Introduction

In order to develop new technological applications to cope with the improvement of people’s living standards, novel approaches based on low-cost, efficient, ecofriendly, and good physical semiconductor properties are becoming a global priority and a very interesting field of research. Nowadays, the most promising materials for such purposes are wide band gap semiconductors, in particular the new class of transparent conducting oxides (TCO) that exhibit both transparency and electronic conductivity simultaneously. In fact, TCOs are characterized by remarkable and multifunctional properties such as larger band gap, higher electron mobility, higher carrier density, and higher breakdown field strength. Thus, several TCO devices can be designed for many applications: high power and high temperature electronic devices, sensor devices, solar cells, displays transparent electronics, short wavelength optoelectronics, spintronics, photocatalysis, etc. [1,2,3,4,5].

Among these TCOs, zinc oxide (ZnO), a direct band gap semiconductor with a significant exciton binding energy and high optical gain at ambient temperature and other distinctive properties, is receiving a great deal of fundamental and applied research interest all over the world [6,7,8,9]. Indeed, it is one of the most promising II–VI binary semiconductors for several potential applications in numerous fields such as energy storage, electronics, optoelectronics, and environment photocatalysis [10,11,12,13,14]. ZnO possess different important characteristics with high transparency, anti-radiation stability, large specific area, strong ultraviolet excitation, good compatibility, piezo-electricity at room temperature, and a high isoelectric point [15, 16]. Furthermore, with its great thermal and mechanical stability, oxidation resistance, affordability, and non-toxicity, ZnO is one of the viable candidates for thermoelectric materials. Such materials are of great importance in space applications (radioisotope thermoelectric generators) and in solar-thermal-electrical-energy production. Indeed, thermoelectric power is being considered as alternative renewable and green energy [17, 18]. This wide band gap semiconductor exists in several interesting polymorphs among which the three main phases are wurtzite (WZ), rock salt (RS), and zinc blende (ZB) structures. The WZ structure, with an iconicity lying between that of covalent and ionic materials, is the most thermodynamically stable phase at the ambient conditions [19,20,21].

Moreover, ZnO can be doped and/or co-doped, n- or p-types, with several metallic and nonmetallic elements from different groups such as transition metal, noble metals, and rare-earth elements, e.g., Al, Ag, B, Bi, Ce, Co, Cr, Cu, Dy, Er, Eu, F, Fe, Ga, In, La, Li, Mg, Mn, Mo, N, Nb, Nd, Ni, P, Pm, Pr, Sb, Sm, Sn, S, and Y [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]. The advantages of the incorporation of dopants, which act as donors or acceptors, reside in the improvement of electrical and optical properties via the generation of excess mobile free carriers (electrons or holes) as well as in the changes introduced in the band gap energy as a result of the generation of lattice defects in the material structure [37]. The control of ZnO different properties, via doping, leads to the design of new devices and more fields of applications. Aluminum, Al, a cheap, durable, abundant, and non-toxic metal belonging to the group III elements, is considered to be the most promising dopant in ZnO due to the improvement of its properties and to mass production applications in photovoltaic devices, organic light-emitting diodes, sensors, optoelectronics and liquid crystal display. Hence, Al-doped zinc oxide, AZO, characterized by a high conductivity and a good optical transmission, has proved to be an excellent TCO material. It is attracting a considerable attention for transparent conducting electrodes in replacement of the widely used commercial indium tin oxide (ITO) electrodes and other applications in solar cells, liquid crystal displays, window layers, etc. [38,39,40,41,42].

Thus, fundamental understanding of different properties of both ZnO and AZO semiconductors, are necessary to improve their performances, as base materials, in electronics and optoelectronics devices as well as in piezoelectric and energy applications. However, the choice of the appropriate method and the adequate conditions to study this material still represents a great challenge to get accurate results in less investigation time. Although, several experimental and theoretical approaches are being developed to determine the AZO properties [43,44,45,46,47], there is still a lack of studies of low Al doping, in particular the 2 at. % Al doping, with a very few experimental studies and an absence of theoretical investigations.

In this context, using Wien2k package [48], we numerically deduce structural, electronic optical and thermoelectric properties of pure wurtzite ZnO, as a reference material, and Al doping concentration of 2.0 at. % of ZnO. Then, keeping in mind the comparative aspects, we carry out a discussion to put into evidence the differences and the improvements in different properties of pure and doped ZnO.

Computational methods

For calculation processing and data collection, we used the code Wien2k package that employs the density functional theory, DFT, based on the full potential linearized augmented-plane wave, FP-LAPW method. In order to deduce the best approximations that improve the results, in less time calculations, we first tested for wurtzite ZnO different approximation: local density approximation, LDA, WC-GGA, PBEsol-GGA, PBE-GGA, etc. Then, to carry out all properties calculations, we opted for generalized gradient approximation, GGA [49], for structural properties determination, and the modified Becke-Johnson, TB-mBJ [50], for electronic and optical properties deductions. The combination of such approximations gives more accurate semiconductor band gap values for several semiconductors and insulators [50, 51]. Prior to any property determination, we confirmed the optimization and the stability of the structure which plays an important role in computational calculations of materials’ physical properties. The X-ray diffraction, XRD, patterns were determined by VESTA and for the thermoelectric properties, we used the BoltzTrap code [52, 53]. Thus, we investigated the most stable WZ-ZnO phase doped with 2% aluminum metal. In the calculation, the basis atom set locations were (1/3, 2/3, 0), (2/3, 1/3, 1/2) for Zn and (1/3, 2/3, 3/8), (2/3, 1/3, 7/8) for O [54, 55], and the initial experimental lattice parameters: a = b = 3.25 and c = 5.207 [56].

The crystal geometry of ZnO that exhibits, in ground state, a stable hexagonal wurtzite structure is considered to be the energetically most favorable as compared to other phases [20]. The unit cells of the wurtzite (B4) phase of pure ZnO is illustrated in Fig. 1a; the Zn and O atoms are marked in the schematic with grey and red circles, respectively. WZ unit cell that belongs to the space group 186 P63mc consists of two zinc atoms and two oxygen atoms. Figure 1b illustrates the adopted hexagonal 2 × 2 × 6 supercell which includes 48 oxygen atoms and 48 zinc atom into which one Zn atom is substituted by an Al atom; thus corresponding to ~ 2% at. Al concentration, i.e., Zn1-xAlxO (with exactly x = 0.02083).

Fig. 1
figure 1

Unit cells of a the wurtzite (B4) phase of ZnO and b the structure of (2 × 2 × 6) supercell Zn0.98Al0.02O

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In order to get a good energy convergence of pure and doped ZnO, the convergence test was performed; it gave a cutoff energy equal to 350 eV according to convergence accuracy. 512 and 200 k-points were used for pure and doped ZnO, respectively. The adopted cutoff wave vector in the interstitial region was RMT x Kmax = 8, where RMT is muffin-tin radius and kmax is the maximum wave vector in the reciprocal lattice. To study the stability of this structure, after doping, we calculated its formation energy that depends on the total energies of pure (Epure) and Al-doped ZnO (Edoped) as well as the chemical potentials of Zn (μZn) and Al (μAl), such as [34]:

$${E}_f={E}_{\textrm{doped}}\left({Al}_{(Zn)}\right)-\left[{E}_{\textrm{pure}}\left(\textrm{ZnO}\right)-{\mu}_{Zn}+{\mu}_{Al}\right]$$
(1)

Thus, the calculated formation energy was found to be −2.5 eV; this negative value indicates and ensures that this material is stable

Results and discussion

Structural properties

In order to study the material stability, all optimization calculations were carried out with structure relaxation. Geometry minimization was also achieved via the calculation of the total energy as a function of unit cell volume; this procedure is necessary to obtain the minimum energy that puts into evidence the stability of both ZnO and AZO. The calculated values of the optimized lattice constants for ZnO are found to be a = b = 3.2856 Å, c = 5.311 Å, and c/a = 1.616, which are consistent with values previously reported experimentally and theoretically [54, 57,58,59,60,61].

The determination of equilibrium volume, V0, ground state energy, E0, bulk modulus at pressure P = 0, B0, pressure derivative of bulk modulus, B0′ are carried out by fitting the total energy as a function of the reduced and extended volume of the unit cell into third-order Birch–Murnaghan’s equation of state. Thus, the total energy, E(V), and pressure P(V) as a function of volume are given by [62]:

$$E(V)={E}_0+\frac{9{V}_0{B}_0}{16}\left\{{\left[{\left(\frac{V_0}{V}\right)}^{\frac{2}{3}}-1\right]}^3B{\prime}_0+{\left[{\left(\frac{V_0}{V}\right)}^{\frac{2}{3}}-1\right]}^2\times \left[6-4{\left(\frac{V_0}{V}\right)}^{\frac{2}{3}}\right]\right\}$$
(2)
$$P(V)=\frac{3{B}_0}{2}\left[\left[{\left(\frac{V_0}{V}\right)}^{\frac{7}{3}}-{\left(\frac{V_0}{V}\right)}^{\frac{5}{3}}\right]\right]\times \left\{1+\frac{3}{4}\left(B{\prime}_0-4\right)\left[{\left(\frac{V_0}{V}\right)}^{\frac{2}{3}}-1\right]\right\}$$
(3)

Thus, the other deduced parameters B, B′, and V regrouped in (Table 1) were also found to be in agreement with published data [62,63,64,65,66] obtained with different experimental techniques (X-ray diffraction, XRD, extended X-ray absorption fine structure, EXAFS and Mossbauer) and computing methods. The similarities in the values for ZnO indicate the reliability and the validity of the present chosen approximations and consequently can safely be used for the 2 % at. Al doping concentrations of AZO. It can also be noticed from Table 1 that the decrease in the AZO cell parameters leads to a decrease in the supercell volume as a result of the large difference in electronegativity and atomic radii of Zn and Al atoms.

Table 1 Deduced and reported [62,63,64,65,66] parameters of pure ZnO and 2 at. % AZO
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Moreover, XRD patterns were also numerically calculated with VESTA code [52]. The obtained results are displayed in Fig. 2 for ZnO and 2 at. % AZO. The peaks indexations, when compared to the standard powder diffraction patterns (JCPDS card no. 36-1451), confirm the hexagonally of these materials. Several intensive peaks, located at different 2θ values, are observed. The ZnO and 2 at. % AZO spectra are very similar in peak positions and quite different in peak intensities. However, a close analysis would shows that, for example, the intensive peaks located at different 2θ values, such as 31.49°, 33.81°, and 35.77° in ZnO spectra shifted towards higher 2θ values, i.e., =, 31.94°, 34.42°, and 36.43° in AZO. The slight shift, also observed experimentally in literature for similar and higher Al-doping concentrations [67, 68], confirms the existence of a decrease in the cell constant with the incorporation of the concentration of Al dopant. In fact, the ionic radius of Al3+ is less that of Zn2+ which leads to a decrease in the cell constants.

Fig. 2
figure 2

XRD patterns of a pure ZnO and b 2 at. % AZO

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Electronic properties

The calculation of the band structure and its corresponding density of states (DOS) generally reflect the electronic tendency of a given material; this would be interesting and practical in device designs. Since DFT approximations underestimate the gap calculations [69] all the following results are obtained by GGA and mBJ combination. Thus, Fig. 3 illustrates the computed energy band structures obtained for pure ZnO (Fig. 3a) and 2 at. % AZO (Fig. 3b). It is clear that the valance band maximum, VBM, is exactly situated beneath the conduction band minimum, CBM, at the same high symmetry Γ (0, 0, 0) point of the first Brillouin zone. Thus, the ZnO is a direct band gap semiconductor, Eg, with a deduced value of its fundamental gap equals to 2.81 eV. This value, though relatively lower than that determined experimentally [70], it is better that other reported approximations, such as GGA-mBJ [71], LDA+GW [72], HSE [73], and LDA mBJ [74] (Table 2).

Fig. 3
figure 3

Calculated electronic band structure of a pure ZnO and b 2 at. % AZO. The Fermi level is set at E = 0 eV

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Table 2 Deduced and reported energy band gap of pure ZnO [8, 21, 70,71,72,73,74], and 2 at. % AZO [75]
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The doping effect is put into evidence in Fig. 3b for 2 at. % AZO; one can easily make several observations: (i) a redistribution of the energy band structure, (ii) a shift of Femi level into the conduction band, (iii) the direct band gap is situated at the same Γ point as pure ZnO, and (iv) the energy band gap is changed. It should be noted that Al dopants in ZnO structure act as donors and consequently AZO behaves as an n-type semiconductor. A hybridization is noticed when going from the Γ center towards M (1/2, 0, 1/2) and from K (2/3, 1/3, 0) towards Γ (0, 0, 0); hybridization occurs between M and K and between Γ and A (1/3, 2/3, 1/2).

Moreover, it can be seen that the Fermi level is located within the conduction band. To explain this physical phenomenon, we know that as the doping of a semiconductor with donor impurities increases, the Fermi level shifts from the energy mid-gap (in the intrinsic case) towards the conduction band. When the doping concentrations reach very high levels, Ef not only attains the CBM but also penetrates in the conduction band leading to what is known as degenerate semiconductors.

This physical phenomenon, also known as Moss–Burstein, occurs when some of the lowest states of the conduction band are filled (i.e., the CB becomes partially filled). This situation is clearly visible in the present 2 at. % AZO energy band structure (Fig. 3b) for which the Fermi level, Ef, is located into the CB. In fact, the optical band gap, from Ef to VBM is larger than the fundamental gap by an amount known as the Moss–Burstein shift [76,77,78]; i.e., this shift widens the optical band gap. Thus, the 2 at. % Al doping of ZnO raises the Fermi level above the CBM and widen the optical band gap to become 3.3 eV. This value is in very good agreement with that determined experimentally, 3.3 eV, for the AZO thin films prepared by sol-gel technique with the same Al doping concentration of 2.0 at. % [75]. A comparable result of 2.26 eV was also reported for 3.0 at. % AZO prepared by the spin coating method [35]. The broadening of the optical band gap of Al-doped zinc oxide compared to pure ZnO films was attributed to Moss–Burstein shift that appears in heavily doped ZnO films. It should be noted that the increase in the optical gap improves simultaneously both electrical conductivity and optical transparency [75, 79] and consequently opens up other prospects in optoelectronics.

The total density of states, TDOS, and partial density of states, PDOS, give an increased understanding of the contribution of different states to the energy bands. Figure 4 illustrates TDOS and PDOS of pure ZnO. The whole DOS spectrum (Fig. 4a) shows the variations in both valence band [−6e–0 V] and conduction band [2.81–16 eV] separated by an energy gap equal to 2.81 eV. The conduction band (Fig. 4b) consists mainly of 4s states of zinc and 2p states of oxygen, followed by a strong hybridization between Zn-3p, Zn-3d, and O-2p. These results are in agreement with literature [42, 80]. For the valance band (Fig. 4c), it can clearly be seen that the DOS curves can be divided into three regions:

Fig. 4
figure 4

Calculated TDOS (a) and PDOS (b and c) of pure ZnO

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  • The first region (from −5.2 to −3.8 eV) shows a strong contribution of Zn-3d together with a weak contribution of O-2p.

  • The second region (−3.8 to −3.2 eV) comes mainly from the Zn-3d.

  • The third region (from −3.2 to 0eV) is the result of a mixture of O-2p and Zn-3d states and a small amount of Zn-3p states. This is principally due a p-d hybridization of O-p and Zn-d states

The obtained results of the total and partial density of states as a function of energy for 2 at. % AZO are shown in Fig. 5. The valence band, in the energy range (−10.4 to −9.6 eV), consists of O-2p, Zn-3d, and Al-2s; whereas, in the range (−9.6 to −3.38 eV), its nature is mainly Zn-3d and O-2p. The bottom of the conduction band is due to a hybridization between the atomic orbitals Zn-4s and O-2p followed by Zn-4p and Al-3s states for higher energies. The Al doping of ZnO leads to a shift of the Fermi level above the conduction band and consequently the donor energy level into the CB. This behavior indicates that 2 at. % Al-doped ZnO is a degenerate n-type semiconductor, which confirms the above band structure results (Fig. 3b). In fact, the incorporation of Al dopants into the ZnO structure produces free electrons leading to higher carrier concentration and consequently higher conductivities.

Fig. 5
figure 5

Calculated TDOS (a) and PDOS (b, c, and d) of 2 at. % AZO

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Optical properties

The semiconductor optical properties form the milestone of any device design and fabrication for thin film applications, in particular in optoelectronics and photoelectronics. The determination and understanding of such properties requires the investigation of some optical parameters, individually or altogether, such as: dielectric function, ε(ω), reflectivity, R(ω), absorption coefficient α(ω) refractive index, n(ω), extinction coefficient k(ω), and conductivity, σ(ω) [42, 71, 81,82,83,84].

Dielectric function

The complex dielectric function, which describes the linear response of macroscopic optical properties of a solid, is usually given by:

$$\varepsilon \left(\omega \right)={\varepsilon}_1\left(\omega \right)+{\varepsilon}_2\left(\omega \right)$$
(4)

where ɛ1(ω) is the real part that represents the dispersion of the incident photons by the materials and ɛ2(ω) the is absorptive imaginary parts of the complex dielectric function. The imaginary part, ε2(ω), is given by the following expression:

$${\varepsilon}_2\left(\omega \right)=\left(\frac{4{\pi}^2{e}^2}{m^2{\omega}^2}\right)\sum\nolimits_{i,j}\int {\left\langle i\left|M\right|j\right\rangle}^2{f}_i\left(1-{f}_i\right)\delta \left({E}_f-{E}_i-\omega \right){d}^3k$$
(5)

where ω is the incident photon energy frequency, e is the electronic charge, M is the dipole matrix, i and j are the initial and final states, respectively, ƒi is the Fermi distribution as a function of the ith state, and Ei is the energy of electron in the ith state.

 The real part ɛ1(ω) can be calculated from the imaginary part by using the Kramers–Kronig’s equation [82,83,84]:

$${\varepsilon}_1\left(\omega \right)=1+\frac{2}{\pi }p\ {\int}_0^{\infty}\frac{\omega^{\prime }{\varepsilon}_2\left({\omega}^{\prime}\right)}{\omega^{\prime 2}-\omega 2}\ {d\omega}^{\prime }$$
(6)

where p indicates the main value of integral. The knowledge of the dielectric function is essential for the determination of other optical parameters: R(ω), n(ω), k(ω), and α(ω). Since wurtzite ZnO possesses a hexagonal symmetry, one has to calculate two different independent principal components for ɛ(ω) so that ɛzz(ω) and ɛxx(ω) correspond to parallel- and perpendicular- polarization of c-axis, respectively. Hence, the comparison between optical parameters should be carried out in both directions.

Figure 6 illustrates the calculated real and imaginary parts of the dielectric function of pure and 2 at. % AZO. From Fig. 6a, which shows the real part of ε1 (ω) of pure ZnO, it can be seen that as the energy increases, we observe a fluctuation at 3.63 eV for ɛ1(ω)xx and ɛ1(ω)zz, whose onset corresponds to a first critical point situated at 2.81 eV. This critical point is also obtained in Fig. 6b for the ɛ2(ω) versus energy curves; it can easily be noticed that the onset of the curves increase occurs at the same critical point situated at 2.81 eV. This point, which represents ΓC–ΓV separation, indicates the onset (threshold) of direct optical transitions between the valence band maximum and the first minimum conduction band. This is known as the threshold of the fundamental absorption that corresponds to the fundamental energy gap, confirming that deduced above, 2.81 eV.

Fig. 6
figure 6

Calculated real and imaginary parts of dielectric function: a and b pure ZnO and c and d 2 at. % AZO

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Moreover, we also notice an important anisotropy in both real and imaginary parts of the dielectric function beyond 8.2 eV, whereas they possess an isotropic behavior for lower energies. This anisotropy in optical properties is expected for low symmetry crystals. However, for 2 at. % AZO, (Fig. 6c and d), we can notice two main and important differences with pure ZnO.

  • At lower energies the appearance of (i) an extra strong fluctuation in ɛ1(ω) versus energy curves and a (ii) high peak for ɛ2(ω) = f(E) curves. This phenomenon is due to a variation in specific molecular bonding, the transition between the occupied states beneath the Fermi level and the non-occupied states above Ef in the conduction band.

The critical point (at 2.81 eV for pure ZnO) is shifted to 3.3 eV in 2 at. % AZO, indicating a transition from the valence band to the partially filled CB; this situation corresponds to the Moss–Burstein effect that leads to the shift of Ef in CB and to a larger optical band gap

Reflectivity

The reflectivity depends on the dielectric function according to the equation [82, 83]:

$$R\left(\omega \right)={\left\lceil \frac{\sqrt{\varepsilon \left(\omega \right)}-1}{\sqrt{\varepsilon \left(\omega \right)}+1}\right\rceil}^2$$
(7)

Figure 7a represents the reflectivity as a function of energy for pure and 2 at. % AZO for R(ω)xx and R(ω)zz. It can be observed that, in all cases, the overall tendency is that, the reflectivity intensifies very slowly up to 11 eV and then strongly increases beyond this value. However, compared to pure ZnO, we notice the appearance in the AZO reflectivity a new high peak with R(ω)zz curve at 0.96 eV, which could be due to the new impurity band introduced by Al doping. This is consistent with the preceding observation and analysis of the imaginary part of the dielectric function.

Fig. 7
figure 7

(a) Relationship between reflectivity with energy for pure ZnO and 2 at. % AZO and (b and c) enlarged reflectivity in 2-10 eV range

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Moreover, the doping effect seems to slightly lower the reflectivity values for both polarization directions, this is better illustrated in the enlarged (Fig. 7b and c). In fact, e.g., at E = 2 eV, R(ω) decreases from RZnO(ω)xx = 0.04212 and RZnO(ω)zz = 0.04138 for ZnO to RAZO(ω)xx = 0.03602 and RAZO(ω)zz = 0.02866 for AZO, respectively. This reflectivity decrease corresponds to ~1.17 for xx-polarization and ~1.45 for zz-polarization. This is a positive effect since it achieves a lower reflectivity. It should be noted that a TCO should have a low reflectivity for its use in various optoelectronic applications [43]. This tendency is in agreement with reported experimental [75] and theoretical works [85]. In fact, the reflectivity of 6 at. % AZO was found to be about twice less than that of pure ZnO [85].

Absorption coefficient

The absorption coefficient, α(ω), an important factor for the evaluation of material optical properties, is a function that can measure the material ability to absorb the energy. This parameter depends on the dielectric function as follows [82].

$$\alpha \left(\omega \right)=\sqrt{2}\omega\ {\left[\sqrt{\varepsilon_1^2\left(\omega \right)+{\varepsilon}_2^2\left(\omega \right)}-{\varepsilon}_1\left(\omega \right)\right]}^{1/2}$$
(8)

Figure 8 illustrates the absorption coefficient as a function of energy for (a) pure and (b) 2% AZO for α(ω)xx and α(ω)zz. Several observations can easily be made, namely:

  • The absorption coefficient of both pure and 2 at. % AZO is very weak in the range of visible spectrum, indicating a high transmission in this region.

  • Pure ZnO, absorbs above 2.81 eV, i.e., in the UV and beyond

  • The anisotropy phenomenon sets in for energies higher than 8.2 eV, beyond which an important absorption is observed around 9.2 eV along zz and at 10.1 eV along xx.

  • the appearance in AZO of a new high peak with α (ω)zz curve at 0.96 eV, confirming the previously obtained peak with reflectivity,

  • For 2 at. % AZO, the absorption coefficient is higher than that of pure ZnO in the range [3.1–6.2eV]. It can also be seen that an important absorption (strong peak) along the zz direction occurs in the IR. Moreover, compared to pure ZnO, a small blue shift phenomenon is obtained in the 2 at. % AZO.

Fig. 8
figure 8

Relationship between absorption coefficient with energy for a pure ZnO and b 2 at. % AZO

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All these observations can for the basis for the study and the design of specific devices as light protectors.

Thermoelectric properties

Thermoelectricity consists of converting thermal energy directly to electrical energy in solid materials. The determination and understanding of materials thermoelectric properties requires the investigation of some interrelated parameters: Seebeck coefficient, S, electrical conductivity, σ, thermal conductivity, κ, power factor, PF, and figure of merit, ZT [18, 53, 85,86,87,88,89,90].

Figure 9 illustrates the Seebeck coefficient as a function of chemical potential for (a) pure ZnO and (b) 2% at. AZO at different temperatures 300 K, 600 K, and 900 K. It is clear that the overall curve behavior is similar: a saturation, a sharp positive peak, a sharp negative peak, and a final saturation. It should be noted that positive S values indicate that the material is p-type whereas negative S values are obtained in n-type semiconductors. However, despite the similarities in curve trends several discrepancies and remarks can be made for both ZnO and AZO as follows.

  • With increasing temperatures, the peak position very slightly shifts towards higher chemical potentials

  • Compared to pure ZnO, the peak positions in AZO shifted towards lower chemical potentials around −1.5 eV.

  • The highest values of Seebeck coefficient, SZnO = 1.329 mV/K and SAZO = 0.688 mV/K for ZnO and 2 at. % AZO, respectively are obtained at 300 K in the p-type region. In the n-region the maximum absolute values are found to be: SZnO = −1.16 mV/K and SAZO = −0.746 mV/K.

  • With increasing absolute temperatures up to 900 K, the S values (peak amplitude) decrease down to SZnO = 0.551 mV/K and SAZO = 0.281 mV/K for ZnO and 2 at. % AZO.

Fig. 9
figure 9

Seebeck coefficient as a function of chemical potential (a and b) and temperature (c and d) of pure ZnO and 2 at. % AZO

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To explain the above observations, let us recall that we deduced that ZnO doping with Al leads to an n-type material and in the present case the 2 at. % AZO is shown to be a degenerate semiconductor as illustrated by its calculated electronic band structure (Fig. 3b). It can be seen that, at the high symmetry Γ (0, 0, 0) point, the conduction band minimum is located below the Fermi level (set at E = 0 eV). Thus, this CBM position could partly be responsible for the shift of Seebeck coefficient peak positions (Fig. 9b) below the chemical potential (at E = 0 eV) for the 2 at. % AZO. Therefore, for this n-type material, we only consider negative values of Seebeck coefficients. Thus, to better illustrate the temperature effects, we plot S as a function of T in the range 300 K-900 K for ZnO (Fig. 9c) and for 2 at. % AZO (Fig. 9d). The highest absolute S values are found to be ׀SZnO׀ = 1.16 mV/K and ׀SAZO׀ = 0.746 mV/K, indicating that pure and doped ZnO represent good thermoelectric materials.

The evolution of electrical conductivity as a function of chemical potential is plotted in Fig. 10 for (a) pure ZnO and (b) 2% at. AZO, in terms of σ/τ where τ is the average time between two successive electron collisions [53] at different temperatures 300 K, 600 K, and 900 K. The curves show that the temperature has very negligible effect on the electrical conductivity for both materials. The lowest conductivity values, for all temperatures, are obtained in the chemical potential range of 0 to approximately 1 eV for ZnO and the range −1 to 1.75 eV) for AZO.

Fig. 10
figure 10

Electrical conductivity (a and b) and thermal conductivity (c and d) as a function of chemical potential for pure ZnO and 2 at. % AZO

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The plots of κ/τ as a function of chemical potential are displayed in (Fig 10c, d). It can be seen that, for T = 300 K, the curve variation is similar to that of electrical conductivity. For ZnO (Fig. 10c), κ/τ decreases sharply in the chemical potential range of −2 to approximately 0 eV, then attains its lowest values in the range 0 to approximately 1 eV) beyond which a final increase is observed. However, the most important remark is that, unlike electrical conductivity, thermal conductivity strongly depends on temperature: as T increases, κ/τ decreases from its highest value of 2 × 1016 W/mKs at T = 300 K to 4.5 × 1015 W/mKs at 900 K. For 2 at. % AZO (Fig. 10d), similar remarks can be formulated: a decrease followed by the lowest saturation and a final sharp increase. However, the lowest saturation region is situated around −1 eV and the maximum values at ~1.2 eV. In fact, the κ/τ maximum decreases from ~ 3 × 1015 W/mKs at T = 300 K to 0.5 × 1015 W/mKs at 900K

Thermoelectric power factor (PF) defines thermoelectric efficiency; it depends on Seebeck coefficient and electrical conductivity, such as PF = σS2. However, the need for a dimensionless parameter requires the use of the figure of merit, ZT, known as the thermal conversion efficiency, which characterizes the materials ability to convert thermal energy into electrical energy. It depends on the combination of three physical parameters: S, σ, and κ as well as the absolute temperature, T, according to the relation ZT = σS2T/κ [17, 90,91,92]

Figure 11a, b represents the power factor, calculated from PF = σS2/τ, as a function of chemical potential for (a) pure ZnO and (b) 2 at. % AZO at different temperatures 300 K, 600 K, and 900 K. It can be seen that, for both cases, the power factor depends on the temperature. For ZnO, the highest peaks are located very near to the Fermi level and the corresponding maximum PF values decrease from 8 × 1011 W/K2ms at 300 K to 2 × 1011 W/K2ms at 900 K. For 2 at. %AZO, the peak positions shifted towards lower chemical potential (−1.75 eV) with the maximum S values decreasing from 2.9 × 1011 W/K2ms at 300 K to 1.4 × 1011 W/K2ms at 900 K.

Fig. 11
figure 11

Power factor (a and b) and figure of merit (c and d) as a function of chemical potential of pure ZnO and 2 at. % AZO

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Figure 11c, d illustrates the figure of merit as a function of chemical potential for (c) pure ZnO and (d) AZO at different temperatures. Both structures show two high peaks whose maximum values approach unity at room temperature. Thus, these materials can be considered as good thermoelectric devices for which ZT should be as close to unity as possible [53, 93, 94]. As the temperature increases up to 900 K, the peaks slightly decrease down to ~0.8.

Conclusions

Structural, electronic, optical, and thermoelectric properties of pure and Al-doped zinc oxide were numerically deduced, analyzed and compared, using Wien2k package. We considered the most stable wurtzite ZnO structure and adopted a hexagonal 2 × 2 × 6 supercell that includes 48 oxygen atoms and 48 zinc atom into which one Zn atom is substituted by an Al atom; thus corresponding to ~2% at. Al concentration. It was first shown that the combination of GGA and mBJ gives more accurate results than other tested approximations and consequently is adopted for this investigation. Several interesting results were deduced. The AZO cell parameters decreased leading to a decrease in the supercell volume as a result of the large difference in electronegativity and atomic radii of Zn and Al atoms. The deduced direct band gap of ZnO, 2.81 eV, increased up to 3.3 eV for 2 at. % AZO.

The dielectric function of pure ZnO showed a fluctuation whose threshold occurs at 2.81 eV representing the direct optical transitions between the valence band maximum and the first minimum conduction band. Whereas, for 2 at. % AZO, two main differences with pure ZnO were noticed (i) at lower energies the appearance of an extra strong fluctuation in ɛ1(ω) and a high peak for ɛ2(ω) curves and (ii) a wider optical band gap of 3.3 eV.

The reflectivity for both materials showed an overall tendency consisting of a very slow intensification of up to 11 eV followed be a very sharp increase beyond this value. However, for the 2 at. % AZO, a new high peak with R(ω)zz curve appeared at 0.96 eV, which could be due to the new impurity band introduced by Al doping. Moreover, the doping effect seems to slightly lower the reflectivity values by ~ 1.17 for xx-polarization and ~ 1.45 for zz-polarization; a similar decreasing tendency for 6 at. % AZO was found to be about twice less than that of pure ZnO [85]. For the optical absorption, besides the overall curve similarities for pure and doped ZnO, it is observed for the latter that (i) the absorption coefficient is higher in the range [3.1–6.2eV], (ii) an important absorption (strong peak) along the zz direction occurs in the IR, and (iii) a small blue shift phenomenon is obtained in the 2 at. % AZO.

For thermoelectric properties, we investigated the effects of chemical potential on Seebeck coefficient, electrical and thermal conductivities, power factor, and figure of merit at temperatures varying from 300 k to 900 K, for ZnO and 2 at. % AZO. The deduced high Seebeck coefficient values, ׀SZnO׀ = 1.16 mV/K and ׀SAZO׀ = 0.746 mV/K, indicate that pure and doped ZnO represent good thermoelectric materials. The trends of variations of electrical and thermal conductivities with chemical potential are quite similar in the range 300 to 900 K for both structures: an initial decrease followed by a low saturation region and then a final increase. However, while the temperature has very negligible effect on the electrical conductivity, it strongly affects the thermal conductivity. In fact as T increases from 300 to 900 K, κ/τ decreases from 2 × 1016 W/mKs to 4.5 × 1015 W/mKs for ZnO and from ~3 × 1015 W/mKs at T = 300 K to 0.5 × 1015 W/mKs 2 at. % AZO. Finally, the value of the figure of merit approaches, at room temperature, unity which is the highest possible value indicating that both ZnO and AZO material can be considered as good thermoelectric device.