2.1 Density Functional Theory
2.1.6 Calculation of physical quantities
Elastic moduli and magnetic exchange parameters are the two important physical parameters, used in this thesis, can be obtained from DFT calculations. In the next sub-sections we briefly describe the approaches adopted to calculate these.
2.1.6.1 Calculation of Elastic moduli
The elastic moduli can be derived from the total energy E(V) calculated as a func- tion of volume V or strain e by DFT based methods. In the following, we describe how they are computed.
Bulk Modulus: The bulk modulus (B) describes the behaviour of the crystal volume or lattice parameters under hydrostatic pressure. It is evaluated from the energy-volume E(V) relations. The bulk modulus can be obtained from the Birch- Murnaghan (B-M) isothermal equation of state which is a relationship between the total energy with crystal volume under hydrostatic pressure [217]. The third order B-M equation of state is given by
E(V) =E0+ 9 16BV0
×
"
V0
V 23
−1
#3
B0′ −
"
V0
V 23
−1
#2"
4V0
V 23
−6
#2
(2.51)
Chapter 2. Methodology
and the related pressure P(V) is given by P(V) = 3B
2
"
V0
V 73
− V0
V
53# ( 1 + 3
4(B0′ −4)
"
V0
V 23
−1
#)
. (2.52) whereV0 is the equilibrium volume,V is the deformed volume, andB0′ is the deriva- tive of the bulk modulus with respect to pressure. The value ofB0′ is almost constant for many substances [218].
Elastic Constants: There are two standard methods for calculating the elastic constants fromab initio calculations. One is the energy-strain method and another is the stress-strain method. In this thesis, we employed the energy-strain method where the total energy is calculated as a function of the applied strain [216]. The applied strains are chosen at constant volume as the total energy depends on the volume much more strongly than on strain. The stain matrix D(e), the elements of which are denoted by e1, e2, ..., e6, transforms the lattice A with basis vectors x, y, z into the deformed lattice A′ with basis vector x′, y′, z′, i.e.
x′ y′ z′
= (D(e) +I)
x y z
=
(1 +e1)x 12e6y 12e5z
1
2e6x (1 +e2)y 12e4z
1
2e5x 12e4y (1 +e3)z
(2.53)
where I is the 3×3 identity matrix. In practice, the e1, e2, ..., e6 are expressed in terms of δ so that the change in E upon strain is written as
E(e1, e2, ..., e6) =E(0) + 1
2V X
i,j=1,6
cijeiej+O(δ3) (2.54)
where cijs are the elastic constants. In general the Eq. 2.54 becomes
E(δ) =E(0) +V Cδ2+O(δ3) (2.55) where C is the particular combination of elastic constants which is obtained by fittingE(δ) by a polynomial ofδ
C = C2
V (2.56)
C2 is the second order coefficient of the polynomial [216].
2.1 Density Functional Theory For a cubic lattice there are three independent elastic constants C11, C12 and C44. The corresponding strains are given in the Table 2.1. The compliances of the cubic phase can be obtained from the elastic constants using the following relations:
S44 = 1 C44
, S11−S12 = 1
C11−C12
, S11+S12 = C11
(C11−C12)(C11+C12),
(2.57)
Table 2.1: Strain for the calculation of elastic constants in the cubic systems. ∆E is the energy change upon strain andV is the equilibrium volume of the lattice.
Strain Parameter ∆E/V
C1 e1 =e2 =e3 =δ 32(C11+ 2C12)δ2 C2 e1 =δ, e2 =−δ, e3 = 1−δδ22 (C11−C12)δ2+O(δ4) C3 e3 = 1−δδ22, e6 =δ 2C44δ2+O(δ4)
Throughout this thesis, the strains have been varied from 0.00 to 0.05 with intervals of 0.01. Total energies are calculated for monoclinic and orthorhombic deformations and the elastic moduli (C11, C12 and C44) are then obtained by fit- ting the variation of total energies on the respective strain tensors to a 4th or- der polynomial equation. In the cubic lattice, the bulk modulus (B) is defined by B = 13(C11+ 2C12). The shear elastic modulus (C′) is defined byC′ = 12 (C11−C12).
For mechanical stability in the cubic lattice, the conditions are
C44>0, C11>|C12|, C11+ 2C12 >0 (2.58) Apart from these, other elastic moduli are relevant when the material is poly- crystalline. In a polycrystalline material, the single-crystal grains are randomly oriented. On a large scale, such materials can be considered to be quasi-isotropic or isotropic in a statistical sense. So suitable averaging methods based on statis- tical mechanics are needed to calculate the polycrystalline elastic moduli. There are two approximation methods to derive the isotropic elastic modulus. They are the Voigt [219] and Reuss [220] averaging methods, which represent the upper and lower bounds respectively of the isotropic elastic modulus. As there are only three independent elastic constants in the cubic crystal, the shear modulus in the Voigt
Chapter 2. Methodology
method is
GV = 1
5(C11−C12+ 3C44) (2.59)
The corresponding one in Reuss method is
GR= 5
S411−4S12+ 3S44 (2.60)
Finally, the shear modulus (G) are typically the average of Voigt and Reuss elastic moduli based on Hill approximation [221, 222] which is expressed as following:
G= 1
2(GR+GV), (2.61)
However, in cases of a number of ferromagnetic Heusler compounds, it was found out that Gv using Voigt formalism is closer to the experimental results [223, 224].
HenceGhas been approximated asGv. The Pugh ratio (Gv/B) and Cauchy pressure (CP) are the two other important quantities characterising the mechanical properties of a material. Pugh ratio Gv/B [225–227], related to the resistance of the material to plastic deformation, measures whether material is more ductile or more brittle.
Compounds having a Pugh ratio greater than 0.57 are considered to be more brittle.
On the other hand, Cauchy pressure CP, calculated as CP=(C12-C44), provides insight to the nature of bonding in a material with cubic symmetry [228]. A positive value of Cauchy pressure indicates more metallic bonding in the system, while a negative value implies a stronger covalent bonding [229].
2.1.6.2 Calculation of the magnetic exchange interactions
The magnetic pair exchange parameters are computed in order to understand the nature of the magnetic interactions of the systems studied in this thesis. They are efficiently calculated using the multiple-scattering Green’s function formalism as implemented in the SPRKKR code [230]. In this approach, the spin part of the Hamiltonian is mapped to a Heisenberg model:
HHeisenberg =−X
µ,ν
X
i,j
Jijµνeµi.eνj (2.62)
2.2 Monte Carlo Simulation (MCS) Method