Elastic and structural instability of cubic Sn3N4 and C3N4 under pressure
Gopal K. Pradhan, Anil Kumar, Umesh V. Waghmare, Sudip K. Deb, Chandrabhas Narayana
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a r l Elastic and structural instability of cubic Sn N and C N under pressure Gopal K. Pradhan, Anil Kumar, Umesh V. Waghmare and Chandrabhas Narayana ∗ Light Scattering Laboratory, Chemistry & Physics of Materials Unit, Theoretical Sciences Unit,Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR), Bangalore 560064, India
Sudip K. Deb
Indus Synchrotron Utilization Division, Centre for Advanced Technology, Indore 452013, India
We use in-situ high pressure angle dispersive x-ray diffraction measurements to determine theequation of state of cubic tin nitride ( γ -Sn N ) under pressure up to about 26 GPa. While we findno evidence for any structural phase transition, our estimate of the bulk modulus ( B ) is 145 ( ± N ata higher pressure of 88 GPa compared to earlier predictions of 40 GPa. Our comparative analysis ofcubic nitrides shows that bulk modulus of cubic C N is the highest (379 GPa) while it is structurallyunstable and should not exist at ambient conditions. PACS numbers: 61.05.C-, 62.50.-p, 62.20.-x, 63.20.D-
Since the discovery of cubic spinel ( γ ) phases in GroupIV nitrides[1–3], they have received a renewed interest inthe past few years due to prediction of superhardness[4]in these class of materials. It is also expected that thesespinel classes of nitrides will exhibit interesting electronicproperties, such as varying electronic band gap energiessuitable for optoelectronic applications. So far, the bi-nary nitrides of tin[1], germanium[2] and silicon[3] havebeen synthesized and shown to adopt the spinel structureunder different experimental conditions. These new poly-morphs have the dense spinel structure with Si/Ge/Snatoms in octahedral as well as tetrahedral coordinationwith nitrogen and the coordination of nitrogen is four.The crystal structure of γ - A N ( γ - A N , A = Si, Ge andSn) (space group F d ¯3 m , u, u, u )positions with A atoms occupying 1/8 of the tetrahe-dral interstitial sites and 1/2 of the octahedral sites[1–3].The anion arrangement produces a rigid-vertex linkageof A -filled regular tetrahedra ( A t N ) and distorted oc-tahedral ( A o N ) sharing half of their edges. The cationsublattice can be described as an array of empty corner-sharing tetrahedra and A -filled truncated tetrahedral.The bulk modulus of the spinel phase is expected to behigh, primarily determined by the nitrogen sublattice,but it should be influenced by the choice of cations. Avery high bulk modulus of 369 GPa[5] has been predictedfor the cubic phase of C N , without any success in syn-thesizing this material. Though both Si N and Ge N have similar bulk moduli, the same for Sn N has beenpredicted to be significantly different. Also, from recentnanoindentation studies[6], tin nitride spinel was foundto be significantly softer and more plastic than its lightercongeners (Si N and Ge N ). Thus, the exact role of thecation in determining the stability and elastic properties is not yet completely understood.Tin nitrides are of special interest due to their promis-ing semiconducting and electrochromic properties[7, 8].The feasibility of utilizing tin nitride thin film as write-once, optical recording media has also been reported[9].Therefore, the precise knowledge of the stability and me-chanical properties of γ -Sn N would be of great interestfor several practical applications. Though the ambientbulk modulus ( B ) of γ -Sn N and γ -Ge N has beenintensively investigated[10–12], there have been no ex-perimental studies to determine the compressibility of tinnitride. With lack of experimental data, first-principlescalculations have been used to investigate the proper-ties of Sn N [6, 13–16]. However, the predicted values ofthe bulk modulus appear to vary between 186 and 218GPa. In this work, we have performed high pressure x-ray diffraction measurements on γ -Sn N to determinethe bulk modulus from the volume as a function of com-pression. We have also carried out first-principles den-sity functional theory (DFT) calculations in the gener-alised gradient approximation (GGA) to achieve a consis-tent understanding of our experimental findings. First-principles calculations have been carried out for othergroup-IV nitrides as well to (a) determine the stabilityof the spinel phase with respect to pressure and possiblephase transitions, and (b) understand the influence ofcation on elastic behaviour. Based on our calculations,we also discuss the stability of cubic phase of C N atambient conditions.Spinel tin nitride (Sn N ) was synthesised by high-pressure solid-state metathesis reactions where tintetraiodide was reacted with lithium nitride and ammo-nium chloride in a piston-cylinder apparatus at 623 Kand 2.5 GPa. Details of the synthesis procedure canbe found elsewhere[17]. In-situ high-pressure angle dis-persive x-ray powder diffraction measurements (up to26 GPa) were performed using the monochromatic syn-chrotron radiation ( λ = 0.68881 ˚A) at XRD1 beam-line at ELETTRA synchrotron source, Italy. The two-dimensional x-ray diffraction (XRD) patterns were col-lected on a MAR345 imaging plate. The sample-to-detector (image plate) distance was calibrated by col-lecting the diffraction pattern of powdered silicon (Si)at ambient conditions. The one-dimensional diffractionpatterns were obtained by integrating along the Debye-Scherrer ring in the two dimensional image patterns usingthe FIT2D software[18]. Polycrystalline Sn N powderwas placed together with gold into the 200 µ m hole ofa stainless steel gasket (T301), preindented to 60 µ m,inserted between the diamonds of a Mao-Bell-type di-amond anvil cell (DAC). Gold was used as an inter-nal pressure calibrant and the pressure in the DAC wasdetermined using the known equation of state of gold[19]. Methanol-ethanol-water (MEW) mixture (16:3:1)was used as pressure transmitting medium. To developa microscopic understanding of the discrepancy betweenthe calculated bulk modulus earlier[6, 13–16] and our ex-perimental estimate, we present new first-principles cal-culations based on DFT with a generalized gradient ap-proximation (GGA)[20] for the exchange correlation en-ergy, as implemented in PWSCF[21] package. We useultrasoft pseudopotentials[22] to describe the interactionbetween the ionic core and valence electrons, and a planewave basis with energy cutoffs of 30 Ry and 240 Ry inrepresentation of wave functions and charge density re-spectively. Integrations over Brillouin zone are sampledwith an 8 × × × × γ -Sn N at var-ious pressures. The cell parameters were obtained byanalysing the XRD profiles by Le-Bail profile fit using theGSAS software[24]. Ambient lattice parameter has beenfound to be a = 9.0205(5) ˚A, which is in close agreementto earlier reported value[17]. Upon increasing the pres-sure, we didn’t observe any new diffraction peaks (seeFig. 1) or sudden discontinuity in the pressure depen-dency of the d values up to the maximum achieved pres-sure of 26 GPa. This indicates that γ -Sn N doesn’tundergo any transition and remains in the cubic crystalstructure up to the highest pressure. Figure 2 shows pres-sure volume compression data. Using the second orderBirch-Murnaghan ( B ′ = 4) equation of state (EOS)[25]to fit the volume compression data, we found the bulkmodulus ( B ) to be 145 ( ±
10 15 20 25 30
Ambient A u A u A u I n t e n s i t y ( a r b . un i t) * FIG. 1. Pressure evolution of X-ray diffraction patterns of γ -Sn N compressed quasi- hydrostatically in MEW up to 26GPa. Diffraction lines due to gold (Au) (used for in- situpressure calibration) and stainless steel gasket (*) are alsoidentified. R e l a t i v e V o l u m e ( V / V ) Pressure (GPa) = 145 ( 1.7) GPa FIG. 2. Relative volume change in γ -Sn N as a functionof pressure at ambient temperature (Solid line is a secondorder Birch-Murnaghan fit to the experimental data (opensymbols)). revisit the theoretical calculations.Our theoretical estimate of the lattice constant of γ -Sn N is 9.136 ˚A, within typical GGA errors of the exper-imental value of 9.0205 ˚A[17]. Bulk modulus estimatedfrom first-principles energies as a function of volume is158 GPa, in reasonable agreement with our measuredvalue of 145 GPa reported here. We note that the earliertheoretical estimate[16] of the bulk modulus is within lo-cal density approximation (LDA) (contrary to the claimof use of GGA in this paper; we have reproduced theirresult with the same code and LDA choice of energy func-tional), and is overestimated to be 187 GPa. Thus, theobserved softness of Sn N reflected in the lower value of B here should be reliable. With the same choice of ex-change correlation energy functional, we systematicallystudied properties of other group IV nitrides in the spinelstructure, and find that the bulk modulus (see Table I)reduces from 379 GPa of carbon nitride to 158 GPa ofSn N . While the lattice constant increases from car-bon to tin by almost 30%, consistent with the size of thecation, the internal structural parameter ( u ) changes byonly 2%. TABLE I. Experimental and calculated equilibrium latticeconstant ( a ) and bulk modulus ( B ) of the γ -phase of differentgroup-IV nitrides.Calc. a Expt. a Calc. B Expt.
B u (˚A) (˚A) (GPa) (GPa) γ -Sn N ± γ -Si N ±
5) 0.2575 γ -C N The local stability of a structure can be confirmedthrough determination of the full phonon dispersion, andby verifying that there are no unstable modes in thestructure. Our results (see Fig. 3) clearly show thatC N in the spinel structure is unstable with respect toboth shear acoustic and optical modes, and should notform. This is consistent with the fact that there are noexperimental reports so far on C N in the spinel form.We also find that the frequency range (band-width) ofacoustic branches reduces from silicon to tin nitrides,confirming their increasing elastic softness. If K is thespring constant or the stiffness of a bond and a is thelattice constant, a simple analysis shows that the bulkor elastic modulus should scale as K/a . Although theslopes of the acoustic modes in these nitrides seem dif-ferent in the phonon dispersion (see Fig. 3), variation of B with 1/ a is almost linear. This means that the bond-stiffness in the nitrides studies here is quite similar, andit is the volume of the unit cell that is responsible largelyfor the monotonous decrease in the bulk modulus fromcarbon to tin nitride. As the spinel structure containscations with tetrahedral and octahedral coordination, itis not straightforward to describe it with a single type ofa bond. Since all anions are symmetry equivalent, ourreference to bond stiffness here is reasonable in the senseof average bond stiffness.Our estimates of the Born effective charges also beara similar feature (see Table II), with two types of effec-tive charges of a cation and degenerate effective chargesof N. While the effective charges of Si N and Sn N aresimilar, those of C N are singularly different. We findthat the effective charge of carbon is most anomalous (i.e.deviates most from its nominal charge of Z*=4); this isnot very surprising, as the anomalous charges (thoughlarger than the nominal values) are known to correlate ω ( c m - ) Γ K X Γ L Sn N ω ( c m - ) Γ K X Γ L Si N -200-1000100200300400500600 ω ( c m - ) Γ K X Γ L C N FIG. 3. Phonon dispersion of a) γ -Sn N , b) γ -Si N and c) γ -C N at equilibrium lattice constant. with structural instabilities (the presence of unstable vi-brational modes) in the system[26]. We also note thatthe effective charges of carbon atoms with 4-fold and 6-fold coordination are rather different, in contrast to rel-atively similar charges of the two symmetry inequivalentcations in Si and Sn nitrides. We find that the bandgap steadily increases from 0.2 eV in Sn N to 3.4 eV inSi N , but reduces substantially to 1.02 eV in C N , con-firming rather different nature of the electronic structureand bonding in C N from the rest. Our analysis basedon projection of energy eigen-functions on the atomic or-bitals shows that carbon in spinel-C N is close to beingsp -hybridized with covalent bonding with N, while the TABLE II. Calculated Born effective charges of the octahe-dral, tetrahedral coordinated cations and N atoms and elec-tronic dielectric constant in the γ -phase of different nitrides.Z* of Z* of Z* of N Dielectrictetrahedral cation octahedral cation constant γ -Sn N γ -Si N γ -C N -1000100200300 ω ( c m - ) P=072 GPaP=081GPaP=088 GPaP=106 GPaP=150 GPa Γ K X Γ L FIG. 4. Phonon dispersion of γ -Sn N as a function of pres-sure cations in Si N and Sn N appear more ionic. In sum-mary, the unexpectedly low measured bulk modulus ofSn N does follow the trend in properties of the groupIV-nitrides and seems to be readily understandable.Our experiments show that Sn N remains stable inthe spinel structure up to 26 GPa. We now use first-principles simulations to explore its stability at muchhigher pressures. Since it is difficult to explore all pos-sible structures as a function of pressure, we determinecomplete phonon dispersion of cubic Sn N as a func-tion of pressure. This is an efficient method to find un-stable modes and detect any structural transition thatthe spinel structure would undergo with pressure, and issimilar in spirit to approach used commonly in ferroelec-tric structural transitions[27]. Our results (see Fig. 4)show that the Sn N remains stable locally in the spinelstructure up to pressures of 88 GPa, and develops an in-stability in the transverse acoustic branch along < > direction. Thus, Sn N in the spinel structure will distortwith a shear-strain above 88 GPa suggesting a transitionfrom cubic to low symmetry phase. At higher pressures,we find that many more unstable modes develop that in-volve optical phonons as well, suggesting that the struc-tures beyond 88 GPa are expected to be more complex.Electronic structure of Sn N is also rather sensitive topressure, but exhibits an unusual trend: its band gap in-creases rather sharply from 0.2 eV at 0 pressure to about2 eV at a pressure of 50 GPa. This is indeed consistentwith the trend in band gaps from Sn N to Si N , wherethe pressure has chemical origin.In summary, in-situ angle dispersive high pressure x-ray studies reveal that bulk modulus of γ -Sn N is 145( ± γ -Sn N , compared to lighter members (Si andGe) in the family. We predict a structural phase transi-tion in γ -Sn N at 88 GPa based on phonon dispersioncalculation as a function of pressure. 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