3.1 Elastic constants and related mechanical properties
As well-known in literature [38-41], cubic materials have three
characteristic independent elastic constants i.e. C11,
C12 and C44. From these constants,
C11 defines the longitudinal elastic behavior of a given
cubic material, whereas C12 represents the off-diagonal
and C44 points out the shear elastic characteristic of
related material. In our work, first principles density functional
theory (DFT) implemented in the CASTEP code has been used to calculate
the above three independent typical cubic elastic constants of
Rh2MnX alloys. Table 2. lists C11,
C12 and C44 values for
Rh2MnX alloys. From Table 2 it is easy to verify the
presence of traditional Born mechanical (dynamical) stability conditions
with C11 > 0, C44> 0, C11 − C12> 0, C11 + C12> 0 and cubic stability i.e. C12< 𝐵 < C12 where (B) is the bulk
modulus of regarding materials [38-41]. Noteworthy to mention here,
bulk modulus defines the resistance amount of a given material under
external deformations where the shear modulus (G) describes materials
resistance to the shape change under a shearing force. The Young’s
modulus (E) describes the resistance of the material to uniaxial
tensions and represents the stiffness degree, i.e., the higher the value
of E, the stiffer is the material [42-47]. Table 2. also shows the
calculated values of bulk modulus, shear modulus, Young’s modulus of
Rh2MnX alloys. In
addition to these three important moduli (B, G and E), the B/G ratio of
any material identifies the brittle or ductile nature of the material.
If the B/G ratio of a material is more or less than the limiting value
of 1.75, then the material displays a ductile or brittle character
[38-47]. It is also clear in Table 2. that the B/G ratio of all
alloys is bigger than the limiting value which means that all of them
are ductile. Moreover, Poisson’s ratio (ν) is the ratio between the
transverse strain (et ) and longitudinal strain
(el ) in the elastic loading direction. It
provides detailed knowledge about the bonding character of solids.
Usually, Poisson’s ratio values are about 0.1 for covalent
materials, 0.25 for ionic materials and vary between 0.28 and 0.42 for
metals [47]. Poisson’s ratio values of Rh2MnX alloys
rise between 0.29 and 0.34 as in Table 2. So, it is apparent that all
Rh2MnX alloys demonstrate metallic bonding. When
compared with previous work of Ref. [26], although the metallic
behavior of Rh2MnTi alloys correlates well with their
findings, our elastic constants, elastic moduli and Vickers hardness
(HV =
0.92(G/B)1.37.G0.708) data somewhat
underestimate their findings for this alloy (Table 2). Also, when we
compare our results with former results of Refs. [27,28] our elastic
and mechanical data slightly overestimate the results of Refs.
[27,28] (Table 2). Therefore, we can ensure that the differences
between the results of this work and Ref. [26] as well as Refs.
[27,28] may mostly arise from the three distinct calculation
methods.
3.2 Electronic and
Magnetic Properties
The electronic band structure of any periodic crystal corresponds to
energy eigenvalues and helps us to figure out the Fermi levels of the
material which determine the conducting, semiconducting, or insulating
aspect of any material. Fig.2 exemplifies the electronic band structure
of Rh2MnTi alloy
as a precursor for spin down (↓) and spin up (↑) states. The band
structure of Rh2MnTi alloy in Fig. 2, display a clear
metallic nature in which the valence and conduction bands intersect each
other along with the chosen high symmetry points. The same result is
also valid for the rest of other concerned alloys like
Rh2MnHf, Rh2MnSc,
Rh2MnSc, Rh2MnzZr and
Rh2MnZn. Further, our analysis on the orbital-selective
partial density of states of Rh2MnTi revealed that the
conductivity of these alloys mainly originates from the large
contributions of the d-electrons of Rh and Mn atoms as seen in Fig. 3.
The same findings are also true again for other alloys having Hf, Sc, Zn
and Zr. Since s, p orbitals do not much contribute to the conductivity
of these alloys we did not provide any results for them. Also, to ensure
this neglected effect of s and p orbitals we also give the
element-specific partial density of states of Rh2MnTi
alloy with the total contributions of alloying elements in Fig.4.
Moreover, in Fig.3 and Fig.4, the density of states plots either for
spin down or spin up components display a non-symmetrical picture due to
a clear polarization between the density of states of these components.
In particular, our results for electronic band structure and partial
density of states of Rh2MnTi alloy are in good agreement
with findings of Ref. [26]
Since Rh2MnX alloys are declared as potential novel
magnetic alloys in its original theoretical work [25], we computed
the magnetic moment (µB) values of these alloys after
spin-polarized calculations. Table 3. compares the calculated total
magnetic moment (µB) values of Rh2MnX alloys with
several former results. Our calculated values for the magnitudes of
µB are in the range of Ti > Zr
> Hf > Sc > Zn and this result
compare well with the findings of Refs. [25-27] with ferromagnetic
ordering.