- Email: [email protected]
The relation between static disorder and photoluminescence quenching law in glasses: A numerical technique
ARTICLE IN PRESS Journal of Luminescence 130 (2010) 1721–1724
Contents lists available at ScienceDirect
Journal of Luminescence journal homepage: www.elsevier.com/locate/jlumin
The relation between static disorder and photoluminescence quenching law in glasses: A numerical technique A.F. Zatsepin a,n, E.A. Buntov a, A.L. Ageev b a b
Department of Physics and Technology, Ural State Technical University—UPI, 19, Mira Street, 620002 Ekaterinburg, Russia Institute of Mathematics and Mechanics, RAS, 16, S. Kovalevskaja Street, 620990 Ekaterinburg, Russia
a r t i c l e in fo
abstract
Article history: Received 9 March 2010 Received in revised form 29 March 2010 Accepted 31 March 2010 Available online 8 April 2010
Photoluminescence (PL) quenching, which usually obeys empirical Street law for glassy solids, is analyzed taking into account the fact that the quenching of each particular emission center is statistically described by the known Mott law. A numerical technique for retrieving energy distributions of luminescent centers from PL quenching measurements is proposed. Mathematical treatment procedure is established and tested for reliability. Calculation results for chalcogenide, quartz and leadsilicate glasses demonstrate the possibility for obtaining both ordinary and polymodal energy distributions of centers and also a good promise for application in PL spectroscopy of disordered solids. & 2010 Elsevier B.V. All rights reserved.
Keywords: Glasses Luminescence quenching Disorder Emission centers Energy distribution
1. Introduction
2. Theory
The study of luminescent characteristics of glassy and amorphous materials is important in order to determine which way the type and degree of structural disorder influence the energy structure, electronic and optical properties [1,2]. Revealing the nature of such ‘‘structure–properties’’ relations on the one hand allows understanding the laws and mechanisms of electronic processes taking place in non-crystalline systems. On the other hand it encourages us to extend materials possible application range within both traditional and novel fields of microelectronics, optics and photonics. However luminescence investigations in the most popular temperature range (80–300 K) cannot provide unambiguous choice between different photoluminescence quenching models [3]. Liquid helium temperatures freeze out phonon modes and thus significantly lower the degree of dynamic disorder which rises from atoms thermal motion. Such a strong cooling provides additional opportunities for discovering the role of static disordering in both dissipative processes and energy structure of luminescent centers inside the optically excited matrix. The main purpose of the present work is to establish the quantitative relation between luminescence quenching law and disorder characteristics of glassy matrix.
Intracenter luminescence quenching for crystals may be described by the classic Mott temperature dependency [1]: Ea 1 , ð1Þ IL ðTÞ ¼ I0 1þ pm exp kT
n
Corresponding author. Tel./fax: + 73433759788. E-mail address: [email protected] (A.F. Zatsepin).
0022-2313/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2010.03.039
where pm is the frequency factor, I0 the luminescence intensity low-temperature limit and k the Boltzmann constant. Here activation energy Ea characterizes the barrier height for nonradiative relaxation channel. On the contrary, for amorphous systems the considerable deviation from the Mott law is often observed, which is due to static disordering. This specific kind of disorder appears during the glass forming process and remains unchanged in a wide temperature range. The empirical Street equation [4] was proposed for such case: 1 T , ð2Þ IL ðTÞ ¼ I0 1þ pS exp T0 where ps is the pre-exponential factor, I0 the low-temperature intensity limit and T0 the thermal parameter. However expression (2) has no strict theoretical basis. The present paper proposes one of the simplest models for static disorder [5] but uses it in a special manner. One can consider the luminescence of amorphous system as a superposition of intracenter radiative transitions in statistical ensemble of similar by type but varied by strain and energy local
ARTICLE IN PRESS A.F. Zatsepin et al. / Journal of Luminescence 130 (2010) 1721–1724
luminescence centers. Since the quenching for identical centers usually obeys the Mott law (1), the expression for IL(T) can be, in general, written as follows: Z 1Z 1 Ea 1 IL ðTÞ ¼ I0 1 þ pm exp gðEa , pm ÞdEa dpm : ð3Þ kT 0 0 Here both Ea and pm change from one luminescent center to other instance of the same type. Therefore g(Ea, pm) function takes into account two-dimensional distribution of similar centers in disordered glass matrix with respect to activation energy and frequency factor. However such a complex equation cannot be properly solved on the basis of single experimental PL quenching curve. Fortunately, Eq. (3) itself gives us the clue: the denominator has a linear dependency on pm and a stronger exponential one for g(Ea). If one replaces variable frequency factor with its effective value, then we can avoid one of the integrals and obtain the following approximate expression: Z 1 Ea 1 1þ pm exp gðEa ÞdEa , ð4Þ IL ðTÞ ¼ I0 kT 0
Model PL quenching: lines - ideal points - with model noise
0.8
PL Intensity (a. u.)
1722
0.6
0.4
103 104
0.2 pm= 105 0.0 0
50
100 150 200 Temperature (K)
250
300
Fig. 1. Initial data Im for Eq. (4) calculated from the Gaussian distribution (7). Points are the same curves with addition of noise.
where g(Ea) is the activation energy distribution for similar luminescent centers, which appears to be delta-function in case of perfect matrix. Expressions (2) and (4) were equated asymptotically in Ref.[6]. As a result the approximate analytical form for g function was found. Unfortunately the approach used is not material-dependent and does not take into account the structure and composition. Moreover, precise analytical solving is impossible for the equation mentioned.
1.0
Initial model Calculated (α = 0.05) Calculated with
0.8
noise (α = 0.5)
0.6 2.1. Solving algorithm
Density of states (a. u.)
0.4 Nevertheless integral equation (4) allows numerical treatment. In order to obtain the discrete version of distribution function g(Ea), we must establish an algorithm for such a treatment. In this equation we move from half-infinite [0, N] to finite [0, Emax] interval of activation energy. Choosing the mesh of energies (see appendix for details) we can get the vector g of distribution values corresponding to these activation energies. First of all let us fix the pm parameter, which is an approximation of unknown correct p0m value. The consequences of inaccurate pm prediction will be discussed later in Section 2.2. Then using the Tikhonov method (see Ref.[7] and also Refs.[8–10]) we can determine the numerical solution. Here in the text we omit extensive details of the algorithm used. The whole mathematical procedure can be found in the appendix. Thus after a series of calculations we finally get the discrete representation ga of activation energy distribution function g(Ea), which depends on effective pm parameter (Eq. (4)) and regularization parameter a (see the appendix).
pm = 5⋅103
0.2
0.0 1.0
0.8
0.6
0.4 pm = 105 0.2 Δpm = 5⋅104
2.2. Model-based testing In order to verify the reliability of the method established, we used a Gaussian model for distribution function: ! 1 ðEa E0 Þ2 : ð5Þ gðEa Þ ¼ pffiffiffiffiffiffiffiffiffi exp 2 w2 w p=2 The values of E0 and w were chosen close to those derived from experiment [3]. Firstly the initial data for Eq. (4) have been calculated using the pre-defined solution (5) (see Fig. 1). Furthermore, the result of the equation solved was compared to initial model (5) under different pm values (Fig. 2). Finally random numbers uniformly distributed over 72.5% of maximum intensity were added to initial data in order to simulate noisy signal.
0.0 0.00
0.05
0.10 0.15 0.20 Activation energy (eV)
0.25
Fig. 2. (a) Distributions calculated from Fig. 1 data in comparison to the Gaussian model for pm ¼ 5 103. Similar results for other pm values not shown. (b) The pm parameter determination error modeling for initial pm ¼ 105 and the 50% error.
Verification was performed for pm ranging from 102 to 105 and as it is seen from Fig. 2, the solution almost perfectly coincides with the pre-defined model function in the absence of noise. The addition of latter leads mainly to deviations near zero activation energies. Deviation at the rest of the distribution curve can be
ARTICLE IN PRESS A.F. Zatsepin et al. / Journal of Luminescence 130 (2010) 1721–1724
3. Results and discussion Using the approach described we have processed some known experimental data on low-temperature luminescence quenching for disordered systems of different nature, namely virgin and neutron-irradiated silica glass (Suprasil), chalcogenide and binary lead-silicate glasses (As2S3 [4,5]; 20PbO 80SiO2 [3]). Photoluminescence spectra are shown in Fig. 3. They all have a more or less complex shape with several overlapped bands caused by either intrinsic electronic states or point defects. Modeling results show that lead-silicate glass luminescence is related with band-to-band
Lead-silicate glass Suprasil
100
80
60
Suprasil (n0)
40
20
Lead silicate
As2S3 glass
Suprasil
0 0
50
100 150 200 Temperature (K)
Suprasil (n0)
300
1.0 As2S3 glass
As2S3
0.8
250
Fig. 4. Experimental PL quenching curves for As2S3, 20PbO 80SiO2, virgin and neutron-irradiated silica (Suprasil).
0.8
glass
Density of states (a. u.)
PL Intensity (a. u.)
1.0
Pb6s2Pb6p transitions. In case of SiO2 there is a wide emission spectrum between 2 and 5 eV with more than three components that are excited by 7.6 eV UV light. In our opinion such luminescence is due to singlet–triplet and singlet–singlet transitions of several oxygen-deficient centers (for example, a and b species of ODC [11]). At the same time neutron irradiation of Suprasil glass dramatically raises the 4.5 eV singlet luminescence band [5] and narrows emission spectrum, making it nearly homogeneous. This means that only one type of ODC dominates after irradiation. Quenching curves for these materials are presented in Fig. 4. Calculations performed (Fig. 5) show that the shape of luminescent centers energy distributions differs from the Gaussian model in the whole energy range even though both half-width and maximum positions are close to those for Gaussians. All curves have asymmetric form with the highenergy tail being longer than the low-energy one. The most narrow energy distribution of luminescence is observed for As2S3 glass. Wider ones are obtained for lead-silicate glass and neutronirradiated Suprasil sample. A considerable fraction of low-energy
PL Intensity (a. u.)
vanished by adjusting the regularization parameter. Thereby we can conclude that in case of random noise lying within the 5% interval the Tikhonov method allows recovery of the initial model with acceptable quality if the value of pm parameter is known and belongs to the 102–105 interval. Meanwhile this approach requires the knowledge of pm values to get the precise solution of Eq. (4). In general case the statistical ensemble of luminescence centers has a distribution of pm just as well as for activation energy. Facing such a problem we have modeled possible errors in pm estimation. During the modeling of quenching curve this parameter was assumed, for instance, equal to 105 while later on the 1.5 times higher value was taken for solution. One can see from Fig. 2 that even in this severe case the solution fits the model fairly well. This fact reveals that the processing algorithm has a much higher sensitivity on the activation energy than on the pm parameter. So we can replace possible distribution of the latter with a single mean value. Furthermore in a first approximation the effective pm value can be estimated, in principle, up to an order of magnitude. The most attractive idea is to somehow derive a rough estimation directly from experimental PL quenching curve. It is worthy of note that distributions obtained for lead-silicate glasses fall quite rapidly with growing activation energy [3]. Moreover, low-energy luminescent centers affect mainly low-temperature quenching points leading to deviation from Eq. (1). It is known that high-temperature tail fits well both Eqs. (1) and (2) [3]. Therefore in this paper we will determine the pm value by the fitting high-temperature part of the PL quenching curve with the Mott law (1). This rough approximation in principle may be further refined by several recursive iterations of Eq. (5) solving.
1723
0.6
0.4
0.2
pm = 170
Lead silicate glass
pm = 113
Suprasil
pm = 260
Suprasil (n0)
pm = 8
0.6
0.4
0.2
0.0 1
2
3 4 Photon energy (eV)
5
6
Fig. 3. Normalized experimental PL spectra for As2S3 glass (2.5 eV excitation [4,5]), 20PbO 80SiO2 glass (4 eV excitation [3]), virgin and neutron-irradiated silica glass (Suprasil, 7.6 eV excitation [5]). All spectra recorded at cryogenic temperatures (7.5–15 K).
0.0 0.00
0.02
0.04
0.06 0.08 0.10 0.12 Activation energy (eV)
0.14
0.16
Fig. 5. Luminescence centers distribution with respect to quenching activation energy calculated for different glass types at a ¼0.7 (see appendix). The pm values are derived from high-temperature region obeying the Mott law (1).
ARTICLE IN PRESS 1724
A.F. Zatsepin et al. / Journal of Luminescence 130 (2010) 1721–1724
centers for all studied materials may point out the dominant role of tunneling transitions at helium temperatures. Contrary to chalcogenide and lead–silicate glasses we see the bimodal distribution in case of virgin silica. Such situation evidently points out the presence of two luminescence center types with different energy distributions. Both of them contribute to the experimental quenching curve [5]. Indeed, the bimodal distribution (Fig. 5) agrees well with corresponding luminescence spectra of unirradiated silica (Fig. 3, [5]). Thus one can see that our treatment allows discrimination of different centers with overlapped emission bands by their quenching character similar to lifetime discrimination in kinetics measurements [12].
4. Conclusion The main result of the present study is the proposed technique of photoluminescence quenching analysis. It allows numerical determination of luminescent centers distribution with respect to activation energy, which characterizes the barrier for nonradiative relaxation channel. The results obtained show that energy distribution of luminescence centers caused by static structural disorder is an essential feature for most of the glassy materials. Such distributions significantly differ from nearly single value typical for crystals. They reflect the structural peculiarities of disordered luminescent matrix and are sensitive to radiation treatments performed as well as to the presence of several emission center types. The technique developed appears to be mathematically stable enough and therefore can find an application in luminescence spectroscopy of non-crystalline solids.
Acknowledgements This work was partially supported by Russian Foundation for Basic Research (Grant no. 08-02-01072-a).
Appendix. Application of Tikhonov’s method for solving the integral equation (4) A.1. Discretization As far as experimental PL quenching dependency IL(T) is merely a finite set of points (In; Tn), nA[0; N], the integral equation (4) allows discretization both in temperature T and in activation energy E dimensions. While Tn sequence is pre-defined by experimental conditions, energy values can be chosen uniformly with constant energy increment DE: Ej ¼j DE, j A [0; M]. In this case M and DE values depend on actual distribution width and energy resolution required. Eq. (4) could then be transformed into the system of N linear algebraic equations: In ¼ I0
M X
DE
gj j DE j ¼ 0 1 þpm exp k Tn
M X
Kn,j gj
ð6Þ
j¼0
Here vector gj approximates activation energy distribution: For gj E g(Ej) and Kn,j is the matrix element for this system. convenience we can normalize Eq. (6). Let g ¼ max Kj,j . Then substitutingK-K^ ¼ K=g, IL -^IL ¼ IL =gI0 we get the following finally in matrix form: ^IL ¼ K^ g
ð7Þ
A.2. Tikhonov’s method for solving equation system (7) Note that in the presence of experimental noise one cannot directly solve (6) because it becomes ill-conditioned (unstable) [8] with growing M. So we need to use a regularization method [9] to get reasonable solution. Applying zero-order Tikhonov’s method [7] for equations system solving we obtain T T ^ a, K^ ^IL ¼ ðK^ K^ þ a EÞg
ð8Þ
where a 40 is the regularization parameter and E^ is the identity T matrix. Here K^ is transpose matrix serving just to make the equation kernel square while Tikhonov’s regularization consists in adding small numbers (a) to the leading diagonal elements. It is known from the theory that the system is stable at big a values and becomes unstable when a approaches zero value. On the other hand the presence of aE term in (8) leads to method error, which rises with growing a. Thus one should choose the a value depending on the experimental data error (higher error level requires stronger regularization). In practice ga can be calculated for the sequence of a values beginning from higher ones: a1 4 a2 4 4 ak unless the appearance of ‘‘nonphysical oscillations’’ of gak elements. Then the optimal ak 1 value is chosen for solution. A.3. Boundary conditions Our calculations showed that it is difficult to retrieve the correct first component of Tikhonov’s solution g0a . Specifically the solution lacks smoothness at this point (g0a differs considerably from g1a ). Therefore we have deliberately introduced the following boundary condition, which should provide the smoothness of Tikhonov’s solution: d2 gð2DEÞ ¼0 dE2
ð9Þ
Due to discretization condition (8) looks like g0 ¼2g1–2g2. ^ Unknown g0 was excluded from system (7) so the matrix Konly influences g1,y,gM elements. Then the modified matrix equation (8) is solved using standard methods of linear algebra. After a values the first one is determined as calculating g1a ,. . .,gM g0a ¼ 2g1a g2a . Thus we finally get the discrete representation ga of activation energy distribution function g(Ea), which depends on effective pm parameter (see Eq. (4)). References [1] N.F. Mott, E.A. Davis, in: Electronic Processes in Non-crystalline Materials, Oxford University Press, 1979, 604 p. [2] Fuxi Gan, in: Optical and Spectroscopic Properties of Glasses, Springer-Verlag, Berlin, Heidelberg, 1992, 283 p. [3] A.F. Zatsepin, et al., Glass Phys. Chem. 36 (2010) 205. [4] R.A. Street, Adv. Phys. 25 (1976) 397. [5] C.M. Gee, M. Kastner, Phys. Rev. Lett. 42 (1979) 1765. [6] R.W. Collins, W. Paul, Phys. Rev. B 25 (1982) 5257. [7] A.N. Tikhonov, V.Ya. Arsenin, in: Methods of Solving Ill-posed Problems, Wiley, New York–London, 1977, 258 p. [8] V.K. Ivanov, V.V. Vasin, V.P. Tanana, in: Theory of Linear Ill-posed Problems and its Applications, USP, Utrecht, 2002, 277 p. [9] H.W. Engl, M. Hanke, A. Neubauer, in: Regularization of Inverse Problems, Kluwer, Dordrecht, 1996, 321 p. [10] C.W. Groetsch, in: The theory of tikhonov regularization for Fredholm equations of the first kind, Pitman Advanced Pub. Program, Boston, 1984, 104 p. [11] A.F. Zatsepin, et al., J. Lumin. 122–123 (2007) 152. [12] A.F. Zatsepin, et al., J. Non-Cryst. Solids 1 (2009) 61.
Contents lists available at ScienceDirect
Journal of Luminescence journal homepage: www.elsevier.com/locate/jlumin
The relation between static disorder and photoluminescence quenching law in glasses: A numerical technique A.F. Zatsepin a,n, E.A. Buntov a, A.L. Ageev b a b
Department of Physics and Technology, Ural State Technical University—UPI, 19, Mira Street, 620002 Ekaterinburg, Russia Institute of Mathematics and Mechanics, RAS, 16, S. Kovalevskaja Street, 620990 Ekaterinburg, Russia
a r t i c l e in fo
abstract
Article history: Received 9 March 2010 Received in revised form 29 March 2010 Accepted 31 March 2010 Available online 8 April 2010
Photoluminescence (PL) quenching, which usually obeys empirical Street law for glassy solids, is analyzed taking into account the fact that the quenching of each particular emission center is statistically described by the known Mott law. A numerical technique for retrieving energy distributions of luminescent centers from PL quenching measurements is proposed. Mathematical treatment procedure is established and tested for reliability. Calculation results for chalcogenide, quartz and leadsilicate glasses demonstrate the possibility for obtaining both ordinary and polymodal energy distributions of centers and also a good promise for application in PL spectroscopy of disordered solids. & 2010 Elsevier B.V. All rights reserved.
Keywords: Glasses Luminescence quenching Disorder Emission centers Energy distribution
1. Introduction
2. Theory
The study of luminescent characteristics of glassy and amorphous materials is important in order to determine which way the type and degree of structural disorder influence the energy structure, electronic and optical properties [1,2]. Revealing the nature of such ‘‘structure–properties’’ relations on the one hand allows understanding the laws and mechanisms of electronic processes taking place in non-crystalline systems. On the other hand it encourages us to extend materials possible application range within both traditional and novel fields of microelectronics, optics and photonics. However luminescence investigations in the most popular temperature range (80–300 K) cannot provide unambiguous choice between different photoluminescence quenching models [3]. Liquid helium temperatures freeze out phonon modes and thus significantly lower the degree of dynamic disorder which rises from atoms thermal motion. Such a strong cooling provides additional opportunities for discovering the role of static disordering in both dissipative processes and energy structure of luminescent centers inside the optically excited matrix. The main purpose of the present work is to establish the quantitative relation between luminescence quenching law and disorder characteristics of glassy matrix.
Intracenter luminescence quenching for crystals may be described by the classic Mott temperature dependency [1]: Ea 1 , ð1Þ IL ðTÞ ¼ I0 1þ pm exp kT
n
Corresponding author. Tel./fax: + 73433759788. E-mail address: [email protected] (A.F. Zatsepin).
0022-2313/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2010.03.039
where pm is the frequency factor, I0 the luminescence intensity low-temperature limit and k the Boltzmann constant. Here activation energy Ea characterizes the barrier height for nonradiative relaxation channel. On the contrary, for amorphous systems the considerable deviation from the Mott law is often observed, which is due to static disordering. This specific kind of disorder appears during the glass forming process and remains unchanged in a wide temperature range. The empirical Street equation [4] was proposed for such case: 1 T , ð2Þ IL ðTÞ ¼ I0 1þ pS exp T0 where ps is the pre-exponential factor, I0 the low-temperature intensity limit and T0 the thermal parameter. However expression (2) has no strict theoretical basis. The present paper proposes one of the simplest models for static disorder [5] but uses it in a special manner. One can consider the luminescence of amorphous system as a superposition of intracenter radiative transitions in statistical ensemble of similar by type but varied by strain and energy local
ARTICLE IN PRESS A.F. Zatsepin et al. / Journal of Luminescence 130 (2010) 1721–1724
luminescence centers. Since the quenching for identical centers usually obeys the Mott law (1), the expression for IL(T) can be, in general, written as follows: Z 1Z 1 Ea 1 IL ðTÞ ¼ I0 1 þ pm exp gðEa , pm ÞdEa dpm : ð3Þ kT 0 0 Here both Ea and pm change from one luminescent center to other instance of the same type. Therefore g(Ea, pm) function takes into account two-dimensional distribution of similar centers in disordered glass matrix with respect to activation energy and frequency factor. However such a complex equation cannot be properly solved on the basis of single experimental PL quenching curve. Fortunately, Eq. (3) itself gives us the clue: the denominator has a linear dependency on pm and a stronger exponential one for g(Ea). If one replaces variable frequency factor with its effective value, then we can avoid one of the integrals and obtain the following approximate expression: Z 1 Ea 1 1þ pm exp gðEa ÞdEa , ð4Þ IL ðTÞ ¼ I0 kT 0
Model PL quenching: lines - ideal points - with model noise
0.8
PL Intensity (a. u.)
1722
0.6
0.4
103 104
0.2 pm= 105 0.0 0
50
100 150 200 Temperature (K)
250
300
Fig. 1. Initial data Im for Eq. (4) calculated from the Gaussian distribution (7). Points are the same curves with addition of noise.
where g(Ea) is the activation energy distribution for similar luminescent centers, which appears to be delta-function in case of perfect matrix. Expressions (2) and (4) were equated asymptotically in Ref.[6]. As a result the approximate analytical form for g function was found. Unfortunately the approach used is not material-dependent and does not take into account the structure and composition. Moreover, precise analytical solving is impossible for the equation mentioned.
1.0
Initial model Calculated (α = 0.05) Calculated with
0.8
noise (α = 0.5)
0.6 2.1. Solving algorithm
Density of states (a. u.)
0.4 Nevertheless integral equation (4) allows numerical treatment. In order to obtain the discrete version of distribution function g(Ea), we must establish an algorithm for such a treatment. In this equation we move from half-infinite [0, N] to finite [0, Emax] interval of activation energy. Choosing the mesh of energies (see appendix for details) we can get the vector g of distribution values corresponding to these activation energies. First of all let us fix the pm parameter, which is an approximation of unknown correct p0m value. The consequences of inaccurate pm prediction will be discussed later in Section 2.2. Then using the Tikhonov method (see Ref.[7] and also Refs.[8–10]) we can determine the numerical solution. Here in the text we omit extensive details of the algorithm used. The whole mathematical procedure can be found in the appendix. Thus after a series of calculations we finally get the discrete representation ga of activation energy distribution function g(Ea), which depends on effective pm parameter (Eq. (4)) and regularization parameter a (see the appendix).
pm = 5⋅103
0.2
0.0 1.0
0.8
0.6
0.4 pm = 105 0.2 Δpm = 5⋅104
2.2. Model-based testing In order to verify the reliability of the method established, we used a Gaussian model for distribution function: ! 1 ðEa E0 Þ2 : ð5Þ gðEa Þ ¼ pffiffiffiffiffiffiffiffiffi exp 2 w2 w p=2 The values of E0 and w were chosen close to those derived from experiment [3]. Firstly the initial data for Eq. (4) have been calculated using the pre-defined solution (5) (see Fig. 1). Furthermore, the result of the equation solved was compared to initial model (5) under different pm values (Fig. 2). Finally random numbers uniformly distributed over 72.5% of maximum intensity were added to initial data in order to simulate noisy signal.
0.0 0.00
0.05
0.10 0.15 0.20 Activation energy (eV)
0.25
Fig. 2. (a) Distributions calculated from Fig. 1 data in comparison to the Gaussian model for pm ¼ 5 103. Similar results for other pm values not shown. (b) The pm parameter determination error modeling for initial pm ¼ 105 and the 50% error.
Verification was performed for pm ranging from 102 to 105 and as it is seen from Fig. 2, the solution almost perfectly coincides with the pre-defined model function in the absence of noise. The addition of latter leads mainly to deviations near zero activation energies. Deviation at the rest of the distribution curve can be
ARTICLE IN PRESS A.F. Zatsepin et al. / Journal of Luminescence 130 (2010) 1721–1724
3. Results and discussion Using the approach described we have processed some known experimental data on low-temperature luminescence quenching for disordered systems of different nature, namely virgin and neutron-irradiated silica glass (Suprasil), chalcogenide and binary lead-silicate glasses (As2S3 [4,5]; 20PbO 80SiO2 [3]). Photoluminescence spectra are shown in Fig. 3. They all have a more or less complex shape with several overlapped bands caused by either intrinsic electronic states or point defects. Modeling results show that lead-silicate glass luminescence is related with band-to-band
Lead-silicate glass Suprasil
100
80
60
Suprasil (n0)
40
20
Lead silicate
As2S3 glass
Suprasil
0 0
50
100 150 200 Temperature (K)
Suprasil (n0)
300
1.0 As2S3 glass
As2S3
0.8
250
Fig. 4. Experimental PL quenching curves for As2S3, 20PbO 80SiO2, virgin and neutron-irradiated silica (Suprasil).
0.8
glass
Density of states (a. u.)
PL Intensity (a. u.)
1.0
Pb6s2Pb6p transitions. In case of SiO2 there is a wide emission spectrum between 2 and 5 eV with more than three components that are excited by 7.6 eV UV light. In our opinion such luminescence is due to singlet–triplet and singlet–singlet transitions of several oxygen-deficient centers (for example, a and b species of ODC [11]). At the same time neutron irradiation of Suprasil glass dramatically raises the 4.5 eV singlet luminescence band [5] and narrows emission spectrum, making it nearly homogeneous. This means that only one type of ODC dominates after irradiation. Quenching curves for these materials are presented in Fig. 4. Calculations performed (Fig. 5) show that the shape of luminescent centers energy distributions differs from the Gaussian model in the whole energy range even though both half-width and maximum positions are close to those for Gaussians. All curves have asymmetric form with the highenergy tail being longer than the low-energy one. The most narrow energy distribution of luminescence is observed for As2S3 glass. Wider ones are obtained for lead-silicate glass and neutronirradiated Suprasil sample. A considerable fraction of low-energy
PL Intensity (a. u.)
vanished by adjusting the regularization parameter. Thereby we can conclude that in case of random noise lying within the 5% interval the Tikhonov method allows recovery of the initial model with acceptable quality if the value of pm parameter is known and belongs to the 102–105 interval. Meanwhile this approach requires the knowledge of pm values to get the precise solution of Eq. (4). In general case the statistical ensemble of luminescence centers has a distribution of pm just as well as for activation energy. Facing such a problem we have modeled possible errors in pm estimation. During the modeling of quenching curve this parameter was assumed, for instance, equal to 105 while later on the 1.5 times higher value was taken for solution. One can see from Fig. 2 that even in this severe case the solution fits the model fairly well. This fact reveals that the processing algorithm has a much higher sensitivity on the activation energy than on the pm parameter. So we can replace possible distribution of the latter with a single mean value. Furthermore in a first approximation the effective pm value can be estimated, in principle, up to an order of magnitude. The most attractive idea is to somehow derive a rough estimation directly from experimental PL quenching curve. It is worthy of note that distributions obtained for lead-silicate glasses fall quite rapidly with growing activation energy [3]. Moreover, low-energy luminescent centers affect mainly low-temperature quenching points leading to deviation from Eq. (1). It is known that high-temperature tail fits well both Eqs. (1) and (2) [3]. Therefore in this paper we will determine the pm value by the fitting high-temperature part of the PL quenching curve with the Mott law (1). This rough approximation in principle may be further refined by several recursive iterations of Eq. (5) solving.
1723
0.6
0.4
0.2
pm = 170
Lead silicate glass
pm = 113
Suprasil
pm = 260
Suprasil (n0)
pm = 8
0.6
0.4
0.2
0.0 1
2
3 4 Photon energy (eV)
5
6
Fig. 3. Normalized experimental PL spectra for As2S3 glass (2.5 eV excitation [4,5]), 20PbO 80SiO2 glass (4 eV excitation [3]), virgin and neutron-irradiated silica glass (Suprasil, 7.6 eV excitation [5]). All spectra recorded at cryogenic temperatures (7.5–15 K).
0.0 0.00
0.02
0.04
0.06 0.08 0.10 0.12 Activation energy (eV)
0.14
0.16
Fig. 5. Luminescence centers distribution with respect to quenching activation energy calculated for different glass types at a ¼0.7 (see appendix). The pm values are derived from high-temperature region obeying the Mott law (1).
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centers for all studied materials may point out the dominant role of tunneling transitions at helium temperatures. Contrary to chalcogenide and lead–silicate glasses we see the bimodal distribution in case of virgin silica. Such situation evidently points out the presence of two luminescence center types with different energy distributions. Both of them contribute to the experimental quenching curve [5]. Indeed, the bimodal distribution (Fig. 5) agrees well with corresponding luminescence spectra of unirradiated silica (Fig. 3, [5]). Thus one can see that our treatment allows discrimination of different centers with overlapped emission bands by their quenching character similar to lifetime discrimination in kinetics measurements [12].
4. Conclusion The main result of the present study is the proposed technique of photoluminescence quenching analysis. It allows numerical determination of luminescent centers distribution with respect to activation energy, which characterizes the barrier for nonradiative relaxation channel. The results obtained show that energy distribution of luminescence centers caused by static structural disorder is an essential feature for most of the glassy materials. Such distributions significantly differ from nearly single value typical for crystals. They reflect the structural peculiarities of disordered luminescent matrix and are sensitive to radiation treatments performed as well as to the presence of several emission center types. The technique developed appears to be mathematically stable enough and therefore can find an application in luminescence spectroscopy of non-crystalline solids.
Acknowledgements This work was partially supported by Russian Foundation for Basic Research (Grant no. 08-02-01072-a).
Appendix. Application of Tikhonov’s method for solving the integral equation (4) A.1. Discretization As far as experimental PL quenching dependency IL(T) is merely a finite set of points (In; Tn), nA[0; N], the integral equation (4) allows discretization both in temperature T and in activation energy E dimensions. While Tn sequence is pre-defined by experimental conditions, energy values can be chosen uniformly with constant energy increment DE: Ej ¼j DE, j A [0; M]. In this case M and DE values depend on actual distribution width and energy resolution required. Eq. (4) could then be transformed into the system of N linear algebraic equations: In ¼ I0
M X
DE
gj j DE j ¼ 0 1 þpm exp k Tn
M X
Kn,j gj
ð6Þ
j¼0
Here vector gj approximates activation energy distribution: For gj E g(Ej) and Kn,j is the matrix element for this system. convenience we can normalize Eq. (6). Let g ¼ max Kj,j . Then substitutingK-K^ ¼ K=g, IL -^IL ¼ IL =gI0 we get the following finally in matrix form: ^IL ¼ K^ g
ð7Þ
A.2. Tikhonov’s method for solving equation system (7) Note that in the presence of experimental noise one cannot directly solve (6) because it becomes ill-conditioned (unstable) [8] with growing M. So we need to use a regularization method [9] to get reasonable solution. Applying zero-order Tikhonov’s method [7] for equations system solving we obtain T T ^ a, K^ ^IL ¼ ðK^ K^ þ a EÞg
ð8Þ
where a 40 is the regularization parameter and E^ is the identity T matrix. Here K^ is transpose matrix serving just to make the equation kernel square while Tikhonov’s regularization consists in adding small numbers (a) to the leading diagonal elements. It is known from the theory that the system is stable at big a values and becomes unstable when a approaches zero value. On the other hand the presence of aE term in (8) leads to method error, which rises with growing a. Thus one should choose the a value depending on the experimental data error (higher error level requires stronger regularization). In practice ga can be calculated for the sequence of a values beginning from higher ones: a1 4 a2 4 4 ak unless the appearance of ‘‘nonphysical oscillations’’ of gak elements. Then the optimal ak 1 value is chosen for solution. A.3. Boundary conditions Our calculations showed that it is difficult to retrieve the correct first component of Tikhonov’s solution g0a . Specifically the solution lacks smoothness at this point (g0a differs considerably from g1a ). Therefore we have deliberately introduced the following boundary condition, which should provide the smoothness of Tikhonov’s solution: d2 gð2DEÞ ¼0 dE2
ð9Þ
Due to discretization condition (8) looks like g0 ¼2g1–2g2. ^ Unknown g0 was excluded from system (7) so the matrix Konly influences g1,y,gM elements. Then the modified matrix equation (8) is solved using standard methods of linear algebra. After a values the first one is determined as calculating g1a ,. . .,gM g0a ¼ 2g1a g2a . Thus we finally get the discrete representation ga of activation energy distribution function g(Ea), which depends on effective pm parameter (see Eq. (4)). References [1] N.F. Mott, E.A. Davis, in: Electronic Processes in Non-crystalline Materials, Oxford University Press, 1979, 604 p. [2] Fuxi Gan, in: Optical and Spectroscopic Properties of Glasses, Springer-Verlag, Berlin, Heidelberg, 1992, 283 p. [3] A.F. Zatsepin, et al., Glass Phys. Chem. 36 (2010) 205. [4] R.A. Street, Adv. Phys. 25 (1976) 397. [5] C.M. Gee, M. Kastner, Phys. Rev. Lett. 42 (1979) 1765. [6] R.W. Collins, W. Paul, Phys. Rev. B 25 (1982) 5257. [7] A.N. Tikhonov, V.Ya. Arsenin, in: Methods of Solving Ill-posed Problems, Wiley, New York–London, 1977, 258 p. [8] V.K. Ivanov, V.V. Vasin, V.P. Tanana, in: Theory of Linear Ill-posed Problems and its Applications, USP, Utrecht, 2002, 277 p. [9] H.W. Engl, M. Hanke, A. Neubauer, in: Regularization of Inverse Problems, Kluwer, Dordrecht, 1996, 321 p. [10] C.W. Groetsch, in: The theory of tikhonov regularization for Fredholm equations of the first kind, Pitman Advanced Pub. Program, Boston, 1984, 104 p. [11] A.F. Zatsepin, et al., J. Lumin. 122–123 (2007) 152. [12] A.F. Zatsepin, et al., J. Non-Cryst. Solids 1 (2009) 61.