1. INTRODUCTION:
Besides a number of exciting physical properties, transition metal
(TM) nitrides compounds adopt several crystal structures, crystal lattices and
form different phases depending on the Rx/RM ratio where
RM and Rx are the atomic radii of metallic and
non-metallic atoms, respectively. Almost all the nitrides of TM are usually
cubic, where the metallic atoms form the face-centered cubic (fcc) sub lattice and non-metallic
atoms occupy interstitial positions, forming NaCl-type
structure and for many of them, the phonon dispersion curves have been measured
by coherent inelastic neutron scattering [1-10]. The general nature of the
optic branches is for most of the TMN, similar to that of ionic compounds with
the rock-salt structure. The longitudinal branches (LO) lie above the
transverse branches (TO), due to the long-range Coulomb interactions that are screened
by the conduction electrons. This metallic screening destroys the Lyddane-Sachs-Teller Splitting [11] and results in a
degeneracy of the LO and TO modes at the center of the Brillouin
zone. However for most of the TM compounds the screening vector is small
compared to the dimensions of the Brillouin zone.
The LO branches therefore recover quickly from the screening
effect and are for most of the TMN found above the TO branches. The only
exceptions are UC [3], UN [5] and as the results of the present measurements
show HfN. The most successful model so far in
reproducing the measured phonon dispersion curves of transition metal carbides
and nitrides with rock salt structure is the breathing shell model [12]. We
shall use breathing shell model (BSM) for the purpose of assessing the role
played by the breathing motion of the valance electrons. Here we used this
model theory to calculate complete phonon dispersion curve of transition metal mononitride (HfN) compound. In
addition, the effect of anion and cation polarizabilities on the phonon spectrum is analysed by this theory. The anomalies in the dispersion of
the acoustic branches and optical branches have been detected which are well
described by experimental results.
2. THEORY OF MODEL APPLIED:
The Breathing Shell Model (BSM) [12] is the isotropic deformation of spherical
electron shells is considered in the form of expansion and /or contraction of
these shells during lattice vibrations. Using the shell model
notations of woods et.al. [13] and introducing
these ideas in the theory of shell model, we get the basic equations of motion
for BSM as follows:
D(q)=(R′+ZCZ)-(R′-ZCY)(R′+G+YCY)-1(R′+YCZ) (1)
Where, R’= (R-QH-1Q†); C and R are the
Coulomb and short-range repulsive interaction matrices respectively. Q is (6×2)
matrix representing the interactions between the ion displacements and
breathing mode variables. The H is a (2×2) matrix and represents the
interactions between the breathing mode variables of different ions in the
lattice. These breathing matrices Q and H are written as [12]
(2)
The explicit expressions for these dynamical matrix.G
and A are the diagonal matrices and represent the core-shell interaction and
shell charge, respectively. The parameter involved in the dynamical matrix D(q) are determined by some
microscopic experimental data, i.e. elastic constant, dielectric constant etc..
In order to minimize the number of parameters, we have assumed the shell
charges, Y1=Y2=Y. The parameters are tabulated in the
Tables.
3. RESULTS AND DISCUSSION:
The phonon spectra obtained above have been used to predict the
phonon dispersion relations wj(q) for
HfN which are measured from inelastic neutron
scattering. This BSM [12] has been used by Jha and co
worker [14, 15] to explain satisfactorily the anomalous phonon properties in
rare earth chalcogenides and pnictides.
(i) Phonon
dispersion relations:
The phonon dispersion relations along
principal symmetry directions Δ, Σ and Λ, using a
phenomenological model theory, which include the short-range breathing motion
of the valance electrons. For this purpose we derive the model
parameters self consistently, using known macroscopic properties which include
equilibrium lattice constant, elastic constants, zone center vibrational frequencies, ionic polarizabilities
and dielectric constants. The input data used for this purpose are given in
Table1. and also output parameters are listed in
Table2. The phonon dispersion curves (PDC) of HfN
displayed in Figs.1 and compared them with their neutron data [8, 16]. The
measured data from (BSM) model are compared to the results of experimental
inelastic neutron scattering values. ○ and
▲ represents the experimental points of longitudinal and transverse
phonons respectively. The simple nature of the optic branches is for most of
the transition metal carbides and nitrides, similar to that of ionic compounds
with the rock-salt structure. The longitudinal branches (LO) lie above the
transverse branches (TO) due to the long-range coulomb interactions that are
screened by the conduction electrons. In optic and acoustic branches are
separated by an energy gap. It is revealed from Fig.1 that in case of HfN the theoretical results of PDC are in very good
agreement with experimental data along (q 0 0), (q q
0) and (q q q) directions for LO, TO and LA, TA
modes. Because of the screening due to conduction electrons the LO and TO modes
are degenerate at the Γ point, whereas the splitting of LO and TO
modes at the L point is due to the long-range coulomb interaction which
is not completely screened at larger q vectors. Strong electron-phonon
interactions might be responsible for such peculiar behaviours.
(ii)
One-Phonon density of nitride compounds:
In Fig. 2, we have plotted one phonon density of state of HfN and compare it with neutron scattering measurement
(experimental)[8]. Our calculated results agree
satisfactorily with the neutron scattering data. Our prediction of a peak at
about 15.1THz (Fig. 1.) due to optical phonon at X-point in HfN
is in accordance with experimental observation [16]. The peaks observed at
50.04 cm-1, 83.4 cm-1, 133.44 cm-1 and 200.16
cm-1 are longitudinal and transverse acoustical phonons, while the
longitudinal and transverse optical peaks are observed at 467.04 cm-1,
500.4 cm-1 and 567.12 cm-1 respectively. From these
results we conclude that the calculated values from BSM model are in good
agreement with the experimental datas [8].
(iii) Specific heat of
nitride compounds:
Einstein took the atoms of a crystal to be independent oscillators
each having the same (circular) frequency ωE
and able to vibrate freely in space. The energy of each oscillator was
quantized in units of ħωE, and
showed that the average energy of a crystal of N atoms at temperature T would
then be given by
Ū= 3Nħ ωE/exp(ħ ωE/kBT)-1
Where, KB is Boltzmann’s constant. The specific heat is
then given by
Cv = 
Cv is
shown as a function of T/θE, where θE is a convenient abbreviation for ħωE/kB
and has the dimensions of temperature. Cv
is zero at T=0 and rises asymptotically to the value 3NkB = 3R when
T» θE. For high temperatures
therefore, quantization is unimportant and the specific heat has the same value
as if each degree of freedom of the system had energy 
kBT, whereas at lower temperatures there is a pronounced deviation
from this result of the theorem of energy.