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Dielectric and baric characteristics of TlS single crystal
Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Physica B journal homepage: www.elsevier.com/locate/physb
Dielectric and baric characteristics of TlS single crystal S.N. Mustafaeva a,n, M.M. Asadov b, A.A. Ismailov a a b
Institute of Physics, ANAS, G. Javid prosp. 33, Az 1143 Baku, Azerbaijan Institute of Chemical Problems, ANAS, G. Javid prosp. 29, Az 1143 Baku, Azerbaijan
art ic l e i nf o
a b s t r a c t
Keywords: Dielectric coefficients Baric characteristics TlS single crystal Ac-conductivity
The investigation of the frequency dependences of the dielectric coefficients and ac-conductivity of the TlS single crystals made it possible to elucidate the nature of dielectric loss and the charge transfer mechanism. Moreover, we evaluated the density and energy spread of localized states near the Fermi level, the average hopping time and the average hopping length. It was shown that the dc-conductivity of the TlS single crystals can be controlled by varying the hydrostatic pressure. This has opened up possibilities for using TlS single crystals as active elements of pressure detectors. & 2014 Elsevier B.V. All rights reserved.
1. Introduction Layered semiconductors attract a lot of attention due to their interesting physical properties. These properties include strong anisotropy of the electric parameters related to special features in the crystalline structure. Physical properties of layered single crystals are very sensitive to external impact, such as temperature, pressure, dc- and ac-voltage. Among these layer semiconductors are thallium chalcogenides, which received a great deal of attention due to their optical, electrical and photoelectric properties [1–4] useful in various semiconductor devices. Thallium monosulfide is a III–VI layer semiconductor compound. It crystallizes in various structures, in particular, in a tetragonal structure of the TlSe type and a monoclinic structure (structural analog of TlGaSe2) [5]. In [6], the electrical and dielectric properties of monoclinic TlS single crystals were studied in the vicinity of high-temperature phase transitions. In [7] data on their low-temperature dc-conduction were given. However, no data are available on highfrequency ac-conduction and electrical properties of TlS single crystals under hydrostatic pressure. In crystals with a layered or chain structure, of considerable importance are defects, such as strongly deformed and even broken bonds, which can act as acceptors. The high density of states near the Fermi level is ascribed to the existence of such defects. Traps created by various defects in crystals play a decisive role in charge transfer phenomena, particularly in alternate electric fields. High-frequency electric measurements of a crystal can give valuable information on the localized states.
n
Corresponding author. E-mail address: [email protected] (S.N. Mustafaeva).
The aim of this work is to study the charge transfer in layered monoclinic TlS single crystals at various frequencies of applied electric field, to establish the ac-conduction mechanism and to determine dielectric and baric coefficients of these crystals.
2. Experimental TlS single crystals were grown, using the slow cooling method, from a melt with excess sulfur (4 at%) with respect to the stoichiometric compound [2]. The grown TlS single crystals have a monoclinic structure (space group C62h ) and the following lattice parameters: a¼ 10.90 Å, b¼10.94 Å, c¼15.18 Å, α¼γ¼ 901 and β¼100.21. The obtained TlS single crystals were found to be p-type. The bulk sample, which is used in the measurements, was prepared by splitting the single crystal along the cleavage plane and hence the resultant surface was mirror-like without any mechanical treatment. TlS samples for electrical measurements had the form of planar capacitors normal to the C axis of the crystals, with indium electrodes. The thickness of the TlS samples was 9 10 2 cm. The dielectric properties of the TlS single crystals were studied by a resonance technique [8] in the frequency range 5 104–3 107 Hz, using a TESLA BM 560 Q-meter. During the measurements, the samples were situated in a shielded chamber. All of the measurements were performed at 300 K in electric fields corresponding to Ohmic current–voltage behavior. The accuracy in determining the sample capacitance and the merit factor (Q¼1/tan δ) of the measuring circuit was limited by reading errors. The capacitors were calibrated with an accuracy of 7 0.1 pF. The reproducibility in the resonance position was 70.2 pF in terms of capacitance and 7 1.0–1.5 scale divisions in terms of Q. The largest 0 deviations from the average were 3–4% in ε and 7% in tan δ.
http://dx.doi.org/10.1016/j.physb.2014.03.095 0921-4526/& 2014 Elsevier B.V. All rights reserved.
Please cite this article as: S.N. Mustafaeva, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.03.095i
S.N. Mustafaeva et al. / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
The samples for baric measurements had the shape of rectangular plates. Indium was used as a contact material to the TlS samples. Dc-electric field from Ohmic region of current–voltage characteristic was applied along to the natural layers of a TlS single crystal. The conductivity was measured by the usual four-probe potentiometric method. The measurements under pressure (up to 0.75 GPa) were performed in a conventional copper–beryllium vessel with a mixture of dehydrated transformer oil and kerosene (1:4) as a pressure transmitting media. This fluid did not cause any irreversible changes in the samples. Pressure was measured with a calibrated manganine gauge with an accuracy not less than 1%.
3. Results and discussion Fig. 1 shows the frequency dependences of real (ε0 ) and imaginary (ε″) parts of complex dielectric permittivity for TlS. As the frequency is raised from 5 104 to 3 107 Hz, ε0 decreases by a factor of 7; ε″ decreases by one order. That is, the ε0 (f) and ε″(f) dispersion curves are characterized by a significant drop over the entire frequency range studied. The observed monotonic reduction in the dielectric permittivity of the TlS single crystal with increasing frequency (Fig. 1, curves 1 and 2) attests to the relaxation nature of the dispersion. The experimental frequency dependence of the dissipation factor tan δ for TlS single crystal at frequencies 5 104–3 107 Hz is characterized with a monotonic descending (Fig. 2). The hyperbolic decrease of tan δ with frequency is evidence of the fact, that conductivity loss becomes the main dielectric loss mechanism [9] at studied frequency range. Frequency-dependent
Fig. 3. Log–log plot of 300-K ac-conductivity against frequency for the TlS single crystal.
300-K ac-conductivity of TlS crystal (Fig. 3) follows the relation
sаc fs with s¼0.5 at f¼ 5 104–106 Hz; and s ¼0.8 at f Z106 Hz.
Conduction-band ac conductivity is known to be mainly frequency-independent up to 1010–1011 Hz. The observed sac f0.8 behavior suggests that the conduction is due to carrier hopping between localized states in the band gap of the material. Such states may be localized near the edges of allowed bands or near the Fermi level [10]. However, under typical experimental conditions, conduction through the states near the Fermi level always prevails over that through the states near band edges. Therefore, the observed sаc f0.8 behavior in TlS attests to hopping transport through the states localized near the Fermi level [10]
sac ðf Þ ¼ ðπ 3 =96Þ е2 кТN 2 F a5 f ½ lnðνph =f Þ4 ;
Fig. 1. Frequency dispersion of the real (1) and imaginary (2) parts of complex dielectric permittivity of the TlS single crystal at T ¼ 300 K.
ð1Þ
where e is the elementary charge and k is the Boltzmann constant, NF is density of localized states near the Fermi level, a is the localization length, νph is the phonon frequency. According to Eq. (1), ac conductivity varies as f[ln(νph/f)]4. Therefore, at frequencies f⪡νph, sac is approximately proportional to f0.8. Using Eq. (1) and our obtained experimental sаc (f) data, we evaluated the Fermi-level density of states, NF ¼ 3.1 1019 eV–1 cm–3. When calculating NF, the localization length was taken as a¼33 Ǻ, which was obtained from experimental results of hopping dcconduction study of TlS [7], and νph ¼ 1012 Hz. According to the hopping conduction theory, the average ac hopping length is determined from the formula R ¼ ð1=2αÞlnðνph =f Þ
ð2Þ
Here, α is the decay constant of the wave function of the localized charge carrier Ψ e–αr (α¼1/a). The value R calculated for TlS single crystal using Eq. (2) is 185 Å. In TlS single crystal R is 5.6 times greater than the average distance between localization centers. Using this value of R and the formula τ 1 ¼ νph expð 2αRÞ;
ð3Þ
we evaluated the mean hop time of charge carriers between localized states in the band gap of TlS: τ¼6.5 10–8 s. Using the relation [10] ΔE ¼ 3=ð2πR3 NF Þ
Fig. 2. The dissipation factor tan δ for TlS single crystal as a function of frequency.
ð4Þ
we estimated the energy spread of localized states ΔE near the Fermi level of TlS single crystal: ΔE ¼2.4 10 3 eV. Fig. 4 presents the pressure dependences of the conductivity of TlS single crystal at temperatures 294, 333 and 345 K (curves 1, 2 and 3). It is evident from Fig. 4 that pressure leads to increase of
Please cite this article as: S.N. Mustafaeva, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.03.095i
S.N. Mustafaeva et al. / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
3
written as Eg(P) ¼Eg(0) |γ|P, where jγj ¼ 0:062–0:066 eV=GPa
4. Conclusions
Fig. 4. Conductivity as a function of pressure for TlS single crystal at temperatures: 294(1), 333(2) and 345 K(3).
conductivity (s) along the layers of TlS single crystals. Measurements permitted to evaluate the pressure behavior of s, which may be written as ln sðPÞ ¼ ln sð0Þ þ βP;
ð5Þ
at T¼ 294–345 K. Assumwhere β¼ d ln s(P)/dP¼ 1.08–1.22 GPa ing that the electrical conductivity changes with pressure according to the equation 1
sðPÞ ¼ sð0Þexpð γP=2kTÞ;
ð6Þ
where γ ¼dEg/dP is the pressure coefficient of the band gap, one can easily find γ ¼ 2kTβ
ð7Þ
One can see that since β40 then it follows from Eq. (7) that γ should be negative to satisfy Eq. (6). Hence, with increasing pressure the forbidden band gap of TlS should decrease. We found that the dependence of the band gap of TlS on pressure may be
The electrical properties (loss tangent, real (ε0 ) and imaginary (ε″) parts of complex dielectric permittivity, and ac-conductivity) of layer TlS single crystals (monoclinic structure) have been studied in the frequency range f ¼5 104–3 107 Hz. The observed monotonic reduction in the dielectric permittivity of the TlS single crystal with increasing frequency attests to the relaxation nature of the dispersion. The hyperbolic decrease of tan δ with frequency is evidence of the fact, that conductivity loss becomes the main dielectric loss mechanism at studied frequency range. The ac conductivity across the layers of TlS crystal varies as f0.8, characteristic of hopping conduction through localized states near the Fermi level. The Fermi-level density of states (NF ¼ 3.1 1019 eV 1 cm 3), the spread of their energies (ΔE ¼2.4 10 3 eV), and the mean hop distance (R¼ 185 Å) and time (τ ¼ 6.5 10–8 s) have been estimated. The effect of hydrostatic pressure (up to 0.75 GPa) on the dc-electric properties of TlS single crystals has been investigated at the temperatures 294, 333 and 345 K. It has been shown that pressure leads to increase of conductivity along the layers of TlS single crystal. The increase of conductivity with pressure is described by the formula s(P)¼ s (0) exp( γP/2kT), and the pressure coefficient of the band gap γ¼dEg/dP was found to be 0.062–0.066 eV/GPa. Obtained results show that TlS single crystals can be used as active elements of pressure detectors. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
H.A. Elshaikh, I.M. Ashraf, A.M. Badr, Egypt. J. Solids 30 (2007) 199. I.M. Ashraf, H.A. Elshaikh, A.M. Badr, Indian J. Phys. 68 (2007) 467. A.M. Badr, H.A. Elshaikh, I.M. Ashraf, J. Mod. Phys. 2 (2011) 12. V. Belyukh, A. Danyluk, K. Glukhov, I. Stakhira, Phys. Solid State 55 (2013) 2317. S. Kashida, K. Nakamura., J. Solid State Chem. 110 (1994) 264. V.P. Aliev, Sh.G. Gasimov, T.G. Mammadov, Phys. Solid State 48 (2006) 2233. S.N. Mustafaeva, M.M. Asadov, A.A. Ismailov, Phys. Solid State 50 (2008) 2040. S.N. Mustafaeva, J. Radioelectron. 5 (2008) 11. V.V. Pasynkov, V.S. Sorokin, Materials of Electron Techniques, Mir, Moscow, 1986. [10] N. Mott, E. Davis, Electron Processes in Noncrystalline Materials, Clarendon Press, Oxford, 1971.
Please cite this article as: S.N. Mustafaeva, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.03.095i
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Dielectric and baric characteristics of TlS single crystal S.N. Mustafaeva a,n, M.M. Asadov b, A.A. Ismailov a a b
Institute of Physics, ANAS, G. Javid prosp. 33, Az 1143 Baku, Azerbaijan Institute of Chemical Problems, ANAS, G. Javid prosp. 29, Az 1143 Baku, Azerbaijan
art ic l e i nf o
a b s t r a c t
Keywords: Dielectric coefficients Baric characteristics TlS single crystal Ac-conductivity
The investigation of the frequency dependences of the dielectric coefficients and ac-conductivity of the TlS single crystals made it possible to elucidate the nature of dielectric loss and the charge transfer mechanism. Moreover, we evaluated the density and energy spread of localized states near the Fermi level, the average hopping time and the average hopping length. It was shown that the dc-conductivity of the TlS single crystals can be controlled by varying the hydrostatic pressure. This has opened up possibilities for using TlS single crystals as active elements of pressure detectors. & 2014 Elsevier B.V. All rights reserved.
1. Introduction Layered semiconductors attract a lot of attention due to their interesting physical properties. These properties include strong anisotropy of the electric parameters related to special features in the crystalline structure. Physical properties of layered single crystals are very sensitive to external impact, such as temperature, pressure, dc- and ac-voltage. Among these layer semiconductors are thallium chalcogenides, which received a great deal of attention due to their optical, electrical and photoelectric properties [1–4] useful in various semiconductor devices. Thallium monosulfide is a III–VI layer semiconductor compound. It crystallizes in various structures, in particular, in a tetragonal structure of the TlSe type and a monoclinic structure (structural analog of TlGaSe2) [5]. In [6], the electrical and dielectric properties of monoclinic TlS single crystals were studied in the vicinity of high-temperature phase transitions. In [7] data on their low-temperature dc-conduction were given. However, no data are available on highfrequency ac-conduction and electrical properties of TlS single crystals under hydrostatic pressure. In crystals with a layered or chain structure, of considerable importance are defects, such as strongly deformed and even broken bonds, which can act as acceptors. The high density of states near the Fermi level is ascribed to the existence of such defects. Traps created by various defects in crystals play a decisive role in charge transfer phenomena, particularly in alternate electric fields. High-frequency electric measurements of a crystal can give valuable information on the localized states.
n
Corresponding author. E-mail address: [email protected] (S.N. Mustafaeva).
The aim of this work is to study the charge transfer in layered monoclinic TlS single crystals at various frequencies of applied electric field, to establish the ac-conduction mechanism and to determine dielectric and baric coefficients of these crystals.
2. Experimental TlS single crystals were grown, using the slow cooling method, from a melt with excess sulfur (4 at%) with respect to the stoichiometric compound [2]. The grown TlS single crystals have a monoclinic structure (space group C62h ) and the following lattice parameters: a¼ 10.90 Å, b¼10.94 Å, c¼15.18 Å, α¼γ¼ 901 and β¼100.21. The obtained TlS single crystals were found to be p-type. The bulk sample, which is used in the measurements, was prepared by splitting the single crystal along the cleavage plane and hence the resultant surface was mirror-like without any mechanical treatment. TlS samples for electrical measurements had the form of planar capacitors normal to the C axis of the crystals, with indium electrodes. The thickness of the TlS samples was 9 10 2 cm. The dielectric properties of the TlS single crystals were studied by a resonance technique [8] in the frequency range 5 104–3 107 Hz, using a TESLA BM 560 Q-meter. During the measurements, the samples were situated in a shielded chamber. All of the measurements were performed at 300 K in electric fields corresponding to Ohmic current–voltage behavior. The accuracy in determining the sample capacitance and the merit factor (Q¼1/tan δ) of the measuring circuit was limited by reading errors. The capacitors were calibrated with an accuracy of 7 0.1 pF. The reproducibility in the resonance position was 70.2 pF in terms of capacitance and 7 1.0–1.5 scale divisions in terms of Q. The largest 0 deviations from the average were 3–4% in ε and 7% in tan δ.
http://dx.doi.org/10.1016/j.physb.2014.03.095 0921-4526/& 2014 Elsevier B.V. All rights reserved.
Please cite this article as: S.N. Mustafaeva, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.03.095i
S.N. Mustafaeva et al. / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
The samples for baric measurements had the shape of rectangular plates. Indium was used as a contact material to the TlS samples. Dc-electric field from Ohmic region of current–voltage characteristic was applied along to the natural layers of a TlS single crystal. The conductivity was measured by the usual four-probe potentiometric method. The measurements under pressure (up to 0.75 GPa) were performed in a conventional copper–beryllium vessel with a mixture of dehydrated transformer oil and kerosene (1:4) as a pressure transmitting media. This fluid did not cause any irreversible changes in the samples. Pressure was measured with a calibrated manganine gauge with an accuracy not less than 1%.
3. Results and discussion Fig. 1 shows the frequency dependences of real (ε0 ) and imaginary (ε″) parts of complex dielectric permittivity for TlS. As the frequency is raised from 5 104 to 3 107 Hz, ε0 decreases by a factor of 7; ε″ decreases by one order. That is, the ε0 (f) and ε″(f) dispersion curves are characterized by a significant drop over the entire frequency range studied. The observed monotonic reduction in the dielectric permittivity of the TlS single crystal with increasing frequency (Fig. 1, curves 1 and 2) attests to the relaxation nature of the dispersion. The experimental frequency dependence of the dissipation factor tan δ for TlS single crystal at frequencies 5 104–3 107 Hz is characterized with a monotonic descending (Fig. 2). The hyperbolic decrease of tan δ with frequency is evidence of the fact, that conductivity loss becomes the main dielectric loss mechanism [9] at studied frequency range. Frequency-dependent
Fig. 3. Log–log plot of 300-K ac-conductivity against frequency for the TlS single crystal.
300-K ac-conductivity of TlS crystal (Fig. 3) follows the relation
sаc fs with s¼0.5 at f¼ 5 104–106 Hz; and s ¼0.8 at f Z106 Hz.
Conduction-band ac conductivity is known to be mainly frequency-independent up to 1010–1011 Hz. The observed sac f0.8 behavior suggests that the conduction is due to carrier hopping between localized states in the band gap of the material. Such states may be localized near the edges of allowed bands or near the Fermi level [10]. However, under typical experimental conditions, conduction through the states near the Fermi level always prevails over that through the states near band edges. Therefore, the observed sаc f0.8 behavior in TlS attests to hopping transport through the states localized near the Fermi level [10]
sac ðf Þ ¼ ðπ 3 =96Þ е2 кТN 2 F a5 f ½ lnðνph =f Þ4 ;
Fig. 1. Frequency dispersion of the real (1) and imaginary (2) parts of complex dielectric permittivity of the TlS single crystal at T ¼ 300 K.
ð1Þ
where e is the elementary charge and k is the Boltzmann constant, NF is density of localized states near the Fermi level, a is the localization length, νph is the phonon frequency. According to Eq. (1), ac conductivity varies as f[ln(νph/f)]4. Therefore, at frequencies f⪡νph, sac is approximately proportional to f0.8. Using Eq. (1) and our obtained experimental sаc (f) data, we evaluated the Fermi-level density of states, NF ¼ 3.1 1019 eV–1 cm–3. When calculating NF, the localization length was taken as a¼33 Ǻ, which was obtained from experimental results of hopping dcconduction study of TlS [7], and νph ¼ 1012 Hz. According to the hopping conduction theory, the average ac hopping length is determined from the formula R ¼ ð1=2αÞlnðνph =f Þ
ð2Þ
Here, α is the decay constant of the wave function of the localized charge carrier Ψ e–αr (α¼1/a). The value R calculated for TlS single crystal using Eq. (2) is 185 Å. In TlS single crystal R is 5.6 times greater than the average distance between localization centers. Using this value of R and the formula τ 1 ¼ νph expð 2αRÞ;
ð3Þ
we evaluated the mean hop time of charge carriers between localized states in the band gap of TlS: τ¼6.5 10–8 s. Using the relation [10] ΔE ¼ 3=ð2πR3 NF Þ
Fig. 2. The dissipation factor tan δ for TlS single crystal as a function of frequency.
ð4Þ
we estimated the energy spread of localized states ΔE near the Fermi level of TlS single crystal: ΔE ¼2.4 10 3 eV. Fig. 4 presents the pressure dependences of the conductivity of TlS single crystal at temperatures 294, 333 and 345 K (curves 1, 2 and 3). It is evident from Fig. 4 that pressure leads to increase of
Please cite this article as: S.N. Mustafaeva, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.03.095i
S.N. Mustafaeva et al. / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
3
written as Eg(P) ¼Eg(0) |γ|P, where jγj ¼ 0:062–0:066 eV=GPa
4. Conclusions
Fig. 4. Conductivity as a function of pressure for TlS single crystal at temperatures: 294(1), 333(2) and 345 K(3).
conductivity (s) along the layers of TlS single crystals. Measurements permitted to evaluate the pressure behavior of s, which may be written as ln sðPÞ ¼ ln sð0Þ þ βP;
ð5Þ
at T¼ 294–345 K. Assumwhere β¼ d ln s(P)/dP¼ 1.08–1.22 GPa ing that the electrical conductivity changes with pressure according to the equation 1
sðPÞ ¼ sð0Þexpð γP=2kTÞ;
ð6Þ
where γ ¼dEg/dP is the pressure coefficient of the band gap, one can easily find γ ¼ 2kTβ
ð7Þ
One can see that since β40 then it follows from Eq. (7) that γ should be negative to satisfy Eq. (6). Hence, with increasing pressure the forbidden band gap of TlS should decrease. We found that the dependence of the band gap of TlS on pressure may be
The electrical properties (loss tangent, real (ε0 ) and imaginary (ε″) parts of complex dielectric permittivity, and ac-conductivity) of layer TlS single crystals (monoclinic structure) have been studied in the frequency range f ¼5 104–3 107 Hz. The observed monotonic reduction in the dielectric permittivity of the TlS single crystal with increasing frequency attests to the relaxation nature of the dispersion. The hyperbolic decrease of tan δ with frequency is evidence of the fact, that conductivity loss becomes the main dielectric loss mechanism at studied frequency range. The ac conductivity across the layers of TlS crystal varies as f0.8, characteristic of hopping conduction through localized states near the Fermi level. The Fermi-level density of states (NF ¼ 3.1 1019 eV 1 cm 3), the spread of their energies (ΔE ¼2.4 10 3 eV), and the mean hop distance (R¼ 185 Å) and time (τ ¼ 6.5 10–8 s) have been estimated. The effect of hydrostatic pressure (up to 0.75 GPa) on the dc-electric properties of TlS single crystals has been investigated at the temperatures 294, 333 and 345 K. It has been shown that pressure leads to increase of conductivity along the layers of TlS single crystal. The increase of conductivity with pressure is described by the formula s(P)¼ s (0) exp( γP/2kT), and the pressure coefficient of the band gap γ¼dEg/dP was found to be 0.062–0.066 eV/GPa. Obtained results show that TlS single crystals can be used as active elements of pressure detectors. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
H.A. Elshaikh, I.M. Ashraf, A.M. Badr, Egypt. J. Solids 30 (2007) 199. I.M. Ashraf, H.A. Elshaikh, A.M. Badr, Indian J. Phys. 68 (2007) 467. A.M. Badr, H.A. Elshaikh, I.M. Ashraf, J. Mod. Phys. 2 (2011) 12. V. Belyukh, A. Danyluk, K. Glukhov, I. Stakhira, Phys. Solid State 55 (2013) 2317. S. Kashida, K. Nakamura., J. Solid State Chem. 110 (1994) 264. V.P. Aliev, Sh.G. Gasimov, T.G. Mammadov, Phys. Solid State 48 (2006) 2233. S.N. Mustafaeva, M.M. Asadov, A.A. Ismailov, Phys. Solid State 50 (2008) 2040. S.N. Mustafaeva, J. Radioelectron. 5 (2008) 11. V.V. Pasynkov, V.S. Sorokin, Materials of Electron Techniques, Mir, Moscow, 1986. [10] N. Mott, E. Davis, Electron Processes in Noncrystalline Materials, Clarendon Press, Oxford, 1971.
Please cite this article as: S.N. Mustafaeva, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.03.095i