FP-LAPW INVESTIGATION OF ELECTRONIC, MAGNETIC, ELASTIC AND THERMAL PROPERTIES OF Fe-DOPED ZIRCONIUM NITRIDE

Full Potential- Linear Augmented Plane Wave (FP-LAPW) method has been employed to study the electronic, magnetic, elastic and thermal properties of Fe-doped Zirconium nitride. In this work, Fe-atoms were doped into the super cell of ZrN in doping concentrations of 12.5%, 25% and 37.5% to replace Zr atoms. Electronic properties such as band structure and DOS were plotted and compared for the doped compounds. Charge density contours were plotted for all the doped compounds. The non-magnetic ZrN doped in different Fe concentrations were found to be ferromagnetic. Magnetic moments have been calculated and compared. Elastic properties have been studied and compared with electronic properties. Appearance of magnetic ordering and its influence with the elastic properties have been reported. Impact of 3d states of Fe in DOS plot on the elastic nature of the compounds has been highlighted. Thermal properties such as Debye temperature and molar heat capacities at low temperature have been determined. Debye temperature is found to decrease with higher doping concentrations. Molar heat capacities are found to increase with higher concentrations of Fe atoms.


INTRODUCTION
The FP-LAPW method has been employed to study the electronic, magnetic, elastic and thermal properties of Iron-doped Zirconium nitride.In this work, Fe-atoms were doped into the super cell of ZrN in doping concentrations of 12.5%, 25% and 37.5% to replace Zr atoms.Electronic properties such as band structure and DOS were plotted and compared for the doped compounds.Charge density contours were plotted for all the doped compounds.The non-magnetic ZrN doped in different Fe concentrations were found to be ferromagnetic.Magnetic moments have been calculated and compared.Elastic properties have been studied and compared with electronic properties.Appearance of magnetic ordering and its influence with the elastic properties have been reported.Impact of 3d states of Fe in DOS plot on the elastic nature of the compounds has been highlighted.Thermal properties such as Debye temperature and molar heat capacities at low temperature have been determined.Debye temperature is found to decrease with higher doping concentrations.Molar heat capacities are found to increase with higher concentrations of Fe atoms.
The scope of this work circles around the important aspects of doping ZrN with Iron and determining its electronic, magnetic, elastic and thermal properties.The electronic structure , magnetic moments, elastic constant values, Debye temperature and molar heat capacities for the doped Zr 0.875 Fe 0.125 N , Z r 0.75 Fe 0.25 N and Zr 0.625 Fe 0.375 N are reported.The band structure of these compounds are plotted and compared.On doping Fe in 12.5%, 25% and 37.5% respectively with the super cell of ZrN (Zr 8 N 8) , it has been found that all the doped compounds uniformly exhibit ferromagnetism.
There is an increase in the values of magnetic moments for doped compounds.
The elastic properties change with respect to increased doping concentration of Fe.The change in elastic properties may have some telling effect on the hardness of the material.Present work also highlights on the thermal properties such as Debye temperature which is found to vary in ferromagnetic state.As ZrN is a ceramic material, the molar heat capacities of doped ZrN compounds are also determined.The heat capacity of the ceramic ZrN is found to be intrinsically increased after doping with Fe.
Zirconium Nitride has recently been under research to unravel its potential application in nuclear reactors as inert matrix fuel(IMF) material.

METHOD OF CALCULATION
The total energies, lattice constants, electronic and magnetic properties, elastic constants and thermal properties were calculated within the Density Functional Theory(DFT) formalism using WIEN2k code (Blaha et al.The importance of this method is that near an atomic nucleus the potential and wave functions are similar to those in an atom which mean that they strongly vary but are nearly spherical.But, between the atoms the potential and wave functions are smoother (Singh 1994).In this method, the potential and charge density are treated without shape approximation and core

Volume Optimization
ZrN is basically a ceramic non magnetic substance.The  A o .Experimentally, the lattice constant was also found to be 4.61 A o (Wyckoff 1973).The electronic band structure calculations were done in magnetic phases for all the three doped compounds.Table 4.1 & Table 4.2

Magnetic Properties
The volume optimization has shown that ZrN is paramagnetic in nature.The magnetic moment of ZrN calculated in magnetic phase is -0.02349 µ B .
After doping ZrN with Fe in concentration percentage of 12.5%, 25% and 37.5% , the magnetic moments are found to increase.With Fe doped in 12.5% in the super cell of Zr 8 N 8 (Zr 0.875 Fe 0.125% N), one Zr atom at (0 0 0) is replaced by an Fe atom.The 3d-bands of Fe atom is responsible for the magnetic ordering of doped compounds.The unpaired 3d orbitals contribute to the total magnetic moments.But these 3d orbitals undergo hybridization as a result of overlapping of 3d bands.It is known that the orbitals that are more exposed can contribute significantly to the magnetic moments in an atom.So, the hybridization results in slight decrement in total magnetic moment due from the unpaired 3d orbitals of doped Fe atom.
Total magnetic moment for the compound Zr 0.875 Fe 0.125% N is 3.29 µ B as given in Table 4.2.An increased doping percentage of 25% which means adding two Fe atoms in the place of two Zr atoms at (0 0 0) and (½ 0 0) yields a magnetic moment of 7.51 µ B for Zr 0.75 Fe 0.25 N.This increase in magnetic moments is almost twice that of Zr 0.875 Fe 0.125 N. The additional 3dpeaks due to 3d-bands are responsible for the increase in magnetic moment.
For the compound Zr 0.625 Fe 0.375 N, there is 37.5% increase in the number of doped Fe-atoms that yields a total magnetic moment of 11.075 B .In this case, the additional peak as shown in Figure 4.7 appears to be little suppressed and not found prominent like that of the previous case.The band structure plot of Zr 0.625 Fe 0.375 N indicates the dispersion of 3d bands.Hence, this compound exhibits only an increase of half the magnetic moment value of its predecessor.Table 4.3 gives the atom-wise contribution of magnetic moments for all the three doped compounds.The appearance of magnetic ordering among the doped compounds causes a significant change in elastic and thermal behaviour of the materials.This is discussed in the sections 4.3.5 and 4.3.6.The increase in magnetic moments with the increase in doping concentration is shown in Figure 4.8.

Elastic Properties
The elastic constants provide the information on the stability and stiffness of the materials.The second order elastic constants (SOEC) C ij are the proportionality constants related to the strain ( i ) and applied stress ( i ) such that i ij i C .The constants C ij is the measure of response of the 80 crystal to the external forces.The elastic constants C ij is obtained from the structures using a common approach (Mehl et al. 1993, Khenata et al. 2006, 2007& Bouhemadou et al. 2007), that is based on the analysis of the changes in the calculated total energy values arising from the changes in strain.A cubic structure has three independent elastic constants C 11 , C 12 , and C 44 .
The bulk modulus is determined with the total energy-unit cell volume data fitted to Murnaghan equation of state (EOS) (Murnaghan 1994).
For cubic systems, the isotropic bulk modulus B (Schrieber et al. 1973) is given by The materials response to linear strain is described by Young's modulus E and is given by The elastic anisotropic factor A is determined using Equation (4.3).
The value of A is equal to unity for the crystals that are isotropic.An anisotropic crystal will have a value ofAgreater than unity.ZrN in the undoped condition has A value less than one.As ZrN is a ceramic material it is more likely that it possesses less metallic characteristics and hence lower in anisotropy.Metals in general exhibit anisotropic crystalline characteristics.
Since, ZrN belongs to a cubic crystal system, it may be closer to isotropic in character.
The values of A for three Fe doped compounds Zr 0.875 Fe 0.125 N, Zr 0.75 Fe 0.25 N, and Zr 0.625 Fe 0.375 N are found to be greater than unity as shown in Table 4.4.Hence, these compounds exhibit high degree of anisotropic behaviour.Moreover, one could find that 3d electrons states of transition metals generally are not delocalized but localized toward the centre of nuclei exhibiting metallic-characteristics.But, the itinerant (delocalized) electrons far from the nuclear-bond only bring about the ductile behavior in materials.
These delocalized electrons cause metal atoms to slide past one another without having to undergo the strong repulsive forces that would otherwise cause the materials to break.In Fe-doped compounds, 3d states are localized towards the nuclei.There are no itinerant electrons to slide with one another under the action of external force and the material is expected to shatter.Hence, our compounds are said to exhibit brittle nature.Doping of Fe atoms largely influence the electronic, magnetic and elastic characters of these compounds.The total DOS plot as seen in Figure 4.7 shows a larger DOS peak for Zr 0.75 Fe 0.25 N.This peak is from 3d orbital of Fe atom which is not so pronounced in the case of Zr 0.875 Fe 0.125 N and Zr 0.625 Fe 0.375 N.This kind of marked behaviour of electronic states of 3d orbital is seen to mildly impact the elastic property of Zr 0.75 Fe 0.25 N. As a result of this, the factors that predict the elastic nature of a material such as   The compound ZrN is known for its hardness as Young's modulus is found to be 296.09GPa in the un-doped state.On doping with Fe, the hardness drops down with the increase in doping concentration.mechanical and classical methods.The Debye temperature corresponds to the upper limit of phonon frequency in a crystal and is calculated using the following relation (Bouhemadou et al. 2008& Wachter et al. 2001).
where h is a Planck's constant, k B is Boltzmann's constant, v a is the atomic volume, N is the number of atoms per formula unit and v m is an average sound velocity.The average sound velocity is approximated (Sun et al. 2004& Jansiukiewiz et al. 2003) as follows: where v t and v l are the transverse and longitudinal sound velocities, respectively and are obtained from the elastic constants as follows:   In general, ceramic materials are known to possess less heat capacity than metals.In metals, the valence electrons of 3d states lying close to the Fermi level would play a vital role in determining heat capacity of materials.This is due to the fact that doping higher concentration of a metal characteristic that of Iron, could reduce inherent ceramic nature in the host ZrN.Generally, ZrN is known to be a ceramic material having high hardness and good thermal property.But a doped ZrN is more likely to have a decreased Debye temperature akin to the results shown in the Table 4.7.The results clearly indicate that there is a tendency to give up its ceramic nature.

SUMMARY
In this work, the electronic, magnetic properties were studied for Fe The extent of application of doped ZrN with higher heat capacities can be explored experimentally in future for possible use in nuclear reactors and in other thermal applications.Also, ZrN is a known refractory compound which is already being used as a coating material.It remains to be seen how a magnetically doped ZrN would influence the materials that are coated.
2001).The band structure calculations were carried out employing full potential linearized-augmented-plane wave plus local orbitals(FP-LAPW) method (Madsen et al. 2001 & Schwarz et al. 2002).The LAPW+lo method expands the Kohn-Sham orbitals in atomic like orbitals inside the Muffin -Tin (MT) atomic spheres and plane waves in the interstitial region.
electrons are treated relativistically.In the present calculations Generalized Gradient Approximations (GGA)(Wu et al. 2006)  were used for the exchange correlation potential and the Kohn-Sham equations were solved(Kohn et al.   1965).The plane wave cutoff for the basis function was set to RK max =7.0.The electronic structures of Fe doped ZrN were studied by constructing super cell of Zr 8 N 8 .The Fe atoms are doped into the super-cells of Zr 1-x Fe x N with x = 12.5%, 25% and 37.5% respectively replacing Zr atoms.Doped Fe atoms occupy structural positions of : (000) in the basis of Zr 1-x Fe x0.125 N, (000) and ( ½ 0 0) in basis of Zr 1-x Fe x0.25 N and (000), (½ 0 0) (¼ ¼ 0) in the basis of Zr 1-x Fe x0.375 N. Unit cells of Zr 8 N 8 were constructed using the configuration of 2×2×2 cubic super-cell.The lattice parameter for primitive unit cell of ZrN was theoretically found out to be 4.60 A o belonging to cubic (Fm3m)space group.The R MT of Zr atom was set at 1.60 bohr while for Fe and N, R MT were set at 1.26 and 0.74 Bohr respectively.The self consistent calculations were found to converge when the total energy of the system is stable within 10 -4 Ry.The integration over the irreducible Brillouin Zones was done on the grid of 200 k points generated for super-cells using the Monkhorst-Pack scheme(Monkhorst et al. 1976).
Figure 4.1 represents the minimum energy of paramagnetic-ZrN which crystallizes in rock salt structure.When ZrN is doped with Fe in the concentration of 12.5%, 25% and 37.5% , the compound settles in magnetic phase for all the doping concentrations.The Figure 4.1 clearly shows the total energies of Zr 0.875 Fe 0.125 N, Zr 0.75 Fe 0.25 N, and Zr 0.625 Fe 0.375 N in magnetic phase.

Figure
Figure 4.1 Total energy as a function of volume for (a) ZrN (b) Zr 0.875 Fe 0.125 N (c) Zr 0.75 Fe 0.25 N(d) Zr 0.625 Fe 0.375 N in rock salt NaCl structure with lattice parameter of 4.6 A o .After doping with Fe atom in the super cell of Zr 8 N 8 , the lattice parameter of super cell changes into 9.49A o , 9 . 1 6 A o and 8.88A o respectively for 12.5%, 25% and 37.5% doping concentrations.

Figure
Figure 4.2 Crystal structures of (a) ZrN (b) Zr 0.875 Fe 0.125 N (c) Zr 0.75 Fe 0.25 N and (d) Zr 0.625 Fe 0.375 N . The crystal structure shown in the Figure 4.2 (b), (c) and (d) are constructed each with a total of 125 atoms of which Fe atoms constitute 12.5%, 25% and 37.5% respectively.The Fe atoms are substitutionally added to replace Zr atoms and the positions occupied by Fe atoms are given below: of Zr and Fe are found out from the crystal structure plotted using software VESTA (Momma et al. 2008) and they are 1.60A o and 1.26 A o respectively.Nitrogen atoms have atomic radius of 0.74 A o .Though the bond length between the Fe-N atoms is not varied when compared to that of Zr-N atoms, both being at 2.22 A o , there is a distinct difference in the way Fe atoms are attached with the nitrogen atoms.In the crystal structure diagram shown in Figure 4.2 (b), (c), and (d), we find that the Fe atoms are slightly pulled apart from the N atoms on both sides.The two dimensional charge density contour plots have been plotted using software VESTA as shown in Figure 4.3.The dense curves with spherical concentrationrepresent an accumulated electronic charge density while the less dense curves represent depleted charge density (Sheik et al.2013).In the intra-atomic bonding, there is an evidence of ionic nature between Zr-N and Fe-N atoms.Generally, in covalent bonding, the bonding is formed between two identical atoms.The charge density plots will be highly delocalized equally since both nuclei will attract the electron density with equal force.But ionic bonding is characterized between two dissimilar atoms with different electron affinities.Hence, the charged density contours around the atoms appear localized, exhibiting spherical symmetry(Sheik et al. 2013).

Figure 4 . 4
Figure 4.4 Band structure and Total DOS of ZrN Electronic configuration of 'Fe 26 ' is

Figure 4
Figure 4.5 (a), at high symmetry point ' ', the 3d orbitals exhibit four-fold 1degenerate and undergo a low dispersion around -3.0 eV.These degenerate bands give rise to energy peaks in DOS plots shown in Fig.(5a) of spin up direction which are not evidenced in DOS of spin down direction.A strong hybridization of atomic orbitals is seen from the energy range of -4.0 eV to -7.2eV that produces sharp peaks in DOS.The partial DOS of different 3d orbitals d z 2 , d x 2 -y 2 , d xy , d xz and d yz are shown in the Figure 4.6(a), (b) & (c).The hybridization results in overlapping of two d z 2 and d x 2y 2 orbitals.This can be vividly seen from the partial DOS plots shown in Figure 4.6(a).This

Figure 4 .
Figure 4.6 Partial DOS of 3d orbital in Fe atom for (a) Zr 0.875 Fe 0 .125 N(b) Zr 0.75 Fe x=0.25 N and (c) Zr 0.625 Fe 0.375 N

Figure
Figure 4.7 Total dos of Zr 0.875 Fe 0.125 N, Zr 0.75 Fe 0.25 N and Zr 0.625 Fe 0.375 N in spin up and spin down directions.

Figure
Figure 4.8 Magnetic moments doped Zr 0.875 Fe 0.125 N, Zr 0.75 Fe 0.25 N and Zr 0.625 Fe 0.375 N with doping concentration general anisotropic in nature.The anisotropy factor A(Sun et al. 2004) has been found using the expression.

b
Chen et al. (2005)    Poisson's ratio which describes the hardness of any material ratio of ZrN and doped compounds are found out using formula given by Equation (4.4).This ratio is equal to about 0.25 for ionic crystals and for covalent it should be around 0.1(Mattesim et al. 2009).The values of Poisson's ratio of these Fe doped compounds suggest that these materials could be ionic in character.As ionic substances have strong tendency to exhibit brittleness, our doped compounds also prove to be brittle in nature.According to empirical formula ofPugh(1954), the ratio of B/G differentiates between the ductile and brittle behaviour.If the ratio of B/G is more than 1.75 then the material is ductile.If B/G value is less than 1.75 then the material is brittle.All the three doped compounds have B/G less than 1.75 and hence they exhibit brittle characteristics.The values of B/G decrease with higher doping concentration.The un-doped ZrN has the highest B/G ratio of 1.40.
crystal anisotropy and Poisson's ratio undergo a slight decrease in their values for Zr 0.75 Fe 0.25 N. In Table 4.4, there is an increasing trend noticeable in the values of A and , going down from the un-doped ZrN.But for Zr 0.75 Fe 0.25 N alone the values decrease indicating some subtle change in the ionicity and hardness of the material.This change may be attributed to the pronounced DOS peak.The elastic constants vary significantly with the increase in the magnetic ordering of the doped compounds.Table 4.5 shows that the values of elastic constants of B, G, and E steadily decrease.Though the Figure 4.9 predicts also the same trend, there is a step decrease instead of a steep one in the value of E from Zr 0.875 Fe 0.125 N to Zr 0.75 Fe 0.25 N.This may be related to the pronounced DOS peak of Zr 0.75 Fe 0.25 N as predicted earlier.

Figure 4 . 9
Figure 4.9 Variation of elastic constants with the doping concentration of Fe Based on the calculated values of Young's modulus E, bulk modulus B, and shear Modulus G, Debye temperature can becalculated.The Debye temperature distinguishes the behaviour of phonons between quantum 11 , C 12 and C 44 are second order elastic constants and is the mass density per unit volume.In the expressions used above in Equations (4.5-4.8),one can understand that Debye temperature is directly related to elastic constants through average elastic wave velocity.This relation is clearly seen in the way average velocity (V m ) and Debye temperature ( D ) increase and decrease with respect to each other.

Figure 4 .
Figure 4.10 Variation of heat capacities with Debye temperature at low temperatures

doped
ZrN compounds.The band structure and DOS involving the 3d bands of doped compounds have been plotted.Crystal structures and the charge density contours of the doped compounds are shown in this work.The charge density contours predict ionic nature for all the compounds.The magnetic moments of the doped compounds were determined and they are found to increase with respect to doping concentration of Fe.Elastic constants of doped ZrN are calculated for the various concentrations of Fe.Increase in magnetic ordering is seen to decrease the elastic constants values of doped material.The role of electronic states of a doped compound influencing the elastic nature is highlighted.After doping Fe in different concentrations, the hardness and elastic strength decrease for ZrN.It is found that brittleness of the compounds increases with doping concentration.The Poisson's ratio determined for the doped compounds also prove the brittle nature.ZrN has high Debye temperature which on doping, decreases significantly.Molar heat capacity, another thermal property determined for these compounds increases with higher doping concentrations of Fe.

Table 4 .6 Calculated values of Debye temperature( D ) in Kelvin for doped compounds
f -Chen et al.

Table 4 .7 Calculated values of heat capacities in J/mol/K for doped compounds at low temperatures Sl.No. Compound
As Fe atoms are doped, this causes the heat capacity of the material to rise.On the other hand, Debye temperature decreases with doping concentration.