First-principles study of vibrational and dielectric properties of Si3N4

First-principles study of vibrational and dielectric properties of␤-Si3N4Yongqing Cai,Litong Zhang,Qingfeng Zeng,Laifei Cheng,and Yongdong XuNational Key Laboratory of Thermostructure Composite Materials,Northwestern Polytechnical University,Xi’an,Shannxi710072,People’s Republic of China͑Received11August2006;published17November2006͒First-principles calculations have been conducted to study the structural,vibrational,and dielectric proper-ties of␤-Si3N4.Calculations of the zone-center optical-mode frequencies͑including longitudinal-optical/transverse-optical splittings͒,Born effective charge tensors for each atom,and dielectric constants,usingdensity functional perturbation theory,are reported.The fully relaxed structural parameters are found to be ingood agreement with experimental data.All optic modes are identified and agreement of theory with experi-ment is excellent.The static dielectric tensor is decomposed into contributions arising from individual infrared-active phonon modes.It is found that high-frequency modes mainly contribute to the lattice dielectric constant.DOI:10.1103/PhysRevB.74.174301PACS number͑s͒:63.20.Ϫe,77.22.Ϫd,77.84.BwSilicon nitride͑Si3N4͒is an industrially important mate-rial due to its chemical,mechanical,and electronic proper-ties.Excellent thermomechanical properties have seen this material used for engine parts,bearings,metal machining, and other industrial applications.1–3Also,Si3N4is considered to be a promising material for microelectronic applications because of its high dielectric constant and large electronic gap.4Over the past three decades,the lattice dynamical proper-ties of Si3N4have been investigated experimentally as well as theoretically by a number of groups.Brillouin zone center ͑BZC͒phonon frequencies of Si3N4have been measured by using Raman and infrared spectroscopies.5,6However,iden-tification of lattice modes in experimental studies is some-times incomplete,partly because of the complexity of the structure and partly because of experimental errors.Earlier theoretical studies of dynamical properties of Si3N4were done by adopting classical methods.7,8Wendel9calculated the zone center frequencies through a forcefield model de-rived using the Hessian biased technique from ab initio cal-culations.Although good agreement with experimental re-sults for crystal structures,lattice expansion parameters,and the thermodynamical properties was obtained,the accuracy of frequency calculation was limited due to the types of in-teractions they took into account might not be enough.Re-cently,first principles calculations of Raman active modes of ␤-Si3N4have been reported.10The calculation was done by using a direct method where the phonon frequencies are cal-culated from Hellmann-Feynman forces generated by the small atomic displacement.Thefirst-principles study of phonon frequencies can be performed within a density functional perturbation theory ͑DFPT͒͑Ref.11͒or direct method approach.12The advan-tage of the direct method is computationally straightforward and allows one to study phonons in the whole Brillouin zone by perturbing the positions of the atoms slightly and calcu-lating the reaction forces.However,one has to conduct a separate calculation using Berry’s phase approach to obtain the longitudinal-optical/transverse-optical͑LO-TO͒splitting of zone center optical modes.In contrast to the direct method of calculating the phonon frequencies,the linear response approach allows one to obtain the effective charges and di-electric tensors directly with no need to artificially increase the cell size in order to accommodate small values of the q vectors as in the direct method approach.The purpose of this paper is to obtain in more detail the dynamical and dielectric properties of␤-Si3N4using the density functional perturbation theory.Our calculations start with the structural optimization and check that our relaxed structural parameters are consistent with the experimental work.The phonon frequencies and Born effective charge ten-sors are then computed from linear response techniques.Ex-cellent agreement is found between the calculated frequen-cies and the measured spectra for both IR-active and Raman-active modes.The“missing”A g mode in Ref.6is identified. Finally,the dielectric properties and longitudinal-transverse ͑LO-TO͒mode splittings are investigated,and our theoretical information is combined to predict the lattice contributions to the bulk dielectric tensor.We thus clarify the dependence of the dielectric response on orientation,and lattice dynami-cal properties andfind that high-frequency modes mainly contribute to the lattice dielectric constant.The calculations are carried using the CASTEP͑Refs.13 and14͒code with norm-conserving pseudopotentials.15The 2s and2p semicore shells are included in the valence band for N,and the3s and3p are included in the valence band for Si.We employ the Ceperley-Alder16local density functional potential as parametrized by V osko et al.17The kinetic en-ergy cutoff for the plane waves is770eV,Brillouin zone integration is performed using a discrete4ϫ4ϫ10 Monkhorst-Pack18k-point sampling for a primitive cell.The first-principles investigation of vibrational and dielectric properties is conducted within a density functional perturba-tion theory͑DFPT͒.11Phonon frequencies at the zone center, Born effective charge tensors,and dielectric tensor are com-puted as second-order derivatives of the total energy with respect to an external electricfield or to atomic displace-ments.Technical details can be found in Refs.19and20.To obtain LO/TO splitting characteristic for␤-Si3N4we intro-duced to the dynamical matrix the nonanalytical term pro-posed by Pick et al.21␤-Si3N4has a space group P63/m.The hexagonal unit cell contains two formula units͑14atoms͒.Figure1shows a ball-stick model of␤-Si3N4.The silicon nitride structure can be described as a stacking of the idealized Si-N layers in anPHYSICAL REVIEW B74,174301͑2006͒ABAB sequence.All Si atoms are equivalent ͑6h sites ͒,but there are two inequivalent nitrogen sites:N 2c at 2c sites and N 6h at 6h sites.The N 2c atoms are in a planar geometry with their three Si nearest neighbors,and the N 6h atoms are in slightly puckered sites surrounded by three Si atoms,whereas Si atoms are at the center of slightly irregular tetra-hedron bonded with one N 2c atom and three N 6h atoms.The structural optimization was done by relaxing both the internal coordinates and the lattice constants by calculating the ab initio forces on the ions,within the Born-Oppenheimer approximation,until the absolute values of the forces were converged to less than 0.01eV/Å.The relaxed lattice and fractional internal parameters of the equilibrium structure are listed in Table I and the initial configuration is taken from Ref.23.It can readily be seen that there is excel-lent agreement between our results and previous experiment.The volumes are slightly underestimated,by 2-3%,as is typi-cal of local density approximation ͑LDA ͒calculations.Itshould be noted that the accuracy in comparison with experi-mental lattice constants is much lower for two reasons:͑a ͒DFT in the LDA is an approximation to the exact n -electron problem.͑b ͒Temperature effects are neglected in the calcu-lations;the calculations are performed for a fictitious classi-cal system at zero temperature.Since the primitive unit cell of ␤-Si 3N 4structure has 14atoms,there are a total of 42modes of vibration.The irre-ducible representation at the center of the Brillouin zone is ⌫optic =4A g ͑R ͒+2E 1g ͑R ͒+5E 2g ͑R ͒+2A u ͑IR ͒+4E 1u ͑IR ͒+3B g +2E 2u ,⌫acoustic =A u +E 1u .Because of inversion symmetry,IR and Raman modes are mutually exclusive.In Table II ,we compare calculated phonon frequencies at the ⌫point with the measured Raman 5,6and infrared 5values.For comparison,we cite also the phonon frequencies from theoretical calculations through direct methods 10and classi-cal model.9Our calculated Raman active mode frequencies present a rms absolute deviation of 8.6cm −1,and a rms rela-tive deviation of 1.9%with respect to the measurements of Ref.6.This is an excellent agreement with both experiments and theory,and makes us very confident in the prediction of the frequencies for the IR active modes.It should benotedFIG.1.͑Color online ͒A ball-stick model of ␤-Si 3N 4.The green ͑gray ͒,red ͑dark gray ͒,and blue ͑light gray ͒spheres represent Si atoms at 6h sites,N atoms at 2c sites,N atoms at 6h sites,respec-tively.N atoms are within the same plane of their three nearest neighbors,whereas Si atoms lie at the center of a slightly irregular tetrahedron.TABLE I.Optimized geometry of ␤-Si 3N 4͑P 63/m ͒Parameter Experiment a Experiment b This work V ͑Å3͒147.016145.881142.700c ͑Å͒7.62727.60707.5572a ͑Å͒ 2.9182 2.9110 2.8852x Si ͑6h ͒0.18420.17400.1742y Si ͑6h ͒0.77040.76600.7678z Si ͑6h ͒0.25000.25000.2500x N ͑2c ͒0.33330.33330.3333y N ͑2c ͒0.66670.66670.6667z N ͑2c ͒0.25000.25000.2500x N ͑6h ͒0.33990.32100.3302y N ͑6h ͒0.03570.02500.0299z N ͑6h ͒0.25000.25000.2500Reference 22.bReference 23.TABLE II.Zone center optic phonon frequencies of ␤-Si 3N 4͑cm −1͒Mode Experiment a ͑144A g ͒Experiment b ͑145A g ͒Theory c Theory d This work Raman E 2g 186185183190181A g 210208201236200E 1g 229230228215225E 2g 451452444518444A g ¯¯457289456E 2g 619620603592610A g 732733715539725E 1g 865866836975859E 1g 9289308971017921A g 939940908547930E 2g10471048101210671035Infrared A u 380378E 2u 447424E 2u 580562A u 910848E 2u 985886E 2u10401021a Reference 5.bReference 6.cReference 10.d Reference 9.CAI et al.PHYSICAL REVIEW B 74,174301͑2006͒that the values of the vibrational properties of ␤-Si 3N 4through the empirical model proposed by Wendel are just qualitative.Because of weak intensity and closeness to a relatively stronger E 2g mode,the A g mode in the intermediate frequency band was misidentified 5,6and is now known to be 457cm −1through theoretical calculation.10Our calculation shows that this mode is at 456cm −1,which is in very good agreement with previous theoretical calculations.The IR modes group into modes with displacements either in the x ,y plane or along the z direction.The E 1u mode has displacement pattern in the x ,y plane,and the A u mode has displacements along z .For IR active mode,the root mean square relative deviation of our results with experimental data is 4.9%,with a slight overall tendency to underestima-tion.For some of the modes,there are large differences with the reported experimental frequencies.5For example,our cal-culated frequency for the E 2u mode is 886cm −1,whereas the experimental value is 985cm −1;the origin of these discrep-ancies is unclear.The LO/TO splittings for the infrared ac-tive mode are presented in Table III .The LO/TO splittings for displacements parallel to the layers occur for the E 2u mode and displacements parallel to c principal axis occur for the A u mode.The large LO/TO splittings of the three highest modes suggest they involve large effective charges and make large contributions to the static dielectric tensor of ␤-Si 3N 4.The Born effective charge tensor quantifies the macro-scopic electric response of a crystal to the internal displace-ments of its atoms.Our results for the dynamical effective charges of ␤-Si 3N 4are presented in Table IV for atoms at the 6h Wyckoff sites in the z =0.75plane and two nitrogen atomsat 2c sites.The effective charges for the other atoms can beobtained from those shown in the table by symmetry consid-erations.It is obvious that both Si and N atoms have lower effec-tive charges than their formal charges,+4and −3for Si and N respectively,relating to the covalency of Si-N bonds.The Si atoms tend to show a more isotropic character than N atoms.As discussed above,the nitrogen atoms at the 2c sites are bonded to three nearest-neighbor silicon atoms in a pla-nar configuration.One might then expect that the largest dy-namical charge transfer would occur for motions of the N atoms in this plane.To check this,we computed the eigen-values in the second column of Table IV .Sure enough,the principal axis associated with the eigenvalue Z 3*=−1.76of the smallest magnitude points normal to the plane of the neighbors.The other two principle axes lie in the plane of the neighbors.Not surprisingly in view of its tetrahedral coordi-nation,the Z *tensor for atom Si is more isotropic,as indi-cated by the smaller spread of the eigenvalues in the table.Crystal symmetry makes the dielectric tensors be com-posed of some independent components.In hexagonal sym-metry such as ␤-Si 3N 4,the calculated electronic ͑␧ϱ͒and static ͑␧0͒dielectric tensors are diagonal and have two inde-pendent components ␧Ќand ␧ʈperpendicular to and along the c axis,respectively.The electronic dielectric constantsare calculated to be ␧Ќϱ=4.19and ␧ʈϱ=4.26.The reportedvalue of ␧ϱis 4.0͑Ref.24͒,and our theoretical values tend to be overestimated in local density approximation ͑LDA ͒cal-culations due to the underestimation of the band gap.25According to the generalized Lyddane-Sachs-Teller ͑LST ͒relation,26␧0=␧ϱ͟m␻LO ,m 2␻TO ,m2,͑1͒which is used separately for each polarization;the static di-electric constants ␧0of both directions are ␧Ќ0=8.79and ␧ʈ0=7.21respectively,showing nearly isotropic character.The dielectric properties of crystalline Si 3N 4have not been stud-ied very much experimentally.Experimental reports of the value of ␧0for polycrystalline Si 3N 4span a wide range from about 8.1to 8.6͑Ref.27͒,our orientationally average static dielectric constant is 8.26,falling comfortably in this range.The static dielectric tensor ␧0can also be separated into contributions arising from purely electronic screening ␧ϱand IR-active phonon modes according to 20␧0=␧ϱ+4␲V 0͚m S m ␻TO ,m 2=␧ϱ+␧ϱ͚m ␻LO ,m 2−␻TO ,m2␻TO ,m2,͑2͒where V 0denotes the volume of the primitive unit cell.The derivation of Eq.͑2͒follows from the relation 20between the mode oscillator strength S m and the corresponding LO-TO splitting,whereas Eq.͑1͒is obtained based on the adiabatic,electrostatic,and harmonic approximation.The contributions to the static dielectric tensor can be examined mode by mode through Eq.͑2͒.The differences between Eq.͑1͒and Eq.͑2͒are small,however.The static dielectric constants of bothdirections from Eq.͑2͒are ␧Ќ0=7.67and ␧ʈ0=6.99,comparedto that of ␧Ќ0=8.79and ␧ʈ0=7.21from Eq.͑1͒.For IR-activeTABLE III.LO/TO splittings for infrared active modes of ␤-Si 3N 4.The first column ͑E =0͒is for no electric field,the second is for the field lying in the plane ͑E ʈa −b ͒,and the third is for E ʈc .⌬␻2=␻LO 2−␻TO 2.Mode Phonon frequency ͑cm −1͒ͱ⌬␻2͑cm −1͒E =0E ʈa −b E ʈc A u 378378395115E 2u 424453424159E 2u 562584562159A u 8488481056629E 2u 8861026886517E 2u102111501021529TABLEIV .The Born effective charges of ␤-Si 3N 4.Z j *͑j =1,2,3͒is the j th eigenvalue of the symmetric part of the Z *tensor.Z ¯is the average of eigenvalues.Only atoms at the 6h Wy-ckoff sites in the z =0.75plane are presented.N 12cN 22cN 16hN 26hN 36hSi 16hSi 26hSi 36hZ 1*−2.95−2.95−1.61−3.15−1.61 3.15 3.58 3.15Z 2*−2.95−2.95−3.15−1.61−3.15 3.58 3.15 3.58Z 3*−1.76−1.76−2.82−2.82−2.82 3.41 3.41 3.41Z¯−2.55−2.55−2.53−2.53−2.53 3.38 3.38 3.38FIRST-PRINCIPLES STUDY OF VIBRATIONAL AND DIELECTRIC PROPERTIES OF ␤-Si 3N 4PHYSICAL REVIEW B 74,174301͑2006͒modes,the mode frequencies and contributions to the dielec-tric tensor are displayed in Table V .It can be seen that the high-frequency modes mainly contribute to the lattice dielec-tric constant.Although the mode contribution to the static dielectric constant is inversely proportional to the square of the mode frequency ͓Eq.͑2͔͒,low frequency modes A u at 378cm −1,E 2u at 424and 562cm −1make relatively small contributions to the static dielectric tensor due to their smaller LO/TO splittings compared to higher frequency modes.When all the modes are summed over,we obtain thetotal lattice dielectric components of ␤-Si 3N 4are ␧Ќlat=3.48and ␧ʈlat=2.73compared to electronic contributions of␧Ќϱ=4.19and ␧ʈϱ=4.26.We can conclude that ␤-Si 3N 4has a lattice dielectric constant smaller than that of the electronic contribution.In conclusion,we have used density functional perturba-tion theory ͑DFPT ͒to investigate the vibrational and dielec-tric properties of ␤-Si 3N 4.Firstly,the structural parameters,including the internal coordinates,are relaxed,and excellent agreement is achieved with experimental results.The vibra-tional modes at the center of the Brillouin zone have been evaluated and compared with experiment.All modes are identified and compared with experiments and previous the-oretical calculations.The calculated zone-center phonon mode frequencies are in good agreement with infrared and Raman experiments.Finally,The Born effective charge ten-sors,and the dielectric permittivity tensors have also been calculated.The mode contributions to the dielectric tensors have been obtained.We find that ␤-Si 3N 4has a lattice dielec-tric constant smaller than that of electronic contribution.The authors acknowledge support from the Natural Sci-ence Foundation of China ͑Contract No.90405015͒,from the National Young Elitists Foundation ͑Contract 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Eng.Mater.287,247͑2005͒.TABLE V .IR active mode frequency,contribution to the com-ponent of dielectric tensor for each IR-active mode.The component of lattice dielectric constant parallel ͑perpendicular ͒to the c axis isdenoted ␧ʈlat ͑␧Ќlat͒.Mode ␻TO ͑cm −1͒␻LO ͑cm −1͒ͱ⌬␻2͑cm −1͒␧Ќlat ␧ʈlat A u 37839511500.39E 2u 4244531590.590E 2u 5625841590.340A u 84810566290 2.34E 2u 8861026517 1.430E 2u102111505291.120CAI et al.PHYSICAL REVIEW B 74,174301͑2006͒。