Description
The characteristics of a material that determine how it reacts when it is subjected to some type of force that attempts to stretch, dent, scratch, or break it.
First Principles Study of Structural, Electronic, Elastic and
Mechanical Properties of GdSn
3
and YbSn
3
Intermetallic
Compounds
Abstract:-
First principles study of structural, electronic, elastic and mechanical properties of ferromagnetic GdSn
3
and non-
magnetic YbSn
3
rare-earth intermetallics, which crystallize in AuCu
3
-type structure, is performed using density
functional theory based on full potential linearized augmented plane wave (FP-LAPW) method. The ground state
calculations are carried out within PBE-GGA, PBE-sol GGA and LSDA approximations for the exchange correlation
potential. The calculated ground state properties such as lattice constants and bulk moduli agree well with the
experiment as well as other theoretical results. We report elastic constants for these compounds for the first time. Both
these compounds are found to be ductile in nature. The computed electronic band structures show metallic character.
We also report mechanical properties of these compounds for the first time. Keywords: Rare-earth; Density functional
theory; Elastic constants.
1. Introduction
Rare earth intermetallic systems present a great diversity of magnetic behaviors, often associated with complex
magnetic phase diagrams as a function of temperature and applied magnetic field. The rare-earth metals have high
magnetic moments and a diverse range of magnetic structures . Their magnetic properties are determined by the
occupancy of the strongly localized '4f' electronic shells, while the outer 's-d' electrons determine the bonding and other
electronic properties. Rare earth based intermetallics are of considerable technological and scientific interest due to
extra ordinary magnetic properties and industrial applications. These properties are governed by different types of
interactions involving the highly correlated and strongly localized '4f' states of rare earth and the 'd' states of transition
metal atoms, which are comparatively weakly correlated and more delocalized and also the valence states of R atoms,
which are expected to be the mediators of indirect exchange coupling [1, 2].
GdSn
3
and YbSn
3
belong to the family of Rare earth tristannide RESn
3
compounds. These systems have attracted
a great deal of interest because of their salient features such as valence fluctuations, magnetic moment formation,
crystal field effects or multiaxial magnetic structures [3]. An experiment has been conducted by Miller and Hall
[4] to synthesize heavy rare-earth element tristannide LnSn
3
(Ln=Tb, Dy, Ho, Er, and Y) compounds using a
high pressure technique. TbSn
3
and YSn
3
have been synthesized at P = 1-7GPa and T = 400°C-1300°C.
These two compounds also have a cubic AuCu
3
type structure with a lattice constant a = 0.466 nm and 0.467 nm, and a
density of 8.45 g/cm
3
and 7.27 g/cm
3
, respectively[5]. Superconducting state in YSn
3
with AuCu
3
-type
structure has been studied by Kawashima et. al. [6]. Their electric field gradients were also calculated. Crystal
fields and magnetic properties of NdSn
3
, NdPb
3
and NdIn
3
have been studied by Lethuillier et. al.[7]. High
magnetic field properties of GdIn
3
have been investigated by Kletowski et. al. [8]. Interband optical transitions in CeSn
3
and LaSn
3
have been studied by measuring and calculating the optical conductivities of these compounds by Kim [9].
Temperature dependence of magnetic fluctuations in the intermediate valence system CeSn
3
has been reported by
Capellmann et. al.[10]. Valency of rare earths in RIn
3
and RSn
3
and ab-initio analysis of
electric field gradients have been investigated by Asadabadi et. al.[11]. The structural properties such as lattice
parameter and bulk modulus of SmSn
3
, EuSn
3
, GdSn
3
, TmIn
3
, YbIn
3
and LuIn
3
using LDA, GGA and GGA +
spin polarized approximations have been calculated using Wien97 code [12]. Observation of the de haas-van
alphen effect in powdered YbSn
3
and CaSn
3
has been reported by Klaasse et. al [13]. YbSn
3
shows a
superconducting transition at around 3.6 K [14].
The rare earth intermetallics GdSn
3
and YbSn
3
compounds crystallize in cubic AuCu
3
-type structure (space
group symmetry of Pm3m (No.221)). In the present work, we report the structural, electronic, elastic and
mechanical properties of GdSn
3
and YbSn
3
compounds using density functional theory. The elastic constants of
these compounds are reported for the first time. A brief description of the computational details is outlined in
Section 2 while Section 3 covers the results, followed by discussion.
2. Computational Method.
The present calculations have been performed using full potential linear augmented plane wave (FP-LAPW)
67
method as implemented in WIEN2k package [15]. In this method, the basis set is obtained by dividing the unit cell into
non-overlapping spheres and an interstitial region. Here PBE-GGA, PBE-sol GGA and LSDA is used
for the exchange and correlation effects [16, 17, 18]. The values of K
max
×R
MT
= 7.0 and l
max
= 10 are kept
throughout the calculation. A dense mesh of 10×10×10 k points is used and tetrahedral method [19] has been
use-d for the Brillouin Zone integration. The calculations are iterated until the total energies are converged below
10
4
Ry. The total energies are calculated as a function of volume and fitted to Birch-Murnaghan equation of
state (third order) [20] to obtain the ground state properties like zero-pressure equilibrium volume.
The elastic moduli require knowledge of the derivative of the energy as a function of the lattice strain. The
symmetry of the cubic lattice reduces the 21 elastic constants to three independent elastic constants namely C
11
,
C
12
and C
44
. The elastic stability criteria for a cubic crystal at ambient conditions are C
11
+2C
12
> 0, C
44
> 0 and C
11
- C
12
>
0 and C
12
< B < C
11
. In the present work, the elastic constants are calculated using the tetrahedral and
rhombohedral distortions on the cubic structure using the method developed by Thomas Charpin integrated it in
the WIEN2k package [15]. The systems are fully relaxed after each distortion in order to reach the equilibrium
state. C
44
is proportional to the shear modulus and can be used as a measure of shear resistance.
3. Result and discussion
3.1. Ground state properties
The spin polarized electronic band structure calculations have been carried out to obtain the total energy of the
GdSn
3
and YbSn
3
intermetallics using the first principles FP-LAPW method. In order to calculate the ground
state properties, the total energies are calculated in AuCu
3
type structure for different volumes around the equilibrium
cell volume V
0
. The calculated total energies are fitted to the Birch-Murnaghan equation of state to determine the
ground state properties like lattice constant (a
0
), bulk modulus (B) and its pressure derivative (B?) at minimum
equilibrium volume V
0
.
B'
E(V ) = E
0
+ BV (V 0 /V ) + ÷ BV
B' (B'÷1)
B'÷1
Pressure is obtained by taking volume derivative of the total
energy
P(V ) = B
V ÷ B'
B'
V
0
and
The calculated ground state properties are given in Table 1, and compared with the experimental data and available
other theoretical calculations [12, 21]. It is seen from Table 1 that our calculated values of lattice parameter using PBE-
GGA are in better agreement with the experimental results than those obtained using PBE- sol GGA and LSDA
calculations. The better prediction of experimental results by GGA has also been observed in most rare-earth metals
and is believed to be due to the fact that the nonlocality of exchange and correlation is better taken into account by
GGA than LDA [22]. Spin polarized calculations does not practically affect the
ground state properties of YbSn
3
due to the presence of almost fully occupied 'f' shell of Ytterbium. The
calculated total magnetic moment (µ
tot
) of both the compounds are given in Table 2. Furthermore, as, to our
knowledge, no experimental data for the bulk modulus and its derivative have been reported yet, our results can
serve as a prediction for future studies.
3.2. Electronic properties:
The self consistent spin polarized band structures (BS) along the high symmetry directions for majority and
minority spins for GdSn
3
and YbSn
3
compounds are presented in Figure 1.The total and partial densities of
states (DOS) for these compounds at ambient pressure are also calculated and presented in Figure 2. The Fermi
level is shown at the origin. It is clear from the figures that the lowest lying bands below -20eV are due to Sn 'd'
like states in GdSn
3
. The bands which lies in between -11 eV and -5 eV are due to Sn 's' like states. There is a
flat band below the Fermi level around -4 eV which are mainly due to Gd 'f' states in majority spin which gets
shifts above the Fermi level in minority spin. The bands just below the Fermi level are due to Sn 'p' and Gd 'd' states.
The strong hybridization of Sn 'p' and Gd 'd' states at the Fermi level shows the metallic character of these compounds.
Due to this metallic character, we found finite DOS (1.325 states/eV) at the Fermi level for
GdSn
3
in which major contribution is due to Sn 'p' states. A peak is observed around 2.5eV above the Fermi
level which is due to 'd' like states of Gd in both the spins. We can see an increase in finite DOS (1.725
states/eV) in minority spin due to delocalization of Gd 'f' states. On the whole, the band profiles are seen to be
almost same for both compounds except for certain changes. The electronic band structure for YbSn
3
shows
similar behavior in both majority and minority spins. The lowest lying band in between -11 eV and -5eV is due
to 's' like states of Sn. There is a flat band lying just below the Fermi level which is due to the 'f' like states of
68
Yb. There is hybridization of Yb 'f' states and Sn 'p' states at the Fermi level shows the metallic character of these
compounds. A peak is observed around 5eV above Fermi level which is due to 'd' like states of Yb in both
the spins. We found finite DOS (0.04 states/ eV) at the Fermi level for YbSn
3
in which major contribution is due
to Sn 'p' states.
3.3. Elastic properties
The elastic constants determine the response of the crystal to external forces, as characterized by bulk modulus, shear
modulus, Young's modulus, and Poisson's ratio, and obviously play an important role in determining the strength and
stability of materials. Elastic properties are also linked to sound velocity and Debye temperature.We
have calculated the elastic constants of the GdSn
3
and YbSn
3
compounds in AuCu
3
structure using PBE-GGA as
exchange correlation at ambient pressure using the method developed by Thomas Charpin and integrated it in the
WIEN2k package [15]. The calculated values of elastic constants are given in Table 2. In the absence of any available
measured data in the literature, these elastic constants could not be compared. It can be noted that our
calculated elastic constants satisfy the stability criterions: C
11
- C
12
> 0, C
44
> 0, C
11
+ 2C
12
> 0, C
12
< B < C
11
,
which clearly indicate the stability of these compounds in AuCu
3
structure.
3.4. Mechanical properties
Elastic constants can be used to determine mechanical properties such as Young's modulus (E), shear modulus
(G
H
), Poisson's ratio (o), and anisotropic ratio (A) for useful applications. These are fundamental parameters
which are closely related to many physical properties like internal strain, thermo elastic stress, sound velocity,
fracture, toughness. We have calculated these properties of these compounds and presented in Table 2. The shear
modulus G
H
describes the material's response to shearing strain using the Voigt-Reuss-Hill (VRH) method [23-
25]; the effective modulus for the polycrystals could be approximated by the arithmetic mean of the two well
known bounds for monocrystals.
The bulk and shear modulus, defined as
B = 1 (C11 + 2C12)
3
and
GH =
C11 ÷ C12 + 3C44 + 5C44 (C11 ÷ C12 )
5 4C44 + 3(C11 ÷ C12 )
2
Another important parameter is the elastic anisotropic factor A, which gives a measure of the anisotropy of the
elastic wave velocity in a crystal and it is given as:
2C44A=
C11 ÷ C12
which is unity for an isotropic material. It also tells more about the structural stability and it is correlated with the
possibility of inducing micro cracks in the materials.
The calculated elastic anisotropic factor for both the compounds is greater than 1, which indicates that these
compounds are not elastically isotropic. As suggested by Pugh [26], if B/G
H
>1.75; a material behaves in a
ductile manner. From Table 2, it can be seen that the highest value of B/G
H
is 2.64 for YbSn
3
indicating it more ductile
than GdSn
3
. Ganeshan et.al. [27] have established a correlation between the binding properties and ductility. The bond
character of cubic compounds is explained with respect to their Cauchy pressure (C
12
-C
44
). The YbSn
3
has a highest
positive Cauchy pressure; resulting strong metallic bonding (ductility) in it as compared to GdSn
3
.
Young's modulus is defined as the ratio of stress and strain, and is used to provide a measure of the stiffness of
the solid, i.e., the larger value of E, the stiffer is the material. It can be seen from Table 2 that the highest value E
occurs for GdSn
3
implying it to be stiffer as compared to YbSn
3
.
Young's modulus (E) and Poisson's ratio (o) is given by
E = 9BGH and o = (3B ÷ E)
3B + GH 6B
The value of Poisson's ratio, found to be ~ 0.33 in metallic materials [28]. It is observed from Table 2 that the value of Poisson's
ratio lies in between 0.325 and 0.332 for GdSn
3
and YbSn
3
. It shows that metallic
contributions to the atomic bonding are dominant for these compounds.
The longitudinal and transverse sound velocities (v
l
and v
t
) are obtained by using these elastic constants as
follows:
C +
2
(2C + C ÷ C )
11
5
44 12
11
C ÷
1
(2C + C ÷ C )
v
l
=
µ
and
vt =
4
4
5
44
µ
12
11
here C
11
, C
12
and C
44
are second order elastic constants andµ is mass density per unit volume, and the average
sound velocity v
m
and Debye temperature is approximately calculated from [29, 30, 31]:
69
÷1
1 2
v
m
=
3
+
3
1 3 and u = h 3n 13 v
3 v
t
v
l
D K 4tV m
B a
w
h
e
r
e
h
i
s
P
l
a
n
c
k
'
s
c
o
nstant, K
B
is Boltzmann's constant, V
a
is the average atomic volume.
We have calculated the average elastic sound velocities, Debye temperatures as well as the densities for both the
compounds by using the calculated elastic constants and are presented in Table 2. The elastic sound velocities
and Debye temperature of GdSn
3
is found to be higher as compared to YbSn
3
.
Conclusion
In conclusion, we have systematically studied the structural, electronic, elastic and mechanical properties of
GdSn
3
and YbSn
3
compounds using FP-LAPW method based on density functional theory, within GGA, PBE-
sol GGA and LSDA approximations. Our results on the structural, electronic, elastic and mechanical properties
for these compounds are in agreement with available theoretical and experimental results. The ground state
properties such as bulk moduli and lattice parameters are computed and compared with the preceding theoretical
and other experimental results, which shows good agreement. The calculated elastic constants (C
11
, C
12
and C
44
)
have shown that GdSn
3
and YbSn
3
are elastically stable in the AuCu
3
structure. Using these elastic constants,
Young's modulus (E), the shear modulus (G
H
), Poisson's ratio (o) and anisotropic ratio (A) are also reported. In present
study we found B/G
H
> 1.75 and C
12
- C
44
> 0; for both the compounds which implies that both the
compounds are ductile in nature. The computed electronic band structures show metallic character. We also
report mechanical properties of these compounds for the first time, which will be tested in the future
experimentally and theoretically.
Acknowledgements
The authors are thankful to UGC (New Delhi) for the financial support for Major Research Project. SPS is
thankful to UGC (SAP).The authors are thankful to Dr. M. Rajagopalan, Emeritus Scientist, Crystal Growth
Centre, Chennai for valuable discussions.
Table 1. Calculated lattice parameter a
0
(Å), Bulk modulus B (GPa), its pressure derivative (B?) and elastic
constants (C
11
, C
12
, C
44
) of GdSn
3
and YbSn
3
in AuCu
3
structure
Solid W o rk a
0
B B' C
11
C
12
C
44
Å ( GP a ) ( GP a ) ( GP a ) ( GP a )
GdSn
3
P B E GGA 4 .6 8 1 5 8 .0 3 4 .6 4 6 5 .5 9 5 3 .5 6 5 0 .7 3
PBE-sol GGA 4.644 6 7 .4 5 3 .9 5 - - -
LSDA 4 .5 9 5 7 1 .6 6 4 .2 9 - - -
a
Expt. 4.678 - - - - -
b b b
Oth. GGA 4.683 64.32 4 .6 9 - - -
b b b
LDA 4.513 62.55 2 .9 5 - - -
YbSn
3
P B E GGA 4 .7 0 2 52.45 4 .4 9 6 2 .5 8 4 4 .1 0 3 0 .7 0
PBE-sol GGA 4 .6 2 1 5 9 .3 8 5 .0 6 - - -
LSDA 4 .5 6 4 6 7 .0 8 4 .5 0 - - -
a
Expt. 4.681 - - - - -
Oth. - - - - - -
a b
Ref. [21], Ref. [12]. Pre. - Present, Expt. - Experiment, Oth. - Other Theoretical calculations.
Table 2. Calculated Young's modulus (E), shear modulus (G
H
), anisotropic factor (A), Poisson's ratio (o), B/ G
H
ratio,
Cauchy's pressure (C
12
- C
44
) , magnetic moment (µ
tot
), density (µ), longitudinal (v
l
), transverse (v
t
)
,
average elastic wave
velocities (v
m
) and Debye temperature (u
D
) of GdSn
3
and YbSn
3
compounds.
Solid µ
to
t
µ x 1033
v
l
v
t
v
m
u
D
E G
H
A o B/G
H
C
12
- C
44
(µ
B
) (kg/m ) (m/s) (m/s) (m/s) (K)
( GP a ) ( GP a ) ( GP a )
GdSn
3
6 0 .4 3 2 2 .8 0 8 .4 4 0 .3 2 2 .5 2 0 2 .8 3 6 .8 1 5.736 4203 2392 2657 147.10
YbSn
3
5 0 .6 6 1 9 .0 1 3 .3 2 0 .3 3 2 .6 4 1 3 .4 0 0 .0 0 5 .9 8 1 3651 1923 2148 119.38
GdSn
3
Spin
GdSn
3
Spin
YbSn
3
Spin
YbSn
3
Spin
Figure 1. Band structure of GdSn
3
and YbSn
3
in both majority and minority spin channels.
71
GdSn
3
Spin
YbSn
3
Spin
GdSn
3
Spin
YbSn
3
Spin
Figure 2. Density of states of GdSn
3
and YbSn
3
in both majority and minority spin channels.
72
doc_733779953.docx
The characteristics of a material that determine how it reacts when it is subjected to some type of force that attempts to stretch, dent, scratch, or break it.
First Principles Study of Structural, Electronic, Elastic and
Mechanical Properties of GdSn
3
and YbSn
3
Intermetallic
Compounds
Abstract:-
First principles study of structural, electronic, elastic and mechanical properties of ferromagnetic GdSn
3
and non-
magnetic YbSn
3
rare-earth intermetallics, which crystallize in AuCu
3
-type structure, is performed using density
functional theory based on full potential linearized augmented plane wave (FP-LAPW) method. The ground state
calculations are carried out within PBE-GGA, PBE-sol GGA and LSDA approximations for the exchange correlation
potential. The calculated ground state properties such as lattice constants and bulk moduli agree well with the
experiment as well as other theoretical results. We report elastic constants for these compounds for the first time. Both
these compounds are found to be ductile in nature. The computed electronic band structures show metallic character.
We also report mechanical properties of these compounds for the first time. Keywords: Rare-earth; Density functional
theory; Elastic constants.
1. Introduction
Rare earth intermetallic systems present a great diversity of magnetic behaviors, often associated with complex
magnetic phase diagrams as a function of temperature and applied magnetic field. The rare-earth metals have high
magnetic moments and a diverse range of magnetic structures . Their magnetic properties are determined by the
occupancy of the strongly localized '4f' electronic shells, while the outer 's-d' electrons determine the bonding and other
electronic properties. Rare earth based intermetallics are of considerable technological and scientific interest due to
extra ordinary magnetic properties and industrial applications. These properties are governed by different types of
interactions involving the highly correlated and strongly localized '4f' states of rare earth and the 'd' states of transition
metal atoms, which are comparatively weakly correlated and more delocalized and also the valence states of R atoms,
which are expected to be the mediators of indirect exchange coupling [1, 2].
GdSn
3
and YbSn
3
belong to the family of Rare earth tristannide RESn
3
compounds. These systems have attracted
a great deal of interest because of their salient features such as valence fluctuations, magnetic moment formation,
crystal field effects or multiaxial magnetic structures [3]. An experiment has been conducted by Miller and Hall
[4] to synthesize heavy rare-earth element tristannide LnSn
3
(Ln=Tb, Dy, Ho, Er, and Y) compounds using a
high pressure technique. TbSn
3
and YSn
3
have been synthesized at P = 1-7GPa and T = 400°C-1300°C.
These two compounds also have a cubic AuCu
3
type structure with a lattice constant a = 0.466 nm and 0.467 nm, and a
density of 8.45 g/cm
3
and 7.27 g/cm
3
, respectively[5]. Superconducting state in YSn
3
with AuCu
3
-type
structure has been studied by Kawashima et. al. [6]. Their electric field gradients were also calculated. Crystal
fields and magnetic properties of NdSn
3
, NdPb
3
and NdIn
3
have been studied by Lethuillier et. al.[7]. High
magnetic field properties of GdIn
3
have been investigated by Kletowski et. al. [8]. Interband optical transitions in CeSn
3
and LaSn
3
have been studied by measuring and calculating the optical conductivities of these compounds by Kim [9].
Temperature dependence of magnetic fluctuations in the intermediate valence system CeSn
3
has been reported by
Capellmann et. al.[10]. Valency of rare earths in RIn
3
and RSn
3
and ab-initio analysis of
electric field gradients have been investigated by Asadabadi et. al.[11]. The structural properties such as lattice
parameter and bulk modulus of SmSn
3
, EuSn
3
, GdSn
3
, TmIn
3
, YbIn
3
and LuIn
3
using LDA, GGA and GGA +
spin polarized approximations have been calculated using Wien97 code [12]. Observation of the de haas-van
alphen effect in powdered YbSn
3
and CaSn
3
has been reported by Klaasse et. al [13]. YbSn
3
shows a
superconducting transition at around 3.6 K [14].
The rare earth intermetallics GdSn
3
and YbSn
3
compounds crystallize in cubic AuCu
3
-type structure (space
group symmetry of Pm3m (No.221)). In the present work, we report the structural, electronic, elastic and
mechanical properties of GdSn
3
and YbSn
3
compounds using density functional theory. The elastic constants of
these compounds are reported for the first time. A brief description of the computational details is outlined in
Section 2 while Section 3 covers the results, followed by discussion.
2. Computational Method.
The present calculations have been performed using full potential linear augmented plane wave (FP-LAPW)
67
method as implemented in WIEN2k package [15]. In this method, the basis set is obtained by dividing the unit cell into
non-overlapping spheres and an interstitial region. Here PBE-GGA, PBE-sol GGA and LSDA is used
for the exchange and correlation effects [16, 17, 18]. The values of K
max
×R
MT
= 7.0 and l
max
= 10 are kept
throughout the calculation. A dense mesh of 10×10×10 k points is used and tetrahedral method [19] has been
use-d for the Brillouin Zone integration. The calculations are iterated until the total energies are converged below
10
4
Ry. The total energies are calculated as a function of volume and fitted to Birch-Murnaghan equation of
state (third order) [20] to obtain the ground state properties like zero-pressure equilibrium volume.
The elastic moduli require knowledge of the derivative of the energy as a function of the lattice strain. The
symmetry of the cubic lattice reduces the 21 elastic constants to three independent elastic constants namely C
11
,
C
12
and C
44
. The elastic stability criteria for a cubic crystal at ambient conditions are C
11
+2C
12
> 0, C
44
> 0 and C
11
- C
12
>
0 and C
12
< B < C
11
. In the present work, the elastic constants are calculated using the tetrahedral and
rhombohedral distortions on the cubic structure using the method developed by Thomas Charpin integrated it in
the WIEN2k package [15]. The systems are fully relaxed after each distortion in order to reach the equilibrium
state. C
44
is proportional to the shear modulus and can be used as a measure of shear resistance.
3. Result and discussion
3.1. Ground state properties
The spin polarized electronic band structure calculations have been carried out to obtain the total energy of the
GdSn
3
and YbSn
3
intermetallics using the first principles FP-LAPW method. In order to calculate the ground
state properties, the total energies are calculated in AuCu
3
type structure for different volumes around the equilibrium
cell volume V
0
. The calculated total energies are fitted to the Birch-Murnaghan equation of state to determine the
ground state properties like lattice constant (a
0
), bulk modulus (B) and its pressure derivative (B?) at minimum
equilibrium volume V
0
.
B'
E(V ) = E
0
+ BV (V 0 /V ) + ÷ BV
B' (B'÷1)
B'÷1
Pressure is obtained by taking volume derivative of the total
energy
P(V ) = B
V ÷ B'
B'
V
0
and
The calculated ground state properties are given in Table 1, and compared with the experimental data and available
other theoretical calculations [12, 21]. It is seen from Table 1 that our calculated values of lattice parameter using PBE-
GGA are in better agreement with the experimental results than those obtained using PBE- sol GGA and LSDA
calculations. The better prediction of experimental results by GGA has also been observed in most rare-earth metals
and is believed to be due to the fact that the nonlocality of exchange and correlation is better taken into account by
GGA than LDA [22]. Spin polarized calculations does not practically affect the
ground state properties of YbSn
3
due to the presence of almost fully occupied 'f' shell of Ytterbium. The
calculated total magnetic moment (µ
tot
) of both the compounds are given in Table 2. Furthermore, as, to our
knowledge, no experimental data for the bulk modulus and its derivative have been reported yet, our results can
serve as a prediction for future studies.
3.2. Electronic properties:
The self consistent spin polarized band structures (BS) along the high symmetry directions for majority and
minority spins for GdSn
3
and YbSn
3
compounds are presented in Figure 1.The total and partial densities of
states (DOS) for these compounds at ambient pressure are also calculated and presented in Figure 2. The Fermi
level is shown at the origin. It is clear from the figures that the lowest lying bands below -20eV are due to Sn 'd'
like states in GdSn
3
. The bands which lies in between -11 eV and -5 eV are due to Sn 's' like states. There is a
flat band below the Fermi level around -4 eV which are mainly due to Gd 'f' states in majority spin which gets
shifts above the Fermi level in minority spin. The bands just below the Fermi level are due to Sn 'p' and Gd 'd' states.
The strong hybridization of Sn 'p' and Gd 'd' states at the Fermi level shows the metallic character of these compounds.
Due to this metallic character, we found finite DOS (1.325 states/eV) at the Fermi level for
GdSn
3
in which major contribution is due to Sn 'p' states. A peak is observed around 2.5eV above the Fermi
level which is due to 'd' like states of Gd in both the spins. We can see an increase in finite DOS (1.725
states/eV) in minority spin due to delocalization of Gd 'f' states. On the whole, the band profiles are seen to be
almost same for both compounds except for certain changes. The electronic band structure for YbSn
3
shows
similar behavior in both majority and minority spins. The lowest lying band in between -11 eV and -5eV is due
to 's' like states of Sn. There is a flat band lying just below the Fermi level which is due to the 'f' like states of
68
Yb. There is hybridization of Yb 'f' states and Sn 'p' states at the Fermi level shows the metallic character of these
compounds. A peak is observed around 5eV above Fermi level which is due to 'd' like states of Yb in both
the spins. We found finite DOS (0.04 states/ eV) at the Fermi level for YbSn
3
in which major contribution is due
to Sn 'p' states.
3.3. Elastic properties
The elastic constants determine the response of the crystal to external forces, as characterized by bulk modulus, shear
modulus, Young's modulus, and Poisson's ratio, and obviously play an important role in determining the strength and
stability of materials. Elastic properties are also linked to sound velocity and Debye temperature.We
have calculated the elastic constants of the GdSn
3
and YbSn
3
compounds in AuCu
3
structure using PBE-GGA as
exchange correlation at ambient pressure using the method developed by Thomas Charpin and integrated it in the
WIEN2k package [15]. The calculated values of elastic constants are given in Table 2. In the absence of any available
measured data in the literature, these elastic constants could not be compared. It can be noted that our
calculated elastic constants satisfy the stability criterions: C
11
- C
12
> 0, C
44
> 0, C
11
+ 2C
12
> 0, C
12
< B < C
11
,
which clearly indicate the stability of these compounds in AuCu
3
structure.
3.4. Mechanical properties
Elastic constants can be used to determine mechanical properties such as Young's modulus (E), shear modulus
(G
H
), Poisson's ratio (o), and anisotropic ratio (A) for useful applications. These are fundamental parameters
which are closely related to many physical properties like internal strain, thermo elastic stress, sound velocity,
fracture, toughness. We have calculated these properties of these compounds and presented in Table 2. The shear
modulus G
H
describes the material's response to shearing strain using the Voigt-Reuss-Hill (VRH) method [23-
25]; the effective modulus for the polycrystals could be approximated by the arithmetic mean of the two well
known bounds for monocrystals.
The bulk and shear modulus, defined as
B = 1 (C11 + 2C12)
3
and
GH =
C11 ÷ C12 + 3C44 + 5C44 (C11 ÷ C12 )
5 4C44 + 3(C11 ÷ C12 )
2
Another important parameter is the elastic anisotropic factor A, which gives a measure of the anisotropy of the
elastic wave velocity in a crystal and it is given as:
2C44A=
C11 ÷ C12
which is unity for an isotropic material. It also tells more about the structural stability and it is correlated with the
possibility of inducing micro cracks in the materials.
The calculated elastic anisotropic factor for both the compounds is greater than 1, which indicates that these
compounds are not elastically isotropic. As suggested by Pugh [26], if B/G
H
>1.75; a material behaves in a
ductile manner. From Table 2, it can be seen that the highest value of B/G
H
is 2.64 for YbSn
3
indicating it more ductile
than GdSn
3
. Ganeshan et.al. [27] have established a correlation between the binding properties and ductility. The bond
character of cubic compounds is explained with respect to their Cauchy pressure (C
12
-C
44
). The YbSn
3
has a highest
positive Cauchy pressure; resulting strong metallic bonding (ductility) in it as compared to GdSn
3
.
Young's modulus is defined as the ratio of stress and strain, and is used to provide a measure of the stiffness of
the solid, i.e., the larger value of E, the stiffer is the material. It can be seen from Table 2 that the highest value E
occurs for GdSn
3
implying it to be stiffer as compared to YbSn
3
.
Young's modulus (E) and Poisson's ratio (o) is given by
E = 9BGH and o = (3B ÷ E)
3B + GH 6B
The value of Poisson's ratio, found to be ~ 0.33 in metallic materials [28]. It is observed from Table 2 that the value of Poisson's
ratio lies in between 0.325 and 0.332 for GdSn
3
and YbSn
3
. It shows that metallic
contributions to the atomic bonding are dominant for these compounds.
The longitudinal and transverse sound velocities (v
l
and v
t
) are obtained by using these elastic constants as
follows:
C +
2
(2C + C ÷ C )
11
5
44 12
11
C ÷
1
(2C + C ÷ C )
v
l
=
µ
and
vt =
4
4
5
44
µ
12
11
here C
11
, C
12
and C
44
are second order elastic constants andµ is mass density per unit volume, and the average
sound velocity v
m
and Debye temperature is approximately calculated from [29, 30, 31]:
69
÷1
1 2
v
m
=
3
+
3
1 3 and u = h 3n 13 v
3 v
t
v
l
D K 4tV m
B a
w
h
e
r
e
h
i
s
P
l
a
n
c
k
'
s
c
o
nstant, K
B
is Boltzmann's constant, V
a
is the average atomic volume.
We have calculated the average elastic sound velocities, Debye temperatures as well as the densities for both the
compounds by using the calculated elastic constants and are presented in Table 2. The elastic sound velocities
and Debye temperature of GdSn
3
is found to be higher as compared to YbSn
3
.
Conclusion
In conclusion, we have systematically studied the structural, electronic, elastic and mechanical properties of
GdSn
3
and YbSn
3
compounds using FP-LAPW method based on density functional theory, within GGA, PBE-
sol GGA and LSDA approximations. Our results on the structural, electronic, elastic and mechanical properties
for these compounds are in agreement with available theoretical and experimental results. The ground state
properties such as bulk moduli and lattice parameters are computed and compared with the preceding theoretical
and other experimental results, which shows good agreement. The calculated elastic constants (C
11
, C
12
and C
44
)
have shown that GdSn
3
and YbSn
3
are elastically stable in the AuCu
3
structure. Using these elastic constants,
Young's modulus (E), the shear modulus (G
H
), Poisson's ratio (o) and anisotropic ratio (A) are also reported. In present
study we found B/G
H
> 1.75 and C
12
- C
44
> 0; for both the compounds which implies that both the
compounds are ductile in nature. The computed electronic band structures show metallic character. We also
report mechanical properties of these compounds for the first time, which will be tested in the future
experimentally and theoretically.
Acknowledgements
The authors are thankful to UGC (New Delhi) for the financial support for Major Research Project. SPS is
thankful to UGC (SAP).The authors are thankful to Dr. M. Rajagopalan, Emeritus Scientist, Crystal Growth
Centre, Chennai for valuable discussions.
Table 1. Calculated lattice parameter a
0
(Å), Bulk modulus B (GPa), its pressure derivative (B?) and elastic
constants (C
11
, C
12
, C
44
) of GdSn
3
and YbSn
3
in AuCu
3
structure
Solid W o rk a
0
B B' C
11
C
12
C
44
Å ( GP a ) ( GP a ) ( GP a ) ( GP a )
GdSn
3
P B E GGA 4 .6 8 1 5 8 .0 3 4 .6 4 6 5 .5 9 5 3 .5 6 5 0 .7 3
PBE-sol GGA 4.644 6 7 .4 5 3 .9 5 - - -
LSDA 4 .5 9 5 7 1 .6 6 4 .2 9 - - -
a
Expt. 4.678 - - - - -
b b b
Oth. GGA 4.683 64.32 4 .6 9 - - -
b b b
LDA 4.513 62.55 2 .9 5 - - -
YbSn
3
P B E GGA 4 .7 0 2 52.45 4 .4 9 6 2 .5 8 4 4 .1 0 3 0 .7 0
PBE-sol GGA 4 .6 2 1 5 9 .3 8 5 .0 6 - - -
LSDA 4 .5 6 4 6 7 .0 8 4 .5 0 - - -
a
Expt. 4.681 - - - - -
Oth. - - - - - -
a b
Ref. [21], Ref. [12]. Pre. - Present, Expt. - Experiment, Oth. - Other Theoretical calculations.
Table 2. Calculated Young's modulus (E), shear modulus (G
H
), anisotropic factor (A), Poisson's ratio (o), B/ G
H
ratio,
Cauchy's pressure (C
12
- C
44
) , magnetic moment (µ
tot
), density (µ), longitudinal (v
l
), transverse (v
t
)
,
average elastic wave
velocities (v
m
) and Debye temperature (u
D
) of GdSn
3
and YbSn
3
compounds.
Solid µ
to
t
µ x 1033
v
l
v
t
v
m
u
D
E G
H
A o B/G
H
C
12
- C
44
(µ
B
) (kg/m ) (m/s) (m/s) (m/s) (K)
( GP a ) ( GP a ) ( GP a )
GdSn
3
6 0 .4 3 2 2 .8 0 8 .4 4 0 .3 2 2 .5 2 0 2 .8 3 6 .8 1 5.736 4203 2392 2657 147.10
YbSn
3
5 0 .6 6 1 9 .0 1 3 .3 2 0 .3 3 2 .6 4 1 3 .4 0 0 .0 0 5 .9 8 1 3651 1923 2148 119.38
GdSn
3
Spin
GdSn
3
Spin
YbSn
3
Spin
YbSn
3
Spin
Figure 1. Band structure of GdSn
3
and YbSn
3
in both majority and minority spin channels.
71
GdSn
3
Spin
YbSn
3
Spin
GdSn
3
Spin
YbSn
3
Spin
Figure 2. Density of states of GdSn
3
and YbSn
3
in both majority and minority spin channels.
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