A11107 3i0blfl PUBLICATIONS

NBSIR 84-2838

Measurement Techniques for

High-Power Semiconductor

Materials and Devices: Annual

Report, January 1, 1982 to

March 31, 1983

US. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Electronics and Electrical Engineering

Semiconductor Materials and Processes Division

Washington, DC 20234

April 1984

-QC100

Prepared for Department of Energy

Division of Electric Energy Systems

Washington, DC 20461

. 1)56

34-2333

1934

c - 5

NATIONAL BUr.I.AJ OF STANDARDS

LIBRARY

NBSIR 84-2838

MEASUREMENT TECHNIQUES FOR

HIGH-POWER SEMICONDUCTOR

MATERIALS AND DEVICES: ANNUAL

REPORT, JANUARY 1, 1982 TO

MARCH 31, 1983

C

qC

| DD

C

W. R. Thurber, J. R. Lowney, and W. E. Phillips

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Electronics and Electrical Engineering

Semiconductor Materials and Processes Division

Washington, DC 20234

April 1984

Prepared for Department of Energy

Division of Electric Energy Systems

Washington, DC 20461

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director
,J , 

i-:- , 1 , " | -/ • ; ' 7';:
TABLE OF CONTENTS

Page

Abstract 1

1 . Introduction and Executive Summary . 1

2. Lifetime-Predicting Computer Program (LT1 ) . . . . 2

3. Time Dependence of the Capacitance-Voltage Relationship of a Heavily Platinum-Doped Silicon Diode 3

References ..... 8

Appendix A. "The Relationship Between Deep-Level Measurements and

Lifetime in Devices" 9

Appendix B. "Analysis of Nonexponential Transient Capacitance in

Silicon Diodes Heavily Doped with Platinum" 14

Appendix C. "Improved Analysis Procedures for Deep-Level

Measurements by Transient Capacitance" 20

Appendix D. Details and Listing of Lifetime-Predicting Computer

Program 26

LIST OF FIGURES

1 . Inverse capacitance squared as a function of reverse voltage for

a 0 V trap-filling bias with time as a parameter 5

2. Percent error in the emission rate obtained by the capacitanceratio method compared with the true emission rate for four

different bias voltage sequences .

iii
PREFACE

This work was conducted as a part of the Semiconductor Technology Program at

the National Bureau of Standards (NBS) . This program serves to focus NBS

research 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 and for use by industry in controlling

device fabrication processes. This research leads to carefully evaluated and

well-documented test procedures and associated technology. Special emphasis

is placed on the dissemination of the results of the research to the electronics community. Application of these results by industry will contribute

to higher yields, lower cost, and higher reliability of semiconductor devices. Improved measurement technology also leads to greater economy in

government procurement by providing a common basis for the purchase specifications of government agencies and, in addition, provides a basis for con- trolled improvements in fabrication processes and in essential device characteristics.

The segment of the Semiconductor Technology Program described in this annual

report was supported by the Division of Electric Energy Systems of the Department of Energy (DoE) under NBS/DoE Task Order DE-A1 01 -76PR0601 0. The

contract was monitored by Mr. Kenneth Klein of DoE. The NBS contacts for

technical information on this project are W. R. Thurber and J. R. Lowney of

the Semiconductor Materials and Processes Division at the National Bureau of

Standards, (301) 921-3625.

The authors would like to acknowledge guidance and support from R. D.

Larrabee and other colleagues at the National Bureau of Standards.

IV
Measurement Techniques for High-Power

Semiconductor Materials and Devices

ANNUAL REPORT

January 1, 1982 to March 31, 1983

W. R. Thurber, J. R. Lowney, and W. E. Phillips

Semiconductor Materials and Processes Division

National Bureau of Standards

Washington, DC 20234

ABSTRACT

This annual report is the final one in a series which describes NBS

research to develop procedures for the effective utilization of

deep-level measurements to detect and characterize defects which

reduce lifetime or contribute to leakage current in power-devicegrade silicon. During this reporting period, the previously

written computer program for predicting excess-carrier lifetime was revised to calculate more accurately lifetimes for high or low

injection conditions and in space-charge regions. Comparisons were made between lifetime measurements on platinum-doped silicon diodes

and the predictions of the computer model. As part of the effort

to extend the procedures to analyze data from nonexponential

transient capacitance measurements, the time dependence of the

capacitance-voltage relationship of a heavily platinum-doped

silicon diode was measured as a function of bias voltage. Included

as appendices are three recent publications resulting from the

work. A listing of the lifetime-predicting computer program is

also an appendix.

Key words: deep-level measurements; deep-level transient spectroscopy; defect characterization; lifetime; power-device grade silicon; transient capacitance techniques.

1. INTRODUCTION AND EXECUTIVE SUMMARY

This annual report describes results of NBS research directed toward the

development of measurement methods for semiconductor materials and devices

which will lead to more effective use of high-power semiconductor devices in

applications for energy generation, transmission, conversion, and conservation. The major effort under this program has been to develop procedures for

the effective utilization of deep-level measurements to detect and characterize defects which reduce lifetime or contribute to leakage current in silicon

used for power devices.

The presence of deep- level impurities in semiconductor power devices is a consequence of their unintentional introduction during crystal growth and

during the wafer fabrication procedure or their intentional introduction in

order to adjust the switching properties of the device. In either case, the

dominant effect of the deep level is to modify the excess-carrier lifetime.

Measurement techniques to detect, characterize, and identify such deep level k are required in order to monitor the presence of unintentional contamination

and to characterize and understand the behavior of intentionally add^l impurities, The use of such techniques for process diagnostics would enhance th.

1
manufacturer's ability to control the yield, reliability, and cost of his

product.

During the period covered by this report, the previously written computer

program for predicting excess-carrier lifetime was revised to calculate lifetimes more accurately for high or low injection conditions and in spacecharge regions. The model accepts input parameters, namely, activation ener- gies and capture cross sections, derived from thermal emission-rate measurements on appropriately doped material. Comparisons were made between lifetime measurements on platinum-doped silicon diodes and the predictions of the

computer model. Reasonable agreement was obtained in magnitude and temperature dependence for both n- and p

- type silicon. The results of this work

were presented at the 1983 Custom Integrated Circuits Conference in

Rochester, New York. The paper which appeared in the proceedings of the

conference is reproduced as Appendix A in this report. A listing of the

computer code for the revised lifetime-predicting program is given in Appendix D.

During the previous reporting period, a method was developed to analyze non- exponential capacitance transients. The method was refined during this re- porting period and published in the Journal of Applied Physics. The paper is

reproduced in Appendix B. The procedures given in the paper were then applied to the more commonly used deep-level transient spectroscopy (DLTS)

technique. A talk covering this work was presented at the Electrochemical

Society's "Defects in Silicon" symposium held in San Francisco in May 1983.

The associated paper was published in the symposium proceedings and is reproduced as Appendix C of this report. As part of the effort to extend the

generalized analysis procedures to DLTS, the time dependence of the

capacitance-voltage relationship of a heavily platinum-doped silicon diode

was measured as a function of bias voltage. Because a description of this

work was not included in the published paper, it appears as Section 3 of this

report.

In response to a request by the American Society for Testing and Materials

(ASTM), a draft document entitled "Standard Practice for Characterizing Semiconductor Deep Levels by Transient Capacitance Techniques" was written and

subsequently revised to incorporate suggestions received from ASTM Committee

F— 1 . The latest version of this draft document is available from the

authors

2. LIFETIME-PREDICTING COMPUTER PROGRAM (LT1

This FORTRAN-77 computer program computes the minority-carrier lifetime in

silicon as a function of temperature and injection level. It can handle up

to three donor and three acceptor species (including the shallow dopants)

which have a single level in the energy gap. Along with these levels, it can handle two multilevel species of the same type as platinum and gold (i.e.,

having both a donor and acceptor level in the gap) . It is assumed that all

these centers act independently, and Shockley-Read-Hall theory of recombination through traps can be used to compute the lifetime.

The necessary trap and dopant parameters (density, energy, degeneracy, and cross sections) are inputs to the program. The formulas used for lifetime

2
prediction are given in references [1] and [2]. It is important to realize

that the excess- carrier recombination lifetime varies greatly with injection

ratio and computational difficulties may arise if more than one trap is present.

The degeneracy factor g, defined in reference [3], requires a word of explanation. For shallow dopants, it is an integer for both donors (g = 2) and

acceptors (g = 4). For traps, it is a positive real number equal to

exp (AS/k), where AS is the change in entropy associated with emission of a carrier and k is the Boltzmann constant. It is related to both the electronic degeneracy of the trap as well as any relaxation of the lattice after

emission. The physics of the degeneracy factor is described in reference

[3]. An alternative method of dealing with the degeneracy factor is to incorporate it into the trap energy, as done in Appendix A. If this method is

used, the trap energy is now called the Gibbs free energy, which is equal to

(Et

- TAS), where Bp is the energy position of the trap level in the gap and

T is absolute temperature.

A discussion of the details of the program, a listing, and sample input and

output are given in Appendix D.

3. TIME DEPENDENCE OF THE CAPACITANCE-VOLTAGE RELATIONSHIP OF A HEAVILY

PLATINUM-DOPED SILICON DIODE

A method was developed to analyze nonexponential capacitance transients measured under conditions such that the trap density is not small compared with

the net shallow dopant density. Charging the traps in only part of the depletion region during the fill pulse is taken into account by the method.

For spatially constant trap and shallow dopant densities, partial charging

increases the nonexponentiality which would otherwise exist for a given ratio

of trap density to shallow dopant density. This capacitance-ratio method was initially developed for isothermal transient capacitance (ITCAP) [4] and was extended to the more commonly used DLTS technique [5]. It is assumed in the

approach that there is a nearly linear relationship between C

-2 and V for the

p-n junction test diode. By making the depletion approximation, it is shown

that the time dependent density of electrons on the defect centers, n t , in

the depletion region of a p

+n junction under reverse bias Vr is [4]:

where C is the junction capacitance, C^ is the capacitance at some reduced

trap-filling bias Vc , A is the junction area, e is the dielectric constant

(permittivity) of silicon, Nd

is the shallow-dopant density (assumed to be

constant), and q is the electronic charge.

However, in order to include the effects related to large ratios of trap to

shallow-dopant density, eq (1) was generalized by replacing the number "2"

with a dimensionless parameter a which may vary with Vr , Vc , and time. Th»* dependence of a on these quantities is related to the ratio of N t /Nd , ).:• very complex. In rearranged form, this new equation is:

3
afv - V 1

^ r c

J C~ ( 2 ) 2 = C

" 2 + . qeA2(N

d - nt)

To examine the relationship in eq (2) as a function of time and voltage,

measurements were made on a platinum-doped, n-type silicon diode on wafer 93B

(wafer containing device no. 2 in Appendices B and C) at -156°C. The results

of C“ 2 vs. vr are shown in figure 1 as a function of reverse bias for a trap-filling bias of 0 V. Points were obtained for four different times:

(1) at t = tf (C = Cf), corresponding to restoration of the reverse bias;

(2) at t = t-j , the first boxcar gate-delay time; (3) at t = t

2

, the second

boxcar gate-delay time; and (4) at t = tf (C = Cf) after the transient has

decayed. Even though the present device is from the same wafer as device

no. 2, approximate fitting of the C"2 VS. Vr relationships suggested a trap

to shallow dopant density ratio of about 0.5 instead of the 0.38 found for

device no. 2. Consequently, the calculated values for nt in [4] cannot be

used for the present diode.

Nonlinear behavior is evident in all four curves in figure 1 at low voltages

with the most pronounced curvature occurring for the data obtained at t = tf.

Therefore, it may be concluded that a varies with voltage for fixed time. It

is impossible to determine a priori the variation of a with time from these

data and eq (2) because of our lack of knowledge of nt . However, with certain assumptions, it was possible to obtain information on a from the curves

in figure 1 . It was assumed that nt = Nfe“ ent

, a good approximation at low

temperatures, where is the total trap density, and en is the emission

rate for electrons. The value of en was obtained from a plot of the

capacitance ratio for this diode, a technique which is discussed below.

First, for Vr = 10 V, a value of a = 2.40 was calculated from the tf (t = °°) curve using eq (2) with = 1.21 x 10 15 cm-3 from reference [4] and the measured C^. Second, Nt = 6.1 x 10 11+ cm"* 3 was calculated from the t

2 (t = 20 ms)

curve using this a. Then, with Nt and known, a was calculated from the

t^ (t = 10 ms) curve and found to be in good agreement with the value from

the tf curve where only Nd entered. Keeping these Nt and Nd values, a was also calculated from the tf (t = 0) curve. The value of a was about 4%

smaller at tf than at t^ . This trend agrees with more precise results obtained from idealized data, which will be given later, and shows the need to

consider the variation of a with time in our analysis.

It is assumed in the capacitance-ratio method [4,5] that a is constant during

the capacitance transient. To obtain the time constant of the transient, a ratio of capacitances Cr is plotted as a function of time. If a changes a significant amount with time, a straight line would not result from the

semi logarithmic plot of Cr

. However, in the following analysis it will be

shown that the capacitance- ratio method is relatively insensitive to moderate

changes in a and gives a surprisingly accurate result for the emission rate.

In the capacitance-ratio method

4
Figure 1 . Inverse capacitance squared as a function of reverse voltage for a

0 V trap-filling bias with time as a parameter. The time t^ is when the reverse bias is restored, t^ is 10 ms later, t^ is 20 ms after t^, and tf

is after complete decay of the transient. The measurements were made on a platinum-doped silicon diode on wafer 93B at -156°C. The time constant

calculated from the capacitance-ratio method for this family of curves is 15 ms.

5
The error in the capacitance-ratio method can be expressed as

e

-e t

n

C + C

r err

( 4 )

where Cerr is the error which must be added to Cr for a calculation of the

true emission rate en

. This error is obtained by evaluating the expression

-e t n. - n. , n t tf

n. . “ ti tf

(5)

where n^ is given by eq (2), rearranged, and where ntf is the value of nt

at t = 00 and nt£ is the value at t = 0. Cerr is the difference between t

general case where a is time dependent and the capacitance-ratio version

err

assumes that a is constant with time. This difference is given

( cb

2 - q2

) («f - °0 C

2C

f

2 + K - a

f1 C

2C. 2 + (a - a.

f i

v i'

C . 2 C

2

) l

1

( c

b

2 - c

2

) “fc

f

2c

b

2 - a. C. 2 C

2 lib + ( a. - a.) C, 2C

2

v l f ' l f

, ( 6 )

where a is given by eq (2), rearranged, and where Of is the value of a at

t = 00 and is the value at t = 0.

The fractional error in the emission rate is

e

r - e in ( 1 + C /C ]

It is not possible to use experimental data directly to determine accurately

the error in the emission rate because ntis not known a priori and the deionization of the shallow dopant (discussed in the appendix in ref. [4])

which varies at low temperature is also unknown. Consequently, idealized

data were generated using the extended Sah-Reddi model given in the appendix

of reference [4] with the parameter values equal to those obtained in reference [4] for device no. 2. The fractional error in emission rates, in percent, is plotted in figure 2 as a function of the product of emission rate

and time. At exactly t = 0, eq (7) is indeterminate and the error must be

evaluated in the limit as t approaches 0. For long times (large en

t) the error goes to zero. It is evident from the trend of the error that it is

important to avoid a small value of reverse bias Vr when the fill pulse is near 0 V. Compared with the other voltage sequences, the 5,0,5 V sequence

has the largest value of a and the most time -dependent a as calculated from

the idealized data using eq (2), rearranged. This is in qualitative agreement with the large change in curvature of the C“2 vs. Vr plots at low bias

voltages in figure 1. Specifically for the 5,0,5 V bias sequence, the change

in a between en

t = 0 and en

t = 2, relative to the value at t = 0, is 9.3 percent and the error in the emission rate at en

t = 1 is 0.16 percent. For the

5,4,5 V sequence, the corresponding change in a is 6.1 percent and the corresponding error in the emission rate is only 0.08 percent. For the 15,0,15 V

6
(%) co 7

Figure

2.

Percent

error

in

the

emission

rate

obtained

by

the

capacitance-ratio

method

compared

with

the

true

emission

rate

for

four

different

bias

voltage

sequences.

This

figure

was

prepared

from

idealized

data

generated

using Nt

/N(j=

0.38,

N^+/N d=

0.77,

and T=

87.7

K.
sequence, the relative change in a is 6.5 percent and for the 15,12,15 V se- quence, a changes by 3.8 percent. The latter two bias sequences have the same error in emission rate, 0.06 percent at en

t = 1. This is a consequence

of the change in a being offset by the effect of Cfc in the Cerr expression,

which is larger at 0 V fill-pulse voltage.

While figure 2 shows that a larger reverse voltage is generally desirable,

the measured capacitance is consequently smaller, and as a result stray ca- pacitance and noise can have a greater effect on the measurement. Also, one may not want to use a large reverse voltage if the emission rate depends on electric field. A small difference between the reverse- and fill-pulse voltages is especially desirable at low reverse bias voltages to minimize the error in the capacitance-ratio method. However, if the difference is too

small, C Cf, Cb , and C are nearly equal so that the results are very sensitive to errors in any of them. Another consideration in minimizing error is

to ignore the initial decay of the transient and only analyze data for which

e n

t > 0.5, for example. In general, the capacitance-ratio method is sufficiently accurate at all times (except for t « 1/en

) to justify its use as a convenient way to correct for a large relative trap density and for incomplete trap filling in the depletion region. The latter effect occurs in

capacitance Vs . voltage depth profiling and other applications where a small

voltage difference is used. Since we have not considered gradients in trap

concentration, however, a more complete analysis would be necessary for profiling experiments.

The DLTS adaptation of the capacitance-ratio method was experimentally demonstrated in the paper in Appendix C. In that work data were collected with

two voltage sequences and two boxcar gate-time ratios for each sequence. The

emission rates obtained from the analysis procedure were essentially identical for the smaller gate-time ratio, but were slightly different for the

larger ratio. This kind of experimental verification, that the emission rate

is independent of changes in experimental parameters, should be done in any

use of the capacitance-ratio method.

REFERENCES

1 . Ghandhi, s. K., Semiconductor Power Devices: Physics of Operation and

Fabrication Technology (Wiley & Sons, New York, 1977), pp. 2-31.

2. Moll, J. L., Physics of Semiconductors (McGraw-Hill, New York, 1964),

pp. 92-1 01 .

3. Engstrom, 0., and Aim, A., Thermodynamical Analysis of Optimal Recombination Centers in Thyristors, Solid-State Electronics 21 , 1571-1576 (1978).

4. Phillips, W. E., and Lowney, J. R., Analysis of Nonexponential Transient

Capacitance in Silicon Diodes Heavily Doped with Platinum, J . Appl . Phys . 54 , 2786-2791 (1983) [in Appendix B]

5. Phillips, W. E., Thurber, W. R., and Lowney, J. R., Improved Analysis

Procedures for Deep-Level Measurements by Transient Capacitance, Defects

in Silicony Proc. Vol. 83-9, W. M. Bullis and L. C. Kimerling, Eds. (Electrochemical Society, Princeton, N. J., 1983), pp. 485-490 [in Appendix C]

8
appendix a

Reprinted from IEEE PROCEEDINGS OF THE

CUSTOM INTEGRATED CIRCUITS CONFERENCE, May 1983

THE RELATIONSHIP BETWEEN DEEP-LEVEL MEASUREMENTS AND LIFETIME IN DEVICES*

Jeremiah R. Lowney, Robert D. Larrabee, and W. Robert Thurber

Semiconductor Materials and Processes Division

National Bureau of Standards

Washington, DC 20234

Abstract

The minority-carrier lifetime in silicon devices affects performance by influencing such

lifetime-related quantities as switching

times, storage times, and reverse-leakage

currents. A computer program has been developed to model the behavior of deep- level re- combination centers that control the minoritycarrier lifetime. The input parameters that

characterize the deep levels can be measured

by capacitive-transient techniques coupled

with a measurement of the low-injection-level

lifetime. These parameters can be used for

deep- level identification and diagnostic purposes and, as input to the program, for predicting lifetime under different injectionlevel conditions. This technique of deeplevel characterization is illustrated for n

+p and p n diodes containing deep levels from

intentionally added platinum.

I . Introduction

Recombination and thermal generation of carriers can be important in limiting and controlling

the behavior of silicon devices and integrated

circuits. The minority-carrier lifetime in the base and emitter of bipolar devices can limit

their gain. Generation (reverse leakage) currents

in charge-storage devices (e.g., CCDs and dynamic

RAMs) adversely affect their refresh times. The

needed switching speeds of some high-speed logic

circuits and high-power devices require intentionally added recombination centers to provide the

required rapid response (e.g., TTL logic circuits).

Therefore, there is a need for a thorough understanding of the theory of carrier recombination

and of thermal generation and its relation to the

diagnostic techniques used to characterize deep

levels which control this behavior. We consider

the Shockley-Read-Hall (SRH) theory ^ of recombination/generation through deep- level traps and, in

particular, the expressions for the high-level,

low-level, and space-charge recombination lifetimes. 2 we then relate the parameters of this

model to the quantities measured by deep- level

transient spectroscopy ( DLTS

3 and by lowin jection- level lifetime measurements. In this

way, deep- level characterization parameters are obtained for identification and diagnostic purposes, for modeling device behavior, and for pre- dicting lifetime under different in jection- level

conditions.

The analysis procedure developed in this paper

is illustrated by application to n+p and p

+

n diodes containing deep levels produced by intentional platinum doping. The minority-carrier capture

cross sections obtained in this way are compared

to the results obtained by other techniques. The

practicality of DLTS measurements on productionline material is demonstrated by results for a commercially available high-voltage rectifier

diode

II. Theory

The Shockley-Read-Hall (SRH) * theory of recom- bination through deep-level traps gives the following expression for the steady-state recombination of electrons and holes, -U, through a singlelevel trap:

(pn-n

i

2

) -U =^ = -=-

(n+n Jv(p+p iK

where n and p are the total electron and hole

densities

T = 1

> o no ' T t n'

T = (n v a

1~ * po ' T t p

n

i = N

c exp[(<V

Pi = N

v exP[(V

( 1)

( 2 )

(3)

(4)

In these expressions NT

is the trap concentrat 1 , v,. is the thermal velocity of the carrier, o t * n , p This work was conducted as part of the Semiconductor Technology Program at NBS and was support •••1 ;nby the Division of Electric Energy Systems, Department of Energy (Task Order A021-EES). Con* n! . the National Bureau of Standards, not subject to copyright.

9
are the carrier capture cross sections, N are

c , v the densities of states in the bands, E

c v are

the band-edge energies, G,p is the Gibbs tree energy of the trap, k is Boltzmann's constant, and T

is absolute temperature.

The minority-carrier recombination lifetime T may be obtained from eq ( 1 )

TP

(n

Q+ni +n ' ) + Tno

G? 0+Pi+n '

(n O+Po+n ' )

, ( 6 )

where the semiconductor is assumed to be n-type,

n^ and p^ are, respectively, the equilibrium electron and hole concentrations and n’ = p' , the excess carrier concentrations. It is very important to realize that the lifetime is a function of the injection ratio, p'/ng, and this ratio should

be measured as part of any lifetime measurement.

For p' << n^, ip is the low-level lifetime given

by:

If n' >> ng, Tp is the high-level lifetime

given by: HL

TP

T + T po no

( 8 )

At intermediate values of p' , the lifetime is not easily measured since it varies as the excesscarrier density decays. It should be noted that

only in the limits of high- and low-injection

level is the lifetime measured by the transient

decay of minority carriers described by eq (6). At intermediate injection levels, the trap occu- pancies may vary during the transient.

The thermal generation rate in a space- charge

region may also be computed from eq ( 1 )

SC where t is called the space-charge generation

lifetime. It is important to recognize that these

three lifetimes (i.e., t50

, iLL, and tHL

) can be very different in value. A center which may not contribute substantially to recombination in the quasi-neutral region of a device may dominate in

the space-charge regions because it has a near mid-gap value of free energy. Because different

centers may contribute to different lifetimes, a fully characterized model that can include all the

recombination centers in the material is required.

We focus now on the low-level lifetime in n- type material since it is usually straightforward

to measure on a device by reverse- recovery ** or by

open-circuit-voltage-decay ^ techniques. At low temperatures, n ^ , p^, and pQ are negligible and

TpLL = Tp0

, a quantity which is usually not strongly dependent on temperature. For deep acceptor levels (electron traps) one obtains tp1*1, *

Tp0

( 1 + n^/n

Q

) as temperature increases somewhat,

but before p. and pQ become significant. Correspondingly, for deep donor levels (hole traps).

one obtains tpLL = + 'tn0 (Pj/n

0

). Midgap levels would obey TpLL » Tp0

( 1 + n /nQ

) + Tno^ p

i^n

o^ as long as pQ

<< nQ

. (Shallow levels are gener- ally not efficient recombination centers because

of their small minority-carrier capture cross sections.) These three cases must be distinguished

and the pertinent parameters measured in order to apply a model of the lifetime to actual devices.

A very accurate method for doing this is deeplevel transient spectroscopy ( DLTS )

The methods of capacitance transient spectroscopy^ and DLTS, ^ in particular, furnish information regarding the energy, density, and capture

cross sections of deep levels. In DLTS, the highfrequency ( «1 MHz) capacitance of a test p-n junction under reverse bias is measured at two se- lected times, t and t^, following a trap-filling

period of reduced bias. After trap filling, the reverse bias is restored, and the traps in the space-charge region thermally emit their trapped

carriers. This changes the bulk space-charge

density in the depletion region and thus changes

the junction capacitance. Measuring the difference in capacitance at two selected times during

the transient provides a convenient way to characterize this exponential decay and to produce a signature for any deep level because this response

will peak at a reproducible temperature for any given values of the times at which the capacitance

is measured. The emission rate, e, (reciprocal response time) at the temperature of the peak response is given by:

(

fc 2/t

i

)

Theoretically, this emission rate for electrons

is given by:

e = ov N exp f-G /kT 1 . (11)

n n t .c

1 T'

All the quantities of eq (11) are, in general, a function of temperature except Boltzmann's constant. It is very important to recognize that

Gt

is the Gibbs free energy of the trap and not

the difference in energy between the trap and the

conduction band edge. It is the Gibbs free energy which is needed in the recombination eqs (4) and

(5), so eq (11) can be used to derive a value for G

t

. However, the majority-carrier capture cross section, o

n (T), is an unknown function of temperature. Thus, an independent measurement of on

(T) is necessary at each teinperature, particularly at the cryogenic temperatures where emission rates

are often measured.

A method for measuring the majority-carrier

capture cross sections has been discussed by —

Brotherton et al. ® The change in charge on the traps is measured as a function of filling time by

determining the change in applied bias needed to maintain a constant high-frequency capacitance

after the filling pulse is removed. A feedback

system capable of measuring capacitance in times

of the order of a microsecond is needed. If the cross section is not measured independently and

customary Arrhenius plots ^ are made to determine

the activation energy from eq (10), it is impor10
tant to remember that the energy obtained is not

the Gibbs free energy needed for lifetime modeling

and is, in general, only a signature of the emission process of the trap. Even if the cross section is temperature independent and the temperature dependencies of v

fc and N

c are taken into account, an Arrhenius plot will not, in general,

yield GT because GT

is itself temperature dependent. However, a characteristic signature of the

deep level can often be obtained in this way.

Another needed parameter is the minoritycarrier capture cross section. Various techniques 9 ' 19 have been described to measure this parameter in a manner similar to that of the majority-carrier cross section. However, the need

to generate minority carriers either by injection

from a second p-n junction 9 (transistor) or by

illumination" makes such measurements very difficult. In addition, there are difficulties in measuring the minority-carrier density that is so generated. The approach used in this paper is to use the measured low-level lifetime to determine

this cross section. This necessitates modeling

the SRH recombination kinetics on a computer and

fitting the low-injection-level lifetime vs tem- perature to the model using measured values for

all parameters except the minority-carrier capture

cross section. This requires knowledge of the

trap density, NT

, which can be determined from the amplitude of the capacitance transient in DLTS or other capacitance transient measurements. 9

III . Applications

As an example of the above technique, we have

considered the case of platinum in silicon. A

computer program was written to evaluate the equilibrium carrier density and lifetime expressions

as a function of both temperature and injection

ratio. Since Pt-doped silicon is thought to have

both a deep-acceptor and a deep-donor level associated with each center, multi-level trap statistics were used. The generalization of eq (1) for

Pt is : "

pn-n. ^ 1 (n t +px +p„x +nx _

, l 1A pA nA ID nD pD

- dn/dt c t—

T +T +T 12 3

where

T, = fpx +n X l(px +n„ x )

1 v nD ID pD n^ nA 1 A pA

T = fnx +p. X IfpX +n X ) 2 pD ID nD n nA 1A pA '

T =(nx +p,X )(nx +p, x 1 3 ^ pD MD nD n pA MA nA ;

)

( 12)

( 13)

( 14)

( 15)

where the meanings of terms are the same as in eq ( 1 ) except that subscripts "D" and "A” have

been added to denote the deep donor and deep ac- ceptor level.

The needed parameters, except for minoritycarrier capture cross sections, have been taken

from Brotherton's work on the Pt impurity

center: 92 E

Tft = Ec - 0.231 - kT In (0.5), ETD =

Ev

+ 0.314 - kT In (3.7), = 7.0 x 10“ 95 Cm

2

,

_ I C O and OpD =8 x 10 cm . These values do not con- tain any temperature dependences since their exact variations are still in question. For our purposes, these cross sections should be adequate

since we are going to fit lifetime data that are measured at temperatures greater than 200 K. At these temperatures the cross sections are expected

to be relatively constant, having values close to those given above. The lifetime data were taken

from Miller et at. 13 The reverse-recovery method

used by Miller is expected to yield the low-level

lifetime although this is not confirmed in their

paper. The trap densities used were based on the

results of the curve fitting that they performed

on their N

T data (measured by thermally stimulated

current techniques for four cases: 2.0 x 10 93

, 7.0 x 10 13

, 2.1 x 10 ", and 6.0 x 10 14 cm

-

at Ptdif fusion temperatures of 850, 900, 950, and

1000°C, respectively.

The results of our fitting their lifetime data

are shown in figures 1 and 2. The minoritycarrier cross section so obtained for n-type material (2 x 10"" cm 2 ) agrees with measurements by

Brotherton, 9 but the minority-carrier capture

cross section for p-type material (2 x 10 " cm 2

) is a factor of four higher than Brotherton '

value. 9 We are not certain whether Brotherton'

measured cross section is accurate for p-type ma- terial because of the difficulty in measuring minority cross sections by transient capacitance

techniques, or whether the trap density is not the same in p-type and n-type material. Only p

+

n diodes were used to determine N

T

in the work of Miller et at. 33 so it is impossible to determine

which of the above possibilities is correct.

The overall agreement between their data and

our computation is shown in figures 1 and 2. At low temperatures, the measured lifetime values are generally below our fitted curves, especially for the more lightly Pt-doped cases. This deviation

is attributed to the presence of another presumably very deep trap in their specimens which preferentially reduces the lifetime at low temperatures. In their fits to the data. Miller et at. used a simplified SRH-based model, which neglected

the terms containing the majority cross sections.

These terms are included in the present model, and therefore, their derived minority cross sections

differ significantly from ours.

Our model now furnishes us the ability to pre- dict the high-level and space-charge-generation

lifetimes. The values for the high-level, and

especially the space-charge generation lifetimes,

differ considerably from the low-level values. At room temperature in n-type silicon, with a Pt- diffusion temperature of 1000®C, the high-level

lifetime was calculated to be 26 ns and the spacecharge lifetime was calculated to be 100 us as compared with a low-level lifetime of 14 ns. The

situation is similar in p-type silicon for which

xHL = 26 ns, -PC - 100 us, and xn LL - 8 ns. In practice, the space-charge lifetime may differ

somewhat from these calculated values heeause f high-electric-field effects in the space-chai i.- region. 9

11
TEMPERATURE |K| Figure 1 . Hole recombination times through the platinum center in 2 (2»cm n-type silicon as a function of temperature. The solid lines represent lifetimes calculated according to eq (12),

and the data points are measured lifetimes from

ref. 13 for four p

+

n diodes diffused with Pt. The diffusion temperatures are: , 850°C; , 900°C; •, 950°C; and#, 1000°C.

Figure 2. Electron recombination times through

the platinum center in 2 (2«cm p-type silicon as a function of temperature. The solid lines represent lifetimes calculated from eq (12), and the data points are measured lifetimes from ref. 13 for four n

+p diodes diffused with Pt. The diffusion temperatures are: , 850°C;A, 900°C; #,

950°C; and#, 1000°C.

The situation in production-line device material is more difficult to assess than in specially

prepared laboratory specimens, e.g., the case of

Pt-doped silicon just discussed. The impurities

are unknown a priori, and a DLTS scan is needed to help identify them. Often there are so many deep

levels that the DLTS signals are hard to interpret

due to overlapping peaks. An example of such a

TEMPERATURE |°C] -175 -150 -125 -100 -75 -50 -25 0

Figure 3. DLTS response of a high-voltage rectifier diode with experimental conditions optimized

to resolve the large negative peak. The indicated energy for each level was obtained from an Arrhenius plot as a signature of each level for comparison purposes. The gate delay times were 89.5 ms and 191 ms. The reverse-bias sequence during the DLTS cycle was -17, -7, -17 V. The energies cor- responding to these peaks are: 0.21, 0.25, 0.29,

0.45, and 0.55 eV in the order of increasing peak

temperature

complex, but resolved, case is shown in figure 3 for a commercial hiqh-voltage rectifier diode.

There are five resolvable peaks which correspond

to different impurities or impurity complexes. To

date, only the highest temperature peak has been

correlated with previous results and has been

attributed to the gold acceptor state.

In this paper, we have shown a method for obtaining the parameters needed to model the recom- bination/generation lifetimes in silicon. The

technique utilizes measurements of low-injectionlevel lifetimes as well as transient-capacitance

results to obtain the needed parameters. This

model can then be used to predict lifetime in devices under different injection conditions. We would also like to emphasize the need to measure

lifetime and DLTS parameters in this way during

the processing of devices as a means of monitoring

deep levels. Some possible deleterious effects

can be traced to the presence of underlying undesirable deep levels. Additional controlled experiments may be necessary to identify the sources of any deep-levels found by comparison with previously established signatures. Test structures (p-n

junctions) can be included on the wafers to aid in the determination of those process steps that lead

to degradation of device performance. Once such problems are diagnosed, processing can be modified

to rid the device of the undesired recombinationaltering deep- levels. If deep levels are intentionally added, the routine measurement of lifetime and DLTS will serve to monitor the deep-level

addition process. 12
1 .

References

W. Shockley and W. T. Read, Phys. Rev. 87 , 835 (1952); R. N. Hall, Phys. Rev. 87, 387

( 1952)

2. S. K. Ghandhi, Semiconductor Power Devices

(New York, Wiley & Sons, 1977), p. 5.

3. D. V. Lang, J. Appl. Phys. 45, 3023 (1974).

4. B. Lax and S. F. Neustadter, J. Appl. Phys.

25, 1148 (1954).

5. B. R. Gossick, J. Appl. Phys. 27, 905 (1956).

6. C. T. Sah, L. Forbes, L. L. Rosier, and A. F. Tasch, Jr., Solid-State Electron. _1_3, 759 (1970).

7. O. Engstrom and A. Aim, Solid-State Electron.

2_1 , 1571 ( 1978).

8. S. D. Brotherton and J. Bicknell, J. Appl.

Phys. 49, 667 (1978).

9.

S. D. Brotherton and P. Bradley, J. Appl.

Phys. 53, 1543 ( 1982)

10. R. H. Wu and A. R. Peaker, Solid-State Elec

tron. 25, 643 (1982).

11. J. L. Moll, Physics of Semiconductors (New

York, McGraw-Hill, Inc., 1964), p. 99.

12. S. D. Brotherton and J. E. Lowther, Phys.

Rev. Lett. 44, 606 (1980).

13. M. D. Miller, H. Schade, and C. J. Neuse, J Appl. Phys. £7, 2569 (1976).

14. W. R. Thurber, W. E. Phillips, and R. D. Larrabee, Measurement Techniques for High Power Semiconductor Materials and Devices : Annual Report Oct. 1, 1980 to Dec. 31, 1981 NBSIR 82-2552 (U.S. G.P.O., Washington, DC,

1982), p. 10.

13
APPENDIX B

Analysis of nonexponential transient capacitance in silicon diodes heavily

doped with platinum

Willie E. Phillips and Jeremiah R. Lowney

Semiconductor Materials & Processes Division, National Bureau ofStandards, Washington, D. C. 20234

(Received 4 October 1982; accepted for publication 4 January 1983) An analysis having improved rigor has been made of the capacitance transient due to thermal

emission from charged defect centers in a semiconductor depletion region. This analysis extends

the range of applicability of capacitance-transient defect characterization techniques to nonexponential transient conditions which occur in diodes with trap densities ofthe same order as the net shallow dopant density or in diodes with somewhat smaller trap densities when defect centers are charged initially in only a part ofthe depletion region. An example ofthe improvement

is shown for three silicon diodes heavily doped with platinum.

PACS numbers: 85.30.De, 71.55.Fr, 72.20.Jv, 85.30.Kk

INTRODUCTION

Defect characterization techniques such as deep-level

transient spectroscopy (DLTS) 1 and isothermal transient ca- pacitance (ITCAP)2 utilize the ability of electrically active defect centers in depletion regions of semiconductors to trap and thermally to reemit charge carriers and thereby to affect the capacitance of the depletion region. It is assumed in conventional DLTS analysis that the measured capacitance

transient varies exponentially with time. There are several causes of nonexponential behavior: the emission rate may

depend on electric field, spatial nonuniformity ofemission or

capture rates may exist because ofcomplexing, the density of

the deep-level defects may not be small compared to the net shallow dopant density, the junction may not be abrupt, several trapping levels may exist (a sum of exponentials is nonexponential), and trap charging may occur in only part of the

depletion region with moderate trap densities (a transient capacitance in series with a constant capacitance). It is the purpose of this paper to use a more rigorous analysis to in- vestigate the capacitance transient in an abrupt-junction,

platinum-doped silicon diode which exhibits no electric field or spatial dependencies, but may have: (1) densities of deep

levels not small compared to that of shallow levels, or (2) trap charging over only a portion ofthe depletion layer with moderate trap densities (e.g., as occurs in profiling measurements). The work ofTomokagee/a/. 3 and that of Carcelle et al. 4 have considered (1), but not (2). THEORY OF ANALYSIS

An analysis of a transient which does not exhibit an

exponential behavior for either or both of the two reasons

listed above can be made in transient capacitance measure- ments as follows. The transient behavior of the junction ca- pacitance when the trap-filling bias is changed to a larger reverse bias, V, , is controlled in «-type silicon by the electron emission rate from defect centers. The net time rate of change of n , , the concentration per unit volume of electrons on the defect centers (i.e., traps) is given by5

dn,

~dt = e„p, -en n, ( 1

)

J. Appl Phys. 54 (5), May 1983

where en is the electron emission rate, e

p

is the hole emission

rate, and p, is the concentration of holes on the defect

centers. Because the trap must hold either an electron or a

hole, the totality condition is satisfied: n,+p,-N„ (2) where N, is the total defect center density. A differential equation in n, can be obtained from Eqs. (1) and (2): dn + ( en +e ]p, — e N,. (3) dt The general solution of Eq. (3) is

n, = C, exp[ - (e„ +ep

)t

] + C2 . (4) The constants C, and C2 can be evaluated at times zero and

infinity. At t — 0, n, = C, + C2 = and at t= oo, n, = C2 = n,f

. Equation (4) can be written - «,/)exp

[ - {e„ + ep

)t ] . (5)

Under ideal one-sided step-junction conditions, the junction

capacitance C at reverse bias Vr is6

c

- 2 = C, 2(K-K)

qeA 2 (Nd - n,) ’

( 6)

where Ch is the junction capacitance at some reduced trapfilling bias, Vc , A is the junction area, e is the absolute dielectric constant (permittivity) of silicon, Nd

is the shallow dopant density (assumed to be spatially uniform), and q is the

electronic charge. (Note: SI units are used throughout this paper.) This equation is valid only if there is no significant compensation of the shallow dopants by the traps as dis- cussed by Sah and Reddi. 7 The results of an extension of

their theory for the present cases is given in the Appendix

and a more general expression leading to a modification of

Eq. (6) is obtained. The result of this analysis has shown that

for sufficiently large reverse biases, there are ranges of bias voltages and times for which Eq. (6) is valid if the factor 2 is replaced by a dimensionless factor a which is constant during each menasurement but changes in value with experi- mental conditions. This generalization of Eq. (6) can be solved for n, to give

14
", a(K-K)ClC 2

qeA 2 (C 2 -C2

)

' (7)

At the initial time (t = 0), the reverse bias is reapplied (at the end of the trap-filling period), n, is n,„ and C is C, . In the

infinite time limit or final steady-state condition n, is n,f and C is Cf

. Substitution of n,, n„, and n,f

into Eq. (5) and solution for the exponential term gives:

exP[ - K +ep

)t

] = (Cl-C 2)(C 2 -C2

) =c (Cl-C^Ct-C2

)

' ( 8 )

The right side of Eq. (8) is a capacitance ratio and will be

called C, . The same equation can also be expressed in terms

of C as: C

2 Cl-C 2 e — I/T

(9) where r = l/(e„ + e

p

). For the special case where the trap density is small compared to the shallow dopant density (i.e., N,/Nd <\), Eq. (9) reduces to the customary exponential form for sufficiently large Cb

. Since C is bracketed by C, and C/( all the denominators in Eq. (9) will be approximately equal if C\ —C2 ~C\ — C

2

. When N, /Nd < 1 and Cb sufficiently large for exponentially, (C 2 — C

2 )/Cj = N,/Nd

. Therefore, one obtains C

2b —C2 zzC 2b — C

2 (1 —N,/Nd ). This expression

shows that in the limit of very low trap density (N,/Nd^0),

the denominators are nearly equal for all Cb > Cf

. For more

moderate trap densities, however, Cb must be sufficiently larger than Cf such that (C 2b — C

2 )/C 2 >N,/Nd

. When the denominators in Eq. (9) are approximately equal, C

2 -C}~{C2 -C}]e~ ,/r

. The difference of squares can be factored and for C, ~C} = C, the sums (C + Cf) and (C, 4- C{) are approximately 2Cf

so thatC~C, + (C, -Cf]e- ,/T

. (10)

Therefore, the conventional analysis in which this exponential transient is assumed is valid for N,/Nd small and for Cb sufficiently large. Under such conditions, a plot of log

[(C — Cf)/(C, — Cj-)] =log C„ against time would be linear with a slope proportional to 1/r. However, when these conditions are violated, as is shown below in the devices studied

here, it is necessary to use the more rigorous analysis leading

to Eq. (8) which predicts that a plot of log Cr against time would be linear with a slope proportional to — 1/r (equal to — 1/r if the natural logarithm is used).

DEVICE FABRICATION

The three devices used in this study are gated p

+ n and

n

¥ p diodes (device No. 10 of test pattern NBS-3)K and were

fabricated on 3- to 5-/2 cm ( 1 1 1 ) n- or p-type silicon wafers. Boron predeposition and diffusion through 0.432-mm diameter openings in 500-nm of field oxide formed a 450-nm deep

p

' n junction. The back side of this device was stripped and

coated with a spun-on platinum emulsion followed by a

drive-in at 850 °C for 2 h in a dry nitrogen ambient for device

No. 1, and at 900 °C for 1 h for device No. 2. The residual platinum was etched from the back side. Contact opening, top metallization (aluminum), back metallization (gold plus

0.6% antimony), top metal definition, and a 10-min 500 °C

microalloy in dry nitrogen completed the fabrication of the

structure. The structure was hermetically packaged in a TO100 header on a ceramic chip close to an electrically isolated temperature-sensing diode which was in good thermal contact with the test device because of high thermal conductivity ofthe ceramic. The n +p diode (device No. 3) was fabricat- ed in a complementary manner with boron replaced by

phosphorus and with a platinum diffusion temperature of 1000 °C for lh.

DEVICE CHARACTERISTICS

The abruptness of the devicejunctions was tested; a plot ofC

~ 2 against Vwas found to be linear at room temperature and to obey the ideal diode equation, Eq. (6). The shallow dopant density Nd on the lightly doped side was calculated from these data. At the temperatures of the measurements

(80-120 K), however, the C

~ 2 versus Vplots were nonideal

as shown in Fig. 1. The theoretical curves (shown as solid

lines) given by Eq. (A5) in the Appendix give a reasonable fit to the data (shown as circles) for devices No. 1 and 2. The

FIG. I. Normalized C

‘ : vs V plots of experimcnlal daia on mi*l

heavily, and very heavily platinum-doped silicon diodes ai lo» irm|.

lures (devices No. 1-3, respectively). The solid lines arc thcoreli, h > based on the present model, Eq |A5), for these three devucs amt t » m i,' , diode.

15
trap and ionized dopant densities of Eq. (A5) NT and Nd were chosen to give a good fit both to these sets of data and to the transient capacitance data. The density ratios used for device No. 1 were N,/Nd = 0. 19 and Nd /Nd = 0.78 with a dopant density (donors) of Nd = 1.08X 1015 cm

-3

. The ratios for device No. 2 were N,/Nd =0.38 and Nd /Nd = 0.77 with Nd = 1.21 X 10 15 cm

-3

. In device No. 3, the dopant density (acceptors) was Na = 3.01 X 10 15 cm

-3 and

the density ratios were N, /Na = 0.90 and N ~ /Na =0.925. The discrepancy between the model and the experimental

data increases with increasing platinum density and is signif- icant for device No. 3. It is not certain whether this discrep- ancy results from an inadequacy of the theory or from an

inhomogeneity of trap density in the specimen near the junction. The spatial dependence of the emission rate was investi- gated in device No. 2 by ITCAP measurements in incre- mental depletion layers of uniform width and electric field distributions. These test were accomplished by ( 1 ) applying a

reverse bias of 2 V to the specimen diode, (2) waiting for equilibrium conditions, (3) partially collapsing the depletion

region by reducing the bias to 0 V, (4) waiting a few seconds

for the traps in the previously depleted region to fill with

majority carriers, and (5) then restoring the depletion bias to

2 V. The capacitance transient response was recorded on an x-y recorder during and following these bias changes (see Fig. 2). The transient response was analyzed to obtain the emission rate. The measurements were repeated at a greater average distance from the junction by increasing the initial value of depletion bias to values greater than 2 V, but maintaining the same width of the active region (i.e., the portion of the space-charge region which upon reduction of the bias

fills with majority carriers). Because the C

-2

versus V plot was linear, an abrupt junction was assumed in which the

total depletion depth varies as the square root of the voltage

across the depletion layer. Thus the bias voltage during the

trap-filling time Vc and the bias voltage during the time of

FIG. 2 Transicnl capacitance response against time of a p' n silicon diode

(device No 2| heavily doped with platinum \N,/Nd = 0. 38). Initial reverse

bias is 20 V, charging reverse bias is 9 35 V, and the temperature is 87.7 K.

C, = 2.36 pF, C, = 2.65 pF, and C, = 3.62 pF.

observing the emission transient Vr can be selected so that the width of the active region is constant but is located at various distances from the junction. The voltage across this active portion ofthe depletion region was held constant as V, and Vc (and thus the depth) were changed. The series of such measurements gave transients with a constant emission rate

in the range of Vr of 2-20 V reverse bias. The tests were

repeated with a larger average electric field (minimum Vr = 5 V) with the same resultant constant emission rate. A

completely independent experiment with sulfur-doped sili- con showed variations with both electric field and position of

the active region. Therefore, for the case of this platinumdoped device, it was concluded that the emission rate is inde- pendent both of these electric field variations in the depletion

region and of the position of the active region relative to the junction at distances greater than the zero-bias depletion

depth.

RESULTS

A typical as-measured isothermal capacitance vari- ation is shown in Fig. 2. The absence of a transient in Cb (the capacitance under trap-charging bias conditions) indicates that a slow-capture region resulting from spatially nonuni- form capture rates

9,10 is not a problem for device No. 2. The

same is true for device No. 1 . A short transient is observed on

device No. 3. When a transient is observed, one can wait

until the traps, in the slow-capture region are filled (as was

done for device No. 3). In Fig. 3, the transient response of Fig. 2 has been re- plotted (shown as circles) from digitized data as the conventional DLTS curve of log Cn against time. The plot is seen to be slightly nonlinear indicating that the measured capaci- tance transient is nonexponential. The instantaneous time

constant (reciprocal of the emission rate) is proportional to the slope of the data and is dependent upon the time at which

it is measured. The initial slope (shown as a dashed line) has a time constant of 12.7 sec. The nonexponentially of the ca- pacitance transient is attributed to violation of the small N,/

Nd approximation (here N,/Nd — 0.38) and to charging of

traps in only part ( — 1/3) of the depletion region. In contrast, a plot of the logarithm of Cr of Eq. (8) against time for the

transient of Fig. 2 is shown as solid dots in Fig. 3. The solid

line is a weighted linear regression fit to the data with a weighting factor of Cr and shows that these data are linear to within experimental error. The + ’s are obtained from Eq. (Al) and are in good agreement with the experimental data. This agreement serves to justify the application of the present model to devices of this kind. The capacitance ratios of a less heavily doped (N,/Nd = 0. 1 9) diode (device No. 1 ) are shown in Fig. 4. Again, the

solid circles are log Cr , the open circles are log C„, and the + ’s are theoretical values of the same quantities as calculat- ed by Eq. (Al). The linearity of the C, values and the good

agreement between the model presented in the Appendix

and the experimental data are again obtained. A more dramatic case is shown in Fig. 5 for the more

heavily doped n

+p diode (device No. 3, N,/Na = 0.90) for two bias sequences. One curve shows data (circles) and the16
FIG. 3. Semilogarithmic digitized and normalized plot of the transient capacitance ratio C„ for the data of Fig. 2 (shown as open circles) against

time. The dashed line is the initial slope. The solid

circles represent the same data with the present more rigorous analysis which leads to the capaci- tance ratio C, given by Eq. (8). The solid line is a weighted linear regression fit to the data of log C,

against time. The + ’s are theoretical values cal- culated by use of Eq. (Al| with C, = 2.36 pF, C/ — 2.65 pF, and cb = 3.62 eV.

ory ( X ’s) for a trap-filling bias of 4 V, and the other shows

data (squares) and theory ( + ’s) for a trap-filling bias of 6 V. The bias during the emission transient for each is 9 V. The

slight difference in the solid weighted linear regression lines through the C, data is attributed to a systematic temperature

difference of about 0.05 K between these runs. The solid

circles and squares which correspond to log Cr at the beginning of the transient are below the fitted line as predicted by

calculations from Eq. (Al). For this reason, experimental

points before the time r/2 are excluded from the linear regression fit but are shown in Fig. 5. Even though the theory

is less successful in predicting log C„ for this case, the agree- ment is very good for log Cr

. The significance of the linearity of the log C, plots in Figs. 3, 4, and 5 is that the present analysis correctly accounts for the large trap density which leads to compensation of the partially deionized dopants, and for the unchanging capacitance in the part of the depletion region in which

the traps are never charged. In Fig. 3, the time constant was

found to be 16.88 ± 0.03 s, which is about 34% greater than

the time constant indicated by the initial slope of the uncorrected capacitance data and about 1 6% greater than that of a

straight line fitted to all of the uncorrected data. The above

uncertainty value of ± 0.03 s is the one standard deviation uncertainty in the statistical fit only and does not include any

instrumentation error. In Fig. 4, the time constant for the

linear regression line is 13.92 ± 0.03 s which is 5% more

than the uncorrected value obtained from a linear regression

line fitted to the open circles. In Fig. 5, the time constant for the 9, 6, 9 V bias sequence (solid squares) is 70.20 ± 0.05 s,

FIG. 4 Semilogarithmic digili/ed and normalized plot of the transient capacitance raiio ( (shown as open circles) and C, (shown as solid ck

cles) against time for a p n silicon diode desn <• No. 1) moderately doped with platinum i \ \ = 0. 19) The solid line is a weight' d linear regres sion fit to the data of log C, against turn 7 fl- are theoretical values calculated h> us* I I , A i with C, = 4.55 pF, < 1.72 pi pF as deduced by curve tilling I xpenme'iii.il \..i ues for this capacitance transient are < 4 pF, Cj = 4 72 pF, and C, o 03 pi Initial r< verse bias is 5 V. churging bias i \1 ^ \ , j m

temperature is 88.3 K

17
TIME (s)

FIG. S. Semilogarithmic digitized and normalized plot of the transient capacitance ratios C„

(shown as open circles and squares) and C, (shown as solid circles and squares) against time

for a n +p silicon diode (device No. 3) very heavily doped with platinum (N,/Na =0.9). Initial reverse bias is 9 V, charging reverse bias is 6 V

for the squares and 4 V for the circles, and the temperature is 1 19 K. Measured capacitance for the squares are C, — 2.20 pF, = 3.17 pF, and Cb = 3.39 pF and for the circles are C, = 2.03 pF, Cf = 3.17 pF, and Cb = 3.62 pF. The 4- ’s are theoretical values for the 9, 6, 9 V biasing sequence (C, = 2.36 pF, Cf = 3.34 pF, and Cb = 3.65 pF), and the X ’s are theoretical values

for the 9, 4, 9 V biasing sequence (C, = 2.07 pF, Cf = 3.34 pF, and Cb = 3.94 pF). The solid

lines are linear regression fits to the two sets of

data with points prior to i = 35 sec excluded.

which is about 4.5 times greater than the time constant indi- cated by the initial slope of the uncorrected capacitance data

(open squares). The time constants reported here for platinum in //-type silicon are in good agreement with an extrapolation of pre- vious characterization measurements on devices similar to device No. 1." Previous measurements on the p-type di- odes 11 were made on heavily platinum-doped devices with

biases in the nonlinear region of the C~2 versus Vcurve and

give an extrapolated time constant less than the time constant reported here.

CONCLUSIONS

It is concluded that the use of the present more rigorous

analysis can correct for the nonexponentiality introduced by

a large density of defect centers which compensate the partially deionized shallow dopants at low temperature. It also corrects for trap charging in only a part of the depletion

region at moderate levels of defect density. It was shown that the conventional exponential approximation is adequate

only for small trap densities and that the correction for charging of these traps in only part of the depletion region decreases with decreasing trap density. The present analysis extends the range of applicability of transient capacitance

techniques to conditions encountered in practical devices

(e.g., N,/Nd not <1) and in more detailed measurements

(e.g., deep level profiling). ACKNOWLEDGMENT

This work was conducted as part of the Semiconductor Technology Program at NBS and was supported by the Division of Electric Energy Systems, Department of Energy

(Task Order A021-EES).

APPENDIX

At low temperatures, the Fermi level lies close to the conduction band edge in n-type silicon or to the valence band

edge in p- type silicon. If platinum is present, the Fermi level

lies below the platinum donor level in p- type material or above the platinum acceptor level in n-type material. These

levels then become charged and the platinum tends to compensate the shallow dopant. At sufficiently low temperatures significant compensation can occur in heavily Pt- doped diodes because the shallow dopants deionize. The re- sult is that ap-n junction diode heavily doped with platinum

has aC'J versus V relationship which is nonlinear at low

temperatures.

In order to model this situation, the theories of Sah and

Reddi7 and of Tomokage et al.

3 have been extended to in- clude effects due to incomplete collapse of the space charge

region and to compensation of the shallow dopant. Consider

the space-charge region ofap +

n or an n +p diode to be divid- ed into three regions. The part of the space-charge region on

the heavily doped side of the junction is neglected as in the previous work. 7,3 The effect of the platinum acceptor in ptype material or the platinum donor in //-type material is ignored as in the previous work7,3 as well. For ap +

n diode

the first region, nearest the metallurgical junction on the

lightly doped //-type side, is taken to have a fixed bulk charge

density of qNd , where Ndis the shallow donor density. The

extent of this region is determined by the crossing of the majority carrier quasi-Fermi level and the platinum donor

level in p- type material or the platinum acceptor level in n- type material when the bias voltage is Vc , during the trap

filling period. The second region is taken to have a time-varying bulk charge density of q(Nd — //,) where >•, is the occupied trap

density. The extent of this region is between the end of the

first region and the crossing of the same quasi-Fermi level and platinum level when the bias voltage is Vr , the maximum

reverse bias during which the transient emission is observed. The third region extends from the end of the second region and has a fixed bulk charge density of q(N d — N,) where Nd

is the average density of ionized donors and N, is the

18
total trap density (i.e., all the traps are occupied and contri- bute to compensation). Since the quasi-Fermi level is close to the band edge in this region, there is some freezeout of the shallow dopant so that Nd is less than Nd

. The extent of this region is determined by the total width of the space charge

region as found by integrating Poisson’s equation twice. Since considerable compensation can occur between the

filled traps and the ionized shallow dopants, this third region can be very wide. The result is a significant decrease in the junction capacitance, which is given by eA /w(t ) at high fre- quencies relative to trap emission rate, where w(t) is the width of the space-charge region and A is the junction area. The result of this extension of the Sah-Reddi model7 is that the high frequency capacitance C (r, V) is given by

c~%K) / 2 y

/2 i_

\qeA 2Nf) (Nd -n,)

X| Py+ [/3Y+N+(Nd -n,)

X(Vr + Vd -Va -0f/Nd+

)]'

,2

\, (Al)

whereP = NT — n, + Nd — Nd and

r = NJ 4> n; -nt

1/2

(A2)

(A3)

The built-in diffusion potential Vd

is taken to be approximately 1 V, i.e., slightly less than the energy gap. The potential </> is the difference between the energy of the quasi-Fermi

level and the trap level. This quantity is always greater than

or equal to zero; if it were less than zero, the traps would be empty and their effect would disappear. The platinum donor

level is approximately 320 meV12 above the valence band, and the platinum acceptor level is approximately 230 meV

below the conduction band. 12 The corresponding values for

4> are approximately 250 meV in p-type material and 160 meV in n-type from estimates of the Fermi energy in material containing ~ 10 15 cm

-3 shallow dopants at ~ 100 K. The voltage Va

is given by

Va = (q/2e)n, leAC ~

( oo , Vc ) - (— ^- YT, (A4)

l \9(A / - Nt )J J where C

~ l (co,V) = ( \ -X

/i \py+ [ pY + {NJ-/Nd ) \qeA lNJ )

X (V + Vd -py1Nd /N d+ )y

l2

\ (A5)

and p = (Nr + Nd -Nd+ )/Nd

. (A6)

In Eq. (Al) it is assumed that

nt =NTe-‘ /T

, (A7) which should be true at the low temperatures at which the measurements were made. Equations (A1)-(A7) are written

for shallow donors and deep acceptor traps, but analogous

equations exist for shallow acceptors and deep donor traps. The values for the trap densities and ionized shallow dopant densities were obtained by fitting approximately both the C

~ 2 versus V data and the total change in capaci- tance during the transient. The total shallow dopant density was obtained from room temperature C

~ 2 versus V plots, which obeyed the ideal linear relationship for an abrupt

junction. The amount of deionization found from these anal- yses was considered very reasonable when compared with

calculations of the neutrality condition at these temperatures for samples containing no traps. A comparison of Eq. (Al) and Eq. (6) shows that they

are in agreement only if y = 0, i.e., <f> = 0. However, if the reverse bias voltages Vc and Vr are sufficiently large and

sufficiently close to each other, the effect of y > 0 is to alter Eq. (6) such that for sufficiently long times, the form of Eq.

(6) remains valid except that the factor 2 must be replaced by

a dimensionless constant a which is in general greater than

2. This approximation is given by Eq. (7). The validity of this approximation is shown by the result of fitting the capaci- tance ratio Cr , given by Eq.(8), which is discussed in the Results section. The dependence of a on bias voltages and

time is very complex, and it is better to use the fundamental

Eq. (Al) to determine the validity of the approximation giv- en by Eq. (7).

'D. V. Lang, J. Appl. Phys. 45, 3023 (1974). "W. E. Phillips and M. G. Buehler, in Semiconductor Measurement Technology. Progress Report, July 1 to September 30, 1976, edited by W, M

Bullis and J. F. Mayo-Wells, NBS Special Publication 400-36 (U S. GPO,

Washington, D.C., 1978), pp. 20-22. 'H Tomokage, H. Nakashima, and K. Hashimoto, Jpn. J. Appl. Phys 21, 67(1982).

4

J. E. Carcelle, P. Cartujo, J. R. Morante, J. Barbolla, J. C Brabant, and M

Brousseau, Rev. Phys. Appl. 15, 843 (1980)

5 M. G. Buehler, Solid-State Electron. 15, 69 (1972). "W. Shockley, Bell Sys. Tech. J. 28, 435 (1949)

7 C. T. Sah and V. G. K. Reddi, IEEE Trans. Electron Devices ED- 1 1 345

(1964). "M. G. Buehler, Semiconductor Measurement Technology Microelectronic

Test Pattern NBS-3 for Evaluating the Resisitivity-Dopant Density Relationship ofSilicon, NBS Special Publication 400-22 |U S GPO. Washing

ton, D.C., 1976).

'’H G. Grimmeiss, L. A. Ledebo, and E. Meijcr, Appl Phys Lett 3* 'o7

(1980).

"’J. A. Borsuk and R M Swanson, J. Appl Phys 52. 6704 1 1981 >

1

F. F. Oeltinger and R. D. Larrabec, editors. Measurement 1, . hmqu* n >/ High Power Semiconductor Materials and Devices Annual Rep, -n < K lober I, 1978 to September JO, 1979, lntcragcm> Report NBSIR BO-206

(U S. GPO , Washington, D C.. 1980), pp 16- 17

|: S. D Brotherton, P Bradley, and J Hickncll. J Appl Phs' 50 W6

(1979).

19
APPENDIX C

Defects in Silicon, Proceedings Volume 83-9, W. M. Bullis and L. C.

Kimerling, Eds., pp. 485-490 (Electrochemical Society, Pennington, N.J.,

1983) .

IMPROVED ANALYSIS PROCEDURES FOR DEEP-LEVEL

MEASUREMENTS BY TRANSIENT CAPACITANCE*

W. E. Phillips, W. R. Thurber, and J. R. Lowney

National Bureau of Standards

Washington, DC 20234

The procedures reported here provide a way to analyze data

from nonexponential transient capacitance measurements

made under conditions such that (a) the trap density is not small compared with the net shallow dopant density or

(b) the traps are charged in only a part of the depletion

layer at moderate levels of trap density. This analysis

requires C”2 vs_ V data to be linear over the voltage range

used, which may be a small range at low temperatures because of the compensation effect of traps. The generalized analysis developed for isothermal transient capacitance ( ITCAP) is extended to the more commonly used DLTS

technique. The use of this analysis to obtain the correct emission rate from a nonexponential transient is illustrated by ITCAP and DLTS measurements for platinum in p

+n

silicon diodes.

The growing popularity of transient-capacitance deep-level

measurements (1) and the convenience of the deep-level transient

spectroscopy (DLTS) method (2) have motivated this effort to extend

the precision of our isothermal transient capacitance (ITCAP) method

(3-4) to DLTS.

We have formulated a generalized approach to ITCAP (4), which

we apply to DLTS. This approach is based upon a nearly linear C~ 2

vs V relationship for a p-n junction test diode. By making the

depletion approximation, it is shown that the density of- electrons

on the defect centers, nt

, in the depletion region of a p

+

n

junction under reverse bias, V

f

, is (4):

o(V - V )C. 2C 2 r c b

, = N , d q eA 2(C 2 - C

2

) D (D

where C is the junction capacitance, Cw is the capacitance at

some reduced trap-filling bias, V , A is the junction area, e is the absolute dielectric constant (permittivity) of silicon, N^ is

the shallow-dopant density (assumed to be constant), q is the

electronic charge, and a is a dimensionless parameter which is determined by the slope of the C~ 2 vs V relationship and which must

remain sufficiently constant between Vr and V

c and throughout

the capacitance transient.

•Contribution of the National Bureau of Standards. Not subject to copyright.

20
Figure 1 shows experimental C~ 2

vs_ V data for two platinum-doped

p

+n silicon diodes of different platinum densities, along with theoretical curves calculated from the theories of Sah and Reddi (5) and

Tomokage et al . (6) extended to include effects due to incomplete

collapse of the space-charge region and to compensation of the shallow dopant by the platinum at low temperatures (4). We can see- in

Fig. 1 that the data are nearly linear over most of the voltage

range, but with a slope that differs from the ideal case. The deviation between the data and the theory at low voltages for device no.

2 is not understood but may be due to inhomogeneity of the trap

density near the junction.

Just after the bias voltage is changed from Vc to Vr

, nt

is initially n

fc , and C is initially C^. After the resulting transient,

n t becomes finally n

fc j and C becomes finally C^. The time constant

of the transient, t, is related to a capacitance ratio given by:

This relationship corrects for two conditions which are usually

neglected in conventional DLTS measurements: 1) that the density of

the deep-level defects may not be small compared to the net shallow

dopant density, and 2) that trap charging may not take place

throughout the depletion region at moderate levels of trap density.

When the ratio of trap to net shallow dopant densities is not small, Nt/Nj > 0.1, both of these conditions will produce a nonexponential

capacitance transient. A test for nonexponentiality has been reported (7) which requires a series of DLTS measurements for which

T = (tj - t^)/ in(t j/tj) is constant but the ratio of gate times

t./t

2 varies over a wide range. A shift in the temperature location

of the DLTS peak as t,/t 2 varies indicates nonexponentiality. If nonexponentiality is found, the correct time constant can be calculated for the above two cases. This was demonstrated by use of the platinum-doped test diodes of Fig. 1.

The diodes of Fig. 1 were gated p

+n diodes fabricated on a 3 to

5 Q*cm <11 1> n-type phosphorus-doped silicon wafer. Boron predeposition and diffusion through 0.432-mm diameter openings In 500 no of field oxide formed a 450-nm deep p

+ region. The backside was stripped and coated with a spun-on platinum emulsion followed by a drive-in diffusion in a dry-nitrogen ambient at 850°C for 2 h for

device no. 1 and at 900°C for 1 h for device no. 2. The residual

platinum was etched from the backside with HF. Contact opening, top

metallization (aluminum), back metallization (gold plus 0.6 percent

antimony), top-metal definition, and a 10-min 500°C microalloy in

dry nitrogen completed the fabrication of these test diodes.

21
A plot of log Cr vs time for ITCAP data for device no. 2 (Nt/Nd =0.38) is shown in Fig. 2 as solid circles. The solid line

is a weighted linear regression fit to these data. The open circles

represent a conventional plot of the same data plotted as log

((C - C

f )/(C^ -Cf

) ). It is nonlinear (nonexponential). The dashed

line shows the initial slope of the uncorrected transient which has

a significantly shorter time constant than the remaining portions of

the curve. The +*s are theoretical values for the corrected and

uncorrected cases, calculated as done in Fig. 1 (4). This demonstrates that these corrections are significant.

Equation (2) can be rearranged as:

C(t, T) C

b

2 C 2c 2 b f C 2-C 2 b f KV c„V\ -t/x

- 1/2

(3)

The DLTS peak occurs when d[c(t , t) - C(t , t) ]/d T = 0 . The

resultant equation can be solved iteratively for t (8):

*1 - t 2

where

B(t) = 1 + A^e-t/ T + A + A^e-^/"1 + A^e-^/* , A x = F(Cb

2 + 3Cf2

) Cf“ 2

, A

2 = F

2(3Cb2 + 3Cf2)Cf“ 2

, A

3 = F

3 (3Cb2 + Cf

2)Cf " 2 , and A 4 = F

4 Cjj^f-2

, where

F = (C±

2 - Cf

2 )/(Cb

2 - Ci 2

) . Fortunately, the iteration converges very rapidly. The correction

is most sensitive to an uncertainty in and least sensitive to an uncertainty in C^.

Figure 3 shows results of constant-voltage DLTS measurements

performed using a double-boxcar system (9) on a diode from the same heavily platinum-doped silicon wafer used for the ITCAP measurements

in Fig. 2. Normalized emission rate values, e/T 2

, where e = t“ which have been corrected using eq (4), are shown in Fig. 3 as well

as the uncorrected values obtained from the usual expression for emission rate which is valid only for exponential transients.

Results are plotted for two different reverse bias sequences (i.e., vr

, V

c

, V

r

). It is evident from the figure that the correction

22
is much larger for the -10, -8, -10 V sequence. This is a consequence of charging only part of the depletion region during the -8 V

trap-filling period. Although the uncorrected curves are significantly different, the corrected curves almost coincide as expected.

This correction of DLTS data requires the measurement of the same three capacitances as a function of temperature as for the

ITCAP measurement. These capacitances, C^, C^, and C^, were measured with a commercial capacitance bridge with a 15-mV, 1—MHz test

signal. The use of 10 MHz for the DLTS measurement is not a problem

because the traps are slow and the stray capacitances are negligible

at these frequencies. The values for and were recorded on an x-y plotter in separate temperature runs. The determination of is much more difficult as it is necessary to capture the leading

edge of the transient. In order to measure the Cj_ associated with

reasonably fast transients, modifications were made to the capacitance bridge to improve its response time. It was then possible,

from oscilloscope photographs, to obtain tC = (Cj -C^ ) for most of

the temperature range normally covered by boxcar DLTS measurements,

and only points in this range are included in Fig. 3. It was found

that, within experimental error, fC was independent of temperature

over the range measured. This is not unexpected as is only

slowly increasing with temperature in the same range. Thus in most

applications the capacitance can be measured at three or four

temperatures and then interpolated to obtain values at the specific

temperatures required. Once these three capacitances are known, eq

(4) can be solved.

As seen in Fig. 3 and from an examination of the equations,

this more general DLTS analysis correction is largest for situations

in which the deep level density is relatively large and the depletion region is incompletly collapsed, as is likely to be the case in

depth-profiling experiments. It is also apparent that the analysis

makes a large but similar correction in the emission rate of each

point such that the slope is changed 2.5% or less. Cross sections

as obtained from the extrapolated Arrhenius plots change about 30%.

Mathematically, the analysis is correcting a constant-voltage DLTS experiment to make it equivalent to constant-capacitance DLTS where

effects of deep-level density and incomplete charging are experimentally minimized. Thus constant-voltage DLTS measurements, which are simpler experimentally and cam extend to faster time constants, can

yield parameters which are at least as accurate as those obtained

from constant-capacitance DLTS. The corrections reported here would

not be required for constant-capacitance DLTS.

We wish to point out, however, that our correction technique is

limited. It should not work as well if either the voltage difference (V

r

- V

c

) is large compared to V

r or the gate times are far apart. The reason is that the slope a in eq (1) will vary significantly over an extended range of these parameters. To examine the effects of a different gate-time ratio on the correction, we mea23
sured the diode of Fig. 3 with the same bias voltage sequences but

with a gate-time ratio of five instead of two. The results are

shown in Fig. 4. In this figure note that the corrected data sets have a definite offset from each other in contrast to the coincident

corrected sets in Fig. 3. Also, the corrected set in Fig. 4 with

the -10, -8, -10 V bias sequence is the one which agrees best with

the corrected sets in Fig. 3. This is reasonable, as the correction

should work best for the data corresponding to the smaller voltage

difference. These two figures illustrate the importance of keeping

gate-time differences and voltage differences small and verifying

the correction by repeating the measurement with a somewhat

different gate-time ratio and voltage difference.

REFERENCES

1. C. T. Sah, L. Forbes, L. L. Rosier, and A. F. Tasch, Jr., SolidSt. Electron. 13 , 759 (1970).

2. D. V. Lang, J. Appl. Phys. 45 , 3023 (1974).

3. W. E. Phillips and M. G. Buehler, Semiconductor Measurement Technology; Progress Report, July 1 to September 30, 1976,

edited by W. M. Bullis and J. F. Mayo-Wells, NBS Special Publ.

No. 400-36 (U.S. G.P.O., Washington, DC, 1978), pp. 20-22.

4. W. E. Phillips and J. R. Lowney, Analysis of Nonexponential

Transient Capacitance in Silicon Diodes Heavily Doped with Platinum, J. Appl. Phys. 54 , 2786 (1983).

5. C. T. Sah and V. G. K. Reddi, IEEE Trans. Electron Devices ED21, 345 (1964).

6. H. Tomokage, H. Nakashima, and K. Hashimoto, Jap. J. Appl. Phys.

22, 67 (1982).

7. W. R. Thurber, R. A. Forman, and W. E. Phillips, J. Appl. Phys.

53, 7397 (1982).

8. W. R. Thurber, W. E. Phillips, and R. D. Larrabee, Measurement

Techniques for High Power Semiconductor Materials and Devices :

Annual Report, October 1, 1980 to December 31, 1981, NBSIR 82-

2552 (U.S. G.P.O., Washington, DC, 1982), pp. 49-52.

9. R. D. Larrabee, W. E. Phillips, and W. R. Thurber, Measurement

Techniques for High Power Semiconductor Materials and Devices ; Annual Report , October 1, 1979 to September 30, 1980, NBSIR SI2325 (U.S. G.P.O., Washington, DC, 1981), pp. 8-9.

24
Fig.1 . Normalized C_2-V plots

of experimental data on moderately and heavily platinum-doped

silicon diodes at 88 K.

Fig. 3. An Arrhenius-type plot

of scaled emission rate vs reciprocal temperature £or a p

+n silicon diode heavily doped with

platinum. Data were obtained

using a DLTS system with an rf bridge operating at 10 MHz with

a gate time ratio (t^tj) of 2 for all points. Corrected values of emission rate were obtained from the general DLTS

analysis discussed in the text

for the two reverse bias se- quences shown.

new

Fig. 2. Normalized semiloqarithmic plot of the digitized

transient capacitance response against time of a p

+

n silicon

diode (device no. 2) heavily

doped with platinum at T = 88 K.

Fig. 4. An Arrhenius-type plot

with all parameters the same as those for Fig. 3 except for the

gate time ratio (tj/t.) which

was five for all points.

25
APPENDIX D

Details and Listing of Lifetime-Predicting Computer Program

The program is divided into five separate routines: MAIN, ZEROIN, TEST, TAU,

and TAUM. The MAIN routine inputs data, outputs results, and performs a few

simple calculations. Subroutine ZEROIN calculates the Fermi energy by using

a variant of the Newton-iteration scheme. It calls subroutine TEST to calculate the net ionized dopant and deep-level density for trial positions of the

Fermi level. Subroutine TAU calculates lifetimes for single-level traps

according to the equations in Ghandhi [1]. Subroutine TAUM calculates lifetimes for multilevel traps according to the equations in Moll [2]

The entry order of input data to the program is: three single-level donors,

three single-level acceptors, first multilevel species (donor, then acceptor), second multilevel species (donor, then acceptor). There are five initial input parameters for each entry: 1) the name of the species (up to 8

characters); 2) the density of the species (cm-3

); 3) the appropriate energy

level of the species in the energy gap (eV) ; 4) the degeneracy factor of the

species appropriate to the level, which combines both electronic and latticerelated components; and 5) a flag, FLAG #1, which if false (F) means that

donor energies are relative to the conduction band and acceptor energies are

relative to the valence band, or if true (T) means that these energies are relative to the opposite bands. These parameters are entered for each species with a format (4A2, IX, E9.4, 2F9.4, A1 ) with a blank line interpreted

as a null entry.

Following these entries, one enters: the electron capture cross section

(cm2

) and hole capture cross section (cm2

) for each species actually entered

above (i.e., there are no blank lines). The format is (2E10.4). Following

these entries, one enters: the starting temperature (K) , the final temperature (K), and the incremental temperature (K) with a format (3F10.0). Finally one enters a second flag, FLAG #2, with format (LI).

This second flag is set to true (T) if it is desired that the lifetime be

computed as a function of injection level. Otherwise, only the low-level

recombination, high-level recombination, and space-charge generation lifetimes are computed. The program makes the assumption that the injected ma- jority and minority carrier densities are equal, which is usually a good

approximation if the trap densities are small compared to the dopant density

so that a change in trap occupancy due to injection of carriers can be neglected. Therefore, this flag should be set to true only in material con- taining low trap density. For material which contains large trap densities,

even if they are not effective recombination centers, a more involved analysis is needed to obtain lifetime as a function of injection ratio. These

restrictions do not apply for the three lifetimes computed when the flag is

false.

The numbers 1 x 10 30 or 1 x 10-30 have been used in the program to avoid zero

denominators. These largest and smallest numbers may be reflected in certain

output quantities.

26
The format 4A2 is used to input and output literals. This format may have to

be changed to suit other computers. The FORTRAN statement CHARACTER is used

to define the literal variables and may not be available with some older

versions of FORTRAN.

The outputs of this program are self-explanatory with the following remarks

An error flag is printed when the ZEROIN search for the Fermi energy fails.

Such an error would imply inconsistent input.

The zero of energy of EF , the Fermi energy, is taken to be the top of the valence band. The zero of energy of all donor levels is the bottom of the

conduction band and for all acceptor levels, the top of the valence band.

For multilevel species, the donor level always precedes the acceptor level in

the list.

The ionized density (ion. den.) refers to the presence of charge associated

with the level. When shallow dopants are ionized, they are charged; when

deep levels are ionized, they are also considered to be charged. An example

of an input and output listing is given following the listing of the code.

27
LIFETIME-PREDICTING COMPUTER PROGRAM

C MINORITY-CARRIER LIFETIME IN SILICON

C

C PROGRAM CAN HANDLE UP TO 3 DONOR AND 3 ACCEPTOR SPECIES

C (SINGLE LEVEL) ALONG WITH TWO MULTILEVEL SPECIES

C OF THE SAME TYPE AS PLATINUM AND GOLD.

C

CC INPUT VARIABLE NAMES

C

c

r LEVEL NAME DENSITY ENERGY DEGEN. FL1

l

c DONOR 1 NAME1 DN (1

EN(1 ) DG(1

FL1 (1

c DONOR 2 NAME2 DN (2) EN (2) DG (2) FL1 (2)

c DONOR 3 NAME3 DN (3) EN(3) DG(3) FL1 (3)

c ACCEPTOR 1 NAME4 DN (4) EN (4) DG (4) FL1 (4)

c ACCEPTOR 2 NAME5 DN(5) EN(5) DG(5) FL1(5)

c ACCEPTOR 3 NAME6 DN (6) EN (6) DG (6) FL1 (6)

c ML DONOR 1 NAME7 DN(7) EN(7) DG(7) FL1 (7)

c ML ACCEP 1 NAME8 DN (8) EN (8) DG (8) FL1 (8)

c ML DONOR 2 NAME9 DN(9) EN(9) DG(9) FL1 (9)

c

r ML ACCEP 2 NAMEIO DNCIO) EN (10) DG (10) FL1 (10)

L

C DONOR ENERGIES ARE REFERENCED TO CONDUCTION BAND

C ACCEPTOR ENERGIES ARE REFERENCED TO VALENCE BAND

c HOWEVER, THE REVERSE IS TRUE IF FL1 IS SET TRUE.

C

C

C

C ELECTRON HOLE

C LEVEL CAPTURE CAPTURE

C CROSS- CROSSC

r SECTION SECTION

C DONOR 1 SE (1 ) SH (1 ) C DONOR 2 SE (2) SH (2)

c DONOR 3 SE(3) SH(3)

c ACCEPTOR 1 SE (4) SH (4)

c ACCEPTOR 2 SE(5

SH(5)

c ACCEPTOR 3 SE (6) SH (6)

c ML DONOR 1 SE(7) SH(7)

c ML ACCEP 1 SE (8) SH (8)

c ML DONOR 2 SE(9) SH(9)

c

c ML ACCEP 2 SE (1 0) SH (10)

c

c A MODIFIED MKS SYSTEM OF UNITS IS USED

c WITH CM REPLACING METERS

C

C T = THE TEMPERATURE IN KELVIN

C EG = THE ENERGY GAP OF SILICON IN ELECTRON VOLTS

28
C EF = THE FERMI LEVEL IN EV ABOVE THE VALENCE BAND

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 FNC = THE DENSITY OF STATES IN THE CONDUCTION BAND IN CM-3

C FNV = THE DENSITY OF STATES IN THE VALENCE BAND IN CM-3

C DI(1) - DI (1 0) = THE DENISTY OF IONIZED SPECIES

C OF STATES NAME1 - NAME 10 IN CM-3

C SE(1) - SE (10) = THE ELECTRON CAPTURE CROSS SECTIONS

C OF STATES NAME1 - NAME10 IN CM+2

C SH (1 ) - SH (2) = THE HOLE CAPTURE CRSS SECTIONS

C OF STATES NAME1 - NAME6 IN CM+2

C TAUNO(I) = HIGH LEVEL ELECTRON LIFETIME PARAMETER OF LEVEL I C TAUPO(I) = HIGH LEVEL HOLE LIFETIME PARAMETER OF LEVEL I

C

C . REFERENCES

C

C 1. GHANDHI, S. K., SEMICONDUCTOR POWER DEVICES:

C PHYSICS OF OPERATION AND FABRICATION TECHNOLOGY,

C PP. 2-31 (JOHN WILEY AND SONS, NEW YORK, 1977)

C

C 2. LARRABEE, R. D., THURBER, W. R., AND BULLIS, W. M.,

C SEMICONDUCTOR MEASUREMENT TECHNOLOGY: A FORTRAN

C PROGRAM FOR CALCULATING THE ELECTRICAL PROPERTIES

C OF EXTRINSIC SILICON, NBS SPECIAL PUBLICATION 400-63,

C OCTOBER, 1980

C

C 3. LARRABEE, R. D., PHILLIPS, W. E., AND THURBER, W. R.,

C MEASUREMENT TECHNIQUES FOR HIGH POWER SEMICONDUCTOR

C MATERIALS AND DEVICES: ANNUAL REPORT, OCTOBER 1, 1979

C TO SEPTEMBER 30, 1980, PP. 33-37.

C

C 4. MOLL, J. L., PHYSICS OF SEMICONDUCTORS (NEW YORK,

C MCGRAW-HILL,1 964) , P.92-101

C

C

C MAIN PROGRAM HANDLES INPUT AND OUTPUT

C AND CALLS THE FOLLOWING SUBROUTINES:

C SUBROUTINE ZEROIN TO DETERMINE THERMAL EQUILIBRIUM CONDITIONS

C SUBROUTINE TAU TO EVALUATE LIFETIME RELATED PARAMETERS

C EXTERNAL TEST

DIMENSION NAME1 (4) ,NAME2 (4) ,NAME3 (4)

DIMENSION NAME4 (4) ,NAME5 (4) ,NAME6 (4)

DIMENSION NAME7 (4) ,NAME8 (4) ,NAME9(4) ,NAME10(4)

DIMENSION DN(10),DI(10),EN(10),DG(10),ENT(10)

DIMENSION SE (1 0) ,SH (1 0)

DIMENSION TAUNOd 0) ,TAUPO (10)

COMMON NAME1 ,NAME2,NAME3,NAME4,NAME5,NAME6

COMMON NAME7,NAME8,NAME9,NAME10

CHARACTER+2 NAME1 ,NAME2,NAME3,NAME4,NAME5,NAME6

CHARACTER*2 NAME7,NAME8,NAME9,NAME1

CHARACTERS AFLl

COMMON DN,D I,EN,DG

29
COMMON T,CN,CP,EG, FNC, FNV, ALPHA, BETA

COMMON TAUNO,TAUPO

LOGICAL FL1,FL2

DIMENSION FL1 (10),AFL1 (10)

CC INPUT THE CHARACTERISTICS OF THE ENERGY LEVELS

C

1 READ (4,50) NAME1,DN(1),EN(1),DG(1),AFL1(1)

50 FORMAT (4A2,1 X,E9.4,2F9.4,A1

READ (4,50) NAME2,DN (2) ,EN(2) ,DG(2) ,AFL1 (2)

READ (4,50) NAME3,DN(3),EN(3),DG(3),AFL1 (3)

READ (4,50) NAME4,DN (4) ,EN(4) ,DG(4) ,AFL1 (4)

READ (4,50) NAME5,DN(5),EN(5),DG(5),AFL1 (5)

READ (4,50) NAME6,DN(6),EN(6),DG(6),AFL1 (6)

READ (4,50) NAME7,DN(7),EN(7),DG(7),AFL1 (7)

READ (4,50) NAME8,DN(8),EN(8),DG(8),AFL1 (8)

READ (4,50) NAME9,DN(9),EN(9),DG(9),AFL1 (9)

READ (4,50) NAME1 0,DN (1 0) ,EN (1 0) ,DG (1 0) ,AFL1 (1 0)

C

C IF DENSITY OF ANY STATE IS ZERO, RENAME IT TO '$$'

C TO INDICATE THAT IT IS NOT PRESENT

C IF (DN(I).EQ.O.) NAME1 (1) = '$$'

IF (DN(2).EQ.O.) NAME2 (1 )='$$'

IF (DN(3) . EQ.O. ) NAME3 (1 )= '$$

IF (DN(4) .EQ.O.) NAME4 (1 )='$$'

IF (DN(5) .EQ.O.) NAME5(1 )='$$'

IF (DN(6) .EQ.O.) NAME6(1)='$$'

IF (DN (7) . EQ.O. ) NAME7(1 )='$$'

IF (DN (8) . EQ.O.) NAME8 (1 )='$$'

IF (DN(9) .EQ.O.) NAME9 (1 )= '$$

IF (DN (10) .EQ.O.) NAME10 (1 ) = '$$ 1 C DO 100 1=1,10

FL1 (I)=. FALSE.

100 CONTINUE

IF (NAME1 (1 ) . EQ. ' $$ 1 ) GO TO 210

IF(AFL1 (l).EQ.'T') FLl (1 )=.TRUE.

READ (4,200) SE(1),SH(1)

200 FORMAT (2E10.4)

210 IF (NAME2(1).EQ.'$$ I ) GO TO 230

IF(AFL1 (2).EQ.'T I ) FLl (2)=. TRUE.

READ (4,200) SE(2),SH(2)

230 IF (NAME3 (1 ) . EQ. '$$ 1 ) GO TO 250

IF(AFL1(3).EQ.'T') FLl (3)=. TRUE.

READ (4,200) SE(3),SH(3)

250 IF (NAME4(1 ) .EQ. '$$' ) GO TO 270

IF(AFL1(4).EQ.'T') FLl (4)=. TRUE.

READ (4,200) SE(4),SH(4)

270 IF (NAME5 (1 ) . EQ . '$$ ' ) GO TO 290

IF(AFL1 (5) .EQ.'T') FLl (5)=. TRUE.

READ (4,200) SE(5),SH(5)

290 IF (NAME6(1).EQ.'$$') GO TO 310

30
IF(AFL1 (6) .EQ. 'T') FL1 (6)=.TRUE.

READ (4,200) SE(6),SH(6)

310 IF (NAME7 (1 ) . EQ. ' $$ ' ) GO TO 315

IFCAFL1 (7).EQ. , T') FLl (7)=.TRUE.

READ (4,200) SE(7),SH(7)

315 IF (NAME8 (1 ) .EQ. '$$ 1 ) GO TO 320

IF(AFL1 (8) . EQ. '

1 ) FLl (8)=. TRUE.

READ (4,200) SE(8),SH(8)

320 IF (NAME9 (1 ) . EQ. 1 $$ 1 ) GO TO 325

I F ( AFLl (9) .EQ. 'T 1 ) FLl (9)=. TRUE.

READ (4,200) SE(9),SH(9)

325 IF (NAME10(1 ).EQ. '$$' ) GO TO 330

IF (AFLl (10) .EQ.'T') FLl (10) = . TRUE.

READ (4,200) SE(10),SH(10)

C

C ENTER TEMPERATURE RANGE OF INTEREST

C START, END, STEP

C330 READ (4,350) TSTART,TEND,TSTEP

350 FORMAT (3F10.0)

IF (TSTEP. EQ.O.) TSTEP = 10.

IF (TEND.EQ.O.) TEND=TSTART

NT=INT(ABS( (TEND-TSTART) /TSTEP) )+1

READ (4,352) FL2

352 FORMAT (LI)

DO 1030 N=1,NT

T=TSTART+(N-1)*TSTEP

C

C PRELIMINARY COMPUTATIONS

C

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

C FROM BARBER, H. D., SOLID-STATE ELECTRONICS 10, 1039 (1967)

C FITTED OVER THE TEMPERATURE RANGE FROM 0 TO 600 KELVIN

EFME= ( ( ( ( (4.54649E-1 7*T-9.66067E-1 4) *T+8.04032E-1 1 ) *T-3.3201 30

2 E-8 ) *T+6 . 83008 E-6 ) *T-0 . 0001 61 708) *T+1 .0627

EFMH= (((((1.11 997E-1 6*T-2. 596730 E-1 3) *T+2 .30049E-10) *T-9.6721

2 E-8) *T+1 . 85678 E-5 ) *T-0. 000523548 )*T+0. 590525

C

C FNC AND FNV ARE THE DENSITIES OF STATES IN THE BANDS

C C0NST=2*(2*PI*K*(FREE ELECTRON MASS) /H**2)** (3/2)

C0NST=4.829E1

FNC=C0NST*(EFME**1 .5)*(T**1 .5)

FNV=CONST* (EFMH**1 .5) * (T**1 .5)

C

C EG IS THE BANDGAP OF SILICON AT THE TEMPERATURE T

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

EG= ( ( (-3.80977E-1 3*T+9 .95E-1 0) *T-8 . 701 1 0E-7) *T+0 . 000032 3741 ) 2 *T+1. 15556

C CAYT=8 .61 73E-5*T

C

C IF THE DEEP LEVEL IS TIED TO THE OPPOSITE BAND

C FROM THE SHALLOW LEVELS ( FLl ( I) =TRUE) , DO THE FOLLOWING

31
DO 355 1=1,10

IFC.N0T.FL1 (I)) GO TO 355

IF(N.EQ.I) ENT (I) =EN (I)

EN(I)=EG“ENT (I)

355 CONTINUE

C

C

C SET UP SEARCH FOR FERMI LEVEL

C EF=0.

EFG=1 .

RE=1 .OE-6

AE=1.0E-6

IFLAG=0

WRITE (7,360) T

360 FORMAT (IX, 'FINDING FERMI LEVEL AT *,F5.0,' KELVIN')

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

WRITE (7,20)

20 FORMAT (IX)

IF (IFLAG.GT.2) GO TO 1010

C

C GUARD AGAINST ZERO DENOMINATORS

C IF (CN.EQ.O.) CN=1 . E-30

IF (CP.EQ.O.) CP=1 . E-30

C

C PRINT OUT PRESENT RESULTS

C WRITE (7,370)

370 FORMAT CO',' NAME ' ,5X, ' DENSITY ENERGY DEGEN. ',

2 'CAPTURE CROSS SECTIONS')

WRITE (7,380)

380 FORMAT (1X,39X,' ELECTRON HOLE')

WRITE (7,20)

IF (NAME1 (1 ) . EQ. ' $$ ' ) GO TO 390

WRITE (7,450) NAME1,DN(1),EN(1),DG(1),SE(1),SH(1)

390 IF (NAME2 (1 ) . EQ. ' $$ ' ) GO TO 400

WRITE (7,450) NAME2,DN(2),EN(2),DG(2),SE(2),SH(2)

400 IF (NAME3(1).EQ.'$$') GO TO 410

WRITE (7,450) NAME3,DN(3),EN(3),DG(3),SE(3),SH(3)

410 IF (NAME4 (1 ) . EQ. ' $$ ' ) GO TO 420

WRITE (7,450) NAME4,DN (4),EN(4) ,DG (4),SE(4) ,SH (4)

420 IF (NAME5 (1 ) . EQ .' $$ ' ) GO TO 430

WRITE (7,450) NAME5,DN(5),EN(5),DG(5),SE(5),SH(5)

430 IF (NAME6 (1 ) . EQ. ' $$ ' ) GO TO 432

WRITE (7,450) NAME6,DN(6),EN(6),DG(6),SE(6),SH(6)

432 IF (NAME7 (1 ) . EQ. ' $$ ' ) GO TO 435

WRITE (7,450) NAME7,DN(7),EN(7),DG(7),SE(7),SH(7)

435 IF (NAME8 (1 ) . EQ. ' $$ ' ) GO TO 437

WRITE (7,450) NAME8,DN (8) ,EN (8) ,DG(8) ,SE (8) ,SH (8)

437 IF (NAME9(1).EQ.'$$') GO TO 440

WRITE (7,450) NAME9,DN(9),EN(9),DG(9),SE(9),SH(9)

440 IF (NAME10(1).EQ.'$$') GO TO 460

32
WRITE (7,450) NAME10,DN(10),EN(10),DG(10),SE(10),SH(10)

450 FORMAT (1 X,4A2,2X,1 PEI 0.3,0PF7.4,3X,0PF5 . 2,5X,2 (1 PEI 0.4,3X)

460 WRITE (7,470)

470 FORMAT CO')

WRITE (7,480)

480 FORMAT (IX, 'THERMAL EQUILIBRIUM CONDITIONS')

WRITE (7,20)

WRITE (7,490)

490 FORMAT (IX,' TEMP. 1000/T EF ' ,7X, ' EG-EF ' ,3X,

2 ' CARRIER DENSITIES')

WRITE (7,500)

500 FORMAT (1X,39X,' ELECTRON HOLE')

WRITE (7,20)

RTEMP=1 000. /T

EFV=EG-EF

WRITE (7,510) T,RTEMP,EF,EFV,CN,CP

510 FORMAT (1 X, F8 .3,1 X, F9.4,2X, F7.4,3X, F7.4,2X,2 (1 PEI 0.3,3X)

WRITE (7,470)

C

C COMPUTE TAUNO(I) AND TAUPO(I)

C DO 520 1=1,10

TAUNO (I) =1 . E30

520 TAUPO (I )=1 . E30

C CONST= (3*K/ (FREE ELECTRON MASS) )**0.5*(T/ (EFFECTIVE MASS))**0.5

C0NST1 =6.7421 762E5* (T/EFME) **0.5

C0NST2=6. 7421 762E5*(T/EFMH) **0.5

DO 540 1=1,10

X=C0NST1 *S E (I) *DN (I)

IF (X.EQ.O.) GO TO 530

TAUN0(I)=1/X

530 X=C0NST2*SH (I) *DN (I

IF (X.EQ.O.) GO TO 540

TAUPO (I) =1 /X

540 CONTINUE

C

C CONTINUE PRINT-OUT OF THERMAL EQUILIBRIUM DATA

C WRITE (7,650)

650 FORMAT (IX,' NAME ION. DEN. % IONIZED',

2 ' TAUNO TAUPO TAUHIGH')

WRITE (7,20)

C HIGH LEVEL LIFETIME IS SUM OF TAUNO AND TAUPO FOR EACH STATE

C NET HIGH LEVEL LIFETIME IS THE RECIPROCAL OF THE

C SUM OF RECIPROCALS OF EACH STATE

SUM=0

IF (NAME1 (1 ) . EQ .

' $$ ' ) GO TO 660

X=TAUNO (1 ) +TAUPO (1

SUM=SUM+1/X

Y=100*DI(1)/DN(1)

WRITE (7,710) NAME1,DI(1) ,Y, TAUNO (1 ), TAUPO (1 ),X

660 IF (NAME2(1).EQ.'$$') GO TO 670

X=T AUNO (2)+TAUP0(2)

33
SUM=SUM+1/X

Y=1 00*DI(2)/DN(2)

WRITE (7,710) NAME2,DI(2),Y,TAUN0(2),TAUP0(2),X

670 IF (NAME3 (1 ) .EQ. '$$' ) GO TO 680

X=TAUN0(3)+TAUP0(3)

SUM=SUM+1 /X

Y=100*DI(3)/DN(3)

WRITE (7,710) NAME3,DI(3),Y,TAUN0(3),TAUP0(3),X

680 IF (NAME4 (1).EQ.'$$') GO TO 690

X=TAUNO (4 ) +TAUPO (4)

SUM=SUM+1/X

Y=100*DI(4)/DN(4)

WRITE (7,710) NAME4,DI (4) ,Y, TAUNO (4) ,TAUP0(4) ,X

690 IF (NAME5(1).EQ.'$$') GO TO 700

X=TAUN0(5)+TAUP0(5)

SUM=SUM+1 /X

Y=1 00*DI(5)/DN(5)

WRITE (7,710) NAME5,DI(5),Y,TAUNO(5),TAUPO(5),X

700 IF (NAME6 (1 ) . EQ. '$$ ' ) GO TO 701

X=TAUN0(6)+TAUP0(6)

S UIWI=S UM+1 /X

Y=100*DI(6)/DN(6)

WRITE (7,710) NAME6,DI (6) ,Y,TAUN0(6) ,TAUP0(6) ,X

701 IF (NAME7 (1 ) . EQ. '$$ 1 ) GO TO 702

H IN=T AUNO (7 ) *TAUNO (8 ) +TAUPO (7 ) *TAUNO (8 ) +TAUPO (7) *TAUPO (8 ) X=HIN/ (TAUNO (8)+TAUP0 (7)

SUM=SUM+1/X

Y=1 00*DI (7) /DN (7)

WRITE (7,710) NAME7,DI(7),Y,TAUN0(7),TAUP0(7),X

Y=100*DI(8)/DN(8)

WRITE (7,710) NAME8,DI (8) ,Y, TAUNO (8) ,TAUP0(8) ,X

702 IF (NAME9 (1 ) . EQ . '$$ ' ) GO TO 720

HIN=TAUNO (9) *TAUNO (1 0)+TAUP0 (9) *TAUNO (1 0)+TAUP0(9) *TAUP0(1 0)

X=HIN/ (TAUNO (1 0) +TAUPO (9)

SUM=SUM+1/X

Y=100*DI(9)/DN(9)

WRITE (7,710) NAME9,DI (9) ,Y, TAUNO (9) ,TAUP0(9) ,X

Y=100*DI(10)/DN(10)

WRITE (7,710) NAME1 0,DI (1 0) ,Y, TAUNO (10) ,TAUP0(1 0) ,X

710 FORMAT (1 X,4A2,1 X,5 (1 PE1 1 .3)

720 WRITE (7,470)

TAUHI=1 /SUM

WRITE (7,730) TAUHI

730 FORMAT (IX, 'THE HIGH LEVEL TRAP RECOMBINATION LIFETIME ',

2 ' IS ' ,1 PEI 0.3, ' SECONDS')

WRITE (7,470)

C

C EVALUATE LIFETIME UNDER DIFFERENT CONDITIONS

C AND PRINT-OUT THE RESULTS

C BUT IF FL2 IS SET TO FALSE,

C ONLY LOW INJECTION LEVEL LIFETIME IS EVALUATED

C ALONG WITH THE SPACE-CHARGE GENERATION LIFETIME.

C FOR FL2 TO BE SET TO TRUE THE TRAP DENSITIES

34
C MUST BE SMALL COMPARED TO THE NET DOPANT DENSITY.

C CNE=CN

CPE=CP

R=CNE

IF (CPE.GT.CNE) R=CPE

R1 =0.01 *R

R2=0.1 *R

SUM1 =0.

SUM2=0.

SUM3=0.

SUM4=0.

SUM5=0.

TAUMAX=1 . E30

C

C TAU1=LIMITING LIFETIME AT ZERO INJECTION LEVEL

C TAU2=LIFETIME AT INJECTION LEVEL OF 0.01

C TAU3=LI FETIME AT INJECTION LEVEL OF 0.10

C TAU4=LIFETIME AT INJECTION LEVEL OF 1.00

C TAU5=LI FETIME IN A DEPLETION LAYER

C TAUl =0.

TAU2=0.

TAU3=0.

TAU4=0.

TAU5=0.

WRITE (7,740)

740 FORMAT (1 X, ' LIFETIME RELATED PARAMETERS')

WRITE (7,20)

WRITE (7,750)

750 FORMAT (IX, ' NAME LOW LEVEL I. R. =0.01

2 ' I . R .=0.10 I. R. =1.00 DEPLETION LAYER')

WRITE (7,20)

C

c DO 810 1=1,10

IF (I . EQ. 1 . AND. NAM El (1 ) . EQ. ' $$ ' ) GO TO 810

IF (I.EQ.2. AND.NAME2 (1 ) . EQ. '$$ ' ) GO TO 810

IF (I . EQ.3. AND.NAME3(1 ) .EQ. '$$') GO TO 810

IF (I . EQ .4. AND .NAME4 (1 ) . EQ . '$$ ' ) GO TO 810

IF (I . EQ. 5 . AND. NAME5 (1 ) . EQ. ' $$ ' ) GO TO 810

IF (I.EQ.6.AND.NAME6(1 ) .EQ. '$$' ) GO TO 810

IF (I . EQ.7. AND .NAME7 (1 ) . EQ. ' $$ ' ) GO TO 810

IF (I.EQ.8.AND.NAME8 (1 ) .EQ. '$$' ) GO TO 810

IF (I . EQ.9. AND.NAME9 (1 ) .EQ. '$$' ) GO TO 810

IF (I. EQ .1 0. AND. NAME1 0 (1 ) . EQ. '$$ ' ) GO TO 810

IF (TAUNO(I) .GE.TAUMAX) GO TO 810

IF (TAUPO(I) .GE.TAUMAX) GO TO 810

IFU.EQ.8.0R.I.EQ.10) GO TO 810

IF(I.GE.7) GO TO 760

CALL TAU(I,CNE,CPE,EN(I),0.,TAU1)

SUM1 =SUM1 +1 /TAUl

I F ( .NOT. FL2) GO TO 755
755

760

770

780

800

810

C820

825

830

840

850

860

CALL TAU(I,CNE,CPE,EN(I),R1,TAU2)

SUM2=SUM2+1/TAU2

CALL TAU(I,CNE,CPE,EN(I),R2,TAU3)

SUM3=SUM3+1/TAU3

CALL TAU(I,CNE,CPE,EN(I)',R,TAU4)

SUM4=SUM4+1/TAU4

CALL TAU(I,0.,0.,EN(I),0.,TAU5)

SUM5=SUM5+1/TAU5

GO TO 780

CALL TAUM(I,CNE,CPE,EN(I),EN(I+1),0.,TAU1)

SUM1 =SUM1+1/TAU1

IF( .NOT. FL2) GO TO 770

CALL TAUM(I,CNE,CPE,EN(I),EN(I+1),R1,TAU2)

SUM2=SUM2+1/TAU2

CALL TAUM(I,CNE,CPE,EN(I),EN(I+1),R2,TAU3)

SUM3=SUM3+1 /TAU3

CALL TAUM(I,CNE,CPE,EN(I),EN(I+1),R,TAU4)

SUM4=SUM4+1 /TAU4

CALL TAUM(I,0. ,0. ,EN(I) ,EN(I+1 ) ,0. ,TAU5)

SUM5=SUM5+1/TAU5

IF (I.EQ.1) WRITE (7,800) NAME1 ,TAU1 ,TAU2,TAU3,TAU4,TAU5

IF (I.EQ.2) WRITE (7,800) NAME2,TAU1,TAU2,TAU3,TAU4,TAU5

IF (I.EQ.3) WRITE (7,800) NAME3,TAU1,TAU2,TAU3,TAU4,TAU5

IF (I.EQ.4) WRITE (7,800) NAME4,TAU1 ,TAU2,TAU3,TAU4,TAU5

IF (I.EQ.5) WRITE (7,800) NAME5,TAU1 ,TAU2,TAU3,TAU4,TAU5

IF (I.EQ.6) WRITE (7,800) NAME6,TAU1,TAU2,TAU3,TAU4,TAU5

IF (I.EQ.7) WRITE (7,800) NAME7,TAU1 ,TAU2,TAU3,TAU4,TAU5

IF (I.EQ.9) WRITE (7,800) NAME9,TAU1,TAU2,TAU3,TAU4,TAU5

FORMAT (1 X,4A2,1X,5 (1 PE1 1 .3)

CONTINUE

WRITE (7,470)

TAUl =1 /SUM1

IF ( .NOT. FL2) GO TO 825

TAU2=1/SUM2

TAU3=1/SUM3

TAU4=1/SUM4

TAU5=1/SUM5

WRITE (7,830) TAUl

FORMAT (IX, 'LOW LEVEL LIFETIME IS *,1PE10.3,

2 ' SECONDS')

WRITE (7,20)

I F ( .NOT. FL2) GO TO 865

WRITE (7,840) TAU2

FORMAT (IX, 'LIFETIME AT 0.01 I.R. IS ',1PE10.3,

2 ' SECONDS')

WRITE (7,20)

WRITE (7,850) TAU3

FORMAT (IX, 'LIFETIME AT 0.10 I.R. IS '1PE10.3,

2 ' SECONDS')

WRITE (7,20)

WRITE (7,860) TAU4

FORMAT (IX, 'LIFETIME AT 1.00 I.R. IS ',1PE10.3,

36
865

870

880

1010

1020

1030

C

c

cc

cc

c

c

cc

c

cc

c

cc

cc

c

c

c

c

c

c

c

c

c

cc

c

c

cc

c

c

c

c

ccc

c

2 ' SECONDS')

WRITE (7,20)

WRITE (7,870) TAU5

FORMAT (IX, 'LIFETIME IN A DEPLETION LAYER IS ',

2 1 PEI 0.3, ' SECONDS')

WRITE (7,880)

FORMAT Cl')

60 TO 1030

WRITE (7,1020) IFLA6

FORMAT (IX, 'ERROR FLAG = ',12)

CONTINUE

END

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

SANDIA MATHEMATICAL PROGRAM LIBRARY

MATHEMATICAL COMPUTING SERVICES DIVISION 5422

SANDIA LABORATORIES

P.O. BOX 5800

ALBUQUERQUE, NEW MEXICO 87115

CONTROL DATA 6600 VERSION 4.5, 1 NOVEMBER 1971

MODIFIED BY D. KAHANER, NBS DIVISION 713.

ABSTRACT:

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

THE GIVEN VALUES B AND C UNTIL THE WIDTH OF INTERVAL

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

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

DESCRIPTION OF ARGUMENTS:

F - NAME OF THE REAL VALUED EXTERNAL FUNCTION.

THIS NAME MUST BE IN AN EXTERNAL STATEMENT

IN THE CALLING PROGRAM.

F MUST BE A FUNCTION OF ONE REAL ARGUMENT.

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

THE VALUE RETURNED FOR B USUALLY IS THE

BETTER APPROXIMATION OF F.

C - THE OTHER END OF THE INTERVAL (B.C).

RE - THE RELATIVE ERROR USED FOR RW

IN THE STOPPING CRITERION.

IF THE REQUESTED RE IS LESS THAN THE MACHINE

PRECISION, THEN RW IS SET TO APPROXIMATELY

THE MACHINE PRECISION.

AE - THE ABSOLUTE ERROR USED IN THE STOPPING

CRITERION. IF THE GIVEN INTERVAL (B,C)

CONTAINS THE ORIGIN, THEN A NONZERO VALUE

SHOULD BE CHOSEN FOR AE.

IFLAG - RETURNS A STATUS OF THE RESULTS INDICATING

WHICH OF THE FOLLOWING CONDITIONS HOLD:

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

B - F(B)*F(C) .LT.O.

37
c

c

cc

c

cc

c

c

c

c

c

c

c

c

c

c

cc

cc

c

c

cc

c

c

c

c

cc

c

c

c

cc

c

c

1c

C - ABS(F(B)).LE.ABS(F(C)).

D - ABS(F(B"OUT")).LE.

MAX(ABS(F(B"IN">),ABS(F(C"IN"))).

E - NUMBER OF EVALUATIONS OF F(X),LE.500.

IFLAG=1 INDICATES NORMAL CASE

ALL CONDITIONS ABOVE HOLD.

IFLAG=2 INDICATES F(B)=0.

CONDITION A MAY NOT HOLD.

IFLAG=3 INDICATES CONDITIONS A, B, C, AND E HOLD, BUT CONDITION D DOES NOT.

THE INTERVAL (B,C> PROBABLY CONTAINS

A SINGLULAR POINT OF THE FUNCTION F. IFLAG=4 INDICATES CONDITIONS A AND E HOLD,

BUT CONDITION B DOES NOT.

A LOCAL MINIMUM OF F (X) IN (B,C)

MAY HAVE BEEN FOUND.

IFLAG=5 INDICATES SEARCH WAS ABORTED

WHEN CONDITION E FAILED.

REFERENCES:

1. L. F. SHAMPINE AND H. A. WATTS, ZEROIN,

A ROOT-SOLVING CODE, SC-TM-70-631 , SEPT., 1970.

2. T. J. DEKKER, FINDING A ZERO BY MEANS

OF SUCCESSIVE LINEAR INTERPOLATION,

CONSTRUCTIVE ASPECTS OF THE FUNDAMENTAL THEORM

OF ALGEBRA*, EDITED BY D. DEJON AND P. HENRICI,

1969.

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

COMPUTING -- AN INTRODUCTION.

SEE NBS SPECIAL PUBLICATION 400-63, PP. 28-30

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

DATA ER/6.0E-8/

INITIALIZE IN PORTABLE MANNER

DATA ER/O.O/

IF (ER.EQ.O.) ER=4.* (R1MACHC4)

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

38
B=C

FB=FC

C=A

FC = FA

C

2 CMB=0.5*(C-B)

ACMB=ABS(CMB)

TOL=RW*ABS (B) +AE

CC TEST STOPPING CRITERION

IF(ACMB.LE.TOL) GO TO 10

C

C CALCULATE NEW ITERATE IMPLICITLY AS B+P/Q

C WHERE P.GE.O. HAS BEEN GUARANTEED.

C THE IMPLICIT FORM IS USED TO PREVENT OVERFLOW.

P=(B-A)*FB

Q=FA-FB

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

P=-P

Q=-Q

C

C UPDATE A AND CHECK FOR SATISFACTORY REDUCTION

C IN THE SIZE OF OUR BOUNDING INTERVAL.

3 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

C

C TEST FOR TOO SMALL A CHANGE

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

C

C INCREMENT BY TOLERANCE

B=B+SIGN(TOL,CMB)

GO TO 7

C

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

5 IF (P.GE. CMB*Q) GO TO 6

C

C INTERPOLATE

B=B+P/Q

GO TO 7

C

C BISECT THE INTERVAL

6 B=0.5* (C+B)

C

C COMPUTATION FOR NEW ITERATE B HAS BEEN COMPLETED.

7 FB=F (B)

IF (FB.EQ.O.) GO TO 11

K0UNT=K0UNT+1

IF (KOUNT.GT.500) GO TO 15

39
o

o

o

c

C DECIDE IF NEXT STEP IS INTERPOLATION OR EXTRAPOLATION

IF (SIGN(1.0,FB).NE.SIGN(1.0,FC)) GO TO 1 C=A

FC=FA

GO TO 1 C

FINISHED. PROCESS RESULTS FOR PROPER SETTING OF I FLAG.

10 IF (FB*FC.GT.O.) GO TO 13

IF (ABS(FB).GT.FX) GO TO 12

IFLAG=1

RETURN

11 IFLAG=2

RETURN

12 IFLAG=3

RETURN

13 IFLAG=4

RETURN

15 IFLAG=5

RETURN

END

C FUNCTION TEST(EF)

C THIS EXTERNAL FUNCTION DEFINES THE PARAMETER 'TEST'.

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

C FUNCTION TEST (EF)

DIMENSION NAME1 (4) ,NAME2 (4) ,NAME3(4)

DIMENSION NAME4(4),NAME5(4),NAME6(4)

DIMENS ION NAME7 (4) ,NAME8 (4) ,NAME9 (4) ,NAME1 0(4)

D IMENSION DN (1 0) ,DI (1 0) ,EN (1 0) ,DG (1 0)

DIMENSION TAUNO(IO) / TAUP0(10)

COMMON NAME1 ,NAME2,NAME3,NAME4,NAME5,NAME6

COMMON NAME7,NAME8,NAME9,NAME10

CHARACTER*2 NAME1 ,NAME2,NAME3,NAME4,NAME5,NAME6

CHARACTERS NAME7,NAME8,NAME9,NAME10

COMMON DN,DI,EN,DG

COMMON T ,CN, CP, EG, FNC, FNV, ALPHA, BETA

COMMON TAUNO,TAUPO

C LEVELS NOT PRESENT HAVE FIRST TWO CHAR. OF NAME = '$$'

C CAYT=8.61 73E-5*T

C

C COMPUTATION OF CN

CN=0.

ALPHA= (EF-EG) /CAYT

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

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

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

CN=FNC/ ((EXP (-ALPHA) )+0. 27)

GO TO 20

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

40
10 CN=FNC*0 .752253* ( (ALPHA*ALPHA+1 .7)**0.75)

C

C COMPUTATION OF CP

20 CP=0.

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

CP=FNV/ ( (EXP (-BETA) )+0. 27)

GO TO 40

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

30 CP=FNV*0.752253* ( (BETA*BET A+1 .7)**0.75)

C

C COMPUTATION OF DI(1)

40 D I (1 ) =0

IF (NAME1 (1 ) . EQ. ' $$ 1 ) GO TO 60

POWER= (EF-EG+EN (1))/CAYT

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

IF (POWER. LE. -80.) GO TO 50

DI(1 )=DN (1 ) / (1 .+ (EXP (POWER) *DG(1)))

GO TO 60

50 DI(1) =DN (1

C

C COMPUTATION OF DI (2)

60 DI (2)=0.

IF (NAME2 (1 ) . EQ. ' $$ 1 ) GO TO 80

POWER= (EF-EG+EN (2) )/CAYT

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

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

DI(2)=DN(2) / (1 .+ (EXP (POWER) *DG (2) )

GO TO 80

70 DI (2)=DN(2)

C

C COMPUTATION OF DI (3)

80 DI (3)=0.

IF (NAME3 (1 ) . EQ. ' $$ ' ) GO TO 100

POWER= (EF-EG+EN (3) ) /CAYT

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

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

DI (3) =DN (3) / (1 . + (EXP (POWER) *DG (3) )

GO TO 100

90 DI(3)=DN(3)

C

C COMPUTATION OF DI (7)

100 DI (7)=0.

T1 =0.

T2=Q.

IF (NAME7 (1 ) . EQ. ' $$ ' ) GO TO 120

P0WER1= (EF-EG+EN (7) ) /CAYT

P0WER2=(EF-EN(8) ) /CAYT

IF(P0WER1 .GE.80.) GO TO 120

IF(P0WER1 .LE.-80.) GO TO 105

T1=EXP(P0WER1)*DG(7)

41
105 I F ( (POWER1 +P0WER2) .GE.80) GO TO 120

IF((PQWER1+P0WER2) .LE.-80) GO TO 110

T2=EXP(P0WER1 +P0WER2) /DG(8)*DG(7)

110 DI (7)=DN (7) / (1 +T1 +T2)

C

C

C COMPUTATION OF DI(9)

120 DI (9)=0.

T1=0.

T2=0.

IF (NAME9 (1 ) . EQ

1 $$ ' ) GO TO 200

P0WER1 = CE F-EG+EN (9) ) /CAYT

P0WER2=(EF-EN(10) ) /CAYT

IF(P0WER1. GE.80.) GO TO 200

IF(P0WER1 .LE.-80.) GO TO 125

T1 =EXP (P0WER1 ) *DG (9)

125 IF((P0WER1+P0WER2) .GE.80) GO TO 200

I F C (P0WER1 +P0WER2) .LE.-80) GO TO 130

T2=EXP(P0WER1 +P0WER2) /DG(1 0) *DG(9)

130 DI (9) =DN (9) / (1 +T1 +T2)

C

C COMPUTATION OF DI(4)

200 DI (4)=0.

IF (NAME4(1 ) .EQ. '$$') GO TO 220

POWER= (EN(4) ~EF) /CAYT

IF (POWER. GT. 80.) GO TO 220

IF (POWER. LE. -80.) GO TO 210

DI (4)=DN (4) / (1 ,+(EXP(P0WER)*DG(4) )

GO TO 220

210 DI (4)=DN(4)

C

C COMPUTATION OF DI(5)

220 DI (5)=0.

IF (NAME5 (1 ) . EQ. '$$ ' ) GO TO 240

P0WER=(EN(5)-EF)/CAYT

IF (POWER. GE.80.) GO TO 240

IF (POWER. LE. -80.) GO TO 230

DI (5)=DN(5) / (1 . + (EXP(POWER)*DG (5) )

GO TO 240

230 DI (5)=DN(5)

C

C COMPUTATION OF DI(6)

240 DI (6)=0.

IF (NAME6(1 ) .EQ. '$$' ) GO TO 260

P0WER=(EN(6)-EF)/CAYT

IF (POWER. GT. 80.) GO TO 260

IF (POWER. LE. -80.) GO TO 250

DI(6)=DN(6) / (1 .+ (EXP (POWER) *DG (6) )

GO TO 260

250 DI (6)=DN (6)

C

C COMPUTATION OF DI(8)

260 DI (8) =0.

42
265

270

C280

285

290

C

CC

C300

C

C

CC

C

C

CC

C

C

C

C

T1=0.

T2=0.

IF (NAME8 (1 ) . EQ. '$$' ) GO TO 280

P0WER1=(EN(8)-EF)/CAYT

P0WER2=(-EN(7)-EF+EG) /CAYT

IFCP0WER1 .GE.80.) GO TO 280

IFCP0WER1 .LE.-80.) GO TO 265

T1 =EXP(P0WER1 ) *DG(8)

IF((P0WER1+P0WER2) .GE.80) GO TO 280

IF( (P0WER1 +P0WER2) .LE.-80) GO TO 270

T2=EXP(P0WER1 +P0WER2) /DG(7)*DG(8)

DI(8)=DN(8) / (1+T1 +T2)

COMPUTATION OF 01(10)

DIC10)=0.

T1 =0.

T2=0.

IF (NAME10C1 ) .EQ. '$$') GO TO 300

P0WER1 = (EN(10)-EF)/CAYT

P0WER2=(-EN(9)-EF+EG)/CAYT

IFCP0WER1 .GE.80.) GO TO 300

IFCP0WER1 .LE.-80.) GO TO 285

T1 =EXP (P0WER1 )*DG (1 0)

IF( (P0WER1 +P0WER2) .GE.80) GO TO 300

IF( (P0WER1 +P0WER2) .LE.-8Q) GO TO 290

T2=EXP (P0WER1 +P0WER2) /DG (9) *DG (1 0)

DI(10)=DN(10)/(1+T1+T2)

TESTING TO SEE HOW CLOSE TO

THE PARAMETER TEST IS THE NET DENSITY OF RESIDUAL CHARGES

WHEN THE FERMI LEVEL IS AT ITS PRESENT LOCATION

TEST= (CN+DI (4)+DI (5)+DI (6) )-(CP+DI (1 )+DI (2)+DI (3)

TEST=TEST+DI(8)+DI(10)-DI(7)-DI(9)

RETURN

END

SUBROUTINE TAU(I,CNE,CPE,ENF,EDN,TAUX)

SUBROUTINE TO COMPUTE SPACE-CHARGE NEUTRAL LIFETIMES

FOR SINGLE LEVEL TRAPS

DESCRIPTION OF ARGUMENTS:

I - INDEX RANGING FROM 1 TO 6 SPECIFYING THE ENERGY LEVEL

CNE - THERMAL EQUILIBRIUM DENSITY OF ELECTRONS

CPE - THERMAL EQUILIBRIUM DENSITY OF HOLES

ENF - POSITION OF THE TRAP LEVEL

EDN - DENSITY OF EXCESS ELECTRON-HOLE PAIRS

TAUX - COMPUTED VALUE OF THE LIFETIME

DIMENSION NAME1 (4) ,NAME2 (4) ,NAME3 (4)

DIMENSION NAME4 (4) ,NAME5 (4) ,NAME6(4)

DIMENS ION NAME7 (4) ,NAME8 (4) ,NAME9(4) ,NAME10(4)

DIMENSION DN(10),DI(10),EN(10),DG(10)

DIMENSION SE(10),SH(10)

DIMENSION TAUNO(IO) ,TAUPO (10)

43
c

cc

10

c

c

c20

C30

40

C

C

C

C

C

C

C

C

C

C

c

c

c

c

c

COMMON NAME1 ,NAME2,NAME3,NAME4,NAME5,NAME6

COMMON NAME7,NAME8,NAME9,NAME10

CHARACTER*2 NAME1 ,NAME2,NAME3,NAME4,NAME5,NAME6

CHARACTERS NAME7,NAME8,NAME9,NAME1

COMMON DN,DI,EN,DG

COMMON T, CN, CP, EG, FNC,FNV, ALPHA, BETA

COMMON TAUNO,TAUPO

COMPUTE GHANDHI'S FACTORS N1 AND Pi CORRECTED FOR DEGENERACY

CAYT=8.6173E-5*T

IF (I.GT.3) GO TO 10

ZN1=FNC*EXP(-(ENF)/CAYT)/DG(I)

ZP1=FNV*EXP(-(EG-ENF)/CAYT)*DG(I)

GO TO 20

ZN1=FNC*EXP(-(EG-ENF)/CAYT)*DG(I)

ZP1=FNV*EXP(-ENF/CAYT)/DG(I)

EVALUATE LIFETIME EXPRESSION

DENOM=CNE+CPE+EDN

IF (DENOM.EQ.O.) GO TO 30

X=TAUNO(I) *(CPE+ZP1+EDN) /denom

TAUX=X+TAUPO (I) * (CNE+ZN1 +EDN) /DENOM

RETURN

X=TAUNO(I) *ZP1 +TAUP0(I)*ZN1

Y=SQRT(FNC) *SQRT(FNV)*EXP(-EG/ (2*CAYT)

TAUX=1 . E30

IF (Y.EQ.O.) GO TO 40

TAUX=X/(2*Y)

RETURN

END

SUBROUTINE TAUM(I,CNE,CPE,END,ENA,EDN,TAUX)

SUBROUTINE TO COMPUTE SPACE-CHARGE NEUTRAL LIFETIMES

FOR MULTILEVEL TRAPS

DESCRIPTION OF ARGUMENTS:

I - INDEX RANGING FROM 7 TO 10 FOR

MULTILEVEL SPECIES.

CNE - THERMAL EQUILIBRIUM DENSITY OF ELECTRONS

CPE - THERMAL EQUILIBRIUM DENSITY OF HOLES

END - POSITION OF THE DONOR TRAP LEVEL

ENA - POSITION OF THE ACCEPTOR TRAP LEVEL

EDN - DENSITY OF EXCESS ELECTRON-HOLE PAIRS

TAUX - COMPUTED VALUE OF THE LIFETIME

DIMENSION NAME1 (4) ,NAME2 (4) ,NAME3 (4)

DIMENSION NAME4C4) ,NAME5 (4),NAME6(4)

DIMENSION NAME7 (4) ,NAME8 (4) ,NAME9 (4) ,NAME1 0(4)

DIMENSION DN(10),DI(10),EN(10),DG(10)

44
DIMENSION SE(10),SH(10)

DIMENSION TAUNO (10) ,TAUPO (10)

COMMON NAME1 ,NAME2, NAME3,NAME4, NAMES, NAME6

COMMON NAME7,NAME8,NAME9,NAME10

CHARACTER+2 NAME1 ,NAME2,NAME3,NAME4,NAME5,NAME6

CHARACTER+2 NAME7,NAME8,NAME9,NAME10

COMMON DN,DI,EN,DG

COMMON T,CN, CP, EG, FNC,FNV, ALPHA, BETA

COMMON TAUNO,TAUPO

C

C COMPUTE GHANDHI'S FACTORS Nl AND Pi CORRECTED FOR DEGENERACY

C CAYT=8.6173E-5*T

ZN1 D=FNC*EXP (- (END) /CAYT) /DG(I)

ZPlD=FNV*EXP(-(EG-END)/CAYT)*DG(I)

ZNl A=FNC*EXP (- (EG-ENA) /CAYT) *DG (1+1

ZP1 A=FNV*EXP (-ENA/ CAYT) /DG (1+1

C

C EVALUATE LIFETIME EXPRESSION

C T1=ZN1A*TAUP0(I+1)

T2= (CPE+EDN) *TAUNO (1+1

T3=ZP1 D+TAUNO (I)

T4= (CNE+EDN) *TAUPO (I)

T5= (CPE+EDN) *TAUNO (I)

T6=ZN1 D+TAUPO (I)

T7= (CNE+EDN) *TAUPO ( 1+1

T8=ZP1 A*TAUNO (1+1

20 DENOM= (CNE+CPE+EDN)*(T1 +T2+T3+T4)

IF (DENOM.EQ.O.) GO TO 30

TAUX= ( (T5+T6) * (T2+T1 ) + (T4+T3) * (T2+T1 ) + (T4+T3) * (T7+T8) ) /D ENOM

RETURN

C 30 X=T6*T1 +T3+T1 +T3*T8

Y=SQRT ( FNC) *SQRT ( FNV) *EXP(-EG/ (2* CAYT)

TAUX=1 . E30

IF (Y.EQ.O.) GO TO 40

TAUX=X/(2*Y)/(T1+T3)

40 RETURN

C END

45
TABLE D-1. PRINT--OUT OF SAMPLE INPUT TO LIFETIME PROGRAM

PHOS 2.0E15 0.045 2. F PLAT 2.0E13 0.314 3.7 T

PLAT 2.0E13 0.231 0.5 T

1.

E-16 1.3E-22

2.

E-14 8.E-16

7.E-15 2. E-1

4

50. 200. 50.

F

TABLE D-2. SAMPLE OUTPUT LISTING FROM LIFETIME PROGRAM

FINDING FERMI LEVEL AT 200. KELVIN

NAME DENSITY ENERGY DEGEN. CAPTURE CROSS; SECTIONS

ELECTRON HOLE

PHOS 2.000E+1

5

0.0450 2.00 1.0000E-16 1 .3000E-22

PLAT 2.000E+13 0.8206 3.70 2.0000E-14 8.0000E-16

PLAT 2.000E+13 0.9036 0.50 7.0000E-1

5

2.0000E-14

THERMAL EQUILIBRIUM CONDITIONS

TEMP. 1000/T EF EG-EF CARRIER DENSITIES

ELECTRON HOLE

200.000 5.0000 0.9788 0.1557 1.974E+15 1 .904E-06

NAME ION. DEN. % IONIZED TAUNO TAUPO TAUHIGH

PHOS 1.994E+15 9.968E+01 5.596E-07 3.486E-01 3.486E-01

PLAT 6.020E-07 3.010E-18 2.798E-07 5.665E-06 9.337E-07

PLAT 1.987E+13 9.937E+01 7.994E-07 2.266E-07 9.337E-07

THE HIGH LEVEL TRAP RECOMBINATION LIFETIME IS 9.337E-07 SECONDS

LIFETIME RELATED PARAMETERS

NAME LOW LEVEL I. R. =0.01 I. R. =0.10 I. R. =1.00 DEPLETION LAYER

PHOS 1.080E+02 O.OOOE+OO O.OOOE+OO O.OOOE+OO 1.733E+12

PLAT 2.280E-07 O.OOOE+OO O.OOOE+OO O.OOOE+OO 8.761 E-01

LOW LEVEL LIFETIME IS 2.280E-07 SECONDS

LIFETIME IN A DEPLETION LAYER IS 8.761 E-01 SECONDS

46
FEDERAL INFORMATION PROCESSING STANDARD SOFTWARE SUMMARY

01. Summary date 02. Summary prepared by (Name and Phone)

Yr. Mo. Day J. R. Lowney and W. R. Thurber. (301) 921-3625

00 V U__2_ 1 4

05. Software title 04. Software date A FORTRAN Program for Calculating the Recombination

Yr. Mo. Day Lifetime of Silicon

06. Short title Lifetime-Predictina ComDuter Proaram (LT1)

08. Software type

Automated Data

| | System

££] Computer Program

| | Subroutine/Module

09. Processing mode

| | Interactive Q Batch

[X] Combination

10. General Computer Systems

j | Support/Utility

[X] Scientific/Engineering

Bibliographic/Textual

Application are

Management/

Business

Process Control Other

a

Specific

Engineering calculation

11. Submitting organization and address National Bureau of Standards

12. Technical contact(s) and phone

J. R. Lowney (301) 921-3625

03. Summary action New Replacement Deletion

Previous HDInternal Software ID

07. Internal Software ID

Semiconductor Materials & Processes Division

Bldg. 225, Rm. A331

Washington, DC 20234

W. R. Thurber (301) 921-3625

R. D. Larrabee (301) 921-3625

13. Narrative This computer program computes the recombination lifetime in silicon due to traps as a function of temperature and carrier density. The trap energy, density, and cross sec- tions are needed as input for each deep level present. Shockley-Read-Hall theory is

used with provision made for multiple species.

14. Keywords

Computer program; deep level; injection ratio; recombination lifetime; silicon; trap

15. Computer manuf'r and model 16. Computer operating system 17. Programing language(s) 18. Number of source program state ments Univac 1100 Exec 9R1 FORTRAN-77 1,000

19. Computer memory requirements 20. Tape drives 21. Disk/Drum units 22. Terminals 60K none none printer

23. Other operational requirements

None

24. Software availability

Available Limited In-house only E

See item 12 for contacts

.

25. Documentation availability

Available Inadequate In house only

1. See item 12 for contacts

2. See NBSIR 84-2838

26. FOR SUBMITTING ORGANIZATION USE

185-101 Standard Form 185 1974 July US. Dopt of CommarCO- (FIPS Pub XT)
U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR 2 REPORT NO. NBSIR 84-2838

. Performing Organ. Report No. 3. Publication Date

April 1984

4. TITLE AND SUBTITLE

Measurement Techniques for High Power Semiconductor Materials and Devices: Annual

Report, January 1, 1982 to March 31, 1983

5. AUTHOR(S)

W. R. Thurber, J. R. Lowney, and W. E. Phillips

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No. NBS/DOE DE-A101-76PR06010

8. Type of Report & Period Covered

Final

1/1/82 to 3/31/83

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

Department of Eneray

Division of Electric Energy Systems

Washington, DC 20461

10.

SUPPLEMENTARY NOTES

[X] Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bliography or literature survey, mention it here) This annual report is the final one in a series which describes NBS research to

develop procedures for the effective utilization of deep-level measurements to detect and characterize defects which reduce lifetime or contribute to leakage current

in power-device-grade silicon. During this reporting period the previously written

computer program for predicting excess-carrier lifetime was revised to calculate

more accurately lifetimes for high or low injection conditions and in space-charge

regions. Comparisons were made between lifetime measurements on platinum-doped

silicon diodes and the predictions of the computer model. As part of the effort to

extend the procedures to analyze data from nonexponential transient capacitance

measurements, the time dependence of the capacitance-voltage relationship of a heavily platinum-doped silicon diode was measured as a function of bias voltage.

Included as appendices are three recent publications resulting from the work. A

listing of the lifetime-predicting computer program is also an appendix.

12.

KEY WORDS (Six to twelve entries ; alphabetical order; capitalize only proper names; and separate key words by semicolon s) Deep-level measurements; deep-level transient spectroscopy; defect characterization;

lifetime; power-device grade silicon; transient capacitance techniques.

13. AVAILABILITY

[]x] Unlimited

Q3] For Official Distribution. Do Not Release to NTIS

J Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

Q£] Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

52

15. Price

$ 10.00

USCOMM-DC 6043-P80