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Evaluation of the model for thermally stimulated luminescence and conductivity; reliability of trap depth determinations
Journal of Luminescence 15 (1977) 1—27 © North-Holland Publishing Company
EVALUATION OF THE MODEL FOR THERMALLY STIMULATED LUMINESCENCE AND CONDUCTIVITY; RELIABILITY OF TRAP DEPTH DETERMINATIONS P. KIVITS
*
and H.J.L. HAGEBEUK
Department of Physics, Eindhoven University of Technology, Eindho yen, The Netherlands Received 9 December 1975 Revised manuscript received 12 February 1976
A review of the basic theory on thermally stimulated luminescence (TSL) and conductivity (TSC) based on a certain simple model is given. Approximate analytical expressions for the shapes of the TSL and TSC curves are derived. Methods from the literature for trap depth determination, some of which can easily be derived from these expressions, are applied to numerically calculated TSL and TSC curves, It turns out that the methods of Bube, Haering and Adams, l-Ioogenstraaten and Unger yield a correct value for the trap depth independent of the values of the retrapping ratio and the frequency factor. The method of Garlick and Gibson is reliable if applied below 15% of the maximum TSL intensity. Some methods, as most of those derived by Chen, yield a correct trap depth for a specific value of the retrapping ratio in the case of TSL only.
I. Introduction Thermally stimulated conductivity (TSC) is the phenomenon that the conductivity of a semiconductor is temporarily enhanced when heated in the dark after excitation at a low temperature. Simultaneously light can be emitted which is known as thermally stimulated luminescence or thermoluminescence (TSL). The TSC and TSL curves which are plots of the thermally stimulated conductivity and luminescence intensity versus temperature, respectively, are often used for determination of the thermal energy depth of trapping levels in the forbidden gap of a semiconductor. In the course of time quite a number of methods have been published to realize this objective. Some methods were developed for TSC, others for TSL. In practice, however, many of the methods are applied to both curves without proving that this is correct. In several papers [1—4]the results of some of these methods are compared experimentally. It was found that different methods do not produce the same trap depth for a given TSL or TSC peak. The real trap was not known in these in*
Present address: Philips Research Laboratories, Eindhoven, The Netherlands.
2
P. Kivits, H.J.L. Hagebeuk / Evaluation of the modelJbr TSL and TSC
vestigations, however. Some workers [5] conclude in their papers that TSL and TSC measurements are a helpful tool for the determination of trapping parameters. Others [6], however, state that no conclusion can be drawn from such experiments without previous knowledge about, for instance, the defect structure of the phosphor. In almost all theories on TSL and TSC the same model is used (section 2). From this model we shall derive expressions for the TSL and TSC curves and from these some formulae for the trap depth in terms of measurable quantities (section 3). A survey of all currently used methods will be given. With a numerical procedure described by Hagebeuk and Kivits [7] TSL and TSC curves can be calculated for various sets of parameters. Nearly all methods for trap depth determination in this paper are applied to these curves. When the trap depth obtained in this way is compared with the value used in the numerical calculations a statement can be made about the reliability of the methods (section 4). The validity of the model itself will not be discussed here. In a separate paper this will be done by extending the model with some additional properties, as recombination via excited states, trap distributions, donor—acceptor pair recombination, cross sections that depend exponentially on reciprocal temperature, thermal quenching of luminescence and a variation of mobility [8].
2. Basic theory for thermally stimulated luminescence and conductivity 2.1. The mathematical model The physical model which one generally uses to describe the TSL and TSC processes is shown in fig. 1. Electrons are thermally excited from trap h1 to conduction band (transition 1) where they contribute to the conduction. From this band they can either be (re)trapped (transition 2), or recombine with a hole trapped at a recombination centre a (transition 3). The latter transition may be accompanied by the emission of light. In this model, the time derivatives of the concentration of free and
conduction band n
h2, h2 a~a vaLence band Fig. 1. Energy level scheme forming the model which is often used for the interpretation of thermally stimulated luminescence and, conductivity experiments.
P. Kivits, H.J.L. Hagebeuk /Evaluation of the model for TSL and TSC
3
trapped electrons are given by the following equations dhj7dt
=
dn/dt
—yhj =
+
13n(h1
—dhj7dt
—
—
hj)
(1)
ctna~
(2)
The following symbols are3);used: E: trap depth (eV);ofh1electrons h2: concentration hj h~:concentration trapped at hof the two trapping (cm— 1 and 3); n:levels concentration of free electrons (cm3); a: concentration of recomh2 (cm centres (cm_3);a+: concentration of holes trapped at recombination centres bination (cm3) *; 13: trapping rate constant (cm3 s~);cv: recombination rate constant (cm3 s1); y: transition probability for electrons from trap to conduction band (s1). It is assumed that the occupation of h 2 does not change during the emptying of the shallower trap. Both traps are then called “thermally disconnected”. The consequences of this assumption are discussed later. Eqs. (1) and (2) govern the process of emptying h1 as a function of time (t). During this process the condition of charge neutrality requires that ,
,
a~=n+hj+h~,
(3)
if no free holes are present. The transition probability y is given by y
=
s exp[—E/kTJ,
(4)
in which k is Boltzmann’s constant and T the absolute temperature. Between the pre-exponential factor s, which is called the frequency factor because of its dimension, and 13 the following relation exists (5) where N~is3/2) the and effective density of states of the conduction band (usually proporg is the degeneracy factor of the trap (taken equal to 1 in the tional to T The coefficients a and 13 are given by following). XSaV,
(6)
13=Shv,
u is the mean thermal velocity of free electrons (proportional to T112) and Sa and Sh are the cross sections for capture of electrons by recombination centres and traps, respectively. According to Lax [9] capture cross sections of deep centres are proportional to T~°~ (m 2.5 or 4). Chen and Fleming [10] assume a range 0 ~ m ~ 4. Following the latter authors we obtain with (5) and (6) that where
s=s 0(T/T0)b
,
a=cw0(T/T0)c, *
—2~b~2
(7)
—7/2~
(8)
,
Here Klasens’ notation has been used, 11.A. Kiasens, J. Phys. Chem. Solids 7 (1958) 175. By h is meant that the centre has trapped an electron, a~means that the centre has lost an electron.
4
P. Kivits, H.J.L. Hagebeuk /Evaluation oft/ic model for TSL and
TSC
where s
0, a0 and T0 are constants. Front (7) and (8) it follows that generally s and cv are rather slowly varying with temperature compared to the exponential increase of ‘y, when L’>> kT as is always the case. Therefore it is often assumed that s and a are independent of the temperature. In our calculations we shall consider both cases. The influence of an exponential dependence with reciprocal temperature of Sa and S1, as is found by Henry and Lang [11], is discussed elsewhere [81 and will not be considered here. The integrated luminescence intensity defined as the miumber of photons emitted by the phosphor per unit time interval is proportional to the change of the occupation of the recombination centre, da~/dt.The conductivity u,~is proportional to the number of free electrons, according to =
neji,~
(9)
312 where is the driftbetween mobilitycharged for electrons. mobility is vibrations proportional to T for thep,1interaction carriers The and acoustical in non-ionic crystals [12]. For scattering by neutral impurities the mobility is independent of temperature [13]. An exponential decrease of p 03/2with T isthe found for are polaron scatwhen carriers scattered tering by optical Further by charged centresphonons [15]. In[14]. the latter casep,~ the Tmobility also depends on the concentration of charged defect centres which may be of importance for the thermally stimulated conductivity where the number of charged centres often increases with temperature [8]. In practice the value of a,~generally cannot be measured using an experimental arrangement with only two contacts on the sample. However, since the evaluation of a,~under transient conditions existing during the presence of a thermally stimulated current from a four probes method as described by Van der Pauw [16], meets with rather complicated experimental problems, one has to realize that contact properties and the formation of space charge may influence the results ‘i’. Since the T-dependence of p, 1 differs for different phosphors we shall restrict ourselves in this paper to the case that p,~is independent of temperature. We assume that at T0, the temperature where the heating starts, hj(T0) = h~(T0)= h2, a’(T0) = h1 + h2 and consequently n(T0) = 0. Finally, we need a relation between Tand t since we are mainly interested in solutions with temperature rather than with time. The heating rate w is given by w
=
dT/dt
.
(10)
Since in experiments usually a constant heating rate is chosen this means that
T=T0+wt. *
(11)
Soinctinies space charge is deliberately created by applying high electrical Odds. Such an cxperimental arrangement can lead to model simplifications since the recombination lifetime may be assumed to be constant (see also section 2.2).
P. Kivits, H.J.L. Hagebeuk I Evaluation of the mode/for TSL and TSC
5
An analytical solution for da’7dt and n from the non-linear differential eqs. (1) and (2) together with (3), (4), (11) and the initial conditions has not been found up to now. Several simplifications have to be introduced to obtain approximate expressions for the TSL and the TSC curve.
2.2. Simplifications of the kinetic’ equations In this section we shall briefly discuss assumptions and approximations that are used in literature in order to simplify the system (1, 2). One currently assumes that often
(dn/dt( << dhldtl
(12)
.
From (3) it can be seen that this implies that the luminescence intensity only depends on the change of the occupation of h1. It is obvious that this will be correct if the recombination lifetime is sufficiently small. This lifetime r is defined by (c.f. eq. (2))
r1/aa~
(13)
.
Hence, r increases with T since a~ decreases. This means that the assumption becomes more unreliable for higher temperatures. If, however, it is additionally assumed that h2 >> h1,hericea~ h~= h2, the recombination lifetime will be short and nearly constant during the emptying of the shallower trap. Eq. (2) then reduces to a linear differential equation. —
Another simplification often used is
n<
(14)
Kelly et al. [6] concluded that approximate solutions based on this assumption “must break down past some definite higher temperature”. However, in our opinion this temperature is not in the region of interest. With (1) it follows that ~dhj1dt+7hj hj
13hj(h1—hfl
<____1_____
(15)
,
cl(h1—hfl
since dhj7dt <0. For higher temperatures h1 still increases. Thus, we may write n/hj
—
hj will be nearly constant while y
(16)
where (4) and (5) are used. In the maximum of the TSL curve E 27 kT (section 4). Hence in normal phosphors the assumption (14) will be justified. In the appendix a numerical example of the ratio n/hj as a function of temperature is given. The model given by (1) and (2) involves essentially a second order mechanism for luminescence independent of the value of 13/a. Nevertheless, first order kinetics can be approximated by taking 13 <
6
P. Kivits, NfL. Hagebeuk /Evaluation of the ,nodelfbr TSL and TSC
tunately, in thermoluminescence literature one often takes j3 0 which does not have any physical relevance as will be seen in section 2.3. As a special case of second order kinetics one usually takes ~3= a, for reasons of mathematical simplicity.
The methods that were originally derived for each of these cases, also known as mono- and bimolecular recombination, or slow and fast retrapping, are called first and second class methods in this paper; when no specific value of the ratio 13/a is assumed the term general class methods is used.
2.3. Approximate expressions for the TSL and TSC curves Using the simplifications discussed in the previous section it is possible to derive approximate expressions for da~/dt(fland n(7) from (1) and (2). However, we shall not use the presence of a thermally disconnected trap since this, together with the assumption
h2 >> h1,
would lead to a too far-reaching simplification of the mathematics. Further we shall drop the suffix of h1 for practical reasons in this and the next section. The set of non-linear differential equations of the first order can be rewritten as
one differential equation of the second order. We obtain 2+ a1 d2a~/dt2 = ‘y(a~) dt a~(l 6)+óh+a’ [y—(l +ö)(a~)~da~/dt ‘
17
(
where 6, the retrapping ratio, is given by 6f3/a.
~l8)
Since the luminescence intensity is proportional to da~/dt,the TSL curve can be numerically calculated from (17) for different values of H, s, w, 6,h and a [7]. From (2) and (3) it follows that n
—(l/aa~)da’]dt
(19)
which makes it subsequently possible to calculate n(fl. As an illustration of the following calculations we show some TSL. and TSC curves for specific sets of parameters in fig. 2.
It can be shown (see appendix) that for normal dope concentrations and in the relevant temperature region, adequate approximations for da’]dt and n are given by da~/dt= —y(a~)2/[a’~(l
—
nya~/a[a~(J—8)+OhJ
6) +
.
oh]
,
(20)
(21)
It should be noted that the same equations are obtained from (1), (2) and (3) when the simplifications (12) and (14) are used. Following the literature we shall now distinguish between the cases 6 = 1 and
P. Kivits, N.J. L. Hagebeuk / Evaluation of the modelfor TSL and TSC I
8—
‘D
I
I
7
I (0
TSC
7
fo
(0
~ 6
TSL
4—
-
ib~ 3-
—
2-
-
170
180
190
200
210 ~T(K1
2s1 w = 0.1 Ks’, h = iO’7 Fig. 2. The TSL and TSC curves forE = 0.516 eV, s = 1.25 X 10’ = iO~cm3 s~,and different values of b. The factor 1/6 along the axes is introduced to reduce all curves to the same scale.
6 << 1. When 6
=
1, eqs. (20) and (21) can be simplified to
da~/dt= —‘y(a~)2/h
(22)
n
(23)
,
=
ya~/cvh.
From fig. 2 it is seen that when 6
=
1, n(T) shows only a very smooth maximum.
This is usually not measured. It might mean that 6 ~ 1 represents a situation that
hardly occurs in phosphors, but we believe that it is more probable that the model under consideration is too simple. When this model is extended with thermally disconnected trapping centres in a relatively low concentration the long high temperature tail of the TSC curve vanishes rapidly [7,8]. In the case 6 << 1, eqs. (20) and (21) tend to
dr
—aiVc 6a~exp[—E/kT] 1 +
(24)
8
P. Kivits, H.J.L. Hagebeuk/Evaluation of the model for TSL and TSC 4
I
0
-2
170
I
I
1
1
I
180
190
200
210 ~ T(K)
4/th versus temperature for the same parameter values as in fIg. 2.
lig. 3. The ratio a
and nN
C
Oexl~L~7~.fl l+6h/a~
(25)
where (4), (5) and (18) have been used. To acquire insight in the behaviour of the nominators of (24) and (25) the value of a~/6h as a function of temperature is shown in fig. 3 for the same parameter values as in fig. 2. Since a~>> Oh over a relevant part of the temperature region, the TSL curve can almost fully be represented by da~/dt= —aON~a’~ exp[—E/kTI
.
(26)
A similar simplification of (25) can only be made in the initial part of the TSC curve since for higher temperatures a+ and Oh are of the same order of magnitude. As already noted, one sometimes takes 6 0. From (24) and (25) it is clearly seen that in this case the TSL cruve as well as the TSC curve vanish. Physically this means that the trap in fact cannot be filled.
In this section we derived approximate expressions for the TSL and TSC curves,
the validity ranges of which are unambiguously formulated. In the next section we shall use these expressions for the derivation of formulae on which some methods
for trap depth determination are’ based.
P. Kivits, H.J.L. Hagebeuk / Evaluation of the modelfor TSL and TSC
9
3. Methods for the determination of the trap depth In literature several expressions exist for the trap depth in terms of measurable quantities. These can be used for the calculation of the trap depth from TSC and TSL curves. The derivation of some of these expressions as presented by their authors, sometimes is very cumbersome. Moreover, assumptions are made which are not obvious. However, it is possible to derive some of the formulae for the trap depth in an appropriate way from the approximating eqs. (20) and (21)as will be shown in section 3.1. Current methods for trap depth determination are summarized in section 3.2. An attempt is made to indicate the underlying principles. As we do not intend to derive or even to mention all methods, some others remain undiscussed. These methods, however, are generally less manageable and therefore not often used in practice [16]. 3.1. Derivation of expressions for the trap depth
From (20) and (21) it follows that (27)
2/[a~(1 —6) +ohl},
1n(—da~/dt)= —E/kT+ln~s(a~) Inn
—E/kT+ln{sa~/a[a~(1—6) +oh]}
(28)
,
where (4) is used. In the initial part of the TSL and TSC curves where a~varies only slowly compared to the exponential increase of y with T one obtains from (27) and (28) that ln(—da~/dt)~~ —H/kr,
(29)
ln nmit
(30)
—E/kT,
since a~(init) h. Thus, a plot of ln(—da’7dt) or ln n versus l/T should yield a
straight line with slope —H/k in the initial part of both curves. This procedure is known as the initial rise method of Garlick and Gibson [17].
Some methods for trap depth determination use the variation of the maximum or the temperature at which this maximum occurs with the heating rate. To acquire insight in the validity of the expressions on which these methods are based we calculated for different heating rates the values of dat/dt, at, TL and nc, a~, T~where the indices Land C stand for the TSL and TSC maxima, respectively. As an example we give some results in table I. It is seen that dat/dt and nC depend strongly on the value of w, contrary to at and a~. We may therefore assume the logarithmic terms at the right hand sides of (27) and (28) to be constants in a first approximation. It then follows that at the TSL and TSC maxima intensity
—
—
ln(—dat/dt) hi nC
=
—E/kTL
E/kTc
+
+ Ci(at),
(31)
C 2(a~),
(32)
P. Kivits, NIL. Hagebeuk / Evaluation of tile model for TSL and TSC
10
Table 1 Some characteristic values of the TSL and TSC curves with E and 6 = 0.01 for different values of the heating rate w
=
0516 eV, s
=
1.25 X 1012 s_I
TL (K)
—h~da~jdt
aL/h
Tc (K)
iic/h
ak/h
l0_2 10_I
175.9 187.9 201.5
6.0 x i0~
183.6 196.6 211.6
10
217.2
0.395 0.396 0.397 0.398
4.4 X 10~ 3.8 x i0~ 3.3 x l0_12 2.9 X lO~
0.0225 0.0223 0.0221 0.0219
1)
w (Ks
where
C
1
5.2 x i0~ 4.6 x 10_2 4.0 x 10_i
228.9
C2 are (nearly
constant) functions of at and a~,respectively. Thus, a l/Tc, respectively, for various heating rates, should give a straight line with slope —E/k. This method for trap depth determination was originally derived by Bube [18] for TSC. It is now found that it can also be applied for TSL independent of the value of the retrapping ratio. It is obvious that from (27) and (28) also equations can be derived for any specific temperature. For instance for the temperature T1 at the lower temperature side of a TSL curve, where the luminescence intensity equals half its maximum value. We obtain from (27) that and
plot of the value of In(_dat/dt) or In nC versus I /TL or
ln(—dat/dt)
=
—E/kT1 + C1(a~)+ In 2
(33)
where a~is the concentration of trapped holes at T1. Since also aI will not strongly depend on the value of w, a plot of ln(—dat/dt) versus 1 /T1 for various heating rates,
should yield a straight line with a slope _E/k. Unger [19] derived that the method of Bube which is based on (31) produces an upper limit for the value of the trap depth and that the method based on (33) yields a lower limit. In this paper we shall refer to the latter as the method of Unger. In the TSL maximum (d/dfl(dat/dt) = 0 and in the TSC maximum dnc/dT = 0. When differentiating (20) and (21) with respect to temperature we must take into account that s and cv depend on temperature as given by (7) and (8), respectively. We shall distinguish between (a) b = 0, c = 0 and (b) b ~ 0, c ~ 0. (a) The case b = 0, c = 0. When 6 <
wE/kT~ sexp[_E/kTL].
(34)
Hence, a plot of ln (w/T~)versus 1 /TL should yield a straight line with a slope —Elk. This procedure is known as the method of Hoogenstraaten [20]. He proved that this method is also valid for ~ = 1. After differentiating (22), derived for 6 = 1, with respect to T, it follows with (5) that in the TSL maximum
wE/kT~= (2at/h)s exp [—LtikTL]
.
(35)
The only difference between (34) and (35) is the pre-exponential factor. As already was mentioned, the value of at does not strongly change for different values of the
P. Kivits, N.J.L. Hagebeuk / Evaluation of the modelfor TSL and TSC
11
heating rate and may readily be assumed to be constant. It is now obvious to define a parameter Seff, called the “effective frequency factor”, according to (36) This parameter is nearly a constant for a given trapping level. It should be noted that either s or Seff,L can be determined by extrapolation to I /TL = 0 from the plot ln(w/T~)versus I /TL as follows from (34) or (35). Hoogenstraaten derived his method for TSL but it is possible to prove analytically that this method is also correct for TSC in the case 6 = 1. From (4) and (23) it follows that in the TSC maximum 5eff,L
=
(2at/h)s.
wE/kT~= 5eff,C exp[—E/kTc],
(37)
with Seff C
=
(a~/h)s.
(38)
Thus, a plot of ln(w/T~)versus 1 /T~should yield a straight line with slope —E/k since 5eff,C may be assumed to be essentially constant. Since except for the initial rise no simple expression for the TSC curve exists when 6 << 1 (section 2.3) it is not easy to prove analytically that Hoogenstraaten’s method yields also a correct value~5iThe~trap depth in this case, as follows from our numerical results in section 4. Chen and Winer [21] derived a method for TSL which produces according to the authors, a correct value of the trap depth independent of the value of 6. The expression which they derived in the case 6 = 1 is 1n[_(dat/dt)k2T~/w2E2] ~E/kTL
+
C 3(at).
(39)
It is easy to derive (39) from (31) and (35) taking 5eff,L (40) C3(at) = C1 (at) 2 in It may thus be concluded that (39) is valid for TSL in both cases 6 = I and 6 << 1. Moreover, it is valid for TSC since it was proved above that the methods of Bube and Hoogenstraaten yield a correct value of the trap depth in those cases. In general the trap depths calculated with these three methods are related according to E (Chen and Winer) = 2E (Hoogenstraaten) E (Bube). (b) The case b ~ 0, c ~ 0. When b ~ 0 and c ~ 0 it is easy to prove that (20) and (21) are also valid. It can further be derived from (4), (7), (8) and (21) that independent of the value of 8 ‘
—
—
In(n/T~c) = —E/kT+ C 4(a~) ,
(41)
where
C4—soT~~a~/a0[a~(l —6)+6h]. Boiko et al. [221 assumed b c = in the TSC maximum and came to 2)= -E/kTc + C ln(nc/Tt~’ 4(a~)
(42)
—
(43)
12
P.
Kivits, Hf. L. Hagebeuk
/ Evaluation of tile model for
TSL and
TSC
Hence, the plot of ln(nc/T~’2)versus l/T~far various heating rates should yield a straight line with slope —E/k independent of the value of 8. We shall refer to this procedure as the first method of Boiko et al. Another method from the same authors derived for the case 0 = 1 is obtained from (4), (7), (8) after differentiating (23). It then follows that in the TSC maximum wE/kT~~2 — w(c
—
b)/T~1 (a~/h)(so/T8)exp[—E/kTcj.
(44)
where (22) has been used. As will be shown in section 4, a rough estimate for the value of the trap depth can be obtained as H 25 kT~.In that case we can neglect the second term at the left hand side of eq. (44) in a first approximation. We then obtain wE/kT~2
(a~/hT~)s 0exp[—E/kT~].
(45)
which is equivalent with (37) when b = 0. The second formula of Boiko et al. [22] can now be obtained taking b = ~-. Hence, 2 = (a~/hT~2)s wE/kT~V exp[—E/kT~1. 0 (46)
The authors state that this method also leads to correct determination of the trap depth in the case 8 << 1. Another method based on (45) is the one of Böer et a!. [231.They obtained wE/k (a~T~/h)s0 exp[—E/kT~],
(47)
which follows from (45) by substituting b = —2. Originally this method was derived for 8 << 1. The authors estimated that when the method is applied on experimental results the calculated trap depth is about 20% too high.
We shall now prove that for TSL expressions similar to (46) and (47) can be obtained. From (22) and (26) it can be derived that 1)(E/kTL + b) s’ exp[—E/kTLI, (48) (w/Tt~ where s’ = so/T8 when 6 <<1 and s’ = (2at/hTob)so when 8 = 1. Neglecting b with respect to E/kTL we come to the equivalents of (46) and (47) in the TSL case. For b it follows that ‘=-~
7/2
WE//CTL and for b = —2,
s exp[—b/kTL},
wE/k—s’exp[--E/kTL],
(49)
(50)
from which it is obvious that both methods yield the same results for TSL and for
TSC. In this section we derived some expressions for the trap depth in terms of measurable quantities. It has been proven that methods originally derived for TSC can also be applied in the TSL case and vice versa, which was not predicted by the original authors. However, these are not the only methods for trap depth determination
P. Kivits, H.J.L. Hagebeuk/Evaluation of the model for TSL and TSC
13
from TSL and TSC curves. In section 3.2 we present a list of 31 methods that can be
used for this purpose. 3.2. A survey of methods from the literature
In the previous section we derived several formulae for the trap depth. Most of the methods based on these expressions use the shift of the TSL or TSC maximum with different heating rates. However, this is not the only way of determining a trap depth from TSL or TSC curves. In this section we give a survey of currently used
methods to realize this objective. For completeness, those methods mentioned in 3.2 are also included. If possible the conditions under which the methods according to the authors are valid are given. It is indicated whether they were originally derived for TSL or TSC. The methods are roughly divided into three groups: (A) Methods making use of heating rate variation. Most of them have been discussed in section 3.1; (B) Methods making use of geometrical approximations; (C) Other methods. Additionally they are ordered in first, second and general class methods (section 2.2).
(A) Methods making use of various heating rates. The following symbols are used: w,: the heating rate in the ith experiment;Lmi: the intensity of the TSL maximum; nmi: the maximum of the TSC curve; Tmj: the temperature at the TSL or TSC maximum; T11: the temperature at the lower side of the TSL or TSC curve where either L
=
~Lmjor n
=
(1) First class methods Method 1, Booth, Bohun and Parfianovitch (TSL) [24,25,26]
E= [kTmiTm2/(Tmi Tm2)] —
w1T,~2/w2T,~1).
Method 2, Böer, Oberldnder and Voigt (TSC) [23]. According to ref. [23] a plot of ln w1 versus IlTmi yields a straight line with a slope —pH/k with 0.7
(3) General class methods Method 4, Hoogenstraaten (TSL) [20]. A plot of ln(T~1/w~) versus 1/Tmj should yield a straight line with a slope E/k (section 3.1). Method 4 is a generalized method I. If one uses more than two heating rates method 4 is of course more accurate than
14
P. Kivits, H.J.L. Hagebeuk/Evaluation of the model for TSL and TSC
method 1. Therefore method 1 will not be included in our discussions. Method 5, Bube, Haering and Adams (TSC) [18,27]: A plot of ln ~mj versus l/Tmi yields a straight line with a slope —E/k (section 3.1). Haering and Adams claim that this method is valid when E/kTm >> 1. An additional condition for larger values of 0 is: h/Ne >> exp(—E/kTm). Method 6, Unger (TSC) [191. Plotting hi nmj versus lIT11 yields a straight line with slope —E/k (section 3.1). Method 7, Schän (TSC) [28]. E= [kTmiTm2I(Tmi
-
Tm2)]
wiT7~/w2T~?)
Method 8, Boiko, Rashba and Trofimenko 1 (TSC) [22]. The plot of ln(nmi/T,~/~) versus l/Tmj yields a straight line with a slope —Elk. Method 9, Boiko, Rashba and Trofimenko 2 (TSC) [22]. The plot of ln(w1/T/~) versus 1 ITmi yields a straight line with a slope —E/k. This method is in fact an extension of method 7. For this reason the method of Schön will not be included in our discussions.
(B) Methods making use of geometrical approximations In fig. 4 a general shape of a TSL or TSC curve is given. It can roughly be approximated by a triangle which is also shown. This principle has been used for the derivation of several expressions for the trap depth. Following the literature, from the
figure several parameters can be defined w’~’T2—T1,
XTm —T1, ~a/X,
l1g=U/W~
‘m
o
I
tJ=T2~Tm,
~i=E/kTm
-
/‘fl\ ~
__~L~ll_ T1
Tm
12
~T Fig. 4. General shape of a TSL or TSC curve with definition of some characteristic quantities.
P. Kivits, H.J.L. Hagebeuk / Evaluation of the model for TSL and TSC
15
As an illustration of the underlying principles of these methods we will now derive a formula forE in the case of TSL when 6 << 1. From (26) it follows that (dü+/dt)Tm
at
r
E
E
1
(~+ldt)a±~LkTmkTlJ
(51)
.
Further at
=
f
w~(da~/dt)dT = (Q/w)
Tm
7 L dT,
(52)
Tm
where L is the measured TSL intensity. A similar equation exists for at triangle approximation we obtain
-
With the
at
=(C’sIW)Lma,
(53)
at
‘(C
(54)
and 5/w)Lm(o+~X).
The left hand side of(51) equals 2. Hence, one finally obtains E=(kTi Tm/k) lfl(2+3X/2u).
(55)
Since all quantities in (55) are measurable, E can be calculated with this formula. Several expressions similar to (55) are possible. Those known to us will now be given. (1) First class methods Method 10, Luschik (TSL) 2~/a. E=kT,
[291
Method 11, Halperin and Braner 1 (TSL) [30] E=(l.72kT,~/X)(l —5.l6/~), with the condition pg
P. Kivits, HI L. Hagebeuk / Evaluation of the modelfor TSL and TSC
16
with the condition
This method is in fact identical with method 10.
Method 13, Chen 1 (TSL) [311 E=2kT
01(l.25 TmIW~ I),
with the condition ~>>
1.
Method 14, Chen2(TSL) [31] E=2.29kT~/w.
which is a simplification of method 13. Method 15, Chen3(TSL) [31] E=(l.548kTf~/X)(l —3.l6/i.~). Method 16, Chen 4 (TSL) [311 E=l.52kTp~/X_3.16kT~
which is a simplification of method 15. Method 17,chen5(TSL) [311 H
=
C6 kT~/o,
with C6
=
0.976 ±0.004.
This is the Luschik method, but corrected for the fact that the area under the glowcurve is not exactly equal to the area of a triangle. Method 18, Chen 6a, 6b (TSC) [32,33] E’3okT,~/wX E=2.8 ~kT~/wX
(a) (b)
(2) Second class methods Method 19, Halperin and Braner 3 (TSL)
E=(2kT~~/X)(l—6/s), with the condition pg~e~(l +2/s).
[301
P. Kivits, H.J.L. Hagebeuk / Evaluation of the modelfor TSL and TSC
17
Method 20, Ha/penn and Braner 4 (TSL) [30] E = 2kT,~/a, with the condition 1g~0.5.
I
Method 21, Ha/penn and Braner 5 (TSL) [30] E(l +w/X)kT,~/a, with the condition
0.5. This method has in fact been derived for the case 6 > 1. Method 22, Chen 7(TSL) [31] Ez2kTm(l.77
TmIW —1).
Method 23, Chen8(TSL) [31] E
=
1 .813 kT~/X
—
4kTm
Method 24, Chen9(TSL) [31] E=C 7 2kT~/u, with C7 =0.853±0.0012. This is the corrected method 20.
(C) Other methods (1) First class methods Method 25, Randall and Wilkins (TSL) [34] E=25kTm. In ref. [34] it is assumed that ‘y = 1 for T = Tm, when 0.5
kTmTi/X.
Although similar to expressions where the triangle approximation is used the formula of Grossweiner is derived by integrating (26). The integral f~~fl dT which I obtained in that way is approximated by a series that may be truncated after the ~‘
18
P. Kivits, H.J.L. Hagebeuk/Evaluation of the modelfor TSL and TSC
first term when z~s>20 ands/w> l0~K~,accordingto the author.
Method 27, Franks and Keating (TSL) [37,38] l/~r(w/Tm)(l.2XO.54)+5.5X l03_~(~0.75)2 with the additional conditions that l0<~<35
and
O.75
Method 28, Dusse/ and Bube (TSC) [39] E’C 8 kTmTi/X. When ~ = 17, 22 or 26, for C8 a value of 1.402, 1.415 or 1.421 has to be taken, respectively. The method is a correction of method 26. (2) Genera! class methods Method 29, Ganlick and Gibson (TSL) [17]. In the initial rise of the curve a slope —Elk is obtained when ln L is plotted versus l/T. In practice, however, it is rather arbitrary what region should be taken as the initial rise. Our numerical calculations presented in section 4 have been applied on three parts of the TSL and TSC curves.
(a) From 0.01 to 0.15 Lm or ~m’ (b) From 0.15 to 0.30 Lm or nm.(c) From 0.15 to 0.50 Lm or nm. Haake [40] concluded that this method becomes more accurate for larger s an lower H values. Method 30, Sandomirskii and Zhdan (TSL} [41] E= l.45SkTmTiIX_0.79kTi with 1.02 < Tm/Ti <2.0, which according to the authors is equivalent to 1.06 < z~< 121.01. The formula is the Grossweiner expression, but corrected with the aid of computer calculations. Method 31, Voight (TSC) [42] l8kTm
4. Reliability of trap depth determinations In section 3 a number of methods for trap depth determinations from TSL or TSC curves are listed. Some methods were extensively discussed, others merely mentioned. In the derivation of expressions on which these methods are based some more or less plausible assumptions or approximations are used. Whether these are
allowed or not can easily be verified for the simple model when TSL and TSC curves are calculated without making simplifications. This can be done with the aid of a numerical procedure described in [7]. Most of the methods mentioned in this paper
are applied on these curves. Subsequently the determined trap depths are compared
P. Kivits, HI L. Hagebeuk / Evaluation of the model for TSL and TSC
19
with the value used in the numerical calculations. We calculated TSL and TSC curves for a phosphor with NC = 1020 cm3 g = 1, h 3 and both h 1 = 1017 cm 2 = 0 and h2 >> h1. In the case h2 >> h~the TSL curves differ mainly from those with h2 = 0 with respect to the shape at the high temperature side, ~‘2 Tm being smaller [8]. As follows from (19) with a~= h2, the TSC curve is identical with the TSL curve when h2 >> h1. For T0 a value of 80 K was taken. Further we chose the following values for the other relevant parameters to cover ,
—
roughly all reasonable sets of values E
0.273,0.341,0.511,0.682 eV
=
6(TSL)= 0.01, 1,3 6(TSC) s w
=
0.01,0.1
6,1.25X 109,1.25X 1012s~ 1.25X 10 “0.01,0.1,1Ks~
The value of a is fixed by this choice because from (5) and (18) it follows that
a—s/8N~
(56)
.
Both s and a are taken independent of temperature, thus b = c = 0 (c.f. eqs. (7) and (8)). We did not calculate TSC curves in the cases 8 = 1 and 3 in the case h 2 = 0 in view of the shapes shown in fig. 2. It is noted that for 6 <0.01 the shape of the TSL curve is equal to the one for 6 = 0.01 (see fig. 2). For each curve with a specific set of values 6, s, w and E, a realtive error c, was calculated according to C
=
(Em
—
Er)/Er
(57)
‘
where Em is the trap depth determined with a certain method and Er is the trap depth used in the numerical calculations. Except for methods 25 and 31 we found that did not strongly depend on the values of s, wand E. Therefore we also calculated an averaged value ~ of the relative error in Em for each value of 6, and its standard deviation ~6 according to
s
wE
s
~wE~
=
(58) -
~
(59)
calculated from (57) for 36 TSL or TSC curves that have been calculated for the mentioned values of E, s and w. The value of and its standard deviation can be considered as a measure for the correctness of a method for one value of the retrapping ratio 6. The summation is carried out over all values of
20
P. Knits, H.J.L. Hagebeuk
/ Evaluation
of the model for TSL and TSC
Table 2 The averaged relative error t/~(%) and standard deviation °b 1%) of the trap depth calculated with literature methods applied on numerically calculated 1St. curves for three values of the retrapping ratio S in the case h 2 = 0. The methods marked with a (*) yield a value br the trap depth which deviates less than 5% from the correct one. The sign is given in those cases where the error is systematic No.
4 5 6 8 9 29a 29b 29c 31
Author(s)
Retrapping ratio ~
General class methods
0.01
Hoogenstraaten Bube, 1-Jacring and Adams Unger Boyko, Rashba and Trofimenko 1 Boyko, Rashba and Trofimenko 2 Garlick and Gibson/IS Garlick and Gibson/3D Garlick and Gibson/SO Voigt (lower limit)
1
3
t). 1 +0.25 +0.1 -6.0 —6.0 -2.7 —5.3 —10.0 28
(0.1 )* (0.08)* (0.02)* (0.8) (0.8) (0.l)* (0.2) (0.3) (28)
+0.25 -m-0.5 +0.2 —5.7 --5.9 ~4.5 —7.3 —13.7 28
(0.1 5)~ +0.7 (0.2)~ (0.1)* +1.0 (O.2)* (0.04)* +0.3(0.1)* (0.7) —5.5 (0.7) (0.8) —5.5 (0.8) (0.1)~ -8.5 (0.1) (0.2) —11.7 (0.2) (0.4) —20.8 (0.4) (28) 28 (28)
+7.9 +2 5 0.5 2 2 I 0.5 —5 --12 19 +6.0 —41 I -0.5
(1.0) (l)* (13) (0.8)* (2)* (4)* (2)r (l.4)* (3) (3) (38) (0.7) (10) (0.61* (0.41*
+8.5 -42 — 20 —32 -—30 --18 —IS --43 3 —6.0 19 10 --72 —15 ---16
(1.4) (3) (14) (1) (2) (5) (4) (2) (3)* (0.3) (38) (1) 116) (1) (2)
(0.1)*
0.05 7 +17 +80 0.1 0.5 0.5
(O.03)* (18) (5) (5) (0.1)* (5)* (4)*
First class methods 2 10 ii 13 14 15 16 17 18 18 25 26 27 28 30
BOer, Oberhinder and Voigt Luschik 1-lalperin and Braner 1 (Then 1 (‘hen 2 (‘hen 3 (‘hen4 (‘henS (‘hen6a (‘lien 6b Randall and Wilkins Grossweiner Pranks and Keating Dussel and Bube Sandomirskii and Zhdan
+9.0 (1.4) --56 (4) --48 ---45
(1) (2)
--33 —57 —15 —2! 19 --27 —80 32 —33
(7) (4) (9) (9) (38) (3) (20) (3) (2)
Second class methods 3 19 20 21 22 23 24
(‘lien and \~‘iner Halperin and Braner 3 llalperin and Braner 4 ltalperin amid BranerS (‘hen 7 (‘hen 8 (‘lien 9
—0.3 18 +103 +180 +44 +20 +73
(15) (3) (5) (0.1) (2.5) (2)
+0.7 (O.l)* —12 +40 - 24 —22 )5
(0.7) (3) (0.6) (0.7) (I)
P. Kivits, HI L. Hagebeuk / Evaluation of the modelfor TSL and TSC
21
Table 3 The averaged relative error c,~(%) and standard deviation 0~ (%) of the trap depth calculated with literature methods applied on TSL or TSC curves for three values of the retrapping ratio b in the case h 2 >> h1. The methods indicated with a (*) yield a value for the trap depth which deviates less than 5% from the correct one. The sign is given in those cases where the error is sys-
tematic. The results of the methods that are not given do not differ more than 1% from the values shown in table 2 No.
Author(s)
Retrapping ratio S
First class methods
0.01
1
3 —31 (1.4) —34 (2) —32 (1) —33 (1.4)
10 13 14
Luschik (‘hen 1 (‘hen 2
+l.7(O.1)~ 0.7 (1)* 1.8 (2)*
—20(0.7) —20 (2) —19 (2)
17 18
(‘hen 5 (‘hen 6a
—0.8 (0.1)* —7 (3)
—22 (0.7) —17 (8)
18
Chen 6b
—13
(3)
—23 (7)
—36
27
l’ranks and Keating
—41
(10)
—57 (16)
—61
(6) (5) (20)
+37 +90
(3) (1)
—31
Second class methods
20
Halperin and Braner 4
+103
21 22 24
I-IalperinandBraner5 (‘hen 7 (‘hen9
+177 +43 +73
(1) (3)
+60 (2) +123(2)
(1) (2)
+16 (1) +37(2)
—2.9 (1)* +17 (2)
The results for TSL are shown in table 2 (h2 = 0) and table 3 (h2 >> h1). In the
-
latter table only those methods are given that yield a value of the trap depth that differs more than 1% from the value in the case h2 = 0. As could be expected from our finding on the shape of the TSL curve when h2 >> h1, all these methods without exception make use of the value of T’2. The results for TSC in the case h2 = 0 are shown in table 4. When h2 >> h1 table 3 can be applied. For those methods not
listed in table 3, the results in table 2 must be used. For most of the results the sign of calculated according to (57) appears to be the same for all values of H, s and w in one retrapping case. This is also indicated in the tables. The methods marked with a (*) yield a trap depth which deviates less than 5% from the value used in the numerical calculations. In both the TSL and the TSC cases the methods of Hoogenstraaten (4), Bube et a!. (5) and Unger (6) produce the best value of the trap depth although the latter is devised to give a lower limit for the trap depth. Also the method of Chen and Winer (3) turns out to be valid when 8 = 0.01, which was not predicted by the authors. The methods of Boiko et al. (8,9) and Boer et al. (2) can easily be corrected in such a way that they become independent of the value of 8. In the latter case the correction factor p should be above the range proposed by the authors. The initial rise method of Garlick and Gibson (29) turns out to be more unreliable
22
P. Kivits, H.I L. Hagebeuk / Evaluation of the model for TSL and TSC
Table 4 The averaged relative error k~(%) and standard deviation o~(%) of the trap depth calculated with literature methods applied on numerically calculated TSC curves in the case h
2 = 0. The methods marked with (*) yield a value for the trap depth which deviates less than 5% from the correct one. The sign is given in those cases where the error is systematic No.
4 5 6 8 9 29a 29b
29c 31
Author(s)
Retrapping ratio b
General class methods
0.01
Hoogenstraaten
0.1
Bube, Haering and Adams
—0.3 (O.2)* +0.07 (O.05)*
Unger Boyko, Rashba and Trofimenko 1 Boyko, Rashba and Trofimenko 2 Garlick and Gibson/iS Garlick and Gibson/30
—6.5 —7 —0.11 —0.6
Garlick and Gibson/50 Voigt (lower limit)
—1.7 (0.t)* 25 (62)
0.01 (0.O4)* (2.5) (2) (0.005)* (0.02)*
+0.09 (0.05)* +0.56 (0.O3)*
+0.1 (0.005)* —8 —8 —0.67 —2.5
(2) (2) (0.O1)* (0.06)*
—5.3 (0.08) 12 (38)
First class methods
2 10
Böer, Oberländer and Voigt Luschik
13
Chen 1
14 16 17
Chen 2 (‘hen 4 Chen 5
+4.9 (4.0)* +41 (6) —19 (2)
—70
(4)
18
(‘hen Ga
+68
(2)
+74
(0.1)
18
(‘hen Gb
+51
(1)
+63
(0.1)
25 26
RandaliandWilkins Grossweiner
21 +47
33 +14
(31) (1)
28
Dussel and Babe Sandomirskii and Zhdan
+37 +28
(84) (1) (5)
30
+8.5 (2.5) —17 (5) +6
(2)
(1)
+11.5 (0.2) —69 (4) —55
(3)
49
(3)
+5.6 (0.4)
+6
(2)
+5.5 (0.4)
Second class methods 3 21
Chen and Winer 1-lalperin and Braner 4 Halperin and Braner 5
—0.6 (0.4)* +67 (10) +68 (13)
—0.38 (0.06)* —38 (8) +43 (9)
22 23
Chen 7 Chen8
+53 ÷67
(4) (7)
—31 +25
(5) (1)
24
(‘hen 9
+42
(9)
—47
(6)
20
for TSL as the slope is calculated nearer to the maximum. In fact only the shape of the TSL curve below 1 5% of the maximum intensity can be used to calculate an accurate value for the trap depth. In the TSC case the reliability range is larger. The
systematically negative value of the error was already predicted by Hofmann [43] who used this fact to develop the fractional glow technique.
P. Kivits, HJ.L. Hagebeuk / Evaluation of the model for TSL and TSC
23
The methods of Voigt (31) and Randall and Wilkins (25) turn out to be inaccurate. These methods are independent of the value of the retrapping ratio but depend strongly on the frequency factor and the heating rate. To obtain a rough estimate for the trap depth we propose E = 27kTm for TSL and H = 2SkTm for TSC. This yields a value with a maximum deviation of 30%. The TSL methods of Chen, applied on TSL curves, give an accurate value forE if 6 = 0.01 (13,14,15,16,17) and if 6 = (22,23,24) when h 2 = 0. Applied on TSC curves these methods lead to incorrect results. In the case where h2 >> h1 the methods 22 and 24 become less accurate for TSL if 6 = 1 although it might be possible that they are correct for values of the retrapping ratio other than those discussed here. The method Chen (18) derived fot TSC gives an inaccurate value if apllied on TSC curves. A better result is obtained in the TSL case if 6 = land h2 =0. The original method of Grossweiner (26) did not result in a correct value for the trap depth in the cases investigated. The data of table 2 suggest that this method may be correct for a value of 6 in the order of 0.1. The correction made by Dussel and Bube (28), although derived for TSC, is more successful in the TSL case if 6 = 0.01. The correction from Sandomirskii and Zhdan (30) yields a good estimate for TSL if 6 = 0.01. The expectation of the authors that the corrected expressions would be generally valid is not confirmed. The formula of Franks and Keating (27) turns out to be incorrect in the cases investigated. The value of x for the calculated TSL curves is 0.74 ±0.02 for 6 = 0.01 in both cases h2 = 0 and h1, 1.10 ±0.04 for 6 = 1 when h2 = 0 and 0.79 ±0.04 when >> h1. Even in the case that the condition 0.75 3 when h2 >> h1. These cases were not included in our calculations. Another method of Halperin and Braner (21) might be correct when the retrapping ratio is very large.
5. Conclusion Many papers have been published on the theory of thermally stimulated luminescence and conductivity. Reviews of this field hardly exist. The methods which have been developed for the determination of the trap depth from TSL or TSC curves are generally applied without sufficient knowledge of whether they are reliable or not. In this paper we have indicated.the mutual connection between several expressions for the trap depth that exist in the literature. After application on numerically calculated TSL and TSC curves it is found that from the considered methods only those of Bube, Haering and Adams, Hoogenstraaten and Unger produce reliable values for the trap depth independent of the values of the frequency factor and the retrapping ratio. An estimate of the latter quantity can be made by comparing the trap depths determined with the TSL methods of Chen since these produce correct
P. Kim’its, H.J.L. Hagebeuk / Evaluation of the model for TSI. and TSC
24
results in the cases 6 = 0.01 or 6 = 1 for TSL. Hence, when the simple model of figure 1 can be used, TSL and TSC measurements are indeed a helpful tool for determining trapping parameters, in agreement with the statement of Bbhm and Scharmann [5]. One of the main results of our TSL and TSC measurements on the ternary compound CdGa 2S4 that will be published elsewhere [44], is that application of some methods mentioned in this paper on the measured TSC curves leads to calculated trap depths that indicate the same value of the retrapping ratio. Hence, one might conclude that this phosphor is a good example of a case where the simple model is applicable. However, this is certainly not the case for CdGa2S4 since the same methods applied on measured TSL curves yield different results. Summarizing we come to the conclusion that very much care must be taken in attaching value to quantitative results of trap depth calculations from TSL and TSC experiments if the model that can be applied is not known, in agreement with the conclusion of Kelly et a!. [6].
For a better understanding of the processes occurring in a semiconductor showing thermally stimulated luminescence or conductivity during heating, elaborate experiments on well-defined phosphors have to be carried out in connection with other measuring techniques.
Acknowledgement The authors wish to thank Wim Batenburg and Noud van Lieshout for most of the numerical work presented in this paper. We further greatly acknowledge Prof. Dr. F. van der Maesen, Prof. Dr. M.J. Steenland and Dr. A.T. Vink for numerous valuable comments on the manuscript.
Table 5 The magnitude of the terms in eq. (17) forE = 0.516 eV, s i’l —3 —6 —3 —m h10 eta ,a10 cm s andbO.Ol. 2 T
—da~/dt
(K)
(cm
1.25 x 1012
a~(I—S)+SIm
~l
w
0.1
X da~’7dt —6 —1
(cm
=
—(1 +b)/ua~)
~/o
~a~
—3 —m
s )
a’d2a4/dt2
=
s )
—6 —m
(cm
s )
—a
(cm
)
17o4.57x 180 190
1013 4.45 x i03° 7.4 x ~ X 1016 2.69 x i~’~2.18 x loam 3.1 x lO~ 8.1 x 1016 4.86 x 1014 1.30 x l0~’ —3.0 x i~’~ 2.7 x 1016
200 210
1.43 x i~’~2.09 x 1028 0.62 x 1012 0.64 x 1027
—6.Ox 1017 —1.0 x 1016
1.5 x iü’~ 1.0 x 1015
—3
—3
(cmn )
(cm
4.7 X 102 3.3 x ~ 1.9 x iø~
4.7
x
102
9.4x ~
3.4 x 1.9 x 3.1 x i04
3.9 x iO~
1.5 x 1O4
P. Kivits, H.IL. Hagebeuk / Evaluation of the model for TSL and TSC
25
Appendix It must be proven that (17) is adequately approximated by (20). We calculated numerically the magnitudes of the terms in (17) for all used sets of parameter values (section 4). As a representative example we give in table 5 the results for the same parameter values as used in fig. 2 and 6 = 0.01. We found that the terms with a~ in
(17) are several orders of magnitude smaller than the other terms and may thus be ignored. Some illustrative analytical proofs can also be given. Consider the denominator of (17).
(A) Firstly, we prove that ‘y/cx<<6h.
(60)
With (4) and (18) this can also be written as h/Ne >> exp[—E/kT],
(61) 3) <1020) in the relewhich is valid for normal dope concentrations (1014
-Q .~
10~
_____________________________________ I
I
C
1&0—11 10-
10-12
-
1013
—
iO1’
—
1o3
ro4
-
-
—15
I
I
10 170
180
190
I
200
210 ~ TOO
11g. 5. The ratio n/h versus temperature for the same parameter values as in fig. 2.
P. Kivits, !-LJ.L. Hagebeuk /Evaluation of the model for TSL and TSC
26
(B) We prove that —[(1
+
6)/cxa~]da~/dt <
First we consider the case 6
—~
+
(1
—
O)a~.
(62)
0. In that case applying (2) and (3), inequality (62) is
equivalent with
n<
(63)
This inequality, or rather n <<1v , is used by several authors to simplify the original differential eq. (1) and (2). It is difficult to prove the validity analytically (section 2.2). Therefore, we calculated numerically n/h as a function of temperature for different sets of parameter values. Some results are given in fig. 5. It is concluded that in the relevant temperature region n <
2n <
(64) /t
References [1] K.H. Nicholas and J. Woods, Brit. J. AppI. Phys. 15 (1964) 783. [2]H. Dittfeld and J. Voigt, Phys. Stat. Sol. 3 (1963) 1941. 13] B.S.F1. Barkhalov and E.L. Lutsenko, Phys. Stat. Sal. (a) 11(1972)433. 14] P.S. Walsh and P. Lightowlcrs, J. Luminescence 4(1971) 393. [5] M. Bohm and A. Scharmann, Phys. Stat. Sol. (a) 5 (1971) 563. [6]P. Kelly, M.J. Laubitz and P. Bräunlich, Phys. Rev. B 4 (1971) 1960. [7]H.J.L. 1-lagebeuk and P. Kivits, Physica 83B (1976) 289 [8] P. Kivits, submitted for publication. [9] M. Lax, J. Phys. (‘hem. Solids 8 (1959) 66. 110] R. (‘hen and R.J. Fleming, J. AppI. Phys. 44(1973)1393. [11] (‘.H. Henry and DV. Lang, Proceedings of the 12th mt. (‘onf. on the Physics of Semiconductors, Stuttgart, 1974, p. 411. 1121 J. Bardeen and W. Shockley, Plays. Rev. 80 (1950) 72. 1131 C. Erginsoy, Phys. Rev. 79(1950)1013. [14] F. Low and D. Pines., Phys. Rev. 98 (1955) 414. 115] F.M. (‘onwell and VU. Weisskopf, Phys. Rev. 77 (1950) 388. 116] I.E. Williams and H. Eyring, 3. Chem. Phys. 15 (1947) 289; I. Broser und R. Broser-Warminsky, Ann. Phys. (Germany) 16(1955)361; 1.3. Saunders,J. Phys. (‘2(1969) 2181; D. Dc Muer, Physica 48 (1970) 1; S. Onnis and A. Rucci, 3. Luminescence 6 (1973) 404. [17]G.F.J. Garlick and Al. Gibson, Proc. Phys. Soc. 60 (1948) 574. [18] R.11. Bube, J. (‘hem. Phys. 23(1955)18. [19] K. Unger, Phys. Stat. Sol. 2 (1962) 1279. 1201 W. Hoogenstraaten, Philips Res. Rept. 13 (1958) 515. 1211 R. (‘hen, and S. Winer, J. Appl. Phys. 41(1970) 5227. [22] 1.1. Boiko, ITt. Rashba and A.P. Trofimenko, Soy. Phys. S.S. 2 (1960) 99.
P. Kivits, H.J.L. Hagebeuk / Evaluation of the model for TSL and TSC
27
[23] K.W. Biter, S. Oberländer und J. Voigt, Ann. Phys. Lpz. 2 (1958) 130. [24] A.H. Booth, Canad. J. Chem. 32 (1954) 214. [25] A. Bohun, Czech. J. Phys. 4 (1954) 91. [26] l.A. Parfianovitch, J. Exp. Theor. Phys. S.S.S.R. 26 (1954) 696. [271 R.R. Haering and E.N. Adams, Phys. Rev. 117 (1960) 451. [28] M. Schitn, Tech. Wiss. Abh. Osram Ges. 7 (1958) 175. [291Ch.B. Luschik, Dok. Akad. Nauk S.S.S.R. 101 (1955) 641. [30] A. Halperin and A. Braner, Phys. Rev. 117 (1960) 408. [31] R. (‘hen, J. AppL Phys. 40 (1969) 570. [32] R. (‘hen, Chem. Phys. L. 11(1971) 371. [33] R. Chen, Chem. Phys. L. 6 (1970) 125. [34] J.T. Randall and M.H.G. Wilkins, Proc. Roy. Soc. A 184 (1945) 366. [35] F. Urbach,Winer Ber. ha 139 (1930) 363. [36] L.I. Grossweiner, J. Appl. Phys. 24(1953)1306. [37] P.N. Keating, Proc. Phys. Soc. 78 (1961) 1408.
[38] J. Franks and P.N. Keating, J. Phys. Chem. Solids 22 (1961) 25. [39] G.A. Dussel and R.H. Bube, Phys. Rev. 155 (1967) 764. [40] C. Haake, J. Opt. Soc. Am. 47 (1957) 649. [411V.B. Sandomirskii and A.G. Zhdan, So!. St. Electr. 13 (1970) 69. [42] J. Voigt, Diplomarbeit Berlin (1958). [43] D. Hofmann, Thesis Berlin (1967). [44] P. Kivits, 1. Reulen, J. Hendricx, F. van Empel and J. van Kleef, submitted for publication.
EVALUATION OF THE MODEL FOR THERMALLY STIMULATED LUMINESCENCE AND CONDUCTIVITY; RELIABILITY OF TRAP DEPTH DETERMINATIONS P. KIVITS
*
and H.J.L. HAGEBEUK
Department of Physics, Eindhoven University of Technology, Eindho yen, The Netherlands Received 9 December 1975 Revised manuscript received 12 February 1976
A review of the basic theory on thermally stimulated luminescence (TSL) and conductivity (TSC) based on a certain simple model is given. Approximate analytical expressions for the shapes of the TSL and TSC curves are derived. Methods from the literature for trap depth determination, some of which can easily be derived from these expressions, are applied to numerically calculated TSL and TSC curves, It turns out that the methods of Bube, Haering and Adams, l-Ioogenstraaten and Unger yield a correct value for the trap depth independent of the values of the retrapping ratio and the frequency factor. The method of Garlick and Gibson is reliable if applied below 15% of the maximum TSL intensity. Some methods, as most of those derived by Chen, yield a correct trap depth for a specific value of the retrapping ratio in the case of TSL only.
I. Introduction Thermally stimulated conductivity (TSC) is the phenomenon that the conductivity of a semiconductor is temporarily enhanced when heated in the dark after excitation at a low temperature. Simultaneously light can be emitted which is known as thermally stimulated luminescence or thermoluminescence (TSL). The TSC and TSL curves which are plots of the thermally stimulated conductivity and luminescence intensity versus temperature, respectively, are often used for determination of the thermal energy depth of trapping levels in the forbidden gap of a semiconductor. In the course of time quite a number of methods have been published to realize this objective. Some methods were developed for TSC, others for TSL. In practice, however, many of the methods are applied to both curves without proving that this is correct. In several papers [1—4]the results of some of these methods are compared experimentally. It was found that different methods do not produce the same trap depth for a given TSL or TSC peak. The real trap was not known in these in*
Present address: Philips Research Laboratories, Eindhoven, The Netherlands.
2
P. Kivits, H.J.L. Hagebeuk / Evaluation of the modelJbr TSL and TSC
vestigations, however. Some workers [5] conclude in their papers that TSL and TSC measurements are a helpful tool for the determination of trapping parameters. Others [6], however, state that no conclusion can be drawn from such experiments without previous knowledge about, for instance, the defect structure of the phosphor. In almost all theories on TSL and TSC the same model is used (section 2). From this model we shall derive expressions for the TSL and TSC curves and from these some formulae for the trap depth in terms of measurable quantities (section 3). A survey of all currently used methods will be given. With a numerical procedure described by Hagebeuk and Kivits [7] TSL and TSC curves can be calculated for various sets of parameters. Nearly all methods for trap depth determination in this paper are applied to these curves. When the trap depth obtained in this way is compared with the value used in the numerical calculations a statement can be made about the reliability of the methods (section 4). The validity of the model itself will not be discussed here. In a separate paper this will be done by extending the model with some additional properties, as recombination via excited states, trap distributions, donor—acceptor pair recombination, cross sections that depend exponentially on reciprocal temperature, thermal quenching of luminescence and a variation of mobility [8].
2. Basic theory for thermally stimulated luminescence and conductivity 2.1. The mathematical model The physical model which one generally uses to describe the TSL and TSC processes is shown in fig. 1. Electrons are thermally excited from trap h1 to conduction band (transition 1) where they contribute to the conduction. From this band they can either be (re)trapped (transition 2), or recombine with a hole trapped at a recombination centre a (transition 3). The latter transition may be accompanied by the emission of light. In this model, the time derivatives of the concentration of free and
conduction band n
h2, h2 a~a vaLence band Fig. 1. Energy level scheme forming the model which is often used for the interpretation of thermally stimulated luminescence and, conductivity experiments.
P. Kivits, H.J.L. Hagebeuk /Evaluation of the model for TSL and TSC
3
trapped electrons are given by the following equations dhj7dt
=
dn/dt
—yhj =
+
13n(h1
—dhj7dt
—
—
hj)
(1)
ctna~
(2)
The following symbols are3);used: E: trap depth (eV);ofh1electrons h2: concentration hj h~:concentration trapped at hof the two trapping (cm— 1 and 3); n:levels concentration of free electrons (cm3); a: concentration of recomh2 (cm centres (cm_3);a+: concentration of holes trapped at recombination centres bination (cm3) *; 13: trapping rate constant (cm3 s~);cv: recombination rate constant (cm3 s1); y: transition probability for electrons from trap to conduction band (s1). It is assumed that the occupation of h 2 does not change during the emptying of the shallower trap. Both traps are then called “thermally disconnected”. The consequences of this assumption are discussed later. Eqs. (1) and (2) govern the process of emptying h1 as a function of time (t). During this process the condition of charge neutrality requires that ,
,
a~=n+hj+h~,
(3)
if no free holes are present. The transition probability y is given by y
=
s exp[—E/kTJ,
(4)
in which k is Boltzmann’s constant and T the absolute temperature. Between the pre-exponential factor s, which is called the frequency factor because of its dimension, and 13 the following relation exists (5) where N~is3/2) the and effective density of states of the conduction band (usually proporg is the degeneracy factor of the trap (taken equal to 1 in the tional to T The coefficients a and 13 are given by following). XSaV,
(6)
13=Shv,
u is the mean thermal velocity of free electrons (proportional to T112) and Sa and Sh are the cross sections for capture of electrons by recombination centres and traps, respectively. According to Lax [9] capture cross sections of deep centres are proportional to T~°~ (m 2.5 or 4). Chen and Fleming [10] assume a range 0 ~ m ~ 4. Following the latter authors we obtain with (5) and (6) that where
s=s 0(T/T0)b
,
a=cw0(T/T0)c, *
—2~b~2
(7)
—7/2~
(8)
,
Here Klasens’ notation has been used, 11.A. Kiasens, J. Phys. Chem. Solids 7 (1958) 175. By h is meant that the centre has trapped an electron, a~means that the centre has lost an electron.
4
P. Kivits, H.J.L. Hagebeuk /Evaluation oft/ic model for TSL and
TSC
where s
0, a0 and T0 are constants. Front (7) and (8) it follows that generally s and cv are rather slowly varying with temperature compared to the exponential increase of ‘y, when L’>> kT as is always the case. Therefore it is often assumed that s and a are independent of the temperature. In our calculations we shall consider both cases. The influence of an exponential dependence with reciprocal temperature of Sa and S1, as is found by Henry and Lang [11], is discussed elsewhere [81 and will not be considered here. The integrated luminescence intensity defined as the miumber of photons emitted by the phosphor per unit time interval is proportional to the change of the occupation of the recombination centre, da~/dt.The conductivity u,~is proportional to the number of free electrons, according to =
neji,~
(9)
312 where is the driftbetween mobilitycharged for electrons. mobility is vibrations proportional to T for thep,1interaction carriers The and acoustical in non-ionic crystals [12]. For scattering by neutral impurities the mobility is independent of temperature [13]. An exponential decrease of p 03/2with T isthe found for are polaron scatwhen carriers scattered tering by optical Further by charged centresphonons [15]. In[14]. the latter casep,~ the Tmobility also depends on the concentration of charged defect centres which may be of importance for the thermally stimulated conductivity where the number of charged centres often increases with temperature [8]. In practice the value of a,~generally cannot be measured using an experimental arrangement with only two contacts on the sample. However, since the evaluation of a,~under transient conditions existing during the presence of a thermally stimulated current from a four probes method as described by Van der Pauw [16], meets with rather complicated experimental problems, one has to realize that contact properties and the formation of space charge may influence the results ‘i’. Since the T-dependence of p, 1 differs for different phosphors we shall restrict ourselves in this paper to the case that p,~is independent of temperature. We assume that at T0, the temperature where the heating starts, hj(T0) = h~(T0)= h2, a’(T0) = h1 + h2 and consequently n(T0) = 0. Finally, we need a relation between Tand t since we are mainly interested in solutions with temperature rather than with time. The heating rate w is given by w
=
dT/dt
.
(10)
Since in experiments usually a constant heating rate is chosen this means that
T=T0+wt. *
(11)
Soinctinies space charge is deliberately created by applying high electrical Odds. Such an cxperimental arrangement can lead to model simplifications since the recombination lifetime may be assumed to be constant (see also section 2.2).
P. Kivits, H.J.L. Hagebeuk I Evaluation of the mode/for TSL and TSC
5
An analytical solution for da’7dt and n from the non-linear differential eqs. (1) and (2) together with (3), (4), (11) and the initial conditions has not been found up to now. Several simplifications have to be introduced to obtain approximate expressions for the TSL and the TSC curve.
2.2. Simplifications of the kinetic’ equations In this section we shall briefly discuss assumptions and approximations that are used in literature in order to simplify the system (1, 2). One currently assumes that often
(dn/dt( << dhldtl
(12)
.
From (3) it can be seen that this implies that the luminescence intensity only depends on the change of the occupation of h1. It is obvious that this will be correct if the recombination lifetime is sufficiently small. This lifetime r is defined by (c.f. eq. (2))
r1/aa~
(13)
.
Hence, r increases with T since a~ decreases. This means that the assumption becomes more unreliable for higher temperatures. If, however, it is additionally assumed that h2 >> h1,hericea~ h~= h2, the recombination lifetime will be short and nearly constant during the emptying of the shallower trap. Eq. (2) then reduces to a linear differential equation. —
Another simplification often used is
n<
(14)
Kelly et al. [6] concluded that approximate solutions based on this assumption “must break down past some definite higher temperature”. However, in our opinion this temperature is not in the region of interest. With (1) it follows that ~dhj1dt+7hj hj
13hj(h1—hfl
<____1_____
(15)
,
cl(h1—hfl
since dhj7dt <0. For higher temperatures h1 still increases. Thus, we may write n/hj
—
hj will be nearly constant while y
(16)
where (4) and (5) are used. In the maximum of the TSL curve E 27 kT (section 4). Hence in normal phosphors the assumption (14) will be justified. In the appendix a numerical example of the ratio n/hj as a function of temperature is given. The model given by (1) and (2) involves essentially a second order mechanism for luminescence independent of the value of 13/a. Nevertheless, first order kinetics can be approximated by taking 13 <
6
P. Kivits, NfL. Hagebeuk /Evaluation of the ,nodelfbr TSL and TSC
tunately, in thermoluminescence literature one often takes j3 0 which does not have any physical relevance as will be seen in section 2.3. As a special case of second order kinetics one usually takes ~3= a, for reasons of mathematical simplicity.
The methods that were originally derived for each of these cases, also known as mono- and bimolecular recombination, or slow and fast retrapping, are called first and second class methods in this paper; when no specific value of the ratio 13/a is assumed the term general class methods is used.
2.3. Approximate expressions for the TSL and TSC curves Using the simplifications discussed in the previous section it is possible to derive approximate expressions for da~/dt(fland n(7) from (1) and (2). However, we shall not use the presence of a thermally disconnected trap since this, together with the assumption
h2 >> h1,
would lead to a too far-reaching simplification of the mathematics. Further we shall drop the suffix of h1 for practical reasons in this and the next section. The set of non-linear differential equations of the first order can be rewritten as
one differential equation of the second order. We obtain 2+ a1 d2a~/dt2 = ‘y(a~) dt a~(l 6)+óh+a’ [y—(l +ö)(a~)~da~/dt ‘
17
(
where 6, the retrapping ratio, is given by 6f3/a.
~l8)
Since the luminescence intensity is proportional to da~/dt,the TSL curve can be numerically calculated from (17) for different values of H, s, w, 6,h and a [7]. From (2) and (3) it follows that n
—(l/aa~)da’]dt
(19)
which makes it subsequently possible to calculate n(fl. As an illustration of the following calculations we show some TSL. and TSC curves for specific sets of parameters in fig. 2.
It can be shown (see appendix) that for normal dope concentrations and in the relevant temperature region, adequate approximations for da’]dt and n are given by da~/dt= —y(a~)2/[a’~(l
—
nya~/a[a~(J—8)+OhJ
6) +
.
oh]
,
(20)
(21)
It should be noted that the same equations are obtained from (1), (2) and (3) when the simplifications (12) and (14) are used. Following the literature we shall now distinguish between the cases 6 = 1 and
P. Kivits, N.J. L. Hagebeuk / Evaluation of the modelfor TSL and TSC I
8—
‘D
I
I
7
I (0
TSC
7
fo
(0
~ 6
TSL
4—
-
ib~ 3-
—
2-
-
170
180
190
200
210 ~T(K1
2s1 w = 0.1 Ks’, h = iO’7 Fig. 2. The TSL and TSC curves forE = 0.516 eV, s = 1.25 X 10’ = iO~cm3 s~,and different values of b. The factor 1/6 along the axes is introduced to reduce all curves to the same scale.
6 << 1. When 6
=
1, eqs. (20) and (21) can be simplified to
da~/dt= —‘y(a~)2/h
(22)
n
(23)
,
=
ya~/cvh.
From fig. 2 it is seen that when 6
=
1, n(T) shows only a very smooth maximum.
This is usually not measured. It might mean that 6 ~ 1 represents a situation that
hardly occurs in phosphors, but we believe that it is more probable that the model under consideration is too simple. When this model is extended with thermally disconnected trapping centres in a relatively low concentration the long high temperature tail of the TSC curve vanishes rapidly [7,8]. In the case 6 << 1, eqs. (20) and (21) tend to
dr
—aiVc 6a~exp[—E/kT] 1 +
(24)
8
P. Kivits, H.J.L. Hagebeuk/Evaluation of the model for TSL and TSC 4
I
0
-2
170
I
I
1
1
I
180
190
200
210 ~ T(K)
4/th versus temperature for the same parameter values as in fIg. 2.
lig. 3. The ratio a
and nN
C
Oexl~L~7~.fl l+6h/a~
(25)
where (4), (5) and (18) have been used. To acquire insight in the behaviour of the nominators of (24) and (25) the value of a~/6h as a function of temperature is shown in fig. 3 for the same parameter values as in fig. 2. Since a~>> Oh over a relevant part of the temperature region, the TSL curve can almost fully be represented by da~/dt= —aON~a’~ exp[—E/kTI
.
(26)
A similar simplification of (25) can only be made in the initial part of the TSC curve since for higher temperatures a+ and Oh are of the same order of magnitude. As already noted, one sometimes takes 6 0. From (24) and (25) it is clearly seen that in this case the TSL cruve as well as the TSC curve vanish. Physically this means that the trap in fact cannot be filled.
In this section we derived approximate expressions for the TSL and TSC curves,
the validity ranges of which are unambiguously formulated. In the next section we shall use these expressions for the derivation of formulae on which some methods
for trap depth determination are’ based.
P. Kivits, H.J.L. Hagebeuk / Evaluation of the modelfor TSL and TSC
9
3. Methods for the determination of the trap depth In literature several expressions exist for the trap depth in terms of measurable quantities. These can be used for the calculation of the trap depth from TSC and TSL curves. The derivation of some of these expressions as presented by their authors, sometimes is very cumbersome. Moreover, assumptions are made which are not obvious. However, it is possible to derive some of the formulae for the trap depth in an appropriate way from the approximating eqs. (20) and (21)as will be shown in section 3.1. Current methods for trap depth determination are summarized in section 3.2. An attempt is made to indicate the underlying principles. As we do not intend to derive or even to mention all methods, some others remain undiscussed. These methods, however, are generally less manageable and therefore not often used in practice [16]. 3.1. Derivation of expressions for the trap depth
From (20) and (21) it follows that (27)
2/[a~(1 —6) +ohl},
1n(—da~/dt)= —E/kT+ln~s(a~) Inn
—E/kT+ln{sa~/a[a~(1—6) +oh]}
(28)
,
where (4) is used. In the initial part of the TSL and TSC curves where a~varies only slowly compared to the exponential increase of y with T one obtains from (27) and (28) that ln(—da~/dt)~~ —H/kr,
(29)
ln nmit
(30)
—E/kT,
since a~(init) h. Thus, a plot of ln(—da’7dt) or ln n versus l/T should yield a
straight line with slope —H/k in the initial part of both curves. This procedure is known as the initial rise method of Garlick and Gibson [17].
Some methods for trap depth determination use the variation of the maximum or the temperature at which this maximum occurs with the heating rate. To acquire insight in the validity of the expressions on which these methods are based we calculated for different heating rates the values of dat/dt, at, TL and nc, a~, T~where the indices Land C stand for the TSL and TSC maxima, respectively. As an example we give some results in table I. It is seen that dat/dt and nC depend strongly on the value of w, contrary to at and a~. We may therefore assume the logarithmic terms at the right hand sides of (27) and (28) to be constants in a first approximation. It then follows that at the TSL and TSC maxima intensity
—
—
ln(—dat/dt) hi nC
=
—E/kTL
E/kTc
+
+ Ci(at),
(31)
C 2(a~),
(32)
P. Kivits, NIL. Hagebeuk / Evaluation of tile model for TSL and TSC
10
Table 1 Some characteristic values of the TSL and TSC curves with E and 6 = 0.01 for different values of the heating rate w
=
0516 eV, s
=
1.25 X 1012 s_I
TL (K)
—h~da~jdt
aL/h
Tc (K)
iic/h
ak/h
l0_2 10_I
175.9 187.9 201.5
6.0 x i0~
183.6 196.6 211.6
10
217.2
0.395 0.396 0.397 0.398
4.4 X 10~ 3.8 x i0~ 3.3 x l0_12 2.9 X lO~
0.0225 0.0223 0.0221 0.0219
1)
w (Ks
where
C
1
5.2 x i0~ 4.6 x 10_2 4.0 x 10_i
228.9
C2 are (nearly
constant) functions of at and a~,respectively. Thus, a l/Tc, respectively, for various heating rates, should give a straight line with slope —E/k. This method for trap depth determination was originally derived by Bube [18] for TSC. It is now found that it can also be applied for TSL independent of the value of the retrapping ratio. It is obvious that from (27) and (28) also equations can be derived for any specific temperature. For instance for the temperature T1 at the lower temperature side of a TSL curve, where the luminescence intensity equals half its maximum value. We obtain from (27) that and
plot of the value of In(_dat/dt) or In nC versus I /TL or
ln(—dat/dt)
=
—E/kT1 + C1(a~)+ In 2
(33)
where a~is the concentration of trapped holes at T1. Since also aI will not strongly depend on the value of w, a plot of ln(—dat/dt) versus 1 /T1 for various heating rates,
should yield a straight line with a slope _E/k. Unger [19] derived that the method of Bube which is based on (31) produces an upper limit for the value of the trap depth and that the method based on (33) yields a lower limit. In this paper we shall refer to the latter as the method of Unger. In the TSL maximum (d/dfl(dat/dt) = 0 and in the TSC maximum dnc/dT = 0. When differentiating (20) and (21) with respect to temperature we must take into account that s and cv depend on temperature as given by (7) and (8), respectively. We shall distinguish between (a) b = 0, c = 0 and (b) b ~ 0, c ~ 0. (a) The case b = 0, c = 0. When 6 <
wE/kT~ sexp[_E/kTL].
(34)
Hence, a plot of ln (w/T~)versus 1 /TL should yield a straight line with a slope —Elk. This procedure is known as the method of Hoogenstraaten [20]. He proved that this method is also valid for ~ = 1. After differentiating (22), derived for 6 = 1, with respect to T, it follows with (5) that in the TSL maximum
wE/kT~= (2at/h)s exp [—LtikTL]
.
(35)
The only difference between (34) and (35) is the pre-exponential factor. As already was mentioned, the value of at does not strongly change for different values of the
P. Kivits, N.J.L. Hagebeuk / Evaluation of the modelfor TSL and TSC
11
heating rate and may readily be assumed to be constant. It is now obvious to define a parameter Seff, called the “effective frequency factor”, according to (36) This parameter is nearly a constant for a given trapping level. It should be noted that either s or Seff,L can be determined by extrapolation to I /TL = 0 from the plot ln(w/T~)versus I /TL as follows from (34) or (35). Hoogenstraaten derived his method for TSL but it is possible to prove analytically that this method is also correct for TSC in the case 6 = 1. From (4) and (23) it follows that in the TSC maximum 5eff,L
=
(2at/h)s.
wE/kT~= 5eff,C exp[—E/kTc],
(37)
with Seff C
=
(a~/h)s.
(38)
Thus, a plot of ln(w/T~)versus 1 /T~should yield a straight line with slope —E/k since 5eff,C may be assumed to be essentially constant. Since except for the initial rise no simple expression for the TSC curve exists when 6 << 1 (section 2.3) it is not easy to prove analytically that Hoogenstraaten’s method yields also a correct value~5iThe~trap depth in this case, as follows from our numerical results in section 4. Chen and Winer [21] derived a method for TSL which produces according to the authors, a correct value of the trap depth independent of the value of 6. The expression which they derived in the case 6 = 1 is 1n[_(dat/dt)k2T~/w2E2] ~E/kTL
+
C 3(at).
(39)
It is easy to derive (39) from (31) and (35) taking 5eff,L (40) C3(at) = C1 (at) 2 in It may thus be concluded that (39) is valid for TSL in both cases 6 = I and 6 << 1. Moreover, it is valid for TSC since it was proved above that the methods of Bube and Hoogenstraaten yield a correct value of the trap depth in those cases. In general the trap depths calculated with these three methods are related according to E (Chen and Winer) = 2E (Hoogenstraaten) E (Bube). (b) The case b ~ 0, c ~ 0. When b ~ 0 and c ~ 0 it is easy to prove that (20) and (21) are also valid. It can further be derived from (4), (7), (8) and (21) that independent of the value of 8 ‘
—
—
In(n/T~c) = —E/kT+ C 4(a~) ,
(41)
where
C4—soT~~a~/a0[a~(l —6)+6h]. Boiko et al. [221 assumed b c = in the TSC maximum and came to 2)= -E/kTc + C ln(nc/Tt~’ 4(a~)
(42)
—
(43)
12
P.
Kivits, Hf. L. Hagebeuk
/ Evaluation of tile model for
TSL and
TSC
Hence, the plot of ln(nc/T~’2)versus l/T~far various heating rates should yield a straight line with slope —E/k independent of the value of 8. We shall refer to this procedure as the first method of Boiko et al. Another method from the same authors derived for the case 0 = 1 is obtained from (4), (7), (8) after differentiating (23). It then follows that in the TSC maximum wE/kT~~2 — w(c
—
b)/T~1 (a~/h)(so/T8)exp[—E/kTcj.
(44)
where (22) has been used. As will be shown in section 4, a rough estimate for the value of the trap depth can be obtained as H 25 kT~.In that case we can neglect the second term at the left hand side of eq. (44) in a first approximation. We then obtain wE/kT~2
(a~/hT~)s 0exp[—E/kT~].
(45)
which is equivalent with (37) when b = 0. The second formula of Boiko et al. [22] can now be obtained taking b = ~-. Hence, 2 = (a~/hT~2)s wE/kT~V exp[—E/kT~1. 0 (46)
The authors state that this method also leads to correct determination of the trap depth in the case 8 << 1. Another method based on (45) is the one of Böer et a!. [231.They obtained wE/k (a~T~/h)s0 exp[—E/kT~],
(47)
which follows from (45) by substituting b = —2. Originally this method was derived for 8 << 1. The authors estimated that when the method is applied on experimental results the calculated trap depth is about 20% too high.
We shall now prove that for TSL expressions similar to (46) and (47) can be obtained. From (22) and (26) it can be derived that 1)(E/kTL + b) s’ exp[—E/kTLI, (48) (w/Tt~ where s’ = so/T8 when 6 <<1 and s’ = (2at/hTob)so when 8 = 1. Neglecting b with respect to E/kTL we come to the equivalents of (46) and (47) in the TSL case. For b it follows that ‘=-~
7/2
WE//CTL and for b = —2,
s exp[—b/kTL},
wE/k—s’exp[--E/kTL],
(49)
(50)
from which it is obvious that both methods yield the same results for TSL and for
TSC. In this section we derived some expressions for the trap depth in terms of measurable quantities. It has been proven that methods originally derived for TSC can also be applied in the TSL case and vice versa, which was not predicted by the original authors. However, these are not the only methods for trap depth determination
P. Kivits, H.J.L. Hagebeuk/Evaluation of the model for TSL and TSC
13
from TSL and TSC curves. In section 3.2 we present a list of 31 methods that can be
used for this purpose. 3.2. A survey of methods from the literature
In the previous section we derived several formulae for the trap depth. Most of the methods based on these expressions use the shift of the TSL or TSC maximum with different heating rates. However, this is not the only way of determining a trap depth from TSL or TSC curves. In this section we give a survey of currently used
methods to realize this objective. For completeness, those methods mentioned in 3.2 are also included. If possible the conditions under which the methods according to the authors are valid are given. It is indicated whether they were originally derived for TSL or TSC. The methods are roughly divided into three groups: (A) Methods making use of heating rate variation. Most of them have been discussed in section 3.1; (B) Methods making use of geometrical approximations; (C) Other methods. Additionally they are ordered in first, second and general class methods (section 2.2).
(A) Methods making use of various heating rates. The following symbols are used: w,: the heating rate in the ith experiment;Lmi: the intensity of the TSL maximum; nmi: the maximum of the TSC curve; Tmj: the temperature at the TSL or TSC maximum; T11: the temperature at the lower side of the TSL or TSC curve where either L
=
~Lmjor n
=
(1) First class methods Method 1, Booth, Bohun and Parfianovitch (TSL) [24,25,26]
E= [kTmiTm2/(Tmi Tm2)] —
w1T,~2/w2T,~1).
Method 2, Böer, Oberldnder and Voigt (TSC) [23]. According to ref. [23] a plot of ln w1 versus IlTmi yields a straight line with a slope —pH/k with 0.7
(3) General class methods Method 4, Hoogenstraaten (TSL) [20]. A plot of ln(T~1/w~) versus 1/Tmj should yield a straight line with a slope E/k (section 3.1). Method 4 is a generalized method I. If one uses more than two heating rates method 4 is of course more accurate than
14
P. Kivits, H.J.L. Hagebeuk/Evaluation of the model for TSL and TSC
method 1. Therefore method 1 will not be included in our discussions. Method 5, Bube, Haering and Adams (TSC) [18,27]: A plot of ln ~mj versus l/Tmi yields a straight line with a slope —E/k (section 3.1). Haering and Adams claim that this method is valid when E/kTm >> 1. An additional condition for larger values of 0 is: h/Ne >> exp(—E/kTm). Method 6, Unger (TSC) [191. Plotting hi nmj versus lIT11 yields a straight line with slope —E/k (section 3.1). Method 7, Schän (TSC) [28]. E= [kTmiTm2I(Tmi
-
Tm2)]
wiT7~/w2T~?)
Method 8, Boiko, Rashba and Trofimenko 1 (TSC) [22]. The plot of ln(nmi/T,~/~) versus l/Tmj yields a straight line with a slope —Elk. Method 9, Boiko, Rashba and Trofimenko 2 (TSC) [22]. The plot of ln(w1/T/~) versus 1 ITmi yields a straight line with a slope —E/k. This method is in fact an extension of method 7. For this reason the method of Schön will not be included in our discussions.
(B) Methods making use of geometrical approximations In fig. 4 a general shape of a TSL or TSC curve is given. It can roughly be approximated by a triangle which is also shown. This principle has been used for the derivation of several expressions for the trap depth. Following the literature, from the
figure several parameters can be defined w’~’T2—T1,
XTm —T1, ~a/X,
l1g=U/W~
‘m
o
I
tJ=T2~Tm,
~i=E/kTm
-
/‘fl\ ~
__~L~ll_ T1
Tm
12
~T Fig. 4. General shape of a TSL or TSC curve with definition of some characteristic quantities.
P. Kivits, H.J.L. Hagebeuk / Evaluation of the model for TSL and TSC
15
As an illustration of the underlying principles of these methods we will now derive a formula forE in the case of TSL when 6 << 1. From (26) it follows that (dü+/dt)Tm
at
r
E
E
1
(~+ldt)a±~LkTmkTlJ
(51)
.
Further at
=
f
w~(da~/dt)dT = (Q/w)
Tm
7 L dT,
(52)
Tm
where L is the measured TSL intensity. A similar equation exists for at triangle approximation we obtain
-
With the
at
=(C’sIW)Lma,
(53)
at
‘(C
(54)
and 5/w)Lm(o+~X).
The left hand side of(51) equals 2. Hence, one finally obtains E=(kTi Tm/k) lfl(2+3X/2u).
(55)
Since all quantities in (55) are measurable, E can be calculated with this formula. Several expressions similar to (55) are possible. Those known to us will now be given. (1) First class methods Method 10, Luschik (TSL) 2~/a. E=kT,
[291
Method 11, Halperin and Braner 1 (TSL) [30] E=(l.72kT,~/X)(l —5.l6/~), with the condition pg
P. Kivits, HI L. Hagebeuk / Evaluation of the modelfor TSL and TSC
16
with the condition
This method is in fact identical with method 10.
Method 13, Chen 1 (TSL) [311 E=2kT
01(l.25 TmIW~ I),
with the condition ~>>
1.
Method 14, Chen2(TSL) [31] E=2.29kT~/w.
which is a simplification of method 13. Method 15, Chen3(TSL) [31] E=(l.548kTf~/X)(l —3.l6/i.~). Method 16, Chen 4 (TSL) [311 E=l.52kTp~/X_3.16kT~
which is a simplification of method 15. Method 17,chen5(TSL) [311 H
=
C6 kT~/o,
with C6
=
0.976 ±0.004.
This is the Luschik method, but corrected for the fact that the area under the glowcurve is not exactly equal to the area of a triangle. Method 18, Chen 6a, 6b (TSC) [32,33] E’3okT,~/wX E=2.8 ~kT~/wX
(a) (b)
(2) Second class methods Method 19, Halperin and Braner 3 (TSL)
E=(2kT~~/X)(l—6/s), with the condition pg~e~(l +2/s).
[301
P. Kivits, H.J.L. Hagebeuk / Evaluation of the modelfor TSL and TSC
17
Method 20, Ha/penn and Braner 4 (TSL) [30] E = 2kT,~/a, with the condition 1g~0.5.
I
Method 21, Ha/penn and Braner 5 (TSL) [30] E(l +w/X)kT,~/a, with the condition
0.5. This method has in fact been derived for the case 6 > 1. Method 22, Chen 7(TSL) [31] Ez2kTm(l.77
TmIW —1).
Method 23, Chen8(TSL) [31] E
=
1 .813 kT~/X
—
4kTm
Method 24, Chen9(TSL) [31] E=C 7 2kT~/u, with C7 =0.853±0.0012. This is the corrected method 20.
(C) Other methods (1) First class methods Method 25, Randall and Wilkins (TSL) [34] E=25kTm. In ref. [34] it is assumed that ‘y = 1 for T = Tm, when 0.5
kTmTi/X.
Although similar to expressions where the triangle approximation is used the formula of Grossweiner is derived by integrating (26). The integral f~~fl dT which I obtained in that way is approximated by a series that may be truncated after the ~‘
18
P. Kivits, H.J.L. Hagebeuk/Evaluation of the modelfor TSL and TSC
first term when z~s>20 ands/w> l0~K~,accordingto the author.
Method 27, Franks and Keating (TSL) [37,38] l/~r(w/Tm)(l.2XO.54)+5.5X l03_~(~0.75)2 with the additional conditions that l0<~<35
and
O.75
Method 28, Dusse/ and Bube (TSC) [39] E’C 8 kTmTi/X. When ~ = 17, 22 or 26, for C8 a value of 1.402, 1.415 or 1.421 has to be taken, respectively. The method is a correction of method 26. (2) Genera! class methods Method 29, Ganlick and Gibson (TSL) [17]. In the initial rise of the curve a slope —Elk is obtained when ln L is plotted versus l/T. In practice, however, it is rather arbitrary what region should be taken as the initial rise. Our numerical calculations presented in section 4 have been applied on three parts of the TSL and TSC curves.
(a) From 0.01 to 0.15 Lm or ~m’ (b) From 0.15 to 0.30 Lm or nm.(c) From 0.15 to 0.50 Lm or nm. Haake [40] concluded that this method becomes more accurate for larger s an lower H values. Method 30, Sandomirskii and Zhdan (TSL} [41] E= l.45SkTmTiIX_0.79kTi with 1.02 < Tm/Ti <2.0, which according to the authors is equivalent to 1.06 < z~< 121.01. The formula is the Grossweiner expression, but corrected with the aid of computer calculations. Method 31, Voight (TSC) [42] l8kTm
4. Reliability of trap depth determinations In section 3 a number of methods for trap depth determinations from TSL or TSC curves are listed. Some methods were extensively discussed, others merely mentioned. In the derivation of expressions on which these methods are based some more or less plausible assumptions or approximations are used. Whether these are
allowed or not can easily be verified for the simple model when TSL and TSC curves are calculated without making simplifications. This can be done with the aid of a numerical procedure described in [7]. Most of the methods mentioned in this paper
are applied on these curves. Subsequently the determined trap depths are compared
P. Kivits, HI L. Hagebeuk / Evaluation of the model for TSL and TSC
19
with the value used in the numerical calculations. We calculated TSL and TSC curves for a phosphor with NC = 1020 cm3 g = 1, h 3 and both h 1 = 1017 cm 2 = 0 and h2 >> h1. In the case h2 >> h~the TSL curves differ mainly from those with h2 = 0 with respect to the shape at the high temperature side, ~‘2 Tm being smaller [8]. As follows from (19) with a~= h2, the TSC curve is identical with the TSL curve when h2 >> h1. For T0 a value of 80 K was taken. Further we chose the following values for the other relevant parameters to cover ,
—
roughly all reasonable sets of values E
0.273,0.341,0.511,0.682 eV
=
6(TSL)= 0.01, 1,3 6(TSC) s w
=
0.01,0.1
6,1.25X 109,1.25X 1012s~ 1.25X 10 “0.01,0.1,1Ks~
The value of a is fixed by this choice because from (5) and (18) it follows that
a—s/8N~
(56)
.
Both s and a are taken independent of temperature, thus b = c = 0 (c.f. eqs. (7) and (8)). We did not calculate TSC curves in the cases 8 = 1 and 3 in the case h 2 = 0 in view of the shapes shown in fig. 2. It is noted that for 6 <0.01 the shape of the TSL curve is equal to the one for 6 = 0.01 (see fig. 2). For each curve with a specific set of values 6, s, w and E, a realtive error c, was calculated according to C
=
(Em
—
Er)/Er
(57)
‘
where Em is the trap depth determined with a certain method and Er is the trap depth used in the numerical calculations. Except for methods 25 and 31 we found that did not strongly depend on the values of s, wand E. Therefore we also calculated an averaged value ~ of the relative error in Em for each value of 6, and its standard deviation ~6 according to
s
wE
s
~wE~
=
(58) -
~
(59)
calculated from (57) for 36 TSL or TSC curves that have been calculated for the mentioned values of E, s and w. The value of and its standard deviation can be considered as a measure for the correctness of a method for one value of the retrapping ratio 6. The summation is carried out over all values of
20
P. Knits, H.J.L. Hagebeuk
/ Evaluation
of the model for TSL and TSC
Table 2 The averaged relative error t/~(%) and standard deviation °b 1%) of the trap depth calculated with literature methods applied on numerically calculated 1St. curves for three values of the retrapping ratio S in the case h 2 = 0. The methods marked with a (*) yield a value br the trap depth which deviates less than 5% from the correct one. The sign is given in those cases where the error is systematic No.
4 5 6 8 9 29a 29b 29c 31
Author(s)
Retrapping ratio ~
General class methods
0.01
Hoogenstraaten Bube, 1-Jacring and Adams Unger Boyko, Rashba and Trofimenko 1 Boyko, Rashba and Trofimenko 2 Garlick and Gibson/IS Garlick and Gibson/3D Garlick and Gibson/SO Voigt (lower limit)
1
3
t). 1 +0.25 +0.1 -6.0 —6.0 -2.7 —5.3 —10.0 28
(0.1 )* (0.08)* (0.02)* (0.8) (0.8) (0.l)* (0.2) (0.3) (28)
+0.25 -m-0.5 +0.2 —5.7 --5.9 ~4.5 —7.3 —13.7 28
(0.1 5)~ +0.7 (0.2)~ (0.1)* +1.0 (O.2)* (0.04)* +0.3(0.1)* (0.7) —5.5 (0.7) (0.8) —5.5 (0.8) (0.1)~ -8.5 (0.1) (0.2) —11.7 (0.2) (0.4) —20.8 (0.4) (28) 28 (28)
+7.9 +2 5 0.5 2 2 I 0.5 —5 --12 19 +6.0 —41 I -0.5
(1.0) (l)* (13) (0.8)* (2)* (4)* (2)r (l.4)* (3) (3) (38) (0.7) (10) (0.61* (0.41*
+8.5 -42 — 20 —32 -—30 --18 —IS --43 3 —6.0 19 10 --72 —15 ---16
(1.4) (3) (14) (1) (2) (5) (4) (2) (3)* (0.3) (38) (1) 116) (1) (2)
(0.1)*
0.05 7 +17 +80 0.1 0.5 0.5
(O.03)* (18) (5) (5) (0.1)* (5)* (4)*
First class methods 2 10 ii 13 14 15 16 17 18 18 25 26 27 28 30
BOer, Oberhinder and Voigt Luschik 1-lalperin and Braner 1 (Then 1 (‘hen 2 (‘hen 3 (‘hen4 (‘henS (‘hen6a (‘lien 6b Randall and Wilkins Grossweiner Pranks and Keating Dussel and Bube Sandomirskii and Zhdan
+9.0 (1.4) --56 (4) --48 ---45
(1) (2)
--33 —57 —15 —2! 19 --27 —80 32 —33
(7) (4) (9) (9) (38) (3) (20) (3) (2)
Second class methods 3 19 20 21 22 23 24
(‘lien and \~‘iner Halperin and Braner 3 llalperin and Braner 4 ltalperin amid BranerS (‘hen 7 (‘hen 8 (‘lien 9
—0.3 18 +103 +180 +44 +20 +73
(15) (3) (5) (0.1) (2.5) (2)
+0.7 (O.l)* —12 +40 - 24 —22 )5
(0.7) (3) (0.6) (0.7) (I)
P. Kivits, HI L. Hagebeuk / Evaluation of the modelfor TSL and TSC
21
Table 3 The averaged relative error c,~(%) and standard deviation 0~ (%) of the trap depth calculated with literature methods applied on TSL or TSC curves for three values of the retrapping ratio b in the case h 2 >> h1. The methods indicated with a (*) yield a value for the trap depth which deviates less than 5% from the correct one. The sign is given in those cases where the error is sys-
tematic. The results of the methods that are not given do not differ more than 1% from the values shown in table 2 No.
Author(s)
Retrapping ratio S
First class methods
0.01
1
3 —31 (1.4) —34 (2) —32 (1) —33 (1.4)
10 13 14
Luschik (‘hen 1 (‘hen 2
+l.7(O.1)~ 0.7 (1)* 1.8 (2)*
—20(0.7) —20 (2) —19 (2)
17 18
(‘hen 5 (‘hen 6a
—0.8 (0.1)* —7 (3)
—22 (0.7) —17 (8)
18
Chen 6b
—13
(3)
—23 (7)
—36
27
l’ranks and Keating
—41
(10)
—57 (16)
—61
(6) (5) (20)
+37 +90
(3) (1)
—31
Second class methods
20
Halperin and Braner 4
+103
21 22 24
I-IalperinandBraner5 (‘hen 7 (‘hen9
+177 +43 +73
(1) (3)
+60 (2) +123(2)
(1) (2)
+16 (1) +37(2)
—2.9 (1)* +17 (2)
The results for TSL are shown in table 2 (h2 = 0) and table 3 (h2 >> h1). In the
-
latter table only those methods are given that yield a value of the trap depth that differs more than 1% from the value in the case h2 = 0. As could be expected from our finding on the shape of the TSL curve when h2 >> h1, all these methods without exception make use of the value of T’2. The results for TSC in the case h2 = 0 are shown in table 4. When h2 >> h1 table 3 can be applied. For those methods not
listed in table 3, the results in table 2 must be used. For most of the results the sign of calculated according to (57) appears to be the same for all values of H, s and w in one retrapping case. This is also indicated in the tables. The methods marked with a (*) yield a trap depth which deviates less than 5% from the value used in the numerical calculations. In both the TSL and the TSC cases the methods of Hoogenstraaten (4), Bube et a!. (5) and Unger (6) produce the best value of the trap depth although the latter is devised to give a lower limit for the trap depth. Also the method of Chen and Winer (3) turns out to be valid when 8 = 0.01, which was not predicted by the authors. The methods of Boiko et al. (8,9) and Boer et al. (2) can easily be corrected in such a way that they become independent of the value of 8. In the latter case the correction factor p should be above the range proposed by the authors. The initial rise method of Garlick and Gibson (29) turns out to be more unreliable
22
P. Kivits, H.I L. Hagebeuk / Evaluation of the model for TSL and TSC
Table 4 The averaged relative error k~(%) and standard deviation o~(%) of the trap depth calculated with literature methods applied on numerically calculated TSC curves in the case h
2 = 0. The methods marked with (*) yield a value for the trap depth which deviates less than 5% from the correct one. The sign is given in those cases where the error is systematic No.
4 5 6 8 9 29a 29b
29c 31
Author(s)
Retrapping ratio b
General class methods
0.01
Hoogenstraaten
0.1
Bube, Haering and Adams
—0.3 (O.2)* +0.07 (O.05)*
Unger Boyko, Rashba and Trofimenko 1 Boyko, Rashba and Trofimenko 2 Garlick and Gibson/iS Garlick and Gibson/30
—6.5 —7 —0.11 —0.6
Garlick and Gibson/50 Voigt (lower limit)
—1.7 (0.t)* 25 (62)
0.01 (0.O4)* (2.5) (2) (0.005)* (0.02)*
+0.09 (0.05)* +0.56 (0.O3)*
+0.1 (0.005)* —8 —8 —0.67 —2.5
(2) (2) (0.O1)* (0.06)*
—5.3 (0.08) 12 (38)
First class methods
2 10
Böer, Oberländer and Voigt Luschik
13
Chen 1
14 16 17
Chen 2 (‘hen 4 Chen 5
+4.9 (4.0)* +41 (6) —19 (2)
—70
(4)
18
(‘hen Ga
+68
(2)
+74
(0.1)
18
(‘hen Gb
+51
(1)
+63
(0.1)
25 26
RandaliandWilkins Grossweiner
21 +47
33 +14
(31) (1)
28
Dussel and Babe Sandomirskii and Zhdan
+37 +28
(84) (1) (5)
30
+8.5 (2.5) —17 (5) +6
(2)
(1)
+11.5 (0.2) —69 (4) —55
(3)
49
(3)
+5.6 (0.4)
+6
(2)
+5.5 (0.4)
Second class methods 3 21
Chen and Winer 1-lalperin and Braner 4 Halperin and Braner 5
—0.6 (0.4)* +67 (10) +68 (13)
—0.38 (0.06)* —38 (8) +43 (9)
22 23
Chen 7 Chen8
+53 ÷67
(4) (7)
—31 +25
(5) (1)
24
(‘hen 9
+42
(9)
—47
(6)
20
for TSL as the slope is calculated nearer to the maximum. In fact only the shape of the TSL curve below 1 5% of the maximum intensity can be used to calculate an accurate value for the trap depth. In the TSC case the reliability range is larger. The
systematically negative value of the error was already predicted by Hofmann [43] who used this fact to develop the fractional glow technique.
P. Kivits, HJ.L. Hagebeuk / Evaluation of the model for TSL and TSC
23
The methods of Voigt (31) and Randall and Wilkins (25) turn out to be inaccurate. These methods are independent of the value of the retrapping ratio but depend strongly on the frequency factor and the heating rate. To obtain a rough estimate for the trap depth we propose E = 27kTm for TSL and H = 2SkTm for TSC. This yields a value with a maximum deviation of 30%. The TSL methods of Chen, applied on TSL curves, give an accurate value forE if 6 = 0.01 (13,14,15,16,17) and if 6 = (22,23,24) when h 2 = 0. Applied on TSC curves these methods lead to incorrect results. In the case where h2 >> h1 the methods 22 and 24 become less accurate for TSL if 6 = 1 although it might be possible that they are correct for values of the retrapping ratio other than those discussed here. The method Chen (18) derived fot TSC gives an inaccurate value if apllied on TSC curves. A better result is obtained in the TSL case if 6 = land h2 =0. The original method of Grossweiner (26) did not result in a correct value for the trap depth in the cases investigated. The data of table 2 suggest that this method may be correct for a value of 6 in the order of 0.1. The correction made by Dussel and Bube (28), although derived for TSC, is more successful in the TSL case if 6 = 0.01. The correction from Sandomirskii and Zhdan (30) yields a good estimate for TSL if 6 = 0.01. The expectation of the authors that the corrected expressions would be generally valid is not confirmed. The formula of Franks and Keating (27) turns out to be incorrect in the cases investigated. The value of x for the calculated TSL curves is 0.74 ±0.02 for 6 = 0.01 in both cases h2 = 0 and h1, 1.10 ±0.04 for 6 = 1 when h2 = 0 and 0.79 ±0.04 when >> h1. Even in the case that the condition 0.75
5. Conclusion Many papers have been published on the theory of thermally stimulated luminescence and conductivity. Reviews of this field hardly exist. The methods which have been developed for the determination of the trap depth from TSL or TSC curves are generally applied without sufficient knowledge of whether they are reliable or not. In this paper we have indicated.the mutual connection between several expressions for the trap depth that exist in the literature. After application on numerically calculated TSL and TSC curves it is found that from the considered methods only those of Bube, Haering and Adams, Hoogenstraaten and Unger produce reliable values for the trap depth independent of the values of the frequency factor and the retrapping ratio. An estimate of the latter quantity can be made by comparing the trap depths determined with the TSL methods of Chen since these produce correct
P. Kim’its, H.J.L. Hagebeuk / Evaluation of the model for TSI. and TSC
24
results in the cases 6 = 0.01 or 6 = 1 for TSL. Hence, when the simple model of figure 1 can be used, TSL and TSC measurements are indeed a helpful tool for determining trapping parameters, in agreement with the statement of Bbhm and Scharmann [5]. One of the main results of our TSL and TSC measurements on the ternary compound CdGa 2S4 that will be published elsewhere [44], is that application of some methods mentioned in this paper on the measured TSC curves leads to calculated trap depths that indicate the same value of the retrapping ratio. Hence, one might conclude that this phosphor is a good example of a case where the simple model is applicable. However, this is certainly not the case for CdGa2S4 since the same methods applied on measured TSL curves yield different results. Summarizing we come to the conclusion that very much care must be taken in attaching value to quantitative results of trap depth calculations from TSL and TSC experiments if the model that can be applied is not known, in agreement with the conclusion of Kelly et a!. [6].
For a better understanding of the processes occurring in a semiconductor showing thermally stimulated luminescence or conductivity during heating, elaborate experiments on well-defined phosphors have to be carried out in connection with other measuring techniques.
Acknowledgement The authors wish to thank Wim Batenburg and Noud van Lieshout for most of the numerical work presented in this paper. We further greatly acknowledge Prof. Dr. F. van der Maesen, Prof. Dr. M.J. Steenland and Dr. A.T. Vink for numerous valuable comments on the manuscript.
Table 5 The magnitude of the terms in eq. (17) forE = 0.516 eV, s i’l —3 —6 —3 —m h10 eta ,a10 cm s andbO.Ol. 2 T
—da~/dt
(K)
(cm
1.25 x 1012
a~(I—S)+SIm
~l
w
0.1
X da~’7dt —6 —1
(cm
=
—(1 +b)/ua~)
~/o
~a~
—3 —m
s )
a’d2a4/dt2
=
s )
—6 —m
(cm
s )
—a
(cm
)
17o4.57x 180 190
1013 4.45 x i03° 7.4 x ~ X 1016 2.69 x i~’~2.18 x loam 3.1 x lO~ 8.1 x 1016 4.86 x 1014 1.30 x l0~’ —3.0 x i~’~ 2.7 x 1016
200 210
1.43 x i~’~2.09 x 1028 0.62 x 1012 0.64 x 1027
—6.Ox 1017 —1.0 x 1016
1.5 x iü’~ 1.0 x 1015
—3
—3
(cmn )
(cm
4.7 X 102 3.3 x ~ 1.9 x iø~
4.7
x
102
9.4x ~
3.4 x 1.9 x 3.1 x i04
3.9 x iO~
1.5 x 1O4
P. Kivits, H.IL. Hagebeuk / Evaluation of the model for TSL and TSC
25
Appendix It must be proven that (17) is adequately approximated by (20). We calculated numerically the magnitudes of the terms in (17) for all used sets of parameter values (section 4). As a representative example we give in table 5 the results for the same parameter values as used in fig. 2 and 6 = 0.01. We found that the terms with a~ in
(17) are several orders of magnitude smaller than the other terms and may thus be ignored. Some illustrative analytical proofs can also be given. Consider the denominator of (17).
(A) Firstly, we prove that ‘y/cx<<6h.
(60)
With (4) and (18) this can also be written as h/Ne >> exp[—E/kT],
(61) 3) <1020) in the relewhich is valid for normal dope concentrations (1014
-Q .~
10~
_____________________________________ I
I
C
1&0—11 10-
10-12
-
1013
—
iO1’
—
1o3
ro4
-
-
—15
I
I
10 170
180
190
I
200
210 ~ TOO
11g. 5. The ratio n/h versus temperature for the same parameter values as in fig. 2.
P. Kivits, !-LJ.L. Hagebeuk /Evaluation of the model for TSL and TSC
26
(B) We prove that —[(1
+
6)/cxa~]da~/dt <
First we consider the case 6
—~
+
(1
—
O)a~.
(62)
0. In that case applying (2) and (3), inequality (62) is
equivalent with
n<
(63)
This inequality, or rather n <<1v , is used by several authors to simplify the original differential eq. (1) and (2). It is difficult to prove the validity analytically (section 2.2). Therefore, we calculated numerically n/h as a function of temperature for different sets of parameter values. Some results are given in fig. 5. It is concluded that in the relevant temperature region n <
2n <
(64) /t
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P. Kivits, H.J.L. Hagebeuk / Evaluation of the model for TSL and TSC
27
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