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First-principles calculations of electronic band structure and optical properties of ternary semiconductors Cd4P2Cl3 and Cd4P2Br3
Computational Condensed Matter 21 (2019) e00390
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Computational Condensed Matter journal homepage: http://ees.elsevier.com/cocom/default.asp
First-principles calculations of electronic band structure and optical properties of ternary semiconductors Cd4P2Cl3 and Cd4P2Br3 Niharendu Barman a, Tanmoy Chaki b, Amit Shankar c, Pradip Kumar Mandal b, * a
Physics Department, Dinhata College, Dinhata, 736135, India Physics Department, University of North Bengal, Siliguri, 734013, India c Physics Department, Kurseong College, Kurseong, 34203, India b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 5 March 2019 Received in revised form 21 April 2019 Accepted 22 April 2019
By first principles DFT calculations the structural, electronic and optical properties of Cd4P2Cl3 and Cd4P2Br3 compounds, a close analogue of CdS, have been determined. Both the ternary compounds are found to be direct band gap semiconductors with band gap energy ~2.4 eV, about 0.1 eV lower than that of CdS. The density of states calculations reveal that the valence band maxima are predominantly contributed by the P-p states and conduction band minima are mainly contributed by the Cd-s states. From the study of optical properties it is found that both the compounds have very high absorption coefficient, exceeding 104 cm1 in the blue region of the visible spectrum. It is found that both the compounds exhibit good transmittance of light with more than 85% in a significant portion of visible spectrum. Such a value of band gap energy together with high absorption coefficient and low reflectance make these compounds suitable for various electronic and optoelectronic devices especially in heterojunction solar cell. © 2019 Elsevier B.V. All rights reserved.
Keywords: Ternary semiconductors FPLAPW method in DFT Electronic structure DOS Optical properties
1. Introduction Cadmium sulphide (CdS), a member of the group II-VI binary semiconductors, has gained substantial attention due to its promising electronic and optical properties. This compound was reported to have direct band gap energy of 2.5 eV which falls within the energy range of the visible light [1,2]. Such a value of band gap energy together with its high absorption coefficient, low reflectance and chemical stability make this compound suitable for various electronic and optoelectronic devices [3,4], particularly in heterojunction solar cells [5,6], light emitting diodes [7], thin film transistors [8,9] and gas detectors [10,11]. Measured electrical (2.24 eV and 1.80 eV) and optical (2.17 eV and 2.19 eV) band gaps of Cd4P2Cl3 and Cd4P2Br3 are found to be similar to that of CdS [12,13]. Based on the experimental and first principles calculations, Roy et al. [14] have inferred that Cd4P2Cl3 and CdS exhibit similar photoluminescence spectrum because of their comparable band gap energy values. A recent optical absorption study [13] also reports that Cd4P2X3 (X ¼ Cl, Br and I) show absorption in visible region with a direct band gap in the range of 2.20e2.24 eV. First
* Corresponding author. E-mail address: [email protected] (P.K. Mandal). https://doi.org/10.1016/j.cocom.2019.e00390 2352-2143/© 2019 Elsevier B.V. All rights reserved.
principles electronic structure calculations suggest that Cd4P2Cl3 and its isomorph Cd4P2Br3 is an excellent photocatalyst for water splitting [13]. From the electrochemical impedance and transient photocurrent measurements it is reported that Cd4P2Br3 shows better catalytic activity than Cd4P2Cl3 for water splitting and CO2 reduction [13]. The same group has also investigated the structures and properties of a family of cadmium phosphochlorides, Cd2P3Cl, Cd4P2Cl3, Cd3PCl3, and Cd7P4Cl6, with varying Cl/Cd and P/Cd ratios and inferred that cadmium phosphochlorides exhibit hydrogen evolution reaction (HER) with Cd7P4Cl6 showing best activity whereas Cd3PCl3 with least activity [15]. These limited studies were primarily focused to understand the electronic structure and electrocatalytic aspects of Cd4P2X3 [13e15], and so far no firstprinciples calculations of their optical properties have been reported. In this paper, we have presented a detailed study of the electronic band structure and optical properties of two isostructural members of Cd4P2X3 compounds, namely the Cd4P2Cl3 and Cd4P2Br3 using “Full potential-linearized augmented plane wave” (FP-LAPW) method based on the density functional theory (DFT). The results of our first principles calculations suggest that these semiconductors are almost transparent with reflectance less than 15% in the energy range up to the absorption edge. The absorption
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edge being 2.41 eV and 2.45 eV for Cd4P2Cl3 and Cd4P2Br3, respectively, their high transmittance capability sustains over a wide window of the visible spectrum. Beyond the absorption edge, these semiconductors behave as a good absorber of light with absorption coefficient exceeding 104 cm1 in the blue region of the visible spectrum. Owing to their high absorbing capability in low wavelength regime and high transmittance in the bulk of the visible spectrum, Cd4P2X3 may be a potential alternative material for the construction of the window layer in heterojunction solar cell. 2. Computational details The structural properties and electronic band structure calculations of Cd4P2X3 (X ¼ Cl and Br) have been performed by the FPLAPW method based on DFT implemented in WIEN2k code [16]. FPLAPW is one of the most efficient schemes to study the ground state properties. In this method, a non-overlapping sphere of radius RMT called muffin-tin sphere is considered around each atom, inside which a linear combination of radial wave functions times spherical harmonics are used as a basis set to expand the wave functions. Whereas, outside the muffin-tin spheres called interstitial region (IT), a plane wave basis set is used. The calculations have been performed by setting RMT of both Cd and Br to 2.50 a.u., while for Cl and P, RMT is set to 2.30 and 2.05 a.u., respectively. The plane wave cut-off parameter RMT kmax has been set to 7.0, where RMT is the smallest muffin-tin sphere; and kmax is the maximum k-vector up to which the plane wave is expanded in the IT region. Fourier series expansion of charge density and potential have been truncated only up to Gmax ¼ 12 a.u.1 because no appreciable difference is noted beyond this limit. For the optimization of structural properties of the materials, the generalized gradient approximation (GGA) parameterized by Perdew-Burke-Ernzerhof (PBE) [17] has been adopted for treating the exchange correlation potential. To
overcome the negligence of band gap due to approximate exchange-correlation potential like GGA, we have also used TranBlaha modified Becke-Johnson (TB-mBJ) exchange potential [18] for determining the electronic and optical properties. To separate the core and valence states, the separation energy of 7.5 Ry has been chosen. The 4p, 4d orbitals of Cd, 3s, 3p of P and 3d, 4s, 4p orbitals of Br and Cl have been treated as valance orbitals whereas lower orbitals have been considered as core states. For integration over the Brillouin zone in reciprocal space, a (4 4 4) k-points grid has been used. The self-consistency has been achieved when energy difference between two successive cycles is less than 104 Ry. 3. Results and discussions 3.1. Structural optimization As stated both Cd4P2Cl3 and Cd4P2Br3 have simple cubic structure with the space group Pa-3, containing a total number of 72 atoms per unit cell as shown in Fig. 1. Out of these 72 atoms 32 are of Cd, 16 are of P, and 24 are of X atoms. Starting from the atomic positions determined by single crystal X-ray diffraction technique [19] the equilibrium atomic positions have been achieved by relaxing the positions of inequivalent atoms until the HellmanFeynman forces on the atoms become less than 1 mRy/a.u. The obtained equilibrium atomic positions are presented in Table 1. The Cd atoms possess two different types of tetrahedral coordination with P and X atoms. One type of Cd atom (labeled as Cd1 in Fig. 1) coordinates two P atoms and two X atoms, forming a CdP2X2 tetrahedron. Another type of Cd atom (labeled as Cd2) has coordination with one P atom and three X atoms forming a CdPX3 tetrahedron. On the other hand, P atoms are also found in two different types which are labeled as P1 and P2. While a P2 atom
Fig. 1. Crystal structure of Cd4P2X3 (X ¼ Cl, Br).
N. Barman et al. / Computational Condensed Matter 21 (2019) e00390 Table 1 Equilibrium atomic positions of Cd4P2Cl3 and Cd4P2Br3. Compound
Cd4P2Cl3
Cd4P2Br3
Atom
Cd1 Cd2 P1 P2 Cl Cd1 Cd2 P1 P2 Br
Wyckoff Site
24d 8c 8c 8c 24d 24d 8c 8c 8c 24d
Atomic positions of the inequivalent atoms x
y
z
0.031014 0.225507 0.109304 0.448927 0.180281 0.036880 0.223144 0.109320 0.449753 0.184361
0.988542 0.225507 0.109304 0.448927 0.180281 0.996210 0.223144 0.109320 0.449753 0.435177
0.250136 0.225507 0.109304 0.448927 0.180281 0.254907 0.223144 0.109320 0.449753 0.268967
bonds with another P2 atom, the P1 atom does not exhibit any such P-P bonding. The existence of two different types of P atoms in Cd4P2Cl3 has already been confirmed by single crystal X-ray diffraction as well as nuclear resonance experiments [14]. We have performed the volume optimization by varying the unit cell volume from the experimental volume to obtain the equilibrium lattice constant. An optimized lattice constant which corresponds to the minimum total energy of the system is obtained by fitting the calculated data points (energy vs. unit cell volume) using Murnaghan equation of state [20]. The optimized lattice constants (a) for these cubic systems are 12.44 Å and 12.68 Å for Cd4P2Cl3 and Cd4P2Br3, respectively. A slightly larger lattice constant of Cd4P2Br3 is essentially due to the larger atomic radius of Br atom as compared to the Cl atom. These values, however, are slightly larger compared with the available experimental values of the lattice parameters which are 12.13 Å and 12.36 Å for Cd4P2Cl3 and Cd4P2Br3, respectively [13]. Such deviation is rather common with the PBE-GGA exchange-correlation potential [21,22]. The curve fitting exercise also provides bulk modulus of 45.02 GPa for Cd4P2Cl3 and 42.39 GPa for Cd4P2Br3 unfortunately no experimental or theoretical data are available for comparison. 3.2. Electronic properties To explore the electronic properties of Cd4P2X3, we have calculated the electronic band structure and density of states (DOS). Our preliminary calculation of electronic band structure of these compounds using PBE-GGA functional suggest that the electronic band structures of both the compounds are typical of a semiconductor with direct band gap energies Eg ¼ 1.36 and 1.43 eV for Cd4P2Cl3 and Cd4P2Br3, respectively. The calculated values of band gap energies disagree largely with the experimental values of Eg ¼ 2.21 eV for Cd4P2Cl3 and 2.24 eV for Cd4P2Br3 obtained from optical absorption measurements [13], corresponding data quoted in Ref. [12] are 2.17 eV and 2.19 eV. From electrical measurements these values were reported respectively as 2.24eV and 1.80eV [12]. It is worth mentioning that the first principle calculation based on PBE-GGA functional often leads to an underestimation of band gap energy, which is primarily because of the approximation used in the exchange-correlation functional like GGA [23e25]. The usage of the modified Becke-Johnson (TB-mBJ) [18] exchange potential has, however, been proved to be a potential route for reliable estimation of the band gap energy [25e27]. Recalculation of band structure adopting TB-mBJ exchange potential resulted in a band gap of 2.41 eV and 2.45 eV for Cd4P2Cl3 and Cd4P2Br3, respectively which are roughly 10% higher than the experimental values but agree well with the previous theoretical results obtained based on hybrid functional calculation [13]. It is worth to note that our theoretically calculated band gap energy reflects the ground state properties,
3
whereas the experimental were made at ambient temperature. This may be the primary reason for such small overestimation of the calculated band gap energy values. Observed shifting of the absorption edge of these compounds towards the lower energy regime upon heating [12] in line with our justification. The electronic band structures of these compounds calculated using TB-mBJ exchange potential are shown in Fig. 2(a) and (b), where the energy bands have been plotted along the high symmetry direction connecting the high symmetry points within the first Brillouin zone. In this figure, the zero of the energy axis is set to the Fermi energy EF. The band structure of both the compounds exhibit qualitatively similar features. Both the valence band maxima and conduction band minima are located at the high symmetry G point signifying both the ternary compounds are direct band gap semiconductors like CdS. Around the high symmetric G point, the lowest conduction band has shape of a parabola, suggesting that the charge carriers in the concerned band have characters of free electrons. The topmost valence band is almost flat, which endows with a high effective mass of the holes. The overall band profile obtained using TB-mBJ exchange potential is in good agreement with the previous theoretical calculation with hybrid functional [13]. The total density of states (DOS) together with the projected density of states (PDOS) on to various atomic orbitals of Cd4P2Cl3 and Cd4P2Br3 are presented in Fig. 3. In this figure, we have presented the DOS obtained for TB-mBJ exchange potential because the estimated band gap energies using this exchange potential shows better agreement with the previously reported experimental values. In both the compounds the DOS at the valence edge is much sharper as compared to the DOS at the conduction band edge. The sharp peak in the DOS just below the Fermi level indicates that the valence electrons are more localized in the close vicinity of the Fermi level. On the other hand, the far less sharp DOS at the conduction region reflects the itinerant character of the conduction electrons. Sharp peak in the DOS of Cd4P2Cl3 (Cd4P2Br3) are observed in the low energy regime from 6.37 to 7.04 eV (from 6.2 to 6.8 eV), which are contributed predominately by the Cd-d and P-s states. It is worth to mention that the P1-s states are strongly hybridized with both the Cd1-d and Cd2-d states, whereas the P2-s states are mainly hybridized with the Cd1-d states. While the Cl-p (Br-p) states contribute dominantly to the DOS of Cd4P2Cl3 (Cd4P2Br3) in the energy range from 2.72 to 4.76 eV (from 1.87 to 4.27 eV), the P-p states in Cd4P2Cl3 (Cd4P2Br3) have dominant contribution to the DOS in the energy range from 2.70 (1.47) eV well up to the Fermi energy. The unoccupied conduction band region is mainly contributed by the Cd-s states. 3.3. Optical properties The frequency dependent complex dielectric function of a solid is given by
ε* ðuÞ ¼ ε1 þ iε2 ;
(1)
where ε1 and ε2 are respectively the real and imaginary part of ε* . The imaginary part ε2 is directly related to the electronic band structure of a solid and contributed by different intraband and interband transitions. The intraband transitions are mainly important for the metallic systems and contribute very little in case of semiconductors [28]. The interband transitions can further be classified into two categories: the direct interband transition and the indirect interband transition. The indirect interband transitions involve the scattering of phonons and are expected to contribute a little to the dielectric function. With all these approximations, the imaginary part of dielectric function can be expressed as [28].
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N. Barman et al. / Computational Condensed Matter 21 (2019) e00390
Fig. 2. The electronic band structure of (a) Cd4P2Cl3 and (b) Cd4P2Br3 obtained using modified Becke-Johnson exchange potential.
Fig. 3. The total and the projected density of states of Cd4P2Cl3 (a, b, c) and Cd4P2Br3(d, e, f).
N. Barman et al. / Computational Condensed Matter 21 (2019) e00390
ε2 ¼
ð X Ve2 2 d3 k jknjpjkn0 j f ðknÞð1 f ðkn0 ÞÞdðEkn 2 2 2pZm u 0 n;n Ekn0 ZuÞ (2)
where, ħu is the energy of the incident photon, p ¼ iħV is the momentum operator, jkn> is the eigenfunction with eigenvalue Ekn, and f(kn) is the Fermi distribution function. The detailed calculation of the matrix elements of the momentum operator can be found in Ref. [29]. On the other hand, the real part of dielectric function is related to its imaginary counterpart through KramersKronig relation given by Ref. [30].
ε1 ¼ 1 þ
2
p
∞ ð
P
0
u’ ε2 ðuÞ ’ du ; u’ 2 u2
(3)
where, P represents the principle part of the integral. The real and imaginary parts of dielectric function were calculated for Cd4P2X3 using equations (2) and (3). The real part n(u) and imaginary part k(u) of the complex refractive index were calculated from the frequency dependent dielectric function using the relations:
1
nðuÞ ¼
√2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 ε1 ðuÞ2 þ ε2 ðuÞ2 þ ε1 ðuÞ
(4)
And
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 ε1 ðuÞ2 þ ε2 ðuÞ2 ε1 ðuÞ √2 1
kðuÞ ¼
(5)
Moreover, the absorption coefficient a and reflectivity R were calculated using the relations:
a¼
2ku c
(6)
and
RðuÞ ¼
ðn 1Þ2 þ k2 ðn þ 1Þ2 þ k2
(7)
Here, c is the velocity of light in free space. The calculations of these optical parameters were performed using TB-mBJ exchange potential. For these calculations, a dense (14 14 14) mesh of kpoints were used for the integration over the first Brillouin zone, and the broadening factor was taken to be 0.1 eV. Fig. 4(a) and (b) show the variation of the real and imaginary parts of the calculated dielectric function for Cd4P2Cl3 and Cd4P2Br3 with the photon energy ranging from zero to 12 eV. The first critical point, which represents the minimum photon energy required for the onset of the response of ε2 , consistently coincides with the band gap energies of the respective compounds. This essentially marks the transition from the topmost valence band regime to the bottom of the conduction band CB at the high symmetry G point. As the photon energy increases a number of peaks become apparent. As evident from the electronic band structures of these compounds the valence bands tend to overlap one another particularly at and around the high symmetric points, which leads to some ambiguity to predict a particular transition responsible for a given peak. Rather a peak is more plausibly contributed by a number of transitions occurring in closely spaced energy values. The indexing of such a huge number of valance
5
bands is evidently a tedious job, even if the valence bands were indexed they would hardly be identified individually in Fig. 2. For the sake of convenience, we demark the entire valance band region in Fig. 2 into a number of well separated sub-regions, each of which is consisted of several number of valence bands. The transitions from few closely spaced bands in these valence band subregions give rise to the peaks in ε2 (u). We shall use the notation VBN to represent the topmost (N ¼ 1), second topmost (N ¼ 2), and third topmost (N ¼ 3) valence band sub-regions. The first peak may be identified to arise due to the direct interband transitions from VB2 to CB at the symmetry point G and/or from VB1 to CB at high symmetric M and R points. From the DOS, we can infer that the P-p electrons are responsible for these transitions. The second peak may be inferred to arise due to the transitions of the P-p electrons from VB2 to CB at the M point, while the third peak is due to the transition from the VB3 to CB at M and R points. The locations of few prominent peaks together with the most plausible interband transitions that contribute dominantly to these peaks are presented in Table 2. The real part ε1 of dielectric function in the zero frequency limit, the so called the static dielectric constant ε1 ð0Þ, is found to be 4.13 and 4.56 for Cd4P2Cl3 and Cd4P2Br3, respectively. Any contribution from the lattice vibrations was not considered in these calculations. Prominent peaks in ε1 (u) curves are 6.21, 6.38 and 5.66 observed at energy values 3.20 eV, 3.79 eV and 4.61 eV for Cd4P2Cl3 which are comparable to the reported dielectric constant value of 5.92 of Cd4P2Cl3 in the optical frequency range [15]. Corresponding values for Cd4P2Br3 are 7.23, 6.92, and 6.88 at 3.22 eV, 3.71 eV and 4.45 eV. Moreover, ε1 becomes negative in high energy regime above 6.96 eV for Cd4P2Cl3 and 7.2 eV for Cd4P2Br3, with the implication that the electromagnetic waves ceases to propagate beyond these energies. As the absorption coefficient a is related to the absorptive part of the dielectric function, their response with energy is quite similar particularly in low energy regime. From the optical absorption spectra shown in Fig. 4(c) and (d), it is clear that in Cd4P2Cl3 (Cd4P2Br3) the absorption edge emerges at a photon energy of 2.41 (2.45) eV with corresponding wavelength of 516 (507) nm, which is originated due to the excitation of electrons from P-p states located at the topmost valence band, to the unoccupied Cd-s states. It is further noted that the absorption coefficient of both the compounds has a high value exceeding 104 cm1 in the blue region of the visible spectrum with corresponding energy between 2.50 eV and 2.75 eV, which indicates that these compounds possess very high absorbing power. The absorption coefficient increases further as the photon energy is shifted towards the ultraviolet region. Among these two compounds, Cd4P2Br3 exhibits slightly better absorption property. From the reflectivity spectra R(u) shown in Fig. 4(e) and (f), one can see that both the compounds have reflectance lying below 15% in the energy range up to the absorption edge. Thus, it turns out that both the compounds exhibit good transmittance of light with more than 85% up to the absorption edge which, in fact, covers a significant portion of visible spectrum. Semiconductor like CdS, which is proposed to be a potential material for the construction of the window layer in heterojunction solar cell, has reflectance of 13e22% near the infrared region [31,32] and transmittance of around 80% for photon energy higher than the absorption edge of this compound [33]. These values are comparable with the corresponding values of the optical parameters of Cd4P2X3. Owing to their high absorbing capability in low wavelength region and high transmittance in the bulk of the solar spectrum, Cd4P2X3 may be a potential alternative material for the construction of the window layer in heterojunction solar cell in addition to their reported HER activity.
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N. Barman et al. / Computational Condensed Matter 21 (2019) e00390
Fig. 4. (a) Real and imaginary parts of the complex dielectric function ε*, (c) the absorption coefficient a and (e) the reflectivity R of Cd4P2Cl3. Corresponding quantities of Cd4P2Br3 are in (b,d,f).
Table 2 Peak positions of ε2(u) together with the possible dominant contribution to the peaks. Compound
Peak position (eV)
Dominant transition(s) (Energy difference in eV)
Cd4P2Cl3
3.39
VB2 VB1 VB1 VB2 VB3 VB3 VB4 VB4 VB2 VB1 VB1 VB2 VB3 VB3 VB4 VB4
4.07 5.05 5.51 Cd4P2Br3
3.44
3.93 4.80 5.32
/ / / / / / / / / / / / / / / /
CB CB CB CB CB CB CB CB CB CB CB CB CB CB CB CB
4. Conclusion In summary, we have carried out a theoretical study based on first principles DFT calculations to determine the structural,
(3.26) (3.18) (3.21) (4.03) (5.11) (5.13) (5.51) (5.54) (3.12) (3.29) (3.21) (3.93) (4.80) (4.80) (5.18) (5.11)
Location of transition in BZ
G
М R M M R R M
G R M M R M R M
electronic and optical properties of Cd4P2Cl3 and Cd4P2Br3 compounds, a close analogue of CdS. Both the compounds are found to be direct band gap semiconductors with band gap energy ~2.4 eV, about 0.1 eV lower than that of CdS. The density of states
N. Barman et al. / Computational Condensed Matter 21 (2019) e00390
calculations reveal that the valence band maxima are predominantly contributed by the P-p states and conduction band minima are mainly contributed by the Cd-s states. From the study of optical properties it is found that both the compounds have very high absorption coefficient, exceeding 104 cm1 in the blue region of the visible spectrum. It is found that both the compounds exhibit good transmittance of light with more than 85% in a significant portion of visible spectrum. Such a value of band gap energy together with high absorption coefficient and low reflectance make these compounds suitable for various electronic and optoelectronic devices in addition to their photocatalytic activity. Acknowledgement PKM acknowledges funding from University of North Bengal (Grant No. 164/R-2018(SF-17).
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Computational Condensed Matter journal homepage: http://ees.elsevier.com/cocom/default.asp
First-principles calculations of electronic band structure and optical properties of ternary semiconductors Cd4P2Cl3 and Cd4P2Br3 Niharendu Barman a, Tanmoy Chaki b, Amit Shankar c, Pradip Kumar Mandal b, * a
Physics Department, Dinhata College, Dinhata, 736135, India Physics Department, University of North Bengal, Siliguri, 734013, India c Physics Department, Kurseong College, Kurseong, 34203, India b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 5 March 2019 Received in revised form 21 April 2019 Accepted 22 April 2019
By first principles DFT calculations the structural, electronic and optical properties of Cd4P2Cl3 and Cd4P2Br3 compounds, a close analogue of CdS, have been determined. Both the ternary compounds are found to be direct band gap semiconductors with band gap energy ~2.4 eV, about 0.1 eV lower than that of CdS. The density of states calculations reveal that the valence band maxima are predominantly contributed by the P-p states and conduction band minima are mainly contributed by the Cd-s states. From the study of optical properties it is found that both the compounds have very high absorption coefficient, exceeding 104 cm1 in the blue region of the visible spectrum. It is found that both the compounds exhibit good transmittance of light with more than 85% in a significant portion of visible spectrum. Such a value of band gap energy together with high absorption coefficient and low reflectance make these compounds suitable for various electronic and optoelectronic devices especially in heterojunction solar cell. © 2019 Elsevier B.V. All rights reserved.
Keywords: Ternary semiconductors FPLAPW method in DFT Electronic structure DOS Optical properties
1. Introduction Cadmium sulphide (CdS), a member of the group II-VI binary semiconductors, has gained substantial attention due to its promising electronic and optical properties. This compound was reported to have direct band gap energy of 2.5 eV which falls within the energy range of the visible light [1,2]. Such a value of band gap energy together with its high absorption coefficient, low reflectance and chemical stability make this compound suitable for various electronic and optoelectronic devices [3,4], particularly in heterojunction solar cells [5,6], light emitting diodes [7], thin film transistors [8,9] and gas detectors [10,11]. Measured electrical (2.24 eV and 1.80 eV) and optical (2.17 eV and 2.19 eV) band gaps of Cd4P2Cl3 and Cd4P2Br3 are found to be similar to that of CdS [12,13]. Based on the experimental and first principles calculations, Roy et al. [14] have inferred that Cd4P2Cl3 and CdS exhibit similar photoluminescence spectrum because of their comparable band gap energy values. A recent optical absorption study [13] also reports that Cd4P2X3 (X ¼ Cl, Br and I) show absorption in visible region with a direct band gap in the range of 2.20e2.24 eV. First
* Corresponding author. E-mail address: [email protected] (P.K. Mandal). https://doi.org/10.1016/j.cocom.2019.e00390 2352-2143/© 2019 Elsevier B.V. All rights reserved.
principles electronic structure calculations suggest that Cd4P2Cl3 and its isomorph Cd4P2Br3 is an excellent photocatalyst for water splitting [13]. From the electrochemical impedance and transient photocurrent measurements it is reported that Cd4P2Br3 shows better catalytic activity than Cd4P2Cl3 for water splitting and CO2 reduction [13]. The same group has also investigated the structures and properties of a family of cadmium phosphochlorides, Cd2P3Cl, Cd4P2Cl3, Cd3PCl3, and Cd7P4Cl6, with varying Cl/Cd and P/Cd ratios and inferred that cadmium phosphochlorides exhibit hydrogen evolution reaction (HER) with Cd7P4Cl6 showing best activity whereas Cd3PCl3 with least activity [15]. These limited studies were primarily focused to understand the electronic structure and electrocatalytic aspects of Cd4P2X3 [13e15], and so far no firstprinciples calculations of their optical properties have been reported. In this paper, we have presented a detailed study of the electronic band structure and optical properties of two isostructural members of Cd4P2X3 compounds, namely the Cd4P2Cl3 and Cd4P2Br3 using “Full potential-linearized augmented plane wave” (FP-LAPW) method based on the density functional theory (DFT). The results of our first principles calculations suggest that these semiconductors are almost transparent with reflectance less than 15% in the energy range up to the absorption edge. The absorption
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N. Barman et al. / Computational Condensed Matter 21 (2019) e00390
edge being 2.41 eV and 2.45 eV for Cd4P2Cl3 and Cd4P2Br3, respectively, their high transmittance capability sustains over a wide window of the visible spectrum. Beyond the absorption edge, these semiconductors behave as a good absorber of light with absorption coefficient exceeding 104 cm1 in the blue region of the visible spectrum. Owing to their high absorbing capability in low wavelength regime and high transmittance in the bulk of the visible spectrum, Cd4P2X3 may be a potential alternative material for the construction of the window layer in heterojunction solar cell. 2. Computational details The structural properties and electronic band structure calculations of Cd4P2X3 (X ¼ Cl and Br) have been performed by the FPLAPW method based on DFT implemented in WIEN2k code [16]. FPLAPW is one of the most efficient schemes to study the ground state properties. In this method, a non-overlapping sphere of radius RMT called muffin-tin sphere is considered around each atom, inside which a linear combination of radial wave functions times spherical harmonics are used as a basis set to expand the wave functions. Whereas, outside the muffin-tin spheres called interstitial region (IT), a plane wave basis set is used. The calculations have been performed by setting RMT of both Cd and Br to 2.50 a.u., while for Cl and P, RMT is set to 2.30 and 2.05 a.u., respectively. The plane wave cut-off parameter RMT kmax has been set to 7.0, where RMT is the smallest muffin-tin sphere; and kmax is the maximum k-vector up to which the plane wave is expanded in the IT region. Fourier series expansion of charge density and potential have been truncated only up to Gmax ¼ 12 a.u.1 because no appreciable difference is noted beyond this limit. For the optimization of structural properties of the materials, the generalized gradient approximation (GGA) parameterized by Perdew-Burke-Ernzerhof (PBE) [17] has been adopted for treating the exchange correlation potential. To
overcome the negligence of band gap due to approximate exchange-correlation potential like GGA, we have also used TranBlaha modified Becke-Johnson (TB-mBJ) exchange potential [18] for determining the electronic and optical properties. To separate the core and valence states, the separation energy of 7.5 Ry has been chosen. The 4p, 4d orbitals of Cd, 3s, 3p of P and 3d, 4s, 4p orbitals of Br and Cl have been treated as valance orbitals whereas lower orbitals have been considered as core states. For integration over the Brillouin zone in reciprocal space, a (4 4 4) k-points grid has been used. The self-consistency has been achieved when energy difference between two successive cycles is less than 104 Ry. 3. Results and discussions 3.1. Structural optimization As stated both Cd4P2Cl3 and Cd4P2Br3 have simple cubic structure with the space group Pa-3, containing a total number of 72 atoms per unit cell as shown in Fig. 1. Out of these 72 atoms 32 are of Cd, 16 are of P, and 24 are of X atoms. Starting from the atomic positions determined by single crystal X-ray diffraction technique [19] the equilibrium atomic positions have been achieved by relaxing the positions of inequivalent atoms until the HellmanFeynman forces on the atoms become less than 1 mRy/a.u. The obtained equilibrium atomic positions are presented in Table 1. The Cd atoms possess two different types of tetrahedral coordination with P and X atoms. One type of Cd atom (labeled as Cd1 in Fig. 1) coordinates two P atoms and two X atoms, forming a CdP2X2 tetrahedron. Another type of Cd atom (labeled as Cd2) has coordination with one P atom and three X atoms forming a CdPX3 tetrahedron. On the other hand, P atoms are also found in two different types which are labeled as P1 and P2. While a P2 atom
Fig. 1. Crystal structure of Cd4P2X3 (X ¼ Cl, Br).
N. Barman et al. / Computational Condensed Matter 21 (2019) e00390 Table 1 Equilibrium atomic positions of Cd4P2Cl3 and Cd4P2Br3. Compound
Cd4P2Cl3
Cd4P2Br3
Atom
Cd1 Cd2 P1 P2 Cl Cd1 Cd2 P1 P2 Br
Wyckoff Site
24d 8c 8c 8c 24d 24d 8c 8c 8c 24d
Atomic positions of the inequivalent atoms x
y
z
0.031014 0.225507 0.109304 0.448927 0.180281 0.036880 0.223144 0.109320 0.449753 0.184361
0.988542 0.225507 0.109304 0.448927 0.180281 0.996210 0.223144 0.109320 0.449753 0.435177
0.250136 0.225507 0.109304 0.448927 0.180281 0.254907 0.223144 0.109320 0.449753 0.268967
bonds with another P2 atom, the P1 atom does not exhibit any such P-P bonding. The existence of two different types of P atoms in Cd4P2Cl3 has already been confirmed by single crystal X-ray diffraction as well as nuclear resonance experiments [14]. We have performed the volume optimization by varying the unit cell volume from the experimental volume to obtain the equilibrium lattice constant. An optimized lattice constant which corresponds to the minimum total energy of the system is obtained by fitting the calculated data points (energy vs. unit cell volume) using Murnaghan equation of state [20]. The optimized lattice constants (a) for these cubic systems are 12.44 Å and 12.68 Å for Cd4P2Cl3 and Cd4P2Br3, respectively. A slightly larger lattice constant of Cd4P2Br3 is essentially due to the larger atomic radius of Br atom as compared to the Cl atom. These values, however, are slightly larger compared with the available experimental values of the lattice parameters which are 12.13 Å and 12.36 Å for Cd4P2Cl3 and Cd4P2Br3, respectively [13]. Such deviation is rather common with the PBE-GGA exchange-correlation potential [21,22]. The curve fitting exercise also provides bulk modulus of 45.02 GPa for Cd4P2Cl3 and 42.39 GPa for Cd4P2Br3 unfortunately no experimental or theoretical data are available for comparison. 3.2. Electronic properties To explore the electronic properties of Cd4P2X3, we have calculated the electronic band structure and density of states (DOS). Our preliminary calculation of electronic band structure of these compounds using PBE-GGA functional suggest that the electronic band structures of both the compounds are typical of a semiconductor with direct band gap energies Eg ¼ 1.36 and 1.43 eV for Cd4P2Cl3 and Cd4P2Br3, respectively. The calculated values of band gap energies disagree largely with the experimental values of Eg ¼ 2.21 eV for Cd4P2Cl3 and 2.24 eV for Cd4P2Br3 obtained from optical absorption measurements [13], corresponding data quoted in Ref. [12] are 2.17 eV and 2.19 eV. From electrical measurements these values were reported respectively as 2.24eV and 1.80eV [12]. It is worth mentioning that the first principle calculation based on PBE-GGA functional often leads to an underestimation of band gap energy, which is primarily because of the approximation used in the exchange-correlation functional like GGA [23e25]. The usage of the modified Becke-Johnson (TB-mBJ) [18] exchange potential has, however, been proved to be a potential route for reliable estimation of the band gap energy [25e27]. Recalculation of band structure adopting TB-mBJ exchange potential resulted in a band gap of 2.41 eV and 2.45 eV for Cd4P2Cl3 and Cd4P2Br3, respectively which are roughly 10% higher than the experimental values but agree well with the previous theoretical results obtained based on hybrid functional calculation [13]. It is worth to note that our theoretically calculated band gap energy reflects the ground state properties,
3
whereas the experimental were made at ambient temperature. This may be the primary reason for such small overestimation of the calculated band gap energy values. Observed shifting of the absorption edge of these compounds towards the lower energy regime upon heating [12] in line with our justification. The electronic band structures of these compounds calculated using TB-mBJ exchange potential are shown in Fig. 2(a) and (b), where the energy bands have been plotted along the high symmetry direction connecting the high symmetry points within the first Brillouin zone. In this figure, the zero of the energy axis is set to the Fermi energy EF. The band structure of both the compounds exhibit qualitatively similar features. Both the valence band maxima and conduction band minima are located at the high symmetry G point signifying both the ternary compounds are direct band gap semiconductors like CdS. Around the high symmetric G point, the lowest conduction band has shape of a parabola, suggesting that the charge carriers in the concerned band have characters of free electrons. The topmost valence band is almost flat, which endows with a high effective mass of the holes. The overall band profile obtained using TB-mBJ exchange potential is in good agreement with the previous theoretical calculation with hybrid functional [13]. The total density of states (DOS) together with the projected density of states (PDOS) on to various atomic orbitals of Cd4P2Cl3 and Cd4P2Br3 are presented in Fig. 3. In this figure, we have presented the DOS obtained for TB-mBJ exchange potential because the estimated band gap energies using this exchange potential shows better agreement with the previously reported experimental values. In both the compounds the DOS at the valence edge is much sharper as compared to the DOS at the conduction band edge. The sharp peak in the DOS just below the Fermi level indicates that the valence electrons are more localized in the close vicinity of the Fermi level. On the other hand, the far less sharp DOS at the conduction region reflects the itinerant character of the conduction electrons. Sharp peak in the DOS of Cd4P2Cl3 (Cd4P2Br3) are observed in the low energy regime from 6.37 to 7.04 eV (from 6.2 to 6.8 eV), which are contributed predominately by the Cd-d and P-s states. It is worth to mention that the P1-s states are strongly hybridized with both the Cd1-d and Cd2-d states, whereas the P2-s states are mainly hybridized with the Cd1-d states. While the Cl-p (Br-p) states contribute dominantly to the DOS of Cd4P2Cl3 (Cd4P2Br3) in the energy range from 2.72 to 4.76 eV (from 1.87 to 4.27 eV), the P-p states in Cd4P2Cl3 (Cd4P2Br3) have dominant contribution to the DOS in the energy range from 2.70 (1.47) eV well up to the Fermi energy. The unoccupied conduction band region is mainly contributed by the Cd-s states. 3.3. Optical properties The frequency dependent complex dielectric function of a solid is given by
ε* ðuÞ ¼ ε1 þ iε2 ;
(1)
where ε1 and ε2 are respectively the real and imaginary part of ε* . The imaginary part ε2 is directly related to the electronic band structure of a solid and contributed by different intraband and interband transitions. The intraband transitions are mainly important for the metallic systems and contribute very little in case of semiconductors [28]. The interband transitions can further be classified into two categories: the direct interband transition and the indirect interband transition. The indirect interband transitions involve the scattering of phonons and are expected to contribute a little to the dielectric function. With all these approximations, the imaginary part of dielectric function can be expressed as [28].
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N. Barman et al. / Computational Condensed Matter 21 (2019) e00390
Fig. 2. The electronic band structure of (a) Cd4P2Cl3 and (b) Cd4P2Br3 obtained using modified Becke-Johnson exchange potential.
Fig. 3. The total and the projected density of states of Cd4P2Cl3 (a, b, c) and Cd4P2Br3(d, e, f).
N. Barman et al. / Computational Condensed Matter 21 (2019) e00390
ε2 ¼
ð X Ve2 2 d3 k jknjpjkn0 j f ðknÞð1 f ðkn0 ÞÞdðEkn 2 2 2pZm u 0 n;n Ekn0 ZuÞ (2)
where, ħu is the energy of the incident photon, p ¼ iħV is the momentum operator, jkn> is the eigenfunction with eigenvalue Ekn, and f(kn) is the Fermi distribution function. The detailed calculation of the matrix elements of the momentum operator can be found in Ref. [29]. On the other hand, the real part of dielectric function is related to its imaginary counterpart through KramersKronig relation given by Ref. [30].
ε1 ¼ 1 þ
2
p
∞ ð
P
0
u’ ε2 ðuÞ ’ du ; u’ 2 u2
(3)
where, P represents the principle part of the integral. The real and imaginary parts of dielectric function were calculated for Cd4P2X3 using equations (2) and (3). The real part n(u) and imaginary part k(u) of the complex refractive index were calculated from the frequency dependent dielectric function using the relations:
1
nðuÞ ¼
√2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 ε1 ðuÞ2 þ ε2 ðuÞ2 þ ε1 ðuÞ
(4)
And
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 ε1 ðuÞ2 þ ε2 ðuÞ2 ε1 ðuÞ √2 1
kðuÞ ¼
(5)
Moreover, the absorption coefficient a and reflectivity R were calculated using the relations:
a¼
2ku c
(6)
and
RðuÞ ¼
ðn 1Þ2 þ k2 ðn þ 1Þ2 þ k2
(7)
Here, c is the velocity of light in free space. The calculations of these optical parameters were performed using TB-mBJ exchange potential. For these calculations, a dense (14 14 14) mesh of kpoints were used for the integration over the first Brillouin zone, and the broadening factor was taken to be 0.1 eV. Fig. 4(a) and (b) show the variation of the real and imaginary parts of the calculated dielectric function for Cd4P2Cl3 and Cd4P2Br3 with the photon energy ranging from zero to 12 eV. The first critical point, which represents the minimum photon energy required for the onset of the response of ε2 , consistently coincides with the band gap energies of the respective compounds. This essentially marks the transition from the topmost valence band regime to the bottom of the conduction band CB at the high symmetry G point. As the photon energy increases a number of peaks become apparent. As evident from the electronic band structures of these compounds the valence bands tend to overlap one another particularly at and around the high symmetric points, which leads to some ambiguity to predict a particular transition responsible for a given peak. Rather a peak is more plausibly contributed by a number of transitions occurring in closely spaced energy values. The indexing of such a huge number of valance
5
bands is evidently a tedious job, even if the valence bands were indexed they would hardly be identified individually in Fig. 2. For the sake of convenience, we demark the entire valance band region in Fig. 2 into a number of well separated sub-regions, each of which is consisted of several number of valence bands. The transitions from few closely spaced bands in these valence band subregions give rise to the peaks in ε2 (u). We shall use the notation VBN to represent the topmost (N ¼ 1), second topmost (N ¼ 2), and third topmost (N ¼ 3) valence band sub-regions. The first peak may be identified to arise due to the direct interband transitions from VB2 to CB at the symmetry point G and/or from VB1 to CB at high symmetric M and R points. From the DOS, we can infer that the P-p electrons are responsible for these transitions. The second peak may be inferred to arise due to the transitions of the P-p electrons from VB2 to CB at the M point, while the third peak is due to the transition from the VB3 to CB at M and R points. The locations of few prominent peaks together with the most plausible interband transitions that contribute dominantly to these peaks are presented in Table 2. The real part ε1 of dielectric function in the zero frequency limit, the so called the static dielectric constant ε1 ð0Þ, is found to be 4.13 and 4.56 for Cd4P2Cl3 and Cd4P2Br3, respectively. Any contribution from the lattice vibrations was not considered in these calculations. Prominent peaks in ε1 (u) curves are 6.21, 6.38 and 5.66 observed at energy values 3.20 eV, 3.79 eV and 4.61 eV for Cd4P2Cl3 which are comparable to the reported dielectric constant value of 5.92 of Cd4P2Cl3 in the optical frequency range [15]. Corresponding values for Cd4P2Br3 are 7.23, 6.92, and 6.88 at 3.22 eV, 3.71 eV and 4.45 eV. Moreover, ε1 becomes negative in high energy regime above 6.96 eV for Cd4P2Cl3 and 7.2 eV for Cd4P2Br3, with the implication that the electromagnetic waves ceases to propagate beyond these energies. As the absorption coefficient a is related to the absorptive part of the dielectric function, their response with energy is quite similar particularly in low energy regime. From the optical absorption spectra shown in Fig. 4(c) and (d), it is clear that in Cd4P2Cl3 (Cd4P2Br3) the absorption edge emerges at a photon energy of 2.41 (2.45) eV with corresponding wavelength of 516 (507) nm, which is originated due to the excitation of electrons from P-p states located at the topmost valence band, to the unoccupied Cd-s states. It is further noted that the absorption coefficient of both the compounds has a high value exceeding 104 cm1 in the blue region of the visible spectrum with corresponding energy between 2.50 eV and 2.75 eV, which indicates that these compounds possess very high absorbing power. The absorption coefficient increases further as the photon energy is shifted towards the ultraviolet region. Among these two compounds, Cd4P2Br3 exhibits slightly better absorption property. From the reflectivity spectra R(u) shown in Fig. 4(e) and (f), one can see that both the compounds have reflectance lying below 15% in the energy range up to the absorption edge. Thus, it turns out that both the compounds exhibit good transmittance of light with more than 85% up to the absorption edge which, in fact, covers a significant portion of visible spectrum. Semiconductor like CdS, which is proposed to be a potential material for the construction of the window layer in heterojunction solar cell, has reflectance of 13e22% near the infrared region [31,32] and transmittance of around 80% for photon energy higher than the absorption edge of this compound [33]. These values are comparable with the corresponding values of the optical parameters of Cd4P2X3. Owing to their high absorbing capability in low wavelength region and high transmittance in the bulk of the solar spectrum, Cd4P2X3 may be a potential alternative material for the construction of the window layer in heterojunction solar cell in addition to their reported HER activity.
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N. Barman et al. / Computational Condensed Matter 21 (2019) e00390
Fig. 4. (a) Real and imaginary parts of the complex dielectric function ε*, (c) the absorption coefficient a and (e) the reflectivity R of Cd4P2Cl3. Corresponding quantities of Cd4P2Br3 are in (b,d,f).
Table 2 Peak positions of ε2(u) together with the possible dominant contribution to the peaks. Compound
Peak position (eV)
Dominant transition(s) (Energy difference in eV)
Cd4P2Cl3
3.39
VB2 VB1 VB1 VB2 VB3 VB3 VB4 VB4 VB2 VB1 VB1 VB2 VB3 VB3 VB4 VB4
4.07 5.05 5.51 Cd4P2Br3
3.44
3.93 4.80 5.32
/ / / / / / / / / / / / / / / /
CB CB CB CB CB CB CB CB CB CB CB CB CB CB CB CB
4. Conclusion In summary, we have carried out a theoretical study based on first principles DFT calculations to determine the structural,
(3.26) (3.18) (3.21) (4.03) (5.11) (5.13) (5.51) (5.54) (3.12) (3.29) (3.21) (3.93) (4.80) (4.80) (5.18) (5.11)
Location of transition in BZ
G
М R M M R R M
G R M M R M R M
electronic and optical properties of Cd4P2Cl3 and Cd4P2Br3 compounds, a close analogue of CdS. Both the compounds are found to be direct band gap semiconductors with band gap energy ~2.4 eV, about 0.1 eV lower than that of CdS. The density of states
N. Barman et al. / Computational Condensed Matter 21 (2019) e00390
calculations reveal that the valence band maxima are predominantly contributed by the P-p states and conduction band minima are mainly contributed by the Cd-s states. From the study of optical properties it is found that both the compounds have very high absorption coefficient, exceeding 104 cm1 in the blue region of the visible spectrum. It is found that both the compounds exhibit good transmittance of light with more than 85% in a significant portion of visible spectrum. Such a value of band gap energy together with high absorption coefficient and low reflectance make these compounds suitable for various electronic and optoelectronic devices in addition to their photocatalytic activity. Acknowledgement PKM acknowledges funding from University of North Bengal (Grant No. 164/R-2018(SF-17).
[14]
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