First Principles Study of the Optical Properties of Alkaline-Earth Metal Nitrides

Computational Materials Science 49 (2010) 400–406

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Computational Materials Science

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First principles study of the optical properties of alkaline-earth metal nitrides

M. Dadsetani *, R. Beiranvand

Physics Department, Faculty of Science, University of Lorestan, Lorestan, Iran article info abstract

Article history: A detailed analysis of the optical properties of alkaline-earth metal nitrides (Be3N2,Mg3N2 and Ca3N2) has Received 10 January 2010 been performed, using the full potential linearized augmented plane wave (FP-LAPW) method within Received in revised form 9 May 2010 density functional theory. The exchange correlation potential is treated by the generalized gradient Accepted 11 May 2010 approximation within Perdew et al. scheme. The real and imaginary parts of the dielectric function Available online 12 June 2010 (x), the optical absorption coefficient I(x) the refractive index n(x), the extinction coefficient k(x) and the electron energy loss function are calculated within the random phase approximation (RPA). Keywords: The calculated results show a qualitative agreement with the available experimental results in the sense Metal nitrides that we can recognize some peaks qualitatively, that is, ones that are due to single particle transitions. Optical properties DFT Furthermore, the interband transitions responsible for the structures in the spectra are specified. The FP-LAPW metal s states and nitrogen p states play the major role in these optical transitions as initial and final states, respectively, for Mg3N2 and Be3N2. In the case of Ca3N2, where Ca has d levels lying near the Fermi level, the Ca d states are mostly final states. The effect of the spin–orbit coupling on the optical properties is also investigated, and it is found to be quite small, especially in the low energy region. The dielectric constants are calculated and compared with the available experimental results. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction In the optical frequency range, Soto et al. [12,17,18] have measured

the electron energy loss spectra of Be3N2 and Mg3N2 to determine The alkaline-earth metal nitrides (M3N2, M = Be, Mg and Ca) the dielectric constant, and de la Cruz et al. [13] have measured form a very important wide band gap semiconductor with a cubic the absorption coefficient, the refractive index, and the extinction bcc structure and the space group of Ia3(206) at normal conditions coefficient of beryllium nitride. But for Mg3N2 and Ca3N2, to our [1–3]. There are two crystallographic structures of Be3N2: a-Be3N2, knowledge, there is not yet any experimental results on these quan- cubic bcc with 40 atoms per cell, and b-Be3N2, hexagonal close tities in the optical frequency range. Our calculated results could packed with 10 atoms per cell [4]. The first structure is stable be- serve as a reference for future experimental work on this compound. tween 20 °C and 1200 °C, which changes to the hexagonal form On these materials, from theoretical point of view, there exist a over 1400 °C. The calcium nitride exists in various structures, number of first principles calculations in terms of their stability, viz., a-Ca3N2 (reddish-brown) [5], b-Ca3N2 (black) [6], c-Ca3N2 and electronic as well as structural properties [4,10,11,19–25].To (yellow) [7], and the different high pressure phase (yellow) [8].It the best of our knowledge, there is no theoretical research on opti- is further known that Mg3N2 and a-Ca3N2 compounds are iso- cal properties (i.e., real and imaginary parts of dielectric functions, structure [9]. They are characterized by their wide band gap, high reflectivity, absorption and electron energy loss function). More- thermal conductivity and large bulk modulus [10,11]. These inter- over, it seems that there is a lack of both experimental and theoret- esting properties make them potential materials for the use in the ical data on the optical properties of alkaline-earth metal nitrides, development of solid state illuminations, optoelectronic applica- and no systematic research on the optical properties of these com- tions, and display and communication devices [12]. They are tech- pounds have been reported. nologically important materials, with a band gap in the ultraviolet In this work, we have investigated a full range of optical prop- region, which can be used as an alternative to the aluminum ni- erties of alkaline-earth nitride, including real and imaginary parts tride, the aluminum–gallium nitride [12], and the beryllium for of the dielectric function, the absorption coefficient, and the energy some optical applications [13]. loss function, using the full potential linearized augmented plane The available experimental studies on the optical properties of wave (FP-LAPW) method with the generalized gradient approxi- these materials are mostly limited to the optical band gap [14–16]. mation (GGA) for the exchange correlation potential, within the density functional theory. The outline is as follows: a brief description of our calculation * Corresponding author. Tel./fax: +98 661 2201335. E-mail addresses: [email protected], [email protected] (M. Dadsetani). method is given in Section 2. In Section 3, we have given the

0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.05.028 M. Dadsetani, R. Beiranvand / Computational Materials Science 49 (2010) 400–406 401 detailed band structure of alkaline-earth nitrides, which is needed WIEN2K [26] implementation of the method which allows the for optical studies. Optical properties through the study of the inclusion of local orbitals in the basis, improving upon linearization imaginary and real parts of the dielectric function, the absorption, and making possible a consistent treatment of the semicore and refraction spectra, and EELS are discussed in Section 4. A brief sum- valance states in an energy window, hence ensuring proper orthog- mary and conclusions are given in Section 5. onality. The exchange correlation potential within the GGA is calculated using the scheme of Perdew et al. [27]. The convergence min 2. Calculation method parameter RMT Kmax (the product of the smallest of the atomic sphere radii RMT and the plane wave cutoff parameter Kmax) which The calculations presented in this work were performed using controls the size of the basis sets in these calculations, was set to 8. the FP-LAPW method. In this method no shape approximation on The maximum l quantum number for the wavefunction expansion the potential or the electronic charge density is made. The calcula- inside the atomic sphere is conﬁned to lmax = 10. The Gmax param- 1 tions of the electronic and optical properties have been done rela- eter was taken to be 14.0 Bohr . Brillouin-zone (BZ) integrations tivistically with and without the spin–orbit coupling. We use the within self-consistency cycles were performed via a tetrahedron

(a) (b) (c) 10 8 4 8 6 2 6 4 4 0 2 2

0 0 -2 -2 -2 -4 -4 -4 -6 Energy (eV) Energy -6 -8 -6

-10 -8 -8 -12 -10 -14 -10 -12 -16

-18 -14 -12 Γ Δ H N Σ Γ Λ P Γ Δ H N ΣΓ Λ P Γ Δ H N ΣΓ Λ P (d) (e) 60 Be N 50 3 2 without so 40 with so 1.0 30 0.8 Be s 20 N1 s 10 0.6 N2 s 1200 0.4

100 Mg3N2 without so 0.2 80 with so 0.01.0 60 0.8 Be p 40 N1 p 0.6 N2 p 20 DOS (State/eV)

DOS (State/eV) 0.4 3000 0.2 250 Ca3N2 without so 0.0 200 with so -15 -10 -5 0 510 150 Energy (eV) 100 50 0 -15 -10 -5 0 5 10 Energy (eV)

Fig. 1. The electronic band structures of Be3N2 (a), Mg3N2 (b) and Ca3N2 (c) along with the total DOS for all three compounds (d) and partial DOS for Be3N2. 402 M. Dadsetani, R. Beiranvand / Computational Materials Science 49 (2010) 400–406 method [28], using 24 K points in the irreducible BZ. For the calcu- contributions to the occupied part of the DOS come from the N lation of the optical properties however, a denser sampling of the 2p and 2s states. BZ was needed, where we used 60 K points. The muffin-tin radius The first structure in the low lying energy side of the DOS con- of alkaline-earth metal (M) and nitrogen atoms (N) were chosen as sists of a narrow peak centered on 10.64 eV for Ca3N2 and M N (RMT; RMT) = (1.6, 1.5), (1.5, 1.5) and (2.3, 2.1) a.u. for Be3N2,Mg3N2 12.00 eV for Mg3N2, and a broad peak about 15 eV for Be3N2. and Ca3N2, respectively. All these values have been chosen in a way This structure originates from Ns states, corresponding to the low- to ensure the convergence of the results. The calculation for the est lying bands in Fig. 1. These bands are lower in energy for Be3N2 optical properties has been done with and without spin–orbit cou- and Mg3N2 than Ca3N2 by around 2.7 and 1.0 eV, respectively. The pling to ascertain if spin–orbit coupling has any dramatic influence structure of the spin–orbit splitting and its amount are not consid- on the optical properties. We find that spin–orbit interaction has a erable in this region. minor influence on the optical properties. The next structure which, in the absence of spin–orbit coupling,

is separated from the ﬁrst by a gap of 5.64 eV for Be3N2, 7.22 eV for Mg3N2, and 8.55 eV for Ca3N2, is a broad peak that is situated between 7.5 eV and the Fermi level, corresponding, to the valence 3. Electronic structure results bands shown in the band structure in Fig. 1. From the partial DOS, it is seen that just below the Fermi level, the bands are dom- Since the optical spectra are calculated from the interband tran- inated by the N 2p, where metals s and p are hybridized. These sitions, ﬁrst we present our calculated electronic structure of these bands are lower in energy for Be3N2 and Mg3N2 than Ca3N2 by compounds. The primitive cell of body centered cubic phase of 5.63 and 2.28 eV, respectively. Spin–orbit coupling has a minor ef- alkaline-earth metal nitrides has 40 atoms, 24 similar M atoms fect on the structure of these peaks. and two types of N atoms, 12 N(1) and 4 N(2). A point worth men- There are two main features in the broad structure above the tioning is that the similar nitrogens in the cell are not equivalent. Fermi level up to around 15 eV, containing, a broad peak around

Our calculated total density of states (DOS) and electronic band 7 eV for Be3N2, 5 eV for Mg3N2 and 2.5 eV for Ca3N2. The distinctive structures for Be3N2,Mg3N2 and Ca3N2 are given in Fig. 1. Due to structure in the DOS above 7.0 eV is a broad peak at around 8 eV for the close similarity between the results obtained for these com- Ca3N2 and 10 eV for Be3N2 and Mg3N2. These structures are mainly pounds, the partial DOS are given only for Be3N2. The results are gi- from the p states of both nitrogen and metal atoms, and 3d states of ven both with and without the spin–orbit coupling. Major calcium.

12.0 12.0 Be N with so Be N c 10.0 3 2 3 2 with so without so 10.0 without so 8.0 8.0 b 6.0 6.0 d 4.0

2.0 4.0 a 0.0 2.0 -2.0 10.0 10.00.0

8.0 Mg3N2 with so Mg3N2 with so without so 8.0 without so 6.0 b

4.0 6.0 (ω)

(ω) c 2 d 1

2.0 ε

ε 4.0 0.0 a 2.0 -2.0

12.0-4.0 12.00.0 10.0 Ca N with so Ca N with so 3 2 10.0 3 2 without so without so 8.0 b 8.0 6.0 4.0 6.0 c 2.0 4.0 0.0 a 2.0 -2.0 d

-4.0 0.0 0 10203040010203040 Energy (eV) Energy (eV)

Fig. 2. Calculated real (left) and imaginary (right) parts of dielectric function. The spectra are slightly broadened. M. Dadsetani, R. Beiranvand / Computational Materials Science 49 (2010) 400–406 403

In Ca3N2 we found an indirect band gap between the top of are four prominent peaks and shoulders appearing in the 2x spec- nitrogen valence 2p bands occurring at the H point and the bottom tra, denoted by a, b, c and d in Fig. 2. Looking at optical matrix ele- of the Ca 3d bands occurring at the C point, confirming the previ- ments, it is obvious that the metal s states and N p states play a ously found results by Imai et al. [20]. It is clearly seen that Be3N2 major role in these optical transitions as initial and final states, and Mg3N2 are two semiconductors with a direct band gap at the respectively for Mg3N2 and Be3N2. In the case of Ca3N2, where Ca Brillouin zone center, confirming the results obtained by Armenta has d levels lying near the Fermi level, this contribution is reason- et al. [10] and in contradiction to results obtained for Mg3N2 by ably big. The 2p electrons of nitrogen and 3d electrons of calcium Fang et al. [21]. are involved in the interband transitions responsible for the peaks

The calculated values of the band gaps for Ca3N2,Be3N2 and b and c in Fig. 2. Spin–orbit coupling has a minor effect on the Mg3N2 are 1.13, 3.26 and 1.50 eV, respectively. The results confirm dielectric function of Mg3N2, as can be seen from Fig. 2, shifting the well known expectation that the theoretical band gaps are on it down by a very small amount. The spin–orbit coupling does the whole underestimated within GGA, in comparison with the not have any significant effect on the results of Be3N2 and Ca3N2. experimental values [17,23,24]. We have included the spin–orbit This is expected, as the spin–orbit coupling changes the eigen- coupling in the calculation of the band gap, but found the effect values only by 0.1 eV, which is not significant in the calculation was negligible. Compared to the experimental and other calculated of optical properties. This has also been found in FP-LMTO calcula- results, our results are in fair agreement with the previous studies. tions for HgI2 [32]. It can be seen that the calculated energy gap values decrease as the In the dispersive part of the dielectric function, 1x, the main anion size increases. Similar behavior was observed in Sr and Ba features are as follows: a peak at around 2–9 eV; a rather steep de- chalcogenides. crease between 3 and 12 eV, after which 1x becomes negative; a minimum at around 8 eV for Ca3N2 and about 12 eV for others; and 4. Optical properties then again a slow increase towards zero at higher energies. The function representing characteristic energy losses (or plas- The optical properties of matter can be described by means of mon oscillations) is one of the most important among those suit- the transverse dielectric function (x). There are two contributions to (x), namely, intraband and interband transitions. The contribution from intraband transitions is important only for metals. The 2.5 interband transitions can further be split into direct and indirect Be3N2 Simulated.exp data transitions. Here we neglect the indirect interband transitions 2.0 Theo. which involve scattering of phonons and are expected to give a Exp small contribution to (x) [29]. To calculate the direct interband 1.5 contribution to the imaginary part of the dielectric function

2(x), we should sum up all possible transitions from the occupied 1.0 to the unoccupied states. Taking the appropriate transition matrix elements into account, the imaginary part of the dielectric function

2(x) is given by random phase approximation (RPA), neglecting 0.5 local ﬁeld effects: Z 2 X 0.03.5 Ve 3 0 2 2ðxÞ¼ 2 2 d k jhjkn pkjijn f ðknÞð1 2phm x nn0 3.0 Mg3N2 Exp. 0 Theo. f ðkn ÞÞdðEkn Ekn0 hxÞð1Þ 2.5 ⁄ where x is the energy of the incident photon, p is the momentum 2.0 h @ operator i @x,|ki is the eigenfunction with eigenvalue Ekn, and f(kn) is the Fermi distribution function. The evaluation of the matrix ele- 1.5 ments of the momentum operator in Eq. (1) is performed over the Energy loss mufﬁn-tin and the interstitial regions separately. A fully detailed 1.0 calculation of the matrix elements is given by Ambrosch-Draxl 0.5 et al. [30].

The real part of the dielectric function 1x follows from the Kra- 3.50.0 mers–Kronig relation [31]. In order to calculate 1x, we need to 3.0 Ca N with so have a good representation of 2x up to high energies. In the pres- 3 2 without so ent work we have calculated 2x up to 75 eV and used this value as 2.5 the truncation energy in Kramers–Kronig relations. This energy range was chosen to produce convergence in the Kramers–Kronig 2.0 transformations. Other optical parameters, such as energy loss spectrum and absorption coefﬁcient can be immediately calculated 1.5 in terms of components of the complex dielectric function [31]. 1.0 Our calculated real and imaginary parts of the dielectric function for the three compounds with and without spin–orbit cou- 0.5 pling are shown in Fig. 2. We note that all the structures in the imaginary part of dielectric function are shifted towards lower 0.0 0 10203040506070 energies as we go from Be to Ca. This trend may be directly inferred from the band structure results, given the shift in the location of Energy (eV) the ﬁrst structure above the Fermi level in DOS curves. It is worth- Fig. 3. Calculated electron energy loss function along with the available experi- while to attempt to identify the transitions that are responsible for mental results [12,18]. In the case of Be3N2, simulated experimental data ([Ref. the structures in 2x, using our calculated band structures. There [12]]) are also presented. 404 M. Dadsetani, R. Beiranvand / Computational Materials Science 49 (2010) 400–406

Table 1 maximum is linked to the bulk plasmon, and the shoulder corre- Comparison of peak positions (eV) of EELS for Be3N2 and Mg3N2. sponds to the surface plasmon. As can be seen in the figure, there Present work Experimenta is a close match between our calculated results and the experiment Shoulder Maximum Shoulder Maximum in the low energy region. Given the overall features of the spectra, there is a reasonable agreement between the two curves for higher Be N 10 23.2 9.8 22.2 3 2 energy, but the locations of the peaks do not coincide. Mg3N2 7.5–11 18.6 5–10 16 The behavior of EELS is similar for all three compounds with a Ref. [12]. some differences in details. The calculated EELS has two major peaks, the bulk plasmon and the surface plasmon for three compounds. The main peak in the EELS (and other optical spectra) able for the description of microscopic and macroscopic properties moves to lower energy and becomes sharper as the metal column of solids. This function is proportional to the probability that a fast is transverse downward. This trend can be linked to the trends ob- electron moving across a medium loses an energy E per unit length. served in the DOS and band structures. Compare the highest lying The most prominent peak in the EELS spectrum is identified as the valance bands for the three systems and also the lowest lying con- plasmon peak, signaling the energy of collective excitations of the duction bands. In Ca3N2 these bands have clearly less dispersion electronic charge density in the crystal. It is possible to have sev- than in Be3N2. This is the reason why main peak moves to lower eral plasmon peaks in a crystal. In Fig. 3, we have given the calcu- energy and becomes sharper as the metal column is transverse lated electron energy loss spectra for the three compounds in the downward. absence of the spin–orbit coupling, together with the available In order to make a more detailed comparison with the experi- experimental results for Be3N2 and Mg3N2 by Soto et al. [12,18]. mental data, the extinction coefficient k(x) and refractive index The results obtained in the presence of spin–orbit coupling are gi- n(x) have been calculated for all three compounds. This compari- ven for Ca3N2. There exist a shoulder peak and a maximum in the son is shown for Be3N2 in Figs. 4 and 5. The spectra contain a lot of experimental curves for Be3N2 and Mg3N2, the shoulder peak and structures and the agreement between our calculation and the the maximum being reproduced in our calculations. The positions measured spectra is truly excellent. The positions of the maxima of shoulder peaks and maxima have been listed in Table 1. The and minima of the experimental curve are extremely well reproduced in the calculation, so are their relative amplitudes.

As seen in Fig. 2, 1x is zero at 11.03, 10.21, and 7.85 eV for Be3N2,Mg3N2 and Ca3N2, respectively. In k(x), i.e., the extinction 3.0 Exp. Be3N2 2.5 Theo. 2.0 Be N Exp. 2.0 3 2 1.5 Theo. 1.5

1.0 1.0

0.5 0.5 0.03.5 with so 3.0 Mg3N2 0.0

) without so 2.0 ω Mg N with so 2.5 ) 3 2

ω without so 2.0 1.6

1.5 1.2

1.0 0.8 Refractive index n( 0.5 0.4

0.04.0 Extinction coefficient k( Ca N with so 0.0 3 2 2.0 without so Ca3N2 with so 3.0 without so 1.6

2.0 1.2

0.8 1.0 0.4

0.0 0.0 020406080 020406080 Energy (eV) Energy (eV)

Fig. 4. Calculated refractive index along with the available experimental results Fig. 5. Calculated extinction coefﬁcient along with the available experimental [13]. results [13]. M. Dadsetani, R. Beiranvand / Computational Materials Science 49 (2010) 400–406 405

Table 2 correction. In order to compensate for the GGA underestimation The calculated and experimental dielectric constant. of the band gap, the calculated 2x have been shifted. The amount Present work (GGA) Exp. of shift in every case is equal to the difference between the calcu- Without so Without so With so With so lated and the experimental band gaps. Also listed are the experi- Without shift With shift Without shift With shift mental values for (0) as measured in Refs. [12,18].

a Our calculated results are bigger than the experimental values, Be3N2 4.72 4.47 4.70 4.46 4.1 b Mg3N2 5.69 4.75 5.68 4.75 5.4 and scissors operator improves the results substantially, and the Ca3N2 6.38 5.56 6.34 5.53 inclusion of spin–orbit coupling decreases the dielectric constant

a by a very small amount. On the whole, our scissors shifted results Ref. [12]. b Ref. [18]. in the presence of spin–orbit coupling have an average deviation of about 10% from the experiment. For the comparison purposes,

there is no experiment for Ca3N2. Fig. 6 shows the calculated frequency-dependent absorption coefficient, we find local maxima at these energies. On the con- coefficients for the three compounds as well as the available exper- trary, in the energy loss spectra (Fig. 3), no maximum is present imental results [13], indicating a close resemblance between Be N at these energies, since x is still large at these energies. Higher 3 2 2 and Mg N . Here, the experiment data are based on the measure- up, however, at around 23 eV for Be N , 18 eV for Mg N , and 10 3 2 3 2 3 2 ment of electron energy loss. Above the band gap, the materials be- and 30 eV for Ca N , all three compounds show a large peak in 3 2 gins to have strong optical dispersions as a consequence of the the energy loss spectrum. These peaks correspond to x going 1 interband transition from metal s and N 2p states to N 2p and Ca through zero once again. At such high energy, x is small, and 2 3d states. Our absorption reaches the maximum value for Be N thus the amplitude of the energy loss function becomes large. 3 2 whose magnitude closely coincides with the experimental peak. In Table 2, we have listed the calculated dielectric constants. For The spectrum of Be N is in very good agreement with the exper- each compound two calculated values are listed: one obtained 3 2 imental spectrum; there are, however, features that need improve- with gap correction using scissor operator and the other with no ment. The low energy peak is too sharp. The indirect interband transitions are not taken into account in the calculation, and these can in part explain the tendency of the calculated spectra to decrease more rapidly towards higher energies than the experimen- 15000 Be N 3 2 Exp. tal ones. These compounds make a transition from absorbing to transmitting above the plasmon energies, for which k ? 0 at high 12000 Theo. energies.

It is interesting to note that in the optical spectra for Ca3N2,a 9000 broad structure around 25 eV is clearly visible. For Be3N2 and Mg3N2, such a structure is not visible. We conclude that the origin 6000 of this structure is due to d states of calcium.

3000 5. Conclusions

60000 We have applied a FP-LAPW method to study the electronic and

Mg N optical properties of the alkaline-earth nitrides Be3N2,Mg3N2 and 5000 3 2 with so without so Ca3N2 within the density functional theory with and without spin–orbit coupling. ) 4000 -1 We have shown that the metal s states and nitrogen p states cm 4 3000 play the major role in these optical transitions as initial and ﬁnal states, respectively, for Mg3N2 and Be3N2. In the case of Ca3N2, ) (10

ω 2000 where Ca has d levels lying near the Fermi level, the Ca d states I( are mostly final states. There is a good agreement between our cal- 1000 culated electron energy loss, refractive index, extinction coefficient and absorption coefficient with experiment especially in the low 100000 energy part of the spectra. Our scissors shifted results of the dielec- Ca N tric constants in the presence of spin–orbit coupling have an aver- 3 2 with so age deviation of about 7% from the experiment. 8000 without so The spin–orbit coupling has a very small (almost negligible) ef-

fect on the optical properties. At high energies, for Mg3N2 it has a 6000 more pronounced effect on the details of the peak positions (over- estimates/underestimates the peaks height) and the average values of the calculated extinction coefﬁcient and imaginary part of 4000 dielectric function. The nature of the fundamental gap in these

materials were found to be direct for Be3N2 and Mg3N2 and indirect 2000 for Ca3N2.

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