The homogeneity of the beginning materials was finalized in the agate mortar and progressively dried up to 673 K for 15 h to avoid ambient humidity and gases, particularly CO2 evaporation. The obtained black powder was ground again for 8 h and pressed into 8 mm pellets in diameter and 1 mm in thick using a 3 ton per cm2 uniaxial pressure and sintered at 1123 K for 15 h to make the blue (LCG) compound.
3.6.4 Conduction mechanism. In order to study the conduction mechanism within the Li2CoGeO4 compound, various theoretical models have been given in the literature such as:- The correlated barrier hopping (CBH) model in which S decreases linearly with the rising temperature.40,41
- The small polaron tunneling (NSPT) model whither the exponent s rises with the increasing temperature.42
- The quantum mechanical tunneling (QMT) model is characterized by an exponent s which increases very slowly with temperature, practically equal to a value of 0.8.43
- The overlapping large polaron tunneling (OLPT) model submitted by Long.44 The exponent s decreases with the temperatures to a minimum value and then increases slightly.
The Fig. 9 (inset) presents the temperature dependence of the exponent s. We can note that the exponent s exhibits two different regions as following:
- Part I, the exponent s decreases linearly with the temperature, which confirms that the (CBH) model is the suitable model.
- Part II, the exponent s increases with the increase in temperature to a minimum value and then increase with the temperature. Thus, the (NSPT) model is the appropriate model.
3.6.4.1 The correlated barrier hopping (CBH) model (region I). This model was first enhanced by Pike for the single polaron jump and was thereafter developed by Eliott for the two-polaron jump.45 Which considers the jump of Li+ charge carriers amidst two sites on a barrier separating them? From the CBH formalism, the next equation gives the exponent s:46 |
| (10) |
where Wm is the binding energy of the carrier in its focused sites, KB is the Boltzmann constant, and τ is a characteristic relaxation time is supposed to be 10−13 s. A very simple estimation of this equation gives the exponent (s): |
| (11) |
For this simple model, the alternating conductivity is given by:47
|
| (12) |
with
n is the number of polarons affected in the hopping process (
n = 1 or 2),
ε′ is the dielectric constant of the compound. NNp is proportional to the square of the concentration of states and
Rω is the hopping distance for the condition (
ωτ = 1) and it is given by:
|
| (13) |
Moreover, NNp can be presented by:
NNp = NT2(bipolaron-hoping) |
The Fig. 10(a) displays the variation of the AC conductivity (Ln (σac)) as a function of (1000/T) in the sample Li2CoGeO4. The AC conductivity of this sample has been explained quite satisfactorily by considering a single conduction mechanism (single polaron). We can notice that theoretical values (lines) are in good agreement with the experimental values (symbols) and implied that this formalism is the simplest assumed formalism to qualify the approximate frequency dependence of AC conductivity. The different parameters used in the fitting procedure are summarized in Table 4. The negative sign of effective energy is connected with the strong interaction between electron and photon. The Fig. 11(a) shows the temperature dependence of Rω at different frequencies. We note that the values of Rω vary is in the interval of distances of 2.74–2.75 Å. These values are of the order of the inter-atomic distance Li–Li: 2.75 Å. Based on these results, it may be thought that the AC conductivity in Li2CoGeO4 is assured by the mobility of polarons because of the movement of Li+ ions located in cavities along the c-axis.
|
| Fig. 10 Variation of Ln (σac) with inverse of temperature at various frequencies for Li2CoGeO4 phase I (a) and phase II (b). | |
Table 4 Parameters used for the adjustment of the CBH model (I) and NSPT model (II)
Phase I |
Phase II |
Frequency (Hz) |
Ueff (eV) |
N(EF) (eV−1 m−3) |
Wm (eV) |
Frequency (Hz) |
N(EF) (eV−1 m−3) |
α (Å−1) |
Wm (eV) |
1 × 102 |
−0.0046 |
9.58 × 1015 |
0.3 |
1 × 102 |
5.61 × 1019 |
1.18 |
0.34 |
1 × 103 |
−0.0031 |
1.83 × 1017 |
|
501 |
2.99 × 1019 |
1.3 |
0.19 |
1 × 104 |
−0.0014 |
5.84 × 1018 |
|
1 × 103 |
2.14 × 1019 |
1.35 |
0.07 |
1 × 105 |
−5.98 × 10−4 |
5.65 × 1019 |
|
7940 |
9.94 × 1018 |
1.47 |
0.04 |
|
| Fig. 11 Variation of the hopping tunneling distance Rω as a function as temperature at different frequencies phase (I) (a) and phase (II) (b). | |
3.6.4.2 The non-overlapping small-polaron tunnelling (NSPT) model (region II). In the NSPT formalism, the exponent s could be determined by the following equation:48 |
| (14) |
when Wm/KBT can take big values, the parameter is contracted to: |
| (15) |
Considering the NSPT approach, σac was given by the following formula:49
|
| (16) |
where
|
| (17) |
α−1 represents the spatial extension of the polaron,
Rω the tunnelling distance, and
N(
EF) the density of states near the Fermi level. The
Fig. 10(b) displays the evolution of the AC conductivity (Ln (
σac)) as a function of (1000/
T) at various frequencies in region II. We can observe that the theoretical calculations (lines) are in good agreement with the experimental data (symbol). The different parameters determined for this model are shown in the
Table 4. Considering the results of adjustment, we represent in
Fig. 11(b) the evolution of the tuning distance of
Rω as a function of the temperature. So, it is remarkable that
Rω increases with the increasing temperature. Consequently, in this model it is evident that the values of
Rω vary in the interval of distances of 1 Å to 3.5 Å. These values are in the same order of magnitude than the interatomic spacing Li–Li distances which was found to be 2.77 Å. From this observation we can conclude that the conduction process is assured by the mobility of small polarons because of the movement of Li
+ ions located in cavities along the [001] axis.