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Multiple acoustic and optical phonon-assisted hopping in oxide glasses containing transition metal ions
Journal of Non-Crystalline Solids 352 (2006) 4229–4231 www.elsevier.com/locate/jnoncrysol
Multiple acoustic and optical phonon-assisted hopping in oxide glasses containing transition metal ions Ryszard J. Barczyn´ski
*
Faculty of Applied Physics and Mathematics, Gdan´sk University of Technology, 80-952 Gdan´sk, Poland Available online 18 September 2006
Abstract Oxide glasses containing transition metal ions usually exhibit electrical conductivity due to polaron hopping. An interpretation of d/c conductivity data in 50P2O550V2O5, 40P2O560V2O5, 50P2O550FeO, and 50TeO240V2O510Fe2O3 semiconducting oxide glasses is attempted by fitting the Gorhan–Bergeron and Emin model parameters. Their model is much more general than those usually employed to interpret experimental electrical conductivity data and takes into account multiphonon interactions with optical as well as acoustic phonons. The Holstein criterion of non-adiabatic hopping is verified as well. Ó 2006 Elsevier B.V. All rights reserved. PACS: 71.23.Cq; 71.38.k Keywords: Phonons; Conductivity; Oxide glasses
1. Introduction Many glasses containing a large amount of transition metal oxides are electronic semiconductors [1]. Their electrical properties are determined by the presence of transition metal ions in two different valence states. Their conductivity was described by Mott as a mechanism of small polaron hopping between such ions [2] mph e2 cð1 cÞ W r¼ expð2aRÞ exp ; ð1Þ kTR kT where mph denotes the phonon frequency, a – the decay constant of the wave function, R – the mean electron hopping range, and c – the part of transition metal ions in the lower valence state. Concentration of charge carriers may be treated as a constant dependent on c(1 c).
*
Tel.: +48 58 347 18 32; fax: +48 58 347 28 21. E-mail address: [email protected]
0022-3093/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.07.014
The activation energy of the hopping process decreases with decreasing temperature, which is a characteristic feature of this mechanism. In the high temperature region it should achieve a constant value of W = WH + Wd/2, where WH is the polaron formation energy and Wd is the mean difference in energy between hopping centers. Mott suggested that the dominant process decreasing the activation energy is an interaction between polarons and optical phonons. The dependence of activation energy on temperature for this case was evaluated by Schnakenberg [3]: 4kT hx0 W ¼ WH tanh ð2Þ þ W d; hx0 4kT where x0 is the mean frequency of optical phonons. Although both constant activation energy and opticalphonons-only models are approximations only, they are usually applied when experimental data on d/c hopping conductivity are concerned. A much more general model of the non-adiabatic hopping conductivity process which takes into account multiple interactions with optical as well as acoustic phonons
R.J. Barczyn´ski / Journal of Non-Crystalline Solids 352 (2006) 4229–4231
4230
1 W d 1 op E0 b 0 þ þ Eb sech2 FA b 2 4 2 b 2 0 W 2d 12 Eop b E0 F h E 0 2 þ F B E0 b 2 4 Eop þ FB b E0 cosech 2 1 op 2 E E0 F h E20 b þ F B 2 b E0 b ; þ þ FB 2 Eop b E0 cosech 2
was proposed by Gorhan–Bergeron and Emin [4]. It describes polarons’ mobility as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 eR2 J ij h2 p W 2d exp l¼ op ac kT 8ðEop h2 2ðEC þ Eac C þ E C ÞkT C ÞkT W d Eop Eac exp exp A A ; ð3Þ 2kT kt kT
W0 ¼
where Jij denotes the integral of wave function overlapping, and energies EC and EA for acoustic (ac) and optical (op) phonons are respectively given by h2 1 X 2Eop hx0 b Eop ¼ cosech ð4Þ x2q;op ; C 4kT N q hx 0 2kT h2 1 X 2Eac hxq;ac b Eac ¼ cosech ð5Þ x2q;ac ; C 4kT N q hxq;ac 2kT 2kT op hx 0 Eop ¼ E tanh ; ð6Þ A hx0 b 4kT 1 X 2kT ac hxq;ac ¼ E tanh : ð7Þ Eac A N q hxq;ac b 4kT
where
In the above equations x0 is the mean frequency of optical phonons (assuming their low dispersion), xq,ac is the freop quency of acoustic phonons at wave vector q, Eac b and E b are contributions to the polaron binding energy due to interaction with acoustic and optical phonons, respectively, and N is the number of vibration modes for phonons of every type. The aim of the present work is to verify the practical possibility of interpretation of d/c conductivity data in some oxide glasses containing transition metal ions by fitting the Gorhan–Bergeron and Emin model parameters. 2. The method In order to simplify numerical calculations and reduce the number of fitting parameters we calculate the activation energy versus temperature numerically from experimental measurements of d/c conductivity: W0 ¼
d lnðrDC Þ; db
b¼
1 : kT
F A ¼ 8Eac b X F B ¼ 2Eac b X F 0A ¼ 8Eac b X F 0B ¼ 2Eac b X
1 X n¼1 1 X n¼1 1 X n¼1 1 X
ð9Þ
AðkÞE2kþ1 b2k1 ; D
ð10Þ
BðkÞE2kþ3 b2k1 ; D
ð11Þ
AðkÞE2kþ1 ð2k 1Þb2k2 ; D
ð12Þ
BðkÞE2kþ3 ð2k 1Þb2k2 : D
ð13Þ
n¼1
X, A(k) and B(k) are, respectively: X ¼
A ð22k 1ÞBk ; AðkÞ ¼ ; N h3 ð2k þ 1Þð2kÞ!22k
BðkÞ ¼
ð2 22k ÞBk : ð3 þ 2kÞð2kÞ!22k
ð14Þ
Bk are Bernoulli numbers, ED is the maximal energy of acoustic phonons’ spectra, and E0 is the mean energy of optical phonons. Fh is a derivative of the cosech function F h ðxÞ ¼
2ðexpðxÞ þ expðxÞÞ : 2 expð2xÞ expð2xÞ
ð15Þ
We can now fit the activation energy given by Eqs. (9)– (14) to data obtained during measurements. We should find op the following parameters: Wd, Eac b , E b , E0, ED and X. Since ac Eb and X are present in (9)–(14) only as multiples, we cannot calculate them independently – we shall only obtain their product. We have used the non-gradient Powell method of non-linear optimization in order to fit the parameters [6].
ð8Þ
In order to obtain useful equations for the activation energy: 1. we assume that phonon spectra are continuous and replace the sums over the phonon frequency spectra with integrals; 2. after Meek [5], we assume parabolic phonon states density to be g(x) = Ax2, where A is a constant which may differ for various glasses; 3. we then replace hyperbolic functions by their appropriate Taylor expansions. From (3)–(7) we obtain an equation for the mobility activation energy:
3. The results The results of parameter fitting to d/c conductivity measurements data obtained in a couple of glass samples are given in Table 1. An example of fitting obtained from the Powell procedure is shown in Fig. 1. The next interesting microscopic parameter of conductivity, the integral of wave function overlapping, Jij, may be calculated by fitting the pre-exponential factor of the full d/c conductivity equation to experimental data (taking into account all the previously calculated parameters). An exemplary result of such fitting is given in Fig. 2. In order to extract Jij from the pre-exponential factor, assumptions are necessary about concentration NC and the hopping distance, R; assuming NC = 1021 cm3 and R = 4 m10 we
R.J. Barczyn´ski / Journal of Non-Crystalline Solids 352 (2006) 4229–4231 Table 1 The resulting fitted model parameters for some glasses containing transition metal oxides
1 2 3 4
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffi 2kTW H hm0 ; J ij < p p
Composition [% mol]
Eop b ½eV
E0 [eV]
ED [eV]
Wd [eV]
where, in the high temperature range, we may assume
50TeO240V2O510Fe2O3 50P2O550FeO 40P2O560V2O5 50P2O550V2O5
0.568 1.292 0.813 0.304
0.084 0.121 0.120 0.074
0.04 0.04 0.034 0.031
0.043 0.255 0.165 0.076
1 1 ac W H ¼ Eop b þ Eb : 2 2
Fig. 1. Fitting of the activation energy (solid line) to experimental data points for 50P2O550FeO glass obtained from the Powell procedure.
4231
ð16Þ
ð17Þ
The Holstein criterion holds well for all our samples and we may conclude that the hopping is non-adiabatic. All the obtained hopping parameters have very probable values, but there are no parameters in the literature acquired by other methods, which could be used for comparison. Thus, we should be very careful with our procedure; particularly, the measurement data used for calculations should be very precise. It is also quite difficult to estimate the real errors of the obtained parameters. The mean deviation of experimental points from the fitted function is small and the numerical procedure allows calculation of parameters with a precision of 3–4 significant digits. At the same time, electrometric measurements of very low currents are very difficult (practically impossible) to be performed with such precision. Moreover, errors are usually systematic and difficult to estimate and deviations of phonon states’ density from that assumed in the parabolic model are difficult to obtain and directly influence the precision of results. We believe that even the second significant digit should be treated with caution. 4. Conclusion A method of interpretation of d/c conductivity data in the framework of the Gorhan–Bergeron and Emin model has been developed. The fitting of model parameters to experimental data in 50P2O550V2O5, 40P2O560V2O5, 50P2O550FeO and 50TeO240V2O510Fe2O3 oxide glasses has been performed. The following microscopic hopping parameters have been evaluated: the phonon spectra characteristics, the mean difference in energy between hopping centers and the integral of wave function overlapping. The Holstein criterion of non-adiabatic hopping has been positively verified for all these glasses. References
Fig. 2. The result of model fitting (solid line) to experimental conductivity data for 50P2O550FeO glass.
obtain Jij = 0.0094, 0.0074, 0.0184 and 0.0015 eV, respectively for 50TeO240V2O510Fe2O3, 50P2O550FeO, 40P2O560V2O5 and 50P2O550V2O5 glasses. We can now check the Holstein criterion for the non-adiabatic character of hopping [7]
[1] [2] [3] [4] [5] [6]
M. Sayer, A. Mansingh, Phys. Rev. B 6 (1972) 4629. N.F. Mott, J. Non-Cryst. Solids 1 (1968) 1. J. Schnakenberg, Z. Physik 185 (1965) 123. E. Gorham-Bergeron, D. Emin, Phys. Rev. B 15 (1977) 3667. P.E. Meek, Philos. Mag. 33 (1976) 897. I. Szepietowski, Master Thesis, Gdan´sk University of Technology, 1994. [7] T. Holstein, Ann. Phys. 281 (2006) 706; T. Holstein, Ann. Phys. 281 (2006) 725.
Multiple acoustic and optical phonon-assisted hopping in oxide glasses containing transition metal ions Ryszard J. Barczyn´ski
*
Faculty of Applied Physics and Mathematics, Gdan´sk University of Technology, 80-952 Gdan´sk, Poland Available online 18 September 2006
Abstract Oxide glasses containing transition metal ions usually exhibit electrical conductivity due to polaron hopping. An interpretation of d/c conductivity data in 50P2O550V2O5, 40P2O560V2O5, 50P2O550FeO, and 50TeO240V2O510Fe2O3 semiconducting oxide glasses is attempted by fitting the Gorhan–Bergeron and Emin model parameters. Their model is much more general than those usually employed to interpret experimental electrical conductivity data and takes into account multiphonon interactions with optical as well as acoustic phonons. The Holstein criterion of non-adiabatic hopping is verified as well. Ó 2006 Elsevier B.V. All rights reserved. PACS: 71.23.Cq; 71.38.k Keywords: Phonons; Conductivity; Oxide glasses
1. Introduction Many glasses containing a large amount of transition metal oxides are electronic semiconductors [1]. Their electrical properties are determined by the presence of transition metal ions in two different valence states. Their conductivity was described by Mott as a mechanism of small polaron hopping between such ions [2] mph e2 cð1 cÞ W r¼ expð2aRÞ exp ; ð1Þ kTR kT where mph denotes the phonon frequency, a – the decay constant of the wave function, R – the mean electron hopping range, and c – the part of transition metal ions in the lower valence state. Concentration of charge carriers may be treated as a constant dependent on c(1 c).
*
Tel.: +48 58 347 18 32; fax: +48 58 347 28 21. E-mail address: [email protected]
0022-3093/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.07.014
The activation energy of the hopping process decreases with decreasing temperature, which is a characteristic feature of this mechanism. In the high temperature region it should achieve a constant value of W = WH + Wd/2, where WH is the polaron formation energy and Wd is the mean difference in energy between hopping centers. Mott suggested that the dominant process decreasing the activation energy is an interaction between polarons and optical phonons. The dependence of activation energy on temperature for this case was evaluated by Schnakenberg [3]: 4kT hx0 W ¼ WH tanh ð2Þ þ W d; hx0 4kT where x0 is the mean frequency of optical phonons. Although both constant activation energy and opticalphonons-only models are approximations only, they are usually applied when experimental data on d/c hopping conductivity are concerned. A much more general model of the non-adiabatic hopping conductivity process which takes into account multiple interactions with optical as well as acoustic phonons
R.J. Barczyn´ski / Journal of Non-Crystalline Solids 352 (2006) 4229–4231
4230
1 W d 1 op E0 b 0 þ þ Eb sech2 FA b 2 4 2 b 2 0 W 2d 12 Eop b E0 F h E 0 2 þ F B E0 b 2 4 Eop þ FB b E0 cosech 2 1 op 2 E E0 F h E20 b þ F B 2 b E0 b ; þ þ FB 2 Eop b E0 cosech 2
was proposed by Gorhan–Bergeron and Emin [4]. It describes polarons’ mobility as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 eR2 J ij h2 p W 2d exp l¼ op ac kT 8ðEop h2 2ðEC þ Eac C þ E C ÞkT C ÞkT W d Eop Eac exp exp A A ; ð3Þ 2kT kt kT
W0 ¼
where Jij denotes the integral of wave function overlapping, and energies EC and EA for acoustic (ac) and optical (op) phonons are respectively given by h2 1 X 2Eop hx0 b Eop ¼ cosech ð4Þ x2q;op ; C 4kT N q hx 0 2kT h2 1 X 2Eac hxq;ac b Eac ¼ cosech ð5Þ x2q;ac ; C 4kT N q hxq;ac 2kT 2kT op hx 0 Eop ¼ E tanh ; ð6Þ A hx0 b 4kT 1 X 2kT ac hxq;ac ¼ E tanh : ð7Þ Eac A N q hxq;ac b 4kT
where
In the above equations x0 is the mean frequency of optical phonons (assuming their low dispersion), xq,ac is the freop quency of acoustic phonons at wave vector q, Eac b and E b are contributions to the polaron binding energy due to interaction with acoustic and optical phonons, respectively, and N is the number of vibration modes for phonons of every type. The aim of the present work is to verify the practical possibility of interpretation of d/c conductivity data in some oxide glasses containing transition metal ions by fitting the Gorhan–Bergeron and Emin model parameters. 2. The method In order to simplify numerical calculations and reduce the number of fitting parameters we calculate the activation energy versus temperature numerically from experimental measurements of d/c conductivity: W0 ¼
d lnðrDC Þ; db
b¼
1 : kT
F A ¼ 8Eac b X F B ¼ 2Eac b X F 0A ¼ 8Eac b X F 0B ¼ 2Eac b X
1 X n¼1 1 X n¼1 1 X n¼1 1 X
ð9Þ
AðkÞE2kþ1 b2k1 ; D
ð10Þ
BðkÞE2kþ3 b2k1 ; D
ð11Þ
AðkÞE2kþ1 ð2k 1Þb2k2 ; D
ð12Þ
BðkÞE2kþ3 ð2k 1Þb2k2 : D
ð13Þ
n¼1
X, A(k) and B(k) are, respectively: X ¼
A ð22k 1ÞBk ; AðkÞ ¼ ; N h3 ð2k þ 1Þð2kÞ!22k
BðkÞ ¼
ð2 22k ÞBk : ð3 þ 2kÞð2kÞ!22k
ð14Þ
Bk are Bernoulli numbers, ED is the maximal energy of acoustic phonons’ spectra, and E0 is the mean energy of optical phonons. Fh is a derivative of the cosech function F h ðxÞ ¼
2ðexpðxÞ þ expðxÞÞ : 2 expð2xÞ expð2xÞ
ð15Þ
We can now fit the activation energy given by Eqs. (9)– (14) to data obtained during measurements. We should find op the following parameters: Wd, Eac b , E b , E0, ED and X. Since ac Eb and X are present in (9)–(14) only as multiples, we cannot calculate them independently – we shall only obtain their product. We have used the non-gradient Powell method of non-linear optimization in order to fit the parameters [6].
ð8Þ
In order to obtain useful equations for the activation energy: 1. we assume that phonon spectra are continuous and replace the sums over the phonon frequency spectra with integrals; 2. after Meek [5], we assume parabolic phonon states density to be g(x) = Ax2, where A is a constant which may differ for various glasses; 3. we then replace hyperbolic functions by their appropriate Taylor expansions. From (3)–(7) we obtain an equation for the mobility activation energy:
3. The results The results of parameter fitting to d/c conductivity measurements data obtained in a couple of glass samples are given in Table 1. An example of fitting obtained from the Powell procedure is shown in Fig. 1. The next interesting microscopic parameter of conductivity, the integral of wave function overlapping, Jij, may be calculated by fitting the pre-exponential factor of the full d/c conductivity equation to experimental data (taking into account all the previously calculated parameters). An exemplary result of such fitting is given in Fig. 2. In order to extract Jij from the pre-exponential factor, assumptions are necessary about concentration NC and the hopping distance, R; assuming NC = 1021 cm3 and R = 4 m10 we
R.J. Barczyn´ski / Journal of Non-Crystalline Solids 352 (2006) 4229–4231 Table 1 The resulting fitted model parameters for some glasses containing transition metal oxides
1 2 3 4
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffi 2kTW H hm0 ; J ij < p p
Composition [% mol]
Eop b ½eV
E0 [eV]
ED [eV]
Wd [eV]
where, in the high temperature range, we may assume
50TeO240V2O510Fe2O3 50P2O550FeO 40P2O560V2O5 50P2O550V2O5
0.568 1.292 0.813 0.304
0.084 0.121 0.120 0.074
0.04 0.04 0.034 0.031
0.043 0.255 0.165 0.076
1 1 ac W H ¼ Eop b þ Eb : 2 2
Fig. 1. Fitting of the activation energy (solid line) to experimental data points for 50P2O550FeO glass obtained from the Powell procedure.
4231
ð16Þ
ð17Þ
The Holstein criterion holds well for all our samples and we may conclude that the hopping is non-adiabatic. All the obtained hopping parameters have very probable values, but there are no parameters in the literature acquired by other methods, which could be used for comparison. Thus, we should be very careful with our procedure; particularly, the measurement data used for calculations should be very precise. It is also quite difficult to estimate the real errors of the obtained parameters. The mean deviation of experimental points from the fitted function is small and the numerical procedure allows calculation of parameters with a precision of 3–4 significant digits. At the same time, electrometric measurements of very low currents are very difficult (practically impossible) to be performed with such precision. Moreover, errors are usually systematic and difficult to estimate and deviations of phonon states’ density from that assumed in the parabolic model are difficult to obtain and directly influence the precision of results. We believe that even the second significant digit should be treated with caution. 4. Conclusion A method of interpretation of d/c conductivity data in the framework of the Gorhan–Bergeron and Emin model has been developed. The fitting of model parameters to experimental data in 50P2O550V2O5, 40P2O560V2O5, 50P2O550FeO and 50TeO240V2O510Fe2O3 oxide glasses has been performed. The following microscopic hopping parameters have been evaluated: the phonon spectra characteristics, the mean difference in energy between hopping centers and the integral of wave function overlapping. The Holstein criterion of non-adiabatic hopping has been positively verified for all these glasses. References
Fig. 2. The result of model fitting (solid line) to experimental conductivity data for 50P2O550FeO glass.
obtain Jij = 0.0094, 0.0074, 0.0184 and 0.0015 eV, respectively for 50TeO240V2O510Fe2O3, 50P2O550FeO, 40P2O560V2O5 and 50P2O550V2O5 glasses. We can now check the Holstein criterion for the non-adiabatic character of hopping [7]
[1] [2] [3] [4] [5] [6]
M. Sayer, A. Mansingh, Phys. Rev. B 6 (1972) 4629. N.F. Mott, J. Non-Cryst. Solids 1 (1968) 1. J. Schnakenberg, Z. Physik 185 (1965) 123. E. Gorham-Bergeron, D. Emin, Phys. Rev. B 15 (1977) 3667. P.E. Meek, Philos. Mag. 33 (1976) 897. I. Szepietowski, Master Thesis, Gdan´sk University of Technology, 1994. [7] T. Holstein, Ann. Phys. 281 (2006) 706; T. Holstein, Ann. Phys. 281 (2006) 725.