Chapter 3. Electronic and Dynamical Properties of Transition Metal Nitrides

Transcription

1 Chapter 3 Electronic and Dynamical Properties of Transition Metal Nitrides

2 Electronic and Dynamical Properties of Transition Metal Nitrides Chapter Introduction Metals share many characteristics, but they don't all have the same reactivity. Some metals don't react at all with other metals, and because of this they can be found in a pure form (examples are copper, gold and platinum). Because copper is relatively inexpensive and has a low reactivity, it's useful for making pipes and wiring. Other metals are highly reactive and combine easily with other elements, such as oxygen. These metals are never found in a pure form, and are difficult to separate from the minerals they are found in. Potassium and sodium are the most reactive metals. They react violently with air and water; potassium can ignite on contact with the water. Transition metal nitrides have been the subject of many theoretical and experimental investigations due to their high scientific interest and technological importance. It has been the topic of intense scientific research and technological developments ever since they were discovered. Particularly, the early binary transition metal nitrides have an attractive mixture of physical and chemical properties. They constitute a diverse class of material properties with interesting technological and fundamental importance because of their strength and durability as well as their optical, electronic and magnetic properties. In most cases, they are metallic and are used as a barrier layers and contacts. Most of the transition metals found fourth in the periodic table to form refractory nitrides and carbides at extremely high melting points. The transition metal nitrides have served as catalyst for the synthesis or decomposition reaction and its nitrides are interesting reaction intermediates. This group of nitrides is characterized by high melting point (except molybdenum nitrides and tungsten nitrides), ultra hardness (comparable to diamond), good electrical and thermal conductivity, and slow resistance to corrosion. Due to its versatile and unique properties, these nitrides are widely used as refractory materials, wear and corrosion resistance surface coatings [1] and thin film ~ 33 ~

3 Electronic and Dynamical Properties of Transition Metal Nitrides Chapter 3 interconnections in integrated circuits [2]. Application of transition metal-nitrides has also been attempted for the electrochemical systems, including fuel cells, electric storage and secondary battery as electrocatalysts with low cost and high efficiency, although not as much as transition metal-carbides. Recently, various investigations have reported on transition metal-nitrides as electrocatalyst in the mass production and synthesis fields, and the attention has been growing exponentially. It has been well known that all electrochemical systems depended heavily on the large amount of transition noble-metals such as Pt, which makes them more expensive. As mentioned earlier, transition metal-nitrides have been considered as a promising electrocatalyst to replace the noble materials due to their unique catalytic properties, low price, and abundance. The electrochemical stability is one of the important considerations of electrochemical properties for long-term stability in real system. Further, relative to parent metals, they behave like noble metals for electrochemical reactions such as oxidation of hydrogen, CO and alcohols, and reduction of oxygen. The combination of transition metal nitrides with other metals, the electro catalytic synergy is often being observed in electrochemical reactions. Thus, combinations with a minute amount of Pt or even non- Pt metals gives higher performance comparable to heavily loaded Pt-based electro catalysts for low temperature fuel cells. By incorporating any of the light elements, boron (B), carbon (C), nitrogen (N), or oxygen (O) with the transition metal (e.g titanium, vanadium, chromium, zirconium, niobium, hafnium, tantalum, platinum or tungsten) compounds of high hardness and compressibility can be synthesized by means of high pressure and high temperature [3]. Recent research has also shown a strong pressure dependence of the oxidation state of metals in these binary compounds, which promises excellent possibilities of ~ 34 ~

4 Electronic and Dynamical Properties of Transition Metal Nitrides Chapter 3 discovering novel members of the transition metal nitride group. For the binary nitrides of transition metals, the oxidation states of the cations are limited by a value of +3, even when the available valence electrons are higher [4]. However, experimental synthesis shows that most transition metal nitrides like hafnium nitrides (Hf 3 N 4 ), as well as the isomorphic Zr 3 N 4 and Ti 3 P 4 ; Ta 3 N 5 [5] and the other noble metal pernitrides PtN 2 [6], IrN 2 and OsN 2 [7] have been synthesized successfully under extreme conditions of pressure and temperature. The large bulk moduli observed experimentally for this class of materials have suggested potential superhard and incompressible solids which could be indispensable for industrial applications and can be used in cutting tools and wear-resistance coatings. In Earth and planetary sciences, laboratory experiments utilizing high pressure and temperature offer the only means to examine directly the conditions of deep planetary interiors. Fundamental questions about phase transformations, crystal structure and the nature of atomic bonding can be answered using high-pressure techniques. Finally, the use of high pressure to synthesize new materials, to study the behavior of existing materials and to tune materials physical properties provides tremendous potential for advances in applied materials research, which has not been possible without development of diamond anvil cell technology resulted in tremendous gains in knowledge of the physical world through scientific investigation of the behavior of matter under a wide range of pressures. ~ 35 ~

5 Platinum Nitride (PtN) and Platinum Pernitride (PtN2) Chapter 3 Figure 3.1 The elements investigated from transition metals The present chapter of thesis describes the structural, electronic, vibrational and thermodynamical properties of variety of transition metal nitrides at ambient and higher pressures. We also report the pressure driven structural phase transition in these transition nitrides to find possible ground state of these compounds. In Fig. 3.1 we mark by yellow color in periodic table to the transition metal elements whose nitrides have been considered here for the above said investigations. 3.2 Platinum Nitride (PtN) and Platinum Pernitride (PtN 2 ) The credit of intense research in the field of transition metal nitrides goes to the successful synthesis of platinum nitride by Gregoryanz et al [8]. New crystalline compound PtN was formed using the laser-heated diamond anvil cell set-up at temperature above 2000 K and pressure of 50 GPa. The system obtained after quenching to atmospheric pressure and room temperature was characterized by Raman spectroscopy and synchrotron x-ray diffraction systematically. The platinum nitride has ~ 36 ~

6 Platinum Nitride (PtN) and Platinum Pernitride (PtN2) Chapter 3 cubic structure, where metal atoms situated on an fcc lattice, while the occupation sites of nitrogen atoms could not be clarified because of the large mass difference between platinum and nitrogen. However, this binary nitride was determined to have a remarkably high bulk modulus (B) of 372 GPa, which is 100 GPa higher than that of bulk Pt ( 270 GPa). This is comparable with the bulk modulus, 382 GPa of the superhard cubic material boron nitride. Theoretical calculations for transition metal nitrides [9] indicate that in general if the bulk modulus of the pure metal is large it will be comparable for the corresponding nitride, and predicted that this enhancement probably arises from a strong directional transition metal nitrogen covalent bond with orbits being close to half-filled [14]. Using a magnetic-susceptibility technique [10], the visual appearance and the absence of the superconducting signal suggest that PtN is either a poor metal or a semiconductor with a small bandgap, which is different to the most of the other supplementary transition metal nitrides VN and NbN. The measured Raman spectrum exhibited a strong longitudinal optic (LO) mode and weak transverse optical (TO) mode. The frequency of modes increases almost linearly with pressure [1]. Since, a platinum nitride was synthesized under high pressure and temperature shown to possess a large bulk modulus, but still the structure of the compound was unknown. Several theoretical investigations of platinum nitride have been carried out [12-19], and the consensus appears to be that the compound does not crystallize in the proposed zincblende crystalline structure [8], because this arrangement violates the requirement of positive strain energy [16-19]. Yu and Zhang [16,19] have theoretically suggested that the platinum nitride is instead stable in the fluorite structure, an arrangement in which the nitrogen atoms occupy all the tetrahedral interstitial sites of the face-centered cubic (fcc) metal lattice and which necessitates a stoichiometry of PtN 2. Later experiments performed by Crowhurst et al. [6] suggest that the platinum nitride has a ~ 37 ~

7 Platinum Nitride (PtN) and Platinum Pernitride (PtN2) Chapter 3 stoichiometry of PtN 2 and that the structure is similar to that of pyrite. Under similar conditions, they could synthesize a recoverable nitride of iridium within the similar stoichiometry; which has a much lower structural symmetry. Thus the recent two experimental and other theoretical calculations created controversy regarding the correct structure of the platinum nitride. The present work aims to clarify this controversy by calculating the structural, electronic and lattice dynamical properties using state of art density functional theoretical calculations Computational Methodology To determine the ground state properties, electronic band structure and phonon properties for platinum nitride in zinc blend (ZB), rocksalt (RS) and pyrite structures, we have performed pseudopotential plane-wave density functional theory (DFT) in the Kohn Sham framework using Quantum-Espresso [20] and ABINIT simulation packages [21]. For the exchange-correlation functional, we have employed local density and the generalized-gradient approximations. For ZB and RS phases the planewave basis sets have been applied up to a kinetic energy cutoff of 90 Ry and 95 Ry respectively. In order to find an appropriate cutoff energy, the total energy as a function of the cutoff energy has been calculated for several different energy cutoff values ranging from 50 to 125 Ry. The total energy convergence criteria were 10-4 Ry for two successive total energy calculations. In the present calculation, we employed fixed lattice constants Å for ZB and RS type crystal structure as an experimental value [8]. The self-consistent calculations are considered to be converged when the total energy of the system is stable within 10 8 Ry. Here, for the self-consistency, the initial potential for the next iteration is constructed using a convergence stabilization scheme. The number of sampling k-points used in the Brillouin zone (BZ) summation of the electronic density and total energy is increased until the total energy converges to ~ 38 ~

8 Platinum Nitride (PtN) and Platinum Pernitride (PtN2) Chapter 3 the desirable tolerance. Thirty six special k-points in the irreducible Brillouin zone are sufficient to achieve the convergence for the total energy. To generate the ( ) mesh in the Brillouin zone, the Monkhorst-Pack scheme [22] has been used. The crystal structure and an associated equilibrium lattice constant have been obtained by minimizing the calculated total energies as function of a lattice constant. Further, to perform the lattice dynamics calculations of PtN, we use the Quantum-Espresso code based upon density functional perturbation theory (DFPT) [23]. In this method, the dynamical matrix, which provides information on lattice dynamics of the system, can be obtained from the ground state electron charge of the nuclear geometry. The kinetic cutoff energy and numbers of k-points mentioned above are found to yield phonon frequencies converged to within 2 5 cm 1. The calculations for platinum nitride called as platinum pernitride due to the formation of nitrogen dimer in pyrite phase is performed using both LDA and GGA on to with Monkhorst Pack k-point mesh of and cutoff energy of 45 Ry while and energy cutoff of 40 Ry respectively in the case of Quantum Espresso calculations. We have also performed these studies using Abinit code [28]. In the case of ABINIT package, the pseudopotential in GGA format is used. The exchange correlation functional are treated within both LDA and GGA pseudopotentials of Troullier and Martins type for Pt and N, which are available on the ABINIT website, are chosen to describe the interaction between the valence electrons and the nuclei and core electrons. The lattice dynamical properties of platinum nitride are calculated within the framework of the self consistent density functional perturbation theory (DFPT) [23] like earlier case. ~ 39 ~

9 Platinum Nitride (PtN) and Platinum Pernitride (PtN2) Chapter Results and Discussion The calculated electronic band structure of PtN in both ZB and RS phases are shown in Fig No energy gap at Fermi level has been found for any structure indicates metallic nature of PtN in both phases because of sequential filling of the d bands. However, no conclusion regarding the conductivity can be drawn at this point due to Figure 3.2 Electronic band structures for PtN; (a) zinc blende and (b) rocksalt phases. The Fermi energy is set to zero as shown by dashed horizontal line. non-availability of any experimental data. The insulating character is predicted for its pyrite structure. The bands near the Fermi level and above 8 ev are mainly due to platinum d and nitrogen p orbitals. The calculated equilibrium values for lattice constant and bulk modulus for ZB and RS phases are given in Table I together ~ 40 ~

10 Platinum Nitride (PtN) and Platinum Pernitride (PtN2) Chapter 3 with the experimental values [8] and other calculations [16-19]. The Table I depicts that the values of lattice constant and bulk modulus are not in very good agreement with the Table I Calculated frequency of the phonon modes at some main symmetry points of the Brillouin zone for PtN in zincblende and rocksalt structures. Lattice structure Present work (cm -1 ) Expt. (cm -1 ) [5][9] Modes Γ-Point L-Point X-Point Other works ZB-PtN RS-PtN TA LA TO LO TA LA TO LO a, 854 b, 628 c a Raman intensity calculation [23] of PtN 2, b Raman frequency [22] of PtN 2, c Raman frequency [21] of PtN 2 values reported in ref [8, 13]. The bulk modulus and equilibrium lattice constants have been obtained from fitting calculated energies over a range of volumes with a second Figure 3.3 Calculated total electronic density of states of ZB-PtN and RS-PtN. The Fermi energy is set to zero as shown by dashed vertical line. order Birch-Murnaghan equationof states. The bulk modulus for RS-PtN is higher than the ZB-PtN phase, but agrees with some of the earlier calculated values [17]. The Table ~ 41 ~

11 Platinum Nitride (PtN) and Platinum Pernitride (PtN2) Chapter 3 I reveals that the ZB-PtN agrees better with the experimental values [5]. For a better understanding of the electronic and optical properties of the PtN compound, the investigation of electronic band structure can be useful. The total density of states of the PtN is presented in Fig 3.3. Fig. 3.3 shows a strong hybridization between the N and Pt states, which is more prominent in the case of zincblende structure. The states between -15 and ev are designated with N (2s) states with a small contribution from Pt. The states above -10 ev are mainly composed of Pt (5d) and N (2p) orbitals. The d- electrons of Pt contribute to the majority of the density of states (DOS) near the Fermi level. In the case of platinum pernitride, we present the bandstructure profile only for the GGA-PWSCF calculations as other calculations show almost similar behavior except with some minor changes The electronic DOS reflects that the platinum pernitride is semiconductor in nature with a very low Figure 3.4 Total charge density of PtN; (a) zinc blend and (b) rocksalt phases. density of states at the Fermi level. The present band profiles agree well with the earlier works [24]. The band gap obtained using LDA-PWSCF, GGA-PWSCF, LDA-ABINIT and GGA-ABINIT is 1.99, 1.87, 1.89 and 1.83 ev, respectively. This indicates that all calculations produce almost same band gap. The strong hybridization indicates the covalent bonding and leads to a separation of the bonding states. In order to visualize the nature of bonding character and to explain the charge transfer in PtN, ~ 42 ~

12 Frequency (cm -1 ) Frequency (cm -1 ) Platinum Nitride (PtN) and Platinum Pernitride (PtN2) Chapter 3 we have calculated the total charge density which is presented in Fig The total charge density is displayed in (110) plane contains Pt and N atoms. This figure reveals that the midpoint between the Pt-atom towards N-atom. The electron density increases at the N atoms and a decrease in the interstitial region between the metallic atoms. This charge rearrangement reflects the electronegative nature of N and transfer of charge from metal to nitrogen atom [8]. The strong bonding is found between metal and nonmetal atoms while weak bonding is found between the non-metals. There is significant charge transfers across the bonding and supports the metal nature of platinum nitride due to an ionic component. However, the zincblende structure a mere common structure for the covalently bonded compound might be due to bonding in GGA (a) LDA (b) X M R M PHDOS (arb. units) 0 X M R M PHDOS (arb.units) Figure 3.5 The phonon dispersion curves and phonon density of states for; (a) GGA pseudo potential and (b) LDA pseudopotentials for platinum pernitride. arising due to notable hybridization of nitrogen and metal states. [21]. Since the Raman scattering is frequently used for phase identification in high pressure synthesis of compounds, we have also calculated the phonon frequencies of the Raman active phonon modes and other phonon modes at the high symmetry points of the Brillouin zone for both zincblende and rocksalt structures of PtN and presented in Table II. Table II also includes the phonon frequencies obtained from Raman ~ 43 ~

13 Platinum Nitride (PtN) and Platinum Pernitride (PtN2) Chapter 3 experiment [5] and calculated results for other hypothetical structures [13-15]. It is observed that the present calculated Raman frequency of rocksalt PtN is different from the experimental Raman data. The present calculated Raman active zone centre phonon mode at 731 cm -1 in the case of zinc blende structure is in general good agreement with Raman band predicted in ref. [15]. However, no conclusive comment can be made for the existence of phase based on phonon frequency calculated at zone centre. Since the calculations on PtN on RS and ZB fail in producing the experimental Raman phonon frequency along with the deviation within the lattice parameter and bulk modulus, we have performed the total energy calculations including the full phonon calculations in two others suggested structures. A detail phonon dispersion curves in fluorite and pyrite phases in addition to the zincblende and rocksalt phases are also required to draw any conclusion for the phase of PtN. The calculation of phonon properties such as phonon dispersion curves, phonon density of states, zone center phonon modes and its pressure dependence particularly to the Raman active modes is vital for platinum nitride as Raman spectroscopy was the major identification tool to characterize this compound [8]. Group theoretical analysis of the optical modes reveals that there are five Raman active (1A g, 1E g, 3T g ), five IR active (5T u ) and four silent (2A u and 2E u ) modes for pyrite PtN 2. The even parity species (Raman active) A g and one of the T g modes can be identified as the in-phase and outphase stretching vibrations of the N 2 dumbbells, respectively. Two E g and the other two T g, Raman active phonon modes represent librational modes. We have calculated the phonon dispersion curves and phonon density of states for platinum pernitride, using LDA and GGA pseudopotentials implemented in PWSCF and ABINIT packages. Fig. 3.5 presents the phonon dispersion curves for pyrite PtN 2 using GGA and LDA implemented in PWSCF package. We do not present the phonon dispersion curves ~ 44 ~

14 Platinum Nitride (PtN) and Platinum Pernitride (PtN2) Chapter 3 using ABINIT as the behavior of the phonon branches is similar except the variation in magnitude, which is evident from the zone center phonon presented in Table II. The absence of any imaginary or negative vibrational mode confirms its dynamical stability at zero pressure. The vibrational profiles can be divided into three regions. The low frequency region (250 cm -1 ) is due to the collective translational vibration of Pt and N atoms. However it is lower than previously reported dispersion curves [21] and is more or less similar in both cases of PWSCF. The intermediate region between 500 and 800 cm -1 corresponds to Pt N bonding, while the high frequency band above 800 cm -1 is due to the N N stretching bond. The phonon density of states presented in right panel along with the dispersion curves reflects all features of phonon dispersion curves. The phonon dispersion curves (not shown here) for fluorite structure also do not show any soft mode or imaginary frequency, which indicates that the fluorite structure for PtN 2 is also dynamically stable. However their zone center frequencies are quite far from the experiment and hence the possibility of fluorite structure is ruled out. Out of five Raman active modes only four modes were clearly detected from Raman spectroscopy [6, 8]. It can be observed from the Table II that there is only a partial match between observed and earlier calculated frequencies [25]. Therefore a detail comparison of the present theoretical calculation with experimental results and other calculations has been carried out in the present study, not only to check the ability of PtN 2 but also to see the success of pseudopotentials and exchange correlation functional implemented in different packages producing the phonon properties. There are no experimental data on IR active phonon modes. ~ 45 ~

15 Platinum Nitride (PtN) and Platinum Pernitride (PtN2) Chapter 3 Table II Calculated zone cantered phonon frequency at high symmetry points of the Brillouin zone for platinum pernitride. Zone centre Phonon frequencies (cm -1 ) A g E g 1 T g 2 T g 3 T g 1 T u 2 T u 3 T u 4 T u 5 T u Expt. [13] Expt. [11] This work, PWSCF-LDA (5) (0.5) (8) (5) This work, PWSCF-GGA This work, ABINIT-LDA This work, ABINIT-GGA Reference [36]DFPT, PWSCF -LDA Reference [36]SDM, PWSCF -LDA Reference [11], GGA Reference [32], PWSCF-GGA Reference [33], LDA 839 (2) 865 (0.3) 863 (0.1) 930 (7) 928 (7) 900 (4) 854 (1) 929 (7) 723 (2) 699 (6) 656 (11) 742 (0.1) 745 (0.2) 741 (0.2) 695 (6) 751 (1) 720 (6) 694 (2) 654 (3.2) 740 (9) 737 (9) 735 (8) 689 (2) 720 (6) (1) (1) (0.6) (7) (7) (4) (0.7) (7) Table II depicts that the present PWSCF calculations are able to produce the E g mode correctly. The observed better value of E g mode frequency is compromised by slightly underestimated and overestimated values of other modes, respectively, from the GGA and LDA calculations. It is further seen from the Table- II that the average deviation is between 2% and 4% for the present PWSCF calculations. In contrast to the lattice constant and bulk modulus values, which are better produced in the case of LDA implemented in PWSCF code, the phonon frequencies are overall better produced using GGA implemented in PWSCF. We mean here that the average deviation is the minimum in the case of GGA. Now from the ABINIT calculations, the LDA implemented within ABINIT produces phonon frequencies better than the GGA ~ 46 ~

16 Raman shift (cm -1 ) Platinum Nitride (PtN) and Platinum Pernitride (PtN2) Chapter 3 implemented in ABINIT as is the case with lattice constant and bulk modulus. However, it is important to note that both the LDA and GGA exchange correlation functionals implemented in ABINIT not able to produce the correct frequency for characteristic E g mode. There is no explanation for this except the formalism, which is different in two program packages. As far as the comparison of present calculations with earlier Pressure (GPa) Figure 3.6 Calculated Raman shifts as a function of uniform pressure for platinum pernitrides. Solid circles are experimental data from [8]. calculations is concerned it seems that the present calculations are over all better in their category. In Fig. 3.6, we present the pressure variation of Raman active modes along with the experimental Raman frequencies. There is a linear increase for all modes similar to the experimental [1] and earlier theoretical results [25] in pressure range 0 50GPa. However in Ref. [25], it is only up to 36GPa and a shift of 60cm -1 is made for an overall good comparison with experiment. This shift is able to produce a good match for all modes other than the E g mode. Fig. 3.6 reveals that the agreement in the present case upto 40GPa is reasonably well except for the lowest T g mode. The only failure of the present calculation seems above 40GPa where the variation is faster and there is a slope change in comparison to the experiment. This may be due to the application of pressure, which is applied isotropically in the present case. Here E g mode matches remarkably well with experiment, which is observed for the first time. ~ 47 ~

17 Hafnium Nitride (HfN) and Zirconium Nitride (ZrN) Chapter Hafnium Nitride (HfN) and Zirconium Nitride (ZrN) It is well known that transition-metal nitrides can crystallize in different crystal lattices and form different phases, depending on the RX/RM ratio where RM and RX are the atomic radii of metallic and nonmetallic atoms, respectively. They form simple crystal structures with MX, M 2 X, M 4 X, and MX 2 stoichiometries, if Hägg s rule, RX/RM < 0.59, is fulfilled. In practice, the structures are not stoichiometric and they exist over a wide concentration range and able to form the face-centered cubic (fcc) sublattice, where nonmetallic atoms occupy interstitial positions to form NaCl-type structure, for example, ZrN, TiN, HfN and NbN. Furthermore, the discovery of ZrN and HfN based new layer structured high T c superconductors [26] as well as the recent synthesis of superconducting nitrides with a new structure Zr 3 N 4 and Hf 3 N 4 [8] has attracted much attention to these materials. Zerr et al. [27] reported on the high-pressure and temperature at 18GPa and 3000K synthesis of zirconium and hafnium nitrides with the stoichiometry M 3 N 4, where M = Zr, Hf. They observed formation of cubic Zr 3 N 4 and Hf 3 N 4 (c-m 3 N 4 ) with a Th 3 P 4 structure, where M cations are eightfold coordinated by N anions. Both these compounds exhibit high bulk moduli around 250 GPa, which indicates high hardness. Previous theoretical as well as experimental reported results are inconsistent for HfN thin films, and are relatively rarely examined, as well as for widely investigated TiN and ZrN. Indeed, the discrepancy in HfN hardness measured by various researchers spans from 16 to 60 GPa [28-29], which makes question the appropriateness of experimental indentation tests particularly in the case of hafniumnitride hard films on silicon substrates. Moreover, a recent theoretical calculation performed by Zhang and He [30] alerted that two widely accepted, but considerably different values of the Young s modulus for TiN are nowadays in common use, namely, 250 and 640 GPa. The bulk modulus of any transition metal nitrides can be assed base ~ 48 ~

18 Hafnium Nitride (HfN) and Zirconium Nitride (ZrN) Chapter 3 on the relationship between the thermal and elastic properties. Moreover, since the thermal stability of metal nitrides at ambient pressure is limited the application of high nitrogen pressure can allow synthesis them at significantly higher temperatures. Recently, these compounds are also being investigated to understand their role in thermoelectric applications [31]. The thermoelectric applications and high melting point of any material is the consequence of some special characteristics of phonon spectra and hence the phonon dispersion curves, thermal properties such as lattice specific heat, internal energy, entropy, etc. are of great importance. Recently, the phonon dispersion curves of HfN and ZrN have been obtained using inelastic neutron scattering which shows anomalous behaviour such as steep slopes of acoustic phonon branches near zone centre of Brillouin zone and a large gap between optic and acoustic phonon branches. This anomalous character of phonon dispersion leads to hardness and high melting point. In addition, there are few recent reports on high pressure Raman spectra of HfN and ZrN [32-33]. In this thesis, we present a systematic study on the pressure dependence of the phonon spectra and thermal properties of two transition metal nitrides using first principle calculations under the frame work of density functional theory (DFT) Computational Methodology The first principles calculations for HfN and ZrN are performed using the plane wave method within (GGA) implemented in the ABINIT code [20]. The cutoff energy for the plane wave basis set for both nitrides HfN and ZrN is 40 Ha. The thermal and phonon properties are subsequently obtained using the linear response approach which is based on the density functional perturbation theory [23]. The basic ingredient phonon dispersions and phonon density of states for the thermal properties has been calculated using 8x8x8 q-grid (29 force-constant matrices) [33]. ~ 49 ~

19 Hafnium Nitride (HfN) and Zirconium Nitride (ZrN) Chapter Results and Discussion The ground state properties such as equilibrium lattice constant, bulk modulus and pressure derivative of bulk modulus for both transition metal nitrides HfN and ZrN are presented in Table-III together with available experimental and other theoretical data. Table III ground states structural parameter of HfN and ZrN System B 1 structure acell /Å Bulk Modulus /GPa B HfN ZrN a 4.54 b 4.54 c 4.52 d a 4.57 b c d a b c a b 250 c e e a Ref [34], b Ref [35], c Ref [36], d Ref [34], e Ref [38]. There is a reasonably good agreement between present theoretical and available experimental data within the experimental error and limitations of pseudopotentials. Figure 3.7(a) Phonon dispersion curves of HfN at ambient pressure Figure 3.7(b) Phonon dispersion curves of ZrN at ambient pressure The predicted lattice constants are slightly higher but it is justified due to the use of GGA pseudo potential and exchange correlation functional which is known to overestimates the lattice constant. ~ 50 ~

20 Hafnium Nitride (HfN) and Zirconium Nitride (ZrN) Chapter 3 Figure 3.8(a) Calculated phonon density of states for HfN at various pressures Figs. 3.7(a) and 3.7(b) present the phonon dispersion curves at ambient condition for HfN and ZrN respectively. Fig. 3.8 presents the pressure variation of phonon density of states (PDOS). We report the phonon DOS only up to 32 GPa as the experimental data on Raman spectra is available up to this pressure. For comparison, we also include the measured high pressure Raman spectra in Figs. 3.8(a) and 3.8(b). These figures reveal that there is an excellent agreement between the Raman spectra and calculated phonon DOS. They also reveal a wide gap separating the low frequency and high frequency region which corresponds to acoustic and optical parts, respectively of the phonon spectra. It is to be pointed that the above two quantities, phonon DOS and Raman spectra are in principle incomparable quantitatively as experimental Raman spectra involves matrix element for the interaction of light with phonons. However, a qualitative description of Raman spectra can be made through phonon DOS. Moreover, our motive here for the comparison of lower and higher frequency region at which the phonon DOS exhibits features related to the acoustic and optical modes, respectively have to gain the information about the presence of acoustic and optic regions as well as its pressure dependence. Figure 3.8(b) Calculated phonon density of states for ZrN at various pressures ~ 51 ~

21 F(J/mol-c) Hafnium Nitride (HfN) and Zirconium Nitride (ZrN) Chapter 3 Figures depict that the lower and higher frequencies increases with pressure. F /kjmol-c x x Temperature(K) 00-GPa 10-GPa 16-GPa 27-GPa 30-GPa 00-GPa 10-GPa 16-GPa 27-GPa 30-GPa E /kjmol-c Temperature /K x x x x x GPa 10-GPa 16-GPa 27-GPa 30-GPa Temperature /K S /Jmol-c -1.K GPa 10-GPa 16-GPa 27-GPa 30-GPa Temperature /K C V /Jmol-c -1.K GPa 10-GPa 16-GPa 27-GPa 30-GPa Temperature /K Figure 3.9 Calculated temperature dependent thermal properties of HfN at various pressures Using the phonon density of states, pressure dependent thermodynamic functions for HfN and ZrN have been calculated and are presented in Figs. 3.9 and 3.10, respectively. The thermodynamical functions Helmholtz free energy ΔF, the internal energy ΔE, constant volume specific heat C v, and entropy S have been calculated using expressions presented chapter 2 (Eqs , within quasi harmonic approximations [39]. Figures show that the shape of plots is similar with slight different range. As temperature increases the calculated ΔF for both compounds decrease gradually, the ~ 52 ~

22 C(J/(mol-c.K)) F /kjmol-c E /kjmol-c -1-1 Hafnium Nitride (HfN) and Zirconium Nitride (ZrN) Chapter GPa 10- GPa 27 GPa 32 GPa GPa 10- GPa 27 GPa 32 GPa S/ Jmol-c -1.K Temperature /K 0- GPa 10- GPa 27 GPa 32 GPa C/Jmol-c -1.K Temperature /K Temperature /K ZrN Temperature(Kelvin) Exp 00- GPa 10 GPa 27 GPa 32 GPa Temperature /K Figure 3.10 Calculated thermal properties of ZrN at various temperature and pressure calculated ΔE and S increases continuously and the constant volume lattice specific heat (C v ) tend to asymptotic limit of 50 J/(mol-c K). The zero temperature values ΔF 0 and ΔE 0 do not vanish due to the zero point motion [46] and can be calculated from asymptotic expressions of corresponding equations at zero temperature. The value of ΔF 0 and ΔE 0 is and kj/mol, respectively. It is to be noted that the comparison between theoretical and experimental data could only be made in the case of ZrN. The comparison between constant volume specific heat C v and experimental constant pressure lattice specific heat, is not a serious drawback as at lower temperature ~ 53 ~

23 Hafnium Nitride (HfN) and Zirconium Nitride (ZrN) Chapter 3 there is not much difference in both specific heats. Furthermore, the lattice dynamical calculations give the constant volume specific heat. The figure depicts that there is a good match between the experimental and calculated specific heat with minor discrepancies which are quite obvious due to above facts. The specific heat plots from Fig reveal that the C v in the case of HfN has more value than ZrN at low temperatures. This can be understood from lower frequency region of the phonon density of states presented earlier. The acoustic phonons of ZrN are softer than HfN due to the fact that the same is derived from Zr displacements which has smaller mass compared to Hf. In addition, the specific heat at lower temperatures is dominated by acoustic phonons and is inversely proportional to the cube of the velocity of sound. At higher temperatures, there are large discrepancies between the calculated and experimental specific heats as the lattice undergoes the thermal expansion and harmonic interactions. As far as the pressure effect on the temperature dependent lattice specific heat for HfN and ZrN is concerned, the pressure causes a decrease in the specific heat and entropy in contrast to the temperature effect. This is in accord with the second law of thermodynamics. This is due to the fact that the acoustic phonon becomes harder and hence the sound velocity increases with the pressure. However, at high temperature it is readily seen that the C v values for Figure 3.11 The phonon contribution to the specific heat of Hafnium nitrides (solid lines) and Zirconium nitrides (dashed lines) all pressures approaches approximately to the classical ~ 54 ~

24 Niobium Nitride (NbN) and Molybdenum Nitride (MoN) Chapter 3 Dulong-Petit values and are identical for both compounds. It is seen that when T < 1500 K, the heat capacity C v is dependent on both temperature T and pressure P. This is due to the anharmonic approximations of Debye model. However, at higher pressures and temperatures, the anharmonic effect on C v is suppressed and is very close to the Dulong-Petit limit 6 NK, where N represents Avogadro s number. 3.4 Niobium Nitride (NbN) and Molybdenum Nitride (MoN) On the basis of structure of transition metal nitrides which can be described as interstitial compounds, where nitrogen atoms are located in the interstitial sites of the densely packed lattices. It has been observed that similar types of transition metal nitrides are able to form solid solutions with other transition metal nitrides. Since, the Figure 3.12 Crystal structures and coordination numbers of Mo and N in MoN: (a) 1 -MoN, (b) 2 -MoN, (c) MoN wurtzite, (d) -MoN, (e) 3 -MoN. The large and small spheres represent Mo and N atoms, respectively [41]. transition metal nitrides have an impressive, interesting and number of outstanding properties including extreme hardness, thermal integrity, high melting points, good chemical resistance, electrical conductivity, and metallic-like appearance, this combination of useful properties has led to numerous thin film applications, such as ~ 55 ~

25 Niobium Nitride (NbN) and Molybdenum Nitride (MoN) Chapter 3 hard coatings, corrosion and abrasion resistant layers, and decorative coatings [3-5]. However, in current age scenario the fastest developing area of application today is microelectronics, where transition metal nitride films are used as barriers between copper and silicon in ultra large scale integration devices. Besides diffusion barriers, transition metal nitrides and metal-silicon-nitrides are also promising gate electrodes for complementary metal oxide semiconductor devices. One of the most interesting and promising mono-nitride, previously extensively studied barrier materials was tantalum nitride, which has high melting point and is very hard and highly conductive. Though, niobium and molybdenum belongs from the same group of mononitrides, they have been comparatively less studied. These nitrides have exceptional thermo-dynamical stability with respect to other metal like copper since they do not form compounds with it. However, there exist several crystalline phases in the phase diagram of Nb-N [70-74]. Most theoretical and experimental studies related to NbN deals with its superconducting properties [75]. The superconductivity with T c ~ 17.3K [39] in NbN was found as long ago as 1941 by Justi et al. [40]. Niobium nitride alloys, for example the Nb-Ti-N system, also show superconducting behavior. The analysis of previous study suggests that the niobium nitride in the rocksalt structure is commonly reported having the highest T c among the other carbides and nitrides at 17.3 K [39]. In similar way, molybdenum nitrides have also attracted considerable attention due to their excellent catalytic properties, resembling those of noble metals in many hydroprosessing reactions. However, in nature the bulk molybdenum nitride was known to exist only in hexagonal forms, including more modifications of mononitride -MoN and Mo 5 N 6 type phases. Experimentally, 1 -MoN has WC type crystal structure (space group P-6m2, No. 187, Z=1), where in the structure, the Mo atom is situated at the center of a trigonal prism formed by nitrogen atoms, and nitrogen has six nearest ~ 56 ~

26 Niobium Nitride (NbN) and Molybdenum Nitride (MoN) Chapter 3 Mo neighbors as shown in Fig. 3.12, 2 -MoN has NiAs type crystal structure (space group P63/mmc, No. 194, Z=2). 2 -MoN can also be obtained by alternating the stacking of the N atom layer between the two equivalent unit cells of 1 -MoN (Fig. 3.12). The 221 superstructure of 2 -MoN is the prototype of 3 -MoN. Additional relaxation of -MoN changes the space group from 194 to 186 and gives 3 -MoN (space group P63mc, No. 186, Z=8). 4 -MoN (space group P63/mmc, No. 194, Z=8) has been synthesized [42, 43]. NbN and MoN mono-nitrides exhibit long history till now from theoretical studies for electron-phonon properties and phase transformation in rock-salt stoichiometric structure. The Density functional theory (DFT) currently being the method of choice for first principles studies of crystalline materials and have been used to address a number of fundamental issues of structure and properties of rocksalt phase of NbN and MoN[44]. From the previous study, the estimated values of band gap, structural properties, vibrational spectrum and electron phonon interaction are multivalued and unclear till date [45]. The experimental and theoretical studies probed a number of structural and vibrational properties of NbN and MoN including pressure induced changes in its phonon spectrum [46, 47]. We found from previous that there are imaginary phonon frequencies along the high symmetry of BZ directions and at several points [44-46]. Hart and Klein [48] have clearly shown that using linear augmented plane wave method the dynamical stability in MoN is not mitigated and the application of pressure fails in fabricating MoN in rocksalt structure due to phonon instability at X-point of the Brillouin zone. According to Weber et al. [49] the phonon anomalies arises due to the electronic transition between the W 3 states which strongly increases a negative contribution to the dynamical matrix for the phonon wave vector at X-point in NbN ~ 57 ~

27 Niobium Nitride (NbN) and Molybdenum Nitride (MoN) Chapter 3 crystal, while Hart and Klein [47] have found that the elastic constant C 44 plays major role and particularly to be responsible for instability which appeared as the negative frequencies in the phonon dispersion curves. In spite of these investigations with clear conclusions a number of open issues remain to be unclear, particularly for the dynamical instability of NbN and MoN in rocksalt structure. Thus previous study clearly reveals that the pressure alone could not bring out the dynamical stability in these compounds [47, 48]. Particularly, In the case of MoN, the application of pressure further increases the instability. Recently, it is shown that the consideration of finite temperature effects leads to the dynamical stability at very low pressure values [49, 50]. Therefore a deep analysis of smearing induced pressure effect on the stability of phonon spectrum for NbN and MoN in rocksalt structure is highly desirable. In the present work, we demonstrate systematic first principles calculations to investigate the electronic band structure and dynamical stability of two mono nitrides NbN and MoN in rocksalt phase as a function of smearing induced electronic temperature and pressure. Our first principles calculations for originally optimized parameters clearly show that the rocksalt phase of NbN and MoN is dynamically unstable. However, the inclusion of smearing parameter or finite electronic temperature effects makes them dynamically stable. We have also investigated the temperature variation of thermodynamic functions Computational Methodology All calculations were carried out within the framework of density functional theory using the pseudopotential plane wave method implemented in Quantum Espresso code [23] similar to some other nitrides discussed above. The generalized gradient approximation, has been used to represents the exchange correlation functional in the DFT formalism for mononitrides. Further, the pressure induced dynamical instability in ~ 58 ~

28 Niobium Nitride (NbN) and Molybdenum Nitride (MoN) Chapter 3 the phonon dispersion curves relies exclusively within the effect of smearing induced pressure on the crystal structure, electronic temperature, and vibrational properties which adopts the GGA approach with Vanderbilt ultrasoft potentials to describe the electronion interaction [51]. Within this ultrasoft pseudopotentail approach the orbital is allowed to be as soft as possible in the core regions so that their plane wave expansion converges rapidly. The rapid convergence affected since the semi core states were treated as valence states for the niobium and molybdenum pseudopotentials and nonlinear corrections were taken into account. The criterion for convergence for the total energy was 10-5 Ry/f.u. To speed up convergence, each eigen value was convoluted with a Mathfessel-Paxton smearing with a width = 0.02 Ry. The cutoff energy set to 40 Ry and 38 Ry for rocksalt NbN and MoN, respectively. The pressure response of NbN and MoN were studied by calculating its P-V equation of states (EOS) for applied pressures up to 50GPa. The lattice parameters were optimized at each value of the external pressure, so that the volume could be determined. Each structure was considered to be converged only when the x, y and z component of the force on nitrogen atom was less than , and the maximum component of the stress tensor was less than 0.01 kbar. The resultant EOS was fitted using the third-order Birch Murnaghan analytical expression to produce the bulk modulus B, and its pressure derivative. The integration over the Brillouin zone (BZ) was performed with a set of reciprocal k-points determined according to the Monkhorst-Pack scheme using 8x8x8 and 6x6x6 mesh, respectively, for NbN and MoN phases. The lattice dynamical properties were calculated using density functional perturbation theory (DFPT) [23]. In order to account of peculiarities of acoustic modes and the force constant calculations were performed using a 4x4x4 q-grid in the case of rocksalt structure of NbN and MoN. ~ 59 ~

29 Niobium Nitride (NbN) and Molybdenum Nitride (MoN) Chapter 3 The phonon densities of states were calculated with tetrahedron method using a high dense (16x16x16) Results and Discussion The ground state properties of rocksalt NbN and MoN particularly at ambient pressure and zero K temperature are calculated by minimizing the total energy with respect to the unit cell volume by means of Murnaghan equation of states [16]. As a result, we obtained the equilibrium value of lattice parameter and unitcell volume. The obtained equilibrium structural parameters such as lattice constant, bulk modulus and pressure derivative of bulk modules for rocksalt NbN and MoN are presented in Table IV Table IV along with the available experimental and previous theoretical data [52-60]. Table IV Lattice parameters (a), bulk moduli (B) and their pressure derivatives (B ) calculated in this work and together with results from other theoretical calculations and experiments. Compound a (Å) Bulk modulus (B) B Theory Other Theory Theory Theory Exp Other Theory Other Theory NbN (Fm3m) a, ( o 4.42 a ) b, 4.37 c, 4.41 d, e ( a ), 302 b, 313 d, 314 f, 287 g, 292 h b MoN (Fm3m) o b, i, j, k, 4.41 l, 4.25 m, n i, 327 b, j, 360 k, 354 l, 384 n 5.16 a Ref. [44]. b Ref. [47]. c Ref. [52]. d Ref. [53]. e Ref. [54]. f Ref. [55]. g Ref.[56]. h Ref. [41]. i Ref. [57]. j Ref. [9]. k Ref. [58]. l Ref. [59]. m Ref. [60]. n Ref. [46]. o Ref. [61]. The Table IV clearly shows a good agreement between present lattice constant (a 0 ), bulk modulus (B) and first derivative of bulk modulus (B ) with experimental as well as previous theoretical calculations for both mononitrides. The calculated electronic band structure in the first Brillouin zone along with electronic density of states is shown in Fig The projected electronic densities of states for NbN and MoN are also ~ 60 ~

30 Niobium Nitride (NbN) and Molybdenum Nitride (MoN) Chapter 3 presented along with the total projected electronic density of states. The total and projected density of states clearly reveals that both rocksalt NbN and MoN have high density of states without any gap at Fermi level (E f ). The density of states for both compounds lie mainly in four energy regions; (i) the lowest region stemming mainly form 2s-N states. (ii) The region at the bottom of the valence-band complex originating (a) (b) Figure 3.13 Electronic bandstructure along with total and partial electronic density of states for (a) NbN and (b) MoN. from s-nb (Mo) and 2p-Nstates. (iii) The region at the top of the valence band complex, it is mainly due to 4d- Nb(Mo) states mixed with 2p-N states; and (iv) the energy region just above E f, dominated by unoccupied 4d- Nb(Mo) states. Further, it is observed from the total and projected electronic density of states that the 2p-N and 4d- Nb(Mo) contributes significantly to the total density of states near Fermi level. This contribution increases further as the atomic number increases. This may be due to the fact that the largest contribution to its cohesive energies from the atomic 4d-orbital [61]. ~ 61 ~

31 Frequency (Thz) Frequency (Thz) Niobium Nitride (NbN) and Molybdenum Nitride (MoN) Chapter 3 20 (a) (b) X L X W L -10 X L X W L Figure 3.14 Phonon dispersion curves in rocksalt structure at ground state along with experimental data [62]; (a) NbN and (b) MoN. Similarly the increasing width of the d-band is the consequence of the increased bond length leads to the major contribution to decrease the stability of system. There is an excellent agreement between the present and previous density of states at Fermi level indicates instability in the structure at equilibrium lattice constant and in ground state. The higher peak of DOS at Fermi level leads to the existence of soft phonon modes in the long wave length region and their collapse causes to a structural transformation. Therefore, a comprehensive analysis of phase stability in terms of phonon dispersion curves throughout the Brillouin zone is essential for these mononitrides. To investigate the dynamical stability, we have calculated the phonon dispersion curves using theoretically obtained equilibrium lattice constant, for all the high symmetry directions in the Brillouin zone for both NbN and MoN mononitrides. The theoretically computed phonon dispersion curves are presented in Figs. 3.14(a) and 3.14(b) for NbN and MoN respectively. The Fig. 3.14(a) and 3.14(b) clearly reveal that the phonon dispersion curves contain negative acoustic phonon frequencies at the high symmetry X-point in the Brillouin zone, which indicates dynamical instability of these ~ 62 ~

32 Niobium Nitride (NbN) and Molybdenum Nitride (MoN) Chapter 3 compounds rocksalt structure hinted earlier in electronic density of states. For further analysis and comparison the experimental neutron scattering and available Raman scattering data are also shown with the calculated phonon dispersion curves. The experimental data in the case of NbN is for stoichiometric NbN 0.93 [62]. In addition the Figure 3.15 Finite temperature phonon dispersion curves and density of states in rocksalt structure along with the experimental data [62]; (a) NbN and (b) MoN PDC in Fig. 3.14(a) clearly a disagreement between experimental and theoretical phonon dispersion zcurves for NbN. The negative acoustic phonon modes are also observed in the case of MoN. These phonon anomalies arise because of electronic transitions between the W 3 states that strongly increase a negative contribution to the dynamical matrix for the phonon wave vector X in the Brillouin zone [49]. To substantiate the hypothesis of finite temperature effects through smearing, we investigated the phonon dispersion curves along high symmetry directions for different values of smearing parameter, starting from 0.02 (originally optimized value) to 0.22 Ry for rocksalt NbN and MoN. Fig presents the phonon dispersion curves of NbN and MoN for the smearing parameter 0.11 Ry and 0.12 Ry, respectively. The smearing parameter value of 0.11 Ry in the case of NbN results in to the pressure of GPa, while the same with the value of 0.12 Ry in the case of MoN results in to the pressure of GPa. We have also incoporated the experimental inelastic neutron scattering ~ 63 ~

33 Niobium Nitride (NbN) and Molybdenum Nitride (MoN) Chapter 3 data in the phonon dispersion curves of NbN presented within Fig. 3.15(a). A reasonably good agreement between the experiment and theory can be observed. The slight deviation between the two results can be attributed to experimental data which is obtained for NbN It is noteworthy that an increase in leads to the disappearance of the phonon anomalies and a better agreement between calculated and experimental dispersion curves. The frequency of phonon modes is positive throughout the Brillouin zone, indicating dynamical stability for both compounds. The value of = 0.11 Ry for NbN leads to the best calculated phonon dispersion curves which is not only in good agreement with experimental dispersion curves obtained using inelastic neutron scattering for NbN 0.93 [140] but also show dynamical stability. There is slight disagreement between the experiment and theory at the X- point in particular the crossing of LA and TA, which is sacrificed due to an overall agreement between theory and experiment. It is important to note that the present value of which is able to give dynamical stability is quite low in comparison to previous study [47]. The earlier calculations of phonon dispersion curves predicted a quite low value of frequency of optical phonons throughout the Brillouin zone even using a quite high value of smearing parameter. There is about 9.28% deviation in the case of frequency of optical phonon modes of NbN between present and experimental data, which of course is lower in comparison to the Ref. [47], where the difference is of the order of 20.7%. This is not a serious drawback looking to the overall success of present calculation. The stabilized phonon dispersion curves in are divided in to the two parts: the lower band ranging up to about 7Thz is mostly attributed to the vibration of Nb or Mo atoms while the higher bands are due to N atoms vibrations. ~ 64 ~

34 Niobium Nitride (NbN) and Molybdenum Nitride (MoN) Chapter 3 Figure 3.16 smearing dependent phonon modes at X-point of the BZ (a) NbN and (b) MoN. The phonon density of states presented in right panel to the phonon dispersion curves reflects all general features of the phonon dispersion curves. A clear gap between acoustical and optical phonons is observed for both compounds. The sharp peaks in the phonon density of states are due to the flat optical branches in the phonon dispersion curves. We find zero phonon density of states below at zero energy confirms that phonons are positive throughout the Brillouin zone and structure is fully stable dynamically. The plot of phonon modes at X-point as a function of smearing parameter presented in Fig and the pressure variation of smearing parameter (inset) illustrate that the cubic phase of NbN and MoN, respectively, can only exist in the pressure range from to GPa and from to 86.3 GPa. We find that the imaginary frequencies at X-point disappear when the electronic temperature Te (smearing parameter ) is raised from x10 4 to x10 4 K and x10 4 to x10 4 K for NbN and MoN respectively. This indicates that a finite electronic temperature (T e ) could indeed lead to the desired stabilization effect in the cubic phase of NbN and MoN and these compounds can be formed in NaCl structure at high temperature and low pressure. ~ 65 ~

35 Niobium Nitride (NbN) and Molybdenum Nitride (MoN) Chapter 3 Figure 3.17 Smearing parameter dependent Fermi energy of (a) NbN and (b) MoN To understand the effect of electronic temperature or smearing parameter, we have presented the variation of Fermi energy with smearing parameter in Figs. 3.17(a) and 3.17(b) for NbN and MoN respectively and observe that the Fermi energy decreases with the increase in for both compounds. This leads to increase in frequencies of lowest acoustic phonon modes and dynamical stability of NbN and MoN in rocksalt phase. Once the phase has been stabilized, the frequencies at X-point remain constant with increasing electronic temperature or pressure, implying that the ion ion interaction is not modified [58]. To have a more comprehensive insight into the influence of phonons exerted on NbN and MoN, we investigated the contribution of phonons to its thermodynamic properties in dynamically stable structure. Temperature dependent thermodynamical functions such as internal energy (ΔE), Helmholtz free energy (ΔF), heat capacity at constant volume (C v ) and entropy (S) for NbN and MoN in rocksalt phase using QHA and the total phonon density of states have been calculated using expressions from [65] and shown in Fig Unfortunately, there are no experimental data available for comparison with the theoretically obtained results. As shown in Fig. 3.18(a), internal energy and entropy have linear behavior with temperature above 150K, but change ~ 66 ~

36 Niobium Nitride (NbN) and Molybdenum Nitride (MoN) Chapter 3 rapidly below 150K. The heat capacity at constant volume C v increases rapidly in the low temperature region (below 100K) and approaches to a constant value in the high Figure 3.18 Temperature variations of thermodynamical functions for NbN and MoN in rocksalt structure at value of mentioned earlier (a) Internal energy, (b) Helmholtz free energy, (c) specific heat at constant volume and (d) entropy. temperature region. Such behavior is with Debye specific heat theory. At higher temperature, C v approaches to 6R, where R= Jmol -1 K -1 is ideal gas constant. Fig presents the variation of Debye temperature ( D ) with temperature for NbN and MoN. Both compounds have similar trends throughout the range. However, the Debye temperature Figure 3.19 Temperature variation of Debye temperature of NbN and MoN in rocksalt structure ~ 67 ~

37 Cadmium Nitride (CdN) Chapter 3 obtained via first and second moment of phonon frequencies is higher for MoN. At room temperature, the Debye temperature is 660K and 680K for NbN and MoN, respectively. 3.5 Cadmium Nitride (CdN) Cadmium nitride, which is a member of the 4d-metal nitrogen compound, has not been synthesized yet. In However, it has recently attracted great attention in the search of high compressibility, high thermal conductivity and high melting temperature [64]. The First principles calculations are performed by Ateser et al. [63] to investigate the structural, elastic, and mechanical properties of CdN for various structures: NaCI, CsCl, ZnS, wurtzite, WC, CdTe, NiAs, and CuS. The calculations indicate that CuS (B18) structure is energetically the most stable among the above considered structures. However, the density functional theoretical calculations carried out by Zhao et al [65] predicted that the rocksalt CdN is energetically more favorable structure. It is observed from the literature that the conclusion on structural and electronic properties on CdN diverge from investigation to investigation. These explorations motivate us to study theoretically the different crystal structures of CdN using ab-initio total energy calculations as a possible candidate crystal structure. In particular, we are interested in studying the thermo dynamical properties along with the structural and electronic properties of CdN in its correct stoichiometry at ambient conditions. We pursue two main goals, first to carry out theoretical calculations of structural and electronic band structure of CdN in rocksalt, NiAs and CuS type crystal structures to investigate its structural and electronic properties in considered crystal structures. Secondly, to perform the ab-initio lattice dynamical calculations in three different crystal structures of CdN as there is not a single report available on these for such an important technological material. The first principles pseudo potentials method of total-energy ~ 68 ~

38 Cadmium Nitride (CdN) Chapter 3 calculations has been shown to be capable of predicting the dynamical properties very accurately [66] Computational Methodology The first-principles calculations are performed using the pseudopotential approximation implemented in Quantum-Espresso code using the ultrasoft pseudopotentials of parameterized by Perdew and Burke [22] within cut off energy of 30Ry. We used 8x8x8 k-points mesh for cubic (rocksalt) and hexagonal (NiAs and CuS) crystal structures respectively. The bulk modulus and the pressure derivative of structural parameters were determined to fit the total energy data by applying third order Murnaghan equation of state [22]. The Lattice dynamical calculations are performed within the framework of DFPT [23]. The phonon spectra obtained using force constant within 4x4x4 q-point mesh for the three considered crystal structures. The vibrational phonon density of states is obtained by applying the tetrahedron method Results and Discussion The optimized structure of CdN in RS, NiAs and CuS phases is depicted in Fig The rocksalt structure has a spacegroup with Cd atoms occupying the 4a Wyckoff positions and N atoms occupy the 4b positions. The NiAs structure has the P63/mmc spacegroup with the Cd and N atom occupy the Figure 3.20 Total energy Vs volume per unit cell per atom. 2a and 2c sites respectively, while CuS structure is hexagonal with all atoms occupying ~ 69 ~

39 Cadmium Nitride (CdN) Chapter 3 Wyckoff positions 2c, 2d, 4f, 2c and 4e. The structure of the three candidate phases were optimized at zero pressure. The equilibrium lattice parameters, bulk modulus, pressure derivative of bulk modulus and Debye temperature are listed in Table V for all three different considered crystal structures of CdN, It is clear from the Fig that the RS structure is energetically the most stable phase at zero pressure. For comparison, we also present other available theoretical values calculated using LDA and GGA exchange correlation functional [65, 64] in Table V, which reveals an overall good agreement between present and previous values. Table V Lattice parameters, bulk modulus, derivative of bulk modulus and Debye Temperature for CdN crystals. Crystal Structure Parametrs NaCl NiAs CuS LDA GGA LDA GGA LDA GGA *, b b *, 5.777, , (Å) 4.64 a, *, 3.31 a, b, c 7.76 a, b b * b b b c(å) a, b b *, B (GPa) a, b a, b, B b 4.76 *, b, 157 b, *,114 b 50.4 b, 37.4 *, 40.7 b b, 4.99 *, b b, 5.59 *, b Debye Temp K( D ) b *, b, c * b, *, b, * Our results, c Ref. [65], a Ref. [61], b Ref. [64], ~ 70 ~

40 Energy (ev) E n e r g y (e V ) Energy (ev) Cadmium Nitride (CdN) Chapter C d N - T o ta l C d - d C d - p C d - s N -s N -p X L X W 0L D O S /s ta te s /e V u n it c e ll C dn -Total C d-d C d-p C d-s N-p N-s K M A L D O S/states/eV/unit cell Total-Edos Cd-d Cd-p Cd-s N-p N-s 5 0 E f K M A L DOS (states/ev unit cell) Figure 3.21 Electronic band structure of CdN along with total and partial electronic density of states. (a) RS, (b) NiAs and (c) CuS structutre The values of bulk modulus indicate that CdN in RS phase is least incompressible among the considered crystal structures (NiAs and CuS). The calculated bulk modulus of RS and NiAs is larger than the pure metal Cd [63,67] except for CuS phase. This indicates that the insertion of N atom into Cd lattice enhances the bulk modulus significantly in contrast, to some of the transition metal nitrides such as LaN [68], HfN ~ 71 ~

41 Cadmium Nitride (CdN) Chapter 3 [67] and PtN [70]. The results show that CdN in CuS phase is one of the least incompressible transition metal nitrides because of having lowest bulk modulus. The Debye temperature is known as an important fundamental parameter closely related with many physical properties such as specific heat. The calculated value of the Debye temperature for B 1 and B 4 (NiAs) agree well with the available data while it diverges for CuS (B 18 ) phase. Fig presents the electronic bandstructure at equilibrium lattice constant along the high symmetry directions in the first Brillouin zone for three considered crystal structures of CdN. We observe that the CdN in three crystal structures exhibiting a metallic nature with no energy gap at Fermi level. The electronic bandstructure exhibits the energy levels originated from atomic states 1 (2s- N), c 15 (2p-N and 4d-TM) at the -point. The bands near the Fermi level and above the Fermi level (5 to 14) ev are mainly due to cadmium d bands while nitrogen p orbital are below the Fermi level (-3 to 0) ev. All electronic band exhibits well pronounced dispersion indicating high mobility of electrons as are additional feature of metallic behavior in the considered three structures. The lowest energy states of RS (Fig. 3.21(a)), NiAs (Fig. 3.21(b)) and CuS (Fig. 3.21(c) type structures are those of N-2p and Cd-4d electron orbital. It is interesting to note that in CdN, the Cd-4d states are located at -7.5eV below the Fermi level, while, in other 4d series of transition metal nitrides particularly in RS and ZB structures all 4d electrons lie near Fermi energy as valance band. The width distribution of the N-2p states is very large near the Fermi level and contributes toward the conduction band due to hybridization effects. The Cd-4d transition metal electron spreads widely below the Fermi energy. The width of the Cd-4d electrons and N-2p electrons are about -4eV to -3eV for all three considered structures. The weak hybridization at Fermi level ~ 72 ~

42 Cadmium Nitride (CdN) Chapter 3 indicates weak bonding and leads to higher lattice parameter and lower bulk modulus. In Fig. 3.21(c), there are unique features particularly in - A line of the Brillouin zone. The electronic bands are dispersion less Figure 3.22(a) Phonon dispersion curves along with total and partial phonon density of states of CdN in RS phase at ambient pressure. near and below the Fermi level. However, in the case of RS and NiAs structures band dispersions are like wave. In contrast to CuS phase the behavior of flattening of bands in the Γ-A direction is responsible for the lower bulk modulus of CdN in CuS phase. Further, the sharp peak near the Fermi level for CdN in CuS (B 18 ) structure shows highly metallic nature due to N-2p valence electrons. This behavior of bandstructure can be attributed to the two reasons. The first point is the expansion of the fcc lattice compared to the pure Cd lattice due to the insertion of nitrogen atoms at the hexagonal interstitial sites. The second point is the interaction between Cd-4d and N-2p electrons. We find from the present study that RS phase is more covalent and partial electronic density of states is more prominent in the region of Fermi level. Now, turning our attention to the phonon dispersion and phonon density of states, an essential ingredient for the calculation of thermo dynamical functions for any compound. It is a well established fact that the mechanical or thermo dynamical stability only reflects the local stability of the crystal structure, however the full phonon dispersion curves believed to shed more kinetic description and information of the structural stability of the compound. ~ 73 ~

43 Cadmium Nitride (CdN) Chapter 3 Moreover, the phonon dispersion relation gives the criterion for the crystal stability and indicates any instability through a soft mode which is responsible for structural changes [71]. The calculated Figure 3.22(b) Phonon dispersion curves along with total and partial phonon density of states of CdN in NiAs phase at ambient pressure. phonon dispersion curves along the high symmetry directions of the Brillouin zone along with the total as well as partial phonon density of states (PHDOS) for all three considered crystal structures are depicted in Figs. Figure 3.22(c) Phonon dispersion curves along with total and partial phonon density of states of CdN in CuS (B18) phase at ambient pressure. ~ 74 ~ 3.22(a-c) for RS, NiAs and CuS (B 18 ) respectively. There are no experimental data available on vibrational modes; from IR or Raman spectroscopy mainly due to unavailability of single crystals of cadmium nitride. The vibrational phonon density of states (DOS) for materials is an important property as it requires the calculation of total phonon dispersion relation throughout the Brillouin zone. It is seen from the Fig that the phonon dispersion curves in the case of RS structure contain six phonon modes with positive phonon frequency throughout BZ. The phonon dispersion curves in NiAs and CuS (B 18 ) crystal structures, there are phonon modes with negative (imaginary) frequency. The positive phonon frequencies throughout BZ suggest that the CdN in RS structure is dynamically stable; however, it is dynamically

44 Cadmium Nitride (CdN) Chapter 3 unstable in the hexagonal type NiAs and CuS (B 18 ) structures due to negative frequencies. A critical assessment of phonon dispersion curves and phonon density of states reveals that there are two regions in which the phonon modes are distributed. In the RS structure of CdN, the phonon branches are distributed almost uniformly up to 400 cm -1 in the high symmetry directions of the Brillouin zone. The top regions about cm -1 consisting of three phonon braches are due to nitrogen atom (lighter) displacements, while the lower region the phonon branches are due to cadmium atom (heavier). It is a fact that the insertion of nitrogen atoms gives clear gap in acoustical and optical phonon branch and frequency results higher in the case of CuS type structure. Similar situation occurs for RS crystal structure except it overlaps at L-point, while more negative phonon branches in NiAs crystal structure. However, for CuScadmium nitride, a clear separation between the elemental contributions is observed. Since the lower bulk modulus and weak hybridization between 2p and 4d orbital band leads to the weaker bonding the optical and acoustical phonon branches crosses at high symmetry L-point in the RS structure and prevents to occur any pseudo gap in the phonon spectra. Further, it is seen that the vibrations are highly dispersive in whole Brillouin zone which is attributed to mainly interaction between Cd and N atom in all three phase structures. The total phonon density of states brings out all special feature of phonon dispersion curves. The sharp peaks in the total phonon density of states are due to the flat phonon branches in dispersion curves. The partial phonon density of sates presented along with the total phonon density of states clearly brings out the contribution of individual atoms ~ 75 ~

45 Cadmium Nitride (CdN) Chapter 3 Figure 3.23 Debye temperature and Specific heat derived from phonon density of states for RS type crystal structures. vibrations in total phonon density of states. The Debye theory of zero phonon density of states is fulfill clearly for CdN in RS phase, which is quite obvious other phases are dynamically unstable and acoustic phonon branches are having negative frequency. The contributions from the lattice vibrations to the specific heat capacity at constant volume and Debye temperature ( D ) for RS CdN crystal structures are presented in Figs as the phonons are real only in RS phase of CdN. Fig clearly shows that the results follow the Debye model at low temperatures and specific heat drops off to zero value fairly quickly for decreasing temperature and an exponential decay. The temperature is limited to 600 K to minimize the potential influence of anharmonicity. It is well known that above melting temperature the theory must be at fault because only a quasi harmonic approximation (QHA) for the lattice vibrations has been taken as a basis, which must ultimately fails on approaching the melting temperature. However, quasi harmonic approximation offers the unique opportunity to recognize and eradicate experimental mistakes and expand the measurements compressibility to the yet unmeasured CdN crystal at different temperatures. At higher temperature, Fig shows that the heat capacity for each crystal structure varied non-linearly up to room temperature (~250K) and almost became constant above the room temperature. It is ~ 76 ~

46 Palladium Pernitride (PdN2) and Iron Pernitride (FeN2) Chapter 3 clearly seen that the C v value approaches to approximately to the classical Dulong-Petit value. Our results of Debye temperature for RS-CdN at zero pressure are consistent with previous theoretical calculations. The D drops with the increasing temperature up to 20K at fixed volume of RS CdN. However, in the temperature range from around 20 K to 250 K, D shows a moderate increase at high temperature while above 250 K, it shows weak temperature dependence and approaches to constant value. 3.6 Palladium Pernitride (PdN 2 ) and Iron Pernitride (FeN 2 ) A class of nitrogen compounds which are related to, but distinct from nitride are called as pernitrides, with or units, where is sometimes called diazenide analogous to carbides. For completeness, there is an additional class of compounds, called azide, containing units. Metal nitrides and pernitrides show a wide range of excellent properties such as hardness, superconductivity, photoluminescence and various types of magnetism [72-73]. Hence, these nitrogen-rich compounds are attracting increasing interest in both experimental and theoretical studies. The experimentally known pernitrides which are not as common as nitrides mostly from alkaline metals, alkaline earth metals, and some transition metals. These metal-rich compounds on the other hand due to the nitrogen-rich side might produce interesting N-N bonded species, more specifically the in pernitrides. Experimentally, first nitrogen-rich nitrides were reported in year 1892[72]. In the 1950s, the first high pressure experiments were performed for the preparation and characterization of BaN 2 [72-73]. The results showed that there were coexistence of Ba 3 N 2 and BaN 2 [72-73]. Later in 2001, once again the same method was used to synthesize the barium pernitride. It crystallizes in the space group C2/c, similar to the ~ 77 ~

47 Palladium Pernitride (PdN2) and Iron Pernitride (FeN2) Chapter 3 ThC 2 structure type [72]. Analogously, SrN 2 [73] was also synthesized the same way as BaN 2. In this reaction, Sr 2 N is used as starting compound which reacts with N 2 under high pressure and high temperature. During this synthesis, first SrN formed and later SrN 2. SrN crystallizes in a monoclinic structure with space group C2/m whereas SrN 2 is formed in the CaC 2 structure type. In 2003, an independent theoretical study was performed on CN 2, SiN 2 and GeN 2 which are not-yet synthesized systems. In this study, the compounds considered need the form A 4+ [N 2 ] 4, where A is C, Si, and Ge, and the anions form the same electronic configuration as S 2 2 in the pyrite structure in FeS 2. The distance between N-N atoms in SiN 2 and GeN 2 is Å and Å respectively, which is closely related to a single bond between nitrogen atoms in a N-N bond, whereas in the case of CN 2 one finds 1.34 Å which is in between the length of a single and double bond between nitrogen atoms. In 2004, a binary noble metal nitride, PtN 2, had been synthesized successfully at 45-50GPa and temperatures more than 2000 K. Initially, it was mysterious and wrongly formulated as PtN. However, afterwards theoretical [74] as well as experimental [6], studies showed that the compound is PtN 2 crystallizes in the pyrite structure. In 2008, palladium nitride was synthesized at a pressure above 58 GPa and a temperature below 1000 K [6]. The other noble metal nitrides (IrN 2, OsN 2, RuN 2, and RhN 2 ) were also formed at high pressure and temperature conditions but PdN 2 decomposes at about 13GPa. There are four iron-rich compounds, among them two closely related daltonide compounds (Fe 4 N and Fe 8 N), the berthollide phase (Fe 3 N) and Fe 2 N, which are exothermic. The next candidate is 1:1 composition of Fe and N, which exists in crystalline powder form or in thin films. It is anti-ferromagnetic and crystallizes in the ZnS-type structure. The next composition is nitrogen-rich i.e., FeN 2 [75] which has ~ 78 ~

48 Palladium Pernitride (PdN2) and Iron Pernitride (FeN2) Chapter 3 been paid less attention. Though, it is ferromagnetic and adopts the space group. The traditional application of this pernitrides have taken advantage of the hard and refractory nature of many of these compound, but numerous recent applications are based on their crystalline structure stability, electronic and vibrational properties only [76]. Since, consensus has been reached concerning the observed crystal structure and stoichiometry i.e. one metal atom for every nitrogen dimmer [76]. Until now the crystalline structure of the newly synthesized palladium per nitride has not been determined. All metal nitrides (PtN 2, IrN 2 and OsN 2 ) studied to date possess uncomplicated behavior due to significant hybridization between 5d electrons and N- 2p electrons at all hydrostatic pressure [54]. They are recoverable to ambient conditions after synthesis at high pressure and high temperature [in the range of 60 GPa and 2000K]. In contrast to these compounds, the synthesized crystalline PdN 2 at high pressure and temperature does not appear to be recoverable at ambient conditions. Daniel et al. [189] have claimed that the equation of states cannot be accurately described within either the local density or generalized gradient approximations however only the Heyd-Scuseria-Ernzerhof exchange-correlation functional (HSE06) provides very good agreement with experimental data [76]. Wessel et al. [77] have recently predicted from theory that iron-perntrides can be synthesized. It is theoretically found that FeN 2 crystallizes in the space group with simple cubic closed-packed iron layers, a bulk modulus of about 192 GPa and an iron saturation moment of approximately 1.68 µ B. However, the thermodynamic calculations of Ref. [77] reveal that the predicted structure is dynamically stable only at 17 GPa. However, the iron pernitride has not been synthesized experimentally yet, experimental groups succeeded in synthesizing binary iron nitride with a 1:1 ratio ~ 79 ~

49 Palladium Pernitride (PdN2) and Iron Pernitride (FeN2) Chapter 3 giving an anti-ferromagnetic phase with a zincblende cubic crystal structure in accordance with related theoretical studies [78]. A number of other properties, such as the bulk modulus of FeN, still remain unverified due to the unavailability of single crystal FeN. Though, iron nitrogen compound with the nitrogen content exceeding the iron content has been synthesized, but a number of issues still remain unclear, most importantly the formation of pernitride itself. This new crystalline phase of iron-nitride revealed thermodynamic stability competing with other phases having composition of FeN+1/2α-N 2, 1/3ε-Fe 3 N+5/6α-N 2 derived from the ThC 2 type structure [77], where, α- N 2, 3ε-Fe 3 N are the alpha phase of nitrogen and the berthollide phase of Fe 3 N. The calculated thermodynamic quantities of Ref [77] reveal that the formation of FeN 2 will be an endothermic process at absolute zero temperature with a formation enthalpy. The pressure and temperature dependent Gibbs free energy of FeN 2 were obtained, including the dynamical contribution. This is accompanied by a small error as nitrogen condenses at moderate temperature and pressure. Thus, the entropy of nitrogen was overestimated and as a result the true value of the Gibbs free energy will be more negative and the formation can take place at a lower pressure than estimated. It is well established that the mechanical or thermodynamic stability only reflects a local stability of structures, whereas the full phonon dispersion curves are believed to shed more kinetic information on the structural stability of the earlier proposed structure. Moreover, phonon dispersion relation gives a criterion for the crystal stability and indicates possible structural changes through soft modes [79]. Therefore, in order to clarify the existence of FeN 2, a systematic lattice dynamical ~ 80 ~

50 Palladium Pernitride (PdN2) and Iron Pernitride (FeN2) Chapter 3 calculation is essential. This will further shed light on the stability of the structure and possible synthesis conditions for iron pernitride. In the present theoretical study we have performed a detailed investigation of structural, electronic and vibrational properties of iron pernitride in a hexagonal space group symmetry and palladium nitride in space group using density functional theory. The schematic diagram of an energetically and mechanically more favourable hexagonally symmetrical crystal structure for spin-polarised FeN 2 with space group symmetry is quite similar to the one obtained in Ref. [77], which confirms that the structure is fully relaxed with optimised coordinates and parameters. The presence of nitrogen dimmers is clearly seen Computational Methodology Our first principles investigation of ground state properties like the lattice constant and bulk modulus are done in the frame work of density functional theory using the ultrasoft pseudopotential method introduced by Vanderbilt implemented in quantum espresso code [22]. In the present approach, exchange correlation is generalised gradient approximations (GGA) using a functional proposed by Perdew Burke- Erzernhof which was found to accurately reproduce ground state parameters, as well as magnetic properties for a wide range of crystals. The neutral atomic configurations were set to be the reference state. The plane waves sets with cut-off energies 38 and 60 Ry were used in order to describe the electronic wave function in a periodic crystal with 500 Ry of augmented charges for PdN 2 and FeN 2 respectively. Integration over the Brillouin zone was carried out using an and 6x6x6 mesh of k-points using the Monkhorst-Pack scheme and occupation numbers were treated within the ~ 81 ~

51 Energy (ev) Energy (ev) Palladium Pernitride (PdN2) and Iron Pernitride (FeN2) Chapter 3 Mathfessel-Paxton scheme with a broadening of σ = 0.02 Ry for both pernitride using first order Hermite-Gauss polynomials. The calculation of the dynamical matrix for phonon calculations using DFPT has been performed on a uniform grid of q-vectors in the Brillouin zone for a 2x2x1 and 4x4x4 without introducing supercells for FeN 2 and PdN 2 respectively [80] Results and Discussions Palladium Pernitrides (PdN 2 ) The structure of pyrite PdN 2 was optimized at zero and different high pressures (11, PdN 2 0 GPa Total p s d 4 2 PdN 2 11 GPa Tot d p s E f -6 X Figure 3.24(a) Electronic band structure for pyrite structure of PdN 2 at zero pressure along with total and projected density of states -6 X M ZA T M Figure 3.24(b) Electronic band structure for pyrite structure of PdN 2 at 11.0GPa along with total and projected density of states ~ 82 ~

52 Volume (Bohr 3 ) Volume (Bohr 3 ) Energy (ev) Energy (ev) Palladium Pernitride (PdN2) and Iron Pernitride (FeN2) Chapter PdN 2 18 GPa Tot # d p 4 2 PdN GPa Tot d p s 0 E f 0 E f X M ZA T M Figure 3.24(c) Electronic band structure for pyrite structure of PdN 2 at 18.0GPa along with total and projected density of states -6 X M ZA T M Figure 3.24(d) Electronic band structure for pyrite structure of PdN 2 at 60.0GPa along with total and projected density of states and 60GPa). The calculated electronic band structures of pyrite PdN 2 crystal structure at zero pressure and higher pressures P T GPa Pressure (GPa) 11GPa, 18GPa and 60GPa presented in Fig. 3.24(a-d) along with total and partial electronic density of states. The electronic structure plots are highly pressure Pressure (GPa) Figure 3.25(a) Equation of state as obtained from GGA-XC. dependent and the gap existing along Γ-X line of Brillouin zone is minimum in the case of 0 and 18GPa. This indicates the gap which opens up with pressure again closes at 18GPa. ~ 83 ~

53 Palladium Pernitride (PdN2) and Iron Pernitride (FeN2) Chapter 3 Figure 3.25(b) Bond length between palladium and nitrogen as function of pressure. Figure 3.25(c) Bond length of nitrogen dimmers as a function of pressure. The projected density of states shown in the right panel of each figure at corresponding high pressure reveals that the valence and conduction orbital bands are determined by N-p and Pd-d electron states. The pressure sharply shifts Fermi energy towards the lower energy region makes Pd-d band crossing the Fermi level at M, ΔT-ZA and Г-X direction in the Brillouin zone makes PdN 2 pyrite from insulator to metallic. Our calculated electronic band structure and density of states are in good agreement [81].This electronic phase transition in pyrite PdN 2 can also lead to first order isostructural transition at applied higher static pressure near the decomposition pressure. Our calculated EOS data is in good agreement with the theoretically calculated EOS of Alberg et al. [82] and experimental x-ray diffraction measurements [81]. ~ 84 ~

54 Palladium Pernitride (PdN2) and Iron Pernitride (FeN2) Chapter 3 Figure 3.26(a) Phonon dispersion curve for pyrite structure of PdN 2 at ambient and 60GPa Figure 3.26(b) zone centre phonon modes for pyrite structure of PdN 2 at higher pressure Figs. 3.25(b) and 3.25(c) present Pd-N and N-N bond lengths up to 15 GPa obtained by optimizing the structure at each pressure. These figures clearly indicate a sharp deep and rise in bond lengths near phase transition pressure (11GPa). Thus the volume dependence of bond length foreshadows of peculiar properties of this compound near the phase transition pressure. The palladium per nitride with pyrite structure is very soft and elastically stable at 0GPa, though thermodynamically unstable, become energetically favorable at high pressure. The lattice dynamical calculations reveal that the pyrite PdN 2 is dynamically stable at ambient and 60GPa [82]. The frequencies of Raman active A g and T g modes, agree well with the experimental data particularly at 60 GPa. The pressure variation of Raman active modes shows a linear variation; however, at higher pressure the optical frequencies decrease and acoustic frequencies increase [8]. Fig. 3.26(b) shows that almost all the zone center phonon branches behave anomalously at calculated electronic phase transition provides clear support in the decomposition of PdN 2 pyrite structure at 11GPa and predict the quenching of crystal structure from high pressure to ambient pressure can be possible to recover crystalline pyrite PdN 2 in ambient pressure. ~ 85 ~

55 Total energy (ev) Palladium Pernitride (PdN2) and Iron Pernitride (FeN2) Chapter Iron Pernitride (FeN 2 ) In the case of FeN 2 the ground state properties for the considered crystal structure under consideration have been calculated using the total energy E and volume V (E-V curves) obtained from the geometry optimization. Fig shows the schematic hexagonal crystal structure of FeN 2 in space group symmetry no Fe and N atoms occupie the 3a and 6c wyckoff positions respectively Figure 3.27 Schematic hexagonal crystal structure of FeN 2 phase in symmetry. Fe and N atoms are in red and with z = The energy volume curves presented in Fig yield the equilibrium lattice constants a = and c = and a = and c = Bohr respectively for spin (SP) and non-spin (NSP) polarized calculations. The bulk modulus and pressure derivative of bulk modulus have been obtained using the equation of state (EOS) from the integration of pressure of the third order Birch- Murnanghan equation of states. Fig clearly reveals that the SP-FeN 2 is Non spin-polarized (NSP) Spin polarized (SP) Volume (Bohr 3 ) Figure 3.28 Theoretically obtained total energy as a function of volume for both spinpolarized (SP) and Non-spin polarized (NSP) FeN 2. energetically stable. The calculated bulk modulus from the geometry optimization is 241 GPa and 299 GPa for SP-FeN 2 and NSP-FeN 2 respectively. The slight higher value of bulk modulus for SP-FeN 2 relative to the previously reported value [77] can be attributed to an increased cutoff energy of 60 Ry (816 ev) as both calculations were performed using GGA. The higher cutoff energy is required for the convergence ~ 86 ~

56 Energy (ev) Energy (ev) Palladium Pernitride (PdN2) and Iron Pernitride (FeN2) Chapter E f 2 0 E f L EDOS L EDOS Figure 3.29(a) Electronic band structure and total electronic density of states of FeN 2 using spinpolarized calculations. Figure 3.29(b) Electronic band structure and total electronic density of states of FeN 2 using nonspin-polarized calculations. of phonon frequencies to within the accuracy of 2-4 cm -1 using self-consistent plane wave methods. When using projected augmented waves (PAW), the cutoff energy of 500 ev (37.75 Ry) used in Ref. [77] is sufficient for total energy calculations. Furthermore, in the absence of experimental data, both are predictive only. The magnetic moment of FeN 2 in the present case is 1.86 μ B, which is within 10% of the previous calculation [77]. To understand the effect of nitrogen incorporation, the electronic structure, electronic density of states (EDOS) and bonding features of SP and NSP FeN 2 are analysed. Figs. 3.29(a) and 3.29(b) display the electronic band structure of FeN 2 for spin and non-spin polarized cases in the full Brillouin zone along with the total electronic density of states. Figures. 3.30(a) and 3.30(b) respectively present the spin-up and spin-down density of states along with the projected density of states respectively for FeN 2. It can ~ 87 ~

57 Energy (ev) Energy (ev) Palladium Pernitride (PdN2) and Iron Pernitride (FeN2) Chapter EDOS dosup dosdw Figure 3.30(a) Total electronic density of states of FeN 2 for spin up and spin down cases ~ 88 ~ p_up p_down d_up d_down s_up s_down PDOS Figure 3.30(b) The projected electronic densities of states of FeN 2 for spin up and spin down cases. clearly be seen from these figures 3.30(a) and 3.30(b) that nearly all bands are similar except the one crossing at the Fermi level along the Г-K, Г-M and A-L directions of the Brillouin zone. It is observed that the SP-FeN 2 bands cross only in the Г-K, Г-M and A-L directions of the Brillouin zone, unlike NSP-case. This may be due to the fact that the finite electronic densities of states at Fermi level shown in the figures indicate high electronic mobility for both cases. This suggests that FeN 2 can be used for special applications which require high electron mobility. Furthermore, a pronounced maximum in the electronic density of states at the Fermi level suggests electronic instability for NSP-FeN 2, while a pronounced minimum at the Fermi level in the density of states plot reveals extraordinary electronic stability for SP-FeN 2. The electronic bandstructure of SP-FeN 2 and NSP-FeN 2 reveals that the lowest occupied band, which lies about 21 ev below the Fermi level, is due to N-2s states. Many band complexes with a width of nearly 20 ev occur above the N-2 bands; this gives predominantly N-2s and Fe-3d states. In the case of SP-FeN 2, we observe covalent bonding between Fe-3d and N-2p states (-10-5 ev). Furthermore, the Fe-3d states (- 5 4 ev) hybridise and form strong covalent bonds. However, the electronic band E f

58 Palladium Pernitride (PdN2) and Iron Pernitride (FeN2) Chapter 3 structure of NSP-FeN 2 is highly dominated by strong hybridization between Fe-3d and N-2p states. From Fig. 3.30(a), we observe that the filled region (-5 0 ev) may be viewed as the bonding states; while unoccupied higher region corresponds to the antibonding states between (0 5 ev) N-2p and N-2p states. Figure 3.31(b) Electronic charge contour map in (110) plane of NSP-FeN 2 in BZ. Contour lines run from 0.00 to 0.09 in steps of 0.01 with the lightest grey corresponding to 0.00 and the darkest grey to For more detailed insight into the bonding character of SP-FeN 2 and NSP-FeN 2, we calculated their charge density; the contour plots are presented in Figs. 3.31(a) and 3.31(b) respectively. It can be seen very clearly that the bonding between Fe atoms and their neighbour N atoms builts very strong directional bonds. Similar conclusions are drawn in Ref [77]. Figure 3.31(a) Electronic charge contour map in (110) plane of SP-FeN 2 in BZ. Contour lines run from 0.00 to 0.09 in steps of 0.01 with the lightest grey corresponding to 0.00 and the darkest grey to An accurate description of phonon frequencies is a powerful test for a theoretical model to date. Despite it being a necessary tool to confirm the dynamical stability of a proposed structure, no serious effort has been made so far to investigate complete phonon dispersion curves and their associated properties. Furthermore, the important information about the dynamical stability of structure and potential relaxation mechanism can be derived from phonon dispersion relations. ~ 89 ~

59 Palladium Pernitride (PdN2) and Iron Pernitride (FeN2) Chapter 3 Figure 3.32 Eigenvector representations of the some selected zone-center phonon modes. The non-availability of any experimental data on vibrational modes from IR or Raman spectroscopy is mainly due to the unavailability of the single crystal of iron pernitride. Recently, Wessel et al. [77] have mentioned about the dynamical stability of FeN 2, however, a detailed analysis of phonon dispersion and associated properties is still lacking. The ability to provide a complete structural and dynamical stability using lattice dynamical calculations motivates us to investigate the phonon dispersion of both magnetic (spin-polarized) and non-magnetic (non-spin polarized) cases. The hexagonal phase of FeN 2 has point group symmetry formed by N-N dimmer bonds with Fe atoms coordinated octahedrally at the corners of the stack layers. The dynamical representation can be expressed as the direct product of the vector representation Γ v and atomic permutation representation for a given group as. Based on lattice structure and symmetry, the number of Raman and infrared active modes can be predicted by group theory. For 9 atoms per unit cell in the case of FeN 2, there are 27 vibrational modes at the Γ-point (zone centre) of the Brillouin zone. The zone centre optical phonon modes are decomposed according to the following representation:. The three acoustic ~ 90 ~