Theoretical calculations on the structural, electronic and optical properties of bulk silver nitrides
Abstract
We present a firstprinciples investigation of structural, electronic and optical properties of bulk crystalline AgN, AgN and AgN based on density functional theory (DFT) and manybody perturbation theory. The equation of state (EOS), energyoptimized geometries, cohesive and formation energies, and bulk modulus and its pressure derivative of these three stoichiometries in a set of twenty different structures have been studied. Band diagrams and total and orbitalresolved density of states (DOS) of the most stable phases have been carefully examined. Within the randomphase approximation (RPA) to the dielectric tensor, the singleparticle spectra of the quasi electrons and quasi holes were obtained via the GW approximation to the selfenergy operator, and optical spectra were calculated. The results obtained were compared with experiment and with previously performed calculations.
pacs:
I Introduction
It is wellknown now that late transitionmetal nitrides (TMNs) usually possess interesting properties leading to a variety of potential technological applications (1); (2); (3). Hence, a significant number of quantum mechanical ab initio calculations of the structural and physical properties of this family of materials have appeared in the literature.
Since Juza and Hahn (4) succeeded to synthesize CuN in 1939, copper nitrides have been produced through various techniques and their properties and applications have been the subject of many theoretical and experimental published works (5). Due to its early discovery, copper nitride may now be considered as the most investigated among the late TMNs (6).
On the other hand, the nitride of silver, the next element to copper in group 11 of the periodic table, has been known for more than two centuries (7); (8). However, despite its earlier discovery, silver nitride may be the least theoretically studied solid in the late TMNs family. Experimental efforts to investigate structural (9); (8), electronic (8) and formation (7); (10); (9); (11) properties of silver nitrides have been made by some researchers.
In 1949, Hahn and Gilbert (9) carried out the first (8) structural study on the reported stoichiometry, AgN. They claimed an fcc structure with and (i.e. Ag atoms in the unit cell). A long time later in 1982, Haisa (12) suggested that the Ag atoms are located at the corners and face centers of the unit cell, while the N atoms, which may be statistically distributed in the octahedral interstices, were given no definite positions (12).
According to the calculated N radius, AgN can be described as an ionic compound, and recent ab initio calculations on the proposed structure revealed insulating characteristics with a fundamental band gap close to . On the other hand, due to the similar lattice of the parent Ag and the easily separated N as N, it can also be argued that this compound is a metal, supporting its black color (8).
Under ordinary conditions (7), it was found that silver can form AgN
(2) 
Even in storage at room temperature, this solid compound decomposes slowly according to Eq. 2 above (14); (8). From a thermochemical point of view, it was found that there is no stable intermediate stage in this decomposition, but there may be a metastable intermediate species (phase) with a remarkably low decomposition rate (7). At this point, it may be worth mentioning that the thermochemistry of silver nitride systems is not fully documented in standard handbook data (7).
In their 1991 work, Shanley and Ennis (7) stated: “Many of the samples … did not survive the minimum handling required to move them, container and all, to the Xray stage. … More vigorously explosive samples propagated throughout their mass leaving no visible residue. Even among supersensitive materials, silver nitride is a striking example of a compound “teetering on the edge of existence”. Under the circumstances, we did not succeed 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 its sensitive explosive behavior, characterization of silver nitride is hindered by its extremely unstable (endothermic) nature (7); (8), and we are presented with an incomplete picture of structural, electronic and optical properties of this material. Surprisingly, this lack of detailed knowledge of many physical properties of silver nitride stimulated only very few published ab initio studies.
In the present work, firstprinciples calculations were carried out to investigate the lattice parameters, equation of state, relative stabilities, phase transitions, electronic and optical properties of silver nitrides in three different chemical formulae and in various crystal structures. Calculation methods are described in Sec. II. In Sec. III, results are presented, discussed and compared with experiment and with previous calculations. The article is concluded with some remarks in Sec. IV.
Ii Calculation Methods
ii.1 Stoichiometries and Crystal Structures
To the best of our knowledge, the only experimentally reported stoichiometries of AgN compounds are AgN (7) and AgN (7). However, previous ab initio studies on AgN compounds considered AgN (8), AgN (15); (8), AgN (15), AgN (15); (16) and AgN (17); (15) in some cubic structures only. Consideration of stoichiometries other than the reported ones is probably due to the fact that many transition metals nitrides (TMs) are known to form more than one nitride (18). Hence, our interest in investigating AgN and AgN is based on this fact.
For AgN, we consider the following seven structures: the facecentered cubic structure of AlFe (D0), the simple cubic structure of CrSi (A15), the simple cubic structure of the antiReO (D0), the simple cubic structure of AgAu (L1), the bodycentered cubic structure of CoAs (D0), the hexagonal structure of FeN, and the trigonal (rhombohedric) structure of RhF.
For AgN, the following four structures were considered: the facecentered cubic structure of NaCl (B1), the simple cubic structure of CsCl (B2), the facecentered cubic structure of ZnS zincblende (B3), the hexagonal structure of NiAs (B8), the hexagonal structure of BN (B), the hexagonal structure of WC (B), the hexagonal structure of ZnS wurtzite (B4), the simple tetragonal structure of PtS cooperite (B17), and the facecentered orthorhombic structure of TlF (B24).
AgN was studied in the following nine structures: the facecentered cubic structure of CaF fluorite (C1), the simple cubic structure of FeS pyrite (C2), the simple orthorhombic structure of FeS marcasite (C18) and the simple monoclinc structure of CoSb (CoSb).
ii.2 Electronic Relaxation Details
In this work, electronic structure spin density functional theory (SDFT) (19); (20) calculations as implemented in the VASP code(21); (22); (23); (24); (25); (26) have been employed. To selfconsistently solve the KohnSham (KS) equations (27)
(3) 
where , and are the band, point and spin indices, receptively, VASP expands the pseudo part of the KS oneparticle spin orbitals on a basis set of planewaves (PWs). Only those PWs with cutoff energy have been included. The Brillouin zones were sampled using centered MonkhorstPack (28) meshes. Any increase in the value or in the density of the mesh produces a change in the total energy less than . For static calculations, partial occupancies were set using the tetrahedron method with Blöchl corrections (29); (30); (31), while the smearing method of MethfesselPaxton (MP) (32) was used in the ionic relaxation, and Fermi surface of the metallic phases has been carefully treated. The PerdewBurkeErnzerhof (PBE) parametrization (33); (34); (35) of the generalized gradient approximation (GGA) (36); (37); (38) was employed for the exchangecorrelation potentials . The implemented projector augmented wave (PAW) method(39); (26) was used to describe the corevalence interactions , where the electrons of Ag and the electrons of N are treated explicitly as valence electrons. While for these valence electrons only scalar kinematic relativistic effects are incorporated, the PAW potential treats the core electrons in a fully relativistic fashion(25). No spinorbit interaction of the valence electrons has been considered.
ii.3 Geometry Relaxation and EOS
At a set of externally imposed lattice constants, ions with free internal parameters were relaxed until all force components on each ion were less than . This is done following the implemented conjugategradient (CG) algorithm. After each ion relaxation step, static total energy calculation with the tetrahedron method was performed, and the cohesive energy per atom was calculated from
(4) 
where is the number of AgN formulas per unit cell, and are the energies of the isolated nonspherical spinpolarized atoms, and are the stoichiometric weights. The obtained as a function of volume per atom were then fitted to a BirchMurnaghan 3rdorder equation of state (EOS)(40) and the equilibrium volume , the equilibrium cohesive energy , the equilibrium bulk modulus
(5) 
and its pressure derivative
(6) 
were determined.
ii.4 Formation Energy
Beside the cohesive energy, another measure of relative stability is the formation energy . Assuming that silver nitrides AgN result from the interaction between the N gas and the solid Ag(A1) via the reaction (compare with Eq. 2)
(7) 
can be given by
(8) 
Here are the stoichiometric weights and is the cohesive energy per atom as in Eq. 4. The cohesive energy and other equilibrium properties of the elemental metallic silver are given in Table 1. The cohesive energy of the diatomic nitrogen () was found to be corresponding to an equilibrium N–N bond length of (For more details, see Ref. (5)).
ii.5 GWA Calculations and Optical Properties
Although a qualitative agreement between DFTcalculated optical properties and experiment is possible, accurate quantitative description requires treatments beyond the level of DFT (41). Another approach provided by manybody perturbation theory (MBPT) leads to a system of quasiparticle (QP) equations, which can be written for a periodic crystal as (42); (43); (44)
(9) 
Practically, the wave functions are taken from the DFT calculations. However, in consideration of computational cost, we used a less dense mesh of points (). The term is the selfenergy which contains all the exchange and correlation effects, static and dynamic, including those neglected in our DFTGGA reference system. In the socalled approximation (45), is given in terms of the Green’s function as
(10) 
where the dynamically (frequency dependent) screened Coulomb interaction is related to the bare Coulomb interaction through
(11) 
with , the dielectric matrix, calculated within the socalled random phase approximation (RPA). We followed the selfconsistent routine on , in which the QP eigenvalues
(12) 
are updated in the calculations of , while is kept at the DFTRPA level. Four updates were performed, and after the final iteration in , is recalculated within the RPA using the updated QP eigenvalues (43); (44); (46). From the real and the imaginary parts of this frequencydependent microscopic dielectric tensor, one can derive all the other frequencydependent dielectric response functions, such as reflectivity , transmitivity , refractive index , extinction coefficient , and absorption coefficient (47); (48); (49):
(13)  
(14)  
(15)  
(16) 
Iii Results and Discussion
The energyvolume equation of state (EOS) for the different structures of AgN, AgN and AgN are depicted in Figs. 1, 2 and 3, respectively. The corresponding calculated equilibrium properties are given in Table 1. In this table, we ordered the studied phases according to the increase in the nitrogen content; then within each series, structures are ordered in the direction of decreasing structural symmetry. For the sake of comparison, we also presented results from experiment and from previous ab initio calculations; and, whenever appropriate, the calculation method and the functional are also given in footnotes of the Table.
The calculated equilibrium properties: cohesive energies, formation energies, volume per atom, volume per Ag atom, and bulk modulus and its pressure derivative which are given Table 1, are visualized in Fig. 4. This kind of visualization allows us to study the effect of nitridation on the parent Ag(A1), since all quantities in this figure are given relative to the corresponding ones of the elemental Ag(A1) given in the first row of Table 1. Moreover, one can easly compare the properties of these phases relative to each other.
iii.1 EOS and Relative Stabilities
Structure  or  atom  
Ag  
A1  Pres.  –  –  –  
Expt. 
() 
–  –  – 




Comp.  –  –  – 







AgN  
D0  Pres.  –  –  –  
A15  Pres.  –  –  –  
D0  Pres.  –  –  –  
Comp. 

–  –  – 


L1  Pres.  –  –  –  
D0  Pres.  –  –  –  
FeN  Pres.  –  –  
RhF  Pres.  –  –  
fcc 
Expt. 

–  – 


AgN  
B1  Pres.  –  –  –  
Comp. 

–  –  – 




–  –  – 



B2  Pres.  –  –  –  
Comp. 

–  –  – 




–  – 



B3  Pres.  –  –  –  
Comp. 

–  –  – 




–  –  – 



B8  Pres.  –  –  
B  Pres.  –  –  
B  Pres.  –  –  
B4  Pres.  –  –  
Comp. 

– 

– 



B17  Pres.  Pres.  –  –  
B24  Pres.  –  
AgN  
C1  Pres.  –  –  –  
Comp. 

–  –  – 



–  –  – 


C2  Pres.  –  –  –  
C18  Pres.  –  
CoSb  Pres. 
Considering in the AgN series, Fig. 1 shows clearly that the relations of AgN in D0, D0 and RhF phases are almost identical, corresponding to equilibrium cohesive energy (Table 1) of , and , respectively. This behavior in the EOS could be traced back to the structural relationships between these three structures, since both D0 and RhF can simply be derived from the more symmetric D0 (see Ref. (5) and references therein). These structural relations may reflect in the EOS’s and in other physical properties, and the three phases may coexist during the AgN synthesis process. Relative to the elemental Ag, these three phases tend not to change the (Fig. 4), lowering it only by , as can been seen from Table 1. It may be worth to mention here that the simple cubic D0 phase is the stable phase of the synthesized CuN (4); (55).
The odd behavior of the EOS of AgN(FeN) with the existence of two minima (Fig. 1) shows that the first minima (to the left) is a metastable local minimum that cannot be maintained as the system is decompressed. Ag ions are in the Wyckoff positions: and ; with to the left of the potential barrier (represented by the sharp peak at ), and to the right of the peak. It may be relevant to mention here that CuN(FeN) was found to behave in a similar manner (5).
The crossings of the less stable A15, D0 and L1 EOS curves with the more stable D0, D0 and RhF ones at the left side of their equilibrium points reveals pressureinduces phase transitions from the latter phases to the former. To show this, we plotted the corresponding relation between enthalpy and the imposed pressure in Fig. 5. Since D0, D0 and RhF phases have identical curves, the corresponding curves are also identical. Hence, only the of D0 is displayed in Fig. 5. A point where the enthalpies of two phases are equal determine the phase transition pressure ; and, indeed, the direction of the transition is from the higher to the lower (56). As depicted in Fig. 5, , and . Thus, D0, D0 and RhF would not survive behind these ’s and A15, D0 and L1 are preferred at high pressure.
Fig. 4 reveals that the AgN group contains the least stable phase among all the twenty studied phases: the hexagonal B. Fig. 2 and Table 1 show that the simple tetragonal structure of cooperite (B17) is the most stable phase in this AgN series. In fact, one can see from Fig. 4 and Table 1 that all the considered AgN phases possess less binding than their parent Ag(fcc), except AgN(B17) which is slightly more stable, with lower . It is interesting to notice that AgN(B17) is more stable than the AgN most stable phases. Moreover, this B17 structure was theoretically predicted to be the groundstate structure of CuN (5), AuN (57), PdN (58) and PtN (59).
Using the fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within LDA and within GGA, Kanoun and Said (16) studied the EOS for AgN in the B1, B2, B3 and B4 structures. The equilibrium energies they obtained from the EOS revealed that B1 is the most stable phase, and the relative stability they arrived at is in the order B1–B3–B4–B2, with a significant difference in total energy between B3 and B4 (see Fig. 2(b) in that article). Within this subset of structures, the numerical values of in Table 1 do have the same order. However, the difference between the equilibrium (B3) and (B4) is only , and the EOS of B3 and B4 match/overlap over a wide range of volumes around the equilibrium point. This discrepancy may be attributed to the unphysical/illdefined measure of stability that Kanoun and Said used, the total energy, while the number of the AgN formula units per unit cell in the B4 structure differs from that in the others
In the CuN nitrogenrichest phase series, we can see from Table 1 and from Fig. 4 that the phases of this group are significantly more stable than all the other studied phases, except C1, which is, in contrast, the second least stable among the twenty studied phases, with more than AgN(B). From Fig. 3, Fig. 4 and Table 1, one can see that in this series, the lower the structural symmetry, the more stable is the phase. It was found that CuN phases have the same trend (5).
iii.2 Volume per Atom and Lattice Parameters
The numerical values of the lattice parameters and the average equilibrium volume per atom for the twenty modifications are presented in Table 1. The middle subwindow of Fig. 4 depicts the values relative to the Ag(fcc). To measure the average of the Ag–Ag bond length in the silver nitride, the equilibrium average volume per Ag atom (), which is simply the ratio of the volume the unit cell to the number of Ag atoms in the unit cell, is visualized in the same subwindow.
From the curve in Fig. 4, we can see that, all AgN and AgN modifications, except the open AgN(B) phase, decrease ; while the AgN phases tend, on average, not to change the number density of the parent Ag(A1).
On the other hand, the curve in Fig. 4 reveals that, relative to the elemental Ag and to each other, tends to increase with the increase in the nitrogen content. Thus, in all these nitrides, the introduced N ions displace apart the ions of the host lattice causing longer AgAg bonds than in the elemental Ag. This cannot be seen directly from the values depicted in the same figure.
For AgN in the B1, B2, B3 and B4 structures, Kanoun and Said (Ref. (16) described in Sec. III.1 above) obtained GGA equilibrium lattice parameters which are in very good agreement with ours. Their obtained LDA lattice parameter values show the common underestimation with respect to their and our GGA values (see Table 1).
Gordienko and Zhuravlev (15) studied the structural, mechanical and electronic properties of AgN(B1) , AgN(B2), AgN(B3), AgN(C1) and AgN(D0) cubic phases. Their DFT calculations were based on pseudopotential (PP) method within LDA, and on linear combinations of atomic orbitals (LCAO) method within both LDA and GGA. For comparison, some of their findings are included in Table 1. Within the parameter subspace they considered, our GGA values of the lattice parameter agree very well with theirs. On the other hand, although their PP values are closer to the GGA ones (ours and theirs), all their LDA values are less than the GGA ones. This confirms the wellknown behavior of LDA compared to GGA (61); (62); (63). Gordienko and Zhuravlev also found that the Ag–Ag interatomic distance increases in the order AgN–AgN–AgN. This agrees with the general trend shown in Fig. 4, since the curve shows an average increase in the same direction.
iii.3 Bulk Modulus and its Pressure Derivative
Fig. 4 reveals that AgN phases tend, on average, to preserve the value of the parent Ag(A1). Increasing the nitrogen content to get AgN phases will increase the value of the parent Ag(A1), except in the case of B. While the nitrogen in AgN tends to lower the value of the parent Ag(A1), the cubic C1 phase posses the highest value. This could be seen from Fig. 3, where the curvature of the curve of C1 is higher compared to the shallow minima of the C2, C18 and CoSb curves.
From the definition of the equilibrium bulk modulus (Eq. 5), one would expect to increase as or decreases. This is because of the minus sign of the former and the inverse proportionality of the latter. That is, roughly speaking, the curve should have a mirror reflectionlike behavior with respect to the and curves. Nevertheless, if or are increasing and the other is decreasing, then the dominant net effect will be of the one with the higher change
It is common to measure the pressure dependence of by its derivative (Eq. 6). Fig. 4 shows that the value of the C2, C18 and CoSb phases increases as these phases are put under pressure. While the values of the rest of the phases shows very low sensitivity to pressure and they tend to slightly lower the bulk modulus, the FeN phase is the most sensitive phase and tends to significantly lower its upon application of pressure. This high sensitivity may indicate that the corresponding minimum on the potential surface is not global, but another local minimum as the one at (Fig. 1).
From the elastic constants they obtained, Gordienko and Zhuravlev(15) calculated the corresponding macroscopic bulk moduli (included in Table 1). They found the highest LDA value for AgN(B1) among all phases they considered, but, in agreement with the present work, they obtained the highest GGA value for AgN(C1). Since LDA relative to GGA overestimates and thus underestimates , each of these two factors (see Subsection III.3) would separately lead to the odd LDA value of GPa which they obtained. Nevertheless, due to this fact, Gordienko and Zhuravlev argued that one should consider the LDA and GGA average value of .
iii.4 Formation Energies
Formation energies in the present work are used as a measure of the relative thermodynamic stabilities of the phases under consideration. That is, the lower the formation energy, the lower the tendency to dissociate back into the constituent components Ag and N.
The obtained formation energies of the twenty relaxed phases are given in Table 1 and depicted graphically in Fig. 4. The latter shows that, relative to each other and within each series, the formation energy (defined by Eqs. 7 and II.4) of the studied phases has the same trend as the cohesive energy
Some of the experimental values of for the synthesized AgN phase (which is claimed to be in an fcc structure) are (11) = , (10) = , (9) = and (7) = ; with an average value of . Among the considered phases in the present work, there is only one phase wich has value that fits in this range, the AgN(C1). Interestingly, this C1 structure has an fcc underlying Bravia lattice; however, the chemical formula differs from that of the synthesized phase.
iii.5 Electronic Properties
The DFT(GGA) calculated band diagrams (i.e. curves) and spinprojected total and orbital resolved (i.e. partial) density of states (DOS) of the most stable phases: D0, RhF, D0, B17, and C18 are presented in Figs. 6, 7, 8, 9 and 10, respectively. Spinprojected total density of states (TDOS) are shown in subfigure (b) in each case. In all the six considered cases, electrons occupy the spinup and spindown bands equally, resulting in zero spinpolarization density of states: . Thus, it is sufficient only to display spinup (or spindown) density of states (DOS) and spinup (or spindown) band diagrams. In order to investigate the details of the electronic structure of these phases, energy bands are plotted along densely sampled highsymmetry string of neighboring points. Moreover, to extract information about the orbital character of the bands, the Ag() and N() partial DOS are displayed at the same energy scale.
Fig. 6(a) shows the band structure of AgN(D0). With its valence band maximum (VBM) at and its conduction band minimum (CBM) at , AgN(D0) presents a semiconducting character with a narrow indirect band gap of . From subfigures 6(c) and (d), it is seen clearly that the Ag()N() mixture in the region from to beneath , with two peaks: a low density peak around and a high density peak around steaming mainly from the bands of silver electrons.
Our obtained PDOS, TDOS and band structure of AgN(D0) agree qualitatively well with Gordienko and Zhuravlev (15); however, using LCAO method within GGA, the value of the indirect of AgN(D0) they predicted is .
To the best of our knowledge, there is no experimentally reported value for AgN. However, Tong (8) prepared AgN samples, and carried out XRD measurements to confirm the fcc symmetry of the prapared samples. Using a TBLMTO code within LDA, Tong then calculated the band structure of AgN and obtained an indirect energy gap of . Nevertheless, we could not figure out the positions of the N ions Tong’s model.
It is a well known drawback of KohnSham DFTbased calculations to underestimate the band gap. Thus the more demanding calculations were carried out, and the obtained value will be presented in Sec. III.6.
Calculated electronic properties of AgN(D0) are displayed in Fig. 8. subfigure 8(a) shows the energy bands of AgN(D0). With its valence band maximum (VBM) at and its conduction band minimum (CBM) at , AgN(D0) presents semiconducting character with a narrow indirect band gap of . From subfigures 8(c) and (d), one can notice clearly the Ag()N() mixture in the region from to below , with two peaks: a low density peak around steaming from an almost equal mixture of Ag() and N(), and a high density peak around steming mainly from the bands of silver electrons plus a relatively very low contribution from the N() states.
Fig. 7 depicts the band diagram and DOS’s of AgN(RhF). In contrast to AgN(D0) and AgN(D0), subfigure 7(a) shows that AgN(RhF) is a semiconductor with a narrow direct band gap of of width located at point. The VBM is at and the CBM is at . From subfigures 7(c) and (d), one can see the Ag()N() mixture is in the region from to beneath , with two peaks: a low density peak around steaming from an almost equal mixture of Ag() and N(), and a high density peak around steaming mainly from the bands of silver electrons plus a relatively very low contribution from the N() states.
The relationship between D0, D0 and RhF structures manifests itself in many common features between the electronic structure of these three AgN nitrides: (i) equal of ; (ii) a deep bound band around below consists mainly of the N states; (iii) a broad valence band with of width that comes mostly from the electrons of Ag plus a very small contribution from N; and (iv) the relatively low TDOS of the conduction bands.
Energy bands , total density of states (TDOS) and partial (orbitalresolved) density of states (PDOS) of AgN(B17) are shown in Figs. 9. It is clear that AgN(B17) would be a true metal at its equilibrium. The major contribution to the very low TDOS around Fermi energy comes from the states of the N atoms as it is evident from subfigure 9(d). Beneath lies a band with of width, in which the main contribution is due to the Ag() states plus a small contribution from the N() states. While the N() states dominate the deep lowest region around , the low density unoccupied bands stem mainly from the N() states. The Fermi surface crosses two partly occupied bands: a lower one in the ,  and  directions, and a higher band in the  and  directions. Thus, is not a continuous surface contained entirely within the first BZ.
It may be worth mentioning here that AgN(B1) (16); (68) and AgN(B3) (16); (69) phases were also theoretically predicted to be metallic.
Although AgN(CoSb) is the most stable phase, but the difference in cohesive energy between AgN(CoSb) and AgN(C18) is less than , and we decided to examine the electronic structure of both phases. With crossing the finite TDOS, Fig. 10 shows that AgN(C18) is metallic at . The orbital resolved DOS’s reveal that the major contribution to the low TDOS at comes from the N() states with tiny contributions from the , and states of Ag, respectively. As one can see from subfigure 10(a), the surface crosses the edges of the first Brillouin zone in the  and  directions.
The calculated electronic properties of AgN(CoSb) are displayed in Fig. 11. Band structure, TDOS and orbital resolved DOS’s have almost the same features as the corresponding ones of AgN(C18). It may be worth to mention here that C1 phase of AgN was also theoretically predicted to be metallic (15).
Compared to the metallic AgN(B17), three new features of these 1:2 nitrides are evident: (i) Deep at there is a highlylocalized mixture of the N()N() states. However, the variation in N() energy with respect to k is smaller than the variation of N() states, resulting in a narrower and higher PDOS. (ii) Below the band that is crossed by there are four bands separated by , , and energy gaps, respectively. (iii) The very tiny contribution of the N() states to the N()Ag() band.
A common feature of all the studied cases is that Ag()orbitals do not contribute significantly to the hybrid bands. Another common feature is the highly structured, intense and narrow series of peaks in the TDOS valance band corresponding to the superposition of N() and Ag() states. In their space, Ag() energies show little variation with respect to ; hence the Van Hove singularitieslike sharp features.
To summarize, we have found that the most stable phases of AgN and AgN are metallic, while those of AgN are semiconductors. A close look at Fig. 9 up to Fig. 6 reveals that as the nitrogen to silver ratio increases from to , the TDOS at decreases; and by arriving at a gap opens. This finding agrees well with Gordienko and Zhuravlev (15). Moreover, it may be worth mentioning here that such behavior was theoretically predicted to be true for copper nitrides as well (5); (70).
iii.6 Optical Properties
Fig. 12 depicts the calculated real and imaginary parts of the frequencydependent dielectric function of AgN(D0) and the corresponding derived optical constants. The optical region
The real part (subfigure 12(a)) shows an upward trend before , where it reaches its maximum value and generally decreases after that. The imaginary part (same subfigure 12(a)) shows an upward trend before and it has three main peaks located at in the optical region, at the right edge of the optical region, and at in the UV range.
Calculated reflectivity and transmitivity are displayed in subfigure 12(b). With , it is evident that AgN(D0) is a good reflector, specially in the red and the infrared regions. In the visible range, the maximum transmitivity is at , which is at the bluegreen edge.
subfigure 12(c) depicts the calculated refraction and extinction coefficients. As they should, these two spectra have, in general, the same qualitative frequency dependence as the real and the imaginary dielectric functions, respectively.
From the absorption coefficient spectrum (subfigure 12(d)), it can be seen that AgN(D0) starts absorbing photons with energy. Hence, it is clear that calculations give a band gap of , which is a significant improvement over the value obtained from DFT. The nonvanishing in the whole optical region agrees with the experiment, since it may explain the observed black color of the synthesized AgN.
To the best of our knowledge, the present work is the first trial to theoretically investigate the optical properties of silver nitride. However, for more accurate optical characterization (e.g. more accurate positions and amplitudes of the characteristic peaks), electronhole excitations should be calculated. This can be done by evaluating the twobody Green function on the basis of our obtained GW oneparticle Green function and QP energies, then solving the socalled BetheSalpeter equation, the equation of motion of (72).
Iv Conclusions
We have succesfully employed firstprinciples calculation methods to investigate the structural, stability, electronic and optical properties of AgN, AgN and AgN. Within the accuracy of the employed methods, the obtained structural parameters, EOS, , and electronic properties show good agreement with the few avialable previous calculations. On the other hand, our obtained results show, at least, partial agreement with three experimental facts: (i) the lattice parameter of AgN(D0) is close to the experimentally reported one; (ii) the positive formation energies reveals the endothermic (unstable) nature of silver nitrides, and (iii) absorption spectrum explains its observed black color. Moreover, the present work may be considered as the first trial to theoretically investigate the optical properties of silver nitride. We hope that some of our obtained results will be confirmed in future experimentally and/or theoretically.
Acknowledgments
All GW calculations and some DFT calculations were carried out using the infrastructure of the Centre for High Performance Computing (CHPC) in Cape Town. Suleiman would like to acknowledge the support he received from Wits, DAAD, AIMS, SUST and the ASESMA group. Many thanks to the Scottish red pen of Ross McIntosh!
Footnotes
 AgN, formerly termed fulminating silver by its discoverers, can be formed from ammoniacal solutions of silver oxide according to the following reaction (1) It can also be formed by means of other reactions (7); (8).
 Ref. (50). This is an average of 56 experimental values, at .
 Ref. (51). Cohesive energies are given at and ; while bulk mudulii are given at room temperature.
 Ref. (51). Cohesive energies are given at and ; while bulk mudulii are given at room temperature.
 Ref. (25) in (52): at room temperature.
 See Refs. (8)–(11) in (52).
 Ref. (53). using the fullpotential linearized augmented plane waves (LAPWÍ) method within LDA.
 Ref. (54): using projector augmented wave (PAW) method within LDA.
 Ref. (54): using projector augmented wave (PAW) method within GGA(PW91).
 Ref. (53). using the fullpotential linearized augmented plane waves (LAPWÍ) method within LDA.
 Ref. (52): using the socalled method of transition metal pseudopotential theory; a modified form of a method proposed by Wills and Harrison to represent the effective interatomic interaction.
 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 highpressure experiments. The given values are estimated at .
 Ref. (54): using projector augmented wave (PAW) method within GGA(PBE).
 Ref. (52): using a semiempirical method in which the experimental static data are fitted to an EOS form where and are adjustable parameters. The given values are estimated at .
 Ref. [(15)]: using pseudopotential (PP) method within LDA.
 Ref. [(15)]: using linear combinations of atomic orbitals (LCAO) method within LDA. ’s are calculated from elastic constants.
 Ref. [(15)]: using linear combinations of atomic orbitals (LCAO) method within GGA. ’s are calculated from elastic constants.
 Ref. [(15)]: using linear combinations of atomic orbitals (LCAO) method within LDA. ’s are calculated from elastic constants.
 Ref. [(15)]: using linear combinations of atomic orbitals (LCAO) method within GGA. ’s are calculated from elastic constants.
 This is the face centered cubic (fcc) structure with (i.e. 4 Ag atoms in the unit cell) suggested by Hahn and Gilbert according to some measurements (Ref. (9)).
 Ref. (9).
 Ref. (8).
 Ref. (12).
 This is the average of the experimental values: (7) = , (11) = , (9) = , and (10) = . We used the conversion relation: or equivalently .
 Ref. [(15)]: using pseudopotential (PP) method within LDA.
 Ref. [(15)]: using linear combinations of atomic orbitals (LCAO) method within LDA. ’s are calculated from elastic constants.
 Ref. [(15)]: using linear combinations of atomic orbitals (LCAO) method within GGA. ’s are calculated from elastic constants.
 Ref. [(15)]: using linear combinations of atomic orbitals (LCAO) method within LDA. ’s are calculated from elastic constants.
 Ref. [(15)]: using linear combinations of atomic orbitals (LCAO) method within GGA. ’s are calculated from elastic constants.
 Ref. [(16)]: using using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within GGA(PBE).
 Ref. [(16)]: using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within LDA.
 Ref. [(16)]: using using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within GGA(PBE).
 Ref. [(16)]: using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within LDA.
 Ref. [(16)]: using using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within GGA(PBE).
 Ref. [(16)]: using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within LDA.
 Ref. [(15)]: using pseudopotential (PP) method within LDA.
 Ref. [(15)]: using linear combinations of atomic orbitals (LCAO) method within LDA. ’s are calculated from elastic constants.
 Ref. [(15)]: using linear combinations of atomic orbitals (LCAO) method within GGA. ’s are calculated from elastic constants.
 Ref. [(16)]: using using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within GGA(PBE).
 Ref. [(16)]: using using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within GGA(PBE).
 Ref. [(16)]: using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within LDA.
 Ref. [(16)]: using using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within GGA(PBE).
 Ref. [(16)]: using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within LDA.
 Ref. [(16)]: using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within LDA.
 Ref. [(15)]: using pseudopotential (PP) method within LDA.
 Ref. [(15)]: using linear combinations of atomic orbitals (LCAO) method within LDA. ’s are calculated from elastic constants.
 Ref. [(15)]: using linear combinations of atomic orbitals (LCAO) method within GGA. ’s are calculated from elastic constants.
 Ref. [(16)]: using using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within GGA(PBE).
 Ref. [(16)]: using using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within GGA(PBE).
 Ref. [(16)]: using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within LDA.
 Ref. [(16)]: using using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within GGA(PBE).
 Ref. [(16)]: using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within LDA.
 Ref. [(16)]: using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within LDA.
 Ref. [(16)]: using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within LDA.
 Ref. [(16)]: using using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within GGA(PBE).
 Ref. [(16)]: using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within LDA.
 Ref. [(16)]: using using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within GGA(PBE).
 Ref. [(16)]: using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within LDA.
 Ref. [(16)]: using using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within GGA(PBE).
 Ref. [(16)]: using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within LDA.
 Ref. [(16)]: using using fullpotential (linearized) augmented plane waves plus local orbitals (FPLAPW+lo) method within GGA(PBE).
 Ref. [(15)]: using pseudopotential (PP) method within LDA.
 Ref. [(15)]: using linear combinations of atomic orbitals (LCAO) method within LDA. ’s are calculated from elastic constants.
 Ref. [(15)]: using linear combinations of atomic orbitals (LCAO) method within GGA. ’s are calculated from elastic constants.
 Ref. [(15)]: using linear combinations of atomic orbitals (LCAO) method within LDA. ’s are calculated from elastic constants.
 Ref. [(15)]: using linear combinations of atomic orbitals (LCAO) method within GGA. ’s are calculated from elastic constants.
 Ref. [(17)]: using the fullpotential linearized augmented plane waves (LAPWÍ) method within LDA.
 Ref. [(17)]: using the fullpotential linearized augmented plane waves (LAPWÍ) method within GGA.
 Ref. [(17)]: using the fullpotential linearized augmented plane waves (LAPWÍ) method within LDA.
 Ref. [(17)]: using the fullpotential linearized augmented plane waves (LAPWÍ) method within GGA.
 In their original article(16), Kanoun and Said stated that “… there are two atom in wurtzite unit cell, and one in all the other cases.” which is a clear typo!
 Since Eq. 5 does not refer to any stoichiometry or any species (that is, it does not consider the way that the change in energy or volume was done), we may take the change in volume (or energy) with respect to itself, with respect to the parent Ag(A1), or with respect to any of the other nineteen considered modifications.
 Surely, this needs not to be so. Compare the definition 4 with the definition 7.
 It is common that one obtains positive DFT formation energy for (even the experimentally synthesized) transitionmetal nitrides. Moreover, the zeropressure zerotemperature DFT calculations have to be corrected for the conditions of formation of these nitrides. Another source of this apparent shortcoming stems from the PBEGGA underestimation of the cohesion in N. We have discussed this point further in Ref. (5).
 Recall that the optical region (i.e. the visible spectrum) is about which corresponds to .
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