Introduction

Transparent thermoelectric materials have been extensively investigated in clean and renewable energy sources. Transparent conducting oxides (TCOs) such as Tl2O3, In2O3, TiO2, SnO2, and ZnO are used in liquid crystal and touch-screen displayers, photovoltaic cells, and thermoelectric devices due to their transparency in the visible light range and attractive electronic properties1,2,3,4. The computed hole mobility for Li3Sb is quite exceptional and comparable with the electron mobility in the best n-type TCOs5.

Among the TCOs with bixbyite structure, Tl2O3 is a degenerate n-type semiconductor with high conductivity and very low resistivity of about 10–4 Ω cm–16. In Tl2O3 transparent conductor7,8,9,10, the carrier concentration changes significantly with oxygen partial pressure10,11,12,13 and semiconducting nature is observed by defect-induced in the crystal structure of compound. The metallic behavior of Tl2O3 probably originates from oxygen vacancy defects9.

The experimental and theoretical studies show that optical band gap of Tl2O3 change from 1.40 to 2.75 eV8,11,14 and 0.33 to 0.6 eV7,14, respectively. This difference in band gap values originate from additional electrons introduced into the material by some n-type defects, and self-doping due to the oxygen vacancy formation7,9. The carrier concentration of Tl2O3 increase by defects and the oxygen vacancies and reach 7.6 × 1020 cm–39. Thallium-doping with the electron configuration of [Xe] 6s24f145d106p1 is suitable way to reduces the band gap and increase the carrier mobility of In2O315.

In the new TCOs, such as ZnSb2O6 (Ga- doping)16 and Ga2O3 (Si- doping)17 compounds, the band gap values for the pure and doped-TCOs increases from 3.38 to 3.56 eV and 4.80 to 4.99 eV, respectively. The widening of the band gap is due to the Burstein-Moss (B. M.) shift.

The band gap value can be altered by change in the carrier concentrations from Eg to \(\mathop {{\acute{E}}_{g} }\limits\) as the Burstein-Moss (B. M.) effect7. The B. M. shifts of the band gap is due to the filling the bottom of the conduction band. The band gap value is given by Ref.18:

$$\mathop {{\acute{E}}_{g} }\limits {\text{ = E}}_{{\text{g}}} { + }\Delta {\text{E}}_{{\text{g}}}^{{{\text{BM}}}} ,$$
(1)

where \(\Delta {\text{E}}_{{\text{g}}}^{{{\text{BM}}}}\) is the Burstein–Moss energy shift given by:

$$\Delta E_{g}^{BM} = \frac{{\hslash }^{2}}{{2m_{vc}^{*} }}(3\pi^{2} n_{e} )^{2/3}$$
(2)

Optical transitions of carriers are interpreted by considering the Burstein-Moss (B. M.) shift in the conduction band13. Optical reflectivity spectra show that there is a strong reflectance in the near-infrared region (1300–1500 nm)14. The B. M. mechanism is observed in transparent conductors like Tl2O3 and narrow band gap materials for using in solar cells and high-temperature thallium oxide-based superconductors19. Narrow band gap materials are typically good thermoelectric materials20. However, development of transparent conductors with high thermoelectric efficiencies is one of effective ways to produce electricity without losing their transparency. Narrow electronic states play a key role in the TCOs for thermoelectric applications.

The structural, optical, and thermoelectric properties are sensitive to the applied pressure21. On the other hand, the effect of pressure has been a powerful method for studying the optoelectronic and thermoelectric properties of materials in the past few decades. The thermoelectric efficiency of a material can be improved by applying pressure22,23,24,25,26,27,28,29.

The efficiency of thermoelectric materials is defined by the figure of merit, ZT = S2σT/κ, where S, σ, T and κ are the Seebeck coefficient, electrical conductivity, absolute temperature, and total thermal conductivity, respectively. It is clear that high ZT value requires high σ, low κ, and relatively high S. The electronic structure of materials is important for the reach high ZT values22.

The experimental results show that Tl2O3 has a cubic bixbyite crystal structure with \(Ia\overline{3 }\) space group and lattice constant of a = 10.5344 Å30. Gomis et al. have studied the structural properties of Tl2O3 under high pressures31. Their results indicate that the crystal structure of Tl2O3 is cubic bixbyite to 22.0 GPa. At room temperature, the amorphous phase of Tl2O3 is predicted above 22 GPa by the theoretical analysis of the elastic properties31.

In the present work, we have investigated the structural, optoelectronic, and thermoelectric transport properties of Tl2O3 under high pressures up to 22.0 GPa by using the DFT as implemented in the Wien2k32 and BoltzTrap codes33. The method of calculations is described in Sect. "Computational details". The optoelectronic and thermoelectric results are presented and discussed in Sect. "Results and discussions" and conclusions is summarized in Sect. “Conclusion”.

Computational details

Pressure -dependent optoelectronic and transport properties of Tl2O3 were calculated by the full-potential linearized augmented plane wave method (FP- LAPW). For calculation of the exchange–correlation potentials, the modified Becke-Johnson (mBJ) functionals34,35,36,37,38 with spin–orbit coupling are used as implemented in the Wien2k package. The electronic wave functions are expanded by the plane-wave cutoff value of Kmax × RMT = 7.0 in the interstitial region (RMT is the smallest atomic muffin tin sphere radius). Gmax, the magnitude of the largest vector in the Fourier expansion of charge density, is selected about 12 (Ry)1/2. The self-consistent field (SCF) calculations are carried out with 600, 1800, and 12,000 k-points for structural, optoelectronic, and thermoelectric calculations, respectively.

The Boltzmann transport equation by employment of the rigid band approximation is used to obtain the thermoelectric properties. In this approximation, a shift of the Fermi level is equivalent to the doping of the compound. The Fermi level is defined as: \({E}_{f}=\frac{{\hslash }^{2}}{2{m}^{*}}{(3{\pi }^{2}n)}^{2/3}\), where n and \({m}^{*}\) are carrier concentration and the effective mass, respectively. The thermoelectric calculations are performed with n = 1 × 1018 to 1 × 1021 cm–3 carrier concentrations at 50 to 800 K. To obtain the best thermoelectric efficiency of Tl2O3 under different pressures and carrier concentrations, we have compared the obtained results under various carrier concentration to each other. It is found that n = 1 × 1018 cm–3 is an effective carrier concentration and it gives us the optimized achievement. Therefore, we have just presented the thermoelectric results with n = 1 × 1018 cm–3. Here, the relaxation time is considered as a constant, and we have used the following relations23:

$${\sigma }_{\alpha \beta }\left(\varepsilon \right)=\frac{1}{N}{\sum }_{i,k}{\sigma }_{\alpha \beta }\left(i,k\right)\frac{\delta \left(\varepsilon -{\varepsilon }_{i,k}\right)}{\delta \left(\varepsilon \right)},$$
(3)
$${\nu }_{\alpha \beta }(T;\mu )=\frac{1}{eT\Omega }\int {\sigma }_{\alpha \beta }\left(\varepsilon \right)\left(\varepsilon -\mu \right)\left\{-\frac{\partial {f}_{\mu }\left(T;\varepsilon \right)}{\partial \varepsilon }\right\}d\varepsilon ,$$
(4)
$${S}_{i,j}={({\sigma }^{-1})}_{\alpha i}{\nu }_{\alpha j},$$
(5)
$${\kappa }_{\alpha \beta }^{0}\left(T;\mu \right)=\frac{1}{{e}^{2}T\Omega }\int {\sigma }_{\alpha \beta }\left(\varepsilon \right){\left(\varepsilon -\mu \right)}^{2}\left\{-\frac{\partial {f}_{\mu }\left(T;\varepsilon \right)}{\partial \varepsilon }\right\}d\varepsilon ,$$
(6)

where μ, fμ and κ0 are chemical potential, Fermi-Dirac distribution function and the electronic part of thermal conductivity (κ0 = κe), respectively. The total thermal conductivity is composed of two electronic κe and lattice κl thermal conductivities (κ = κe + κl). A direct method for study of lattice thermal conductivity κl, is molecular dynamic simulation. A long simulation time need to calculate the κl and prediction of κl for Tl2O3 with the large unit cell is very expensive and it is impossible by Wien2k and BoltzTrap codes. On the other hand, there is not experimental evidence for κl of Tl2O3 and compared to the previously reported results for Tl2O show that κl of Tl2O is below 1 W/mK39 and it has a negligible contribution from the total thermal conductivity κ. Nevertheless, here we have calculated the electronic part of thermal conductivity κe.

For a semiconductor with a narrow band gap, the dependence of the Seebeck coefficient (S) on the effective mass of carriers is calculated as follows:

$$S=\frac{8{\pi }^{2}{k}_{B}^{2}}{3e{h}^{2}}{({m}_{1}^{*}{m}_{2}^{*}{m}_{3}^{*})}^{1/3}{N}_{v}^{2/3}T{(\frac{\pi }{3n})}^{2/3},$$
(7)
$${m}_{i}^{*}=\frac{\int {e}^{-\frac{dE\left(k\right)}{{k}_{B}T}}{m}_{b}^{*}\left(k\right)d(k)}{\int {e}^{-dE(k)/{k}_{B}T}dk},$$
(8)
$${m}_{b}^{*}=\frac{{\hslash }^{2}}{\frac{{\partial }^{2}E(k)}{\partial {k}^{2}}},$$
(9)

where ℏ, Nv, \({m}_{i}^{*}\) and \({m}_{b}^{*}\) are the Planck constant, band degeneracy, the mass components along three perpendicular directions (i = x, y, z) and band mass, respectively40.

For study of optical spectra, the complex dielectric function is calculated. Dielectric function is the linear response of a system to electromagnetic waves and can be described by both the real ε1(ω) and imaginary ε2(ω) parts which are calculated as following relations41:

$${\varepsilon }^{\alpha \beta }\left(\omega \right)={\varepsilon }_{1}^{\alpha \beta }\left(\omega \right)+i{\varepsilon }_{2}^{\alpha \beta }\left(\omega \right),$$
(10)
$${\varepsilon }_{2}^{\alpha \beta }\left(\omega \right)=\frac{{\hslash }^{2}{e}^{2}}{\pi {m}^{2}{\omega }^{2}}\sum_{c\text{,}v}\int dk\langle {c}_{k}|{p}^{\alpha }|{v}_{k}\rangle \langle {v}_{k}|{p}^{\beta }|{c}_{k}\rangle \delta \left({\varepsilon }_{ck}-{\varepsilon }_{vk}-\omega \right),$$
(11)
$$\varepsilon_{1}^{\alpha \beta } \left( \omega \right) = \delta_{\alpha \beta } + \frac{2}{\pi }P \int \limits_{0}^{\infty } \frac{{\omega Im\mathop {{\acute{\varepsilon}}_{{\alpha \beta \left( {\mathop {\acute{\omega}} } \right)}} }}}{{\mathop {\acute{\omega}} ^{2} - \omega^{2} }}d\mathop {\acute{\omega}} ,$$
(12)
$${\alpha }_{\alpha \alpha }\left(\omega \right)=\frac{\sqrt{2 }\omega }{c}\sqrt{\left|{\varepsilon }_{\alpha \alpha }\right|-{Re\varepsilon }_{\alpha \alpha }\left(\omega \right),}$$
(13)
$${Eloss}_{\alpha \alpha }\left(\omega \right)=Im\left(\frac{-1}{{\varepsilon }_{\alpha \alpha }\left(\omega \right)}\right).$$
(14)

Results and discussions

Optoelectronic results

The electronic band structure (B. S) of Tl2O3 was calculated using the mBJ and mBJ + SO potentials and the obtained B. S spectrum at zero pressure is shown in Fig. 1 by mBJ. A direct band gap \({E}_{g}^{F}\) of 0.49 eV is appeared between top of the valence band Γ2 and bottom of the conduction band Γ3. The experimental results show that there is a Bursntein-Moss shift \({E}_{g}^{B. M}\) about 0.74 eV in Tl2O3 compound between Γ3 and Γ4 points13. Therefore, widening of the band gap is about 1.23 eV. On the other hand, the optical band gap value at zero pressure was obtained by optical absorption calculations and compared with the electronic band structure. The results of the electrical and optical calculations for pure Tl2O3 and other TCOs have been summarized in Table 1. Obtained band gap value is in close agreement with the experiment. The bottom of the conduction band exhibit Tl-6s dangle band around the Γ point. The presence of parabolic curve, like free electrons in s state, justify the conductor behavior of Tl2O3.

Figure 1
figure 1

Calculated band structure of Tl2O3 and band gap engineering by mBJ at zero pressure. Obtained band gap is compared with the calculated absorption spectra at zero pressure.

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Table 1 Obtained optoelectronic and thermoelectric properties (at 300 K) of Tl2O3 and compared with the other TCO materials.
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The calculated band structure of the Tl2O3 under pressure is plotted in Fig. 2. The band gap’s nature is direct at Γ direction under all pressures. Obtained band gap values at zero pressure are achieved 0.49 and 0.45 eV for mBJ and mBJ + SO, respectively, which are in close agreement with the band gaps reported by other (see Table 1)8,9,35,36,37,38,39,42,43,44,45,46. The variation of band gap values with pressures is shown in Fig. 3. It is observed that the band gap value linearly increases with pressure and band gap values by mBJ are more than mBJ + SO at all pressures. We have fitted two linear relations between band gap values and pressures as:

Figure 2
figure 2

Calculated band structures of Tl2O3 under pressure by (a) mBJ and (b) mBJ + SO.

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Figure 3
figure 3

The variation of the band gap value with respect to the pressure for Tl2O3.

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$${\text{E}}_{\text{g}}^{\text{mBJ}}\text{= 0.01891 P+ 0.5114}\text{; } {\text{E}}_{\text{g}}^{\text{mBJ+SO}}\text{= 0.01869 P+ 0.4665}.$$
(15)

The results indicate that it is possible to control the band gap value and the other optoelectronic and thermoelectric properties by pressure. Nevertheless, we to predict the transport properties of electrons under applied pressure by the calculation of the effective mass of carries from the curvature of band structure plots. To clarify the effective mass of electrons under pressure, the curvature of electronic bands in the bottom of the conduction band are investigated at Γ → P direction by Eq. (9). As shown in Fig. 4, with increasing pressure, the effective mass of electrons is increased and the electron mobility decreases. It should be noted that by increasing pressure, the lattice constants and hence bond lengths decrease and the interaction between electrons and ions increases which results in the change the effective mass of electrons.

Figure 4
figure 4

The variation of the effective mass of electrons with respect to the pressure for Tl2O3.

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The calculated total and partial densities of states (DOSs) of Tl2O3 are presented in Fig. 5a at zero pressure. Three main regions are observed in DOS spectra. The first region is observed in − 19 to − 16 eV. It is a mixture of the O-2 s states with a few contributions of Tl-6 s, 4f, and 6p states. The second region is created in the − 10 to 0 eV near the Fermi level. It is considerably dominated by Tl-5d, 6 s, 6p, and 4f states and O-2p states while in the third region at the bottom of the conduction band, Tl-6 s and O-2p states play a key role in electronic behavior of Tl2O3.

Figure 5
figure 5

Density of states of Tl2O3 (a) at zero pressure and (b) different pressures by mBJ.

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Pressure dependence of the DOSs are plotted in Fig. 5b. It is clear that by increasing pressure, the valence band maximum is nearly constant while the conduction band minimum is shifted upward. It is found that the effect of hydrostatic pressure significantly shifts the Tl- 6s state in the bottom of the conduction band. Electronic results show that pressure tuning is an effective way to control the physical properties of TCOs.

In Fig. 6a and b, the real and imaginary parts of the dielectric function are plotted at different pressures. Obtained static dielectric constants at zero photon energy, ε1(0), are 4.8, 4.62, 4.4, 4.3, and 4.2 at 0.0, 4.9, 10.5, 15.0 and 22.0 GPa, respectively which compared with similar TCOs in Table 131,32,34. It is also found that ε1(0) decreases with increasing pressure. Tl2O3 can be used in semiconductor devices because of the inverse relationship between the static dielectric constants and semiconductor band gap which verifies by the Penn model47 (\({\upvarepsilon }_{1}\left(0\right)\approx 1+{(\frac{\mathrm{\hslash }{\upomega }_{{\text{p}}}}{{{\text{E}}}_{{\text{g}}}})}^{2}\)).

Figure 6
figure 6

Calculated optical spectra for Tl2O3 at various pressures (a) the real part of dielectric function, (b) the imaginary part of dielectric function (c) the electron energy loss spectrum EELS and (d) Absorption.

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By increasing pressure, the main peaks, corresponding to the different pressures, are shifted towards higher energies. For the first and second main peaks, the intensity of peaks decreases with an increase in pressure while for the other main peaks near to the zero value ε1(ω) the intensity of peaks increases with pressure. After some fluctuations, it reaches to zero value at 0.0 GPa in the range of 21.0 and 24.0 eV. It is clear that the negative value of ε1(ω) is almost appeared with increasing pressure. In this region, the material exhibits a high reflection with metallic nature. After 25 eV, we can to see the positive value of ε1(ω) and the overall trend of ε1(ω) remains the same at the high energy of incident photons under the effect of pressure.

As shown in Fig. 6b, the main peaks of the imaginary part of the dielectric function, ε2(ω), are corresponding to the optical transitions of electrons from the valence band to the conduction band around the EF. The optical band gap or threshold energy of ε2(ω) spectrum is shown as E1, correspond to direct inter-band transitions from Γ2 → Γ4 in Fig. 1. By increasing pressure, this value increase. The second strong peak E2 is corresponding to the optical transitions Γ1 → Γ4, as it is observed in the experimental absorption measurements9. Other main peaks are related to the electronic transitions from the deep electronic states in the valence band to the conduction band35.

The electron energy loss spectrum, EELS, is a useful optical spectrum for prediction of Plasmon energies. The electron density in the material plays a key role in the excitation of the plasmons. Figure 6c shows the calculated EELS spectra. EELS is zero at low energies blew the optical band gap without scattering of electrons. By enhancing the energy of incident photons, EELS increases slowly due to the inelastic scattering of electrons. EELS spectrum reaches to the maximum value then decreases. The maximum energy value of EELS corresponds to the Plasmon energy. Obtained Plasmon energies are 24, 24.2, 24.55, 24.69 and 25.38 eV at 0.0, 4.9, 10.5, 15.0 and 22 GPa, respectively. It is clear that the Plasmon energy is shifted to the higher value of EELS with increasing pressure.

In Fig. 6d, the pressure- dependent of absorption spectra of Tl2O3 are displayed at the energy range of 0–35 eV. By increasing pressure, the absorption value increase from 146.72 to 163.47 × 104 cm–1 and a blue shift is observed towards higher values in the energy of incident photons correspond to the energy range of 15.32- 20.54 eV in the ultraviolet UV spectrum. As shown in the inset of Fig. 6d, there is not absorption for photons with energies lower than optical band gap and absorption start with an increase in photon energy. It is clear that optical band gap increase with an increase in pressure.

Thermoelectric results

As shown in Fig. 7, the Seebeck coefficient (SC) of Tl2O3 is calculated as a function of carrier concentration, pressure, and temperatures for both n- and p-types Tl2O3. The Seebeck coefficient is negative for n-type Tl2O3, and it is positive for p-type. It is observed that the SC increase with increasing temperature then it decreases with temperature as defined in Eq. (7). The SC is directly proportional to the temperature and it is inversely proportional to the carrier concentration. Therefore, the SC increase with temperature at low temperature. On the other hand, at high temperatures, the carrier concentration n is thermally exited hence by increasing n, the SC decreases. Obtained SC, at room temperature (300 K) for n- type Tl2O3, are − 268, − 275, − 281, − 286 and − 292 μV/K at 0.0, 4.9, 10.5, 15.0 and 22 GPa, respectively. The results show that with increasing pressure, the SC for n-type one increases and it is almost similar for p-type one in the temperature range of 50–350 K.

Figure 7
figure 7

The variation of Seebeck coefficient with respect to the temperature and pressure at n = 1 × 1018 cm–3 carrier concentration for (a) n- and (b) p -types of Tl2O3.

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As shown in Fig. 4 and defined in Eq. (7), by increasing pressure, the effective mass of electrons increases which results an increased SC of n- type Tl2O3. For p- type Tl2O3 with increase in pressure, the hole states in the valence band are nearly constant (Fig. 2) and hence its changes with pressure is almost small. It is also seen that the more effect of pressure on SC is at high temperatures.

The calculated electrical conductivity (σ/τ) is plotted in Fig. 8. In this plot, the variation of σ/τ with n = 1 × 1018 cm–3 carrier concentration, temperature and pressure are studied. The electrical conductivity is almost constant at low temperature 450 K then it increases by increasing temperature and excitation of carrier concentration as following relation:

$$\sigma \, = \, ne\mu ,$$
(16)

where μ is the mobility. The figure shows that the electrical conductivity (σ/τ) remains almost constant with enhancing temperature at high pressure (22.0 GPa).

Figure 8
figure 8

The variation of electrical conductivity with respect to the temperature and pressure at n = 1 × 1018 cm–3 carrier concentration for (a) n- and (b) p -types Tl2O3.

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The pressure and temperature dependences of electronic part of thermal conductivity (κe/τ) are displayed in Fig. 9 for n- and p- type Tl2O3 at n = 1 × 1018 cm-3 carrier concentration. Electronic thermal conductivity depends upon carrier concentration and temperature, as follows:

Figure 9
figure 9

The variation of thermal conductivity with respect to the temperature and pressure at n = 1 × 1018 cm–3 carrier concentration for (a) n- and (b) p -types of Tl2O3.

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$${k}_{e}=\frac{n{\pi }^{2}T{k}_{b}^{2}}{3{m}^{*}{v}_{e}}.$$
(17)

It is clear that by the increase in temperature, the electrical conductivity increases and it remains constant at all pressures at low temperatures. An increasing trend is observed just in n-type Tl2O3 which correspond the electronic states in the conduction band, with the pressure effect at low temperatures. At high temperatures, with the increase in pressure, the effective mass of carries increase, so the electronic thermal conductivity decreases. The maximum value of κe/τ for n-type Tl2O3 is obtained about 6.5 × 1013 W/Kms at 800 K and zero pressure.

In Fig. 10, the variation of figure of merit (ZT) are shown with temperature and pressure at n = 1 × 1018 carrier concentration. ZT increases with increasing in temperature and pressure. At room temperature, the ZT values of 0.82 and 0.98 K achieved at 22.0 GPa, for n and p-types Tl2O3, respectively. High pressure as well as temperature have a strong effect on the figure of merit (ZT) of Tl2O3. In Table 1, obtained results are compared with the thermoelectric properties of TCOs. These thermoelectric findings imply that this TCO material can be used in the fabrication of thermoelectric devices under pressure.

Figure 10
figure 10

The variation of figure of merit (ZT) with respect to the temperature and pressure at n = 1 × 1018 cm–3 carrier concentration for (a) n- and (b) p -types of Tl2O3.

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Conclusion

The effect of pressure on the structural, optoelectronic and thermoelectric properties of Tl2O3 were investigated by using the FP-LAPW method and the modified Becke-Johnson (mBJ) functional with spin–orbit coupling. In band structure, obtained direct band gap of Tl2O3 is 1.23 eV at zero pressure. The results show that the band gap and the effective mass of carries increases with pressure. The bottom of the conduction band composed of Tl-6s dangle band and the hydrostatic pressure significantly shifts Tl-6s state. By increasing pressure, the blue shift observed in optical responses such as the real and the imaginary parts of the dielectric functions, electron energy loss and absorption spectra. For n- type Tl2O3, the Seebeck coefficient increases with pressure and reaches to − 292 μV/K at room temperature but thermal conductivity decrease under the effect of pressure. By increasing pressure, it is possible to control the high Seebeck coefficient and the low thermal conductivity for Tl2O3 compound. The maximum figure of merit ZT of p-type Tl2O3 is 0.98. The results indicate that pressure and temperature tuning play significant roles in the design of thermoelectric devices based on Tl2O3.