arXiv:1212.6507v1 [cond-mat.mtrl-sci] 28 Dec 2012 entesbeto aytertcladexperimental and theoretical many of subject have the vari- applications and through been properties produced their and been techniques have ous nitrides copper 1939, riso hsfml fmtrashv perdi the in appeared have materials of family literature. this of mechanical erties quantum of number initio significant a Hence, Discussion and Results III. oavreyo oeta ehooia applications technological potential leading of properties variety interesting a to possess usually (TMNs) V Conclusions IV. I aclto Methods Calculation II. .Introduction I. ic uaadHahn and Juza Since ti elkonnwta aetasto-ea nitrides transition-metal late that now well-known is It an electronic structural, the on calculations Theoretical .FrainEnergy Formation D. References Acknowledgments Energies Formation D. .ESadRltv Stabilities Relative and EOS A. Structures Crystal and Stoichiometries A. .Bl ouu n t rsueDerivative Pressure its and Modulus Bulk C. EOS and Relaxation Geometry C. .Vlm e tmadLtieParameters Lattice and Atom per Volume B. Details Relaxation Electronic B. .Eetoi Properties Electronic E. Properties Optical and Calculations GWA E. .OtclProperties Optical F. acltoso h tutrladpyia prop- physical and structural the of calculations rsaln Ag crystalline r eecluae.Terslsotie eecmae wit compared were obtained results sel the The calculations. to approximation performed calculated. GW the were via tra single-p obtained carefully the were been tensor, holes have dielectric quasi phases the stable th to most (RPA) the of approximation of derivative diagram (DOS) Band pressure states studied. its of been have and structures energy-optimiz modulus different (EOS), twenty bulk state and of equation energies, The theory. bation epeetafis-rnilsivsiaino structural, of investigation first-principles a present We .INTRODUCTION I. 2 eateto hsc,SdnUiest fSineadTech and Science of University Sudan Physics, of Department 1 colo hsc,Uiest fteWtaesad Johann Witwatersrand, the of University Physics, of School CONTENTS 4 ucee osnhsz Cu synthesize to succeeded 3 ,ANadAgN and AgN N, oamdS .Suleiman H. S. Mohammed 2 ae ndniyfntoa hoy(F)admn-oypert many-body and (DFT) theory functional density on based Dtd aur ,2013) 1, January (Dated: ivrnitrides silver 3 in N 1 – 13 12 12 12 ab 3 2 1 9 8 8 7 4 4 3 3 3 2 2 . ,2, 1, ∗ aeb oeresearchers. some by made rd a o ecniee stems investigated most the as TMNs considered late the be among now may tride Ag oxide ver ulse works published ie gaosi h ntcl) ogtm ae in later time long A cell). unit the in Haisa atoms 1982, Ag 4 (i.e. e a omAg form can ver the in positions definite distributed the no statistically given while were be cell, interstices, unit may octahedral the which of centers atoms, face N and corners the at Ag stoichiometry, reported the on study tural Ex- family. structural TMNs investigate least late to the efforts the perimental be in may solid nitride studied silver theoretically discovery, earlier its spite oe sdet h nryrlae uigtedecompo- the during released energy reaction: ethanol the sition to or due water is under power handle to easy formation and mato aln ae droplet the water from even falling touch, a slightest the of to impact due explode may It supporting metal, a color is black compound its this that argued be also mn ocpe ngop1 fteproi al,has table, periodic the centuries of two than 11 more group for known in been copper to ement lie nfcsrcuewith structure fcc an claimed h aetA n h aiysprtdNa N as N separated easily the and Ag parent to the close gap band fundamental insulat- 1 a recent revealed with structure and characteristics proposed ing compound, the ionic on calculations an as described vni trg tro eprtr,ti oi com- solid Eq. this to temperature, according slowly room decomposes at pound storage in Even . 35 n ailP Joubert P. Daniel and n14,Hh n Gilbert and Hahn 1949, In codn otecluae ais Ag radius, N calculated the to According nteohrhn,tentieo ivr h etel- next the silver, of nitride the hand, other the On ne riayconditions ordinary Under 3 ,i neteeysniieepoiecompound explosive sensitive extremely an is N, eV ril pcr fteqaieetosand electrons quasi the of spectra article lcrncadotclpoete fbulk of properties optical and electronic nteohrhn,det h iia atc of lattice similar the to due hand, other the On . n oa n ria-eovddensity orbital-resolved and total and s dgoere,chsv n formation and cohesive geometries, ed 7 xmnd ihnterandom-phase the Within examined. s he tihoere nastof set a in stoichiometries three ese , 8 -nryoeao,adotclspec- optical and operator, f-energy xeietadwt previously with and experiment h h lc ealcloigsldoutcome, solid metallic-looking black The . 12 7 8 ugse htteA tm r located are atoms Ag the that suggested , . 9 ooy hrom Sudan. Khartoum, nology, 5 – sug ot Africa. South esburg, 3 2Ag pia rpriso bulk of properties optical d u oiserydsoey oprni- copper discovery, early its to Due . 11 N 13 rpriso ivrntie aebeen have nitrides silver of properties 3 N 6 rmamnaa ouin fsil- of solutions ammoniacal from . −→ 1, † 9 A N + 6Ag 7 are u h first the out carried a twsfudta sil- that found was it , 4 = 14 u ti relatively is it but , . 369 2 . 8 7 h explosive The . , and A ˚ 8 ur- oee,de- However, . 8 , 9 electronic , 1 3 above a be can N 3 binitio ab .They N. 2 Z tcan it , 8 4 = struc- 7 8 , , (1) 14 14 12 / 3 8 . . . 2

From a thermochemical point of view, it was found that the face-centered cubic structure of NaCl (B1), the sim- there is no stable intermediate stage in this decomposi- ple cubic structure of CsCl (B2), the face-centered cubic tion, but there may be a metastable intermediate species structure of ZnS zincblende (B3), the hexagonal structure 7 (phase) with a remarkably low decomposition rate . At of NiAs (B81), the hexagonal structure of BN (Bk), the this point, it may be worth mentioning that the ther- hexagonal structure of WC (Bh), the hexagonal structure mochemistry of silver nitride systems is not fully docu- of ZnS wurtzite (B4), the simple tetragonal structure of mented in standard handbook data7. PtS cooperite (B17), and the face-centered orthorhombic In their 1991 work, Shanley and Ennis7 stated: “Many structure of TlF (B24). of the samples ... did not survive the minimum han- AgN2 was studied in the following nine structures: the dling required to move them, container and all, to the face-centered cubic structure of CaF2 fluorite (C1), the X-ray stage. ... More vigorously explosive samples prop- simple cubic structure of FeS2 pyrite (C2), the simple agated throughout their mass leaving no visible residue. orthorhombic structure of FeS2 marcasite (C18) and the Even among supersensitive materials, silver nitride is a simple monoclinc structure of CoSb2 (CoSb2). striking example of a compound “teetering on the edge of existence”. Under the circumstances, we did not suc- ceed in developing data on the proportion of silver nitride required for explosive behavior in these mixtures.” Thus, beside the potential hazard to lab workers due to B. Electronic Relaxation Details its sensitive explosive behavior, characterization of silver nitride is hindered by its extremely unstable (endother- 7,8 In this work, electronic structure spin density func- mic) nature , and we are presented with an incomplete tional theory (SDFT)19,20 calculations as implemented picture of structural, electronic and optical properties of in the VASP code21–26 have been employed. To self- this material. Surprisingly, this lack of detailed knowl- consistently solve the Kohn-Sham (KS) equations27 edge of many physical properties of silver nitride stimu- lated only very few published ab initio studies. In the present work, first-principles calculations were ~2 n(r′) 2 + dr′ + V (r) carried out to investigate the lattice parameters, equation r r′ ext ( − 2me ∇ of state, relative stabilities, phase transitions, electronic Z | − | (2) and optical properties of silver nitrides in three different σ,k r σ,k r σ,k σ,k r chemical formulae and in various crystal structures. Cal- +VXC [n( )] ψi ( )= ǫi ψi ( ), culation methods are described in Sec. II. In Sec. III, ) results are presented, discussed and compared with ex- periment and with previous calculations. The article is where i, k and σ are the band, k-point and spin in- concluded with some remarks in Sec. IV. dices, receptively, VASP expands the pseudo part of the σ,k r KS one-particle spin orbitals ψi ( ) on a basis set of plane-waves (PWs). Only those PWs with cut-off en- II. CALCULATION METHODS ergy Ecut 600 eV have been included. The Brillouin zones were≤ sampled using Γ-centered Monkhorst-Pack28 A. Stoichiometries and Crystal Structures 17 17 17 meshes. Any increase in the Ecut value or in× the density× of the k-mesh produces a change in the To the best of our knowledge, the only experimen- total energy less than 3 meV/atom. For static calcula- tally reported stoichiometries of Ag-N compounds are tions, partial occupancies were set using the tetrahedron 7 7 29–31 Ag3N and AgN3 . However, previous ab initio stud- method with Bl¨ochl corrections , while the smearing 8 8,15 32 ies on Ag-N compounds considered Ag4N , Ag3N , method of Methfessel-Paxton (MP) was used in the 15 15,16 15,17 Ag2N , AgN and AgN2 in some cubic struc- ionic relaxation, and Fermi surface of the metallic phases tures only. Consideration of stoichiometries other than has been carefully treated. The Perdew-Burke-Ernzerhof the reported ones is probably due to the fact that many (PBE) parametrization33–35 of the generalized gradi- transition metals nitrides (TMs) are known to form more ent approximation (GGA)36–38 was employed for the 18 σ,k r than one nitride . Hence, our interest in investigating exchange-correlation potentials VXC [n( )]. The imple- 26,39 AgN and AgN2 is based on this fact. mented projector augmented wave (PAW) method For Ag3N, we consider the following seven structures: was used to describe the core-valence interactions Vext(r), 10 1 2 3 the face-centered cubic structure of AlFe3 (D03), the sim- where the 4d 5s electrons of Ag and the 2s 2p elec- ple cubic structure of Cr3Si (A15), the simple cubic struc- trons of N are treated explicitly as valence electrons. ture of the anti-ReO3 (D09), the simple cubic structure While for these valence electrons only scalar kinematic of Ag3Au (L12), the body-centered cubic structure of relativistic effects are incorporated, the PAW potential 25 CoAs3 (D02), the hexagonal structure of ǫ-Fe3N, and the treats the core electrons in a fully relativistic fashion . trigonal (rhombohedric) structure of RhF3. No spin-orbit interaction of the valence electrons has For AgN, the following four structures were considered: been considered. 3

C. Geometry Relaxation and EOS E. GWA Calculations and Optical Properties

At a set of externally imposed lattice constants, ions Although a qualitative agreement between DFT- with free internal parameters were relaxed until all force calculated optical properties and experiment is possible, ˚ components on each ion were less than 0.01 eV/A. This is accurate quantitative description requires treatments be- done following the implemented conjugate-gradient (CG) yond the level of DFT41. Another approach provided algorithm. After each ion relaxation step, static total en- by many-body perturbation theory (MBPT) leads to a ergy calculation with the tetrahedron method was per- system of quasi-particle (QP) equations, which can be formed, and the cohesive energy per atom Ecoh was cal- written for a periodic crystal as42–44 culated from

AgmNn Ag N Esolid Z mEatom + nEatom ~2 r′ AgmNn − × 2 ′ n( ) QP E = , (3) + dr + Vext(r) ψ k (r) coh Z (m + n)  − 2m∇ r r′ i, × ( Z | − | ) (8) where Z is the number of Ag N formulas per unit cell, QP QP QP QP m n + dr′Σ(r, r′; ǫ )ψ (r′)= ǫ ψ (r). EAg and EN are the energies of the isolated non- i,k i,k i,k i,k atom atom Z spherical spin-polarized atoms, and m,n = 1, 2 or 3 are the stoichiometric weights. The obtained E as a func- coh QP r tion of volume V per atom were then fitted to a Birch- Practically, the wave functions ψi,k ( ) are taken from Murnaghan 3rd-order equation of state (EOS)40 and the the DFT calculations. However, in consideration of k equilibrium volume V0, the equilibrium cohesive energy computational cost, we used a less dense mesh of - r r′ QP E0, the equilibrium bulk modulus points (10 10 10). The term Σ( , ; ǫi,k ) is the self- energy which× contains× all the exchange and correlation ∂P ∂2E effects, static and dynamic, including those neglected in B0 = V = V 2 (4) − ∂V V =V0 − ∂V V =V0 our DFT-GGA reference system. In the so-called GW approximation45, Σ is given in terms of the Green’s func- and its pressure derivative tion G as

2 ′ ∂B 1 ∂ ∂ E B0 = = V (V 2 ) (5) ′ ′ ′ ′ ∂P P =0 B0 ∂V ∂V V =V0 ΣGW = j dǫ G(r, r ; ǫ,ǫ )W (r, r ; ǫ), (9)   Z were determined. where the dynamically (frequency dependent) screened Coulomb interaction W is related to the bare Coulomb D. Formation Energy interaction v through Beside the cohesive energy, another measure of relative stability is the formation energy Ef . Assuming that sil- W (r, r′; ǫ)= j dr ε−1(r, r ; ǫ)v(r , r′), (10) ver nitrides Ag N result from the interaction between 1 1 1 m n Z the N2 gas and the solid Ag(A1) via the reaction (com- pare with Eq. 1) with ε, the dielectric matrix, calculated within the so- solid n gas ⇋ solid called random phase approximation (RPA). We followed mAg + N2 AgmNn , (6) 2 the GW0 self-consistent routine on G, in which the QP eigenvalues Ef can be given by

solid solid Ef (AgmNn )= Ecoh(AgmNn ) QP QP QP ǫi,k = Re ψi,k HKS VXC +ΣGW0 ψi,k (11) mE (Agsolid)+ n E (Ngas) − coh 2 coh 2 D E . (7) − m + n are updated in the calculations of G, while W is kept Here m,n = 1, 2, 3 are the stoichiometric weights and at the DFT-RPA level. Four updates were performed, solid Ecoh(AgmNn ) is the cohesive energy per atom as in Eq. and after the final iteration in G, ε is recalculated within solid 43,44,46 3. The cohesive energy Ecoh(Ag ) and other equilib- the RPA using the updated QP eigenvalues . From rium properties of the elemental metallic silver are given the real εre(ω) and the imaginary εim(ω) parts of this in Table I. The cohesive energy of the diatomic nitro- frequency-dependent microscopic dielectric tensor, one gas gen (Ecoh(N2 )) was found to be 5.196 eV/atom corre- can derive all the other frequency-dependent dielectric sponding to an equilibrium N–N bond− length of 1.113 A˚ response functions, such as reflectivity R (ω), transmitiv- (For more details, see Ref. 5). ity T (ω)=1 R (ω), refractive index n (ω), extinction − 4

1.8 coefficient κ (ω), and absorption coefficient α (ω)47–49: ¡

1.9 ¡ Ag N 1 2 3 [εre (ω)+ jεim (ω)] 2 1 R (ω)= 1 − (12) 2.0 A15 2 ¡ [εre (ω)+ jεim (ω)] +1 D0 3 1 2.1 1 2 ¡ L1 1 2 2 2 2

n (ω)= εre (ω)+ εim (ω) + εre (ω) (13) /atom) Fe3 N

√2 eV 2.2 ¡ (   1 RhF3 1   1 2 coh

2 2 2 E 2.3 D02 κ (ω)= ε (ω)+ ε (ω) εre (ω) (14) ¡ √2 re im − D09   1   1 2.4 2 ¡ 2 2 2 α (ω)= √2ω ε (ω)+ ε (ω) εre (ω) (15) re im − 2.5   ¡   12 14 16 18 20 22 24 26

3 V (A /atom) III. RESULTS AND DISCUSSION

The energy-volume equation of state (EOS) for the dif- FIG. 1. (Color online.) Cohesive energy Ecoh(eV/atom) ver- ˚3 ferent structures of Ag N, AgN and AgN are depicted sus atomic volume V (A /atom) for Ag3N in seven different 3 2 structural phases. in Figs. 1, 2 and 3, respectively. The corresponding cal- culated equilibrium properties are given in Table I. In this table, we ordered the studied phases according to 1.8 the increase in the nitrogen content; then within each se- £ AgN ries, structures are ordered in the direction of decreasing 2.0 structural symmetry. For the sake of comparison, we also £ B1 presented results from experiment and from previous ab B2 2.2 initio calculations; and, whenever appropriate, the cal- £ B3 /atom) culation method and the XC functional are also given in B81 eV

( 2.4 footnotes of the Table. £ Bk E

The calculated equilibrium properties: cohesive Bh 2.6 energies, formation energies, volume per atom, volume £ B4 per Ag atom, and bulk modulus and its pressure B17 2.8 derivative which are given Table I, are visualized in £ B24 Fig. 4. This kind of visualization allows us to study 10 15 20 25

¢ 3 the effect of nitridation on the parent Ag(A1), since V (A /atom) all quantities in this figure are given relative to the corresponding ones of the elemental Ag(A1) given in FIG. 2. (Color online.) Cohesive energy Ecoh(eV/atom) ver- the first row of Table I. Moreover, one can easly com- sus atomic volume V (A˚3/atom) for AgN in nine different pare the properties of these phases relative to each other. structural phases.

1.0 ¤

1.2 ¤

A. EOS and Relative Stabilities 1.4 ¤

1.6 ¤ AgN2 Considering Ecoh in the Ag3N series, Fig. 1 shows 1.8 ¤ C1 clearly that the E(V ) relations of Ag3N in D09, D02 and 2.0 ¤ C2

RhF3 phases are almost identical, corresponding to equi- /atom)

3.35 C18 ¤ eV

librium cohesive energy (Table I) of 2.513, 2.514 and ( 3.40 CoSb2 ¤ − − E

2.514 eV/atom, respectively. This behavior in the EOS 3.45 ¤ − 3.50 could be traced back to the structural relationships be- ¤ 3.55 tween these three structures, since both D02 and RhF3 ¤

3.60 ¤

can simply be derived from the more symmetric D09 (see 3.65 ¤ 3.70 Ref. 5 and references therein). These structural relations ¤ 8 10 12 14 16 18 20

may reflect in the EOS’s and in other physical properties, ¥ 3 V (A /atom) and the three phases may co-exist during the Ag3N syn- thesis process. Relative to the elemental Ag, these three phases tend not to change the Ecoh (Fig. 4), lowering FIG. 3. (Color online.) Cohesive energy Ecoh(eV/atom) ver- ˚3 it only by 0.03 eV/atom, as can been seen from Ta- sus atomic volume V (A /atom) for AgN2 in four different ble I. It may∼ be worth to mention here that the simple structural phases. 5

TABLE I. Calculated and experimental zero-pressure properties of the twenty studied phases of Ag3N, AgN and AgN2: Lattice ◦ ◦ 3 constants (a(A),˚ b(A),˚ c(A),˚ α( ) and β( )), equilibrium atomic volume V0(A˚ /atom), cohesive energy Ecoh(eV/atom), bulk ′ modulus B0(GP a) and its pressure derivative B0, and formation energy Ef (eV/atom). The presented data are of the current work (Pres.), experimentally reported (Expt.) and of previous calculations (Comp.). ˚ ˚ ˚ ◦ ◦ ˚3 ′ Structure a(A) b(A) c(A) α( ) or β( ) V0(A /atom) Ecoh(eV /atom) B0(GPa) B0 Ef (eV /atom) Ag Pres. 4.164 – – – 18.06 −2.484 88.188 5.793 a b b c d A1 Expt. (4.08570 ± 0.00018) – – – −2.95 100.7 , 101 6.12 −3.59g, −2.66h, 142f 5.00l, 5.70j, Comp. 4.01f – – – −2.67i 5.97k Ag3N D03 Pres. 6.322 – – – 15.79 −2.055 98.356 5.457 1.107 A15 Pres. 5.065 – – – 16.24 −1.976 92.280 5.470 1.186 Pres. 4.328 – – – 20.27 −2.513 71.980 5.386 0.649 D0 9 Comp. 3.995q, 4.169r , 4.292s – – – 95.7r , 87.1s L12 Pres. 3.972 – – – 15.67 −2.081 100.743 5.530 1.081 D02 Pres. 8.662 – – – 20.31 −2.514 72.230 5.335 0.648 ǫ-Fe3N Pres. 5.967 – 5.560 – 21.43 −2.469 64.737 2.335 0.692 RhF3 Pres. 6.126 – – α = 59.989 20.31 −2.514 72.237 5.396 0.648 fcct Expt. 4.369u, 4.29v, 4.378x – – 2.587 ± 0.364w AgN Pres. 4.617 – – – 12.30 −2.253 147.600 5.145 1.587 B1 Comp. 4.57q, 4.506r, 4.619s – – – 219.2r , 162.3s 4.653p 4.476o, 4.606p – – – 197.18o , 147.40p 4.883o Pres. 2.873 – – – 11.86 −2.021 146.157 5.260 1.819 B2 2.833q, 2.806r , 2.876s – – – 138.96p 4.823p Comp. 2.78o, 2.87p – – 204.10o 5.451o Pres. 4.950 – – – 15.16 −2.122 109.639 5.210 1.718 B3 4.88q, 4.816r, 4.946s – – – 100.11p 5.825p Comp. 4.79o, 4.94p – – – 151.05o 4.542o B81 Pres. 3.544 – 4.929 – 13.40 −1.996 130.485 5.240 1.844 − Bk Pres. 3.521 – 9.368 – 25.15 1.891 57.077 5.110 1.949 − Bh Pres. 3.096 – 3.023 – 12.55 2.121 141.385 5.285 1.719 Pres. 3.501 – 5.734 – 15.22 −2.113 105.992 5.467 1.727 B4 Comp. 3.41o, 3.54p – 5.52o, 5.69p – 143.68o , 110.12p 4.82o, 4.663p B17 Pres. Pres. 3.158 – 5.560 – 13.86 −2.517 132.556 5.185 1.323 B24 Pres. 4.337 4.601 5.091 – 12.70 −2.202 138.704 5.132 1.638 AgN2 Pres. 5.157 – – – 11.43 −1.959 164.844 4.996 2.333 C1 5.124q, 5.055r , 5.172s – – – 181.3r , 164.5s Comp. 5.013m, 5.141n – – – 215m, 164n C2 Pres. 5.617 – – – 14.77 −3.626 30.058 6.894 0.666 C18 Pres. 3.440 4.513 5.508 – 14.25 −3.680 35.878 7.269 0.612 CoSb2 Pres. 5.976 5.651 10.261 β = 151.225 13.90 −3.699 35.117 7.822 0.593 a Ref.50. This is an average of 56 experimental values, at 20◦C. b Ref.51. Cohesive energies are given at 0 K and 1 atm = 0.00010 GPa; while bulk mudulii are given at room temperature. c Ref. (25) in 52: at room temperature. d See Refs. (8)–(11) in 52. f Ref.53. using the full-potential linearized augmented plane waves (LAPW) method within LDA. g Ref.54: using projector augmented wave (PAW) method within LDA. h Ref.54: using projector augmented wave (PAW) method within GGA(PW91). i Ref.54: using projector augmented wave (PAW) method within GGA(PBE). j Ref.52: using a semiempirical estimate based on the calculation of the slope of the shock velocity vs. particle velocity curves obtained from the dynamic high-pressure experiments. The given values are estimated at ∼ 298 K. k 52 ′ Ref. : using a semiempirical method in which the experimental static P − V data are fitted to an EOS form where B0 and B0 are adjustable parameters. The given values are estimated at ∼ 298 K. l Ref.52: using the so-called method of transition metal pseudopotential theory; a modified form of a method proposed by Wills and Harrison to represent the effective interatomic interaction. m Ref. [17]: using the full-potential linearized augmented plane waves (LAPW) method within LDA. n Ref. [17]: using the full-potential linearized augmented plane waves (LAPW) method within GGA. o Ref. [16]: using full-potential (linearized) augmented plane waves plus local orbitals (FP-LAPW+lo) method within LDA. p Ref. [16]: using using full-potential (linearized) augmented plane waves plus local orbitals (FP-LAPW+lo) method within GGA(PBE). q Ref. [15]: using pseudopotential (PP) method within LDA. r Ref. [15]: using linear combinations of atomic orbitals (LCAO) method within LDA. B0’s are calculated from elastic constants. s Ref. [15]: using linear combinations of atomic orbitals (LCAO) method within GGA. B0’s are calculated from elastic constants. t This is the face centered cubic (fcc) structure with Z = 4/3 (i.e. 4 Ag atoms in the unit cell) suggested by Hahn and Gilbert according to some measurements (Ref. 9). u Ref. 9. v Ref. 8. w This is the average of the experimental values: (+314.4 ∓ 2.5) kJ/mol7 = (3.25853 ± 0.02591) eV/atom, +199 kJ/mol11 = 2.062 eV/atom, +255 kJ/mol9 = 2.643 eV/atom, and +230 kJ/mol10 = 2.384 eV/atom. We used the conversion relation: 1 eV/atom = 96.521 kJ/mol or equivalently 1 kJ/mol = 0.010364 eV/atom. x Ref. 12.

cubic D09 phase is the stable phase of the synthesized first minima (to the left) is a metastable local mini- 4,55 Cu3N . mum that cannot be maintained as the system is de- compressed. Ag ions are in the 6g Wyckoff posi- The odd behavior of the EOS of Ag3N(Fe3N) with tions: (x, 0, 0), (0, x, 0), ( x, x, 0), ( x, 0, 1 ), (0, x, 1 ) the existence of two minima (Fig. 1) shows that the − − − 2 − 2 6

Ag N AgN AgN 3 3 2 2 Bulk mudulus pressure derivative 1

§ 0 0

B 1 ¦

2 ¦

3 ¦ 80 60 Bulk mudulus 40 20

(GPa) 0 0

B

¦20

¦40 ¦60 30 Volume per metal atom 25 20

/atom) Volume per atom

3 15 ¨ A

( 10 Ag

0 5 V

, , 0 0

V ¦5 0.5 0.0 /atom)

eV ¦ 0.5 ( coh

¦1.0 Cohesive energy per atom E

2.5 Formation energy per atom 2.0

/atom) 1.5 eV ( f 1.0 E 0.5 3 9 2 2 1 2 3 k h N B2 B1 B3 B4 C2 C1 B B 3 L1 B17 B24 C18 A15 B8 D0 D0 D0 Fe RhF CoSb

FIG. 4. (Color online.) Calculated equilibrium properties of the twenty studied phases of silver nitrides. All quantities are given relative to the corresponding ones of the fcc crystalline elemental silver given in the first row of Table I. The vertical dashed lines separate between the different stoichoimetries. 7

E(V ) EOS for AgN in the B1, B2, B3 and B4 struc- 0.8 tures. The equilibrium energies they obtained from the D03 0.6 A15 E(V ) EOS revealed that B1 is the most stable phase,

D09 D0 A15 (26.0 GPa) and the relative stability they arrived at is in the order 0.4 L1 9 2 B1–B3–B4–B2, with a significant difference in total en- 0.2 ergy between B3 and B4 (see Fig. 2(b) in that article).

0.0 Within this subset of structures, the numerical values of /atom)

eV E in Table I do have the same order. However, the dif-

( coh

H 0.2 © D0 D0 (19.5 GPa) 9 3 ference between the equilibrium Ecoh(B3) and Ecoh(B4)

0.4 © is only 0.009 eV , and the E(V ) EOS of B3 and B4 D0 L1 (17.8 GPa) 9 2 match/overlap∼ over a wide range of volumes around the 0.6 © equilibrium point. This discrepancy may be attributed to 0.8 © 16 18 20 22 24 26 28 30 the unphysical/ill-defined measure of stability that Ka- P (GPa) noun and Said used, the total energy, while the number of the AgN formula units per unit cell in the B4 structure FIG. 5. (Color online.) Enthalpy H vs. pressure P equation differs from that in the others60. Nevertheless, it may be of state (EOS) for some Ag3N phases in the range where D09 worth mentioning here that AgN(B3) was found to be → → → 15 A15, D09 D03 and D09 L12 phase transitions occur. elastically unstable . In the CuN2 nitrogen-richest phase series, we can see from Table I and from Fig. 4 that the phases of this 1 1 and (x, x, 2 ); with x 3 to the left of the potential bar- group are significantly more stable than all the other ∼ 3 rier (represented by the sharp peak at 18.2 A˚ /atom), studied phases, except C1, which is, in contrast, the sec- 1 ∼ ond least stable among the twenty studied phases, with and x = 2 to the right of the peak. It may be relevant to mention here that Cu3N(Fe3N) was found to behave 0.017 eV/atom more than AgN(Bk). From Fig. 3, Fig. in a similar manner5. 4 and Table I, one can see that in this series, the lower the structural symmetry, the more stable is the phase. It The crossings of the less stable A15, D03 and L12 EOS 5 was found that CuN2 phases have the same trend . curves with the more stable D09, D02 and RhF3 ones at the left side of their equilibrium points reveals pressure- Comparing the relative stability of Ag3N, AgN and induces phase transitions from the latter phases to the AgN2, we find from Table I and from Fig. 4 that AgN2 former. To show this, we plotted the corresponding rela- in its least symmetric phase, the simple monoclinc struc- tion between enthalpy H = E(V )+ P V and the imposed ture of CoSb2, is the most energetically stable phase with Ecoh = 3.699 eV/atom. pressure P in Fig. 5. Since D09, D02 and RhF3 phases − have identical E(V ) curves, the corresponding H(P ) curves are also identical. Hence, only the H(P ) of D09 is displayed in Fig. 5. A point where the enthalpies of two B. Volume per Atom and Lattice Parameters phases are equal determine the phase transition pressure Pt; and, indeed, the direction of the transition is from The numerical values of the lattice parameters and the the higher H to the lower H56. As depicted in Fig. 5, average equilibrium volume per atom V0 for the twenty Pt(D09 L12)=17.8 GPa, Pt(D09 D03)=19.5 GPa modifications are presented in Table I. The middle sub- → → and Pt(D09 A15) = 26.0 GPa. Thus, D09, D02 and window of Fig. 4 depicts the V0 values relative to the → RhF3 would not survive behind these Pt’s and A15, D03 Ag(fcc). To measure the average of the Ag–Ag bond and L12 are preferred at high pressure. length in the silver nitride, the equilibrium average vol- Ag Fig. 4 reveals that the AgN group contains the least ume per Ag atom (V0 ), which is simply the ratio of the stable phase among all the twenty studied phases: the volume the unit cell to the number of Ag atoms in the hexagonal Bk. Fig. 2 and Table I show that the sim- unit cell, is visualized in the same subwindow. ple tetragonal structure of cooperite (B17) is the most From the V0 curve in Fig. 4, we can see that, all stable phase in this AgN series. In fact, one can see AgN and AgN2 modifications, except the open AgN(Bk) from Fig. 4 and Table I that all the considered AgN phase, decrease V0; while the Ag3N phases tend, on av- phases possess less binding than their parent Ag(fcc), erage, not to change the number density of the parent except AgN(B17) which is slightly more stable, with Ag(A1). Ag 0.033 eV/atom lower Ecoh. It is interesting to notice On the other hand, the V0 curve in Fig. 4 reveals that AgN(B17) is 0.003 eV/atom more stable than the that, relative to the elemental Ag and to each other, V Ag ∼ 0 Ag3N most stable phases. Moreover, this B17 structure tends to increase with the increase in the nitrogen con- was theoretically predicted to be the ground-state struc- tent. Thus, in all these nitrides, the introduced N ions ture of CuN5, AuN57, PdN58 and PtN59. displace apart the ions of the host lattice causing longer Using the full-potential (linearized) augmented plane Ag-Ag bonds than in the elemental Ag. This cannot be waves plus local orbitals (FP-LAPW+lo) method within seen directly from the V0 values depicted in the same LDA and within GGA, Kanoun and Said16 studied the figure. 8

For AgN in the B1, B2, B3 and B4 structures, Ka- according to Eq. 4, the value of B0 is proportional to the noun and Said (Ref. 16 described in Sec. IIIA above) absolute change in Ecoh, while it is far more sensitive to −1 obtained GGA equilibrium lattice parameters which are any change in V0 because it is proportional to (∆V0) . in very good agreement with ours. Their obtained LDA It is common to measure the pressure dependence of ′ lattice parameter values show the common underestima- B0 by its derivative B0 (Eq. 5). Fig. 4 shows that the tion with respect to their and our GGA values (see Table B0 value of the C2, C18 and CoSb2 phases increases as I). these phases are put under pressure. While the B0 val- Gordienko and Zhuravlev15 studied the structural, me- ues of the rest of the phases shows very low sensitivity to chanical and electronic properties of AgN(B1) , AgN(B2), pressure and they tend to slightly lower the bulk modu- AgN(B3), AgN2(C1) and Ag3N(D09) cubic phases. Their lus, the Fe3N phase is the most sensitive phase and tends DFT calculations were based on pseudopotential (PP) to significantly lower its B0 upon application of pressure. method within LDA, and on linear combinations of This high sensitivity may indicate that the correspond- atomic orbitals (LCAO) method within both LDA and ing minimum on the potential surface is not global, but GGA. For comparison, some of their findings are included another local minimum as the one at 16.2 A˚3/atom (Fig. in Table I. Within the parameter subspace they consid- 1). ered, our GGA values of the a lattice parameter agree From the elastic constants they obtained, Gordienko very well with theirs. On the other hand, although their and Zhuravlev15 calculated the corresponding macro- PP a values are closer to the GGA ones (ours and theirs), scopic bulk moduli (included in Table I). They found all their LDA values are less than the GGA ones. This the highest LDA B0 value for AgN(B1) among all phases confirms the well-known behavior of LDA compared to they considered, but, in agreement with the present work, 61–63 GGA . Gordienko and Zhuravlev also found that the they obtained the highest GGA B0 value for AgN2(C1). Ag–Ag interatomic distance increases in the order Ag3N– Since LDA relative to GGA overestimates Ecoh and thus AgN–AgN2. This agrees with the general trend shown in underestimates V0, each of these two factors (see Subsec- Ag Fig. 4, since the V0 curve shows an average increase in tion IIIC) would separately lead to the odd LDA value the same direction. of 219.2 GPa which they obtained. Nevertheless, due to this fact, Gordienko and Zhuravlev argued that one should consider the LDA and GGA average value of B0. C. Bulk Modulus and its Pressure Derivative

D. Formation Energies Fig. 4 reveals that Ag3N phases tend, on average, to preserve the B0 value of the parent Ag(A1). Increasing the nitrogen content to get AgN phases will increase the Formation energies in the present work are used as B0 value of the parent Ag(A1), except in the case of Bk. a measure of the relative thermodynamic stabilities of While the nitrogen in AgN2 tends to lower the B0 value of the phases under consideration. That is, the lower the the parent Ag(A1), the cubic C1 phase posses the highest formation energy, the lower the tendency to dissociate B0 value. This could be seen from Fig. 3, where the back into the constituent components Ag and N2. curvature of the Ecoh(V ) curve of C1 is higher compared The obtained formation energies Ef of the twenty re- to the shallow minima of the C2, C18 and CoSb2 curves. laxed phases are given in Table I and depicted graphically From the definition of the equilibrium bulk modulus in Fig. 4. The latter shows that, relative to each other (Eq. 4), one would expect B0 to increase as Ecoh or V0 and within each series, the formation energy Ef (defined decreases. This is because of the minus sign of the for- by Eqs. 6 and 7) of the studied phases has the same mer and the inverse proportionality of the latter. That trend as the cohesive energy65. That is, all phases have is, roughly speaking, the B0 curve should have a mirror the same relative stabilities in the Ef space as in the Ecoh reflection-like behavior with respect to the Ecoh and V0 space. However, while Ag3N phases tend to have equal curves. Nevertheless, if Ecoh or V0 are increasing and Ecoh as the AgN phases, all Ag3N modifications have a the other is decreasing, then the dominant net effect will lower Ef than the AgN ones. Hence, silver nitride is more be of the one with the higher change64. For example, likely to be formed in the former stoichiometry. However, Fig. 4 shows that in going from D03 to A15, both Ecoh all the twenty obtained Ef values are positive; which, in and V0 increase resulting in a negative change in B0. In principle, means that all these phases are thermodynam- 66 going from A15 to D09, Ecoh is decreasing while V0 is in- ically unstable (endothermic) . creasing, but, in the end, the latter won the competition Some of the experimental values of Ef for the syn- and lowered the value of B0. This argument stays true thesized Ag3N phase (which is claimed to be in an throughout the three series. When there is no significant fcc structure) are +199 kJ/mol11 = 2.062 eV/atom, 10 9 change in both Ecoh and V0, there is no significant change +230 kJ/mol = 2.384 eV/atom, +255 kJ/mol = 7 in B0. This is the case when one goes from C18 to CoSb2. 2.643 eV/atom and (+314.4 2.5) kJ/mol = (3.25853 ∓ ± A close look at the B0 curve in Fig. 4, reveals that the 0.02591) eV ; with an average value of 2.587 0.364 eV . ± huge decrease in Ecoh between C1 and C2 defeats the Among the considered phases in the present work, there relatively small increase in V0. This is simply because, is only one phase wich has Ef value that fits in this range, 9 the AgN2(C1). Interestingly, this C1 structure has an fcc of 0.130 eV . From sub-figures 8(c) and (d), one can no- underlying Bravia lattice; however, the chemical formula tice clearly the Ag(d)-N(p) mixture in the region from differs from that of the synthesized phase. 7.249 eV to 0.091 eV below EF , with two peaks: a −low density peak− around 1.3 eV steaming from an al- most equal mixture of Ag(d−) and N(p), and a high density E. Electronic Properties peak around 4.3 eV steming mainly from the bands of silver d electrons− plus a relatively very low contribution σ k from the N(p) states. The DFT(GGA) calculated band diagrams (i.e. ǫi ( ) curves) and spin-projected total and orbital resolved (i.e. Fig. 7 depicts the band diagram and DOS’s of partial) density of states (DOS) of the most stable phases: Ag3N(RhF3). In contrast to Ag3N(D09) and Ag3N(D02), D09, RhF3, D02, B17, and C18 are presented in Figs. 6, sub-figure 7(a) shows that Ag3N(RhF3) is a semiconduc- 7, 8, 9 and 10, respectively. Spin-projected total density tor with a narrow direct band gap of 0.129 eV of width located at Γ point. The VBM is at 0.089 eV and the of states (TDOS) are shown in sub-figure (b) in each case. − In all the six considered cases, electrons occupy the spin- CBM is at 0.040 eV . From sub-figures 7(c) and (d), up and spin-down bands equally, resulting in zero spin- one can see the Ag(d)-N(p) mixture is in the region from polarization density of states: ζ(ǫ)= n↑(ǫ) n↓(ǫ). Thus, 7.286 eV to 0.089 eV beneath EF , with two peaks: − −a low density peak− around 1.366 eV steaming from an it is sufficient only to display spin-up (or spin-down) den- − sity of states (DOS) and spin-up (or spin-down) band almost equal mixture of Ag(d) and N(p), and a high den- sity peak around 4.382 eV steaming mainly from the diagrams. In order to investigate the details of the elec- − tronic structure of these phases, energy bands are plotted bands of silver d electrons plus a relatively very low con- along densely sampled high-symmetry string of neighbor- tribution from the N(p) states. ing k-points. Moreover, to extract information about the The relationship between D09, D02 and RhF3 struc- orbital character of the bands, the Ag(s,p,d) and N(s,p) tures manifests itself in many common features between partial DOS are displayed at the same energy scale. the electronic structure of these three Ag3N nitrides: (i) σ k equal Eg of 0.13 eV ; (ii) a deep bound band around Fig. 6(a) shows the band structure ǫi ( ) of ∼ ∼ 14.6 eV below EF consists mainly of the N(2s) Ag3N(D09). With its valence band maximum (VBM) − at (R, 0.086 eV ) and its conduction band minimum states; (iii) a broad valence band with 7.2 eV of width − ∼ (CBM) at (Γ, 0.049 eV ), Ag3N(D09) presents a semicon- that comes mostly from the 4d electrons of Ag plus a very ducting character with a narrow indirect band gap Eg small contribution from N(2p); and (iv) the relatively low of 0.134 eV . From sub-figures 6(c) and (d), it is seen TDOS of the conduction bands. σ k clearly that the Ag(d)-N(p) mixture in the region from Energy bands ǫi ( ), total density of states (TDOS) 7.286 eV to 0.086 eV beneath EF , with two peaks: a and partial (orbital-resolved) density of states (PDOS) low− density peak− around 1.5 eV and a high density peak of AgN(B17) are shown in Figs. 9. It is clear that around 4.0 eV steaming mainly from the bands of silver AgN(B17) would be a true metal at its equilibrium. The d electrons. major contribution to the very low TDOS around Fermi Our obtained PDOS, TDOS and band structure of energy EF comes from the 2p states of the N atoms as it Ag3N(D09) agree qualitatively well with Gordienko and is evident from sub-figure 9(d). Beneath EF lies a band Zhuravlev15; however, using LCAO method within GGA, with 7.3 eV of width, in which the main contribution ∼ the value of the indirect Eg of Ag3N(D09) they predicted is due to the Ag(4d) states plus a small contribution from is 0.25 eV . the N(2p) states. While the N(2s) states dominate the To the best of our knowledge, there is no experimen- deep lowest region around 13.5 eV , the low density un- 8 tally reported Eg value for Ag3N. However, Tong pre- occupied bands stem mainly from the N(2p) states. The pared Ag3+xN samples, and carried out XRD measure- Fermi surface crosses two partly occupied bands: a lower ments to confirm the fcc symmetry of the prapared sam- one in the X-M, Γ-Z-A and Γ-X-R directions, and a ples. Using a TB-LMTO code within LDA, Tong then higher band in the X-M-Γ and M-A directions. Thus, calculated the band structure of Ag3N and obtained an EF is not a continuous surface contained entirely within indirect energy gap of 1.35 eV . Nevertheless, we could the first BZ. not figure out the positions of the N ions Tong’s model. It may be worth mentioning here that AgN(B1)16,68 It is a well known drawback of Kohn-Sham DFT-based and AgN(B3)16,69 phases were also theoretically pre- calculations to underestimate the band gap. Thus the dicted to be metallic. more demanding GW calculations were carried out, and Although AgN2(CoSb2) is the most stable phase, but the obtained Eg value will be presented in Sec. IIIF. the difference in cohesive energy between AgN2(CoSb2) Calculated electronic properties of Ag3N(D02) are dis- and AgN2(C18) is less than 0.02 eV/atom, and we de- played in Fig. 8. sub-figure 8(a) shows the energy bands cided to examine the electronic structure of both phases. σ k ǫi ( ) of Ag3N(D02). With its valence band maximum With EF crossing the finite TDOS, Fig. 10 shows that (VBM) at (H, 0.091 eV ) and its conduction band mini- AgN2(C18) is metallic at 0 K. The orbital resolved − mum (CBM) at (Γ, 0.039 eV ), Ag3N(D09) presents semi- DOS’s reveal that the major contribution to the low conducting character with a narrow indirect band gap Eg TDOS at EF comes from the N(2p) states with tiny 10

15 15 15 15 TDOS up Ag s N s 10 10 10 10 TDOS down Ag p N p 5 5 5 Ag d 5

(eV) 0 0 0 0 F

E



5 5 5 5 E

10 10 10 10

15 15 15 15

10 5 0 5 10 0 1 2 3 4 5 6 7 8 9 0.0 0.1 0.2 0.3 0.4 0.5 0.6

M X M R X R

TDOS (arb. units) PDOS (arb. units) PDOS (arb. units) (a) (b) (c) (d)

FIG. 6. (Color online.) DFT calculated electronic structure for Ag3N in the D09 structure: (a) band structure along the high-symmetry k-points which are labeled according to Ref. [67]. Their coordinates w.r.t. the reciprocal lattice basis vectors are: M(0.5, 0.5, 0.0), Γ(0.0, 0.0, 0.0), X(0.0, 0.5, 0.0), R(0.5, 0.5, 0.5); (b) spin-projected total density of states (TDOS); (c) partial density of states (PDOS) of Ag(s,p,d) orbitals in Ag3N; and (d) PDOS of N(s,p) orbitals in Ag3N,.

15 15 15 15 TDOS up Ag s N s 10 10 10 10 TDOS down Ag p N p 5 5 5 Ag d 5

(eV) 0 0 0 0 F E

 5 5 5 5

    E

10 10 10 10

   

15 15 15 15

   

10 5 0 5 10 0 1 2 3 4 5 6 7 8 9 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 

F B Z L Y 

  Q  TDOS (arb. units) PDOS (arb. units) PDOS (arb. units) (a) (b) (c) (d)

FIG. 7. (Color online.) DFT calculated electronic structure for Ag3N in the RhF3 structure: (a) band structure along the high-symmetry k-points which are labeled according to Ref. [67]. Their coordinates w.r.t. the reciprocal lattice basis vectors are: F (0.5, 0.5, 0.0), Q(0.375, 0.625, 0.0), B(0.5, 0.75, 0.25), Z(0.5, 0.5, 0.5), Γ(0.0, 0.0, 0.0), L(0.0, 0.5, 0.0), Y (0.25, 0.5, −.25), Σ(0.0, 0.5, −.5); (b) spin-projected total density of states (TDOS); (c) partial density of states (PDOS) of Ag(s,p,d) orbitals in Ag3N; and (d) PDOS of N(s,p) orbitals in Ag3N.

TDOS up Ag s N s 5 5 5 5 TDOS down Ag p N p Ag d 0 0 0 0 (eV) F

E 5 5 5 5

     E

10 10 10 10

   

15 15 15 15

  

N P H P N H 

 5 5 1 4 5 7

10  0 10 0 2 3 6 8 9 0.0 0.2 0.4 0.6 0.8 1.0  (a) TDOS (arb. units) PDOS (arb. units) PDOS (arb. units) (b) (c) (d)

FIG. 8. (Color online.) DFT calculated electronic structure for Ag3N in the D02 structure: (a) band structure along the high-symmetry k-points which are labeled according to Ref. [67]. Their coordinates w.r.t. the reciprocal lattice basis vectors are: Γ(0.0, 0.0, 0.0), N(0.0, 0.0, 0.5), P (0.25, 0.25, 0.25), H(0.5, −.5, 0.5); (b) spin-projected total density of states (TDOS); (c) partial density of states (PDOS) of Ag(s,p,d) orbitals in Ag3N; and (d) PDOS of N(s,p) orbitals in Ag3N. 11

15 15 15 15 TDOS up Ag s N s 10 10 TDOS down 10 Ag p 10 N p 5 5 5 Ag d 5

(eV) 0 0 0 0 F

E 

E 5 5 5 5

   

10 10 10 10

   

15 15 15 15

  

X M Z M X R Z 

 A 5 5 4 1.5

10  0 10 0 2 6 8 10 0.0 0.5 1.0 2.0 2.5  (a) TDOS (arb. units) PDOS (arb. units) PDOS (arb. units) (b) (c) (d)

FIG. 9. (Color online.) DFT calculated electronic structure for AgN in the B17 structure: (a) band structure along the high-symmetry k-points which are labeled according to Ref. [67]. Their coordinates w.r.t. the reciprocal lattice basis vectors are: X(0.0, 0.5, 0.0), M(0.5, 0.5, 0.0), Γ(0.0, 0.0, 0.0), Z(0.0, 0.0, 0.5), A(0.5, 0.5, 0.5), R(0.0, 0.5, 0.5); (b) spin-projected total density of states (TDOS); (c) partial density of states (PDOS) of Ag(s,p,d) orbitals in AgN; and (d) PDOS of N(s,p) orbitals in AgN.

10 10 10 10 TDOS up Ag s N s 5 5 TDOS down 5 Ag p 5 N p 0 0 0 Ag d 0

(eV) 5 5 5 5

    F E

 10 10 10 10

    E

15 15 15 15

   

20 20 20 20

    10 5 0 5 10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0 1 2 3 4 5 6 7 8

X Y Z U R R T Z  

 S S TDOS (arb. units) PDOS (arb. units) PDOS (arb. units) (a) (b) (c) (d)

FIG. 10. (Color online.) DFT calculated electronic structure for AgN2 in the C18 structure: (a) band structure along the high- symmetry k-points which are labeled according to Ref. [67]. Their coordinates w.r.t. the reciprocal lattice basis vectors are: Γ(0.0, 0.0, 0.0), X(0.0, 0.5, 0.0), S(−.5, 0.5, 0.0), Y (−.5, 0.0, 0.0), Z(0.0, 0.0, 0.5), U(0.0, 0.5, 0.5), R(−.5, 0.5, 0.5), T (−.5, 0.0, 0.5); (b) spin-projected total density of states (TDOS); (c) partial density of states (PDOS) of Ag(s,p,d) orbitals in AgN2; and (d) PDOS of N(s,p) orbitals in AgN2.

15 15 15 15 TDOS up Ag s N s 10 10 10 10 TDOS down Ag p N p 5 5 5 Ag d 5 0 0 0 0 (eV)

F 5 5 5 5

    E

10 10 10 10

    E

15 15 15 15

   

20 20 20 20

   

B E Z Y B D E Y

A A C 5 5 1 4 5 7 1 4 5 7

10  0 10 0 2 3 6 0 2 3 6 8  (a) TDOS (arb. units) PDOS (arb. units) PDOS (arb. units) (b) (c) (d)

FIG. 11. (Color online.) DFT calculated electronic structure for AgN2 in the CoSb2 structure: (a) band structure along the high-symmetry k-points which are labeled according to Ref. [67]. Their coordinates w.r.t. the reciprocal lattice basis vectors are: Γ(0.0, 0.0, 0.0), B(−.5, 0.0, 0.0), A(−.5, 0.5, 0.0), E(−.5, 0.5, 0.5), Z(0.0, 0.0, 0.5), Y (0.0, 0.5, 0.0), D(−.5, 0.0, 0.5) and C(0.0, 0.5, 0.5); (b) spin-projected total density of states (TDOS); (c) partial density of states (PDOS) of Ag(s,p,d) orbitals in AgN2; and (d) PDOS of N(s,p) orbitals in AgN2. 12 contributions from the 5s, 4d and 3p states of Ag, re- it is evident that Ag3N(D09) is a good reflector, specially spectively. As one can see from sub-figure 10(a), the EF in the red and the infrared regions. In the visible range, surface crosses the edges of the first Brillouin zone in the the maximum transmitivity T (ω) is at 2.54 eV Z-U-R-S-T -X and T -Z directions. 489 nm, which is at the blue-green edge. ∼ ≡ The calculated electronic properties of AgN2(CoSb2) sub-figure 12(c) depicts the calculated refraction n(ω) are displayed in Fig. 11. Band structure, TDOS and and extinction κ(ω) coefficients. As they should, these orbital resolved DOS’s have almost the same features as two spectra have, in general, the same qualitative fre- the corresponding ones of AgN2(C18). It may be worth quency dependence as the real εre(ω) and the imaginary to mention here that C1 phase of AgN2 was also theoret- εim(ω) dielectric functions, respectively. ically predicted to be metallic15. From the absorption coefficient α (ω) spectrum (sub- Compared to the metallic AgN(B17), three new fea- figure 12(d)), it can be seen that Ag3N(D09) starts ab- tures of these 1:2 nitrides are evident: (i) Deep at sorbing photons with 0.9 eV energy. Hence, it is clear ∼ 22.7 eV there is a highly-localized mixture of the that GW0 calculations give a band gap of 0.9 eV , which ∼N(s −)-N(p) states. However, the variation in N(2s) en- is a significant improvement over the value∼ obtained from ergy with respect to k is smaller than the variation of DFT. The non-vanishing α (ω) in the whole optical region N(2p) states, resulting in a narrower and higher PDOS. agrees with the experiment, since it may explain the ob- (ii) Below the band that is crossed by EF there are four served black color of the synthesized Ag3N. bands separated by 11.4 eV , 1.6 eV , 0.38 eV To the best of our knowledge, the present work is the and 0.28 eV energy∼ gaps, respectively.∼ (iii)∼ The very first trial to theoretically investigate the optical proper- tiny∼ contribution of the N(p) states to the N(2p)-Ag(4d) ties of silver nitride. However, for more accurate optical band. characterization (e.g. more accurate positions and am- A common feature of all the studied cases is that plitudes of the characteristic peaks), electron-hole exci- Ag(p)-orbitals do not contribute significantly to the hy- tations should be calculated. This can be done by eval- brid bands. Another common feature is the highly struc- uating the two-body Green function G2 on the basis of tured, intense and narrow series of peaks in the TDOS our obtained GW one-particle Green function G and QP valance band corresponding to the superposition of N(2p) energies, then solving the so-called Bethe-Salpeter equa- 72 and Ag(4d) states. In their k-space, Ag(4d) energies tion, the equation of motion of G2 . show little variation with respect to k; hence the Van Hove singularities-like sharp features. To summarize, we have found that the most stable IV. CONCLUSIONS phases of AgN and AgN2 are metallic, while those of Ag3N are semiconductors. A close look at Fig. 9 up to We have succesfully employed first-principles calcula- Fig. 6 reveals that as the nitrogen to silver ratio increases tion methods to investigate the structural, stability, elec- from x = 1 to x = 1/2, the TDOS at EF decreases; tronic and optical properties of Ag3N, AgN and AgN2. and by arriving at x = 1/3 a gap opens. This finding Within the accuracy of the employed methods, the ob- 15 ′ agrees well with Gordienko and Zhuravlev . Moreover, tained structural parameters, EOS, B0, B0 and electronic it may be worth mentioning here that such behavior was properties show good agreement with the few avialable theoretically predicted to be true for copper nitrides as previous calculations. On the other hand, our obtained 5,70 well . results show, at least, partial agreement with three ex- perimental facts: (i) the lattice parameter of Ag3N(D09) is close to the experimentally reported one; (ii) the pos- F. Optical Properties itive formation energies reveals the endothermic (unsta- ble) nature of silver nitrides, and (iii) absorption spec- Fig. 12 depicts the GW calculated real and imagi- trum explains its observed black color. Moreover, the nary parts of the frequency-dependent dielectric function present work may be considered as the first trial to the- εRPA(ω)ofAg3N(D09) and the corresponding derived op- oretically investigate the optical properties of silver ni- tical constants. The optical region71 is shaded in each tride. We hope that some of our obtained results will be sub-figure. confirmed in future experimentally and/or theoretically. The real part εre(ω) (sub-figure 12(a)) shows an up- ward trend before 2.3 eV , where it reaches its maxi- mum value and generally∼ decreases after that. The imag- ACKNOWLEDGMENTS inary part εim(ω) (same sub-figure 12(a)) shows an up- ward trend before 1.0 eV and it has three main peaks All GW calculations and some DFT calculations were located at 2.6 eV∼ in the optical region, 3.3 eV at carried out using the infrastructure of the Centre for the right edge∼ of the optical region, and at ∼ 4.1 eV in High Performance Computing (CHPC) in Cape Town. the UV range. ∼ Suleiman would like to acknowledge the support he re- Calculated reflectivity R(ω) and transmitivity T (ω) ceived from Wits, DAAD, AIMS, SUST and the AS- are displayed in sub-figure 12(b). With 0.6 R(ω) 0.8, ESMA group. Many thanks to the Scottish red pen of ≤ ≤ 13

(a) (b) (c) (d) 1.0 4.5

( ) ( ) reflectivity refractive index 1.6 absorption coefficient

% $ % 12 $ rε im 4.0 transmitivity extinction coefficient 1.4 0.8 3.5 5

) 10 1.2 # 10 ) )

( 3.0

*

& & (

0.6 ( im 8 1 1.0

' 2.5

" ) ,T , 0.8 , ) 6 ) 2.0

)

& 0.4 &

( # ( 0.6 / cm /

( 1.5 n ) 4 R

rε & ( " 0.4 0.2 1.0 2 ( 0.5 0.2

0 0.0 0.0 0.0 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

(eV) (eV) (eV) (eV) ! ! ! !

FIG. 12. (Color online.) The GW calculated frequency-dependent optical spectra of Ag3N(D09): (a) the real εre(ω) and the imaginary εim(ω) parts of the dielectric function εRPA(ω); (b) reflectivity R(ω) and transmitivity T (ω); (c) refraction n(ω) and extinction κ(ω) coefficients; and (d) absorption coefficient α(ω). The shaded area highlights the optical region.

Ross McIntosh!

∗ Corresponding author: [email protected] 11 R. Juza and H. Hahn, † Homepage: http://www.wits.ac.za/staff/daniel.joubert2.htm Zeitschrift fr anorganische und allgemeine Chemie 1 D. Aberg,˚ P. Erhart, J. Crowhurst, J. M. 244, 133 (1940), ISSN 1521-3749, Zaug, A. F. Goncharov, and B. Sadigh, http://dx.doi.org/10.1002/zaac.19402440205 Physical Review B 82, 104116 (Sep 2010), 12 M. Haisa, Acta Crystallographica Section A 38, 443 (Jul http://link.aps.org/doi/10.1103/PhysRevB.82.104116 1982), http://dx.doi.org/10.1107/S0567739482000990 2 13 X. P. Du and Y. X. Wang, Ag3N, formerly termed fulminating silver by its discov- Journal of Applied Physics 107, 053506 (2010), erers, can be formed from ammoniacal solutions of silver http://link.aip.org/link/?JAP/107/053506/1 oxide according to the following reaction 3 (aq) −→ M. G. Moreno-Armenta, W. L. Prez, and N. Takeuchi, 3Ag2O+2NH3 2Ag3N+5H2O. (16) Solid State Sciences 9, 166 (2007), ISSN 1293-2558, It can also be formed by means of other reactions7,8. http://www.sciencedirect.com/science/article/pii/S129325580600285814 J. L. Ennis and E. S. Shanley, 4 R. Juza and H. Hahn, Journal of Chemical Education 68, A6 (1991), Zeitschrift fr anorganische und allgemeine Chemie http://pubs.acs.org/doi/pdf/10.1021/ed068pA6, 241, 172 (1939), ISSN 1521-3749, http://pubs.acs.org/doi/abs/10.1021/ed068pA6 http://dx.doi.org/10.1002/zaac.19392410204 15 A. Gordienko and Y. Zhuravlev, Journal of Struc- 5 M. S. H. Suleiman, M. P. Molepo, and D. P. Joubert, ArXiv tural Chemistry 51, 401 (2010), ISSN 0022-4766, e-prints(Nov. 2012), arXiv:1211.0179 [cond-mat.mtrl-sci] http://dx.doi.org/10.1007/s10947-010-0061-8 6 Y. Du, A. Ji, L. Ma, Y. Wang, 16 M. Kanoun and S. Goumri-Said, Physics Letters A and Z. Cao, Journal of Crystal Growth 362, 73 (2007), ISSN 0375-9601, 280, 490 (2005), ISSN 0022-0248, http://www.sciencedirect.com/science/article/pii/S03759601060153 http://www.sciencedirect.com/science/article/pii/S002202480500426417 R. Yu and X. F. Zhang, 7 E. S. Shanley and J. L. Ennis, Physical Review B 72, 054103 (Aug 2005), Industrial & Engineering Chemistry Research 30, 2503 http://link.aps.org/doi/10.1103/PhysRevB.72.054103 (1991), http://pubs.acs.org/doi/pdf/10.1021/ie00059a023, 18 A. F. Wells, Structural Inorganic Chemistry, 5th ed. http://pubs.acs.org/doi/abs/10.1021/ie00059a023 (Oxford University Press, 1984) ISBN 9780198553700, 8 J. Tong, Darstellung, Strukturen und Eigen- http://books.google.co.za/books?id=lQfwAAAAMAAJ schaften ausgew¨ahlter Perowskit-Materialien 19 U. von Barth and L. Hedin, und Molek¨ulkristalle, Ph.D. thesis, Max-Planck- Journal of Physics C: Solid State Physics 5, 1629 (Feb Institut f¨ur Festk¨orperforschung, Stuttgart (2010), 1972), http://iopscience.iop.org/0022-3719/5/13/012/ http://elib.uni-stuttgart.de/opus/volltexte/2010/5816/20 M. Pant and A. Rajagopal, Solid State Communications 9 H. Hahn and E. Gilbert, 10, 1157 (1972), ISSN 0038-1098, Zeitschrift fr anorganische Chemie http://www.sciencedirect.com/science/article/pii/003810987290934 258, 77 (1949), ISSN 1521-3749, 21 G. Kresse and J. Hafner, http://dx.doi.org/10.1002/zaac.19492580109 Physical Review B 47, 558 (Jan 1993), 10 R. Anderson and N. Parlee, High http://link.aps.org/doi/10.1103/PhysRevB.47.558 Temperature Science 2, 289 (1970), 22 G. Kresse and J. Hafner, http://www.osti.gov/energycitations/product.biblio.jsp?query_id=0&page=0&osti_id=4085944Physical Review B 49, 14251 (May 1994), 14

http://link.aps.org/doi/10.1103/PhysRevB.49.14251 (Cambridge University Press; Cambridge, 2006) 23 G. Kresse and J. Furthm- 44 J. Harl, The Linear Response Function in Density Func- ller, Computational Materials Science tional Theory: Optical Spectra and Improved Description 6, 15 (1996), ISSN 0927-0256, of the Electron Correlation, Ph.D. thesis, University of Vi- http://www.sciencedirect.com/science/article/pii/0927025696000080enna (2008), http://othes.univie.ac.at/2622/ 24 G. Kresse and J. Furthm¨uller, 45 L. Hedin, Phys. Rev. 139, A796 (Aug 1965), Physical Review B 54, 11169 (Oct 1996), http://link.aps.org/doi/10.1103/PhysRev.139.A796 http://link.aps.org/doi/10.1103/PhysRevB.54.11169 46 G. Kresse, M. Marsman, and J. Furthmuller, 25 J. Hafner, Journal of Computational Chemistry “Vasp the guide,” (2011), available on-line at 29, 2044 (2008), ISSN 1096-987X, http://cms.mpi.univie.ac.at/vasp/vasp/. Last ac- http://dx.doi.org/10.1002/jcc.21057 cessed October 2012. 26 G. Kresse and D. P. Joubert, 47 M. Fox, Optical Properties of Solids, Oxford Master Se- Physical Review B 59, 1758 (Jan 1999), ries in Physics: Condensed Matter Physics (Ox- http://link.aps.org/doi/10.1103/PhysRevB.59.1758 ford University Press, 2010) ISBN 9780199573363, 27 W. Kohn and L. J. Sham, http://books.google.co.za/books?id=-5bVBbAoaGoC Physical Review 140, A1133 (Nov 1965), 48 M. Dressel and G. Gr¨uner, http://link.aps.org/doi/10.1103/PhysRev.140.A1133 Electrodynamics of solids : optical properties of electrons in matter 28 H. J. Monkhorst and J. D. Pack, (Cambridge University Press, Cambridge New York, 2002) Physical Review B 13, 5188 (Jun 1976), ISBN 0521592534 http://link.aps.org/doi/10.1103/PhysRevB.13.5188 49 A. Miller, in Handbook of Optics, Volume 1: Fundamentals, Techniques, 29 O. Jepson and O. Anderson, Solid State Communications Optical Society of America (McGraw-Hill, Inc., New York, 9, 1763 (1971), ISSN 0038-1098, NY, USA, 2010) ISBN 0070479747, 9780070479746 http://www.sciencedirect.com/science/article/pii/003810987190313950 J. Donohue, The structures of the elements, 30 G. Lehmann and M. Taut, physica status solidi (b) A Wiley-interscience publication (John Wi- 54, 469 (1972), ISSN 1521-3951, ley & Sons Inc., 1974) ISBN 0471217883, http://dx.doi.org/10.1002/pssb.2220540211 http://books.google.co.za/books?id=Q-rvAAAAMAAJ 31 P. E. Bl¨ochl, O. Jepsen, and O. K. Ander- 51 C. Kittel, Introduction to Solid State Physics, eigth ed. sen, Physical Review B 49, 16223 (Jun 1994), (John Wiley & Sons, Inc., 2005) ISBN 9780471415268, http://link.aps.org/doi/10.1103/PhysRevB.49.16223 http://books.google.co.za/books?id=kym4QgAACAAJ 32 M. Methfessel and A. T. Paxton, 52 S. Raju, E. Mohandas, and V. Raghunathan, J. Phys. Physical Review B 40, 3616 (Aug 1989), Chem Solids 58, 1367 (1997) http://link.aps.org/doi/10.1103/PhysRevB.40.3616 53 M. J. Mehl and D. A. Papaconstantopou- 33 J. P. Perdew, K. Burke, and M. Ernzerhof, los, Physical Review B 54, 4519 (Aug 1996), Physical Review Letters 77, 3865 (Oct 1996), http://link.aps.org/doi/10.1103/PhysRevB.54.4519 http://link.aps.org/doi/10.1103/PhysRevLett.77.3865 54 E. Zarechnaya, N. Skorodumova, S. Simak, B. Johans- 34 J. P. Perdew, K. Burke, and M. Ernzerhof, son, and E. Isaev, Computational Materials Science Physical Review Letters 78, 1396 (Feb 1997), 43, 522 (2008), ISSN 0927-0256, http://link.aps.org/doi/10.1103/PhysRevLett.78.1396 http://www.sciencedirect.com/science/article/pii/S09270256080000 35 M. Ernzerhof and G. E. Scuseria, 55 U. Hahn and W. Weber, The Journal of Chemical Physics 110, 5029 (1999), Physical Review B 53, 12684 (May 1996), http://link.aip.org/link/?JCP/110/5029/1 http://link.aps.org/doi/10.1103/PhysRevB.53.12684 36 A. D. Becke, Physical Review A 38, 3098 (Sep 1988), 56 G. Grimvall, Thermophysical Properties of Materials http://link.aps.org/doi/10.1103/PhysRevA.38.3098 (North Holland, 1986) http://books.google.co.za/books?id=TCWZlgbB 37 J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. 57 M. S. H. Suleiman and D. P. Joubert, in Jackson, M. R. Pederson, D. J. Singh, and C. Fi- South African Institute of Physics 57th An- olhais, Physical Review B 46, 6671 (Sep 1992), nual Conference (SAIP 2012), No. 298 (2012) http://link.aps.org/doi/10.1103/PhysRevB.46.6671 http://indico.saip.org.za/confSpeakerIndex.py?view=full&letter=s 38 J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. 58 M. S. H. Suleiman and D. P. Joubert, in Jackson, M. R. Pederson, D. J. Singh, and C. Fi- South African Institute of Physics 57th An- olhais, Physical Review B 48, 4978 (Aug 1993), nual Conference (SAIP 2012), No. 299 (2012) http://link.aps.org/doi/10.1103/PhysRevB.48.4978.2 http://indico.saip.org.za/confSpeakerIndex.py?view=full&letter=s 39 P. E. Bl¨ochl, Physical Review B 50, 17953 (Dec 1994), 59 J. von Appen, M.-W. Lumey, and R. Dron- http://link.aps.org/doi/10.1103/PhysRevB.50.17953 skowski, Angewandte Chemie International Edition 40 F. Birch, Physical Review 71, 809 (Jun 1947), 45, 4365 (2006), ISSN 1521-3773, http://link.aps.org/doi/10.1103/PhysRev.71.809 http://dx.doi.org/10.1002/anie.200600431 41 M. Gajdoˇs and K. Hummer and G. Kresse 60 In their original article16, Kanoun and Said stated and J. Furthm¨uller and F. Bechstedt, that “... there are two atom in wurtzite unit cell, and Physical Review B 73, 045112 (Jan 2006), one in all the other cases.” which is a clear typo! http://link.aps.org/doi/10.1103/PhysRevB.73.045112 61 Z. Wu and R. E. Cohen, 42 W. G. Aulbur, L. J¨onsson, and J. W. Physical Review B 73, 235116 (Jun 2006), Wilkins (Academic Press, 1999) pp. 1 – 218, http://link.aps.org/doi/10.1103/PhysRevB.73.235116 http://www.sciencedirect.com/science/article/pii/S008119470860248962 V. N. Staroverov, G. E. Scuseria, J. Tao, and J. P. 43 J. Kohanoff, Electronic Structure Calculations for Solids Perdew, Physical Review B 69, 075102 (Feb 2004), and Molecules : Theory and Computational Methods http://link.aps.org/doi/10.1103/PhysRevB.69.075102 15

63 J. P. Perdew and S. Kurth, in further in Ref. 5. A Primer in Density Functional Theory, Lecture Notes 67 C. J. Bradley and A. P. Cracknell, The Mathematical in Physics (Springer, 2003) ISBN 9783540030836, Theory of Symmetry in Solids: Representation Theory http://books.google.co.za/books?id=mX793GABep8C for Point Groups and Space Groups (Oxford: Clarendon 64 Since Eq. 4 does not refer to any stoichiometry or Press, 1972) any species (that is, it does not consider the way 68 D. Engin, C. Kemal, and C. Y. Oztekin, that the change in energy or volume was done), we Chinese Physics Letters 25, 2154 (2008), may take the change in volume (or energy) with re- http://stacks.iop.org/0256-307X/25/i=6/a=063 spect to itself, with respect to the parent Ag(A1), or 69 R. de Paiva, R. A. Nogueira, and J. L. A. with respect to any of the other nineteen considered Alves, Physical Review B 75, 085105 (Feb 2007), modifications. http://link.aps.org/doi/10.1103/PhysRevB.75.085105 65 Surely, this needs not to be so. Compare the defini- 70 M. G. Moreno-Armenta and G. Soto, tion 3 with the definition 6. Solid State Sciences 10, 573 (2008), ISSN 1293-2558, 66 It is common that one obtains positive DFT for- http://www.sciencedirect.com/science/article/pii/S12932558070029 mation energy for (even the experimentally synthe- 71 Recall that the optical region (i.e. the visible spec- sized) transition-metal nitrides. Moreover, the zero- trum) is about (390 ∼ 750) nm which corresponds to pressure zero-temperature DFT calculations have to (3.183 ∼ 1.655) eV . be corrected for the conditions of formation of these 72 M. Rohlfing and S. G. Louie, nitrides. Another source of this apparent shortcom- Physical Review B 62, 4927 (Aug 2000), ing stems from the PBE-GGA underestimation of http://link.aps.org/doi/10.1103/PhysRevB.62.4927 the cohesion in N2. We have discussed this point