Comment on the Initial‐Rise Method for Determining ...

Comment on the Initial‐Rise Method for Determining Trap Depths P. Bräunlich Citation: Journal of Applied Physics 38, 2516 (1967); doi: 10.1063/1.1709939 View online: http://dx.doi.org/10.1063/1.1709939 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/38/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Measurement of vacuum pressure with a magneto-optical trap: A pressure-rise method Rev. Sci. Instrum. 86, 093108 (2015); 10.1063/1.4928154 Reply to “Comment on ‘Determination of the depth of 50% of maximum ionization, I 50 , for electron beams by the divided difference method’ ” [Med. Phys.31, 2068–2074 (2004)] Med. Phys. 31, 3160 (2004); 10.1118/1.1803412 Comment on “Determination of the depth of 50% of maximum ionization, I 50 , for electron beams by the divided difference method” [Med. Phys.31, 2068–2074 (2004)] Med. Phys. 31, 3158 (2004); 10.1118/1.1803411 Method for Determining the Response of Temperature Sensors to a Rapid Temperature Rise Rev. Sci. Instrum. 36, 1880 (1965); 10.1063/1.1719485 Comparison of Methods for Determining Electron Trap Distributions J. Appl. Phys. 35, 3067 (1964); 10.1063/1.1713181

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2516

KWEI,

SCHONHORN,

structure may indeed be larger3 than that obtainable from the usual molding conditions, but the organization of the molecules is likely to be less perfect. One therefore anticipates a lower E. value since it is also known that the modulus of a bulk specimen is reduced by quenching. Our experimental result showing a lower E' value, when compared with a film of the same thickness prepared by the usual method, is consistent with the above considerations. For the isothermally crystallized PE sample (p= 0.954), it is possible that the oriented surface layer does not extend as deeply as in other specimens and therefore the modulus is low. However, we have not carried out a systematic investigation. We plan to use ultrasonic reflection and transmission methods and point-probe techniques to measure the mechanical properties of the transcrystalline zone in the direction normal to the surface. It is hoped that the anisotropy of the mechanical properties of oriented spherulitic structures can be studied by these methods. CONCLUSIONS

The dynamic Young's moduli of molded films of polyethylene and polypropylene decrease with in-

AND

FRISCH

creasing thickness. The dependence of the modulus on the thickness of the film can be described quantitatively by a model in which the surface transcrystalline region and the bulk material respond in a "parallel" combination to an applied stress. The modulus of the transcrystalline region in each polymer is higher than that of the bulk phase. The thickness of a surface region is about 12 IJ. (tIs) in the case of polypropylene and about 15 IJ. in polyethylene. The dynamic moduli of polyethylene films are also dependent on the thermal history and the nature of the mold surface. Our results also carry the implication that when a crystalline polymer is used as a component of a composite material, e.g., as an adhesive, it is perhaps necessary to examine more closely the nature and the properties of the interfacial region. ACKNOWLEDGMENTS

The authors gratefully acknowledge the cooperation of F. J. Padden, Jr. in sectioning and photographing the polymers. We are also indebted to F. W. Ryan and H. M. Zupko for their valuable assistance in the course of this work.

VOLUME 38, NUMBER 6

JOURNAL OF APPLIED PHYSICS

MAY 1967

Comment on the Initial-Rise Method for Determining Trap Depths P. BRAuNLIcH The Bendix Corporation, Research Laboratories Division, Southfield, Michigan

(Received 28 December 1966; in final form 17 February 1967) For the determination of the trap depth from thermoluminescence and thermally stimuated currentglow curves commonly the initial-rise method by Garlick and Gibson is used. This method provides accurate results only if the condition Rh( To) [2/z-1J:::;j( To) is fulfilled. R is the retrapping factor, z the filling ratio of the traps at the temperature To, j the concentration of unoccupied recombination centers, and h the density of trapped electrons.

INTRODUCTION

Thermoluminescence (TL) and thermally stimulated current (TSC) measurements have been widely used as tools for the determination of trap parameters in luminescent and photoconducting materials. A number of techniques for the evaluation of TL and TSC data have been developed for this purpose. Formulas used to calculate the trap depth, E, were derived from the different reaction-kinetics models which describe the thermally stimulated recombination processes. The most common method to determine E is what is called the initial-rise method, first advocated by Garlick and Gibson. 1 This method is based on the fact that the rise of the low-temperature side of a glow peak is proportional to expo (- E/kT) for temperatures well below the peak temperature T m' Other methods utilize the shift of T m with increasing 1 G. F. J. Garlick, and A. F. Gibson, Proc. Roy. Soc. (London) A60, 574 (1948).

heating rates q= dT/ d(l--9; the characteristic difference in T m as defined by the TL peak and the TSC peak5 •1O ; the half-width of a specific peakll- 15 ; or just a simple linear relation between E and T m,1&-17 Another method A. H. Booth, Can. J. Chern. 32, 214 (1954). A. Bohun, Czech. J. Phys. 4, 91 (1954). 4M. Schon, Tech.-Wiss. Abhand!. Osram-Ges. 7,175 (1958); Paper in Schottky Halbleiterprobleme (Frederick Vieweg und Sohn, Braunschweig, 1958), Bd. IV, 282. 6 P. Braunlich, Ann. Physik 12, 262 (1963). 6 K. W. Boer, S. Oberlander, and J. Voigt, Ann. Physik 2, 130 (1958). 7 R. R. Haering and E. N. Adams, Phys. Rev. 117,451 (1960). 8 W. Hoogenstraaten, Philips Res. Repts. 13, 515 (1958). 9 J. J. Boiko, E. E. Rashba, and A. P. Trofimenko, Soviet Phys.-Solid State 2, 99 (1960). 10 P. Briiunlich and A. Scharmann, Phys. Status Solidi 18, No. 1,307 (1966). 11 L. J. Grossweiner, J. App!. Phys. 24,1306 (1953). 12 P. N. Keating, Proc. Phys. Soc. (London) 78, 1408 (1961). 13 Ch. B. Lushik, Dok!. Akad. Nauk. SSSR, 101,641 (1955). 14A. Halperin and A. A. Braner, Phys. Rev. 117,408 (1960). 16K. Unger, Abhand!. Deutsch. Akad. Wiss. 7,170 (1960). 16 J.T. Randall and M. H. F. Wilkins, Proc. Roy. Soc. (London) Al84, 366 (1945). 17 J. Voigt, Diplomarbeit, Berlin (1958). 2

3

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INITIAL-RISE

METHOD

FOR

is the Fermi-level analysis which can be used for those photoconductors where trapped and free electrons are in a quasi thermal equilibrium.Is- 2o Thermally stimulated processes seem to be the most suitable phenomena to study the energy of traps since they involve directly the thermal release of trapped carriers. Despite efforts of the past 20 years to improve glow-curve techniques for studying the traps, no method has been devised which does not involve, in one way or another, either experimental or theoretical difficulties which limits its application. The expectations which were based on these techniques have not fully materialized. A search of the literature discloses considerable disagreement in the results published by different investigators. Examples of this are the publications by Nicholas and Wood21 and Dittfeld and Voigt22 in which they report on determination of trap parameters in pure CdS crystals using several different methods. Certainly, differences in the materials used may be cause for some disagreement, but most of the discord is due to experimental difficulties and a lack of consistent theoretical interpretation. Obviously, a specific method can be applied to a material only if the prevailing reaction-kinetics process can be described by the model upon which the derivation of the method is based. In general, however, there is no definite information on the special type of recombination kinetics in a particular material. By means of various electrical and optical experiments, some of the numerous models discussed in the literature thus far23 can be excluded, but usually, many others remain as a possible description of the kinetics. Also, calculations of glow curves based on different models fail in the effort to establish with certainty the desired information, since some of the parameters which govern the kinetic process cannot be determined experimentally. A good fit between the measured and calculated curves is therefore obtained by variation of the remaining undetermined parameters. To add to the dilemma, it has been shown that a particular glow-peak can be calculated equally well by using entirely different models describing the reaction kinetics.24 It is apparent that more basic information is needed on the processes studied. To overcome the principal theoretical difficulties it is desirable to have a method for determination of the important parameter E, which is independent of 18 R. H. Bube, J. Chern. Phys. 23, 18 (1955); Phys. Rev. 106, 703 (1957); Phys. Rev. 8, 566 (1950). 19 R. H. Bube, G. A. Dussel, C. Ho. and L. D. Miller, J. App!. Phys. 37, 2 (1966). 20 I. Broser and R. Broser-Warminsky, Ann. Physik 16, 361 (1955); Brit. J. App!. Phys. Supp!. 4, 90 (1955). 21 K. H. Nicholas and J. Woods, Brit. J. App!. Phys. 15, 783 (1964). 22 H. J. Dittfeld and J. Voigt, Phys. Status Solidi 3, 1941 (1963). 23 P. Braunlich, Proceedings of the Advanced Research Institute on Application of Thermoluminescence to Geological Problems, Spoleto, Italy, Sept. 1966 (in print). 24 P. Braunlich and A. Scharmann, Z. Physik 177, 320 (1964).

DETERMINING TRAP

DEPTHS

Ec

f>

CIL

l'

I Ev

h

2517

}~

H

A

111111111111111111111111

FIG. 1. Simplified energy-level scheme as used for TL and TSC theories.

the particular recombination kinetics. At present, no such technique is known. Most theoretical calculations and techniques for the evaluation of glow curves are based on a simple model (Fig. 1), which considers only transitions between traps H and recombination centers A on one side, and the conduction band on the other. The recombination kinetics is governed by the following parameters: trap depth E; attempt-to-escape frequency ao; the filling ratio, z=h(To)IH, of the traps H with electrons h at the temperature To; the initial concentration f( To) of unoccupied recombination centers; and the so-called retrapping factor R=(3/Y, where (3 is the co efficient for the trapping transitions and'Y that br the recombination transitions. In addition, there are the parameters q=dTldt and T, which are determined by the equipment used. For adequate calculations, all of these parameters have to be known. The formulas for the determination of E derived from this model require information about some of these same parameters. In general, it is possible to estimate some of them within certain acceptable limits or to perform the experiment under certain simplifying conditions. From the over-all glow spectrum one can find, for example, whether the ratio f( To) Ih( To) is large or approaches the value of unity. The filling ratio, z, can be adjusted in many cases to values close to a saturation (z= 1) or to a partial trap filling (z< 1). The retrapping factor, R, usually remains as an unknown parameter. Its influence on the shape of a glow peak has been demonstrated1o and it is known that it can vary between a value near zero to 104 or higher, depending on the material and the particular trap level.4 Since a safe estimation of R is not readily available in many cases it is desirable to have a method for the determination,of the trap depth, E, which does not depend on R. It is generally believed that the initial-rise method is independent of the retrapping. 21 ,25 Although there are

2. C. H. Haake, J. Opt. Soc. Am. 47, 649 (1957).

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2518

~

P.

BRAUNLICH

Various techniques have been used to separate these peaks or to obtain the desired information despite the overlapping. 8 ,15,19,25 For the sake of simplification we will assume, for this analysis, that the peak being considered is well separated from the others. Depending on its position in the glow spectrum, one can roughly distinguish between two cases for Eq. (3):

0.1

~

~

0.01

0.011

+-----\---+---t---+--+-----\--3.0 2.4 2.6 2.2 1.6

I.'

2.'

2.0

FIG. 2. TL glow curves (plotted vs l/T) calculated from Eqs. (1), (2) and (3a) with 0=1010 sec-I, E/k=9400° K, q=21° sec', h(To)/H=l.

some remarks in the literature about the influence of R on the initial rise,1O,24,26 no relationship has been clearly established. The purpose of this paper is to discuss such a relationship, and to give the limits within which the initial-rise technique is independent of R. THE INFLUENCE OF R ON THE INITIAL RISE OF A GLOW PEAK

According to the model shown in Fig. 1, two differential equations have to be solved. These are the equations for the rate of change, dn/dt, of the free electrons, n, and for the rate of change, dh/ dt, of the trapped electrons, h:

dn/dt=ah-f3n(H-h) -'Ynf,

( 1)

dh/dt= -ah+f3n(H-h) ,

(2)

f~h.

(3a)

f»h.

(3b)

Here, f is the density of unoccupied recombination centers of all types and h is the concentration of electrons trapped in the isolated peak being considered. It is usually assumed that n«h and dn«dh. Thermoluminescence-glow curves from Eqs. (1), (2), and (3a) have been calculated using these simplifications. 27 When R=O, the Randall-Wilkins formula 16 is obtained, and when R= 1, the Garlick-Gibson formula results.! Curves for other values of the retrapping factors have to be calculated numerically. A formula for the TSC glow curve cannot be derived for Eq. (3a) and the described assumptions. If Eq. (3b) is valid, both TL and TSC glow curves can be calculated for R= 0 and R= 1.21 Assuming the temperaturedependence of "I and !J. can be neglected compared to that of a, the solution yields TL and TSC peaks which do not have the often-observed different maximum temperatures, thus providing a further criterion to decide which of the relations, Eqs. (3a) or (3b), is valid. In order to prove that R can influence the initial rise of a glow peak, we have applied the initial-rise method

Elk

where dt= dTq-l and a= ao exp( - E/kT). The TL intensity I is given as I ex 'Ynf and the conductivity as

·8

u= enp,.

Since the crystal is electrically neutral, the total concentration of electrons in the conduction band and in the traps must be equal to the concentration of unoccupied recombination centers. Assuming that we have N different types of recombination centers and M different types of independent traps in a crystal, this ,condition may be expressed by the equation N

M

Lfi=n+ Lhj • i~1

(3)

j=1

Each trap level gives rise to a separate glow peak with these peaks often being superimposed on each other, thus complicating the evaluation of a glow spectrum. 26H. Gobrecht and D. Hofmann, ]. Phys. Chern. Solids 27, ,509 (1966).

0.1

10 ------ R :aP... 'Y

FIG. 3. E/k determined by initial-rise method from the calculated TL glow curves shown in Fig. 2. 27

p. Br.aunlich, Diplomarbeit" Giessen (1961),

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I KIT I A L - R I S E 1\1 E THO D

FOR

for determination of trap energy to calculated TL glow curves for different values of R according to Eqs. (1), (2), and (3a). The calculations were made with a trap depth Elk=9400oK~0.81 eV. The obtained values, plotted vs liT, are shown in Fig. 2. By applying the initial-rise method in a range where leT) is smaller than 0.021 ( T m), slopes were found which are shown in Fig. 3 as a function of R. The results indicate that these values are in agreement with Elk only if 0::; R::; 1. The reason for this is quite obvious, since with an increase in R, it takes a longer time to empty the traps. The peak temperature, T m, and the half-width increase, the intensity at T m decreases and the initial rise departs from expo ( - ElkT) earlier. Because of the broad peaks, the increasing part of the curve, logl vs 11 T, seems still to be linear. Its slope, however, is equal to Elk only at very low intensities which often can not be evaluated, thus resulting in values for the trap depth which are too small. Obviously, the shape of the peak does not change with R as long as the recombination rate, 'Ynf, is larger than the retrapping rate, (3n(H- h). The calculations (Figs. 2 and 3) indicate that the initial-rise method provides sufficient accuracy within the limits f3n(H-h) ::;'Ynj. This leads to a simple criterion for its applicability. With z=h(To)IH it becomes Rh[h( To) 1hz -lJ::;j.

(4)

DE T E R YI: I

h( To) If( To)~h( To) If( T)

> h( T) If( T).

SUMMARY

The initial-rise method by Garlick and Gibson is a simple method for the determination of trap depths from thermoluminescence and thermally stimulated current-glow curves. It is, however, incorrect to believe that for its application in the case where the model in Fig 1, or its extension to traps with an energetic distribution, describe the kinetics, the knowledge of no further constant is necessary. The obtained values for E are smaller than the real ones unless a condition of the form Rh( To) (2Iz-1) ::;f( To) is fulfilled. A rough criterion of the applicability is provided by the condition: R·h( To) If( To) «1. It is essential to check this in all cases where trapped and free electrons are in quasi thermal equilibrium (R» 1). It follows, therefore, that for proper application of this method, adequate information at least on the parameters Rand h( To) If( To) is needed.

(5)

f3 E

It is sufficient to insert h(To)lf(To) instead of hlf in the relations (4) to (6) because of h(To)lf(To)? h(T)lf(T) for T> To. This can be proved as follows: Letus first assumef( To)~h( To). Thenish( To) If( To)~ he T) If( T) for all T> To. In case h ( To)