTIONS

Several options are built into this program to allow one to modify the output

listing without changing the program pev se. These options are exercised

when a control card is inserted into the data card deck so that it is read by

the program as a D1 data card. The program recognizes that it is a control

card and not a D1 data card by the name. When such a control card is encountered, the program sets up the option desired and then goes back and expects

the next card to be a D1 data card. If no control cards are included in the

deck of input data cards, the program simply generates an output listing as outlined previously.

This is considered to be the normal (or no-option) output format and is the

format that should suffice for most applications of this program.

The control cards consist of a single (or double) word name selected from a reserved name list designed not to conflict with dopant state names. Some of

the control cards additionally contain numerical arguments that are needed to

specify the parameters of the option desired. In either event, the format of

all control cards is identical to that of the dopant state data cards (i.e.,

2A6, ElO.4, 2F10.4), so the names and any numerical arguments must be placed

on the control card accordingly.

2.1.1 DOUBLE ON

This option doubles the amount of data printed on the output listing at

each temperature. The format of the output listing consists of two

lines of output information at each temperature instead of just one (thus the name DOUBLE ON). The first line of information is identical to a no-option run* and the second line contains the parameters

listed below. The corresponding control card consists of the two words

DOUBLE ON in the first nine positions with the rest of the card blank.

This control card turns on this option for all subsequent sets of dopant

state data cards, and it remains on until a DOUBLE OFF control card is

read. This means that the DOUBLE ON control card must be the first card

of a group of data card sets for which the double output is desired.

The DOUBLE OFF control card must be the card which follows the last data

card in the group (i.e., the first card of a run for which single output

only is desired) . In addition to all the parameters listed in a no- option run, the DOUBLE ON option will cause the program to list the following parameters:

1. The energy gap of silicon in electron volts.

2. A parameter called TEST which indicates how close the calculated

Fermi level position comes to solving the detailed charge balance

equation. TEST is the number of residual charges per cubic

centimeter,

3. The voltage equivalent of the temperature in volts (i.e., kT/q)

.

4. The position of the Fermi level with respect to the conduction band

in electron volts.

*A run is herein defined as the evaluation of one set of five dopant state

data cards over the full temperature range.

5
5. The ionized density of the dopant state A3 in reciprocal cubic

centimeters (there was insufficient room for this to be included on

the single-line no-option listing)

.

6. The electrical conductivity in mho per centimeter.

7. The mobility of electrons in the conduction band in square centimeters per volt-second, as follows:

A. the upper limiting mobility of electrons set by lattice and

electron-electron scattering,

B. the mobility obtained by adding the effects of ionized impurity

scattering (corrected for electron-electron scattering) to A,

above, and

C. the mobility obtained by adding the effects of neutral impurity

scattering to B, above.

8. The mobility of holes in the valence band in square centimeters per

volt-second as follows:

A. the upper limiting mobility of holes set by lattice and holehole scattering,

B. the mobility obtained by adding the effects of ionized impurity

scattering (corrected for hole-hole scattering) to A, above,

and

C. the mobility obtained by adding the effects of neutral impurity

scattering to B, above.

These parameters were selected to complement the parameters of a no- option run and are listed on the line immediately below them on the output listing, as shown in section 5.3. The sets of two-line data corre- sponding to each temperature are separated by two blank lines in order

to make it more obvious which of the two coliamn headings applies to each

line. Some of the parameters listed here are also listed in the CONSTANTS option discussed below, and some are modified versions of parameters listed in the no-option listing (e.g., the position of the Feirmi

level with respect to the conduction band instead of with respect to the

valence band). The three mobility values listed for electrons and holes

provide information needed for giving one a picture of the changing role

of the various scattering mechanisms with temperature. Although the ma- jority carrier mobility is calculated as accurately as possible, less

attention has been paid to the calculation of minority carrier mobility.

(See sec. 3.2 below.)

2.1.2 DOUBLE OFF

This control card turns off the DOUBLE ON option as explained above.

The control cards DOUBLE ON and DOUBLE OFF simply set and clear a switch, respectively, and affect nothing but the format of the output

listing. Therefore, it is permissible to use any of the other control

card options while in the DOUBLE ON mode of output listing. The default

condition (i.e., if neither DOUBLE ON nor DOUBLE OFF control cards

appear in the deck of data cards) is DOUBLE OFF.

2.1.3 CONSTANTS

This option generates a listing of some of the parameters used in the

evaluation that are not functions of dopant density. These parameters

6
are evaluated at the same temperatures as a no-option run. The corre- sponding control card consists of the single word CONSTANTS in the first

nine positions with the rest of the card blank. The CONSTANTS option

does not relate in any way to any dopant state data cards that might

follow. Consequently, the next card could be another control card or the D1 data card of another run. The CONSTANTS control card can be located anywhere in the deck of cards to be read by the program as long as it is encountered when the program expects to find a D1 data card. The

CONSTANTS control card causes the program to list the following

parameters:

1. 100 0/( temperature ) in inverse kelvin,

2. the temperature in kelvin,

3. the voltage equivalent of the temperature in volts (i.e., kT/q),

4. the band gap in silicon in electron volts,

5. the relative dielectric constant of silicon,

6. the upper limiting mobility of electrons in the conduction band set

by lattice scattering in square centimeters per volt-second,

7. the upper limiting mobility of holes in the valence band set by

lattice scattering in square centimeters per volt-second,

8. the temperature in kelvin raised to the 3/2 power,

9. the relative effective mass of electrons in the conduction band,

10. the relative effective mass of holes in the valence band,

11. the density of states in the conduction band in reciprocal cubic

centimeters (designated FNC in Program and on output listings),

and

12. the density of states in the valence band in reciprocal cubic cen- timeters (designated FNV in Program and on output listings).

These parameters are printed across a single line on the output listing

in the order given. Since these parameters are functions of temperature, each such line corresponds to a different temperature. The

resulting table is provided with descriptive column headings as shown in

the sample output listing in section 5.5. The last five items on the

above list are useful for interpreting Arrhenius plots of the logarithm

of carrier density vs. 1000/T in terms of the activation energies of the

dopant states present (e.g., as might be done in interpreting the

results of measurements of Hall coefficient as a function of temperature )

.

2.1.4 DOLOOP XX ARG(l) ARG(2)

This option causes the program to generate multiple listings for a single set of five dopant state data cards. Each successive listing has

the density of a selected dopant state incremented by a specified

amount. The following information must be provided on the DOLOOP XX

control card:

A. DOLOOP XX is the name; XX specifies which dopant state is to have

its density incremented. XX is selected from the list Dl, D2, Al,

A2, and A3. If anything else appears in the XX position on this

control card, an error message to this effect will be printed on the

output listing, and the DOLOOP XX control card will be ignored.

7
B. ARG(l) is a numerical argument which specifies the amount by which

one would like to increase (or decrease) the dopant density each

time. Its dimensions are (centimeters)"^. In accordance with

the format statement pertaining to data cards (i.e., 2A6, E10.4,

2F10.4), it is expressed in E format in positions 13 to 22 on the

control card.

C. ARG(2) is a dimensionless argviment which specifies the total number

of listings desired. In accordance with the format statement

pertaining to data cards, it is expressed in F format (with decimal

point) in positions 23 to 32 on the control card.

The DOLOOP XX option refers only to the next set of five dopant state

data cards, and everything reverts back after completing the number of

listings specified. Only one dopant state can be incremented at a time

(i.e., no provision has been made for nested do loops).

2.1.5 STOP

This option causes a FORTRAN stop instruction to be executed and is used

to exit the program. The corresponding control card consists of the single word STOP in the first four positions on the card with the rest of

the card blank. Normally, the STOP control card is the last card in the

deck of cards to be read by the program.

2.2 WARNING AND ERROR MESSAGES

There are several warning and error messages that are printed on the output

listing when unusual circumstances are encountered during the execution of

the program. A warning message informs the user that the indicated condition

was encountered and suggests that its effects on the validity of the next

data line(s) be considered. Error messages inform the user of an event that

precludes the continuation of the current run and, after printing the error

message, the program proceeds to the next set of five dopant state data

cards. The messages are:

1. BORN APPROXIMATION FOR IONIZED SCATTERING NOT VALID FOR MAJORITY CARRIER

This warning message is printed when the log term, BN, in the modified

Brooks-Herring ionized impurity formula is less than or equal to 10.0.

In the Born approximation, the scattering is treated as a small perturbation on the motion of the incident carrier. This assumption is violated

at high carrier densities and low temperatures.

2. (EF-EG)/CAYT = XX ERROR IN N MAY EXCEED 1 PERCENT

In this warning message, XX is replaced by the dimensionless ratio of the

difference in energy between the Fermi level and the conduction band

divided by the voltage equivalent of the temperature (i.e., kT/q) . This

message is printed when XX > -2 and is a warning that the calculation of

electron density may be in error by greater than 1 percent due to the

expressions used to approximate the Fermi -Dirac integral [5] . This oc- curs only at high dopant densities when the material is degenerate or nearly so.

8
3. EF/CAYT = XX ERROR IN P MAY EXCEED 1 PERCENT

This warning message is identical in function to that of the one immediately preceding, except that it pertains to hole density instead of electron density.

4. INVALID ARGUMENT OF DOLOOP COMMAND

This error message is printed when the argument on the DOLOOP XX control

card (i.e., XX) is not selected from the list D1, D2, A1, A2, or A3.

After printing this message, the program proceeds to process the run without the DOLOOP option (i.e., it produces one run instead of the number called for on the invalid DOLOOP XX control card).

5. CHARGE BALANCE EQUATION COULD NOT BE SOLVED, IFLAG = X

This error message is printed when the algorithm for the determination of

the Fermi level position cannot find a solution to the charge balance

equation. When this message is printed, X is replaced by the value of an integer flag (i.e., IFLAG) that specifies the reason why the charge balance equation could not be solved. The significance of the values of

IFLAG are discussed in section 3.1 below.

6. LOOPSWITCH IS OUT OF RANGE

This is an error message that should never be printed if the program is

functioning properly. LOOPSWITCH is a program variable whose value is

determined by the XX portion of the DOLOOP XX control card and the program does not contain a statement capable of causing it to be out of

range. This message was included as a check on the proper operation of

the DOLOOP option.

2.3 ADDITIONAL INFORMATION

The program listing in section 4 of this report is copiously documented with

comments to guide the user through the details of the program. The complete

program, listed in section 4, consists of a main program followed by a series

of three subroutines and one function. The function TEST and the subroutine

ZEROIN are used for finding the position of the Fermi level, and the subroutines MOB and SICI are used for performing the mobility computations. Notice

that the mobility algorithm requires knowledge of many of the parameters calculated by the Fermi level locating routine; thus, it cannot be used without

first finding the location of the Fermi level.

All of these routines have been written in FORTRAN IV for use on the Univac

1108 computer at the National Bureau of Standards. This is a 36-bit machine

with 26-bit mantissa floating-point numbers. It may be necessary to use double precision to obtain a satisfactory accuracy on machines with significantly smaller word lengths. It may also be necessary to modify the alphanumeric format for data cards and the IF statements that test alphanumeric

variables for machines that do not store six alphanumeric characters per

word.

9
There are several places within the program where a zero denominator might

occur during execution (e.g., when computing the Hall coefficient for a very

small carrier density at low temperatures). The present program tests for

this circumstance in all places where this has occurred in actual use. When

this condition is encountered, a suitable flag is set, the division in question is circumvented, and the printing of all affected parameters is suppressed. In addition, all parameters with a zero value are omitted from the

output listing so as not to clutter the output pages with multiple zero entries. This has been accomplished by preceding each WRITE statement with an appropriate IF statement that suppresses the printing of undesired output.

Since this may slow down the printing speed or increase the cost of execution

in some systems, the reader may wish to modify this portion of the program to

achieve the same result with object-time format statements [5] computed

during execution and stored in an array for use by the appropriate WRITE

statement.

Copies of the main program are available on cards from the authors.

3. THEORY AND METHOD OF COMPUTATION

The computations of the position of the Fermi level and carrier mobilities

are key parts of this program. The various equations used in these computations are discussed below. Numbers cited in the following sections refer to

statement numbers in the complete program given in section 4. If two numbers

are given, the first refers to electrons or donor states, and the second to

holes or acceptor states.

3.1 LOCATION OF THE FERMI LEVEL

For uniformly doped silicon in thermal equilibrium (as assumed herein), the

summation of all negative charges must equal the summation of all positive

charges (i.e., the material must be electrically neutral). The negative

charges consist of electrons in the conduction band and any negatively

charged impurity states (acceptors). The positive charges consist of holes

in the valence band and any positively charged impurity states (donors). The

Fermi level is located at just that point that makes the total net charge

equal zero. The FORTRAN parameter TEST defined in statement 140 of the function TEST is a measure of how precisely the algorithm has located the position of the Fermi level to obtain charge neutrality.

TEST = (Net negative charge) - (Net positive charge)

TEST = (CN + DI3 + DI4 + DI5) - (CP + DI 1 + DI2) (140)*

where the symbols represent densities in reciprocal cubic centimeters as follows :

CN, electrons in the conduction band,

CP, holes in the valence band,

DI 1 , ionized donor species 1,

*The statement numbers in this section refer to the function TEST unless

otherwise indicated.

10
DI2, ionized donor species 2,

DI3, ionized acceptor species 3,

DI4, ionized acceptor species 4, and

DI5, ionized acceptor species 5.

The densities of electrons and holes in the conduction and valence bands,

respectively, are computed using Blakemore's approximations to the FermiDirac integral [5]

n = N [exp(- n) + 0.27]"^ (8,26)

for values of n in the range from -80 to +1,

2 3/4 n = 0.752253 N (n + 1.7)^ (10,30)

for values of n greater than +1, and

n = 0 (2,20)

for values of ri less than -80. In these equations, n is the electron or hole

density, N is the density of states in the appropriate band, n = q(Ep - Eg) At for electrons (4), and -qEp/kT for holes (22), Ep is the Fermi

energy, in electron volts. Eg is the width of the forbidden energy gap in

electron volts (the zero of energy is assumed at the valence band edge), and

kT/q is the voltage equivalent of the temperature (353 of Main Program).

The density of states is given by

2 3/2 *3/2 3/2 N = 2(2tt k m A ) m 1

o

A ooo *3/2 3/2

= 4.829 X 10 m T

(354,355) Main Program

where m is the effective mass of the appropriate carrier relative to the

free electron mass, m^. The relative electron effective mass, m^, is

given by [7]''"

m* = 1.0627 - 1 .61708 x lO'^^T + 6.83008 x 10~^T^ - 3.32013 x 10"^"^

e -11 4 -14 5 -17 6 + 8.04032 X 10 T - 9.66067 x 10 T + 4.54649 x 10 T

(350) Main Program

and the relative hole effective mass, m^, is given by [7]

This and the following two equations have been carried to high order in

kelvin temperature for purposes of computational accuracy, without regard to

how high an order is really necessary to describe the measured data to

within experimental accuracy.

11
m* = 0.590525 - 5.23548 x ^0~\ + 1.85678 x 10~^T^ - 9.67212 x 10~\"^

h

+ 2.30049 X 10~^°T^ - 2.596730 x 10~^\^ + 1.11997 x lo'^^T^ . (352) Main Program

The temperature dependence of the forbidden energy gap of silicon is taken

into account in accordance with a fit to the data of Macfarlane et at. [8]

-5 -7 2 -10 3

E = 1.15556 + 3.23741 x 10 T - 8.7011 x 10 T + 9.95401 x 10 T

g

- 3.80977 X 10~^^T^

(356) Main Program

The density of ionized impurities is computed from the following:

N.It= N [1 + g exp(A)] (40 ff)

where is the total density of the dopant impurity, g is the degeneracy

factor, and A = q(Ep - Eg + E^)AT for donor states and q(E^ - Ep)AT

for acceptor states, and E^^ is the value of the activation energy, relative

to the appropriate band edge. This treatment of ionized impurity states is

only valid for simple donor and acceptor states that have the following properties:

1. Each impurity state gives rise to only one electronic state. The

present version of this program cannot handle the case of coupled

multiple levels associated with a single impurity species unless the

levels are widely separated and can be treated independently.

2. The donor levels are neutral when occupied by an electron and have a temperature-independent activation energy referenced to the bottom

of the conduction band.

3. The acceptor levels are neutral when occupied by a hole and have a temperature-independent activation energy referenced to the top of

the valence band.

These properties are appropriate for the colximn 3 and column 5 impurities in

the periodic table (e.g., boron and phosphorus) and may, or may not, be

appropriate for other impurities with larger activation energies or multiple

energy levels. Impurity states satisfying these criteria are called dopant

states in this report and are characterized by the following three parameters

which are specified on the input data cards for each state:

1. the concentration of the dopant state per unit volume of silicon,

2. the limiting activation energy at low concentrations relative to the

appropriate band edge, and

3. the degeneracy of the dopant state (an integer larger than unity).

The decrease in activation energy with increasing dopant density or increasing compensation is taken into account with the use of an equation based on

12
the work of Penin et at. [9] for phosphorus-doped silicon. Penin's expression gives zero activation energy at 3 x 10^^ cm~-^ regardless of the

initial activation energy, E^g • Since deeper levels are expected to re- quire greater densities to obtain zero activation energy [10], Penin's ex- pression has been modified to retain the same decrease in activation energy,

in eV, for a given increase in dopant density. To accomplish this, Penin's

factor of 8 X 0.045, where 0.045 is the phosphorus activation energy, is

fixed at 0.36. Thus, since E^^q greater for a deep level, the dopant

density for zero activation energy is correspondingly greater.

The resulting equation for calculating the decrease in activation energy with

dopant density and compensation is

= - + 7") (-r ,/a*) - a , A AO 2 r d 2

d

(70 ff) Main Program

where E^q is the activation energy for low densities of the dopant, =

[3/{47rNi ) ] is the average distance between the dominant dopant atoms of a given class (donor or acceptor), a* = 21 x 10"^ cm is the effective Bohr

radius, a = 3.6 x 10"^ eV cm, and N2 is the total density of compensating

impurities of the other class (acceptor or donor). This ec[uation is used in

the computer program for all dopant states, but its applicability to deep

levels has not been verified.

Since the reduction in activation energy with increasing donor density is due

in large part to the lowering of the bottom of the conduction band, the total

donor density is used in the calculation of the reduction for each donor

state. Likewise, when more than one acceptor state is present, the total

acceptor state density is used to calculate the reduction in activation energy of each acceptor state. However, when two or more donors (or acceptors)

are present, each in significant quantity, this approach probably overestimates the reductions in activation energy of each acceptor state.

The computation of the reduction in activation energy is not included as part

of the routine for finding the Fermi level, but has been included in the Main

Program as part of the initialization procedure. The routine for finding the

Fermi level position assumes that the values of activation energy previously

placed in COMMON are the correct values. The complete program outputs both

the limiting value of activation energy initially inputted and the calculated

lowered value of activation energy (see sec. 5).

The correct position of the Fermi level is found by the subroutine ZEROIN

[11-13] which searches for a zero of the parameter TEST. ZEROIN is listed as part of the complete program in section 4. The subroutine is well documented

with descriptive comments before the code and explanations of all significant

steps as they occur in the body of the code. The subroutine looks for a zero between the variables EF and EFG which are given initial values of 0.0 and

1.2 eV, respectively, by the calling program. In searching for the root,

ZEROIN uses the secant rule unless tests indicate that bisection would be

advantageous. The interval between EF and EFG is narrowed until the stopping

criterion is satisfied. The stopping criterion is

13
EF-EFG

< RE X EF + AE ,

where RE is the maximum allowable relative error and AE is the maximum allowable absolute error in eV. In the calling program these are both assigned

the value 1 x 10"^ so that they have equal weight in the stopping criterion when EF = 1.0 eV. Initially, the subroutine compares RE with the roundoff error of the machine, ER, specified in a DATA statement, which can be

easily changed to correspond to the user's computer, and selects the larger

of the two for the stopping criterion.

At the conclusion of the computations, EF is the better approximation to the zero of TEST as the subroutine, by interchange if necessary, makes

I TEST(EF) I < I TEST(EFG) | . Before the return to the main program, the parameter IFLAG is set to indicate the status of the results. The normal case with the root located within the stopping criterion is denoted by IFLAG = 1. A possible, but unlikely, result is TEST(EF) = 0 without the stopping criterion being met (current value of EFG not close enough to EF ) . For this situation IFLAG =2. If the stopping criterion is met, but | TEST(EF) | is larger

than the absolute value of TEST evaluated with either initial argument, IFLAG

=3. If the stopping criterion is satisfied, but TEST(EF) x TEST (EFG) > 0,

there is apparently no root in the interval and IFLAG = 4. ZERQIN permits up

to 500 evaluations of TEST before terminating and setting IFLAG =5. If upon

return to the main program IFLAG is not 1 or 2, its value is printed for diagnostic purposes. The program then proceeds to the next temperature. For a successful evaluation (IFLAG = 1 or 2), the residual value of TEST, called

TESTF, is printed in the DOUBLE ON output listing.

3.2 COMPUTATION OF CARRIER MOBILITIES AND RELATED PARAMETERS

The scattering mechanisms which are included in the calculation of the carrier mobility are lattice, ionized impurity, neutral impurity, and electronelectron (or hole-hole) interactions. The mobility calculations are sum- marized in the tables which follow. In table 1, the symbols used in the

mobility formulas are defined and the corresponding computer program name is

given. Table 2 gives the mathematical formulas for calculating electron

mobility along with the statement number of the corresponding expression in

the computer program. Table 3 gives this information for the calculation of

hole mobility.

The temperature dependence of the electron mobility due to lattice scattering

is taken from the work of Norton et al. [14], and the magnitude is based on the results at 300 K of Thurber et at. [15]. The temperature dependence of

this mobility changes slope at about 110 K, and this is the reason for the

different equations in table 2. For holes, the lattice mobility for temperatures below 72 K is from Braggins [15], whereas at higher temperatures the

results of Ludwig and Watters [17] are used.

The ionized impurity scattering mobility is calculated by the Brooks-Herring

[18,19] expression, with the modifications discussed by Li and Thurber [20]

and Li [21], For p-type material, individual mobilities are computed for

each band, and then the total mobility is found by weighting the mobility in

each band by the hole density in that band. The effect of carrier-carrier

14
Table 1. Symbols Used in Mobility Formulas.

rid L itcllla L l Ud i Symbol

r 1 u y 1 uM t Name Definition*

u SUMTD Total donor density (cm" 3)

SUMTA Total acceptor density (cm"^)

III U L N11 I ^ff'\r0 mnhil'il'v/ "Frip plpp1">"nn<; ( rvn^ /\! •

ULP Lattice mobility for holes (cm^/V-s)

UIN Ionized impurity mobility for electrons (cm^/V-s)

1 1 T P Tnm* 7pH imniir^'i mnh"i 1 i I'v "for h^lp<^ irm^/V*'^)

Y GAMMA Factor giving effect of carrier-carrier scattering

on ionized impurity mobility

1 CION Density of ionized impurities (cm" 3)

G(b) GN Log term in Brooks-Herring ionized impurity mobility

formula for electrons

U r Inn "hp v^m "in Rv^nnl^Q — Mpv^K'inn inni'yprl TmniiP"it"V/ mn hi 1 i "I" v formula for holes

SUMID Total ionized donor density (cm" 2) V SUMIA Total ionized acceptor density (cm" 3)

n' PN^TAR LiliO 1 r\r\ F*F"f L. \ ICl-LIVC pr "t" 1 \/p plprtK^nn CICV^LlUll <^r^ppninn OV^ICCIIIIiy Hpn*^i1"\/ UCIIOIUjf frm"^^ ^V-IH J

P

C"P*fQ^"t""iv/£i ki/~\np c^v^ppnnnn HpnCT'hx/ tTTcCLiVc no 1 c bcrcciiiiiy uciibiLy ^lim j

n T Fl tTT LI mes relative qiciccliil LuiisLaiiu Lniicb \jc i - mittivity of free space

k Boltzmann constant

h Planck constant (li = h/2-n)

q Electronic charge

mo Free electron mass < Longitudinal effective mass (in units of mp)

*

Densi ty-of-states effective mass (in units of m^)

* Conductivity effective mass (in units of m^)

UNIS Neutral impurity scattering mobility (cm^/V-s)

\

CNUT Density of neutral impurities (cm"^)

n CN Electron density (cm"^)

P CP Hole density (cm" 3)

* Quantities in SI units unless otherwise stated.

15
Table 2. Formulas for Calculating Electron Mobility.

Scatteri ng Mechani sm Formula Sta tement

Number

1 fl f ti TP + electron-electron

Nn111] <^ 3ij X IQl^ 1 \J cm-3 Lrlll T < 108 K ui = 1 89 X 107T-1-52 ri _ O 08 Nn/f2 x 1017)1 1 u ; J QO T > 108 K M|_ = 7.98 X 108T-2-32 [i _ o.08 Nd/(2 x 1017)] 10

Nd > 3 X 1017 cm-3

T £ 108 K ML = l -89 X 107T-1-52 (0.884) 22

T > 108 K vi|_ = 7.98 x io8t-2-32 (0.884) 30

Ionized impurity +

el ectron-electron

yj = Y

X 7.3 X lOl^jl. 5/NiG(b)

Y

= [1 - exp(-n'/ND+)]

n' = n + p + Nd+ (1 - Nd+/Nd)

G(b) = ln(b + 1 ) - b/(b + 1

)

24k e m* (kT)2

b = ° ^ X 10-6

q2li2n'

90

86

42

80

52

Combined lattice

and ionized impurity

P| T = Pi f(X)

LI L f(X) = 1 + X2 [CiXcosX + sinX(SiX - J)]

X2 = 6

104

104

98

Neutral impurity _ n no r2 fkTvl/2 1 , Em 1/2

P

_ 1.136 X 10-19 m^

t.. — — ......

114

112

k2 mo

_ 2TT3n3 mi

no

Total electron y = (Vp, t + l/y.,)"i 116

mobility ^

*These statement numbers refer to subroutine MOB.

16
Table 3. Formulas for Calculating Hole Mobility.

Scattering

Mechanism Formula Statement

Number*

Lattice + N„ < 3 X 1017 cm-3

A hole-hole

T < 72 K = 1.6 X lo^T-i-S^ [1 - 0.08 Na/(2 x lO^^)] 118

T > 72 K = 2.3 X lon-2-7 [1 . 0.08 Na/(2 x lO^^)]

>_ 3 X 1017

120

T < 72 K = 1.6 X 107T-1-54 (0.884) 132

1 > 79 V - 9 "3 in9T-2 7 DQ/I ^ 1 C K ~ • lU 1 • ^U.cScS'tj 1 /I n 1 f u

Ionized impurity

+ hnl p-hnl

p

yCpj^ +

('"d2/'"dl

^ ^ ' ^13 (md3/>)^-^exp(-0.044/kT)]

1 96

[1 + (mj3/mJ^)i-5 + (m33/m;^)i-5 exp(-0.044/kT)]

Y

= [1 - exp(-pVNj)] 198

P'

= p + n + N+ (1 - N+/N„) ^ A A A

152

for i = 1, 2, 3

^li

23-5 (4TTKe )2 m*.0-5(kT)i-5

0 dl ^ ^Q_2

ui-5 q3 m*. Nj G(b.)

192

G(b. ) = ln(b. + 1) - b./(b. + 1) 190

24Ke m*. (kT)2

0 di

q2ti2p'

10-5 162

Combined lattice

and ionized impuri ty

- f(X)

f(X) = 1 + X2[CiXcosX + sinX(SiX - J)]

X2 = 6 P^^^I

214

214

208

Neutral impurity ,(m* /m* )i-5 + exp(-0.044/kT)]

Nl ^N2^"'d2""dl^ ' ^N3^"'d3""dl

^N

[1 + (m*2/mdi)^-5 + (md3/'^dl)^"^ exp(-0.044/kT)]

for i = 1 , 2, 3

^Ni

= [| (|I-)

1/2 . E 1/2 + - (^) 3 ^ kT^

1.136 X 10-19 m*.

<^ m.

2Tr3q 3m

di

^"^ 5N.,4TrKe h3

N 0

X 10-

224

222

221

220

Total hole

mobil ity

p = (I/pli + Vp,^)-i 226

These statement numbers refer to subroutine MOB . 17
scattering in reducing the lattice and ionized impurity mobilities is taken

into account following the work of Li and Thurber [20]. The mobility due to

both lattice and ionized impurity scattering is combined according to the

mixed-scattering formula of Debye and Conwell [22] involving sine and cosine

integrals.

The expressions used to calculate the carrier mobilities contain the dielectric constant of silicon as a parameter. The value of 11.7 for the relative

dielectric constant of silicon at 300 K was taken from Dunlap and Watters

[23]. The temperature variation was calculated from the index of refraction

data of Cardona et at. [24], assuming that the index of refraction is the

square of the relative dielectric constant. The resulting expression for the

program parameter DIEL evaluated in subroutine MOB statement No. 2 is:

DIEL = 4 (absolute dielectric constant) and

DIEL = (1.2711 X 10"^) exp(7.8 x 10~^T) farads per meter.

The neutral impurity scattering mobility is calculated from the Erginsoy

equation [25] , with the modification proposed by Sclar [26] . The net mobility is found by combining reciprocally the neutral impurity mobility with the

result for the combined lattice and ionized impurity mobility.

Both electron and hole mobilities are calculated, even though usually only

one carrier is important. An effective conductivity mobility is calculated

by summing the electron and hole mobilities weighted by the density of each

carrier. The effective conductivity mobility computed this way is the same

as the majority carrier conductivity mobility in extrinsic specimens. In

intrinsic specimens, it corresponds to a hypothetical mobility for the total

carrier density (i.e., electron plus hole density). The effective conductivity mobility is the only value of mobility listed in the no-option output

format. Additional mobility values are available in the DOUBLE ON output

format. The expressions for calculating the conductivity, resistivity, and

Hall coefficient include appropriately weighted contributions from both

charge carriers.

18
4. Annotated Program Listing

C ELECTRICAL PROPERTIES OF SILICON

C

C PROGRAM CAN HANDLE 2 DONOR LEVELS AND 3 ACCEPTOR LEVELS

C

C

C VARIABLE NAMES

C

C LEVEL NAME DENSITY ENERGY DEGEN. ION. CONC.

C

C DONOR 1 NAME1 DNI ENINI DGl DI1

C DONOR 2 NAME2 DN2 ENIN2 DG2 DI2

C ACCEPTOR 1 NAME3 DN3 ENIN3 DG3 DI3

C ACCEPTOR 2 NAME4 DN4 ENIN4 DG4 DI4

C ACCEPTOR 3 NAMES DN5 ENIN5 DG5 DI5

C

C

C STATEMENT NUMBERS ENDING IN ZERO ARE PART OF THE PROGRAM LOGIC

C OTHER NUMBERS WERE ADDED FOR IDENTIFICATION PURPOSES

C A MODIFIED MKS SYSTEM OF UNITS IS USED WITH CM REPLACING METERS

C

C T = THE TEMPERATURE IN KELVIN

C EG = THE ENERGY GAP OF SILICON IN ELECTRON VOLTS

C EXCITON ENERGY OF 0.01 EV IS NOT INCLUDED IN EG

C EF = FERMI LEVEL IN EV (DETERMINED IN PROGRAM BY TRIAL AND ERROR)

C EF IS TAKEN TO BE ZERO AT THE VALENCE BAND EDGE

C CN = THE CONCENTRATION OF ELECTRONS IN THE CONDUCTION BAND IN CM-3

C CP = THE CONCENTRATION OF HOLES IN THE VALENCE BAND IN CM-3

C MOB = MOBILITY ON THE OUTPUT PAGES, BUT NOT IN THE PROGRAM:

C IN THE PROGRAM, THE FOLLOWING NOTATION IS USED:

C UDN = THE ELECTRON MOBILITY IN CM**2/ (VOLT-SEC

)

C UDP = THE HOLE MOBILITY IN CM**2/ (VOLT-SEC)

C UD = THE CONDUCTIVITY MOBILITY IN CM**2/ (VOLT-SEC

)

C RH = THE HALL COEFFICIENT IN CM**3/C0UL0MB

C SIGMA = THE CONDUCTIVITY IN MHO/CM

C RHO = THE RESISTIVITY IN OHM-CM

C ON THE OUTPUT PAGE THE FOLLOWING NOTATION HAS BEEN USED:

C Dl ION = THE DENSITY OF IONIZED DONOR 1 IN CM-3

C D2 ION = THE DENSITY OF IONIZED DONOR 2 IN CM-3

C A1 ION = THE DENSITY OF IONIZED ACCEPTOR 1 IN CM-3

C A2 ION = THE DENSITY OF IONIZED ACCEPTOR 2 IN CM-3

C A3 ION = THE DENSITY OF IONIZED ACCEPTOR 3 IN CM-3

C FNC AND FNV ARE THE DENSITY OF STATES IN THE CONDUCTION AND

C VALENCE BANDS RESPECTIVELY IN CM-3

C

C INPUT DATA CARDS:

C ORDER OF CARDS IS THE SAME AS IN THE TABLE ABOVE

C ORDER OF DATA ON CARD = NAME, DENSITY, ENERGY, DEGEN.

C TO ELIMINATE A LEVEL FROM THE PROGRAM, NAME IT 'DUMMY'

C THE LAST DATA CARD SHOULD READ SIMPLY 'STOP'

C

C THE FIRST DATA CARD OF ANY SET CAN BE USED TO CONTROL THE PROGRAM

C SEE THE LIST OF IF STATEMENTS FOLLOWING STATEMENT 10 FOR SUCH USE

C 'DOUBLE ON' TURNS ON 2 OUTPUT DATA LINES AT EACH TEMP.

19
C 'DOUBLE OFF' RESTORES SINGLE OUTPUT LINE FORMAT

C 'CONSTANTS' GENERATES A PAGE OF THE PARAMETERS THAT DO NOT

C CHANGE WITH DOPING LEVEL

C 'DOLOOP XX' (WHERE XX = D1 - A3) PRODUCES MULTIPLE RUNS

C CARD FORMAT = DOLOOP XX, INC. DENSITY, NO. OF RUNS

C 'STOP' IS USED TO TERMINATE THE SESSION WITH THE COMPUTER

C EXTERNAL TEST

DIMENSION NAME1 (2) ,NAME2 (2) ,NAME3(2) ,NAME4(2) ,NAME5 (2)

COMMON NAME1 ,NAME2,NAME3,NAME4,NAMES

COMMON DN1 ,DN2,DN3,DN4,DN5,SUMTD,SUMTA

COMMON DI1,DI2,DI3,DI4,DI5

COMMON EN1,EN2,EN3,EN4,EN5

COMMON DG1,DG2,DG3,DG4,DG5

COMMON T,CN, CP, EG, EFME,EFMH,FNC,FNV,CAYT,TPOWR,Q, ALPHA, BETA

COMMON UD,UDN,ULN,UDP,ULP,SIGMA,RHO,RH

COMMON 0UT8,0UT11,ISTAT

Q=1.6022E-19

C INITIALIZE THE SETTINGS OF THE OPTION SWITCHES

LOOPSW=0.

D0UBSW=6H0FF

C

C READ THE Dl DATA (OR CONTROL) CARD

10 READ 20, NAME1,DN1,ENIN1,DG1

IF (NAME1 (1) .EQ.6HD0UBLE) GO TO 730

IF (NAME1 (1) .EQ.6HC0NSTA) GO TO 50

IF (NAMEKI) .EQ.6HD0L00P) GO TO 740

IF (NAME1 (1).EQ.6HST0P ) GO TO 760

CC READ THE REMAINING FOUR DATA CARDS

READ 20, NAME2,DN2,ENIN2,DG2

READ 20, NAME3,DN3,ENIN3,DG3

READ 20, NAME4,DN4,ENIN4,DG4

READ 20, NAMES, DNS, ENIN5,DG5

20 FORMAT (2A6,E10.4,2F10.4)

C IF (LOOPSW.EQ.O.) GO TO 50

C PERFORM THE NECESSARY DOLOOP INSTRUCTIONS

DO 720 NRUN=1,NDIT,1

IF (NRUN.EQ.1.) GO TO SO

IF (L00PSW.EQ.1.) DN1=DN1+DELDEN

IF (L00PSW.EQ.2. ) DN2=DN2+DELDEN

IF (L00PSW.EQ.3.) DN3=DN3+DELDEN ^

IF (L00PSW.EQ.4.) DN4=DN4+DELDEN

IF (LOOPSW.EQ.S.) DN5=DNS+DELDEN

IF (LOOPSW.GT.S.) GO TO 30

GO TO SO

30 PRINT 40

40 FORMAT (1 H0,26HLOOPSWITCH IS OUT OF RANGE)

GO TO 10

C RUNN0=1. DURING FIRST RUN, THEN 2. UP TO 77 KELVIN, THEN 3. TO

C ROOM TEMPERATURE, THEN 4. ABOVE ROOM TEMPERATURE

50 RUNN0=1

.

20
C CALULATE REDUCTION IN ACTIVATION ENERGIES

C EQUATION OF RENIN ET AL. FOR PHOSPHORUS-DOPED SILICON IS

C GENERALIZED FOR USE WITH OTHER ACTIVATION ENERGIES

C SEE PAPER IN SOVIET PHYS. -SOLID STATE 7, 2580 (1966)

SUMTD=DN1+DN2

C GUARD AGAINST ZERO DIVISOR IN CALCULATION OF RDD

IF (SUMTD.LE.O.) RDD=1.E20

IF (SUMTD.LE.O.) GO TO 60

RDD=1 .E8*(3./(4.*3.1416*SUMTD))**0.3333

60 SUMTA=DN3+DN4+DN5

C GUARD AGAINST ZERO DIVISOR IN CALCULATION OF RDA

IF (SUMTA.LE.O.) RDA=1.E20

IF (SUMTA.LE.O.) GO TO 70

RDA=1 . E8* (3. / (4. *3 . 1 41 6*SUMTA) )**0. 3333

70 EN1=ENIN1

C FOLLOWING STATEMENT COMPUTES THE REDUCTION IN ACTIVATION ENERGY

C OMIT THIS STATEMENT IF FIXED ENERGY LEVEL ENIN1 IS DESIRED

C SAME COMMENTS APPLY TO OTHER DOPANT STATES

EN1 =ENIN1 -0.36* (0.5+21 ./RDD )*EXP(-RDD/21 .)-3.6E-8*SUMTA**0. 3333

IF (EN1-0.0) 80,90,90

C IF ANY ACTIVATION ENERGY IS NEGATIVE, SET IT EQUAL TO ZERO

80 EN1=0.0

90 EN2=ENIN2

EN2=ENIN2-0. 36* (0. 5+21 ./RDD)*EXP(-RDD/21 .)-3.6E-8*SUMTA**0. 3333

IF (EN2-0.0) 100,110,110

100 EN2=0.0

110 EN3=ENIN3

EN3=ENIN3-0.36*(0.5+21./RDA)*EXP(-RDA/21 .)-3.6E-8*SUMTD**0.3333

IF (EN3-0.0) 120,130,130

120 EN3=0.0

130 EN4=ENIN4

EN4=ENIN4-0.36* (0.5+21 . /RDA)*EXP(-RDA/21 . )-3.6E-8*SUMTD**0. 3333

IF (EN4-0.0) 140,150,150

140 EN4=0.0

150 EN5=ENIN5

EN5=ENIN5-0. 36*(0. 5+21 ./RDA)*EXP(-RDA/21 .)-3.6E-8*SUMTD**0. 3333

IF (EN5-0.0) 160,170,170

160 EN5=0.0

C THE FIRST TEMPERATURE WILL BE 4.2 KELVIN

170 T=4.2

C T WILL BE INCREMENTED IN EQUAL 1000/T INTERVALS

RTEMP=1000./T

C

C SET UP THE OUTPUT PAGE

WRITE (6,180)

180 FORMAT (1H1,32HELECTRICAL PROPERTIES OF SILICON)

IF (NAME1 (1).EQ.6HC0NSTA) GO TO 320

C READ-OUT ALL PERTINENT INPUT DATA

WRITE (6,190)

190 FORMAT (1 H0,26HDOPANT STATES FOR THIS RUN)

WRITE (6,200)

200 FORMAT (1H0,77H DOPANT DENSITY ENERGY (INPUT) E 2NERGY(CALC.) DEGENERACY)

21
IF (NAME1 (1) .EQ.6HDUMMY ) 60 TO 210

WRITE (6,250) NAME1 ,DN1 ,ENIN1 ,EN1 ,DG1

210 IF (NAME2(1) .EQ.6HDUMMY ) 60 TO 220

WRITE (6,250) NAME2,DN2,ENIN2,EN2,D62

220 IF (NAME3(1) .EQ.6HDUMMY ) 60 TO 230

WRITE (6,250) NAME3,DN3,ENIN3,EN3,D63

230 IF (NAME4(1) .EQ.6HDUMMY ) GO TO 240

WRITE (6,250) NAME4,DN4,ENIN4,EN4,DG4

240 IF (NAME5(1) .EQ.6HDUMMY ) GO TO 260

WRITE (6,250) NAMES, DN5,ENIN5,EN5,D65

250 FORMAT (1X,2A6,6X,1PE9.3,7X,0PF7.4,8X,F8.4,10X,F8.4)

260 WRITE (6,270)

270 FORMAT (IHO)

C PROVIDE COLUMN HEADIN6S FOR OUTPUT TABLE

WRITE (6,280)

280 FORMAT (1H0,129H TEMP 1000/T EF N P

2 COND. MOB RH RHO D1 ION D2 ION A1

3I0N A2 ION)

WRITE (6,290)

290 FORMAT (1H ) IF (D0UBSW.NE.6H0N ) GO TO 350

WRITE (6,300)

300 FORMAT (1H ,130H EG TEST KT/Q EG-EF A3

2I0N SIGMA MOB-(LAT) +IONIZED +NEUTRAL MOB+(LAT) +IONI

3ZED +NEUTRAL)

WRITE (6,310)

310 FORMAT (IHO)

GO TO 350

C COLUMN HEADINGS FOR 'CONSTANTS' OPTION

320 WRITE (6,330)

330 FORMAT (1H0,128H TEMP 1000/T KT/Q EG DI

2EL M08-(LAT) MOB+(LAT) T**3/2 ME MH FN

3C FNV)

WRITE (6,340)

340 FORMAT (IHO)

C EFME AND EFMH ARE THE EFFECTIVE MASSES OF ELECTRONS AND HOLES

C 1039 (1967). EFME AND EFMH EQS FITTED OVER TEMP RANGE 0 TO 600 K

350 EFME= ( ( ( ( (4 . 54649E-1 7*T-9 . 66067E-1 4) *T+8 . 04032E-1 1

) *T-3 . 3201 30E-8

)

2*T+6.83008E-6)*T-. 0001 61 708)*T+1. 0627

352 EFMH=( ( (((1.11 997E-16*T-2. 596730E-13)*T+2.30049E-10)*T-9.67212E-8)

2*T+1 .85678E-5)*T-.000523548)*T+. 590525

353 CAYT=8.6173E-5*T

C CONST = 2*(2*PI*K*(FREE ELECTRON MASS) /H**2)**(3/2)

C0NST=4.829E15

TP0WR=T**1.5

C FNC AND FNV ARE THE DENSITY OF STATES IN THE CONDUCTION AND

C VALENCE BANDS RESPECTIVELY

354 FNC=C0NST*(EFME**1 .5)*TP0WR

355 FNV=C0NST*(EFMH**1 .5)*TP0WR

C EG FROM FIT OF DATA FROM MACFARLANE, PHYS. REV. Ill, 1245 (1958)

356 EG=(( (-3. 80977E-13*T+9.95E-10)*T-8. 701 10E-7)*T+. 0000323741 )*T+1. 15

2556

C IN 'CONSTANTS' OPTION, WE NEED NOT DETERMINE EF

22
IF (NAME1 (1) .EQ.6HC0NSTA) 60 TO 530

EF=0.0

EFG=1 .2

RE=1 .OE-6

AE=1 .OE-6

IFLAG=0

C CALL SUBROUTINE TO FIND VALUE OF EF WHICH MAKES TEST VERY SMALL

CALL ZEROIN (TEST,EF,EFG,RE,AE,IFLAG)

IF (IFLAG.GT.2) GO TO 690

C

C CALL SUBROUTINE TO CALCULATE MOBILITY

CALL MOB

C ERROR AND/OR WARNING DIAGNOSTIC MESSAGES

IF (ALPHA. LT. -2.) GO TO 370

WRITE (6,360) ALPHA

360 FORMAT (20X,1 5H (EF-EG) /CAYT = ,F8.4,

2 33H ERROR IN N MAY EXCEED 1 PERCENT)

370 IF (BETA.LT.-2.) GO TO 390

BETAN=-BETA

WRITE (6,380) BETAN

380 FORMAT (20X,1 OHEF/CAYT = ,F8.4,

2 33H ERROR IN P MAY EXCEED 1 PERCENT)

390 IF (ISTAT.GE. 10000) WRITE (6,400)

400 FORMAT (20X,72HB0RN APPROXIMATION FOR IONIZED SCATTERING NOT VALID

2 FOR MAJORITY CARRIER)

C

C OUTPUT THE RESULTS AND SET UP THE NEXT RUN

WRITE (6,410) T,RTEMP,EF

410 FORMAT ( F1 1 . 4,2X, F9 . 4, F1 1 . 6)

IF (CN.NE.O.) WRITE (6,420) CN

IF (CP.NE.O.) WRITE (6,430) CP

IF (UD.NE.O.) WRITE (6,440) UD

IF (RH.NE.9.999999E19.AND.RH.NE.0.) WRITE (6,450) RH

IF (RH.NE.9.999999E19.AND.RH0.NE.0.) WRITE (6,460) RHO

IF (DI1.NE.0.) WRITE (6,470) DI1

IF (0I2.NE.0.) WRITE (6,480) DI2

IF (DI3.NE.0.) WRITE (6,490) DI3

IF (DI4.NE.0.) WRITE (6,500) DI4

420 FORMAT (1 H+,33X,1 PE1 1 . 3)

430 FORMAT (1H+,44X,1PE11 .3)

440 FORMAT (1 H+,55X,1 PEI 1 . 3)

450 FORMAT (1 H+,66X,1 PEI 1 .3)

460 FORMAT (1 H+,77X,1 PEI 1 . 3)

470 FORMAT (1H+,88X,1PE1 1 .3)

480 FORMAT (1 H+,99X,1 PEI 1 . 3)

490 FORMAT (1 H+,1 10X,1 PEI 1 .3)

500 FORMAT (1H+,121X,1PE10.3)

IF (D0UBSW.NE.6H0N ) GO TO 590

0UT3=EG-EF

TESTF=TEST(EF)

WRITE (6,510) EG,TESTF,CAYT,0UT3

510 FORMAT (F11.6,1PE11.3,1PE11.4,2X,F9.6)

IF (DI5.NE.0.) WRITE (6,430) DI5

23
IF (RH.NE.9.999999E19.AND.SIGMA.NE.0.) WRITE (6,440) SIGMA

IF (ULN.NE.O.) WRITE (6,450) ULN

IF (0UT8.NE.0.) WRITE (6,460) OUTS

IF (UDN.NE.O.) WRITE (6,470) UON

IF (ULP.NE.O.) WRITE (6,480) ULP

IF (0UT11 .NE.O.) WRITE (6,490) 0UT1

1

IF (UDP.NE.O.) WRITE (6,500) UDP

WRITE (6,520)

520 FORMAT (1H0)

GO TO 590

C OUTPUT INSTRUCTIONS FOR THE 'CONSTANTS' OPTION

530 RDIEL=11.4294*EXP(7.8E-5*T)

IF (T.GE.108.) GO TO 540

ULN=1.89E7*T**(-1.52)

GO TO 550

540 ULN=7.98E8*T**(-2.32)

550 IF (T.GE.72.) GO TO 560

ULP=1.6E7*T**(-1.54)

GO TO 570

560 ULP=2.3E9*T**(-2.7)

570 WRITE (6,580) T,RTEMP,CAYT,EG,RDIEL,ULN,ULP,TPOWR,EFME,EFMH,FNC,FN

2V

580 FORMAT (0P2F11 .4,1PE11 .4,2X,0PF9.4,F11 .4,1P7E11 .3)

C SET UP NEXT TEMPERATURE

C THE MAXIMUM TEMPERATURE OF INTEREST IS 500 DEGREES KELVIN

590 IF (T-500.) 600,710,710

600 IF (RUNNO-2.) 610,620,630

610 RUNN0=2.

C SECOND TIME AROUND, THE TEMPERATURE WILL BE 20 DEGREES KELVIN

T=20.

RTEMP=1000./T

GO TO 350

C AFTER THE SECOND TIME AROUND, 1000. /T IS DECREMENTED BY 2.

620 RTEMP=RTEMP-2.

T=1000./RTEMP

C AN EVALUATION AT LIQUID NITROGEN TEMPERATURE IS DESIRED

IF (T.GE.77.) GO TO 640

GO TO 350

630 IF (T.EQ.77.) GO TO 650

C AN EVALUATION AT ROOM TEMPERATURE (300 KELVIN) IS DESIRED

IF (RUNN0.EQ.4.) GO TO 680

IF (T.EQ.300.) GO TO 670

C FROM 77 KELVIN TO ROOM TEMPERATURE, 1000/T IS DECREMENTED BY 1.

RTEMP=RTEMP-1.

T=1000./RTEMP

IF (T.GT.300.) GO TO 660

GO TO 350

640 T=77.

RTEMP=1000./T

RUNN0=3.

GO TO 350

650 RTEMP=12.

T=1000./RTEMP

24
GO TO 350

660 T=300.

RTEMP=1000./T

GO TO 350

670 RTEMP=3.

T=1000./RTEMP

RUNN0=4.

GO TO 350

C ABOVE ROOM TEMPERATURE 1000, /T IS DECREMENTED BY 0.5

680 RTEMP=RTEMP-0.5

T=1000./RTEMP

GO TO 350

690 WRITE (6,700) IFLAG

700 FORMAT (1 H0,51 HCHARGE BALANCE EQUATION COULD NOT BE SOLVED, IFLAG=

2,11)

GO TO 590

710 IF (D0UBSW.NE.6H0N ) GO TO 720

C INITIALIZE CERTAIN VARIABLES FOR NEXT RUN

ULN=0.

0UT8=0.

UDN=0.

ULP=0.

0UT11=0.

UDP=0.

720 CONTINUE

IF (NAME1 (1 ) .EQ.6HC0NSTA) 60 TO 10

LOOPSW=0.

HEADSW=6H0FF

GO TO 10

C ROUTINE TO SET-UP OR CANCEL DOUBLE OUTPUT LINE FORMAT

C DOUBSW IS A SWITCH USED TO CONTROL THE OUTPUT FORMAT

730 D0UBSW=6H0N

IF (NAME1 (2) .EQ.6H0N ) GO TO 10

IF (NAME1 (2) .EQ.6H ON ) GO TO 10

C DEFAULT VALUE IS TAKEN TO MEAN 'DOUBLE OFF'

D0UBSW=6H0FF

GO TO 10

C ROUTINE TO SET-UP DOLOOP OPTION

740 DELDEN=DN1

NDIT=ENIN1

IF (NAME1 (2) .EQ.6H D1 ) L00PSW=1

.

IF (NAME1 (2).EQ.6H D2 ) L00PSW=2.

IF (NAME1 (2) .EQ.6H A1 ) L00PSW=3.

IF (NAME1 (2).EQ.6H A2 ) L00PSW=4.

IF (NAME1 (2) .EQ.6H A3 ) L00PSW=5.

IF (LOOPSW.EQ.O.) WRITE (6,750)

750 FORMAT (1H0,34HINVALID ARGUMENT OF DOLOOP COMMAND)

GO TO 10

760 STOP

END

25
FUNCTION TEST (EF)

C

C THIS EXTERNAL FUNCTION DEFINES THE PARAMETER 'TEST'.

C SUBROUTINE ZEROIN LOCATES THE VALUE OF EF WHICH MINIMIZES 'TEST'.

DIMENSION NAME1 (2) ,NAME2 (2) ,NAME3 (2) ,NAME4(2) , NAMES (2)

COMMON NAME1 ,NAME2,NAME3,NAME4, NAMES

COMMON DNI ,DN2,DN3,DN4,DNS,SUMTD,SUMTA

COMMON DI1,DI2,DI3,DI4,DIS

COMMON EN1,EN2,EN3,EN4,ENS

COMMON D61,DG2,DG3,DG4,DG5

COMMON T,CN, CP, EG, EFME,EFMH,FNC,FNV,CAYT,TPOWR,Q, ALPHA, BETA

COMMON UD,UDN,ULN,UDP,ULP, SIGMA, RHO,RH

COMMON 0UT8,0UT11,ISTAT

C

C COMPUTATION OF CN

2 CN=0.

4 ALPHA=(EF-EG)/CAYT

IF (ALPHA. GT.1.) GO TO 10

IF (ALPHA. LE. -80.) 60 TO 20

C THE FOLLOWING EQUATION IS USED FOR ALPHA BETWEEN -80 AND +1

8 CN=FNC/((EXP(-ALPHA))+0.27)

GO TO 20

C THE FOLLOWING EQUATION IS USED FOR ALPHA GREATER THAN +1

10 CN=FNC*0.7522S3*((ALPHA*ALPHA+1 .7)**0.7S)

C COMPUTATION OF CP

20 CP=0.

22 BETA=-EF/CAYT

IF (BETA. GT.1.) GO TO 30

IF (BETA. LE. -80.) GO TO 40

C THE FOLLOWING EQUATION IS USED FOR BETA BETWEEN -80 AND +1

26 CP=FNV/((EXP(-BETA))+0.27)

60 TO 40

C THE FOLLOWING EQUATION IS USED FOR BETA GREATER THAN +1

30 CP=FNV*0.7S22S3*((BETA*BETA+1 .7)**0.7S)

C COMPUTATION OF DI1

40 DI1=0.

IF (NAME1 (1).EQ.6HDUMMY ) GO TO 60

P0WER=(EF-EG+EN1)/CAYT

IF (POWER. 6E. 80.) 60 TO 60

IF (POWER. LE. -80.) 60 TO SO

DI1=DN1/(1 .+(EXP(P0WER)*D61))

60 TO 60

50 DI1=DN1

C COMPUTATION OF 012

60 DI2=0.

IF (NAME2(1) .EQ.6HDUMMY ) 60 TO 80

P0WER=(EF-EG+EN2)/CAYT

IF (POWER. GE. 80.) 60 TO 80

IF (POWER. LE. -80. ) 60 TO 70

DI2=DN2/(1 .+(EXP(P0WER)*D62))

60 TO 80

70 DI2=DN2

C COMPUTATION OF DI3

26
80 DI3=0.

IF (NAME3(1 ) .EQ.6HDUMMY ) GO TO 100

P0WER=(EN3-EF)/CAYT

IF (POWER. GE. 80.) GO TO 100

IF (POWER. LE. -80.) 60 TO 90

DI3=DN3/(1 .+(EXP(P0WER)*D63))

GO TO 100

90 DI3=DN3

C COMPUTATION OF DI4

100 DI4=0.

IF (NAME4(1 ) .EQ.6HDUMMY ) GO TO 120

P0WER=(EN4-EF)/CAYT

IF (POWER. GT. 80.) GO TO 120

IF (POWER. LE. -80. ) GO TO 110

DI4=DN4/(1 .+(EXP(P0WER)*DG4))

GO TO 120

110 DI4=DN4

C COMPUTATION OF DI5

120 DI5=0.

IF (NAME5(1 ) .EQ.6HDUMMY ) GO TO 140

P0WER=(EN5-EF)/CAYT

IF (POWER. GE. 80.) GO TO 140

IF (POWER. LE. -80.) GO TO 130

DI5=DN5/(1 . + (EXP(P0WER)*D65))

GO TO 140

130 DI5=DN5

C TESTING TO SEE HOW CLOSE TO NEUTRALITY WE HAVE COME

C THE PARAMETER TEST IS THE NET DENSITY OF RESIDUAL CHARGES

C WHEN THE FERMI LEVEL IS AT ITS PRESENT POSITION

140 TEST=(CN+DI3+DI4+DI5)-(CP+DI1+DI2)

RETURN

END

27
SUBROUTINE ZEROIN(F,B,C,RE,AE,IFLAG)

C

C SANDIA MATHEMATICAL PROGRAM LIBRARY

C MATHEMATICAL COMPUTING SERVICES DIVISION 5422

C SANDIA LABORATORIES

C P. 0. BOX 5800

C ALBUQUERQUE, NEW MEXICO 87115

C CONTROL DATA 6600 VERSION 4.5, 1 NOVEMBER 1971

CC MODIFIED TO RUN AT NBS BY D. KAHANER DIV 713

CC ABSTRACT

C ZEROIN SEARCHES FOR A ZERO OF A FUNCTION F(X) BETWEEN

C THE GIVEN VALUES B AND C UNTIL THE WIDTH OF THE INTERVAL

C (B,C) HAS COLLAPSED TO WITHIN A TOLERANCE SPECIFIED BY

C THE STOPPING CRITERION, ABS(B-C) .LE. 2.*(RW*ABS(B)+AE)

.

CC DESCRIPTION OF ARGUMENTS

C F - NAME OF THE REAL VALUED EXTERNAL FUNCTION. THIS NAME

C MUST BE IN AN EXTERNAL STATEMENT IN THE CALLING

C PROGRAM. F MUST BE A FUNCTION OF ONE REAL ARGUMENT.

C B - ONE END OF THE INTERVAL (B,C) . THE VALUE RETURNED FOR

C B USUALLY IS THE BETTER APPROXIMATION TO A ZERO OF F. C C - THE OTHER END OF THE INTERVAL (B,C)

C RE - RELATIVE ERROR USED FOR RW IN THE STOPPING CRITERION.

C IF THE REQUESTED RE IS LESS THAN MACHINE PRECISION,

C THEN RW IS SET TO APPROXIMATELY MACHINE PRECISION.

C AE - ABSOLUTE ERROR USED IN THE STOPPING CRITERION. IF THE

C GIVEN INTERVAL (B,C) CONTAINS THE ORIGIN, THEN A

C NONZERO VALUE SHOULD BE CHOSEN FOR AE.

C IFLAG - RETURNS A STATUS OF THE RESULTS INDICATING WHICH

C OF THE FOLLOWING CONDITIONS HOLD.

C A - ABS(B-C) .LE. 2.*(RW*ABS CB)+AE)

C B - F(B) * FCC) .LT. 0.

C C - ABSCFCB)) .LE. ABSCFCO)

C D - ABSCFCB )) .LE. MAXCABSCFCB )),ABSCFCC )))

C OUT IN IN

C E - NUMBER OF EVALUATIONS OF FCX) .LE. 500

C =1 INDICATES NORMAL CASE. ALL CONDITIONS ABOVE HOLD.

C =2 INDICATES FCB) = 0. CONDITION A MAY NOT HOLD.

C =3 INDICATES CONDITIONS A, B, C, AND E HOLD BUT D DOES

C NOT. CB,C) PROBABLY CONTAINS A SINGULAR POINT OF F.

C =4 INDICATES CONDITIONS A AND E HOLD BUT B DOES NOT.

C A LOCAL MINIMUM OF FCX) I.M CB,C) MAY HAVE BEEN FOUND

C =5 INDICATES SEARCH WAS ABORTED WHEN CONDITION E FAILED

C

C REFERENCES

C 1. L F SHAMPINE AND H A WATTS, ZEROIN, A ROOT-SOLVING CODE,

C SC-TM-70-631, SEPT 1970.

C 2. T J DEKKER, FINDING A ZERO BY MEANS OF SUCCESSIVE LINEAR

C INTERPOLATION, *CONSTRUCTIVE ASPECTS OF THE FUNDAMENTAL

C THEOREM OF ALGEBRA*, EDITED BY B DEJON AND P HENRICI, 1969.

C 3. L F SHAMPINE AND R C ALLEN, NUMERICAL COMPUTING—

28
AN INTRODUCTION, 1973.

INITIALIZE FOR UNIVAC 1108

DATA ER/6.0E-8/

INITIALIZE IN PORTABLE MANNER

DATA ER/0.0/

IF (ER .EQ. 0.0) ER = 4. * (R1MACH (4) )

RW=AMAX1 (RE,ER)

IC=0

ACBS=ABS(B-C)

A=C

FA=F(A)

FB=F(B)

FC = FA

K0UNT=2

FX=AMAX1 (ABS(FB),ABS(FC))

IF (ABS(FC) .GE. ABS(FB)) GO TO 2

PERFORM INTERCHANGE

A=B

FA=FB

B=C

FB=FC

C=A

FC = FA

CMB=0.5*(C-B)

ACMB=ABS(CMB)

TOL=RW*ABS(B)+AE

TEST STOPPING CRITERION

IF (ACMB .LE. TOL) GO TO 10

CALCULATE NEW ITERATE IMPLICITLY AS B+P/Q

WHERE WE ARRANGE P .GE. 0.

THE IMPLICIT FORM IS USED TO PREVENT OVERFLOW.

P=(B-A)*FB

Q=FA-FB

IF (P .GE. 0.) GO TO 3

P=-P

Q=-Q

UPDATE A AND CHECK FOR SATISFACTORY REDUCTION

IN THE SIZE OF OUR BOUNDING INTERVAL.

A=B

FA=FB

IC=IC+1

IF (IC .LT. 4) GO TO 4

IF (8.*ACMB .GE. ACBS) GO TO 6

IC=0

ACBS=ACMB

29
C TEST FOR TOO SMALL A CHANGE

4 IF (P .GT. ABS(Q)*TOL) 60 TO 5

C

C INCREMENT BY TOLERANCE

B=B+$IGN(TOL,CMB)

60 TO 7

C

C ROOT 0U6HT TO BE BETWEEN B AND (C+B)/2.

5 IF (P .6E. CMB*Q) 60 TO 6

CC INTERPOLATE

B=B+P/Q

60 TO 7

C 6 B=0.5*(C+8)

C BISECT

C

C HAVE COMPLETED COMPUTATION FOR NEW ITERATE B

7 FB=F(B)

IF (FB .EQ. 0.) 60 TO 11 K0UNT=K0UNT+1

IF (KOUNT .6T. 500) 60 TO 15

C

C DECIDE WHETHER NEXT STEP IS INTERPOLATION OR EXTRAPOLATION

IF (SI6N(1 .0,FB) .NE. SI6N (1 .0, FC) ) 60 TO 1 C=A

FC = FA

60 TO 1 C

C

C FINISHED. PROCESS RESULTS FOR PROPER SETTIN6 OF IFLA6

C 10 IF (FB*FC .6T. 0.) 60 TO 13

IF (ABS(FB) .6T. FX) 60 TO 12

IFLA6 = 1 RETURN

11 IFLA6 = 2

RETURN

12 IFLA6 = 3

RETURN

13 IFLA6 = 4 I RETURN

15 IFLA6 = 5

RETURN

END

30
SUBROUTINE MOB

C

C SUBROUTINE TO EVALUATE CARRIER MOBILITIES AND RELATED ITEMS

C THIS SUBROUTINE CALLS SUBROUTINE SICI (SI,CI,X)

DIMENSION NAME1 (2) ,NAME2 (2) ,NAME3(2) ,NAME4(2) ,NAME5 (2)

COMMON NAME1,NAME2,NAME3,NAME4, NAMES

COMMON DN1,DN2,DN3,DN4,DN5,SUMTD,SUMTA

COMMON DI1,DI2,DI3,DI4,DI5

COMMON EN1,EN2,EN3,EN4,EN5

COMMON DG1,DG2,DG3,DG4,DG5

COMMON T,CN, CP, EG, EFME,EFMH,FNC,FNV,CAYT,TPOWR,Q, ALPHA, BETA

COMMON UD,UDN,ULN,UDP,ULP,SIGMA,RHO,RH

COMMON 0UT8,0UT11,ISTAT

C UPON RETURNING, ISTAT INDICATES THE STATUS OF THE COMPUTATIONS

C EACH DIGIT (POSITION) OF ISTAT IS USED INDIVIDUALLY

C ISTAT=0

C DIEL = 4*PI*(ABS0LUTE DIELECTRIC CONSTANT IN FARADS/METER)

C RELATIVE DIELECTRIC CONSTANT AT 300 KELVIN = 11.7

C DUNLAP AND WATTERS, PHYS.REV. 92, 1396-1397 (1953)

C TEMPERATURE DEPENDENCE CALCULATED FROM INDEX OF REFRACTION DATA

C CARDONA ET AL, J. PHYS. CHEM. SOLIDS. 8, 204-206 (1959)

2 DIEL=1 .2711E-9*EXP(7.8E-5*T)

C

C SEE PAPERS BY LI AND THURBER, SSE 20, 609 AND LI, SSE 21, 1109

SUMID=DI1+DI2

SUMIA=DI3+DI4+0I5

CI0N=DI1+DI2+DI3+DI4+DI5

CNUT=DN1+DN2+DN3+DN4+DN5-CI0N

C WE MUST GUARD AGAINST ZERO DIVISORS IN THE FOLLOWING COMPUTATIONS

C SOME OF THESE ZERO DEVISORS WILL BE FLAGGED BY SETTING ISTAT

IF (CION.LE.O.) CI0N=1.

IF (CNUT.LE.O.) CNUT=1.

C CALCULATION OF ELECTRON MOBILITY

C LATTICE SCATTERING (WITH CORRECTION FOR ELECT . -ELECT. SCATTERING)

C EQ FOR CORRECTION DEPENDS ON DONOR DENSITY

C EQS FOR TEMPERATURE DEPENDENCE OF ULN FROM NORTON, ET AL.

C SEE PAPER IN PHYS. REV. B8, 5632 (1973)

C VALUE OF ULN AT 300 K FROM THURBER, ET AL. TO BE PUBLISHED IN J ECS

IF (SUMTD.GE.3.E17) 60 TO 20

IF (T.GE.108.) GO TO 10

8 ULN=1 .89E7*T**(-1 .52)*(1 .-0.08*SUMTD/2.E17)

GO TO 40

10 ULN=7.98E8*T**(-2.32)*(1 .-0.08*SUMTD/2.E17)

GO TO 40

20 IF (T.6E.108.) 60 TO 30

22 ULN=1 .89E7*(T**(-1 .52))*0.884

GO TO 40

30 ULN=7.98E8*(T**(-2.32))*0.884

C ADD ON THE EFFECTS OF IONIZED IMPURITY SCATTERIN6

C WE MUST ALLOW FOR THE FACT THAT SUMTD MIGHT VANISH IN CNSTAR EQ.

40 IF (SUMTD. LE.O.) CNSTAR=CN+CP

IF (SUMTD. LE.O.) 60 TO 50

31
42 CNSTAR=CN+CP+(1 .-SUMID/SUMTD)*SUMID

C BN IS LOG TERM IN MODIFIED BROOKS-HERRING IONIZED IMPURITY MOB. EQ

C BN = 6*DIEL*MASS*(1 .3806E-23)**2*T*T/(PI*Q*Q*HBAR*HBAR*CNSTAR)

C WE MUST ALLOW FOR THE FACT THAT CNSTAR MIGHT VANISH IN EQ FOR BN

C BN IS SET TO A LARGE NUMBER AND FIRST POSITION OF ISTAT

C IS SET TO UNITY

50 IF (CNSTAR. LE.O.) BN=9.999999E1

9

IF (CNSTAR. LE.O.) ISTAT=ISTAT+1

IF (CNSTAR. LE.O.) GO TO 80

52 BN=1 .14E24*DIEL*T*T/CNSTAR

C IF BN IS SMALL, BORN APPROXIMATION IS NOT VALID. IN THIS CASE,

C FIFTH POSITION OF ISTAT IS INCREMENTED.

IF (BN-10.) 60,60,80

60 ISTAT=ISTAT+10000

IF (CION-1.0) 70,80,80

70 UDN=ULN

0UT8=UDN

GO TO 110

80 GN=AL0G(8N+1 .)-BN/(BN+1 .)

C CALCULATE GAMMA FOR EFFECT OF E-E SCATTERING ON ION. IMP. MOBILITY

IF (CNSTAR. LE.O.) GAMMA=0.632

IF (CNSTAR. LE.O.) GO TO 90

IF (SUMID.LE.O.) GAMMA=0.432

86 IF (SUMID.GT.O.) GAMMA=(SUMID/CNSTAR)*(1 .-EXP(-CNSTAR/SUMID)

)

IF (GAMMA. LT. 0.432) GAMMA=0.432

90 UIN=GAMMA*7.3E17*TP0WR/(CI0N*GN)

IF (UIN.LE.O.) RATI0N=1.E6

IF (UIN.LE.O.) 60 TO 100

98 RATI0N=6.*ULN/UIN

100 XN=SQRT (RATION)

C COMBINE LATTICE AND IONIZED IMPURITY SCATTERING BY THE MIXEDC SCATTERING FORMULA OF DEBYE AND CONWELL, PHYS. REV. 93,

C 698-706 (1954). SUBROUTINE SICI PERFORMS THIS FUNCTION

CALL SICI (SI,CI,XN)

104 UDN=ULN*(1 .0+RATI0N*(CI*C0S(XN)+SIN(XN)*(SI-1 .5707963)))

0UT8=UDN

C ADD ON THE EFFECTS OF NEUTRAL IMPURITY SCATTERING

C NEUTRAL IMPURITY MOBILITY AS PER ER6INS0Y, PHYS REV 79,1013 (1950)

C UNIS = 0.02*PI**3*Q**3*MASS/(5*DIEL*CNUT*H**3)

C UNIS CONTAINS GEOMETRIC MEAN MASS WITH VALUE OF £FME/6**0. 667

C SEE BROOKS, ADV. IN ELECTRONICS AND ELECTRON PHYSICS 7, 161 (1955)

110 UNIS=4.85E11*EFME/(DIEL*CNUT)

C SCLAR'S CORRECTION TO UNIS, SEE PHYS REV 104, 1559 (1956)

C SEE ALSO PAPER BY LI AND THURBER, SSE 20, 609 (1977)

112 EI=0.215*EFME/((DIEL/1 .1121E-10)**2*CAYT)

114 UNIS=UNIS*0.82*(2*SQRT(1/EI) /3+SQRT(EI) /3.

)

116 UDN=1/(1 /UDN+1/UNIS)

C THAT COMPLETES THE COMPUTATION OF THE ELECTRON MOBILITY

C CALCULATION OF HOLE MOBILITY

C LATTICE SCATTERING WITH CORRECTION FOR HOLE-HOLE SCATTERING

C EQ FOR ULP FOR T.LT.72 FROM BRAGGINS, PHD THESIS,

C SYRACUSE UNIVERSITY, 1975

C EQ FOR ULP FOR T.GE.72 FROM LUDWIG AND WATTERS,

32
C PHYS REV 101, 1699 (1956)

IF (SUMTA.GE.3.E17) GO TO 130

IF (T.GE.72.) GO TO 120

118 ULP=1.6E7*T**(-1 .54)*(1 .-0.08*SUMTA/2.E17)

GO TO 150

120 ULP=2.3E9*T**(-2.7)*(1 .-0.08*SUMTA/2. E17)

GO TO 150

130 IF (T.GE.72.) GO TO 140

132 ULP=1.6E7*(T**(-1.54))*0.884

GO TO 150

140 ULP=2.3E9*(T**(-2.7))*0.884

C ADD ON THE EFFECTS OF IONIZED IMPURITY SCATTERING

C WE MUST ALLOW FOR THE FACT THAT SUMTA MIGHT VANISH IN CPSTAR EQ.

150 IF (SUMTA. LE.O.) CPSTAR=CN+CP

IF (SUMTA. LE.O.) GO TO 160

152 CPSTAR=CP+CN+(1 .-SUMIA/SUMTA)*SUMIA

C BP IS LOG TERM IN BROOKS-HERRING IONIZED IMPURITY MOBILITY

C BP = 6*DIEL*MASS*(1 .3806E-23)**2*T*T/(PI*Q*Q*HBAR*HBAR*CPSTAR)

C MASS FOR BP OF EACH BAND FROM LI, NBS SPECIAL PUBLICATION 400-47

C NO TEMPERATURE DEPENDENCE FOR MASS, 300 K VALUE USED

C IS SET TO 1 C WE MUST ALLOW FOR THE FACT THAT CPSTAR MIGHT VANISH IN EQ FOR BP

C BP IS SET TO A LARGE NUMBER AND THIRD POSITION OF ISTAT

160 IF (CPSTAR. LE.O.) BP1 =9 . 999999E1

9

IF (CPSTAR. LE.O.) BP2=9. 999999E1

9

IF (CPSTAR. LE.O.) BP3=9 . 999999E1

9

IF (CPSTAR. LE.O.) ISTAT=ISTAT+1 00

IF (CPSTAR. LE.O.) GO TO 190

162 BP1=1 .16E24*0.568*DIEL*T*T/CPSTAR

BP2=1 . 1 6E24*0 . 41 2*DIEL*T*T/ CPSTAR

BP3=1 .16E24*0.246*DIEL*T*T/CPSTAR

C IF BP IS SMALL, BORN APPROXIMATION IS NOT VALID. IN THIS CASE,

C FIFTH POSITION OF ISTAT IS INCREMENTED.

IF (BP1-10.) 170,170,190

170 ISTAT=ISTAT+10000

IF (CION-1.0) 180,190,190

180 UDP=ULP

0UT11=UDP

GO TO 220

1 90 GP1 =ALOG (BP1 +1 . ) -BP1 / (BP1 +1 . ) GP2=AL0G (BP2+1 . ) -BP2/ (BP2+1 .

)

GP3=AL0G (BP3+1 . )-BP3/ (BP3+1 . ) C UIP = 0.01*(2**3.5)*DIEL*DIEL*SQRT(MD)*(1 .38E-23*T)**1 .5/

C ((PI**1 .5)*Q*Q*Q*MC*CI0N*GP)

192 UIP1=2.66E35*DIEL*DIEL*TP0WR*SQRT(0.568) /(0.448*CI0N*GP1)

UIP2=2.66E35*DIEL*DIEL*TP0WR*SQRT(0.412)/(0.532*CI0N*GP2)

UIP3=2.66E35*DIEL*DIEL*TP0WR*SQRT (0.246) /(0.253*CI0N*GP3)

C WEIGHT INDIVIDUAL MOBILITIES BY HOLE DENSITY IN THAT BAND

196 UIP= (UIP1+UIP2* (0.41 2/0. 568) **1.5+UIP3*EXP(-0.044/CAYT)* (0.246/0.

5

268)**1 .5)/(1 . +(0. 412/0. 568)**1 .5+EXP(-0. 044/CAYT)*(0. 246/0. 568)**1

3.5)

C CALCULATE GAMMA FOR EFFECT OF H-H SCATTERING ON ION. IMP. MOBILITY

IF (CPSTAR. LE.O.) GAMMA=0.632

33
IF (CPSTAR.LE.O.) 60 TO 200

IF (SUMIA.LE.O.) GAMMA=0.432

198 IF (SUMIA.GT.O.) 6AMMA=(SUMIA/CPSTAR)*(1 .-EXP(-CPSTAR/SUMIA))

IF (GAMMA. LT. 0.432) GAMMA=0.432

200 UIP=GAMMA*UIP

IF (UIP.LE.O.) RATI0P=1.E6

IF (UIP.LE.O.) GO TO 210

208 RATI0P=6.*ULP/UIP

210 XP=SQRT(RATIOP)

C COMBINE LATTICE AND IONIZED IMPURITY SCATTERING BY THE MIXEDC SCATTERING FORMULA OF DEBYE AND CONWELL, PHYS. REV. 93,

C 698-706 (1954). SUBROUTINE SICI PERFORMS THIS FUNCTION

CALL SICI (SI,CI,XP)

214 UDP=ULP*(1 .0+RATI0P*(CI*C0S(XP)+SIN(XP)*(SI-1 .5707963)))

0UT11=UDP

C ADD ON THE EFFECTS OF NEUTRAL IMPURITY SCATTERING

C NEUTRAL IMPURITY MOBILITY AS PER ERGINSOY, PHYS REV 79,1013 (1950)

C UNIS = 0.02*PI**3*Q**3*MASS/(5*DIEL*CNUT*H**3)

C VALENCE BAND OF SILICON HAS THREE SEPARATE COMPONENTS

C PROCEDURE TAKEN FROM LI, NBS SPECIAL PUBLICATION 400-47

C FOR THIS CALCULATION, MASS VALUES OF LI AT 300 K ARE USED

220 UNIS1=1 .60E12*0.568/(DIEL*CNUT)

UNIS2=1.60E12*0.412/(DIEL*CNUT)

UNIS3=1 .60E12*0.246/(DIEL*CNUT)

C SCLAR'S CORRECTION TO UNIS, SEE PHYS REV 104, 1559, (1956)

221 EI1=0.71*0.568/((DIEL/1 .1121E-10)**2*CAYT)

EI2=0.71*0.412/((DIEL/1 .1121E-10)**2*CAYT)

EI3=0.71 *0.246/((DIEL/1.112lE-10)**2*CAYT)

222 UNIS1=UNIS1*0.82*(2.*SQRT(1 ./EI1)/3.+SQRT(EI1) /3.)

UNIS2=UNIS2*0.82*(2.*SQRT(1 ./EI2)/3.+SQRT(EI2)/3.)

UNIS3=UNIS3*0.82*(2.*SQRT(1 ./EI3) /3. +SQRT (EI3) /3.

)

C WEIGHT INDIVIDUAL MOBILITIES BY HOLE DENSITY IN THAT BAND

224 UNIS=(UNIS1+UNIS2* (0.41 2 /0.568)**1.5+UNIS3*EXP(-0.044/CAYT)* (0.246

2/0. 568)**1 .5)/(1. +(0.412/0. 568)**1 .5+EXP(-0.044/CAYT)*(0. 246/0. 568

3)*1.5)

226 UDP=1 /(I /UDP+1 /UNIS)

C THAT COMPLETES THE COMPUTATION OF THE HOLE MOBILITY

IF (CN.LE.O. .AND.CP.LE.O. ) GO TO 230

RH=((CP*UDP*UDP-CN*UDN*UDN)/(Q*(CP*UDP+CN*UDN)))/(CP*UDP+CN*UDN)

SIGMA =Q*(CN*UDN+CP*UDP)

RH0=1 ./SIGMA

UD=(CN*UDN+CP*UDP)/(CN+CP)

C END OF THE MOBILITY AND RELATED COMPUTATIONS

C

C UPON RETURNING, ISTAT INDICATES THE STATUS OF THE COMPUTATIONS

C FIRST DIGIT UNITY, CNSTAR WAS EQUAL TO ZERO

C THIRD DIGIT UNITY, CPSTAR WAS EQUAL TO ZERO

C FIFTH DIGIT 1 OR 2, BORN APPROXIMATION NOT VALID

C 9.999999E19 IS A LARGE NUMBER INTRODUCED TO AVOID ZERO DIVISORS

C230 RETURN

END

34
SUBROUTINE SICI (SI,CI,X)

C

C THIS ROUTINE COMBINES LATTICE AND IONIZED MOBILITY SCATTERING

C ACCORDING TO THE MIXED-SCATTERING FORMULA OF DEBYE AND

C CONWELL, PHYS. REV. 93, 693-706 (1954)

C

C TEST ARGUMENT RANGE

Z=ABS(X)

IF (Z-4.) 10,10,50

C Z IS NOT GREATER THAN 4.

10 Y=Z*Z

S I=X* (((((( . 979421 54E-1 1 *Y-. 22232633E-8)*Y+. 30561 233E-6)*Y-. 283414

260E-4)*Y+.16666582E-2)*Y-.55555547E-1)*Y+1.)

C TEST FOR LOGARITHMIC SINGULARITY

IF (Z) 30,20,30

20 CI=-1.E38

RETURN

30 CI=0. 57721 566+ALOG (Z)-Y*( ( ( ( (-.13869851 E-9*Y+.26945842E-7)*Y-.3095

22207E-5 ) * Y+ . 231 46303E-3) *Y- . 1 041 6642E-1 ) *Y+ . 24999999)

40 RETURN

C Z IS GREATER THAN 4.

50 SI=SIN(Z)

Y=COS(Z)

Z=4./Z

U=((((((((.40480690E-2*Z-.022791426)*Z+.055150700)*Z-. 07261 641 8)*Z

2+. 0498771 59)*Z-. 333251 86E-2)*Z-. 0231461 68)*Z-.11349579E-4)*Z+. 0625

300111)*Z+.25839886E-9

V=( (((((( ( (-. 0051 086993*Z+. 0281 91 786)*Z-. 065372834) *Z+. 079020335)*

2Z-. 0440041 55 )*Z-.0079455563)*Z+. 02601 2930)*Z-.37640003E-3)*Z-. 0312

3241 78) *Z-.66464406E-6)*Z+. 25000000

CI=Z*(SI*V-Y*U)

SI=-Z*(SI*U+Y*V)+1 .5707963

C TEST FOR NEGATIVE ARGUMENT

IF (X) 60,40,40

C X IS LESS THAN -4.

60 SI=-SI

RETURN

END

35
5. Illustrative Output Data Cards and Examples

5.1 Data Cards Used to Generate Output Listings

The output examples to follow illustrate a no-option run, the DOUBLE ON

format, the DOLOOP XX option, and the CONSTANTS option. Following is a listing of the complete deck of dopant state data and control cards that

used to generate these output examples. Lines separate the card sets for

each example.

PHOSPHORUS 1 .00000E15 0. 045 2.

DUMMY

DUMMY

BORON 1 .00000E13 0. 045 4.

DUMMY

DOUBLE ON

PHOSPHORUS 1 .00000E18 0. 045 2.

DUMMY

DUMMY

BORON 1 .00000E13 0. 045 4.

DUMMY

DOUBLE OFF

DOLOOP A2 5 .OOOOOEU 4.

PHOSPHORUS 1 .00000E15 0. 045 2.

DUMMY

DUMMY

BORON 0.045 4.

DUMMY

CONSTANTS

STOP

36
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45
6. CONCLUSIONS

A FORTRAN program has been presented for calculating the electrical properties of silicon containing five dopant states. The charge-balance equation

is solved to arrive at the Fermi energy which determines the carrier densities. An effective conductivity mobility is calculated for both electrons

and holes, taking into account the significant scattering mechanisms, and

then the resistivity is computed from the mobilities and carrier densities.

An attempt has been made to include in the algorithm the temperature and dopant density dependence of as many parameters as possible. In some cases

(e.g., the temperature dependence of the indirect energy gap of silicon),

there are numerous experimental results in the literature [27] . In cases where there is significant disagreement in the literature (e.g,, the variation of dopant activation energy with dopant density), there is room for improvement. In a few cases where little or nothing is known (e.g., the variation of the Hall scattering factor with temperature and dopant density), no attempt has been made to include these effects. Therefore, the user of this

program is urged to review the literature regularly for additional or more comprehensive measurements that would improve the accuracy of the computed

results. The authors invite any suggestions and comments in this regard.

ACKNOWLEDGMENTS

The authors would like to thank Thomas F. Leedy, William J. Keery, and Sheng

S. Li (University of Florida) for many helpful discussions and suggestions

during the development of this program. They would also like to thank David

K. Kahaner for a critical review of the manuscript and for pointing out the

availability of the subroutine ZEROIN and Jane Walters for preparing the

manuscript for publication.

REFERENCES

1. Larrabee, R. D., Interpretation of Hall Measurements, Sem'iQonduetov

Chavaotevization Teohniques, Proceedings Volume 78-3, P. A. Barnes and

G. A. Rozgonyi, Eds, pp. 71-76 (The Electrochemical Society, Princeton,

1978)

.

2. Bullis, W. M. , and Stricter, F. J., Electrical Properties of n-Type

Silicon Doped with Gold, J. Appl. Phys. 39, 314-318 (1968).

3. Thurber, W. R. , Lewis, D. C. , and Bullis, W. M, , Resistivity and Carrier

Lifetime in Gold-Doped Silicon, AFCRL Report AFCRL-TR-73-0 1 07 (January

31, 1973).

4. Larrabee, R. D. , "X" Levels in Indium-Doped Silicon, Pvoo. IRIS Specialty Group MeetingJ Annapolis, Maryland, June 13-15, 1978.

5. Blakemore, J. S. , Semieondudtov Statistics, pp. 354-365 (Pergamon Press,

New York, 1962).

6. McCracken, D. D. , ^ Guide to Fortran IV Programming, pp. 87-89 (John

Wiley and Sons, New York, third corrected printing, August, 1968).

46
7. Barber, H. D. , Effective Mass and Intrinsic Concentration in Silicon,

Solid-state Eleotvonias 10, 1039-1051 (1967).

8. Macfarlane, G. G. , McLean, T. P., Quarrington, J. E., and Roberts, V.,

Fine Structure in the Absorption -Edge Spectrum of Si, Fhys. Rev. 111 , 1245-1254 (1958).

9. Penin, N. A., Zurkin, B. G. , and Volkov, B. A., Influence of the Concentration of Donors and Acceptors on the Electrical Conductivity of

Heavily Doped n-Type Silicon, SoV. Phys. - Solid State 1_, 2580-2584

(1966)

.

10. Sclar, N. , Resistivity and Deep Impurity Levels in Silicon at 300 K,

IEEE Trans. Eleotvon Devices ed-24 , 709-712 (1977).

11. Shampine, L. F., and Watts, H. A., ZEROIN - A Root-Solving Code, Sandia

Report SC-TM-70-631, September 1970.

12. Dekker, T. J., Finding a Zero by Means of Successive Linear Interpolation, Constructive Aspects of the Fundxxmental Theorem of Algebra, B.

Dejon and P. Henrici, Eds., pp. 37-51 (Wiley-Interscience, London,

1969)

.

13. Shampine, L. F., and Allen, R. C. , Jr., Numerical Computing: An Introduction, pp. 87-108, 242-246 (Saunders, Philadelphia, 1973).

14. Norton, P., Braggins, T., and Levinstein, H. , Impurity and Lattice Scattering Parameters as Determined from Hall Mobility Analysis in n-type

Silicon, Phys. Rev. B 8, 5632-5653 (1973).

15. Thurber, W. R. , Mattis, R. L., Liu, Y. M., and Filliben, J. J.,

Resistivity-Dopant Density Relationship for Phosphorus-Doped Silicon, J.

Electrochem. Soc. 127, 1807-1812 (1980).

16. Braggins, T. T., Analysis of Carrier Concentration and Mobility in

Boron-Doped Silicon, Ph.D. Thesis, University Microfilms International,

Ann Arbor, Michigan (1975).

17. Ludwig, G. W. , and Watters, R. L. , Drift and Conductivity Mobility in

Silicon, Phys. Rev. 101 , 1699-1701 (1956).

18. Brooks, H., Theory of the Electrical Properties of Germanium and

Silicon, Advances in Electronics and Electron Physics, L. Marton, Ed.,

Vol. 7, pp. 85-183 (Academic Press, New York, 1955).

19. Herring, C, and Vogt, E., Transport and Deformation Potential Theory

for Many-Valley Semiconductors with Anisotropic Scattering, Phys. ReV.

101, 944-961 (1956).

20. Li, S. S., and Thurber, W. R. , The Dopant Density and Temperature

Dependence of Electron Mobility and Resistivity in n-Type Silicon,

Solid-state Electronics 20, 609-516 (1977).

47
21. Li, S. S., Semiconductor Measurement Teohnology: The Theoretical and

Experimental Study of the Temperature and Dopant Density Dependence of

Hole Mobility, Effective Mass, and Resistivity in Boron-Doped Silicon,

NBS Special Publication 400-47 (November 1979).

22. Debye, P. P., and Conwell, E. M. , Electrical Properties of N-Type Germanium, Phys. Rev. 93, 693-706 (1954).

23. Dunlap, W. C. , Jr., and Watters, R. L. , Direct Measurement of the Dielectric Constants of Silicon and Germanium, Phys. Rev. 92, 1396-1397

(1953).

24. Cardona, M. , Paul, W. , and Brooks, H. , Dielectric Constant of Germanium

and Silicon as a Function of Volume, J. Phys. Chem. Solids 8, 204-206

( 1959)

.

25. Erginsoy, C. , Neutral Impurity Scattering in Semiconductors, Phys. Rev.

79, 1013-1017 (1950).

26. Sclar, N. , Neutral Impurity Scattering in Semiconductors, Phys. Rev.

104, 1559-1568 (1956).

27. Neuberger, M. , and Welles, S. J., Silieony Electronic Properties Information Center, Hughes Aircraft Company, Culver City, California, October

1969, AD 698342.

48
FEDERAL INFORMATION PROCESSING STANDARD SOFTWARE SUMMARY

31. Summary date

Yr. Mo. Day

Software date

Yr. Mo. Day

be. Short title

02. Summary prepared,. by (Name and Phone) R. D. Larrabee", W. R. Thurber' and W. M. Bui 1 is

05. Software title A FORTRAN Program for Calculating the Electrical

Parameters of Extrinsic Silicon

58. Software type

Automated Data System

||^ Computer Program r] Subroutine/Module

09. Processing mode

I I Interactive

I I Combination

03. Summary action New Replacement Deletion H(

Previous Internal Software ID

07. Internal Software ID RDL/WRT-3

10. General

Application area

Computer Systems

Support/Utility

\Y\ Scientific/Engineering

I I Bibliographic/Textual

Management/

Business

Process Control Q Other

Specific Scientific Functional

Eval uat ion

.1 Submitting organization and address National Bureau of Standards

Electron Devices Division

Washington, DC 20234

12. Technical contact(s) and phone R. D. Larrabee (301) 921-3625

W. R. Thurber (301) 921-3625

W. M. Bullis (301) 921-3786

,3. Narrative Many electrical properties of silicon are strongly dependent upon the specific

pature and density of the active impurities present. Calculation of these electrical

oroperties hinges on the solution of the charge balance equation to determine the

position of the Fermi level for each specific case of interest. A FORTRAN program is sresented that performs this determination and then calculates some of the often-used

slectrical parameters of silicon as a function of temperature. Results obtained from

this program have proven useful in interpreting Hall effect data, determining the

degree of ionization of the separate dopant states as a function of temperature,

Dredicting the behavior of specimens when the dopant picture is intentionally (or

conceptually) changed, and understanding the variations in the relative roles of the

different scattering mechanisms on carrier mobility as the temperature is changed.

4 Keywords

Carrier density; computer program; electrical properties of silicon

nobility; resistivity; silicon.

Hall effect;

5 Computer manuf'r and model UNIVAC 110816. Computer operating system UNIVAC 1100

17. Programing language(s) FORTRAN IV18. Number of source program state- ments 700

9. Computer memory requirements 20. Tape drives None

21. Disk/Drum units None22. Terminals

Pr i nter

3. Other operational requirements

l>lone

4 Software availability

Available Limited a

>ee item 12 for contacts

In-house only

25. Documentation availability

Available Inadequate In-house only m n n

1. See item 12 for contacts.

2. NBS Special Publication 400-63 available

from the U. S. Government Printing'

Office, Stock No.

5. FOR SUBMITTING ORGANIZATION USE

^('301) 921-3625

|/ (301) 921-3786

standard Form 185

1974 July

U.S. Dept. of Commerce—NBS

(PIPS. Pub. 30)
NBS-n4A (REV. 2-8C) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBS SP 400-63

2. Performing Organ. Report No 3. Publication Date

October 1980

4. TITLE AND SUBTITLE

Semiconductor Measurement Technology : A FORTRAN Program for Calculating the

Electrical Parameters of Extrinsic Silicon

5. AUTHOR(S)

R. D. Larrabee, W. R. Thurber, and W. M. Bullis

6. PERFORMING ORGANIZATION (/f jo/nt or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No,

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City. State. ZIP)

Same as No . 6

10. SUPPLEMENTARY NOTES

[x] Document describes a computer program; SF-185, PIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual sunrtmary of most si gnifi cant information. If document includes a significant

bi bl io graphy or literature survey, mention it tiere) Many electrical properties of silicon are strongly dependent upon the specific nature

and density of the active impurities present. Calculation of these electrical

properties hinges on the solution of the charge balance equation to determine the

position of the Fermi level for each specific case of interest. A FORTRAN program is

presented that performs this determination and then calculates some of the often-used

electrical parameters of silicon as a function of temperature. Results obtained from

this program have proven useful in interpreting Hall effect data, determining the

degree of ionization of the separate dopant states as a function of temperature,

predicting the behavior of specimens when the dopant picture is intentionally (or

conceptually) changed, and understanding the variations in the relative roles of the

different scattering mechanisms on carrier mobility as the temperature is changed.

12. KEY WORDS (S/x to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Carrier density; computer program; electrical properties of silicon; Hall effect;

mobility; resistivity; silicon.

13. AVAILABILITY

[X^ Unlimited Q For Official Distribution. Do Not Release to NTIS

Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D C

20402.

Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

54

15. Price

$3.75

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