J Orzechowski, A Mandowski - Modelling excitation kinetics in the framework of semi-localized transitions model - страница 1
ВІСНИК ЛЬВІВ. УН-ТУ
Серія фіз. 2009. Bun. 43. С. 135-142
VISNYKLVIV UNIV. Ser. Physics. 2009. Is. 43. P. 135-142
PACS number(s): 78.60.Kn
MODELLING EXCITATION KINETICS IN THE FRAMEWORK OF SEMI-LOCALIZED TRANSITIONS MODEL
J. Orzechowski1, A. Mandowski2
1Stanislaw Staszic High School ul. Szkolna 12, 27-200 Starachowice, POLAND 2Institute of Physics, Jan Diugosz University Armii Krajowej 13/15, 42-200 Czestochowa, POLAND email: firstname.lastname@example.org
Recently the model of semi-localized transitions (SLT) was proposed combining both localized and delocalized transitions occurring in thermoluminescence (TL) and related phenomena. This paper presents formulation and exemplary application of the SLT model to the analysis of excitation processes, which are very important in constructing dose dependence characteristics of TL. It shows how various factors determine initial conditions for TL. Possible consequences for the analysis of TL and related phenomena in terms of standard kinetic models are discussed also.
Key words: thermoluminescence, semi-localized transition, traps, recombination.
Thermoluminescence (TL) is a widely known technique, frequently used for characterization of traps and recombination centers (RCs) in insulating materials . Commercial applications include dosimetry of ionizing radiation and dating of archaeological and geological objects. Prior to the measurement a sample is excited at appropriately low temperature to fill traps and RCs with charge carriers. Then, the sample is heated, usually with a constant rate. During heating emitted luminescence is recorded. The luminescence is related to recombination of charge carriers, which were thermally released from traps and then moved to RCs.
Theoretical description TL is usually based on two models: the model of localized transitions (LT) and the simple trap model (STM). The first one (the LT model) allows trapped charge carriers to recombine solely to adjacent RC. The second one (STM) assumes transition of charge carriers via conduction band. These transitions are delocalized. Recently  a more general model was proposed combining both localized and delocalized transitions. This is the model of semi-localized transitions (SLT). Analytic equations for SLT are constructed using enumeration technique which transforms concentrations of carriers to concentrations of states of localized trap -recombination centre (T-RC) pairs. The model allows explaining some unusual properties observed in TL measurements (e.g. the occurrence of very high frequency factors ).
This paper discussed the possibility of applying SLT model to the analysis of excitation processes, which is performed before the main TL measurement. This stage of
© Orzechowski J., Mandowski A., 2009
measurement is very important for dose dependence characteristics. Example approach of this kind is proposed. Possible consequences for TL will be analysed.
Typical TL experiment is performed with linear heating rate scheme, i.e. the temperature T = T0 + pt, where T0 is the initial temperature, t denotes time and p is the heating rate. The first "classical" model - STM - assumes spatially uniform distribution of separate traps and recombination centres (RCs). Charge carrier transitions taking place during heating occur via conduction band after thermal release. The kinetics of trapping and recombination is governed by the following set of differential equations :
-nt = щуг exJ,-E^\- псАг (Ж, - щ \ /=Lp (1a)
-m s = Bsmsnc, s=1..k, (1b)
Z ms = Z n, + Пс + M, (1c)
s=1 г =1
where пг, and ms denote the concentrations of trap states, electrons trapped in 'active' traps and holes trapped in recombination centres, respectively. M stands for the number of electrons in the thermally disconnected traps (deep traps), i.e. traps that are not emptied during the experiment. Аг and Bs stand for the trapping and recombination probabilities, respectively, and v is the frequency factor. Luminescence intensity is proportional to (-m), i.e.
L = -—. (2)
Another important case is the model of laclized transitions (LT). Halperin and Braner . They assumed that traps and RCs are closely correlated in space forming pairs that can be considered as independent units - i.e. all charge transfer takes place within groups of one kind, each having one trapping state, one excited state and one RC. Correct equations for this case were given by Land :
-П = nv exp ^Tij - Ane, (4a)
-m = Bne, (4b)
m = n + ne, (4c)
where ne denotes the concentration of electrons in the excited state. Because the transport of charge carriers does not take place through the conduction band, the TL peak should not be accompanied by thermally stimulated conductivity. It is assumed that A and B are constants.
The SLT model for TL kinetics was first introduced in  (some earlier attempts were erroneous and will be not mentioned here). The model unifies STM and LT models. The basic rules are the following: it is assumed that a solid consists of trap-recombination centre (T-RC) pairs. The energy band diagram representing the general SLT model is given in figure 1 (for detailed description cf.  and ).
A C — n
h h h
Fig. 1. Energy band diagram for the SLT model for heating stage
Each pair may contain not more than one electron and one hole. Traps have two localized energy levels: the ground level and the excited level. Therefore, we have the following set of the allowed states of T-RC pair:
■ h0 = ■
• h0 = ■
• h = ■
E0 =• 0
■ e = ■
• e0 =■
■ e; = •
Where the numbers inside brackets
denote occupation of all states in a single T-RC unit, i.e. ne is the number of electrons in
the local excited level, n is the number of electrons in trap level and h denotes the number of holes in RC level. The symbol (6) is a time-dependent variable denoting concentration (in cm-3) of all T-RC units having ne, n and h charge carriers in the
respective trap levels. Therefore, the variables H"m (t) and E"m (t) denote the concentrations of states (i.e. corresponding T-RC units) with full and empty RCs, respectively. Due to previous assumption the states H11 and E11 will not be taken into account. Allowed
transitions between various states of a single T-RC pair may be represented here by the following diagram:
H° —^ h° —^ h°°
1 ° K °
Ci Ci \* Ci (7)
^1 <-—— £/° <—-— jc/°
The bar variables A and B denote probability densities (in s-1) of localized transitions for trapping and recombination, respectively. Similarly, K and C denote probability densities (in cm3s-1) of delocalized transitions for trapping from the conduction band to the excited state of a trap and recombination, respectively. Thermal activation probabilities for transition of a single charge carrier from the ground trap level to the excited level, and subsequent transition from the excited level to the conduction band are given by D and V, respectively (all in s-1). The difference between D1 and D2 as well as between V1 and V2 will be discussed later. The corresponding set of differential equations is written below:
H° =- (D1 + Cnc) H° + AH ° (8a)
H = D1H1° - (A + B + V + Cnc) H° + KncH°° (8b)
H°° = Vx -(K + C)ncH°° (8c)
E°° = CncH° - D2E° + AE° (8d)
El = Cnfl\ + D2E° - (A + V2) E° + KncE°° (8e)
= BH + CncH°° + V2 E - KncE°° (8f)
hc =-Cnc (tf 1° + H + H°°) + V1H + V2E - Knc (H°° + E°°) (8g)
The coefficients D and V are responsible for thermal activation, i.e.
D(,)="exp (m) (9) V(t)="Vexpi;m) <l0)
where E, EV, v and "v are constants. When the states H"m and E"m have different activation energies, the coefficients split up into D1 (with E1, v1), V1 (with EV1, vV1) and
D2 (with E2, v2), V2 (with EV2, vV2). It leads to the 'cascade detrapping' phenomenon 
resulting in very narrow TL peaks characterized by enormously high frequency factors and overestimated activation energies. Luminescence intensities related to recombination from localized excited levels (LB ) and from the conduction band (LC - delocalized) are
defined as follows:
-Cnc ( h° + H ° + H °°)
As the luminescence comes from different transitions it may be characterized by different spectral distribution. Any differences may be observed by applying spectrally resolved measurements. Classical limits of SLT and 'downconversion' to STM and LT was discussed in . A comparison of these three models is shown in fig. 2.
Fig. 2. TL curves calculated in the framework of SLT model for various retrapping coefficients: r = 0 , r = 1 and r = 102. The curves b correspond to TL intra-pair direct transitions intensity (11), the curves c corresponds to radiative transitions
from the conduction band LC (12). The curves a correspond to the sum of both intensities Ltot = Lb + LC . The calculations were performed for the following parameters: E = 0,9 eV, EV = 0,7 eV, v = vV = 1010 s-1, and the heating rate P = 1K/s . Relevant solutions for the LT model (setting V = 0 ) were shown as full circles (•)
ourselves to two hole transitions from the valence band: Kv represents trapping of free hole to RC for electrons and Ke represents trapping of free hole to deeper electron trap. The diagram is shown in fig. 2.
n n n
Fig. 2. Energy band diagram for the SLT model for excitation stage
To construct differential equations for the excitation stage, first we have to write a diagram illustrating transitions between various states of T-RC pairs take into account Kv and Ke transitions. The diagram is shown below:
The diagram allows to write corresponding set of differential equations:
= -(Cnc + Kehv) H° + AHl + KvhvE0°
= -( A + B + Cnc) H0 + KncH00 + KVhVEl0
= KehvH° - (C + K)ncH00 + KyhE
= CncH° - (Ky + Ke )hyE° + AEl0
= CnH'o - (A + Kyhy)E0 + KnX
= BH1, + Cnfll + KXE° - (Kyhy + Knc)E°
= G - Cnfll - CnflO - (C + K)nfll - KncEl
-- G - KXH - (Ky + Ke )hyE° - KyhyE'1 - KyhyEl