# Theoretical Study of the Electronic, Magnetic, Mechanical and Thermodynamic Properties of the Spin Gapless Semiconductor CoFeMnSi

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{B}follows the Slater–Pauling rule as M

_{total}= Z

_{total}− 24, where M

_{total}is the total magnetic moment per formula unit and Z

_{total}is the total valence electron number, i.e., 28 for CoFeMnSi material. We have also examined the mechanical properties of CoFeMnSi and computed its elastic constants and various moduli. Results show CoFeMnSi behaves in a ductile fashion and its strong elastic anisotropy is revealed with the help of the 3D-directional-dependent Young’s and shear moduli. Both mechanical and dynamic stabilities of CoFeMnSi are verified. In addition, strain effects on the electronic and magnetic properties of CoFeMnSi have been investigated, including both uniform and tetragonal strains, and we found that the spin-gapless feature is easily destroyed with both strain conditions, yet the total magnetic moment maintains a good stability. Furthermore, the specific behaviors under various temperatures and pressures have been accessed by the thermodynamic properties with a quasi-harmonic Debye model, including bulk modulus, thermal expansion coefficient, Grüneisen constant, heat capacity and Debye temperature. This comprehensive study can offer a very helpful and valuable reference for other relative research works.

## 1. Introduction

_{2}MnAl [2,12,15,16,17], Ti

_{2}CoSi [5], Ti

_{2}Vas [5], Zr

_{2}MnAl [18,19], Zr

_{2}MnGa [18,20] and Cr

_{2}ZnSi [13,21,22], but also, a few experimental syntheses and measurements confirming the presence of the spin-gapless semiconducting behaviors, such as Mn

_{2}CoAl [14], Ti

_{2}MnAl [2]. Conventionally, Heusler compounds represent a huge family of intermetallic alloys and they can be mainly divided into two groups [23,24]: full-Heusler with general formula X

_{2}YZ and half-Heusler with general formula XYZ, in which Z is an sp main group element and X and Y are the transition metal elements. Consequently, various Heusler compounds can be simply designed by substituting with an element from the same group in the periodic table. For a full-Heusler alloy, when the highly ordered cubic structure is considered under normal conditions, there are two typical structural configurations [12,25]: the first one is the Cu

_{2}MnAl-type, also known as the L2

_{1}structure; the second one is the Hg

_{2}CuTi-type, also known as the XA structure.

_{2}YZ and Fe

_{2}YZ. In particular for CoFeMnZ, Dai et al. [31] firstly synthesized CoFeMnSi by an arc-melting method and confirmed its LiMgPdSb-type structure by X-ray diffraction. Different atomic orderings have been considered and, with the lowest energy configuration, it shows half-metallic properties. Afterwards, several studies followed, such as Alijani et al. [32] and Klaer et al. [33], who studied electronic, structural and magnetic properties in CoFeMnZ (Z = Al, Ga, Si, Ge) from both theoretical calculations and experimental measurements and found that there is a small amount of disorder present in the crystal structure. Then, Feng et al. [34] detailed a thorough theoretical study of the possible different disorders and their effect on the electronic and magnetic properties. Immediately after, Bainsla et al. [28] revealed the spin-gapless semiconducting behavior in EQH CoFeMnSi from experimental results. Recently, Fu et al. [35] prepared CoFeMnSi in a bulk sample and studied its magnetic and transportation properties. They found a semiconductor-like transporting characteristic in CoFeMnSi with a Curie temperature of 763 K.

## 2. Computational Methodology

^{6}4s

^{2}), Mn (3d

^{5}4s

^{2}), Co (3d

^{7}4s

^{2}) and Si (3s

^{2}3p

^{2}). A cutoff energy of 500 eV is set for the plane-wave basis set and a Monkhorst–Pack special 12 × 12 × 12 k sampling point mesh is selected in the Brillouin zone. The reciprocal space integrations are performed with a k-mesh of 120 points in the irreducible wedge of the Brillouin zone by using the tetrahedron method. The total energy convergence tolerance is set within 1 × 10

^{−6}eV/atom during the self-consistent field cycle. The quasi-harmonic Debye model employed [40,41,42] has been adopted for studying the thermodynamic properties, and the dependencies of several parameters on pressure (0–10 GPa) and temperature (0–500 K) were computed, including unit cell volume, bulk modulus, heat capacity, Grüneisen constant, thermal expansion coefficient and Debye temperature.

## 3. Results and Discussions

#### 3.1. Crystal Structure and Equilibrium Lattice

#### 3.2. Electronic and Magnetic Properties

_{u}are above the Fermi energy level in the spin-down channel and, thus, not occupied, leading to an e

_{u}- t

_{u}(nonbonding - bonding) energy band gap formation. These results are consistent with a previous study [34]. According to the generalized electron filling rule and Slater–Pauling rule [43,44], the total occupied states for CoFeMnSi are 16 for the spin-up channel and 12 for the spin-down channel, respectively, and thus, a net spin magnetic moment is expected and it should be equal to 4.00 μ

_{B}, as it is the difference between the two spin directions. At the equilibrium state, the total and partial magnetic moments are calculated and presented in Table 1. The integral value of the total magnetic moment (4 μ

_{B}) matches the above-mentioned theoretical analysis and it follows the Slater–Pauling rule in the form of M

_{total}= Z

_{total}− 24, where Z

_{total}is the total valence electron number, 28 for CoFeMnSi and M

_{total}is the total magnetic moment. It can also be observed from Table 1 that Mn atoms provide the main contribution to the total magnetic moment, while Co and Fe carry relative moments aligned parallel to Mn atoma. The large magnetic moment of Mn atoms is from their strong spin-splitting effect, as revealed by the partial density of state in the literature [34].

#### 3.3. Mechanical Property and Dynamic Stability

_{11}, C

_{12}and C

_{44}, in which C

_{12}and C

_{44}reflect the elasticity in terms of the shape and C

_{11}characterizes the elasticity in terms of the length [29,30,46,47]. All these elastic constants for CoFeMnSi have been calculated with the stress–strain method [36] and the derived values for CoFeMnSi are summarized in Table 2. By applying the Voigt–Reuss–Hill approximation [48], several other mechanical parameters, such as the bulk modulus B, the shear modulus G and the Young’s modulus E, can be calculated with the following formulae as:

_{V}(B

_{R}) and G

_{V}(G

_{R}) stand for the lower (upper) limit of the Voigt (Reuss) boundary and they are derived from the elastic constants as follows:

#### 3.4. Strain Effects

_{B}. Whereas, the magnetic moments of Mn and Fe atoms display relatively large variations under uniform strain—increase for Mn and decrease for Fe. There is a negligible change in the moment for both Si and Co atoms.

_{B}when the c/a ratio is varied from 0.9 to 1.1. Whereas, the partial moments of Mn and Fe atoms exhibit small variations: increase for Fe and decrease for Mn when the structure is changed from cubic to tetragonal.

#### 3.5. Thermodynamic Property

_{0}, bulk modulus B, thermal expansion coefficient α, heat capacity C

_{V}, Grüneisen constant γ and Debye temperature θ

_{D}are calculated for the temperature range 0–500 K and pressure from 0 to 10 GPa.

_{0}of CoFeMnSi under different pressures and temperatures. All the volumes are normalized with respect to the equilibrium volume V0 at 0 K without pressure. With increasing temperature, the cell volume continuously increases. This is apparently expected because of the thermal expansion. However, the changing rate is not constant: small at temperature from 0 to 500 K and then larger at higher temperatures. On the contrary, the cell volume continuously shrinks in a linear manner with increasing pressure due to pressure compression. Overall, the change in the volume with temperature is much smaller than with pressure under the current studied conditions.

^{−5}K

^{−1}.

_{V}is another crucial physical parameter for materials, and it can reflect important details about the lattice vibration and the phase transition. The calculated variation of C

_{V}with pressure and temperature is depicted in Figure 16. It is clearly observed that the effect of the temperature on C

_{V}is much stronger than that of pressure. With temperature increase, C

_{V}slightly increases at low temperature and then grows rapidly at high temperature. With higher temperature, C

_{V}is expected to saturate to the Dulong–Petit limit. The variation of C

_{V}with temperature shows a very small difference among different pressures. With increasing pressure, C

_{V}decreases slightly. The calculated heat capacity for CoFeMnSi at 0 GPa and 300 K is 80.85 J·mol

^{−1}·K

^{−1}.

_{D}with temperature and pressure has been investigated and the results are shown in Figure 18. It is found that θ

_{D}remains almost constant at the low temperature range from 0 to 100 K and then slowly decreases. The rate of change becomes smaller at higher pressure. For a given temperature, θ

_{D}increases with pressure in a linear manner. The calculated Debye temperature of CoFeMnSi at 0 GPa and 300 K is 629.37 K. Our calculation of the thermodynamic properties of CoFeMnSi can provide a valuable reference for further work and also inspire future investigations.

## 4. Conclusions

_{B}obeys the well-known Slater–Pauling rule, i.e., M

_{total}= Z

_{total}−24, with Z

_{total}is the total valence electron number and M

_{total}is the total magnetic moment. Moreover, the mechanical properties of CoFeMnSi have been computed under an equilibrium state and several elastic constants and moduli are obtained. It is found that CoFeMnSi behaves in a ductile manner and it exhibits very strong elastic anisotropy, as revealed by the 3D surface plots of the directional-dependent moduli. Both mechanical and dynamic stabilities of CoFeMnSi have been verified. Furthermore, the strain effects on the electronic and magnetic properties of CoFeMnSi have been investigated, including both uniform and tetragonal strains, and it is found that the spin-gapless feature is easily lost with both strain conditions yet the total magnetic moment maintains a relative good stability. Finally, the specific thermodynamic properties under various pressures and temperatures have been assessed by applying the quasi-harmonic Debye model, including bulk modulus, thermal expansion coefficient, Grüneisen constant, heat capacity and Debye temperature. The considered temperature range was from 0 to 500 K and pressure was from 0 to 10 GPa. This study comprehensively reveals different physical aspects of CoFeMnSi and can offer a very valuable reference for its real-world application.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The calculated total energy of CoFeMnSi with respect to different lattice constants. The inset is the corresponding crystal structure.

**Figure 2.**The calculated spin-polarized electronic band structure of CoFeMnSi at the equilibrium lattice constant.

**Figure 3.**Schematic representation of the different energy levels in both spin directions for CoFeMnSi.

**Figure 4.**The calculated distributions of the electronic spin density

**(a)**and charge density difference

**(b)**in the (110) plane of CoFeMnSi at the equilibrium lattice.

**Figure 5.**The surface plots for the calculated directional dependent Young’s modulus and shear modulus of CoFeMnSi.

**Figure 6.**The calculated 2D projection of the Young’s modulus and shear modulus of CoFeMnSi in different planes.

**Figure 8.**The calculated CBM and VBM of CoFeMnSi in both spin directions under different uniform strains.

**Figure 9.**The calculated total and atomic spin magnetic moments of CoFeMnSi under different uniform strains.

**Figure 11.**The calculated CBM and VBM of CoFeMnSi in the spin-down direction under different tetragonal strains.

**Figure 12.**The calculated total and atomic spin magnetic moments of CoFeMnSi under different tetragonal strains.

**Figure 15.**The thermal expansion coefficient of CoFeMnSi against different temperatures and pressures.

**Table 1.**The calculated equilibrium lattice constant and the corresponding total and atom-resolved magnetic moments of CoFeMnSi.

Compound | Lattice [Å] | Magnetic Moment [μ_{B}] | |||||
---|---|---|---|---|---|---|---|

Total | Fe | Mn | Co | Si | |||

CoFeMnSi | Current | 5.611 | 4.00 | 0.34 | 3.08 | 0.69 | −0.11 |

Reference [31] | 5.653 * | 3.99* | 0.576 | 2.649 | 0.878 | −0.07 | |

Reference [28] | 5.658 * | 4.01 | 0.53 | 2.72 | 0.82 | ||

Reference [45] | 5.67 * | 3.49 * | |||||

Reference [32] | 5.611 | 4.00 | 0.52 | 2.70 | 0.89 | −0.11 |

**Table 2.**The calculated elastic constants (Cij), bulk modulus (B), shear modulus (G), Young’s modulus (E), Poisson’s ratio (ν), Pugh’s ratio (B/G) and anisotropy factor (η) for CoFeMnSi.

Compound | C_{11} [GPa] | C_{12} [GPa] | C_{44} [GPa] | B [GPa] | G [GPa] | E [GPa] | υ | B/G | η | |
---|---|---|---|---|---|---|---|---|---|---|

CoFeMnSi | Current | 332.2 | 188.9 | 157.5 | 236.7 | 114.8 | 296.5 | 0.29 | 2.06 | 2.19 |

Reference [32] | 317.0 | 189.0 | 167.0 | 231.0 | 114.0 | 293.0 | 0.29 | 2.04 | 2.60 |

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**MDPI and ACS Style**

Tan, X.; You, J.; Liu, P.-F.; Wang, Y.
Theoretical Study of the Electronic, Magnetic, Mechanical and Thermodynamic Properties of the Spin Gapless Semiconductor CoFeMnSi. *Crystals* **2019**, *9*, 678.
https://doi.org/10.3390/cryst9120678

**AMA Style**

Tan X, You J, Liu P-F, Wang Y.
Theoretical Study of the Electronic, Magnetic, Mechanical and Thermodynamic Properties of the Spin Gapless Semiconductor CoFeMnSi. *Crystals*. 2019; 9(12):678.
https://doi.org/10.3390/cryst9120678

**Chicago/Turabian Style**

Tan, Xingwen, Jiaxue You, Peng-Fei Liu, and Yanfeng Wang.
2019. "Theoretical Study of the Electronic, Magnetic, Mechanical and Thermodynamic Properties of the Spin Gapless Semiconductor CoFeMnSi" *Crystals* 9, no. 12: 678.
https://doi.org/10.3390/cryst9120678