- Email: [email protected]
On weakly dispersive multiple-trapping transport
Journal of Non-Crystalline Solids 355 (2009) 1414–1418
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol
On weakly dispersive multiple-trapping transport W. Tomaszewicz * Department of Physics of Electronic Phenomena, Gdansk University of Technology, Narutowicza 11/12, 80-952 Gdansk, Pomerania, Poland
a r t i c l e
i n f o
Article history: Available online 15 June 2009 PACS: 72.20.Jv 72.80.Ng Keyword: Electrical and electronic properties
a b s t r a c t Equations for multiple-trapping carrier transport, corresponding to the time-of-flight method are approximately solved under the assumption that the majority of carriers are in a thermal quasi-equilibrium. The solutions show a Gaussian shape of the carrier packet. The mean velocity of the carrier sheet for the dispersive transport regime decreases in time and its dispersion grows faster than the square root of time. The accuracy of the obtained formulas is verified by Monte Carlo calculations for exponential and Gaussian trap distributions. A satisfactory agreement is obtained up to the effective carrier transit time, provided that the trap density falls-off sufficiently fast in the energy gap. A new method of determining energetic trap profiles in disordered solids from the time-of-flight measurements is proposed. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction The time-of-fight (TOF) technique is a straightforward, frequently applied method of investigating the carrier transport in low-conductivity solids, both crystalline and amorphous. The sample is sandwiched between two electrodes with a constant voltage applied, and the excess carriers are generated by a short light pulse. The carrier motion in the sample induces a current transient in the measuring circuit. Information about the carrier transport mechanism can be inferred from the transient form as well as from its dependence on the experimental parameters. As regards disordered solids, there exist two basic carrier transport mechanisms – multiple-trapping (MT) and hopping. Transitions of carriers between extended states and localized states (traps) gap occur in case of MT, whereas straightforward carrier transitions between localized states take place in case of hopping. For both mechanisms, the carrier transport may be either Gaussian or dispersive. The former transport regime is characterized by constant velocity and a Gaussian carrier sheet shape in a solid, the latter – by a gradual decrease in the mean velocity and an extremely large dispersion of the carrier packet. The first successful theory of dispersive transport was developed by Scher and Montroll [1], who attributed this phenomenon to very slow equilibration of charge carriers over localized states. The Scher–Montroll theory initiated extensive investigations on dispersive transport (see the reviews [2,3] for earlier works). In spite of this, some problems seem to be still unresolved. In particular, this concerns the simplified description of MT dispersive
transport, given by Tiedje and Rose [4] and by Orenstein and Kastner [5]. Their main idea was that the majority of trapped carriers for specific trap distributions are in a thermal quasi-equilibrium with the free carriers. This approach has been utilized in many subsequent papers. However, its validity has been questioned by Arkhipov et al. [6], since it does not describe the carrier packet broadening. The main aim of this paper is to resolve this controversy. 2. Transport equations The present investigations are based on a standard MT model, assuming very small trap occupancy, electric field uniformity in the sample as well as negligible carrier diffusion. It should be borne in mind that the first assumption may be incorrect at the final stage of the carrier transport, due to the gradual filling of deeper traps. However, it is difficult to provide an analytical description of the MT transport taking into account the trap occupancy saturation. Only some special cases have been studied so far [7]. In the following formulas, the free and trapped carrier densities are denoted by nðz; tÞ and nt ðz; tÞ, respectively, where z ¼ x=l0 E is the reduced space variable (x is the space variable, l0 – the free carrier mobility and E – the electric field strength) and t is the time variable. The MT carrier transport can be described by the continuity equation:
o onðz; tÞ ½nðz; tÞ þ nt ðz; tÞ þ ¼ 0; ot oz
ð1Þ
and the equation relating the free and trapped carrier densities [8]: * Tel.: +48 58 347 16 50; fax: +48 58 347 28 21. E-mail address: [email protected] 0022-3093/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2009.05.039
nt ðz; t Þ ¼
Z 0
t
Uðt0 Þnðz; t t0 Þdt 0 :
ð2Þ
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Here, the function UðtÞ determines the probability that the carrier is trapped in a time unit and remains in the trap until time t. This function is given by the formula:
UðtÞ ¼ C t
Z
1
Nt ðeÞ exp ½t=sr ðeÞde;
ð3Þ
0
where C t is the carrier capture coefficient, Nt ðeÞ is the trap density at the energy level e per unit energy, and 1 0
sr ðeÞ ¼ m exp ðe=kT Þ
ð4Þ
is the mean lifetime of the trapped carrier (m0 is the frequency factor, k – the Boltzmann constant and T – the sample temperature). The energy e is measured from the edge of the allowed band. The current intensity IðtÞ, registered in the TOF experiment, equals the conduction current intensity in the sample, averaged over its thickness. Therefore
IðtÞ ¼
Z s0
I0 n 0 s0
nðz; tÞdz;
ð5Þ
0
where n0 is the density of generated carriers, averaged over sample thickness, s0 ¼ d=l0 E and I0 ¼ en0 l0 ES are respectively the carrier time-of-flight and the initial current intensity in a trap-free sample (with d being the sample thickness, e – the elementary charge and S – the sample cross-sectional area). For times smaller than the effective carrier transit time se through the sample, Eq. (5) may be rewritten as
IðtÞ ¼ I0
dzðtÞ ; dt
t < se ;
ð6Þ
where the dash denotes averaging over the spatial carrier distribution. The transit time se is implicitly given by the formula
zðse Þ ¼ s0 :
The last factors in integrands in Eqs. (11) and (12) may be approximated by the unit step function, H½e0 ðtÞ e, provided that the functions N t ðeÞsr ðeÞ and N t ðeÞs2r ðeÞ vary sufficiently slowly with energy. Then,
ð7Þ
H1 ðtÞ 1 þ C t Z
ss ðtÞ C t
Z
e0 ðtÞ
e0 ðtÞ
0
Nt ðeÞs2r ðeÞde:
An approximate solution of the equations identical to (1) and (10) has been already obtained in the paper [9], dealing with non-isothermal carrier transport, and has the form of
n0 s0 HðtÞ
nðz; tÞ
( exp
fðtÞ ¼
nðz; t t 0 Þ nðz; tÞ t 0
onðz; tÞ : ot
ð9Þ
This results in an approximate equation, describing carrier trapping/detrapping processes,
h
i onðz; tÞ ; nt ðz; tÞ H1 ðtÞ 1 nðz; tÞ ss ðtÞ ot
ð10Þ
H1 ðtÞ ¼ 1 þ C t
ss ðtÞ ¼ C t
Z 0
Z
Z
1
Nt ðeÞsr ðeÞ½1 exp ½t=sr ðeÞde;
ð11Þ
Nt ðeÞs2r ðeÞf1 ½1 þ t=sr ðeÞ exp ½t=sr ðeÞgde:
ð12Þ
0 1
ð15Þ ð16Þ
t
Hðt 0 Þdt0 ;
ð17Þ
ss ðt0 ÞH3 ðt0 Þdt0 :
ð18Þ
0
nðtÞ ¼
Z
t
0
Thus, the carrier packet in the considered approximation has a Gaussian shape. The ‘centroid’ and the RMS spread of carrier distribution are given respectively by the formulas:
zðtÞ ¼ fðtÞ;
ð19Þ
rðtÞ ¼ ½2nðtÞ1=2 :
ð20Þ
The above results constitute a straightforward extension of those obtained for the Gaussian carrier transport [10,11]. In such a case the functions HðtÞ and ss ðtÞ are constant which implies that zðtÞ / t and rðtÞ / t 1=2 . Inserting the free carrier density (15) into the integral (5) the following formula for the current transient intensity is obtained:
( " #) I0 HðtÞ s0 fðtÞ ; IðtÞ ¼ 1 þ erf 2 2n1=2 ðtÞ
ð21Þ
where erfð. . .Þ is the error function. The initial current decay and the effective carrier transit time se , corresponding approximately to the transition to faster final current decay are given by
IðtÞ I0 HðtÞ;
where the functions:
) ½z fðtÞ2 ; 4nðtÞ
2½pnðtÞ1=2 h i nt ðz; tÞ H1 ðtÞ 1 nðz; tÞ;
The progress of carrier thermalization in trapping states is characterized by the demarcation level e0 ðtÞ [4,5,8], defined implicitly by the formula sr ½e0 ðtÞ ¼ 1:8t, which gives:
The level separates the traps approximately with equilibrium (e < e0 ðtÞ) and non-equilibrium (e > e0 ðtÞ) occupancy. In the case of weakly dispersive transport, when the approximate thermal equilibrium between free carriers and the majority of the trapped carriers is established, Eq. (2) describing the trapping/detrapping kinetics can be simplified. If the trap density in the energy gap decreases sufficiently fast, the main contribution to the integral in Eq. (3) should proceed from the energy region e < e0 ðtÞ. The exponential function argument in the integrand is then much larger than unity for almost all values of energy e, and the function UðtÞ should differ significantly from zero only for very small time values. The free carrier density in Eq. (2) can be then replaced by the initial terms of its Taylor series,
ð14Þ
4. Solution of transport equations
where the functions
ð8Þ
ð13Þ
It should be stressed that both integrals are calculated over the energy interval 0 6 e 6 e0 ðtÞ, where the trapped carriers are in a thermal quasi-equilibrium. Equations equivalent to Eq. (10), with the last term having been omitted, have been obtained in [4,5] under the assumption of an exact thermal equilibrium between free carriers and carriers trapped in the energy region e 6 e0 ðtÞ. As has been already indicated, this approach has been criticized [6], as it does not describe the spatial carrier dispersion. The mentioned term approximately takes into account the deviations of carrier densities from their equilibrium values. It will be seen that the presence of the term results in a finite spread of the carrier packet.
3. Thermal quasi-equilibrium approximation
e0 ðtÞ ¼ kT lnð1:8m0 tÞ:
Nt ðeÞsr ðeÞde;
0
fðse Þ ¼ s0 :
t < se ;
ð22Þ ð23Þ
It follows from Eqs. (15)–(17) as well as Eqs. (22),(23) that the carrier packet’s effective mobility determined by the trapping/ detrapping events is given by leff ðtÞ ¼ l0 HðtÞ. The effective carrier
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W. Tomaszewicz / Journal of Non-Crystalline Solids 355 (2009) 1414–1418
mobility concept in the case of dispersive transport has been introduced in [4,5] and its validity is confirmed by this study.
zðtÞ
5. Functions zðtÞ and rðtÞ for model trap distributions
rðtÞ
Formulas determining ‘centroid’, zðtÞ, and RMS, rðtÞ, of the carrier packet for some model distributions of traps are given in this section. These functions almost completely characterize the carrier transport in the initial time interval, t 6 se . In particular, the zðtÞ function determines the current transient form for t < se , as well as the transit time se (see Eqs. (6),(7)). The mentioned functions are calculated for the exponential distribution,
Nt ðeÞ ¼
Ntot e e0t ; exp kT c kT c
e P e0t ;
ða 1Þst t "
as0r
;
a > 1;
#1=2 ða 1Þ3 st 1:8t ð3aÞ=2 ; 0:9ð2 aÞð3 aÞ a s0r " #1=2 2ða 1Þ3 st t 1=2 rðtÞ ; a > 2: ða 2Þ a s0r
ð30Þ 1 < a < 2;
ð31Þ
ð32Þ
Since the carrier packet ‘centroid’ velocity is constant, the carrier transport may be in fact considered Gaussian. However, the
ð24Þ
and for a special case of the Gaussian distribution,
" # 2Ntot e e0t 2 ; Nt ðeÞ ¼ 1=2 exp p kT c kT c
e P e0t :
ð25Þ
In both expressions N tot stands for the total density of traps, the characteristic temperature T c determines the trap density decrease rate with energy and e0t denotes the lower limit of the trap distribution (N t ðeÞ ¼ 0 for e < e0t ). The ‘cut-off’ in the trap density is introduced for calculational convenience in numerical simulations of the carrier transport. The simulation results are presented in the next section. The majority of formulas are derived under the assumptions that t s0r (with s0r ¼ sr ðe0t Þ) and that HðtÞ 1 in case of approximate formulas. The functions zðtÞ and rðtÞ for exponential trap distribution (24) and a dispersive transport regime corresponding to the value of parameter a ¼ T=T c < 1 may be calculated exactly. This can be done, making use of the general solutions of MT equations in terms of Laplace transforms [10,12], as well as of some results given in [1]. The obtained formulas have the form:
zðtÞ ¼
a st t ; a < 1; 2 0 s C ð1 þ aÞCð1 aÞ r
ð26Þ
Fig. 1. Current transients calculated for exponential trap distribution (24) and several values of a ¼ T=T c . Other calculation parameters: s0 =st ¼ 104 , m0 s0 ¼ 106 , e0t =kT ¼ 20. Points and lines denote numerical and analytical results, respectively, whereas arrows indicate effective transit times se .
"
rðtÞ ¼
#1=2 a 2 1 st t ; a < 1; 2 Cð1 þ 2aÞ C ð1 þ aÞ Cð1 þ aÞCð1 aÞ s0r ð27Þ
where st ¼ 1=C t N tot denotes the mean trapping time of free carriers and Cð. . .Þ is the Euler gamma function. On the other hand, the functions zðtÞ and rðtÞ, calculated from Eqs. (19),(20), are as follows:
a ð1 aÞst 1:8t ; a < 1; 1:8a2 s0r " #1=2 a ð1 aÞ3 1:8t rðtÞ s ; t 1:8a3 ð2 aÞ s0r
zðtÞ
ð28Þ 0:5 < a < 1:
ð29Þ
Therefore, the exact and approximate expressions for zðtÞ and
rðtÞ differ solely in respect of multiplicative coefficients, which proves to some extent that the present approach is correct. It is seen that the carrier packet ‘centroid’ velocity, dzðtÞ=dt, decreases in time, which is a characteristic feature of dispersive transport. The carrier transport regime for exponential distribution of traps with a > 1 is commonly regarded as Gaussian. In this case the exact formulas for zðtÞ and rðtÞ probably cannot be obtained. The approximate formulas have the form:
Fig. 2. Current transients computed for Gaussian trap distribution (25) and several values of a ¼ T=T c . Other calculation parameters: s0 =st ¼ 104 , m0 s0 ¼ 106 , e0t =kT ¼ 20. The notations are as in Fig. 1.
W. Tomaszewicz / Journal of Non-Crystalline Solids 355 (2009) 1414–1418
carrier packet dispersion for 1 < a < 2 increases faster than the square root of time, contrary to the pure Gaussian transport case. The formulas determining the considered functions for Gaussian trap distribution (25) are the following:
zðtÞ
rðtÞ
exp 1=4a st t ; ½1 þ erf ð1=2aÞs0r 1=2 exp 1=8a2 f2½1 þ erf ð1=aÞg st t 1=2 2
½1 þ erf ð1=2aÞ
3=2
s0r
ð33Þ
;
ð34Þ
where the parameter a ¼ T=T c . These formulas are derived under more restrictive assumptions than before, that is t exp½ð1þ aÞ=a2 s0r . The carrier transport in this time region is Gaussian for an arbitrary value of a. However, if the density of traps decays slowly with energy, so that a < 1, the time of carrier thermalization might be very long. In this case a gradual transition from a disper-
1417
sive to Gaussian transport regime should occur. The idea of such transition has been primarily introduced in [13]. 6. Numerical results Monte Carlo simulations of MT carrier transport are performed in order to verify the accuracy of formulas determining the current transients and the related quantities. The utilized procedure is similar to that described in [14]. Numerical results (denoted by points) in the following figures are compared with analytical results (denoted by lines). In calculations, the exact formulas (11),(12) for the functions HðtÞ and ss ðtÞ are used. The integrals (17),(18), determining the functions fðtÞ and nðtÞ, are computed numerically. Figs. 1 and 2 show several current transients, obtained for exponential (24) and Gaussian (25) trap distributions, respectively. The arrows mark the effective transit time se , calculated from Eq. (23).
Fig. 3. Spatial carrier distribution for several times calculated for exponential trap distribution (24) with a ¼ 0:75. The remaining parameters are as in Fig. 1. Points and lines mark numerical and analytical results, respectively.
Fig. 4. Energetic distribution of carriers for two times computed for exponential trap distribution (24) with a ¼ 0:75. The remaining parameters are as in Fig. 1. Points and lines denote numerical and analytical results, respectively. Sloping full and dashed lines refer, to quasi-equlibrium and non-equilibrium carrier distributions, respectively.
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W. Tomaszewicz / Journal of Non-Crystalline Solids 355 (2009) 1414–1418
As expected, the accuracy of analytical solutions for both distributions improves with increase in parameter a. The relative difference between current intensities in the time interval t < se , calculated numerically and analytically for smaller values of a, remains constant and decreases for exponential and Gaussian distributions, respectively. The accuracy of analytical results in the time region t P se is better for the Gaussian distribution. This is explained by the fact that in the case of exponential distribution, the ratio of carrier densities, captured in the energy intervals e < e0 ðtÞ and e > e0 ðtÞ (with e0 ðtÞ e0t kT), is constant and the carrier transport continues to remain dispersive. In the Gaussian distribution case the mentioned ratio increases in time and a transition from a dispersive to Gaussian transport regime takes place. Fig. 3 presents the time evolution of the total carrier density, ntot ðz; tÞ ¼ nðz; tÞ þ nt ðz; tÞ, in the sample for exponential trap distribution with a ¼ 0:75. In this case the spatial carrier distribution, computed numerically, differs remarkably from the Gaussian distribution, given by Eqs. (15),(16). Nevertheless, the positions of ‘centroids’ of both distributions are nearly the same, which explains the similarities of current transients, calculated analytically and numerically for t < se . Fig. 4 shows the trapped carrier densities per energy unit, n0t ðz; t; eÞ, averaged over sample thickness, for exponential distribution of traps with a ¼ 0:75. It should be expected that the energetic distribution of trapped carriers is given by the relationships: n0t ðt; eÞ / N t ðeÞsr ðeÞ for e < e0 ðtÞ and n0t ðt; eÞ / N t ðeÞ for e > e0 ðtÞ. In the case of exponential trap distribution the relationships take the form of n0t ðt; eÞ / expðe=kT e=kT c Þ for e < e0 ðtÞ and n0t ðt; eÞ / expðe=kT c Þ for e > e0 ðtÞ. The sloping full and dashed lines in the figure correspond to these relationships accordingly, whereas the horizontal lines mark the position of demarcation level e0 ðtÞ. It is seen that the energetic distribution of trapped carriers, computed numerically, is in good agreement with the above predictions. 7. Conclusions and final remarks In this paper, an approximate description of current transients, registered by the TOF method, is given for the case of MT quasiequilibrium carrier transport. The obtained formulas are verified
by comparison with some exact formulas, as well as with the results of MT transport simulations for model trap distributions. The agreement is quite satisfactory, provided that the trap density decreases sufficiently fast with energy. The main aim of TOF measurements in disordered solids is to determine the energetic distribution of traps and their parameters. Several methods of analysis of experimental data may be used for this purpose (see, for example, the review [15]). Basing on the results obtained in this paper, yet another method can be proposed. Making use of Eqs. (8), (13) and (22) and assuming that HðtÞ 1, the following formula is obtained:
d Q0 1:8s0 C t Nt ½e0 ðtÞ; dt IðtÞ
t < se :
ð35Þ
Here, Q 0 is the total charge generated in the sample equal to the area under the curve IðtÞ versus t. The above formula enables us to calculate the function l1 0 C t N t ½e0 ðtÞ. The frequency factor value m0 , required to determine demarcation energy e0 ðtÞ (see Eq. (8)), may be found from the TOF measurements at several temperatures. For a proper value of m0 , all calculated trap distributions should coincide, provided that the temperature dependencies of l0 , C t and m0 are negligible. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
H. Scher, E.W. Montroll, Phys. Rev. B 12 (1975) 2455. G. Pfister, H. Scher, Adv. Phys. 27 (1978) 747. J.M. Marshall, Rep. Prog. Phys. 46 (1983) 1235. T. Tiedje, A. Rose, Solid State Commun. 37 (1980) 49. J. Orenstein, M.A. Kastner, Phys. Rev. Lett. 46 (1981) 1421. V.I. Arkhipov, M.S. Iovu, A.I. Rudenko, S.D. Shutov, Solid State Commun. 62 (1987) 339. V.I. Arkhipov, V.M. Login, A.I. Rudenko, Fiz. Tekh. Poluprovodn. 20 (1986) 1309. W. Tomaszewicz, B. Jachym, J. Non-Cryst. Solids 65 (1984) 193. W. Tomaszewicz, J. Phys.: Condens. Matter 4 (1992) 3967. F.W. Schmidlin, Phys. Rev. B 16 (1977) 2362. V.I. Arkhipov, A.I. Rudenko, Philos. Mag. B 45 (1982) 177. J. Noolandi, Phys. Rev. B 16 (1977) 4466, 4474. E. Muller-Horsche, D. Haarer, H. Scher, Phys. Rev. B 35 (1987) 1273. J.M. Marshall, Philos. Mag. 36 (1977) 959. J.M. Marshall, Philos. Mag. 80 (2000) 1705.
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol
On weakly dispersive multiple-trapping transport W. Tomaszewicz * Department of Physics of Electronic Phenomena, Gdansk University of Technology, Narutowicza 11/12, 80-952 Gdansk, Pomerania, Poland
a r t i c l e
i n f o
Article history: Available online 15 June 2009 PACS: 72.20.Jv 72.80.Ng Keyword: Electrical and electronic properties
a b s t r a c t Equations for multiple-trapping carrier transport, corresponding to the time-of-flight method are approximately solved under the assumption that the majority of carriers are in a thermal quasi-equilibrium. The solutions show a Gaussian shape of the carrier packet. The mean velocity of the carrier sheet for the dispersive transport regime decreases in time and its dispersion grows faster than the square root of time. The accuracy of the obtained formulas is verified by Monte Carlo calculations for exponential and Gaussian trap distributions. A satisfactory agreement is obtained up to the effective carrier transit time, provided that the trap density falls-off sufficiently fast in the energy gap. A new method of determining energetic trap profiles in disordered solids from the time-of-flight measurements is proposed. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction The time-of-fight (TOF) technique is a straightforward, frequently applied method of investigating the carrier transport in low-conductivity solids, both crystalline and amorphous. The sample is sandwiched between two electrodes with a constant voltage applied, and the excess carriers are generated by a short light pulse. The carrier motion in the sample induces a current transient in the measuring circuit. Information about the carrier transport mechanism can be inferred from the transient form as well as from its dependence on the experimental parameters. As regards disordered solids, there exist two basic carrier transport mechanisms – multiple-trapping (MT) and hopping. Transitions of carriers between extended states and localized states (traps) gap occur in case of MT, whereas straightforward carrier transitions between localized states take place in case of hopping. For both mechanisms, the carrier transport may be either Gaussian or dispersive. The former transport regime is characterized by constant velocity and a Gaussian carrier sheet shape in a solid, the latter – by a gradual decrease in the mean velocity and an extremely large dispersion of the carrier packet. The first successful theory of dispersive transport was developed by Scher and Montroll [1], who attributed this phenomenon to very slow equilibration of charge carriers over localized states. The Scher–Montroll theory initiated extensive investigations on dispersive transport (see the reviews [2,3] for earlier works). In spite of this, some problems seem to be still unresolved. In particular, this concerns the simplified description of MT dispersive
transport, given by Tiedje and Rose [4] and by Orenstein and Kastner [5]. Their main idea was that the majority of trapped carriers for specific trap distributions are in a thermal quasi-equilibrium with the free carriers. This approach has been utilized in many subsequent papers. However, its validity has been questioned by Arkhipov et al. [6], since it does not describe the carrier packet broadening. The main aim of this paper is to resolve this controversy. 2. Transport equations The present investigations are based on a standard MT model, assuming very small trap occupancy, electric field uniformity in the sample as well as negligible carrier diffusion. It should be borne in mind that the first assumption may be incorrect at the final stage of the carrier transport, due to the gradual filling of deeper traps. However, it is difficult to provide an analytical description of the MT transport taking into account the trap occupancy saturation. Only some special cases have been studied so far [7]. In the following formulas, the free and trapped carrier densities are denoted by nðz; tÞ and nt ðz; tÞ, respectively, where z ¼ x=l0 E is the reduced space variable (x is the space variable, l0 – the free carrier mobility and E – the electric field strength) and t is the time variable. The MT carrier transport can be described by the continuity equation:
o onðz; tÞ ½nðz; tÞ þ nt ðz; tÞ þ ¼ 0; ot oz
ð1Þ
and the equation relating the free and trapped carrier densities [8]: * Tel.: +48 58 347 16 50; fax: +48 58 347 28 21. E-mail address: [email protected] 0022-3093/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2009.05.039
nt ðz; t Þ ¼
Z 0
t
Uðt0 Þnðz; t t0 Þdt 0 :
ð2Þ
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W. Tomaszewicz / Journal of Non-Crystalline Solids 355 (2009) 1414–1418
Here, the function UðtÞ determines the probability that the carrier is trapped in a time unit and remains in the trap until time t. This function is given by the formula:
UðtÞ ¼ C t
Z
1
Nt ðeÞ exp ½t=sr ðeÞde;
ð3Þ
0
where C t is the carrier capture coefficient, Nt ðeÞ is the trap density at the energy level e per unit energy, and 1 0
sr ðeÞ ¼ m exp ðe=kT Þ
ð4Þ
is the mean lifetime of the trapped carrier (m0 is the frequency factor, k – the Boltzmann constant and T – the sample temperature). The energy e is measured from the edge of the allowed band. The current intensity IðtÞ, registered in the TOF experiment, equals the conduction current intensity in the sample, averaged over its thickness. Therefore
IðtÞ ¼
Z s0
I0 n 0 s0
nðz; tÞdz;
ð5Þ
0
where n0 is the density of generated carriers, averaged over sample thickness, s0 ¼ d=l0 E and I0 ¼ en0 l0 ES are respectively the carrier time-of-flight and the initial current intensity in a trap-free sample (with d being the sample thickness, e – the elementary charge and S – the sample cross-sectional area). For times smaller than the effective carrier transit time se through the sample, Eq. (5) may be rewritten as
IðtÞ ¼ I0
dzðtÞ ; dt
t < se ;
ð6Þ
where the dash denotes averaging over the spatial carrier distribution. The transit time se is implicitly given by the formula
zðse Þ ¼ s0 :
The last factors in integrands in Eqs. (11) and (12) may be approximated by the unit step function, H½e0 ðtÞ e, provided that the functions N t ðeÞsr ðeÞ and N t ðeÞs2r ðeÞ vary sufficiently slowly with energy. Then,
ð7Þ
H1 ðtÞ 1 þ C t Z
ss ðtÞ C t
Z
e0 ðtÞ
e0 ðtÞ
0
Nt ðeÞs2r ðeÞde:
An approximate solution of the equations identical to (1) and (10) has been already obtained in the paper [9], dealing with non-isothermal carrier transport, and has the form of
n0 s0 HðtÞ
nðz; tÞ
( exp
fðtÞ ¼
nðz; t t 0 Þ nðz; tÞ t 0
onðz; tÞ : ot
ð9Þ
This results in an approximate equation, describing carrier trapping/detrapping processes,
h
i onðz; tÞ ; nt ðz; tÞ H1 ðtÞ 1 nðz; tÞ ss ðtÞ ot
ð10Þ
H1 ðtÞ ¼ 1 þ C t
ss ðtÞ ¼ C t
Z 0
Z
Z
1
Nt ðeÞsr ðeÞ½1 exp ½t=sr ðeÞde;
ð11Þ
Nt ðeÞs2r ðeÞf1 ½1 þ t=sr ðeÞ exp ½t=sr ðeÞgde:
ð12Þ
0 1
ð15Þ ð16Þ
t
Hðt 0 Þdt0 ;
ð17Þ
ss ðt0 ÞH3 ðt0 Þdt0 :
ð18Þ
0
nðtÞ ¼
Z
t
0
Thus, the carrier packet in the considered approximation has a Gaussian shape. The ‘centroid’ and the RMS spread of carrier distribution are given respectively by the formulas:
zðtÞ ¼ fðtÞ;
ð19Þ
rðtÞ ¼ ½2nðtÞ1=2 :
ð20Þ
The above results constitute a straightforward extension of those obtained for the Gaussian carrier transport [10,11]. In such a case the functions HðtÞ and ss ðtÞ are constant which implies that zðtÞ / t and rðtÞ / t 1=2 . Inserting the free carrier density (15) into the integral (5) the following formula for the current transient intensity is obtained:
( " #) I0 HðtÞ s0 fðtÞ ; IðtÞ ¼ 1 þ erf 2 2n1=2 ðtÞ
ð21Þ
where erfð. . .Þ is the error function. The initial current decay and the effective carrier transit time se , corresponding approximately to the transition to faster final current decay are given by
IðtÞ I0 HðtÞ;
where the functions:
) ½z fðtÞ2 ; 4nðtÞ
2½pnðtÞ1=2 h i nt ðz; tÞ H1 ðtÞ 1 nðz; tÞ;
The progress of carrier thermalization in trapping states is characterized by the demarcation level e0 ðtÞ [4,5,8], defined implicitly by the formula sr ½e0 ðtÞ ¼ 1:8t, which gives:
The level separates the traps approximately with equilibrium (e < e0 ðtÞ) and non-equilibrium (e > e0 ðtÞ) occupancy. In the case of weakly dispersive transport, when the approximate thermal equilibrium between free carriers and the majority of the trapped carriers is established, Eq. (2) describing the trapping/detrapping kinetics can be simplified. If the trap density in the energy gap decreases sufficiently fast, the main contribution to the integral in Eq. (3) should proceed from the energy region e < e0 ðtÞ. The exponential function argument in the integrand is then much larger than unity for almost all values of energy e, and the function UðtÞ should differ significantly from zero only for very small time values. The free carrier density in Eq. (2) can be then replaced by the initial terms of its Taylor series,
ð14Þ
4. Solution of transport equations
where the functions
ð8Þ
ð13Þ
It should be stressed that both integrals are calculated over the energy interval 0 6 e 6 e0 ðtÞ, where the trapped carriers are in a thermal quasi-equilibrium. Equations equivalent to Eq. (10), with the last term having been omitted, have been obtained in [4,5] under the assumption of an exact thermal equilibrium between free carriers and carriers trapped in the energy region e 6 e0 ðtÞ. As has been already indicated, this approach has been criticized [6], as it does not describe the spatial carrier dispersion. The mentioned term approximately takes into account the deviations of carrier densities from their equilibrium values. It will be seen that the presence of the term results in a finite spread of the carrier packet.
3. Thermal quasi-equilibrium approximation
e0 ðtÞ ¼ kT lnð1:8m0 tÞ:
Nt ðeÞsr ðeÞde;
0
fðse Þ ¼ s0 :
t < se ;
ð22Þ ð23Þ
It follows from Eqs. (15)–(17) as well as Eqs. (22),(23) that the carrier packet’s effective mobility determined by the trapping/ detrapping events is given by leff ðtÞ ¼ l0 HðtÞ. The effective carrier
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mobility concept in the case of dispersive transport has been introduced in [4,5] and its validity is confirmed by this study.
zðtÞ
5. Functions zðtÞ and rðtÞ for model trap distributions
rðtÞ
Formulas determining ‘centroid’, zðtÞ, and RMS, rðtÞ, of the carrier packet for some model distributions of traps are given in this section. These functions almost completely characterize the carrier transport in the initial time interval, t 6 se . In particular, the zðtÞ function determines the current transient form for t < se , as well as the transit time se (see Eqs. (6),(7)). The mentioned functions are calculated for the exponential distribution,
Nt ðeÞ ¼
Ntot e e0t ; exp kT c kT c
e P e0t ;
ða 1Þst t "
as0r
;
a > 1;
#1=2 ða 1Þ3 st 1:8t ð3aÞ=2 ; 0:9ð2 aÞð3 aÞ a s0r " #1=2 2ða 1Þ3 st t 1=2 rðtÞ ; a > 2: ða 2Þ a s0r
ð30Þ 1 < a < 2;
ð31Þ
ð32Þ
Since the carrier packet ‘centroid’ velocity is constant, the carrier transport may be in fact considered Gaussian. However, the
ð24Þ
and for a special case of the Gaussian distribution,
" # 2Ntot e e0t 2 ; Nt ðeÞ ¼ 1=2 exp p kT c kT c
e P e0t :
ð25Þ
In both expressions N tot stands for the total density of traps, the characteristic temperature T c determines the trap density decrease rate with energy and e0t denotes the lower limit of the trap distribution (N t ðeÞ ¼ 0 for e < e0t ). The ‘cut-off’ in the trap density is introduced for calculational convenience in numerical simulations of the carrier transport. The simulation results are presented in the next section. The majority of formulas are derived under the assumptions that t s0r (with s0r ¼ sr ðe0t Þ) and that HðtÞ 1 in case of approximate formulas. The functions zðtÞ and rðtÞ for exponential trap distribution (24) and a dispersive transport regime corresponding to the value of parameter a ¼ T=T c < 1 may be calculated exactly. This can be done, making use of the general solutions of MT equations in terms of Laplace transforms [10,12], as well as of some results given in [1]. The obtained formulas have the form:
zðtÞ ¼
a st t ; a < 1; 2 0 s C ð1 þ aÞCð1 aÞ r
ð26Þ
Fig. 1. Current transients calculated for exponential trap distribution (24) and several values of a ¼ T=T c . Other calculation parameters: s0 =st ¼ 104 , m0 s0 ¼ 106 , e0t =kT ¼ 20. Points and lines denote numerical and analytical results, respectively, whereas arrows indicate effective transit times se .
"
rðtÞ ¼
#1=2 a 2 1 st t ; a < 1; 2 Cð1 þ 2aÞ C ð1 þ aÞ Cð1 þ aÞCð1 aÞ s0r ð27Þ
where st ¼ 1=C t N tot denotes the mean trapping time of free carriers and Cð. . .Þ is the Euler gamma function. On the other hand, the functions zðtÞ and rðtÞ, calculated from Eqs. (19),(20), are as follows:
a ð1 aÞst 1:8t ; a < 1; 1:8a2 s0r " #1=2 a ð1 aÞ3 1:8t rðtÞ s ; t 1:8a3 ð2 aÞ s0r
zðtÞ
ð28Þ 0:5 < a < 1:
ð29Þ
Therefore, the exact and approximate expressions for zðtÞ and
rðtÞ differ solely in respect of multiplicative coefficients, which proves to some extent that the present approach is correct. It is seen that the carrier packet ‘centroid’ velocity, dzðtÞ=dt, decreases in time, which is a characteristic feature of dispersive transport. The carrier transport regime for exponential distribution of traps with a > 1 is commonly regarded as Gaussian. In this case the exact formulas for zðtÞ and rðtÞ probably cannot be obtained. The approximate formulas have the form:
Fig. 2. Current transients computed for Gaussian trap distribution (25) and several values of a ¼ T=T c . Other calculation parameters: s0 =st ¼ 104 , m0 s0 ¼ 106 , e0t =kT ¼ 20. The notations are as in Fig. 1.
W. Tomaszewicz / Journal of Non-Crystalline Solids 355 (2009) 1414–1418
carrier packet dispersion for 1 < a < 2 increases faster than the square root of time, contrary to the pure Gaussian transport case. The formulas determining the considered functions for Gaussian trap distribution (25) are the following:
zðtÞ
rðtÞ
exp 1=4a st t ; ½1 þ erf ð1=2aÞs0r 1=2 exp 1=8a2 f2½1 þ erf ð1=aÞg st t 1=2 2
½1 þ erf ð1=2aÞ
3=2
s0r
ð33Þ
;
ð34Þ
where the parameter a ¼ T=T c . These formulas are derived under more restrictive assumptions than before, that is t exp½ð1þ aÞ=a2 s0r . The carrier transport in this time region is Gaussian for an arbitrary value of a. However, if the density of traps decays slowly with energy, so that a < 1, the time of carrier thermalization might be very long. In this case a gradual transition from a disper-
1417
sive to Gaussian transport regime should occur. The idea of such transition has been primarily introduced in [13]. 6. Numerical results Monte Carlo simulations of MT carrier transport are performed in order to verify the accuracy of formulas determining the current transients and the related quantities. The utilized procedure is similar to that described in [14]. Numerical results (denoted by points) in the following figures are compared with analytical results (denoted by lines). In calculations, the exact formulas (11),(12) for the functions HðtÞ and ss ðtÞ are used. The integrals (17),(18), determining the functions fðtÞ and nðtÞ, are computed numerically. Figs. 1 and 2 show several current transients, obtained for exponential (24) and Gaussian (25) trap distributions, respectively. The arrows mark the effective transit time se , calculated from Eq. (23).
Fig. 3. Spatial carrier distribution for several times calculated for exponential trap distribution (24) with a ¼ 0:75. The remaining parameters are as in Fig. 1. Points and lines mark numerical and analytical results, respectively.
Fig. 4. Energetic distribution of carriers for two times computed for exponential trap distribution (24) with a ¼ 0:75. The remaining parameters are as in Fig. 1. Points and lines denote numerical and analytical results, respectively. Sloping full and dashed lines refer, to quasi-equlibrium and non-equilibrium carrier distributions, respectively.
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As expected, the accuracy of analytical solutions for both distributions improves with increase in parameter a. The relative difference between current intensities in the time interval t < se , calculated numerically and analytically for smaller values of a, remains constant and decreases for exponential and Gaussian distributions, respectively. The accuracy of analytical results in the time region t P se is better for the Gaussian distribution. This is explained by the fact that in the case of exponential distribution, the ratio of carrier densities, captured in the energy intervals e < e0 ðtÞ and e > e0 ðtÞ (with e0 ðtÞ e0t kT), is constant and the carrier transport continues to remain dispersive. In the Gaussian distribution case the mentioned ratio increases in time and a transition from a dispersive to Gaussian transport regime takes place. Fig. 3 presents the time evolution of the total carrier density, ntot ðz; tÞ ¼ nðz; tÞ þ nt ðz; tÞ, in the sample for exponential trap distribution with a ¼ 0:75. In this case the spatial carrier distribution, computed numerically, differs remarkably from the Gaussian distribution, given by Eqs. (15),(16). Nevertheless, the positions of ‘centroids’ of both distributions are nearly the same, which explains the similarities of current transients, calculated analytically and numerically for t < se . Fig. 4 shows the trapped carrier densities per energy unit, n0t ðz; t; eÞ, averaged over sample thickness, for exponential distribution of traps with a ¼ 0:75. It should be expected that the energetic distribution of trapped carriers is given by the relationships: n0t ðt; eÞ / N t ðeÞsr ðeÞ for e < e0 ðtÞ and n0t ðt; eÞ / N t ðeÞ for e > e0 ðtÞ. In the case of exponential trap distribution the relationships take the form of n0t ðt; eÞ / expðe=kT e=kT c Þ for e < e0 ðtÞ and n0t ðt; eÞ / expðe=kT c Þ for e > e0 ðtÞ. The sloping full and dashed lines in the figure correspond to these relationships accordingly, whereas the horizontal lines mark the position of demarcation level e0 ðtÞ. It is seen that the energetic distribution of trapped carriers, computed numerically, is in good agreement with the above predictions. 7. Conclusions and final remarks In this paper, an approximate description of current transients, registered by the TOF method, is given for the case of MT quasiequilibrium carrier transport. The obtained formulas are verified
by comparison with some exact formulas, as well as with the results of MT transport simulations for model trap distributions. The agreement is quite satisfactory, provided that the trap density decreases sufficiently fast with energy. The main aim of TOF measurements in disordered solids is to determine the energetic distribution of traps and their parameters. Several methods of analysis of experimental data may be used for this purpose (see, for example, the review [15]). Basing on the results obtained in this paper, yet another method can be proposed. Making use of Eqs. (8), (13) and (22) and assuming that HðtÞ 1, the following formula is obtained:
d Q0 1:8s0 C t Nt ½e0 ðtÞ; dt IðtÞ
t < se :
ð35Þ
Here, Q 0 is the total charge generated in the sample equal to the area under the curve IðtÞ versus t. The above formula enables us to calculate the function l1 0 C t N t ½e0 ðtÞ. The frequency factor value m0 , required to determine demarcation energy e0 ðtÞ (see Eq. (8)), may be found from the TOF measurements at several temperatures. For a proper value of m0 , all calculated trap distributions should coincide, provided that the temperature dependencies of l0 , C t and m0 are negligible. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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