1 Introduction

Thin films of chalcogenide glassy materials gained great interest over the past several decades due to their applications in potential in several technological devices. The common feature of this class of glassy materials is the presence of the localized states in the mobility edge as a result of the presence of the short-range order as well as the various inherent defects [1,2,3]. Previous works have been established that the physical properties of the chalcogenide glassy materials are highly dependent on the atomic ratios of the elements present in the chemical formula [4, 5].

The Ge–In–Se ternary system belongs to the chalcogenide glassy materials. Where it forms glasses through average compositions of a range extending from a coordination number Z = 2.4 to Z = 2.67 which are considered critical values according to Philips [6]. The system possesses interesting physical properties, such as high infrared transmission spectra, high refractive index, and fast characterization, which make the system attractive for many technological applications, such as IR detector [7], telecommunications [8], acousto-optic devices [9], switching and memory devices [10]. The thermal stability of the thermally evaporated Ge15Se85-xlnx deposited films increases with increasing the In content [11], while the amorphous to the crystalline state of Se70In15Ge15 has occurred at 413 K [12]. The optical properties of the ternary system Ge30Se70−xInx [13], GexSe92-xIn8 [14], Ge20Se80−xInx [15], Ge10Se90−xInx [16], and Ge26InxSe74-x [17] have been reported. In addition, the DC electrical conductivity of Ge20In5Se74 [18], Ge20Se80-xInx [19,20,21], Ge40-xInxSe60 [22], and Se70In15Ge15 [12] has been reported, while there are only two reports concerning the AC conductivity and dielectric properties of the ternary Ge–In–Se system. The first reports the frequency and temperature dependence of the dielectric properties of Se90Ge10–xInx thin films at Low Temperature (78–260 K) [23], while the other reports the AC conductivity and dielectric properties of Ge20Se75In5 films (300–423 K) [24].

In the previous work, the authors have studied the linear and non-linear optical properties of the amorphous Ge26InxSe74-x (1 ≤ × ≤ 5) thin films [17], whereas in the present work they decided to keep the chemical formula of the studied ternary system and study the effect of the In addition on the DC and AC conductivity for the Ge26InxSe74-x system.

2 Experimental method

Bulk ingot materials with a chemical composition Ge26InxSe74-x (x = 1, 2, 3 and 5) have been synthesized from the pure constitute elements (5 N, Sigma-Aldrich) in sealed evacuated quartz tubes by the conventional melt quenching technique [17].

Amorphous thin films with exact chemical compositions (a) Ge26.6In0.7Se72.7 (b) Ge25.4In2.4Se72.2 (c) Ge27.8In2.9Se69.3 (d) Ge25.8In4.8Se69.4, respectively, as previously detected from the EDX analysis were deposited at room temperature onto clean glass substrates using a high vacuum coating unit (Type Edwards, E306A) in a vacuum pressure of – 2 × 10−4 Pa [17]. The amorphous structure of the prepared films is confirmed by X-ray diffractometer (XRD) Type Philips X'Pert as reported in previous work [17].

Silver paste coplanar electrodes of a width of 0.5 cm were thermally evaporated onto terminals of the investigated films to serve Ohmic electrodes for electrical measurements. The DC electrical resistance, R, was measured as a function of temperature, T, in the temperature range 295–523 K by means of a high impedance electrometer (Type Keithley 6517 A). The DC electrical conductivity σDC was calculated using the relation, σDC = d/RA (where d is the sample thickness, A is the cross-sectional, and R is the measured sample resistance). Whereas, a programmable RLC bridge (Type Hioki IM 3536) was used to measure the impedance, Z, the capacitance, C, and the loss tangent, tan δ, as a function of frequency in the frequency range 4 Hz–8 MHz and in the temperature range 303–523 K. The dielectric constant, ε1 has been calculated from the relation ε1 = Cd/ε0A, where ε0 is the electrical permittivity of vacuum (ε0 = 8.854 × 10−12 F.m−1). The dielectric loss ε2 was calculated from the relation ε2 = ε1 tan δ, where δ = 90–φ, where φ is the phase angle. The AC conductivity σAC was calculated according to the relation σAC = ωε0ε2 [24].The sample temperature was measured using a chromel–alumel thermocouple.

3 Results and discussion

3.1 DC conductivity

The temperature-dependence of DC conductivity of the investigated samples with exact chemical compositions (a) Ge26.6In0.7Se72.7 (b) Ge25.4In2.4Se72.2 (c) Ge27.8In2.9Se69.3 (d) Ge25.8In4.8Se69.4 films in the temperature range 295–523 K are shown in Fig. 1. The figure depicts that the conductivity increases with increasing temperature through the entire temperature range indicating a semiconductor behavior, with two different conduction mechanisms. The first is in the high-temperature range (T > 380 K) that can be represented through the thermally activated process across the extended states. While the other is in the low-temperature range (at T < 380 K) that can be represented through a less thermally activated process and represented by Mott’s formula for the hopping conduction through the localized states. The variation of the conductivity through the extended states follow the Arrhenius relation [20, 25]:

$$\sigma_{DC} = \sigma_{0} \exp \,( - \Delta E_{DC} /KT)$$
(1)

where σo is the pre-exponential factor, ∆EDC is the activation energy and K is the Boltzman’s constant. The values of the pre-exponential factor, σ0, the activation energy, ∆EDC, and the room temperature conductivity, σRT, for such regions as a function of In content which is calculated from Fig. 1 are listed in Table 1. It is noticed that, by increasing the In content, the conductivity increases, while the activation energy decreases. This decrease in ∆EDC is due to the reduction of average binding energy and formation of defect centers by adding In [19, 26, 27]. On the other hand, the value of ∆EDC was found less than the half value of the optical bandgap calculated in our previous work [17] for the samples of the same composition, which indicates the presence of impurities within the gap. Therefore, the value of ∆EDC in the present work indicates that the Fermi level is most probably displaced from the center of the gap towards the valence band [12, 16]. The calculated values of ∆EDC of investigated films are in agreement with other works, as shown in Table 1.

Fig. 1
figure 1

Plot of the DC conductivity vs. reciprocal temperature for a Ge26.6In0.7Se72.7, b Ge25.4In2.4Se72.2, c Ge27.8In2.9Se69.3, and d Ge25.8In4.8Se69.4

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Table 1 The conductivity at room temperature, the pre-exponential factor and activation energies calculated from DC conductivity for Ge26InxSe74-x thin films
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The measured conductivity is the sum of two components:

$$\sigma_{DC} = \sigma_{hop} + \sigma_{ext}$$
(2)

where σhop and σext, are the conduction contribution due to hopping between the nearest localized states and the conduction contribution between the extended states, respectively.

In the high-temperature region, the linearity denoted that σDC is a thermally activated process [27] according to Eq. 1. As the temperature decreases, the activated Arrhenius behavior is replaced by a power law relationship between the logarithm of conductivity and temperature. The low-temperature variable range hopping conductivity is characterized by Mott’s variable-range hopping relation [14, 25, 28]:

$$\sigma_{hop} = \left( {\frac{{\sigma_{h0} }}{\sqrt T }} \right)\exp \left[ { - \left( {\frac{{T_{o} }}{T}} \right)^{1/4} } \right]$$
(3)

where T0 is the hopping parameter that is given as

$$T_{0} = \left( {\frac{{18\alpha^{3} }}{{KN(E_{F} )}}} \right)$$
(4)

where σho is the pre-exponential factor of the hopping conduction, N (EF) is the density of states at the Fermi level, α−1 is the decay length of a localized wave function at the Fermi level which is taken as 10−9 m for electrons and K is the Boltzman’s constant.

The linear variation of σhop√T vs. (1/T) 1/4 as shown in Fig. 2 confirms that in this region the transport is due to variable range hopping of charge carriers in the localized states near the Fermi level and is characterized by relation (3). The values of T0, σh0 as well as the density of states at the Fermi level N(EF) determined from Fig. 2 are listed in Table 2.

Fig. 2
figure 2

ln (σDC√T) vs. (1/T)1/4 for a Ge26.6In0.7Se72.7, b Ge25.4In2.4Se72., c Ge27.8In2.9Se69.3, and (d) Ge25.8In4.8Se69.4 thin films

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Table 2 Various parameters calculated from DC conductivity for Ge26InxSe74-x thin films in temperature range 295–347 K
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Two other hopping parameters can be calculated according to Mott [28, 29], which are the hopping distance R (cm) and the average hopping energy W (eV), given by

$$R = \left[ {\frac{9}{{8\pi \alpha N(E_{F} )KT}}} \right]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$4$}}}}$$
(5)

and

$$W = \frac{3}{{4\pi R^{3} N(E_{F} )}}$$
(6)

The calculated values of R and W for the investigated compositions are also listed in Table 2. It is observed that both values of R and W decrease by increasing In content. In addition, W ˃ KT and αR ˃ 1 indicate that the variation of conductivity at a low temperature of the investigated samples obey the condition of Mott’s model of variable-range hopping (VRH) process [25, 28, 30].on the other hand, the value of N(EF) increases by adding In and this confirms the increase of conductivity due to the increase of localized states in the gap. The determined values of R, W, and N(EF) for Ge26In5Se69 are in agreement with those reported for Ge20In8Se72 [14].

3.2 AC conductivity

  • a) Frequency dependence of AC conductivity

The measurements of the AC conductivity provide significant information about the conduction mechanism of glassy systems. The real part of AC conductivity is due to trapped charges which are active only at higher frequencies and can be calculated from Jonscher’s universal power law [31,32,33]:

$$\sigma_{AC}^{^{\prime}} \left( \omega \right) = \sigma_{DC} + \sigma_{AC} \left( \omega \right) = \sigma_{DC} + A\omega^{S}$$
(7)

where σDC is the DC conductivity, A is a temperature-dependent constant that determines the strength of the polarizability and S is the frequency exponent that 0 < s < 1.

Figure 3 shows the frequency dependence of AC conductivity for the investigated Ge26InxSe74-x films compositions at room temperature. It is observed that for all compositions, there are two distinct regions; the low-frequency region, where the conductivity is frequency independent and attributed to the DC conductivity resulting from the effect of the electrode polarization, followed by the dispersion region at which the conductivity increases rapidly at high frequencies obeys the power-law relation [33, 34]. This increment of the AC conductivity is attributed to the hopping or tunneling of charge carriers as the applied electric field frequency increases [35]. In addition, the AC conductivity increases by increasing In content. This is due to the formation of localized states in the band tail whose density increases by increasing In content and hence increases conductivity [36].

Fig. 3
figure 3

Frequency dependence of AC conductivity for a Ge26.6In0.7Se72.7, b Ge25.4In2.4Se72.2, c Ge27.8In2.9Se69.3, and d Ge25.8In4.8Se69.4 thin films at room temperature

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Figure 4 shows the frequency dependence of AC conductivity for Ge26InxSe74-x thin films at different temperatures. It is found that the conductivity increases by increasing frequency according to Eq. 7. In addition, the conductivity increases by increasing temperature for all investigated samples due to thermally activated polaron hopping [37]. The values of the frequency exponent S for different compositions and temperatures are calculated from the slopes of the linear part in Fig. 4.

Fig. 4
figure 4

Frequency dependence of AC conductivity for a Ge26.6In0.7Se72.7, b Ge25.4In2.4Se72.2, c Ge27.8In2.9Se69.3, and d Ge25.8In4.8Se69.4 thin films at different temperatures

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Figure 5 represents the temperature dependence of the frequency exponent S for Ge26InxSe74-x thin films. It is clear that the value of S decreases as the temperature increases for all compositions. It is also clear that S decreases with the increase of the In content in the investigated compositions. This means that the obtained experimental results can be interpreted according to the correlated barrier hopping CBH model [38, 39].

Fig. 5
figure 5

Temperature dependence of the frequency exponent S for a Ge26.6In0.7Se72.7, b Ge25.4In2.4Se72.2, c Ge27.8In2.9Se69.3, and d Ge25.8In4.8Se69.4

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In the CBH model, it is proposed that electrons transfer over a barrier between two defect sites by thermal activation, where each site has a coulombic potential well associated with it. For two adjacent sites separated by distance R, the coulomb wells overlap and hence the effective barrier WM will reduce to W which is given by [38, 39]

$$W = W_{M} - \frac{{ne^{2} }}{{\pi \varepsilon^{^{\prime}} \varepsilon_{0} R}}$$
(8)

where WM is the maximum potential barrier height, e is the electronic charge, ε' is the real part of dielectric constant, ε0 is the permittivity of the free space and n is the number of electrons that hop science n = 1 or 2 for a single polaron and bipolaron processes, respectively. The frequency exponent S according to this model obeys the relation [36, 40]:

$$S = 1 - \frac{{6K_{B} T}}{{W_{M} + K_{B} T{\text{ln}}\left( {1/\omega \tau_{0} } \right)}}$$
(9)

where kB is the Boltzman constant, T is the absolute temperature and τ0 is the characteristic relaxation time. At lower temperatures, WM >  >  > KBTln(ωτ0) hence the value of S is approximately given by [36]:

$$S = 1 - \frac{{6K_{B} T}}{{W_{M} }}$$
(10)

For the case of single polaron hopping, the value of WM is typically one-quarter of optical bandgap Eg, while WM is equal to Eg for bipolaron hopping [41]. The values of frequency exponent S, potential barrier height WM, and theoretical optical bandgap Eg at room temperature for Ge26InxSe74-x are tabulated in Table 3. It is clear that the values of WM are approximately a quarter of the optical bandgaps of Ge26InxSe74-x thin films, which indicates that the single polaron hopping is the dominating conduction mechanism in such films [41, 42]. In addition, the values of theoretical bandgaps at room temperature for different compositions are compatible with those determined experimentally in the earlier work [17].

  • b) Temperature dependence of AC conductivity

Table 3 Frequency exponent S, potential barrier height WM and experimental optical bandgap at room temperature for Ge26InxSe74-x thin films
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The AC conductivity exhibits temperature-dependence and obeys the relation:

$$\sigma_{AC} = \sigma_{0} \exp \left( {\frac{{ - \Delta E_{AC} }}{KT}} \right)$$
(11)

where ΔEAC is the activation energy for the AC conductivity. Figure 6 shows the plot between lnσAC(ω) and 1000/T which shows semiconductor behavior as the AC conductivity increases by increasing temperature for all compositions. In addition, the AC conductivity shows the same behavior as the DC conductivity that increases by increasing In content. This is owing to the short-range order of the investigated samples as well as the formation of defects which increases the density of localized states in the band tail [43]. In other words, the formation of the hetero-polar In–Se bond at the expense of the homo-polar Se–Se bond causes the reduction of band energy and consequently, increases the AC conductivity [44, 45]. In addition, there are two slopes for all compositions in the frequency range 10 Hz–103 Hz, while there is one slope at higher frequencies. From the slope of the straight line, AC activation energy ΔEAC can be calculated. Table 4 demonstrates the values of ΔEAC for Ge26InxSe74-x at different frequencies. It is noticed that the values of ΔEAC decrease by increasing frequency for all investigated samples. This decrease may be due to the increase of the applied electric field that enhances the electronic jump between the localized states [46, 47] and the small value of ΔEAC confirms that hopping conduction is the dominant mechanism [48]. In addition, the value of ΔEAC is lower than that of ΔEDC for all compositions. This is due to the charge carriers that jump large paths at DC conduction but they did not need to do this in presence of the AC conduction [24].

Fig. 6
figure 6

Temperature dependence of AC conductivity for a Ge26.6In0.7Se72.7, b Ge25.4In2.4Se72.2, c Ge27.8In2.9Se69.3, and d Ge25.8In4.8Se69.4

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Table 4 AC activation energies at different frequencies for Ge26InxSe74-x thin films
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4 Conclusions

The DC and AC conductivity of amorphous chalcogenide thin films of chemical composition Ge26In0.7Se72.7 (a), Ge25.4In2.4Se72.2 (b), Ge27.8In2.9Se69.3 (c) and Ge25.8In4.8Se69.4 (d) were studied at different temperatures and frequencies.

The variation of the DC conductivity in the temperature range 295–523 K exhibits a semiconductor behavior in the entire temperature range with two conduction mechanisms. The first was observed in the high-temperature range (T > 380 K), where the conduction is due to the thermally activated process through the extended states with single activation energy. The corresponding electrical parameters (activation energy, ΔE, and pre-exponential factors, σo) of the Arrhenius relation were calculated for each composition and found that the activation energy decreases from 0.709 to 0.466 eV by increasing In content. While the second is in the low-temperature range (T < 380 K), where the conduction is due to be less thermally activated and can be represented by the hopping conduction through the localized states according to the Mott variable range hopping model. The density of the localized states and the hopping parameter were calculated.

The AC conductivity, measured in the temperature range 295–523 K and over the frequency range 4–8 MHz, reveals that the conductivity in the dispersion region follows the Jonscher’s power-law, σ(ω)α ωs . In addition, the value of the exponent s decreases with increasing temperature as well as with increasing In content. So, the results were interpreted according to the correlated barrier hopping CBH model. The potential barrier height, WM, and the theoretical optical bandgap, Eg, at room temperature for the investigated compositions were evaluated.