Conductivity and dielectric studies of Li₂SnO₃

Abstract

Lithium stannate (Li₂SnO₃) has been prepared by solution evaporation method. The precursor obtained is sintered at 800°C for 5, 6, and 7 h, respectively. X-ray diffractogram confirmed that the sample obtained after sintering is Li₂SnO₃. The pelletized Li₂SnO₃ after heating at 500 °C for 3 h is used for electrochemical impedance spectroscopy characterization. Impedance measurements have been carried out over frequency range from 50 Hz to 1 MHz and temperature range from 563 to 633 K. The conductivity–temperature relationship is Arrhenian. Several important parameters such as activation energy, ionic hopping frequency and its rate, carrier concentration term, mobile ion number density, ionic mobility, and diffusion coefficient have been determined. The characteristics of log conductivity and log ionic hopping rate against temperature for the system suggest that the conduction and ionic hopping processes are thermally activated. The values of activation energy for conduction and relaxation processes as well as activation enthalpy for ionic hopping are about the same.

Introduction

Research on tin-based oxide compounds as electrode material for lithium-ion batteries have been intensified ever since Idota and co-workers employed amorphous tin composite oxide as anode material with its capacity higher than graphite [1]. Tin composite oxide is one promising material as anode for lithium-ion batteries since it can improve the electrochemical performance by reducing capacity fading compared to pure tin itself [2]. Lithium stannate or lithium tin oxide (Li₂SnO₃) is known to be a promising breeding material for nuclear fusion reactors [3, 4] and as one of the starting materials to synthesize zinc stannate (ZnSnO₃) [5] other than being electrode materials for lithium-ion batteries [6–9]. From literature review, there have been several reports published on the Li₂SnO₃ structure [10–13] and also on the enthalpy and heat capacity of Li₂SnO₃ [14, 15]. Li₂SnO₃ has also been reported to exhibit good catalytic properties among other lithium-based oxides such as Li₂MnO₃, Li₂CeO₃, LiFeO₂, LiZnO₂, and LiAl₅O₈ [16].

Conductivity studies on various lithium-based oxides viz. Li₂TiO₃ [17], LiCeO₂ [18], LiSmO₂ [19], LiV₂O₅ [20], LiBiP₂O₇ [21], and others [17, 22] have been reported in the literature. However, to the best of our knowledge, there is no report on conductivity of Li₂SnO₃ that has been published in the literature except Zhang and co-authors who have prepared thick films of Li₂SnO₃ and examined its conductivity under vacuum condition and at different humidity atmospheres [23]. Studies on the conductivity of lithiated electrode materials are important in order to gain a better insight and understanding on the ionic conduction mechanism especially in its usage for lithium-ion batteries. Electrochemical impedance spectroscopy (EIS) is one simple and non-destructive way to investigate the ion transport of the material at the microscopic level. From conductivity data, various important parameters such as activation energy, hopping frequency, hopping rate of the ions, charge carrier concentration term, ionic mobility, diffusion coefficient, number density of mobile ions, activation enthalpy, and entropy for conduction can be determined. These parameters will add more information to the literature on Li₂SnO₃. The potential application of Li₂SnO₃ as an electrode material in lithium-ion batteries serve as a motivation for this study and the information gathered as stated above would be advantageous in the design of lithiated tin-based oxide for anode in lithium batteries.

In the present work, solution evaporation technique was employed for the synthesis of Li₂SnO₃. The Li₂SnO₃ was prepared by evaporating the solutions of lithium and tin acetates and sintering the product precursor. X-ray diffraction (XRD) was carried out to prove that the product obtained after sintering is Li₂SnO₃. EIS was performed on the pelletized Li₂SnO₃ at high temperatures ranging from 563 to 633 K. The dielectric and modulus studies of Li₂SnO₃ are also presented in this paper.

Experimental

Sample preparation

Tin (II) acetate (Sn(C₂H₃O₂)₂; 7.1034 g) (CAS: 638-39-1, Aldrich, Russia) was dissolved in 600 ml distilled water until the solution appeared muddy-like in color. Ethanol (10 ml) was then added to the solution to ensure complete dissolution before 6.1212 g lithium acetate dihydrate (C₂H₃O₂Li·2H₂O) (CAS: 6108-17-4, Sigma-Aldrich, USA) was added to the solution. At this point, 10 ml nitric acid was added until pH = 1 and a milky solution obtained. The mixture was continuously stirred and heated to ensure complete dissolution. Heating was continued until a white solid was obtained. The white solid was ground to powder and sintered at 1,073 K for 5, 6, and 7 h, separately. After sintering, the samples were once again ground into fine powder using pestle and mortar.

X-ray diffractogram

The samples were characterized by XRD with Cu-Kα X-radiation of wavelength λ = 1.5406 Å using Siemens D5000 diffractometer to prove them to be Li₂SnO₃. XRD data in the range of 2θ = 15° to 80° were collected with a step size of 0.05°. Diffraction occurs based on Bragg's Law:

$$ 2\;d\sin \theta = n\;\lambda $$

(1)

where d is the interplanar spacing, θ is the Bragg angle of the diffraction peak, n is the order of reflection, and λ is the wavelength of x-ray.

Electrochemical impedance spectroscopy

For impedance measurements, 1.0 g of each sample was pelletized to a thickness of about 0.1 cm and diameter of 1.5 cm. The applied pressure was 400 bar. The pellets were then heated at 500°C for 3 h to minimize the grain boundary effect [24]. The impedance of the pelletized samples was determined by complex impedance spectroscopy using the HIOKI 3531 Z Hi-tester bridge over the frequency range from 50 Hz to 1 MHz at temperatures ranging from 563 to 633 K. The pellets were sandwiched between stainless steel electrodes. From the complex impedance plot, the bulk resistance, R _b can be determined. The ionic conductivity, σ was then calculated using the following equation:

$$ \sigma = \frac{t}{{{R_b}A}} $$

(2)

where t is thickness of the pellet in centimeters and A is area of the pellet in square centimeter.

Results and discussion

X-ray diffractogram

Figure 1 shows the XRD patterns of Li₂SnO₃ using acetates of lithium and tin as starting materials and sintered at 1,073 K for several sintering times. The product obtained was identified as monoclinic Li₂SnO₃ following the JCPDS 31-0761 data. These results are found to be closely matched to the X-ray diffractogram of Li₂SnO₃ reported by Zhang et al. [8]. From the XRD, pattern of Li₂SnO₃ sintered for 8 h, four additional peaks with low intensity can be observed at 2θ = 20.45°, 22.75°, 26.65°, and 51.85°. This is in good agreement with the JCPDS file no. 76-1149 in which peaks at 2θ = 20.25°, 22.84°, 26.50°, and 52.50° have been assigned to ($\overline 1 11$), (111), (022), and (223) crystal planes that are also assigned to monoclinic Li₂SnO₃. Hence, it can be concluded that no impurity peak can be observed in the XRD patterns indicating purity of the material. Also, it can be observed that sharper peaks appeared as sintering time increases. Table 1 compares the standard d interplanar spacing from JCPDS data with the calculated d spacing of the product obtained in this work using Eq. 1. It can be observed that the calculated values of d spacing are well-matched to the standard d interplanar spacing and thus indicate the formation of Li₂SnO₃.

Fig. 1

XRD diffractograms of Li₂SnO₃ sintered at temperature of 1,073 K for various sintering hours

Table 1 List of standard and calculated d interplanar spacing of Li₂SnO₃

Crystallite size (t) of the samples can be calculated using Scherrer's equation:

$$ t = \frac{{0.9\lambda }}{{\beta \cos \theta }} $$

(3)

Here, β is the full width at half maximum height of the most intense diffraction peak representing the (002) plane in radians. Crystallite size of the Li₂SnO₃ sample at different sintering times was calculated and tabulated in Table 2. It is noted that crystallite size decreases as sintering time increases.

Table 2 Crystallite size (t) for Li₂SnO₃ sintered at temperature of 800 °C as a function of sintering times

Conductivity studies

Figure 2 depicts the complex impedance plot at 573 K obtained for Li₂SnO₃ samples sintered at 800 °C for various sintering times. As can be seen from the figure, the Cole-Cole plot revealed semicircle shape which is due to bulk resistance, R _b , and bulk capacitance, C _b , of the sample. It arises because of the ion migration in the bulk of the material and is due to different relaxation times of the ions [25]. The fact that the center of the semicircle is located below the real axis reveals the non-Debye nature of the material [26]. The intercept of the semicircle with the real axis gives the bulk resistance, R _b . The bulk capacitance at the maximum of the semicircle can be calculated using the equation below:

$$ \omega {R_b}{C_b} = 1 $$

(4)

Fig. 2

Cole-Cole plot at 573 K for Li₂SnO₃ samples sintered at 1,073 K for 5, 6, 7, and 8 h, respectively. The inset figure shows its equivalent circuit

From the figure, C _b in the range between 21 and 41 pF are obtained. According to literature [21], the bulk capacitance in the order of pF indicates that the conduction process occurs through the bulk or grain interior of the material. The shape of the impedance plot is due to the response of the surface of a material to an input signal that is equivalent to the response of an electrical circuit. Hence, the impedance data can be fitted to the response of an equivalent circuit when subjected to an ac signal as shown in the inset of Fig. 2. CPE represents the constant phase element and is used instead of a capacitor to account for the depressed semicircle. The impedance of CPE is given by [27]:

$$ {Z_{\text{CPE}}} = 1/k{\left( {jw} \right)^p}\;{\text{where}}\;0 \leqslant p \leqslant 1 $$

(5)

or

$$ {Z_{\text{CPE}}} = k\left[ {\cos \left( {p\pi /2} \right) - j\;\sin \left( {p\pi /2} \right)} \right]/{\omega^p} $$

(6)

Here k ⁻¹ corresponds to the capacitance value of the CPE element, ω is angular frequency (ω = 2πf where f is frequency), and p is related to the deviation from the vertical axis in the Z″ versus Z′ plot. CPE behaves in an intermediate way between resistor and capacitor. The real and imaginary parts of the impedance associated to the equivalent circuit can be calculated using the expressions given below:

$$ Z\prime = \frac{{{R_b} + R_b^2{k^{{ - 1}}}{\omega^p}\cos \left( {\pi p/2} \right)}}{{1 + 2{R_b}{k^{{ - 1}}}{\omega^p}\cos \left( {\pi p/2} \right) + {R_b}^2{k^{{ - 2}}}{\omega^{{2p}}}}} $$

(7)

$$ Z\prime \prime = \frac{{R_b^2{k^{{ - 1}}}{\omega^p}\sin \left( {\pi p/2} \right)}}{{1 + 2{R_b}{k^{{ - 1}}}{\omega^p}\cos \left( {\pi p/2} \right) + R_b^2{k^{{ - 2}}}{\omega^{{2p}}}}} $$

(8)

Table 3 lists the parameters involved in the circuit. It can be observed that k ⁻¹ capacitance values are in pF range which is comparable to C _b obtained from Eq. 4. The good fit between the experimental data and calculated line indicates that the equivalent circuit recommended suited well.

Table 3 The parameters of circuit elements for Li₂SnO₃ samples heated at 573 K for 5, 6, 7, and 8 h, respectively

From Eq. 2 and the Cole-Cole plot, it can be noted that the highest conducting sample is sintered at 8 h with conductivity value of 3.57 × 10⁻⁸ S cm⁻¹ at 573 K. At the same temperature, Li₂TiO₃ has been reported to exhibit conductivity of ∼6.00 × 10⁻⁸ S cm⁻¹ [17]. Vītiņš et al. [17] strongly agreed that lithium ions are the main contribution for Li₂TiO₃ conductivity due to its insignificant electronic conductivity and the immobile oxygen ions in cubic close pack arrangement. Lithium-ion conduction is thought to occur by interstitial diffusion. Therefore, it conforms that Li⁺ ions are responsible for Li₂SnO₃ conductivity since both Li₂SnO₃ and Li₂TiO₃ have the same structure.

Figure 3 presents the Cole-Cole plot for Li₂SnO₃ sample sintered for 7 h at various temperatures. It can be observed that R _b decreases as temperature increases leading to increment in conductivity. From Eq. 2, it can be concluded that the lower value of R _b gives the higher conductivity. Hence, highest conductivity of 1.42 × 10⁻⁷ S cm⁻¹ is obtained at 633 K. Impedance plot for Li₂SnO₃ sample at temperatures of 563, 573, and 583 K depicted in Fig. 3a can be best represented by the equivalent circuit comprising a parallel combination of resistance, R _b , and CPE as shown in the inset figure. For temperatures higher than 583 K, the Cole-Cole plot in Fig. 3b shows a few data points after the semicircle which implies the beginning of formation of a spike. Hence, the equivalent circuit becomes a parallel combination of resistance, R _b , and CPE with another CPE in series as depicted in the inset Fig. 3b. CPE can also be used to represent the spike [27]. The expressions for the real and imaginary parts of the impedance related to this new equivalent circuit are given below:

$$ Z\prime = \frac{{{R_b} + R_b^2k_1^{{ - 1}}{\omega^{{{p_1}}}}\cos \left( {\pi {p_1}/2} \right)}}{{1 + 2{R_b}k_1^{{ - 1}}{\omega^p}\cos \left( {\pi {p_1}/2} \right) + R_b^2k_1^{{ - 2}}{\omega^{{2{p_1}}}}}} + \frac{{\cos \left( {\pi {p_2}/2} \right)}}{{k_2^{{ - 1}}{\omega^{{{p_2}}}}}} $$

(9)

$$ Z\prime \prime = \frac{{R_b^2k_1^{{ - 1}}{\omega^{{{p_1}}}}\sin \left( {\pi {p_1}/2} \right)}}{{1 + 2{R_b}k_1^{{ - 1}}{\omega^{{{p_1}}}}\cos \left( {\pi {p_1}/2} \right) + R_b^2k_1^{{ - 2}}{\omega^{{2{p_1}}}}}} + \frac{{\sin \left( {\pi {p_2}/2} \right)}}{{k_2^{{ - 1}}{\omega^{{{p_2}}}}}} $$

(10)

Fig. 3

Complex impedance plot for Li₂SnO₃ sample (sintering time of 7 h) at temperature range between a 563 and 583 K and b 593 and 633 K. The inset figure shows their corresponding equivalent circuit

All the parameters for the circuit elements are summarized in Table 4.

Table 4 The parameters of the circuit elements for Li₂SnO₃ sample (sintering time of 7 h) at various temperatures

Temperature dependence of conductivity for Li₂SnO₃ systems with different sintering times is displayed in Fig. 4. From the plot of log σ versus 1,000/T for Li₂SnO₃ pellets, the temperature dependence of ionic conductivity obeys Arrhenius rule:

$$ \sigma = {\sigma_0}\exp \left[ {\frac{{ - {E_A}}}{{kT}}} \right] $$

(11)

Fig. 4

Log σ versus 1,000/T for various Li₂SnO₃ samples

Here σ ₀ is a pre-exponential factor, E _A is the activation energy of conduction, k is Boltzmann constant, and T is temperature in Kelvin. The activation energy can be obtained from the slope of the graph. From the graph, it can be observed that the conductivity increases linearly with temperature implying that conduction is a thermally activated process. The regression value, R ² is found to be 0.99 for all samples.

The conductivity at 563 K and activation energy values for all samples studied is listed in Table 5. The low conductivity obtained in this work may be due to the strong octahedral coordination between Li⁺ and O²⁻ ions that provides difficulty for the Li⁺ ions to detach itself from the oxygen ions to jump to neighboring sites. This could possibly account for the high activation energy obtained.

Table 5 Conductivity at 563 K and activation energy of Li₂SnO₃ for different sintering hours

Figure 5 depicts the graph of log σ(ω) versus log ω for Li₂SnO₃ sample sintered for 7 h. Here, σ(ω) is the total direct current (d.c.) and alternate current (a.c.) conductivity. From the figure, it can be seen that there is plateau at the low-frequency region and extrapolating it on the y-axis gives the value of d.c. conductivity. These σ _dc values are found to be in good agreement with the values depicted in Fig. 4. On the other hand, the sample exhibits strong frequency dispersion of conductivity in the high frequency region as temperature increases. Hence, it can be inferred that a.c. conductivity is dominant in the high-frequency region. It is also clear from the figure that the σ(ω) spectra in the high-frequency region converge at different temperatures which implies that a.c. conductivity is temperature independent at high frequencies. Such behavior has also been observed in the LiDyO₂ [22] and LiNiVO₄ [28] systems.

Fig. 5

Log σ(ω) total versus log ω for Li₂SnO₃ sample (sintering time 7 h) at various temperatures

It is well-known that Almond and West have proposed that the σ(ω) data can be used to estimate the ionic hopping rate, ω _p [29]. Extrapolating at twice the value of d.c. conductivity from the vertical axis horizontally towards the graph and then extrapolating downwards vertically to the horizontal axis will give ω _p in logarithmic scale as shown in Fig. 5. The hopping rate of ions, ω _p can also be given by the relation below [30]:

$$ {\omega_p} = \frac{{\sigma T}}{K} $$

(12)

where

$$ K = n{e^2}{a^2}\gamma \;{k^{{ - 1}}} $$

(13)

Here, K is charge carrier concentration term, e is electron charge, γ is correlation factor which is set equal to 1, and a is the jump distance, i.e., the distance between two adjacent sites for the ions to hop. From literature [30, 31], a is taken as 3 Å for all materials. n is the number density of mobile ions or charge carriers. n can be determined using Eq. 13. The ionic mobility, μ, and diffusion coefficient, D, can then be calculated using equations as given below:

$$ \mu = \frac{\sigma }{{ne}} $$

(14)

$$ D = \frac{{kT\sigma }}{{n{e^2}}} $$

(15)

The values of σ, ω _p , K, n, μ, and D at all temperatures studied for Li₂SnO₃ sample sintered for 7 h are tabulated in Table 6. As can be seen from the table, it can be concluded that no significant change in K and n within the temperature range studied. This implies that the concentration of charge carriers is temperature independent. Savitha and co-workers [32] deduced that if K is independent of temperature, then all the ions in the system are mobile and thus can be best represented by the strong electrolyte model. Therefore, it can be concluded that the conduction mechanism in Li₂SnO₃ is attributed to the hopping of charge carriers. From the table, it can be observed that the mobility of ions, μ increases with increasing temperature. This suggests that the conductivity of the samples can be attributed to the increase in ionic mobility since number density of the mobile ions is considered to be constant over temperature range studied. Also, it is noted that the diffusion coefficient, D increases with temperature. The diffusion coefficient, D of Li₂SnO₃ is comparable to the diffusion constant of Li₄Ti₅O₁₂ which is of the same magnitude order of 10⁻¹² cm² s⁻¹, but lower than that of Li_0.8Mn₂O₄ (10⁻⁹ cm² s⁻¹) [17] and even lower than that of LiV₂O₅ (10⁻⁴ cm² s⁻¹) [24]. The low value of diffusion coefficient of Li⁺ ions in Li₂SnO₃ may also be due to strong Li⁺-O²⁻ octahedral coordination that imposes difficulty for the ions to diffuse.

Table 6 Parameters of σ, ω _p , K, n, μ, and D at various temperatures for Li₂SnO₃ sample sintered for 7 h

The hopping rate of ions in a material is a valuable piece of information to elucidate the ionic conduction. The ionic hopping rate, ω _p can also be represented by the following equation:

$$ {\omega_p} = {\omega_e}\exp \left( {\frac{{ - {H_m}}}{{kT}}} \right) $$

(16)

Here, ω _e is the effective attempt frequency and H _m is activation enthalpy for hopping or migration of ions [29]. Figure 6 depicts the plot of log ω _p against 10³/T for Li₂SnO₃ samples sintered for 5, 6, 7, and 8 h, respectively. From the graphs plotted, H _m and ω _e can be obtained from the slope and intercept at the vertical axis of the graph, respectively. The linear variation of log σ with temperature for all the samples suggests that the ionic hopping process for Li₂SnO₃ is thermally activated. The effective attempt frequency, ω _e can also be expressed as:

$$ {\omega_e} = {\omega_0}\exp \left( {\frac{{{S_m}}}{k}} \right) $$

(17)

where ω ₀ is the true attempt frequency and S _m is activation entropy for hopping or migration of ions [29]. The true attempt frequency, ω ₀ , can be estimated using harmonic potential expression given by [29]:

$$ {\omega_0} = {\left( {\frac{{{H_m}}}{{2m{a^2}}}} \right)^{{1/2}}} $$

(18)

Fig. 6

Variation of log ω _p versus 1,000/T for Li₂SnO₃ samples sintered for 5, 6, 7, and 8 h, respectively

Here, m is the mass of the mobile ion (Li⁺). Substituting both the obtained values of ω _e and ω ₀ (see Eqs. 16 and 18) into Eq. 17 gives the value of S _m /k. According to Almond and West [29], the hopping rate of ions is strongly dependent on the entropic term, S _m /k. Table 7 lists the transport properties for all Li₂SnO₃ samples. For all samples, the values of ω _p , n, and μ are taken at 593 K.

Table 7 Transport parameters of ω _p , ω _e , H _m , E _A , ω ₀ , S _m /k, n, and μ for Li₂SnO₃ samples of various sintering times

From Table 7, it can be noted clearly that H _m value is in good agreement with E _A value for all samples studied. According to literature, if H _m and E _A are closely matched, it is said that the concentration of carriers is temperature independent and thus the enthalpy of carrier formation can be neglected [19, 33]. If H _m and E _A values are different, this implies that certain amount of energy has been used in the formation of a free charge carrier [21]. The negative value of the entropic term is also observed in LiGaO₂ system [29]. The trend of n follows the conductivity trend with sintering time.

Dielectric studies

The conductivity of ionic conductors can also be understood from dielectric studies. The dielectric property of a material is attributed to the formation of dipoles which arises due to the migration of ions along the sites. The complex permittivity of a system is defined by

$$ \varepsilon * = \varepsilon \prime - i\varepsilon \prime \prime $$

(19)

where ε′ is dielectric constant (the real part of complex permittivity) and ε″ is dielectric loss (the imaginary part of complex permittivity). Using the measured impedance values, the dielectric constant and dielectric loss can be calculated from the equations:

$$ \varepsilon \prime = \frac{{Z\prime \prime }}{{\omega {C_0}\left( {Z\prime {\prime^2} + Z{\prime^2}} \right)}} $$

(20)

$$ \varepsilon \prime \prime = \frac{{Z\prime }}{{\omega {C_0}\left( {Z\prime {\prime^2} + Z{\prime^2}} \right)}} $$

(21)

Here Z″ and Z′ is the imaginary and real parts of the complex impedance. C ₀  = ε ₀ A/t and ε ₀ is permittivity of free space.

The frequency dependence of the dielectric constant, ε′, at various temperatures for Li₂SnO₃ sample at sintering time of 7 h is shown in Fig. 7a. It can be noted that the ε′ shows frequency dependence in the frequency region investigated for all temperatures. This is attributed to the occurrence of electrode polarization and space charge [34]. Also, it can be seen clearly from the figure that the dielectric constant increases with increasing temperature which is the result of lithium-ion migration [35]. Figure 7b displays the variation of dielectric loss, ε″, with frequency for Li₂SnO₃ sample (sintering time 7 h) at different temperatures. Similar to Fig. 7a, it can be seen that ε″ decreases with the increase in frequency but increases with the increase in temperature. When the frequency is increased, there is no time for charge carriers to accumulate at the electrode–pellet interface due to the increasing rate of reversal of the electric field and the ions are mainly confined in the bulk of the material resulting in the decrease of ε″ [36]. However, both dielectric constant and dielectric loss exhibit a small increase in the high frequency region (inset Fig. 7a and b). Dielectric loss occurs as a result of collisions among the mobile charge carriers [37]. The large value of ε″ can be attributed to the motion of free charge carriers within the material [38].

Fig. 7

Variation of a dielectric constant, ε′ and b dielectric loss, ε″ with frequency for Li₂SnO₃ sample (sintering time 7 h) at various temperatures. The inset figures show dielectric constant and dielectric loss in frequency range between 640 kHz and 1 MHz, respectively

Other than permittivity, the dielectric relaxation can also be studied in terms of electrical modulus. The electrical modulus of a system can be expressed as below:

$$ M * = M\prime + iM\prime \prime $$

(22)

where M′ and M″ are real and imaginary parts of the modulus. From Eqs. 20 and 21, the imaginary and real parts of modulus can be obtained using the relations below:

$$ M\prime \prime = \frac{{\varepsilon \prime \prime }}{{\varepsilon {\prime^2} + \varepsilon \prime {\prime^2}}} $$

(23)

$$ M\prime = \frac{{\varepsilon \prime }}{{\varepsilon {\prime}^2 + \varepsilon \prime {\prime}^2}} $$

(24)

Figure 8 depicts the graph of imaginary part of the modulus, M″ versus log ω for Li₂SnO₃ sample (sintering time 7 h) at various temperatures. The imaginary component of the electrical modulus, M″ is associated to energy loss that occurs in the conduction process [39]. It can be seen clearly that the peaks shift towards higher frequency with increasing temperatures. This indicates that the dielectric relaxation process is thermally activated in which the ionic transport is by hopping [28]. Also, it can be noted that the height of the peak increases slightly as temperature increases. The region on the left of the M″ peak, i.e., the low-frequency region determines the frequency range in which the mobile ions are able to hop from one site to an adjacent site [39]. On the contrary, the high-frequency region represents the charge carriers being confined in potential wells that they can only move in short distance [39]. The appearance of peaks indicates the presence of relaxation time for the sample at selected temperatures. The relaxation time is defined as the time of the transition for the mobile ions to move from long distance to short distance. The occurrence of relaxation time, τ can be obtained from the peak of M″ using the relation τ = 1/ω _peak . The existence of the peak also confirms that the material is an ionic conductor [40].

Fig. 8

Variation of imaginary component of modulus, M″ with frequency for Li₂SnO₃ sample (sintering time 7 h) at various temperatures

Figure 9 shows the plot ln τ versus 1,000/T for Li₂SnO₃ sample with sintering time of 7 h. The regression value R ² is 0.98. It can be observed that the value of τ decreases with increasing temperature. The plot can be represented as

$$ \tau = 1.77 \times {10^{{ - 2}}}\exp \left( { - 0.86/kT} \right) $$

(25)

where the activation energy for relaxation process is 0.86 eV. This value is comparable to the activation energy of conductivity which is 0.80 eV. It is understood that if the activation energy value for the conductivity is similar to that of relaxation process, the mobile ion has to overcome the same barrier while conducting as well as relaxing [41]. The small difference may be due to the fact that the relaxation process involved only the hopping energy of the carriers between localized states.

Fig. 9

Temperature dependence of relaxation time for Li₂SnO₃ sample sintered for 7 h

Conclusions

The impedance of the pelletized lithium stannate (Li₂SnO₃) heated at 500 °C for 3 h is measured. The plot of log conductivity against reciprocal temperature obeys the Arrhenius rule. Transport parameters for conduction mechanism such as activation energy, ionic hopping frequency and its rate, carrier concentration term, mobile ion number density, ionic mobility, and diffusion coefficient have been determined. The ionic transport of Li₂SnO₃ is by hopping. Both the conductivity and ionic hopping are thermally activated processes. Variation of imaginary part of modulus, M″ as a function of frequency shows the shifting of peaks towards high frequency as temperature increases which implies that there is a distribution of ionic relaxation time. The nearly same value of activation energies for conduction and relaxation processes indicates that the energy required for the lithium ion to jump to adjacent sites in both processes are the same.