Ab-initio study of tuning the electronic and magnetic properties of $\mathrm{Ni_{2}MnGa}$ Heusler alloy by Co and Mn compound doping

Energy is the necessary evil of the modern civilization. Reducing the production of greenhouse gas while keeping on with the ever-increasing energy demand is a great challenge to mankind. Renewable energy technologies are one of the most trusted and tested methods to limit greenhouse emissions and global warming. For caloric materials, which is one of the major parts of renewable energy, the efficiency depends on ordering temperature, e.g., Curie temperature($T_{C}$) for magnetocaloric (MC) materials. Hence, to achieve the goal of 4/5^th of the world’s electricity by 2050 [1] from renewable sources, it is extremely important to be able to tune the $T_{C}$ of the related materials [2].

Since the theoretical prediction of half-metallic properties in $\mathrm{NiMnSb}$ by Groot [3] and experimental observations by [4, 5, 6], Heusler alloys(HA’s) have drawn massive attention from the scientific community [7, 8, 9] ascribed to their wide usage in MC devices [10], magnetic shape memory alloys [11], spintronics [12], and giant magnetoresistance devices [13, 14, 15]. The full HA’s, with generic symbol $X_{2}YZ$ stabilizes in the $L2_{1}$ structure [16] with a completely ordered phase. The $Y-Z$ disordered structure has the $B_{2}$ ground state [17], while the complete $X-Y-Z$ disordered system has the $A_{2}$ [18]. Wyckoff positions of 4a(0, 0, 0), 4b($\frac{1}{2},\frac{1}{2},\frac{1}{2}$), and 8c($\frac{1}{4},\frac{1}{4},\frac{1}{4}$) are occupied by $Z$, $Y$, and $X$, respectively [11, 19, 20]. The ternary full HA, $\mathrm{Ni_{2}MnGa}$, exhibits both magnetic and structural phase transition [21]. In this context, Ni-Mn-Ga systems show properties like shape memory effect and magnetic field-induced strain, which is advantageous in actuators and sensors [11, 22]. They also show favorable MC (both conventional and inverse) properties, suitable to replace the century-old Joule-Thomson cooling [23, 24]. Ni-Mn-Ga system exhibits magneto structural phase transition with a huge change in isothermal magnetic entropy($\Delta S_{m}\approx-18$ J kg^-1 K^-1) but a relatively narrow range of working temperature in full-width half maximum($T_{fwhm}$) around 290 K for an applied field change of 5 T, typical to the materials with first-order phase transition(FOPT) [25]. However, the thermal hysteresis of FOPT and the Curie temperature is predominant to magnetic refrigeration.

The electronic structure and magnetic properties have been reported for Ni-doped $\mathrm{Co_{2}MnAl}$ [26] and Co-doped $\mathrm{Ni_{2}MnAl}$ [27] systems, with the dopant at the $X$ position only. Off-stoichiometric $\mathrm{Ni_{2-\mathit{x}}Q_{\mathit{x}}MnGa}$  and $\mathrm{Ni_{2-\mathit{x}}Q_{\mathit{x}}MnGa_{1-\mathit{z}}R_{z}}$, where Q, and R are the dopant elements. For the past few decades, $\mathrm{Ni_{2-\mathit{x}}Q_{\mathit{x}}MnGa}$/Al has been under study to understand their martensite phase transition($T_{M}$) and Curie temperature($T_{C}$) [28, 29]. In $\mathrm{Ni_{2+\mathit{x}}Mn_{1-\mathit{x}}Ga}$, observed that the MC effect is the highest around $0.18\leqslant x\leqslant 0.27$ concentration as magnetic and structural phase change occurs in this region [30]. Partially substituting Ni with Co is generally known to enhance the ferromagnetic coupling and hence the $T_{C}$. This increases the possibility that the quaternary system undergoes a martensitic transition together with a meta-magnetic phase transition [31].

Generally, the structural and magnetic properties are highly impacted by the off-stoichiometric combination of the main group element($Z$) and transition metals($X$ and $Y$). The effect of $Y$ replacing $Z$, mostly $\mathrm{Ni_{2}MnZ_{1-y}Mn}$, $Z$ = Ga, In, etc., is well studied [14, 32, 33]. Substituting the Mn element in the $Z$ position can stabilize the cubic state of the system [14]. The Mn_Z atoms interact antiferromagnetically between the surrounding Mn_Z and normal Mn_Y since the distance between Mn_Y-Mn_Z is shorter than Mn_Y-Mn_Y and Mn_Z-Mn_Z [34]. Here, Mn_Z has been denoting the Mn atom in the Ga site($Z$), and Mn_Y is the Mn’s normal position($Y$). The doping of the Co element in the $X$ position can increase the ferromagnetic interaction between the atoms and the Curie temperature of the material [31]. Additionally, even a small concentration of Co atoms in the $X$ position significantly impacts the Curie temperature. Also, the cell volume effect and valence electron concentration per atom ratio($e/a$) affect $T_{C}$ inversely, i.e., as the $e/a$ ratio increases, the $T_{C}$ decreases and vice-versa [35, 36].

From the above discussion, investigation of the effect of doping both the $X$ and the $Z$ sites with $d$ elements, resulting to a disordered $\mathrm{Ni_{2-\mathit{x}}Co_{\mathit{x}}MnGa_{1-\mathit{z}}Mn_{\mathit{z}}}$, is interesting due to site preferences and their compound effect of electronic, magnetic, and thermodynamic properties. We have investigated the effect of Co and Mn in the $X$ and $Z$ site and observed the effect in a restricted cubic phase. The cubic phase has minimum energy in the $L2_{1}$ structure. Increasing the Co concentration above 10% retains the cubic state as the ground state in a $B2$ structure [37]. At around $x=0.25$, the cubic (austenite) phase collapses to the tetragonal (martensitic) phase[27]. We expected Mn doping in the Ga site and Co doping in the Ni site to enhance the material’s magnetic moment and Curie temperature. The $x$ concentration varies from a value greater than 10%, i.e., $x=0.12~{}\text{to}~{}0.24$. Additionally, the $z$ concentration varies from $0~{}\text{to}~{}0.5$ since the $e/a$ ratio increases up to 8, comparatively for the pure system $\mathrm{Ni_{2}MnGa}$($e/a$ = 7.5). Therefore, the $z$ concentration is up to 0.5. This disorder system will be referred to as NiMnGa($x,z$) hereafter, where $x,z$ is the concentration of Co in the Ni site and Mn in the Ga site, respectively, for brevity. For example, $\mathrm{Ni_{1.88}Co_{0.12}MnGa_{0.74}Mn_{0.26}}$ will be denoted as NiMnGa($0.12,0.26$).

Notably, in this work, we have reported systematic studies of the variation of electronic structure, magnetic exchange interactions, and tuning the Curie temperature($T_{C}$) of this disordered NiMnGa($x,z$) system. We have especially looked into patterns of the effect of variation of site occupancy upon a specific substitution for a wide range of concentrations. The results are interpreted from the outcome of electronic structure calculations. This approach enables us to understand the microscopic origin of the macroscopic property($T_{C}$). The study of the substitution of a single $3d$ element on $\mathrm{Ni_{2}MnGa}$ is, as discussed above, very scattered. The investigation of double doping in $\mathrm{Ni_{2}MnGa}$ is rare and far between. However, they offer exciting phenomena, not only from the perspective of fundamental understanding but also for materials engineering with target properties. For example, the tuning of $T_{C}$ is technologically relevant as it is one of the crucial factors that act as the working temperature range of many devices, namely, MC, spintronics, etc. [10, 12].

The appropriate method for handling off-stoichiometric compositions is Green’s function-based formalism with coherent potential approximation(CPA), as in the case of $\mathrm{Ni_{2-\mathit{x}}Q_{\mathit{x}}MnGa_{1-\mathit{z}}R_{z}}$ [38]. We have performed ab-initio calculations using multiple scattering Green’s function formalism as implemented in spin polarised relativistic Korringa-Kohn-Rostoker(SPRKKR) code [39, 40, 41]. The Perdew-Burke-Ernzerhof within generalized gradient approximation is used as the exchange-correlation functional [42]. First Brillouin zone integrations were performed with 2500 grids of $k$-points, and energy convergence criteria were set as $10^{-5}$ Ry in the calculation in the range. We have implemented full potential spin-polarized scalar relativistic implementation of SPRKKR with angular momentum cut-off $\ell_{max}=2$ as suitable for our system.

The lattice parameter with a minimum energy of NiMnGa($x,z$) is calculated using the following procedure: (i) Obtain the lattice parameter from the materials project database [43]; (ii) Calculated the self-consistent field of the system, with varying lattice parameter ranging from 94% to 106%, with identical calculation for each lattice parameters. (iii) Fit the lattice parameter vs energy plot obtained in the last step using a 4^th order polynomial. The parameter corresponding to the minima of the curve is the optimized lattice parameter. For NiMnGa($x,z$), we have taken the optimized lattice parameter of previous calculations that has a minimum $x,z$ change as the starting point and followed the steps above. We have shown the optimization curve of NiMnGa($0,0$) in Figure (1(a)). Other minimization energy curves are not shown here for brevity. In the present study, doping concentrations are varied by changing the Wyckoff site occupancy of the corresponding element in the input structure. The resultant structure is used for further calculations.

The magnetic exchange energy($\mathcal{J}^{\nu\mu}_{ij}$) was calculated to understand the properties of magnetic interactions. The Heisenberg model is defined as,

where $\nu,\mu$ represent atoms in different sublattices, $i,j$ is a different lattice point, $\mathbf{e}^{\nu}_{i}$ is the magnetic orientation of $i^{th}$ atom at $\nu$ sublattices. The $\mathcal{J}_{i,j}^{\nu\mu}$ is calculated by the energy difference due to an infinitesimal change of magnetic direction, as formulated by Lichtenstein [44].

Finally, the $T_{C}$ is estimated using mean field theory, yielding

where $\mathcal{J}_{l}$ is the largest eigenvalue of the determinant, as described in [34]. It must be remembered that mean field calculations generally overestimate the $T_{C}$.

We have calculated the electronic and magnetic properties of NiMnGa($x,z$) for ($x=0.0,0.12\text{~{}and~{}}0.24$) and ($z=0.0,0.15,0.26,0.35\text{ and~{}}0.5$). The optimized lattice parameter of each sample was calculated using the method described in the sec. (2) and tabulated in Tables (1, 2, 3). Figure (1(d)) shows the variation of lattice parameters with $x$ and $z$, which is mostly linear.

Figures (1(a)-1(c)) shows the calculations of the pure $\mathrm{Ni_{2}MnGa}$. Figure (1(a)) shows the lattice parameter optimization as described in section (2). Our calculated density of states(DOS) (Figure (1(b))) and magnetic exchange interactions($\mathcal{J}_{ij}$) (Figure (1(c))) with the Mn atom at the center for $\mathrm{Ni_{2}MnGa}$ matches the previous findings [45]. The $\mathcal{J}_{ij}$ is highest for Ni-Mn interactions, with the value $\approx$ 5 meV. All the interactions are predominantly ferromagnetic in this case. The complete table of optimized lattice parameters, the total and individual magnetic moment per atom, and Curie temperature of the pure system is tabulated in Table (1).

The change of lattice parameter with $x,~{}z$ in the cubic domain is shown in Figure (1(d)). This trend shows that doping Co at the $X$ site decreases the lattice parameter but doping at the $Z$ site increases the lattice parameter.

The lattice parameter increases when Mn is doped at the Ga site ascribed to its larger atomic radius i.e., $\text{r}_{\text{Mn}}$(1.61 Å) $>$ $\text{r}_{\text{Ga}}$(1.36 Å). In contrast, doping of Co at the Ni site decreases its lattice constants even though the atomic radius of Co(1.52 Å) is larger than Ni(1.49 Å). This can be associated with the following reasons: Firstly, the exchange interaction between the Co and Mn is higher than the Ni and Mn, and this can be well understood from Figures ((1(c)), (3b.1-3b.2). Secondly, the $d$-$d$ hybridization of Co and Mn is relatively stronger than Ni and Mn [47]. Then the lattice parameter starts to decrease as the Co concentration in the Ni site increases, as shown in Figure (1(d)).

In the pristine($\mathrm{Ni_{2}MnGa}$) DOS, we observed a pseudogap in the minority channel around $\approx$ 1 eV below the Fermi-level. This pseudogap originated by the hybridization between $3d$ orbitals of Ni and $3p$ orbitals of Ga. The gap is terminated by a peak just below the Fermi-level originating from the hybridization of the same orbitals and drives the system to Jahn-Teller instability. In NiMnGa($x,0$) and NiMnGa($x,z$) systems, the doping at the Ni site stabilizes the Jahn-Teller instability as the peak smears out and the pseudogap becomes narrower with higher $x$ value, as shown in Figure (33a.2) for NiMnGa($x,0$) and Figure (6) for NiMnGa($x,z$). In NiMnGa($0,z$) systems the pseudogap almost remains the same throughout the doping range as shown in Figure (55a.4).

The magnetic moments of NiMnGa($x,0$), NiMnGa($0,z$), and NiMnGa($x,z$) are shown in Table (1), Table (2), and Table (3), respectively. From the tables and Figure (8(a)), which shows the variation of the total magnetic moment as a function of $x$ and $z$ concentrations, we see the general features: (i) Magnetic moment is increasing linearly with doping. (ii) Magnetic moment increases faster by doping in $Z$ site than doping in $X$ site. This value is due to the overall magnetic moments of $X$ site occupants, i.e., Ni and Co have weaker magnetic moment than Mn_Z. (iii) Mn_Z has a higher magnetic moment than the Mn_Y site. The antiferromagnetic interaction between the Mn_Y and Mn_Z atoms is stronger since the inter-atomic distance from Mn_Y to Mn_Z are smaller compared to Mn of the same sites(Mn_Y to Mn_Y, and Mn_Z to Mn_Z). The variation in $x$ and $z$ affects the atomic moments in various sites and the total magnetic moments. General observations are: (i) Atomistic moments of $X$ sites are most susceptible to $x$ and $z$ concentrations, while $Y$ and $Z$ sites moments remain almost unchanged. (ii) Both Ni and Co moment increases with $x,z$ concentration but more with $z$ concentrations.

The Curie temperature($T_{C}$) has been calculated using mean field approximations(MFA). As expected, MFA overestimates the $T_{C}$. For pure $\mathrm{Ni_{2}MnGa}$, our calculated $T_{C}\approx$ 392 K is in good agreement with experimental findings 382 K [48](365 K is reported by [32, 33]). For NiMnGa($x,z$), the qualitative variation of $T_{C}$ increases monotonously, with the quantitative value matching the previous findings [27]. $T_{C}$ for NiMnGa($0,z$) first increases from 392 K to 741 K for $z=0.15$. Higher doping of Mn_Z decreases the $T_{C}$ monotonously. For Co-doped systems, both NiMnGa($x,0$) and NiMnGa($x,z$) increase monotonously and give approximately the same $T_{C}$ irrespective of Co concentrations. To get an understanding of the variation of $T_{C}$, we have calculated the magnetic pair interaction $\mathcal{J}_{ij}$ as shown in Figures ((1(c)), (33b.2), (55b.4), and (7)) for $\mathrm{Ni_{2}MnGa}$, NiMnGa($x,0$), NiMnGa($0,z$) and NiMnGa($x,z$) respectively with Mn_Y at the center. When the Co atom is present, i.e., $x\neq 0$(as in NiMnGa($x,0$) and NiMnGa($x,z$) case), Co and Mn_Y is the dominant interaction. The difference is that for NiMnGa($x,0$), the maximum interaction is almost constant (33b.2), but for NiMnGa($x,z$), the interaction keeps increasing. The same variation is evident from Figure (8(b)) for $x$ = 0.15 and 0.24. For NiMnGa($0,z$), there is a ferro-antiferro competition between Ni-Mn_Y and Mn_Y-Mn_Z. This brings the $T_{C}$ down after the initial increase from pure $\mathrm{Ni_{2}MnGa}$.

We have investigated the electronic and magnetic properties of $\mathrm{Ni_{2-\mathit{x}}Co_{\mathit{x}}MnGa_{1-\mathit{z}}Mn_{\mathit{z}}}$, with $0\leqslant x\leqslant 0.24$, and $0\leqslant z\leqslant 0.5$. The study of replacing both $X$ and $Z$ is substantially sparse compared to any one of them. We have used DFT and MFA to study the compound effect of dual-doping. Electronic structures, DOS, and magnetic properties including magnetic exchange interactions and moments are calculated using DFT as implemented in the SPRKKR package. The Curie temperature($T_{C}$) is calculated using MFA. Our calculation shows the existence of strong Mn-Mn antiferromagnetic interaction between Mn at Ga and Mn at its sub-lattice. Its also noticed that changing concentration at Mn_Z does not change the magnetic exchange substantially. This is an interesting result, as opposed to the findings in the case of $\mathrm{Ni_{2}Mn_{1+\mathit{x}}Sn_{1-\mathit{x}}}$[34].

The $T_{C}$’s are obtained using MFA simulations using the magnetic exchange interaction values obtained from ab-initio calculations. The numerical results for the pure system are close to that reported by experiments.

We point out the calculations are done in cubic($c/a=1$) phases only. Though, in the dual-doping case, we have shown the variation of magnetic exchange interaction parameters and magnetic Curie temperatures. These findings will help to design the functional properties like MCE and shape memory alloys by alloying $\mathrm{Ni_{2}MnGa}$ suitably.

We acknowledge the High Performance Computing Center (HPCC), SRM IST for providing the computational facility to carry out this research work effectively.