Charge trafficking in luminescence during isothermal relaxation

Journal of Luminescence 50 (1991) 265—271 North-Holland

265

Charge trafficking in luminescence during isothermal relaxation W.F. Hornyak and A. Franklin Physics Department, University of Maryland, College Park, MD 20742, USA Received 20 March 1991 Revised 24 June 1991 Accepted 25 June 1991

Model studies have been carried out for several isothermal luminescent anneal processes. It is found to be possible to ascribe the same form to the relaxation time rate of these processes if the critical parameter influencing the relaxation mechanism is assumed to have an associated distribution of comparable width in each case. It is suggested that the well known stretched exponential law evident in many condensed matter relaxation phenomena might also provide a natural basis for this behavior.

1. Introduction

2. Generalized first-order relaxation

Current emphasis on improving the precision in thermoluminescent (TL) dating requires a better understanding of the basic physical processes operating. For example, such knowledge is necessary in properly allowing for fading, thermally assisted tunneling, bleaching, and charge transfer processes. Extracting such information from experimental data usually requires resort to some form of theoretical modeling of the process. Earlier, it had been found that a closed form theoretical expression, the so-called “after effect” function gave a very good accounting for an isothermal luminescent decay process with electron detrapping through the conduction band [1]. At the outset of the present study, we inquire intocharge the generalized on theto nature of trafficking requirements processes in order arrive at a relaxation rate following this type of time-dependent behavior. In this context, we report on some preliminary studies for various lumines-

Suppose a material exists possessing some suitable property, q, which is undergoing isothermal annealing obeying a first-order process, such that dq/d t = (1) —

with a lifetime r0. This would lead to a time development q(t) ssq =q0 e’~”~o, (2) where q0 is the value of q at t = 0. Assume that a time-independent distribution in lifetimes exists which may be designated as h(r). Then the corresponding gross property, Q, as a function of time would be t’~dT, (3) Q(t) Q = fh(T) e 0 with Q 0=

cence processes using only weak approximations in the relevant differential equations. 0022-2313/91/$03.50 © 1991

—

f h(r) 0

dT,

the value of Q at

t

=

0.

(4)

Assume further that the distribution h(’r) can be

Elsevier Science Publishers B.V. All rights reserved

266

W.F. Hornyak, A. Franklin

/

Charge trafficking in luminescence during isothermal relaxation

expressed as a normal statistical distribution about zero in a parameter z such that h(T) dr = N e~2z2dz, with z = In r/r

=

b Q0—J

2z2

[

+~

exp —b

—

ezI1 dz.

t

—

—

—

—

j

The above derivation closely follows the resuits obtained by Wagner [2] in analyzing the charge relaxation in dielectrics. The integral may be written as G(x, y) the “after effect function” and is to be found tabulated in ref. [3]. In this notation dQ _____ yG(x, y), (8) dt T0V~ —

—

with x = t/T0 and y = b. The time development behavior of dQ/dt given by eq. (8) is shown in fig. 1 with y as a parameter. The curves have been normalized to the same value of dQ/d t at x = 3 X iO~ in order to better exhibit shape differences. The curve for y = co corresponds to a delta-function narrow distribution width for z and is the simple

exponential decay represented by eq. (2): —

dt

=

0.4

~‘ ~

= ~

10-s

~0:6

STANDARD SLOPE

io-~

—

dQ

y

~

—

—

y -05

(6)

The time rate of change of Q in turn becomes dQ Q 2z2 Z e 0b expi —b dz dt T0~ T0 (7)

—

y 08 y=1

0.1

T 0

—

y—~

(5)

0. This corresponds to the assumption that the natural logarithms of the lifetimes, r, are statistically distributed about a mean value, in T0. Imposing the normalization of eq. (4) gives N = Q0 b/ i/~. The time development of Q following from eq. (3) becomes

Q

1

~

10~

10.2

0.1

1

10

100

x = tI

the time development of the isothermal relaxation rate is very Fig. 1. A family of “after effect function” curves showing that strongly influenced by the parameter y. The time dependence is contained in the variable x = t/1-0. The curves have been arbitrarily normalized at x = 0.003. The curve for y corresponds to an exponential decay. Curves for y <0.4 fit a simple decay form of the type of a constant multiplied by the factor (1 + cx)~ in the region 0.1
or a reciprocal time dependence. As fig. 1 shows, even for y as large as 0.2 a broad region of x

centered at x = 1 also exhibits an essentially reciprocal time dependence possessing a slope of —

1 in the log—log plot. It should also be noted

that except for y precisely zero (which would correspond to a distribution having an unphysical infinite width), the actual resulting functions have no divergence at x = 0 and are completely integrable. In general, for all values of y 0.4 the shape of the decay curve near x = 1 may be successfully fitted by a modified f’-type curve of the form

T0

For very broad distributions yielding y there results in view of LimG(x, y) = 1/x

<<

1,

dQ =

—

—

constant( 1

+ cx)

~,

(11)

dt with

the expression dQ Qo —

dt

—

~

~.t 1, c 10 and, as before, x = t/r0. In the parameter range y > 0.4 excepting y

co,

1 ,

(10)

=

no simple equivalent shape is available to

adequately range of x. fit the exact behavior even in a limited

WF. Hornyak, A. Franklin

/

Charge trafficking in luminescence during isothermal relaxation

3. Isothennal relaxation via the conduction band Isothermal relaxation involving passage of electrons from a trap site through the conduction band prior to recombination at a luminous center if unaccompanied by retrapping results in a sim-

pie first-order process. In the simplest situation, there is a single type of trap site and a single type of recombination center. If the population of the latter is designated m, then under the reasonable quasi-equilibrium approximation concerning the

electron population in the conduction band, n,~, and assuming a narrow trap level with an activation energy, E0, and a corresponding lifetime, T0,

there follows in analogy with eq. (2) m=moe~/To.

(12)

The dependence of lifetime, T, Ofl activation energy, E, is generally assumed to be of the Arrhe-

nius form; thus, in this case: = s eE~~~cT. (13) Now, assume that the trapped electrons sess a distribution of activation energies, E, posand hence of lifetimes, h(r), which is taken ad hoc to be in accordance with the assumptions involved with eq.(5). (With the absence of retrapping, it ~5 unnecessary to distinguish between the energy distribution of available trap sites and the initial distribution of trapped electrons ~(E, t) at t = 0.) Then the parameter z becomes

1/’r

The electron density distribution ‘q( E, t) in the energy interval between E and E + dE is ~(E, t) dE = e~ q(E, 0) dE and n = J~°-q(E, t) d E is the total number of trapped electrons at time, t. Only at t = 0 is q(E, t) the Gaussian distribution of eq. (14). Generally, as time progresses, ~(E, t) becomes more asymmetrically skewed favoring larger values of E.

Finally, in exact analogy with eq. (8) the thermoluminescent (TL) emission rate becomes the “after effect function”: m0

(TL) = —dm/dt

yG(x, y),

=

=ln

T/70

(E

(15)

~

with x=t/r0

and

y=b,

and where m and n have been taken as equal in

view of the approximations assumed for n~. This result was first noted by Medlin [4] and examined in detail by Hornyak and Franklin [1]. In the latter paper, the width parameter was y2 in 2/(kT)2 and hence = a(kT)2. is notable to point out that relaxterms of a It constant a= b ation rates of the form given by eq. (10) or (11), i.e. the ‘t’-type’ decay modes, do not require very broad distributiqns. At an anneal temperature of 500 K and when y = 0.3, a value capable of generating this type of behavior only corresponds to a full width at half maximum (FWHM) of 5.5 kT or 0.25 eV. With E 0

z

267

1.38 eV (ap-

propriate for such an anneal temperature), this

E0)/kT,

and hence h(r) dT=~?j(E,0) dE2(E—E 2/(kT)2 dE, (14) m0b e —b 0) kT~ where ~(E, 0) is the activation energy distribu-

width corresponds to only 17% of the mean activation energy. On the other hand, at the other CONDUCTION BAND

— —

tion of the trapped electrons corresponding to h(T) at t = 0. Thus, the above assumption embodied in eq. (5) is seen to have resulted in assigning a Gaussian normal distribution to the activation energies, E, centered about a mean value of E

aCE)

0

and possessing a width parameter b. The situation is illustrated in fig. 2, where a( E) = s e —E/kT is the probability per electron in the activation energy range between E and E + dE for being thermally elevated into the conduction band.

VALENCE BAND Fig. 2. A simple kinetic model for thermoluminescence with detrapping of electrons through the conduction band. The initial electron occupation distribution of activation energies, E, is ~7(E) taken to be a Gaussian normal distribution. The quantity a(E) is s e

268

W.F. Hornyak, A. Franklin

extreme, a width corresponding to y

/ Charge trafficking in luminescence during isothermal relaxation =

1 or 0.072

eV and only 5.2% of E0 is sufficient to give a decay rate discernably different from a simple exponential law corresponding to y = 00• Activation energy distributions having a rectangular shape have also been studied [5]. For a comparable percentage width to its Gaussian counterpart, a very similar relaxation decay rate results. The behavior of the combined isothermal and glow curve characteristics of commercial TLD-400 (CaF2 : Mn) ribbon could be successfully accounted for using a Gaussian normal distribution with a value of y

=

0.475 for the isothermal decay

at 500 K [1]. Referring to fig. 1, this situation is seen to involve a decay rate not easily amenable to a simple simulating expression.

sponding to the rate constant r0 rate of n pairs would be dn/d t = n/r0

A typical model for quantum mechanical athermal tunneling relaxation assumes a local cation—electron pair recombining through penetration of a simple rectangular potential barrier (e.g., Hama and Gouda [6]); refer to fig. 3. Direct application of the WKB approximation leads to a

barrier tunneling rate constant (including a frequency factor v~) = e~’, (16) with J3 = (2/h)~I2MV0,where V0 is the height of the potential barrier, h is Planck’s constant divided by 2’rr, and M is the effective electron mass. If a first order process is assumed then for a fixed exact separation distance of r0 corre-

E

i

r Fig. 3. A diagram illustrating the quantum mechanical tunneling process for transmission through a one-dimensional rectangular barrier with barrier height, V0, and width, r.

the relaxation

—

and n=n0e

(17)

.

Now, assume that the cation—electron pair formation process led to a distribution in separation distances, r, giving a lifetime distribution, h(’r). Again, as an ad hoc assumption, take eq. (5) to hold, then T

z=ln— =f3(r—r0),

and hence ~

~

h(r) 4. Quantum mechanical athermal tunneling

1,

dr,

(18)

dT=

which corresponds to a Gaussian normal distribution in r centered about the mean value of r0 and having an associated width parameter, b.

Finally, the observed TL rate with time again results in the “after effect function” form: dn (TL)

=

—

n0 =

T~V~yG(x,

y),

(19)

with T0’ = ye —~r0, x = t/r0, and y = b. Hama and Gouda [6], in studying the isothermal luminescence from solid organic phosphors, observed a relaxation rate of the form given by eq. (11) with the associated parameters j.i and c in the proper range. They deduce a value of r0 45 A and a distribution width in r of the order of 5 A, albeit for a somewhat skewed bell-shaped curve obtained as a Laplace transform of the empirical fit to the data. However, the assumption implied in this transformation is that eq. (11) fitted in a limited range of t/’r0 holds for all values of time. With this consideration in mind, their findings are essentially in good agreement with the above analysis. 5. Photostimulated luminescence The isothermal relaxation model discussed in section 3 involves bimolecular transitions for the

W.F. Hornyak, A. Franklin

\ ~

269

the excited state may undergo TL emitting transitions by recombining with a luminous hole state, population nh, or be retrapped back into the initial trap state. The reaction rate for the former

~‘°

ELECTRON

z

STATES

T~

HOLE

Charge trafficking in luminescence during isothermal relaxation

_____

EXCITED ~\STATE~,~\

STATE

/

1L? INCIDENT hv

process is yn~and sne for the latter. It is generally assumedstate thatis amuch thermally assisted at the excited less than the rate photon room temperature carrying trapped electrons to transfer rate. With these assumptions, the rates of change of n~and ~h are

CONFIGURATION COORDINATE

Fig. 4. A configuration coordinate diagram appropriate for a cation—electron correlated pair with a common excited state. The exact excited substate reached by the photon-interactive electron is not explicitly shown.

dn~ =ng4~u—(s+y)ne,

(20a)

—

d~

dnh

—

(20b) If, in analogy with the usual assumption in the —yn~.

—

essentially free Fermi electrons, n 0, in the conduction band. A relatively large separation is assumed between the original trap site source of the electron and the final recombination hole site. Thus, the rate of recombination of electrons in the conduction band and any one of the hole centers is ~2cO)c0m)m = n0-ym, where i~ is the average of an electron the conduction 0mvelocity is the cross-section forinelectron capture band, by a hole center, and y = iI3CUm. The cation—electron pair model discussed in section 4 is a simplified version of an exciton model for a correlated, close-spaced charge pair produced during some excitation process with emphasis on the physical nature of the pair-wise recombination process. The relaxation of such systems is clearly monomolecular. In the present instance, we wish to examine the general photostimulated luminescence process for such exciton models. Assume that some initiating excitation process produced an electron—hole pair in the vicinity of some defect complex; refer to the configuration coordinate diagram of fig. 4. The electron and the luminescent hole state share a common excited state. An irradiating light source continuously exposes the sample to photons of energy, hi’, capa-

model discussed in section 3, dfle/dt is taken negligibly small, then

ble of carrying the trapped electrons, ng, over the barrier leading to the excited state with occupation number, ~ The photon flux is 4, and the cross-section for transfer to the excited state is ~ thus, the transfer rate is n54w-. The electrons in

dt with the rate constant

‘~ =

(s

+

~)

and since dnh ~7

— —

dflg

+

dfle ~7

—

dng (21)

=

dt (s + ,~) ~ Equation (21) is a first-order equation. An additional approximation usually made is that y =~zs. The excess energy hi’ E * according to the Frank—Condon principle would carry the electron to a higher excited state. The resulting broader wave function associated with such excited states would lead to a larger overlap integral with the hole state and hence an increased y. The not unlikely possibility, confirmed experimentally at least for zircon by Tenipler [7],is for y to depend on E * through the relationship ~ = ,ç * e pE for a single narrow energy electron trap. Equation (21) becomes —

—

*

=

—

~e_13E*~ffflg

=

—n 5/T,

l/T =

—

s

4o

e

3E

(22)

270

WF. Hornyak, A. Franklin

/

Charge trafficking in luminescence duringisothermal relaxation

If, as before, a distribution in rate, constants is assumed to be of the form T

z

=ln—

13(E*

then h(T) dT

=

Es’),

9!bp e~2p2~*~o)2 dE*.

(23)

densed matter physics (Plonka [81,and the references contained therein) that a very simple stretched exponential law can be used to fit successfully a wide variety of empirical data over many orders of magnitude in relaxation time. This law states that the equation corresponding to eq.(2) is q=qoe_(t~’To)’,

For a fixed value of hi’, this expression leads to a Gaussian normal distribution of electron trap state energies. Finally, for the photoluminescence rate (PL) there results dnh

~

dng

nog

(24)

with

0<~<1,

(25)

where it may be demonstrated that the parameter

~ corresponds to a measure of the width of an implied distribution of lifetimes. The straightforward time derivative provides the decay rate equation dq_ cit

~It~’’ ‘r —qo——~—) 0 T0

e_0//To)C.

(26)

5* x

=

—4o- e~~t and

y

=

b.

S

Once again, eq. (24) is seen to correspond to the “after effect function” behavior. The photostimulated luminescence of zircon has been subjected to a detailed experimental study [7].Although the results generally suggest a

far more complicated system to be operating than that described above, isothermal decay of the sort exhibited in fig. 1 do appear in curves the data. Some success was achieved in fitting a portion of this data with essentially the model described above using a rectangular distribution function with a width z~ E * rather than the present Gaussian shape.

6. The stretched exponential law The previous sections describe three completely diverse relaxation models all of which exhibit an “after effect function”-type time-dependent decay rate when in each case the critical relevant variable is taken to yield a normal statistical distribution in lifetimes. The most important feature of these distributions is the associated relative width. Rectangular distributions of comparable relative width result in very similar decay rates. It is well known in other branches of con-

It was demonstrated [9] that this stretched

exponential law and equations derived from eq. (8) with Gaussian distributions of the same relative width were in some cases virtually indistin-

guishable over many orders of magnitude in elapsed time. This investigation also established the correspondence between the width parameters ~=and through an approximate equation 170~ (1 y ~)1• (27) —

e This situation raises the possibility that luminescent phenomena may also join the category of condensed matter relaxation phenomena properly obeying a stretched exponential law. It should not escape our attention that eq. (8) was in fact first derived for the charge relaxation in a dielectric! A theoretical basis for arriving at a stretched exponential law starting with a statistically amenable distribution in lifetimes has been given by Klafter and Shiesinger [101. A broadly based review of dispersive transport processes that form the basis of such relaxation laws is given by Sher et al. [11].

7. Conclusions A number of disparate luminescent isothermal relaxation models are found to exhibit time development decay functions of like kind when distri-

WF. Homyak, A. Franklin

/

Charge trafficking in luminescence during isothermal relaxation

butions in lifetimes of comparable relative widths are assumed. The shape of these decay functions is most sensitively related to the assumed relative width rather than the precise shape of the lifetime distribution function. The possibility is raised that the ubiquitous stretched exponential law of condensed matter physics may provide the natural basis for such lifetime distributions. On the one hand verifying a universal relaxation law to be operating would be intellectually satisfying but on the other hand it would reduce

the model specificity that could be deduced from empirical data. An essential feature of the present model analyses is the assumption of the operation of first-order processes. For bimolecular processes

the introduction of retrapping violates this assumption and indeed for very strong retrapping converts the process to one of second-order. Computer assisted analyses are now in progress

to assess the effect of progressively increasing the retrapping channel strength.

271

Acknowledgements We wish to acknowledge the many useful discussions research. with Reuven Chen on aspects of this References [1] W.F. Hornyak and A. Franklin, NucI. Tracks 14 (1988) 81. [2] W.W. Wagner, Ann. Phys. 40 (1913) 817. [3] E. Janke and F. Emde, Tables of Functions with Formulae and Curves (Dover, New York, 1943). [41W.L. Medlin, Phys. Rev. 123 (1961) 502. [51W.F. Homyak and R. Chen, J. Lumin. 44 (1989) 73. [6] Y. Hama and K. Gouda, Radiat. Phys. Chem. 21(1983) 185. [7] R.H. Templer, Thesis, Oxford University (1986). [81 A. Plonka, Lecture Notes in Chemistry 40 (Springer, New York, 1986). [9] W.F. Hornyak, R. Chen and A. Franklin, J. Lumin. 46 (1990) 251. [10] J. Klafter and M.F. Shlesinger, Proc. NatI. Acad. Sci. USA 83 (1986) 848. [11] H. Sher, M.F. Schlesinger and J.T. Bender, Phys. Today 44 (1991) 26.