W Tomaszewicz, P Grygiel - Thermally stimulated currents in disordered solids at step heating - страница 1
ВІСНИК ЛЬВІВ. УН-ТУ
Серія фізична. 2007. Вип.40. С.219-231
VISNYKLVIV UNIV. Ser.Physic. 2007. N40. P.219-231
PACS number(s): 72.20.Jv, 72.80.Ng
THERMALLY STIMULATED CURRENTS IN DISORDERED SOLIDS
AT STEP HEATING
W. Tomaszewicz1, P. Grygiel
Department of Physics of Electronic Phenomena
Gdansk University of Technology G. Narutowicza 11/12, PL-80952 Gdansk, Poland 1e-mail: email@example.com
In the paper, thermally stimulated currents (TSCs) in disordered solids due to step sample heating are investigated theoretically. The Gobrecht-Hofmann method of TSC analysis is extended to the case of meaningful carrier retrapping. Two kinds of experiments concerning TSCs measured in samples with either coplanar or sandwich electrodes are considered. The main factor limiting TSC is then, respectively, either carrier recombination assumed to be monomolecular or bimolecular, or carrier neutralization on the collecting electrode. In all the cases simple formulas are given, which make possible to determine the energetic distribution of traps from the dependence of released charge on activation energy of the initial TSC rise during sequential heating cycles. Their accuracy is estimated by analysing TSC curves, calculated numerically for exponential trap distribution.
Key words: thermally stimulated currents, step heating, carrier trapping, carrier recombination, multiple-trapping model, amorphous solids.
The TSC experiments consist in photogenerating of excess charge carriers in a sample cooled to a low temperature, and registering current flowing in measuring circuit in course of subsequent sample heating. The resulting TSC is due to carrier release from trapping states of investigated solid and yields information about their energy distribution and/or density. Two kinds of experiments can be distinguished: (1) Sample with coplanar electrodes is illuminated by weakly absorbed light. TSC is then determined by carrier trapping/detrapping and recombination ('TSC recombination peak', TSCRP). (2) Sample having sandwich electrodes is excited by strongly absorbed light. In this case TSC is due to one-sign carrier transport and neutralization on collecting electrode ('TSC transport peak', TSCTP).
In majority of TSC measurements the sample is heated with a constant rate after the end of excitation. Sometimes, more complex heating schemes are utilized. One of them is the step heating, when the sample is linearly heated to some temperature, rapidly cooled to initial temperature, heated to higher temperature, and so on (cf. Fig. 1). Such TSC measurements were performed for many disordered solids, e.g., amorphous hydrogenated silicon [1-3] and photoconducting polymers [4-8]. However, the
© Tomaszewicz W., Grygiel P., 2007
Fig. 1. Time dependence of the sample temperature in the case of step heating
corresponding theory is still incomplete. In principle all the methods of analysis of TSCs registered at fractional heating (see , p. 172-177) should be also applicable to the case of step heating. In these methods, however, carrier retrapping is ignored, which limits their usefulness and may yield incorrect results. More detailed descriptions of TSCs at step heating, including carrier retrapping, concerned mainly the initial TSC rise [10-13].
In this paper, we present the extension of the Gobrecht-Hofmann method to the case of meaningful carrier retrapping. The method was primarily used for analysis of thermally stimulated luminescence at fractional heating , and subsequently adopted for analysis of thermostimulated currents, e.g., [7-9]. It consists in determination of the energetic distribution of traps from the dependence of released charge on activation energy of the initial TSC rise, during sequential heating cycles. Both TSCRPs, corresponding to monomolecular and bimolecular carrier recombination, as well as TSCTPs are considered. The analytical results are compared with the numerical ones, obtained for exponential trap distribution.
The present study is based on earlier descriptions of TSCRPs  and TSCTPs  in disordered materials. Although they concerned linear sample heating, the obtained results should apply after slight changes for other heating schemes. The validity of the multiple-trapping model and the existence of continuous distribution of trapping states in the energy gap were assumed. The main simplifying assumptions were: (1) negligibly small trap occupancy, and (2) strongly non-equilibrium distribution of trapped carriers. The second assumption is adequate, if the trap density varies slowly in energy gap compared to the Boltzmann factor.
According to [15-16], the intensities of TSCRP and TSCTP can be expressed as
Here, t is the time variable, Q(...) is the collected charge, and s0(t) is the demarcation energy level, defined implicitly by the equation
Tr (s, t ) = — exp
is the mean carrier dwell-time in the traps of depth є, measured from the edge of allowed band (v0 - the frequency factor, k - the Boltzmann constant and T(t) - the sample temperature).
The time dependence of demarcation energy є0(Ґ) in the case of step heating,
T (t) = T0 + p(t - tm_1), tm_1 < t < tm, m = 1,2,3,...(^ = 0) (4)
(T0 - the initial temperature, p - the heating rate), was investigated in . At the onset of the (m+1)th heating cycle є0(Ґ) = єm and dє0(t)/dt <x exp[-єm/kT(t)], where the energy
Єm = k (cX - T* ) , (5)
c* = 0,967ln (46K-v0/p), T* = 180K, (6)
and Tm = T(tm).The accuracy of Eqs. (5)-(6) is illustrated by Fig. 2. From Eq. (1) it follows that the initial TSC increase has thermally activated character,
I (t)x exp [-Є,,, / kT (t)] , t = tm . (7)
Therefore, the TSC measurements at step heating enable us to determine the approximate values of demarcation energy Єщ = є0(^) and of frequency factor v0.
The formulas for the collected charge Q^0(t)] were derived in [15-16]. The charge expresses in terms of the function
0(t) = Ct j Nt (є)<Іє, є0 <Є0 (t)<Єt, (8)
where Ct is the carrier capture coefficient, Щє) is the trap density per energy unit, and є/0 and є are the trap distribution limits. The O(t) function determines the probability
that the carrier is captured in time unit in the trap of depth є > e0(t) and stays in the trap up to time t. In what follows, we shall present the formulas for Q^m) and the resulting formulas which make possible to determine the trap density. For simplicity, we shall write here є instead of Єщ. TSCRP, monomolecular recombination The collected charge is given by
q 00 =
(cf. ), where QM is the total collected charge corresponding to the area under TSC curve and tr is the mean time of carrier recombination. Making use of Eq. (8) one obtains:
xRCt j Nt (є ')іє
If Qfc) << QM , Eq. (10) simplifies to
Differentiating Eq. (10) over energy one gets:
Q» dQ (є)
trCN (є) can be
Q ( ) d
When Q(є) = QM , Eq. (12) can be approximated by
j_ dQ (є)
Qoo' іє .
TSCRP, bimolecular recombination
The Ф(є) function is interrelated with the collected charge by the formula
Ф(є) = CrrnQ^ exp
(see ). Here, CR is the carrier recombination coefficient, n, is the initial density of generated carriers, and the charge
(with e - the elementary charge, ц0 - the free carrier mobility, E - the electric field strength, and S - the sample cross-section area, perpendicular to the carrier flow direction). From Eqs. (8) and (14) it follows that
When Q^) << QR , Eq. (16) may be simplified to
By differentiation of Eq. (16) one gets the formula
which for Q( ) << QR simplifies to
Q2 ( ) d
The collected charge is expressed by
1 - exp [-ТоФ(є)]
Q (є) = Qa
is the free carrier time-of-flight (d - sample thickness). According to Eqs. (8) and (20)
ToCt j Nt (є ')<Іє' = F
where the function y = F1(x) (0 < x < 1) is defined as inverse of the function
1 - exp(-y)
The properties of the F1(... ) function are discussed in Appendix. According to them, for Q^) << QM Eq. (22) can by approximated by
T0Ct J Nt (є ')іє ' =
and for Q^) = QM by
Differentiation of Eq. (22) gives the formula
T0Q J Nt (є')<Іє' = 2
T0CtNt (є) = —F2
where the function
The behaviour of the F2(...) function is also considered in Appendix. From Eq. (26) the following approximate formulas result. For Qfc) << QM
Q» dQ (є)
and for Qfc) = QM
Q2 (є) Іє ' „ 2 dQ (є)
Мы, (>. ^* ■ - (34)
One should indicate some similarities between formulas describing TSCRP at monomolecular and bimolecular recombination and TSCTP. Let us consider first initial behaviour of TSCs, when Q(p) << QM or Q(p) << QR. According to [15-16] the following relationships hold:
Q» = (30)
(TSCRP, monomolecular recombination),
Qr = (31)
(TSCRP, bimolecular recombination), and
Q» = І0 T0 (32)
І0 = en0^0ES (33) is the current intensity, corresponding to the motion of all generated carriers in the allowed band. Thus, Eqs. (11), (17) and (24) may be rewritten as
с є і
and Eqs. (12), (19) and (28) as
C , ч 1 dQ (є)
-LNt (є)-—---. (35)
І0 л) Q2 (є) dp 1 '
These identities result from the fact that initial increase of TSC is independent of carrier recombination or neutralization at collecting electrode.
Let us consider now final behaviour of TSCRPs at monomolecular recombination and TSCTPs, when Q(p) - QM . Then, from Eqs. (13) and (29) the following relationship results:
/ ч dQ (є)
Nt (є)ос v '. (36) tK ' dp
The relationship implies that carrier retrapping does not influence the final decay of TSC (cf. [15-16]). Since the Gobrecht-Hofmann method ignores the carrier retrapping, one can conclude that the method applies solely to TSCRPs at monomolecular recombination and TSCTPs in the final temperature region.
The given formulas enable us to determine the energetic trap distribution with accuracy to multiplicative coefficients from the TSC courses measured in entire temperature region. In the case of TSCRPs the kinetics of carrier recombination must be known; for bimolecular recombination the knowledge of the value of (C/Цо is also necessary (see Eq. (15)). This value may be determined from the dependence of TSCRP on the initial carrier density, n0.
As already indicated, from the TSC measurements the dependence of the collected charge after m heating cycles on the corresponding demarcation energy, Q(pm), can be determined. Two methods of calculating the trap density can be proposed:
(A) The total trap density in the energy region pm < є < pt may be found directly from Eqs. (10), (16) and (22) or corresponding approximate equations. The trap density Nt(pm) can be next calculated by numerical differentiation of the total trap density.
(B) The trap density Nt(pm) may be calculated from Eqs. (12), (18) and (26). In these equations, the charge derivatives must be approximated by finite differences, viz.
AQm = Q (Bm )~ Q (Bm-1 ) , Авт = Bm -Bm-v