3.1. Electronic Structure
Using the modified Becke Johnson (mBJ) potential for high symmetry points [A→Γ→M→L→A→H→K→Γ] of the irreducible Brillouin zone (BZ), the electronic band energy diagrams for Up and Down states of Mn
xZn
1−xTe (x = 8% and 16%) are depicted in the
Figure 2. The spin-polarized calculations have been performed, in which the majority and minority spin electrons were treated separately. We applied the Generalized Gradient Approximation Plus U (GGA+U). Here the Fermi level is indicated by dashed line separating the valance band maximum and conduction band minimum.
It is obvious that these crystals have different band dispersion in the k-space. From the band structure, it was established that all the titled compounds were direct band semiconductors with energy band gap magnitudes equal to 2.20 and 2.0 eV for MnxZn1−xTe (x = 8% and 16%), respectively. These calculated band gap values were in sufficiently good agreement with the experimental values calculated from UV-vis diffuse reflectance spectra.
For all the band structures, principal contributions to the valance band minimum originated from 3d orbitals of Mn and p Zn orbitals, while the unoccupied states originated from 3p orbitals of S with a small admixture of other states of different atoms.
The spin polarized total density of states (TDOS) and partial density of states (PDOS) in a wide energy range is extended within −12.0 eV ~ 14.0 eV for the electronic states of Mn
xZn
1−xTe (x = 8% and 16%) compounds depicted in the
Figure 3 and
Figure 4. We saw only minor changes at the conduction band minimum for both percentages.
Following
Figure 3, it is clear that for the 8% doped crystal the density of states (DOS) is comparable to the 16% doped crystals. Also, the energy separation between occupied and unoccupied states was established to be higher for 8% with respect to 16%.
The (PDOS) for the electronic states of MnxZn1−xTe (x = 8% and 16%) can be subdivided into three separate energy regions. For the compounds MnxZn1−xTe (x = 8% and 16%), a major contribution is s orbitals of Te with a small admixture coming from Zn s/p/d. For an energy interval extending within −7.2 … 6.5 eV, the PDOS, for the compounds MnxZn1−xTe (x = 8% and 18%) had a major contribution due to d and p originated Zn band states, along with the small contribution Te-p and Mn-s atoms.
In Mn, doped ZnTe Mn-s/p/d stats played a crucial role for reported energy range (−4.0–4.0 eV). A strong hybridization was observed between Zn s/p and d states along with the Mn s/p/d states. Its role in conduction bands became more prominent than the valence band.
3.2. Optical Function Dispersion
Study of the optical properties is important for understanding the electronic structure of the materials. These can be attained from the complex dielectric function
which is expressed as
The imaginary part
is found from the momentum dipole transition matrix elements between the occupied and the unoccupied electronic states, and has been computed using Equation (1) [
28],
The dispersion of the real part of the dielectric function
was computed using the imaginary part by using Kramer’s- Kronig relations.
The symbol P represents the principal value of the integral.
With the help of real and imaginary part dispersions for dielectric function, other optical properties were calculated. The complex index of refraction is written as
Here
is refractive index and
is extinction coefficient can be obtained from dielectric function.
At low frequency (i.e.,
ω = 0), the real part of the refractive index is called the static refractive coefficient
.
So, from the complex dielectric function dispersion which contains both real and imaginary parts all the other optical functions were also calculated, such as absorption coefficient
, energy loss function
L(
ω) and reflectivity
. As the reflectivity is the percentage of reflected ray intensity on the incident ray intensity of electromagnetic waves on the system, it can be expressed as
The absorption coefficient is the power absorbed in a unit length of solid, and is calculated by using this formula
The energy loss function has been calculated as
The optical absorption spectra may be considered an effective experimental tool to identify the hyperfine band electronic structure of crystalline solid state materials. At the beginning we explored the optical absorption spectra for Mn
xZn
1−xTe (x = 8% and 18%). To describe the optical function dispersions of the titled crystals possessing tetragonal symmetry, only two tensor components (
εxx(
ω) =
εyy(
ω) and
εzz(
ω)) are sufficient following the general symmetry. But in the present work, we consider an average value of the two tensor components, both for the real part as well as in the imaginary part dispersion of the dielectric constant. The optical properties (the real (
ε1ave (
ω)) and imaginary (
ε2ave (
ω)) part of dielectric functions and other associated optical properties) have been investigated at the equilibrium constant at energies up to 25.0 eV and are illustrated in
Figure 5 and
Figure 6. Real and imaginary parts of the dielectric function dispersion for Mn
xZn
1−xTe (x = 8% and 18%) are shown in
Figure 5.
The determination of the imaginary part
ε2ave (
ω) confirms that the threshold energy of the dielectric function (i.e., first optical critical point) exists at energy equal to about at 2.6 eV for Mn
xZn
1−xTe (x = 8% and 18%) compounds. This point is MV-MC splitting or ΓV-ΓC transition, which is in accordance with the threshold for direct optical inter-band transitions known as the fundamental absorption edge existing in the middle of the highest CBM and VBM BZ points. The spectra demonstrated a relatively fast increase in the fundamental absorption edge caused by the abrupt surpassing of many points which are contributing towards
ε2ave (
ω). The number of spectral peaks increased as the energy increased and we have detected the highest peaks at energy about 3.5 eV for Mn
xZn
1−xTe (x = 8% and 18%). Weak anisotropy was noticed in spectra of Mn
xZn
1−xTe (x = 8% and 18%) within the energy range situated between 3.2 ~ 7.0 eV. So, applying the Kramer-Kronieg relations [
19,
20], the real part dispersion is computed following the imaginary part dispersion (this is shown in
Figure 6).
The principle spectral peaks of the real part, having a magnitude of < 7.0 for MnxZn1−xTe (x = 8% and 18%) compounds, were observed at energies about 3.5 eV. But the spectra curves were decreasing as the energy was increasing and crossed the zero line at about 6.0 eV for MnxZn1−xTe (x = 8% and 18%). The calculated values of static dielectric constant were found to be equal to about 3.8 and 3.7 for MnxZn1−xTe (x = 8% and 18%) at the conditions of equilibrium lattice constant. We also studied the dispersions of associated optical constant like refractive index nave (ω), absorption spectra Iave (ω), reflectivity Rave (ω), and energy loss function Lave (ω). So, following the results obtained it is clear that the Mn doping may effectively vary the resonance position of the spectral maxima for the titled crystals.
The principal method to find out how deeply light penetrates into the material is the determination of absorption coefficient I (
ω) dispersion. The highly localized inter-band transitions mainly gave rise to absorption spectra. The calculated absorption coefficients dispersions are plotted in
Figure 7. The spectra showed similar behaviors for the investigated materials, but also illustrated a few differences with significant optical anisotropy. The absorption edges were closely related to their band gaps and have been found to be equal to 2.5 eV for Mn
xZn
1−xTe (x = 8% and 18%) compounds. In addition, as the energy increased, the absorption coefficient values obtained also were increasing and the maximum spectral peaks were situated in the spectral range situated within 5.0 ~ 9.0 eV. The spectra showed a sharp drop at 15.0 eV, which may be caused by electronic inter-band transitions, appearing only when a photon of specific energy is resonantly absorbed. The sharp decrease in the spectrum also shows a possibility of forbidden in dipole approached inter-band transitions in the band structure. Generally, it is necessary to have a direct band gap type for any crystalline material that may be considered as a promising candidate for photovoltaic, photoelectrical and even photo-thermal applications [
20,
21].
The refractive index n(ω) dispersion is shown in
Figure 8. In this work we pay particular attention only to the average refractive index magnitudes. It is crucial there exists weak anisotropy between Mn
xZn
1−xTe (x = 8% and 18%). In
Figure 8, we have established that the highest refractive index peaks for Mn
xZn
1−xTe (x = 8% and 18%) compounds are situated near the spectral energy equal to about 3.5 eV. After the energy limit mentioned, the titled peaks were substantially spectrally shifted toward lower energies as we moved from 8% to 16%, but as the energy increased from 3.5 eV a decrease in the spectral peaks was observed.
Figure 9 shows that with the energy increases, we have an inverse relation with energy loss function (as shown in
Figure 10), but it still shows minimal reflectivity in the visible region. It is also very important that at higher energies, all the three materials have a slightly higher reflectivity.
3.3. Thermoelectric Properties
We calculated and analyzed the electrical conductivity (σ), thermal conductivity (κ), Seebeck coefficient (S), power factor (PF), and figure of merit (ZT) as a function of temperature in the range 100–800 K. The average values of the electrical conductivity (σ
ave) were computed for three compounds, as shown in
Figure 11. It is evident from the figure that all of the compounds showed different behavior with increasing temperature. In fact, average electrical conductivity (σ
ave (T)) of Mn
xZn
1-xTe (x = 8 and 16%) is shown in
Figure 11, and clearly demonstrates the fact that doping by 8% lead to a significant increase of electrical conductivity and temperature for spin up case, and for 8% spin down case we had a lower value of electrical conductivity obtained even at higher temperature. When the doping was enhanced up to 16% then there was a linear increase of conductivity versus electrical conductivity and temperature for up case, but at down case there was no change in electrical conductivity even at higher temperatures. These results may serve as an independent confirmation of the semi-conducting nature of these doped chalcogenides. At 100 K the values of σ
ave (T) for Mn
xZn
1−xTe (x = 8% and 16%) considered were found to be equal to 0 and 0.45 × 10
18(Ω s)
−1 for up and down cases, while for 16% the value of σ
ave (T) is 0 × 10
18 and 0.5 × 10
18(Ω s)
−1. With increase of temperature, that is, at 800 K, there was a substantial rise in σ
ave (T). The maximum value for Mn
xZn
1−xTe (x = 8%) is 4.8 × 10
18(Ω ms)
−1 and 0.4 × 10
18(Ω ms)
−1 for up and down case and for Mn
xZn
1−xTe (x = 16%) was 5.6 × 10
18(Ω ms)
−1 and 0 × 10
18 (Ω ms)
−1 for up and down, respectively. At the same temperature (800 K) the compound having doping 16% showed the maximum value of electrical conductivity.
The calculated value of the Seebeck coefficient
Save (T) is plotted as a function of T (100–800 K) as shown in
Figure 12. The Seebeck coefficient has an inverse relation with carrier concentration, which is represented by the following formula
In the equation above,
n is carrier concentration while m* means effective mass, K
B is Boltzmann constant, h is Planks constant. The Seebeck coefficient of Mn
xZn
1−xTe (x = 8% and 16%) compound shows a significant anisotropy in the all-inclusive within 100–800 K temperature range. These changes in
Save (T) are caused by the band anisotropies of the electronic band structure. As the value of the Seebeck coefficient was positive, it demonstrated p-type semiconductor features (see
Figure 12). With the increase of T from 100–800 K, the Seebeck coefficient of Mn
xZn
1−xTe (x = 8%) decreased from 1.9 × 10
−3–8 × 10
−4 μVK
−1 up case and for down case there was no change in the Seebeck coefficient as in the variation in the temperature. For Mn
xZn
1−xTe (x = 16%)
Save (T) for 100–350 K, they had an inverse relationship with temperature.
The calculated average electronic thermal conductivity σ
ave (T) (the inset figure shows the zoom up state) is shown in
Figure 13 for Mn
xZn
1−xTe (x = 8% and 16%). It is clear from
Figure 6 that σ
ave (T) for these three chalcogenides increased with T and demonstratied a huge anisotropy within the entire temperature range from 100 K to 800 K (the inset figure shows the zoom 16% down state).
Figure 6 confirms that the crystal Mn
xZn
1−xTe (x = 8%) showed minimal changes of σ
ave (T) for down 8% with respect to Up. σ
ave (T) was enhanced with increasing temperature from 100–800 K and achieved its maximal value, that is, 2.8 × 10
14 W/mKs for Mn
xZn
1−xTe (x = 8% Up) and 0.5 × 10
14 W/mKs Mn
xZn
1−xTe (x = 8% Dn) when the rate of doping was increased up to 16%. Then the value of thermal conductivity was increased in the up case, as shown
Figure 13. For Mn
xZn
1−xTe (x = 16%) the σ
ave (T) displays the highest value at 800 K with respect to the Mn
xZn
1−xTe (x = 8%) 3 × 10
14 W/mKs (Up) and 0.25 × 10
14 W/mKs (Dn).
Figure 14 presents the calculated average power factor
S2σ
ave (T) for all materials (the inset figure shows the zoom 16% down state). It was observed that the PF increases linearly for Mn
xZn
1−xTe (x = 8% Up) with temperature, while for the down case PF only slowly varied with temperature. At 100 K, Mn
xZn
1−xTe (x = 8%) had a magnitude 0.27 × 10
11 W/mK
2s (Up) and 0.0 × 10
11 W/mK
2s (Dn) case, when temperature increases to 800 K. Power factors in case of up and down have magnitudes 2.6 × 10
11 W/mK
2s and 5.5 × 10
10 W/mK
2s, respectively. For Mn
xZn
1−xTe (x = 16%), power factor increased for the up case with increase in temperature, as shown in
Figure 14. Hence, this might reflect the fact that this crystal may be promising for cooling devices and Mn
xZn
1−xTe (x = 8% and 16%) shows higher greater value of
S2σ
ave (T) at higher temperatures only for up case.
The figure of merit ZT =
S2σT/k has been calculated by including the electrical conductivity and Seebeck coefficient times T over thermal conductivity, as shown in
Figure 15. For temperature range within 100 … 800 K, ZT for Mn
xZn
1−xTe (x = 8% and 16%) compounds exhibit different behavior, firstly both materials demonstrate a decrease with increasing T up to 800 K and after we had almost temperature-independent behavior. From the results obtained the figure of merit ZT = 1 (Up 8%) and 0.80 (Dn 8%) and for 16% its values were 0.90 and 0.60 for Up and down cases, respectively. The larger value of ZT generally is caused by lower thermal conductivity and higher electrical conductivity. In general, our calculations confirm that at higher temperatures Mn
xZn
1−xTe (x = 8% and 16%) possess better thermoelectric efficiencies and have more potential for thermoelectric devices.
It is necessary to emphasize that a huge role for the chalcogenides begins to be played by the anharmonic phonons, as was shown using photo-induced optical methods [
29,
30,
31].