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Abstract
Recently, boron arsenide (BAs) has been measured with high thermal conductivity in the experiments, great encouragement for low-power photoelectric devices. Hence we systematically investigate the direct and indirect optical absorptions of BAs and BSb by using first-principles calculations. We obtain the absorption onset corresponding to the value of indirect bandgap by considering the phonon-assisted second-order indirect optical absorption. The temperature-dependent calculations also capture the redshift of absorption onset, enhancement, and smoothness of optical absorption spectra. Moreover, in order to introduce the first-order absorption in the visible range, the doping effect of congeners is studied without the assist of phonon. It is found that the decrease of local direct bandgap derives from either the decrease of bonding-antibonding repulsion of p orbital states by the heavier III group elements or the similar influence of lighter V group elements on the s orbital states. Thus, the doping of congeners can improve the visible optical absorptions.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The photoelectric devices have become the indispensable microdevices in the modern industrial society. As the future development trend of the high efficient and low-power photoelectric devices, heat dissipation is the main factor to restrict the performance of devices. For instance, due to the thermal effects, the increasing temperature in the photovoltaic solar cells affects negatively the conversion efficiency and lifetime [1]. Thus the heat dispersing photoelectric materials become increasingly important. As the known famous carbon-based materials, diamond and graphite have a record thermal conductivity of $\sim$ 2000 W/(m$\cdot$K) at room temperature [2–4], but the high cost and anisotropy severely limit their photoelectric applications. Hitherto, many works are committed to find the new structures and materials for heat dissipation in photoelectric devices [5,6]. Recently, with a predictive first-principles calculation of the Boltzmann transport approach, it is found that semiconducting BAs and BSb with zinc-blende face-centered cubic (FCC) structure have a remarkable thermal conductivity [7,8], such as 2000 W/(m$\cdot$K) in BAs. For the reason, the strong covalent bonding and the large mass ratio give rise to the acoustic bunching as well as the large frequency gap between acoustic phonons and optic phonons. But the high-order anharmonicity [9,10] is found to bring the thermal conductivity of BAs down to 1400 W/(m$\cdot$K). Subsequently, the experimental data of high-quality BAs [11–13], $\sim$1300 W/(m$\cdot$K) at room temperature, agrees well with the theoretical predictions.
Although the high thermal conductivity of BAs and BSb is urgently needed physical property for the photoelectric devices, unfortunately, BAs and BSb have the indirect bandgap in the visible range and large local direct bandgap over 3.0 eV [14–16], similar to the classical semiconductor silicon [17,18]. The first-order visible optical absorption process, one-photon direct electronic transition, is forbidden obviously as stated in the previous works [19–21]. Thus it is required to take into account the second-order indirect absorption process [22–25], which has helped silicon apply to a solar cell with tolerable photoelectric conversion efficiency. When the photon energy is greater than the indirect bandgap minus (plus) the absorbed (emitted) phonon energy, the phonons can ensure the momentum conservation in the indirect electronic transition to achieve the second-order optical absorption process, which makes many indirect-bandgap semiconductors have the application potential in photoelectric devices [26–30]. Then, due to the powerful cooling performance of BAs and BSb and the comparability of electronic structure to silicon, they both hold plenty of promise in the next-generation photoelectric devices. Thereout, it is essential to study the optical properties of BAs and BSb seriously, and a full description of optical absorption requires the calculation of electron-phonon coupling. Furthermore, the executable methods are also necessary to improve the optical absorptions in the visible range. Some common methods, such as pressure [31,32] and doping [33–35], influence the electronic structure and modify the physical properties, so may also have a positive impact on the optical absorptions.
In the present work, based on the first-principles calculations, we investigate the first-order and second-order optical absorptions of BAs and BSb, which are both predicted to have a high thermal conductivity [7,8]. Our results show that the main optical absorption peak appears in the ultraviolet region around 7.0 eV. The absorption onset is in coordination with the indirect bandgap, and also redshifts with the rising temperature. The absorption spectra of BAs and BSb in the visible range are similar to those observed in other indirect-bandgap semiconductors [23]. The further study about the doping of congeners shows that the enhanced first-order optical absorption in the visible range results from the reduction of the local direct bandgap.
2. Methods
The technical details of the calculations are as follows. All calculations in this work were carried out in the framework of density functional theory (DFT) with General gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) implementation [36], as performed in the QUANTUM ESPRESSO [37]. The ion and electron interactions were treated with the norm-conserving pseudopotentials [38]. The hybrid Hartree-Fock+DFT functional of HSE was used in order to obtain accurate bandgap and the direct optical absorption. By requiring the convergence of results, the kinetic energy cutoff of $600$ eV and the Monkhorst-Pack $k$-mesh of 30$\times$30$\times$30 (20$\times$20$\times$20) were used in all calculations about the ground-state properties of the primitive cell (cubic crystal cell). The phonon spectra and electron-phonon coupling were calculated on a 30$\times$30$\times$30 $q$-grid by using the density functional perturbation theory (DFPT) [39] and maximally localized Wannier functions [40–43]. Phonon-assisted optical absorption process could be analyzed by the second-order time-dependent perturbation theory with electron-phonon coupling [22–26] and the absorption coefficient was expressed as,
In addition, the first-order absorption was calculated with the following standard model. In the semiconductor, the dielectric function $\varepsilon (\omega )$ was contributed by the interband transition process. The imaginary part of the interband contribution could be calculated using the results from DFT calculations as [44]
3. Results
The lattice structure of BAs and BSb is FCC structure with the space group of F$\bar {4}$3m, where B atoms locate at the tetrahedral center of the four nearest neighbor V-group atoms with the B-V-B bond angle of 109.47$^\circ$. The lattice constants of BAs and BSb are 4.80 Å and 5.30 Å , respectively, agreement well with the previous computational works [45–48] and the experimental value of 4.78 Å in BAs [49] or 5.30 Å in BSb [50]. Similar to the band structure of silicon, they are both indirect-bandgap semiconductors, as shown in Figs. 1(a) and 1(b). The valence band maximum (VBM) locates at $\Gamma$ point and conduction band minimum (CBM) locates at $\Gamma$-X line (BAs) or X point (BSb). The bandgap of BAs and BSb in the calculation of PBE functional are 1.31 and 0.80 eV, respectively. In order to correct the underestimated effect of PBE functional, the hybrid functional of HSE is used to obtain accurate bandgap, which increases to 1.84 (1.24) eV in BAs (BSb). It is found that the experimental bandgap of BAs (1.82 eV) [16] is closer to the result of HSE functional.
Fig. 1. (a) Band structures of BAs and (b) BSb with diatomic primitive cell calculated using PBE functional. Both materials have VBM at $\Gamma$ point, and the CBM of BAs locates at $\Gamma$-X (0.5,0.0,0.5) line, different from the X point in BSb. (c) Phonon spectra of BAs and (d) BSb. Two transverse modes are degenerate alone two lines.
In the study about phonon, the primitive cell of BAs or BSb includes two atoms thus there should be six branches in the phonon spectra. But the symmetries alone $\Gamma$-L and $\Gamma$-X lines lead to the double degenerate transverse acoustic modes and transverse optical modes [Figs. 1(c) and 1(d)]. The heavy Sb atom also brings about a lower frequency in the whole spectrum of BSb than that of BAs. And it worth noting that the LO-TO splitting of BSb (14.3 cm$^{-1}$) is significantly greater than that of BAs (3.1 cm$^{-1}$), because of the bigger difference of ionicity between B and Sb elements.
3.1 Phonon-assisted optical absorptions
In the calculations about first-order absorption, the direct optical absorption spectra $\alpha (\omega )$ can be obtained by the dielectric function as $\alpha (\omega )=\sqrt {2}\omega \times$ $[\sqrt {\varepsilon _1(\omega )^2+ \varepsilon _2(\omega )^2}-\varepsilon _1(\omega )]^{1/2}$, where $\varepsilon _1(\omega )$ and $\varepsilon _2(\omega )$ are the real and imaginary parts of frequency-dependent complex dielectric function $\varepsilon (\omega )$. As shown in Figs. 2(a) and 2(b), the direct optical absorption increases sharply between 4.5 and 6.5 eV, and the main peak appears in the ultraviolet region around 7.0 eV. There is almost no optical absorption in the visible range (1.62$\sim$3.11 eV), limited by the large local direct bandgap in BAs and BSb, consistent with the conclusions of previous works [19,32]. The direct absorption onset occurs at the minimum direct bandgap [Figs. 2(c) and 2(d)], corresponding to the transition from the valence state (bonding p states: $\Gamma _{8v}$) to the conduction state (antibonding p states: $\Gamma _{7c}$) at $\Gamma$ point [45]. But, after considering the phonon-assisted indirect optical absorption under the temperature of 100 K, the redshift of absorption onset is shown obviously [Figs. 2(c) and 2(d)]. The small deviation between absorption onset and the value of indirect bandgap is attributed to the smearing treatment of $\delta$ function of energy conservation in Eq.(1) [43]. The optical absorption spectra of BAs and BSb in the visible range ($<$ local direct bandgap) are similar to the changing curve of silicon [43]. So the results indicate that BAs and BSb can be applied to photoelectric devices even though the existence of indirect bandgap.
Fig. 2. (a) The direct optical absorption spectra of BAs and (b) BSb in the standard model calculations with HSE functional. (c) The full optical absorption spectra of BAs and (d) BSb with the phonon-assisted indirect absorption process in the low-energy region (logarithmic scale). It is observed that the absorption onset transfer from the local direct bandgap to the global indirect bandgap.
Figure 3 shows the temperature-dependent optical absorption spectra of BAs. Note that the temperature effects on electronic states, including the lattice expansion and electron-phonon renormalization [51–54], are not yet considered in our calculations. But the temperature-dependent results still capture two features. One is the redshift of $\sim$0.1 eV from 100 K to 400 K, similar to the universal phenomenon in other materials [26–29]. However, the incomplete temperature effects in our calculations result in the relative smaller redshift. The other one is the enhancement and smoothness of optical absorption spectra at high temperatures. The reason is that there are more exciting phonons to assist the optical absorptions with the increase of temperature. The attainable range of the final electronic state from the same initial electronic state is broadened and the absorption cross section increases. So the temperature gives rise to the above effects, which are also present in BSb certainly.
Fig. 3. The full optical absorption spectra of BAs in low-energy region under different temperatures. The inset details are the optical absorption spectra around the values of indirect and direct bandgap.
3.2 Doping effect of congeners
From the preceding results, although the phonons assist the indirect optical absorption in the visible range, the second-order process is still weaker than the first-order process and especially has a small correction for the absorption of high-frequency photons when the first-order absorption is the dominant process in the high energy spectral region. Therefore, except for the phonon-assisted indirect absorption process, we also explore the way to improve the visible optical direct absorption process. Consider that the doping of congeners has been successfully used to improve the performance of III-V semiconductor [33–35], here we systematically investigate its influence on the optical absorption of BAs and BSb. And note that the calculations about the doping effect do not refer to the electron-phonon coupling. The FCC crystal (4 times primitive cell) is adopted as the periodic unit in order to study the doping effect. For B$_4$As$_4$ and B$_4$Sb$_4$, one atom (B, As, and Sb) is substituted by one atom of the same main group. The calculation shows that the heavier atomic doping leads to the increase of lattice constant, such as 5.15 Å of B$_3$As$_4$In, contrary to the case of lighter atomic doping. In the electronic structures of doping cases, BAs-based systems are all the semiconductors, as summarized in Table 1. However, there are only three semiconducting BSb-based systems. The band structures of four classic cases after doping are plotted in Fig. 4. It is obvious that the four systems have a much smaller ’direct’ bandgap than that of prototypical structure seemingly. But, what needs to pay attention is that the X point (0.5, 0.0, 0.5) in the reciprocal space of primitive cell is folded at $\Gamma _c$ point (0.0, 0.0, 0.0) in the reciprocal space of FCC crystal cell when L point (0.5, 0.0, 0.0) is folded at R$_c$ point (0.5, 0.5, 0.5). Thus the fold leads to the pseudo direct bandgap and the doping of congeners hasn’t really changed the character of indirect bandgap in B$_4$As$_4$ or B$_4$Sb$_4$. Among all the doping systems, B$_3$Sb$_4$Al has the smallest bandgap in the HSE functional [Table 1], but it is metallicity with the PBE calculations, which also appears in B$_4$As$_3$N [Fig. 4].
Fig. 4. (a) The band structures of B$_3$As$_4$Al, (b) B$_4$As$_3$N, (c) B$_3$Sb$_4$Al and (d) B$_4$Sb$_3$P with FCC crystal cell calculated using PBE functional. The inset details in (b) and (c) shows that B$_4$As$_3$N and B$_3$Sb$_4$Al transforms into semiconductor in the HSE functional. The corner mark ’c’ of symmetry points signifies that the K points are in the reciprocal space of FCC crystal cell.
Table 1. Bandgap of B$_4$As$_4$ and B$_4$Sb$_4$ with the doping of congeners in the calculation of HSE functionals.
Figure 5 plot the direct optical absorption spectra of all semiconducting doping systems. By comparing the B$_3$As$_4$X (X=Al, Ga, In), as shown in upper panels in Fig. 5, it is observed that the enhancement of absorptions increases with the increasing atomic number of doping elements, which can be deduced simply from the band structures of III-As compounds [46,55,56]. From light to heavy III group elements, III-As compounds become from indirect bandgap (BAs) to direct bandgap at $\Gamma$ point (GaAs), even metal (InAs). It is palpable that B$_3$As$_4$X has a smaller and smaller local direct bandgap at $\Gamma _c$ point. The deeper reason is that the direct bandgap at $\Gamma$ point is mainly affected by the III-As bond length, when there are the small differences between p orbital energies of III group elements [Fig. 6]. And the larger atomic radius of the heavier atom can increase the lattice constant as well as the B-As bond length, and result in the decrease of $\Gamma _{8v}-\Gamma _{7c}$ bonding-antibonding repulsion of p states [45]. So doping can improve the optical absorption of B$_3$As$_4$X in the visible range. However, B-V compounds are all indirect bandgap semiconductor with large local direct bandgap at $\Gamma$ point thus there is a little doping effect on optical absorption, except for B$_4$As$_3$N, as shown in middle panels in Fig. 5. The band structure of B$_4$As$_3$N shows that the CBM locates at R$_c$ point (0.5,0.5,0.5) [Fig. 7], different from the $\Gamma _c$-X$_c$ line of B$_4$As$_3$P and B$_4$As$_3$Sb. The s orbital energy of N atom deviates from other V group elements [Fig. 6], thus the doping of N atom has a greater impact on the s orbital electronic states of B$_4$As$_3$N. Also consider the longer B-N bond of 1.95 Å in B$_4$As$_3$N than that in FCC BN crystal (1.57 Å), so that B$_4$As$_3$N has the smaller energy difference between bonding and antibonding states of s orbit, which leads to the CBM at R$_c$ point, as shown in Fig. 7. Hence B$_4$As$_3$N also has higher optical absorption in the visible range. Finally, the lower panels in Fig. 5 show that three semiconducting BSb-based systems can all increase the visible optical absorptions. The increasing absorption of B$_3$Sb$_4$Al results from the small local direct bandgap at $\Gamma _c$, similar to B$_3$As$_4$X. For the other two, due to the large difference between the s orbits of Sb and P (As) [Fig. 6], B$_4$Sb$_3$P and B$_4$Sb$_3$As have the CBM locating at R$_c$ point, similar to B$_4$As$_3$N. In short, three cases all have a small local direct bandgap, which increases the optical absorption slightly.
Fig. 5. (a) The direct optical absorption spectra of B$_3$As$_4$Al, (b) B$_3$As$_4$Ga, (c) B$_3$As$_4$In, (d) B$_4$As$_3$N, (e) B$_4$As$_3$P, (f) B$_4$As$_3$Sb, (g) B$_3$Sb$_4$Al,(h) B$_4$Sb$_3$P and (i) B$_4$Sb$_3$As in the standard model calculations. The black line is the optical absorption spectra of BAs or BSb. The inset details are zoom-in view of absorption spectra in low-energy region (0.5$\sim$3.5 eV)
Fig. 6. Atomic s and p orbital energies of III and V groups elements. The black lines and red lines show the III and V groups elements, respectively, with s and p orbits labelled as square and triangle.
Fig. 7. Projected band structures of B$_4$As$_3$N. The thickness of red and green lines denotes the relative amount of s and p orbits, respectively.
4. Conclusion
In the present work, we have systematically investigated the direct and indirect optical absorptions of BAs and BSb and the doping effect of congeners by using first-principles calculations. Due to the indirect bandgap and the large local direct bandgap of BAs and BSb, the first-order optical absorption in the visible range is forbidden and the second-order optical absorption becomes the dominant process. Therefore, considering the assist of phonon in the indirect electronic transition, we obtain the optical absorption onset corresponding to the value of indirect bandgap. Moreover, the redshift of absorption onset, enhancement, and smoothness of optical absorption spectra are also captured in the temperature-dependent results. In view of the lower second-order absorption than first-order absorption and the importance of introducing first-order visible optical absorptions, so the doping effect of congeners on optical absorptions is studied without the electron-phonon coupling. It is found that the smaller local direct bandgap in some doping cases can improve the first-order optical absorption in the visible range for two different reasons. One is that the heavier III group elements can increase B-As bond length and decrease the bonding-antibonding repulsion of p states, thus lead to small local direct bandgap (B$_3$As$_4$Al, B$_3$As$_4$Ga, B$_3$As$_4$In and B$_3$Sb$_4$Al). The other one is lighter V group elements mainly affect the bonding and antibonding states of s orbit, such as CBM at R$_c$ point, also generate small local direct bandgap (B$_4$As$_3$N, B$_4$Sb$_3$P and B$_4$Sb$_3$As).
In the future, the inclusion of full-scale temperature effects, lattice expansion, and electron-phonon renormalization, may improve the predictions of temperature-dependent optical absorptions and thus better guide the experiments. Furthermore, it’s also important to notice that electron-hole Coulomb interaction is not taken into account since the bandgap of HSE calculations is in good agreement with the experiments about III-V semiconductors [16,57–59]. In fact, the band edge wave functions in the indirect bandgap semiconductors locate at different K points in reciprocal space, leading to the small overlap of wave functions, as well as the small Coulomb interaction between electrons and holes. Therefore, the combination of phonon-assisted spectra and HSE calculations can simulate the optical properties of BAs and BSb well, and it is not strictly needed to account for excitonic effect.
Funding
National Natural Science Foundation of China (11904312, 11904313); Department of Education of Hebei Province (QN2018012, ZD2018015); Natural Science Foundation of Hebei Province (A2019203507).
Disclosures
The authors declare no conflicts of interest.
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