Appl Phys A (2009) 97: 345–350 DOI 10.1007/s00339-009-5243-x
Ab initio calculations of yttrium nitride: structural and electronic properties S. Zerroug · F. Ali Sahraoui · N. Bouarissa
Received: 13 November 2008 / Accepted: 16 April 2009 / Published online: 5 May 2009 © Springer-Verlag 2009
Abstract Using first principles total energy calculations within the full-potential linearized augmented plane wave method, we have studied the structural and electronic properties of yttrium nitride (YN) in the three phases, namely wurtzite, caesium chloride and rocksalt structures. The calculations are performed at zero and under hydrostatic pressure. In agreement with previous findings, it is found that the favored phase for YN is the rocksalt-like structure. We predict that at zero pressure YN in the rocksalt structure is a semiconductor with an indirect bandgap of 0.8 eV. A phase transition from a rocksalt to a caesium chloride structure is found to occur at ∼134 GPa. Besides, a transition from an indirect (Γ − X) bandgap semiconductor to a direct (X − X) one is predicted at pressure of ∼84 GPa. For the electron effective mass of rocksalt YN, these are the first results, to our knowledge. The information derived from the present study may be useful for the use of YN as an active layer in electronic devices such as diodes and transistors. PACS 71.15.Nc · 71.20-b · 71.20.Be · 71.20.Lp · 71.20.Nr · 72.80.Ga
1 Introduction Yttrium is a chemical element that has the symbol Y and resembles the lanthanides. It is a silver-metallic transition S. Zerroug · F. Ali Sahraoui Laboratoire d’Optoélectronique et Composants, Département de Physique, Université Ferhat Abbas, Sétif 19000, Algeria N. Bouarissa () Department of Physics, Faculty of Science, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia e-mail:
[email protected]
metal, which is common in rare-earth minerals, relatively stable in air, and strongly resembles scandium in appearance. It can appear to gain a slight pink luster on exposure to light. By alloying Y with nitrogen, we may produce compounds with excellent combinations of properties like hardness, chemical inertness, electrical and thermal conductivity [1, 2]. In the industry, YN may be important as electric contact, diffusion barrier, buffer layer, among other uses [3]. Besides, combinations of GaN and YN could also be useful in the growth of YN/GaN heterostructures or YGaN alloys [4]. To provide a basis for understanding future device concepts and applications, knowledge of the fundamental properties of the devices is required. Furthermore, information on the pressure dependence of these properties is very important for the application. As a matter of fact, pressuretuning studies of many physical properties as a function of volume have proved invaluable in systems ranging from the electronic structure of semiconductors to photo-physics of solid state molecular structures [5]. In spite of the prospective applications of YN, only limited experimental and theoretical studies have been reported so far on its physical properties [1–4, 6]. Thus, many fundamental properties of the material of interest remain to be determined precisely. In this paper, we present results of structural and electronic properties of YN using the full potential linearized augmented plane wave (FP-LAPW) method within the density functional theory (DFT) with the generalized gradient approximation (GGA). Attention has been given to the pressure dependence of the studied properties. The structure of YN is very simple: face centered cubic (fcc), or hexagonal closed packed (hcp) [1, 3]. The N atoms are located in interstitial sites of the metal-lattice in octahedral geometry [3]. Nevertheless, the preference for the cubic symmetry has been reported for many transition metal nitrides [3] due
346
to the ionic character of the N-atoms. Three phases, namely wurtzite, caesium chloride and rocksalt are considered here.
2 Computational method The calculations were performed in the frame work of DFT. We have employed the FP-LAPW method as implemented in the WIEN 2K code [7]. In dealing with electron– electron interaction, the exchange and correlation effects for the structural properties were treated using the GGA based on Perdew et al. [8] form, while for the electronic properties, the exchange-correlation functional of Engel and Vosko (EV-GGA) [9] was applied. The motivation for the use of the two different forms of the GGA is that the GGA by Perdew et al. used in the present work has turned out to be much more successful for the calculation of the structural properties than other approximations. However, it is well known that the GGA usually underestimate the energy gap. This is mainly due to the fact that it has simple form which is not sufficiently flexible for accurately reproducing both exchange-correlation energy and its charge derivative. Engel and Vosko by considering this short coming constructed a new functional form of the GGA which was able to better reproduce the exchange potential at the expense of less agreement as regards exchange energy. This approach, which is called the EV-GGA, yields a better band splitting and some other properties which mainly depend on the accuracy of the exchange-correlation potential [10, 11]. The 4s 2 , 4p 6 , 5s 2 , and 4d 1 electrons of Y and 2s 2 and 2p 3 electrons of N, are included in the valence. In the FP-LAPW method, the wave function and potential are expanded in spherical harmonic functions inside non-overlapping spheres surrounding the atomic sites (muffin-tin spheres) and a plane wave Fig. 1 Total energy versus volume for YN
S. Zerroug et al.
basis set in the remaining space of the unit cell (interstitial region) is used. A plane wave cut-off of kmax = 8.5/RMT (where RMT is the smallest muffin-tin radius in the unit cell) was used. The muffin-tin (MT) radii were taken to be 1.9 and 1.8 a.u. for Y and N, respectively. 60, 35 and 47 special k-points were used in the irreducible part of the Brillouin zone for YN in the wurtzite, CsCl and rocksalt structures, respectively.
3 Results and discussions 3.1 Structural properties Figure 1 shows the variation of the total energy as a function of the volume for YN in the three phases: wurtzite, CsCl and rocksalt. Note that the lowest energy state is that of the rocksalt structure. The rocksalt structure is favored over the wurtzite and CsCl ones. The preference for the rocksalt structure (the cubic symmetry) may be traced back to the ionic character of the N-atoms. The curves of Fig. 1 are fit to the Murnaghan’s equation of state so as to determine the equilibrium structural parameters. Table 1 shows the calculated equilibrium lattice constant a0 , structural parameter c0 /a0 , bulk modulus B0 , and pressure derivative of the bulk modulus B0 for the YN in the wurtzite, CsCl and rocksalt structures. Also shown for comparison are the available experimental and previous theoretical data. Note that for the a0 of YN in the rocksalt structure, the agreement between our result and the experimental data quoted in [3] is better than 2%, while the agreement with the theoretical result of Soto et al. [3] is within 1%. Moreover, our result is in perfect agreement with that reported in [6]. This is consis-
Ab initio calculations of yttrium nitride: structural and electronic properties
347
Fig. 2 Enthalpy versus pressure for YN
Table 1 Equilibrium lattice constant a0 , structural parameters c0 /a0 , bulk modulus B0 , and pressure derivative of B0 (B0 ) for wurtzite, CsCl and rocksalt structures analyzed for YN Structure
a0 (A0 )
c0 /a0
B0 (GPa)
B0
Reference
Wurtzite
3.77
1.58
114
3.61
This work
3.51396
1.57
3.78
1.58
115
3.73
Theor. [6]
CsCl Rocksalt
Theor. [3]
3.00
153
3.46
This work
3.01
136
4.11
Theor. [6]
4.93
151
3.86
This work
4.877
Expt. [3]
4.90921 4.93
Theor. [3] 157
3.50
Theor. [6]
tent with the general trend of the GGA approach [10, 11]. For the wurtzite structure, our results concerning a0 and c0 are somewhat larger than those reported by Soto et al. [3] but are in excellent agreement with the results of Mancera et al. [6]. In addition, the calculated value of c0 /a0 obtained in the present work for the relaxed structure is close to its
ideal value of 83 . As regards CsCl structure, our result regarding a0 is compared to that reported by Mancera et al. [6] and showed very good agreement. For the fact that there is no experimental data concerning the B0 and B0 for all structures being considered here, our results are compared to the theoretical data reported in [6]. The agreement between our results and those of [6] is better than 13% and 16% for B0 and B0 , respectively. It is to be noted that the B0 of the rocksalt and CsCl structures is larger than that of the wurtzite structure reflecting thus the stronger chemical bonds in rocksalt and CsCl YN as compared to wurtzite
YN which might be a consequence of repulsive interactions between nearer N–N in octahedral sites of hexagonal closepacked structure. In order to get information about the transition pressure (pt ) between rocksalt and CsCl structures, we have calculated the Gibbs free energy (G), (G = E0 + PV − TS) for these two phases. Since the present calculations are performed at T = 0 K, the Gibbs free energy becomes equal to the enthalpy, i.e., H = E0 + PV. The variation of the enthalpy H as a function of pressure for both phases of interest is plotted in Fig. 2. The transition pressure is obtained from the enthalpy curve crossings. Note that a transition from rocksalt to CsCl may occur at a pressure of ∼134 GPa. At this pressure the enthalpies of both structures are equal. Recently, Mancera et al. [6] have predicted a high pressure phase transformation to a CsCl structure at ∼138 GPa which is in good agreement with our finding. Generally, in the group-III nitrides there is a local dynamical stability of the rocksalt structure against a Cmcm-like distortion. However, such a distortion is observed in other IIIA–VA compounds which also exhibit the rocksalt structure [12, 13]. The local stability of the rocksalt phase of the group-III nitrides may be attributed to the absence of p electrons in the core of the N atoms which favors the transfer of charge toward them, resulting thus in a larger ionicity than in the rest of IIIA–VA compounds [13]. 3.2 Electronic properties The calculated electronic band-structure of rocksalt YN at zero pressure is shown in Fig. 3. The zero energy reference is at the top of the valence band. The valence band is due to N 2p electrons with a small contribution from Y 4d elec-
348
S. Zerroug et al.
Fig. 3 Electronic band-structure of rocksalt YN
trons. The bottom of the valence state is a singlet originating from the bonding Y(5s)–N(2s) like orbitals that exhibit a weak dispersion. The valence band top is formed by the triply degenerate hybridized Y(5p,4d)–N(2p)-like orbitals in an antibonding manner. The N states contribute preferably to the bottom of the valence band, while a substantial number of Y states contribute to the highest valence band. While the valence band energies always exhibit the behavior to be expected from bonding combinations of hybridization between the N and Y states, the behavior of conduction band states is much more complicated. The conduction bands are more delocalized and more ‘free-electron like’ than the valence bands. The free electron behavior results in more dispersive bands and band crossing. The higher conduction states arise from the hybridization of the Y(5p,4d)- and N(2p,3s)-like orbitals. The d levels on each Y atom interact with the p levels on each neighboring N atom forming p–d σ bonds and p–d π bonds. The maximum of the valence band is at the zone center Γ , whereas the minimum of the conduction band is at the X high-symmetry point. Hence, YN is an indirect (Γ − X) bandgap semiconductor with a fundamental bandgap of 0.80 eV. Our value is in good agreement with that of ∼0.85 eV reported by Stampfl et al. [14, 15] using Screened-exchange local density approximation approach, but is much larger than the value of ∼0.3 eV reported by Sato et al. [3] using the FP-LAPW method. It is to be noted that the GGA calculation by Stampfl et al. [14] predicted that YN is a semimetal. The reason of the discrepancy between our result concerning the energy bandgap with respect to the GGA and that reported in [14] is mainly due to the use of the EV-GGA in our case which constructed a new functional form of the GGA as has been reported in Sect. 2 of the present paper.
The pressure-induced energy shifts for the optical transition related to the energy bandgaps for rocksalt YN are shown in Fig. 4. Note that as pressure increases both bandgap energies (Γ − X) and (X − X) decrease, while the bandgap (Γ − Γ )increases. The increase of (Γ − Γ ) bandgap with rising pressure is generally the behavior commonly observed in binary tetrahedral semiconductors [16– 19]. The extent of the bandgap transition is examined by our calculations. Our results show that YN goes over from an indirect bandgap (Γ − X) to a direct bandgap (X − X) at pressure of 83.84 GPa which corresponds to an estimated crossover bandgap energy of ∼0.16 eV, provided that no structural phase transformations have occurred up to 100 GPa. The linear pressure coefficients are important parameters for fundamental research and device applications. These parameters are calculated in the present study for rocksalt YN. Our results showed that the linear pressure coefficients of the direct (Γ − Γ ) and (X − X) and the indirect (Γ − X) energy bandgaps are 8.41 × 10−3 , −12.96 × 10−3 and −8.32 × 10−3 eV/GPa, respectively. Due to the lack of both experimental and theoretical data concerning the pressure coefficients of rocksalt YN, to the best of our knowledge, our results are predictions and may serve as a reference for future work. The electron effective mass is an important material parameter describing most carrier transport properties. Values of effective masses can often be determined by, for instance, cyclotron resonances or transport measurements [20]. On the theoretical side, effective masses may be obtained from the electronic band structure [21]. A theoretical effective mass in general turns out to be a tensor with nine components. Thus, it is a direction dependent quantity. However, for a simple case, i.e., for the band extremum occurring at the wave vector k = 0 and the parabolic E(k) relationship
Ab initio calculations of yttrium nitride: structural and electronic properties
349
Fig. 4 Bandgap energies versus pressure for rocksalt YN
Fig. 5 Electron effective mass at the X-high symmetry point versus pressure in rocksalt YN
(as in the case of the AIII B V compounds), where E is the carrier energy, the effective mass becomes a scalar and is a value independent of direction. In our case (see Fig. 3), the curve representing the first conduction band is almost parabolic in the vicinity of the X-point which is the band minimum. Hence, the electron effective mass is taken to be a scalar quantity. Since YN in the rocksalt structure is found to be an indirect (Γ − X) bandgap semiconductor in the whole pressure range 0–83.84 GPa, the electron effective mass has been calculated at the X-high-symmetry point for various pressures up to 80 GPa. We have adopted a parabolic line fit to the conduction band dispersion in the vicinity of the minima. The electron effective mass was obtained for the lowest conduction band at X from the second derivative of the band energy with respect to the wave vector k.
At zero pressure, we predict a value of ∼0.27m0 , where m0 is the free electron mass. Figure 5 shows the electron effective mass (in units of m0 ) at X calculated as a function of pressure up to 80 GPa. Note that the electron effective mass increases monotonically with increasing pressure. This is believed to be due to the fact that applied high pressure leads to a reduction in the volume, causing enormous changes to the inter-atomic bonding since the atoms become more closely packed. This makes the interaction between the electron wave packet with the periodic lattice larger or in other words the binding force between the electron and the lattice becomes stronger. It will be then difficult for the electrons to move, meaning thereby that the electron has acquired a larger effective mass than that it acquired at zero pressure. As pressure increases, the interaction be-
350
tween the electron and the lattice becomes stronger, leading thus to an increase in the electron effective mass. The same qualitative behavior has been reported for the electron effective mass at X valley in cubic Al1−x Gax N as a function of composition [22]. This trend is consistent with the decrease of the electron mobility in semiconductors under compression.
4 Conclusions In summary, we have presented structural and electronic properties of wurtzite, CsCl and rocksalt YN as calculated using the ab initio total energy calculations within the FPLAPW method in the framework of DFT. Quantities such as equilibrium lattice constants, bulk modulus and its pressure derivative, bandgap energies and electron effective mass have been calculated and compared with previously available data. Generally, a good agreement was obtained between our results and those reported in the literature. Our results predicted that the rocksalt structure is more favorable for YN than the wurtzite and CsCl ones. In this case, the fundamental gap at zero pressure is found to be indirect with a value of 0.80 eV. A phase transition from rocksalt to CsCl YN has been found to occur at a pressure of ∼134 GPa, which agree well with previous calculations. Our calculations revealed that a transition from an indirect (Γ − X) bandgap semiconductor to a direct (X −X) one may occur at a pressure of 83.84 GPa. The electron effective mass at the X point for rocksalt structure has been calculated for the first time at various pressures up to 80 GPa. While our results concerning the linear pressure coefficients of bandgap energies and the electron effective mass are predictions, it might be worthwhile checking with future works. Therefore, more experimental measurements and first-principles calculations are needed in order to obtain more accurate and reliable results.
S. Zerroug et al.
References 1. V.A. Gubanov, A.L. Ianovsky, V.P. Zhukov, Electronic Structure and Refractory Carbides and Nitrides (Cambridge University Press, Cambridge, 1994) 2. G. Soto, E. Aparicio, M. Avalos-Borja, J. Alloys. Compd. 389, 42 (2005) 3. G. Soto, M.G. Moreno-Armenta, A. Reyes-Serrato, Comput. Mater. Sci. 42, 8 (2008) 4. W. De La Cruz, J.A. Díaz, L. Mancera, N. Takeuchi, G. Soto, J. Phys. Chem. Solids 64, 2273 (2003) 5. J.V. Badding, Annu. Rev. Mater. Sci. 28, 631 (1998) 6. L. Mancera, J.A. Rodríguez, N. Takeuchi, J. Phys. Condens. Matter 15, 2625 (2003) 7. P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, in WIEN 2K, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties, ed. by K. Schwarz (Techn. Universität, Wien, 2001), ISBN-3-9501031-1-2 8. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996) 9. E. Engel, S.H. Vosko, Phys. Rev. B 47, 13164 (1993) 10. Z. Charifi, H. Baaziz, F. El Haj Hassan, N. Bouarissa, J. Phys. Condens. Matter 17, 4083 (2005) 11. S. Zerroug, F. Ali Sahraoui, N. Bouarissa, Eur. Phys. J. B 57, 9 (2007) 12. V. Ozoli¸nsˆ, A. Zunger, Phys. Rev. Lett. 82, 767 (1999) 13. A. Mujica, A. Rubio, A. Muñoz, R.J. Needs, Rev. Mod. Phys. 75, 863 (2003), and references therein 14. C. Stampfl, W. Mannstadt, R. Asahi, A.J. Freeman, Phys. Rev. B 63, 155106 (2001) 15. C. Stampfl, R. Asahi, A.J. Freeman, Phys. Rev. B 65, 161204 (2002) 16. S. Ves, U. Schwarz, N.E. Christensen, K. Syassen, M. Cardona, Phys. Rev. B 42, 9113 (1990), and references therein 17. K.J. Chang, S. Froyen, M.L. Cohen, Solid State Commun. 50, 105 (1984) 18. F. Benmakhlouf, A. Bechiri, N. Bouarissa, Solid State Electron. 47, 1335 (2003) 19. N. Bouarissa, Eur. Phys. J. B 26, 153 (2002) 20. W. Nakwaski, Physica B 210, 1 (1995), and references therein 21. N. Bouarissa, J. Phys. Chem. Solids 67, 1440 (2006) 22. A. Bhouri, F. Ben Zid, H. Mejri, A. Ben Fredj, N. Bouarissa, M. Said, J. Phys. Condens. Matter 14, 7017 (2002)
