Electronic structure and magnetic properties in $T_2\text{AlB}_2$ ($T$ = Fe, Mn, Cr, Co, and Ni) and their alloys

TL;DR

This study uses density functional theory to analyze the electronic structure and magnetic properties of $T_2$AlB$_2$ compounds with various transition metals, revealing how doping affects magnetization and Curie temperature.

Contribution

It provides detailed insights into the magnetic behavior and electronic structure of $T_2$AlB$_2$ compounds and their alloys, highlighting the effects of different dopants on magnetic properties.

Findings

Fe$_2$AlB$_2$ has an easy axis along the $a$ crystallographic direction.

Mn$_2$AlB$_2$ exhibits ferromagnetism in the $ab$ plane and antiferromagnetism along $c$.

Doping with Co or Ni reduces magnetization and Curie temperature, with electron doping having a stronger effect.

Abstract

The electronic structure and intrinsic magnetic properties of $\text{Fe}_2\text{AlB}_2$-related compounds and their alloys have been investigated using density functional theory. For $\text{Fe}_2\text{AlB}_2$, the crystallographic $a$ axis is the easiest axis, which agrees with experiments. The magnetic ground state of $\text{Mn}_2\text{AlB}_2$ is found to be ferromagnetic in the basal $ab$ plane, but antiferromagnetic along the $c$ axis. All $3d$ dopings considered decrease the magnetization and Curie temperature in $\text{Fe}_2\text{AlB}_2$. Electron doping with Co or Ni has a stronger effect on the decreasing of Curie temperature in $\text{Fe}_2\text{AlB}_2$ than hole doping with Mn or Cr. However, a larger amount of Mn doping on $\text{Fe}_2\text{AlB}_2$ promotes the ferromagnetic to antiferromagnetic transition. A very anisotropic magnetoelastic effect is found in $\text{Fe}_2\text{AlB}_2$: the magnetization has a much stronger dependence on the lattice parameter $c$ than on $a$ or $b$, which is explained by electronic-structure features near the Fermi level. Dopings of other elements on B and Al sites are also discussed.

Tables

Table 1. Table 1: Calculated atomic spin magnetic m i subscript 𝑚 𝑖 m_{i} , spin magnetization M 𝑀 M , and Curie temperature T C subscript 𝑇 C T_{\text{C}} in Fe 2 AlB 2 .

Fe2AlB2		$m_{\text{i}}$ (${\mu }_B/\mathrm{atom}$)				$M$		$T_{\text{C}}$
Method		Fe	Al	B		(${\mu }_B/f.u.$)		(K)
FP-GGA		1.43	$-0.04$	$-0.01$		2.73
ASA-GGA		1.38	$-0.08$	$-0.04$		2.62		329
FP-LDA		1.31	$-0.01$	$-0.03$		2.54
ASA-LDA		1.20	$-0.06$	$-0.03$		2.29		232

Table 2. Table 2: Pairwise exchange parameters J i ​ j subscript 𝐽 𝑖 𝑗 J_{ij} for the Heisenberg Hamiltonian H = − ∑ J i ​ j ​ 𝐞 ^ i ⋅ 𝐞 ^ j 𝐻 ⋅ subscript 𝐽 𝑖 𝑗 subscript ^ 𝐞 𝑖 subscript ^ 𝐞 𝑗 H=-\sum J_{ij}\,\hat{\bf e}_{i}\cdot\hat{\bf e}_{j} , and 𝐞 ^ i subscript ^ 𝐞 𝑖 \hat{\bf e}_{i} is the unit vector pointing along the direction of the local spin moment at site i 𝑖 i . The experimental lattice parameters and atomic positions are used. For Fe 2 AlB 2 , both GGA and LDA results are shown. For Mn 2 AlB 2 , the PBE functional is used, and both FM and AFM spin configurations are considered.

Fe2AlB2				$R_{ij}$			${\widehat{𝐑}}_{ij}$				$J_{ij}$(meV)
Lbl.		No.		(Å)	($a$)		$x$	$y$	$z$		GGA	LDA
$J_{ab}$		2		2.721	0.931		0.5	$0.785$	0		$2.95$	$1.41$
$J_c$		2		2.870	0.982		0	$0$	0.982		$7.36$	$4.29$
$J_a$		2		2.923	1		1	$0$	0		$14.58$	$10.63$
$J_b$		1		3.222	1.102		0	$1.102$	0		$25.41$	$20.37$
$J_{abc}$		4		3.955	1.353		0.5	$-0.785$	0.982		$-0.43$	$0.01$
$J_{ac}$		4		4.097	1.401		1	$0$	0.982		$-2.76$	$-1.49$
$J_{bc}$		2		4.315	1.476		0	$1.102$	0.982		$-2.63$	$-2.37$
$J_{ab}^{\prime }$		2		4.350	1.488		1	$1.102$	0		$-4.62$	$-4.62$
$J_{ab}^{\prime \prime }$		2		4.949	1.693		1.5	$-0.785$	0		$-2.03$	$-1.43$
Mn2AlB2				$R_{ij}$			${\widehat{𝐑}}_{ij}$				$J_{ij}$(meV)
Lbl.		No.		(Å)	($a$)		$x$	$y$	$z$		FM	AFM
$J_{ab}$		2		2.747	0.941		$0.5$	$-0.797$	$0$		$2.10$	4.44
$J_c$		2		2.890	0.990		$0$	$0$	$-0.990$		$-0.81$	3.02
$J_a$		2		2.920	1		$1$	$0$	$0$		$1.47$	3.29
$J_b$		1		3.213	1.100		$0$	$1.100$	$0$		$1.56$	3.59
$J_{abc}$		4		3.987	1.365		$-0.5$	$-0.797$	$0.990$		$-0.44$	1.52
$J_{ac}$		4		4.108	1.407		$-1$	$0$	$0.990$		$-0.27$	1.37
$J_{bc}$		2		4.322	1.480		$0$	$1.100$	$0.990$		$-0.15$	0.56
$J_{ab}^{\prime }$		2		4.342	1.487		$-1$	$1.100$	$0$		$0.33$	0.23
$J_{ab}^{\prime \prime }$		2		4.960	1.698		$-1.5$	$-0.797$	$0$		$0.32$	0.86

Table 3. Table 3: Lattice parameters, internal atomic positions, y 4 ​ j subscript 𝑦 4 𝑗 y_{4j} and y 4 ​ i subscript 𝑦 4 𝑖 y_{4i} , on-site atomic magnetic moment of T 𝑇 T atoms, m T subscript 𝑚 𝑇 m_{T} ( μ B subscript 𝜇 B \mu_{\mathrm{B}} ), relative total energy ( meV / f . u . formulae-sequence meV 𝑓 𝑢 \mathrm{m}\mathrm{eV}\mathrm{/}{f.u.} ), and critical temperatures (Curie temperature in Fe 2 AlB 2 or Néel temperature in Mn 2 AlB 2 ), T C subscript 𝑇 C T_{\text{C}} ( K kelvin \mathrm{K} ) in T 2 subscript 𝑇 2 T_{2} AlB 2 with T = Fe 𝑇 Fe T=\text{Fe} , Mn, Cr, Co, and Ni. T 𝑇 T atom occupies the 4 ​ j 4 𝑗 4j site (0 y 4 ​ j subscript 𝑦 4 𝑗 y_{4j} 1/2) and B atom occupies the 4 ​ i 4 𝑖 4i (0 y 4 ​ i subscript 𝑦 4 𝑖 y_{4i} 0) site.

$T$	$a$	$b$	$c$	$y_{4j}$	$y_{4i}$	$m_T$	$\Delta E$	$T_{\text{C}}$
Cr	2.921	11.034	2.929	0.3521	0.2057	0
Expt. [Chaban and Kuz’ma, 1973]	2.937	11.07	2.971	0.352	0.220
Expt. [Lu et al., 2016]	2.937	11.047	2.968
Mn-NM	2.890	11.050	2.817	0.3562	0.2060	0	0
Mn-FM	2.892	11.056	2.826	0.3551	0.2060	0.42	$-21.5$
Mn-AFM	2.887	11.109	2.830	0.3547	0.2061	0.75	$-63.6$	296
Expt. [Cenzual et al., 1991]	2.92	11.08	2.89	0.355	0.209
Expt. [Chai et al., 2015]	2.936	11.12	2.912
Fe-NM	2.951	11.261	2.698	0.3531	0.2065	0	0
Fe-AFM	2.941	11.212	2.739	0.3559	0.2070	1.06	$-97.9$
Fe-FM	2.915	11.017	2.851	0.3537	0.2063	1.37	$-164.1$	298
Expt. [Jeitschko, 1969]	2.923	11.034	2.870	0.3540	0.2071
Co	2.962	11.314	2.689	0.3541	0.2073	0.21
Ni	2.979	11.041	2.843	0.3586	0.2101	0

Full text

identifier

I Introduction

Magnetic cooling, which is based on the magnetocaloric effect (MCE) and was discovered one century ago, has long been used in scientific laboratories to attain extremely low temperatures. A major breakthrough came in the late 1990s when Pecharsky and Gschneidner discovered giant MCE around room temperature (RT) in a new class of magnetic materials Pecharsky and Gschneidner (1997). This discovery has rekindled research interest in utilizing MCE for much broader applications, such as domestic appliances, which usually operate around RT. If successful, this more energy-efficient and environment-friendly magnetic cooling technique may replace conventional compressor-based refrigeration and revolutionize the cooling industry. This new era may arrive in the near future—only if one can find or engineer a proper MCE material which has large MCE under a magnetic field that can be generated by permanent magnets, and is also abundant, affordable, and has a good lifespan.

Fe2AlB2 is one of the promising candidates for this purpose and has attracted great attention since the recent discovery of its substantial MCE around RT Tan et al. (2013). The reported entropy change has a value of $\Delta{S_{m}}=$4.17.7\text{,}\mathrm{J}\text{,}{\mathrm{kg}}^{-1}\text{,}{\mathrm{K}}^{-1} in the presence of an external field $B=$25\text{\,}\mathrm{T}. Although Fe2AlB2 does not have the largest MCE of all materials, it does not contain any rare, expensive, or toxic elements. Moreover, its volume barely changes during the magnetic transition Lewis et al. (2015), which may ensure Fe2AlB2 has a good life span for refrigerator applications operating at high cycle frequencies Du et al. (2015a, b).

The MCE often peaks at the Curie temperature $T_{\text{C}}$ of the material; however, real applications require materials with a large MCE over a certain operating temperature range. This likely needs to be achieved by using composite materials with multiple compositions, so the system can have MCE over the whole operating temperature range for specific applications. It seems the first logical selection of doping would be the substitution of Fe with other $3d$ transition-metal elements $T$. Pure Mn2AlB2 Becher et al. (1966) and Cr2AlB2 Chaban and Kuz’ma (1973) can be formed and share the same structure of Fe2AlB2. Combining theory with experiments, Kádas et al. studied the phase stability in $T_{2}$AlB2 with $T=\text{Cr}$, Mn, Fe, Co, and Ni. They found that although compounds are metastable with $T=\text{Co}$ and Ni, Fe2-xCoxAlB2, Fe2-xNixAlB2, or even (Fe2-x-yCoxNiy)AlB2 could be stable Kádas et al. (2017). However, the magnetic properties of those alloys or even their parent compounds are not well understood. For example, the magnetic ground state of Mn2AlB2 had been reported to be ferromagnetic (FM) Becher et al. (1966), but recent experiments concluded that it should be either nonmagnetic (NM) or antiferromagnetic (AFM) Chai et al. (2015). To provide guidance on tuning Fe2AlB2, a better understanding of the magnetic properties of pure $T_{2}$AlB2 and their alloys is desired.

The high melting temperature of FeB makes it a difficult impurity to remove from Fe2AlB2 samples. The rapid cooling by melt spinning had been used to greatly improve the Fe2AlB2 purity by suppressing the growth of FeB Du et al. (2015a). On the other hand, extra Al is often added during the synthesis to decrease the formation of FeB impurities Du et al. (2015a); Hirt et al. (2016). With a higher Al content, Al13Fe4 becomes the main impurity phase. Reported magnetization values measured at low temperature vary between 1.0 and $1.32\text{\,}\mu_{\mathrm{B}}\mathrm{/}{Fe}$ and the $T_{\text{C}}$ values vary between 282 and $320\text{\,}\mathrm{K}$ Tan et al. (2013); Du et al. (2015a); Chai et al. (2015); ElMassalami et al. (2011). The variation of experimental values may be due to the existence of an impurity phase in the Fe2AlB2 sample.

Experimentally, substitution of Mn Chai et al. (2015); Du et al. (2015a, b) or up to $15\text{\,}\mathrm{\char 37\relax}$ of Co Hirt et al. (2016) on Fe sites had been reported; a large amount of Mn or Co doping makes the structures unstable Du et al. (2015a) or phase inhomogeneous Chai et al. (2015). Both dopings decrease the magnetization and $T_{\text{C}}$ in Fe2AlB2. A spin-glass state had been found in Fe1.5Mn0.5AlB2 at low temperature Du et al. (2015b). With Co doping, $T_{\text{C}}$ and the Co content are linearly correlated, which makes Co content a convenient parameter to tune the material to have MCE between RT and $200\text{\,}\mathrm{K}$ Hirt et al. (2016). A few studies have been reported to investigate the electronic structures and structural, electronic, and magnetic properties of $T_{2}$AlB2 ElMassalami et al. (2011); Cheng et al. (2014); Chai et al. (2015); Cedervall et al. (2016).

In this work, using density functional theory (DFT), we investigate the intrinsic magnetic properties including magnetization, exchange parameters, Curie temperature, and magnetocrystalline anisotropy in $T_{2}$AlB2 and their alloys. The magnetoelastic effect in Fe2AlB2 and the dopings of various elements on B and Al sites are also discussed. Electronic structures are studied to understand the magnetic properties.

II Computational details

II.1 Crystal structure

Fe2AlB2 crystallizes in the orthorhombic Mn2AlB2-type ($Cmmm$, space group no. 65) structure. The primitive cell contains one formula unit (f.u.) while the conventional cell contains two. The crystal structure is shown in Fig. 1. We double the conventional cell along the $c$ axis to denote the first few exchange parameters. B atoms occupy the $4i(m2m)$ site, forming a zigzag chain in the $ab$ plane and along the $a$ axis. Fe atoms occupy the $4j(m2m)$ sites and form Fe6B prisms with neighboring B atoms. The structure can be derived from the FeB structure by inserting one layer of Al atoms perpendicular to the $b$ axis, between each pair of planes containing the Fe6B prisms Jeitschko (1969). Each Al atom, which occupies the $2a(mmm)$ site, is surrounded by eight Fe atoms. Together they form a body-centered-tetragonal cell elongated along the $b$ axis. For this Fe-Al cage, the Fe-Al bond length is $2.61\text{\,}\AA$ and the Fe-Fe distances are $2.872.923.22\text{\,}\AA$, along the $c$, $a$, and $b$ directions, respectively. Lying in the $ab$ plane, the nearest Fe-Fe bond has a length of $2.72\text{\,}\AA$. The structure can also be derived by stacking the pure Fe plane and the Al-B plane alternatively along the $c$ axis. This view is probably more convenient to understand the magnetic properties, such as exchange coupling and magnetoelastic effect, both of which are very anisotropic, behaving very differently along the $c$ axis than along the $a$ or $b$ axis, as we will discuss later.

II.2 Computational methods

Electronic structure and most magnetic properties are calculated using a standard linear muffin-tin orbital (LMTO) basis set Andersen (1975) generalized to full potentials (FP) Methfessel et al. (2000). This scheme employs generalized Hankel functions as the envelope functions. Calculations are carried out within the generalized gradient approximation (GGA) to DFT with the exchange-correlation parametrization of Perdew, Burke, and Ernzerhof (PBE) Perdew et al. (1996), unless LDA von Barth and Hedin (1972) (local density approximation, with the exchange-correlation parametrization of von Barth and Hedin) is specified.

The magnetocrystalline anisotropy energy (MAE) is calculated using the force theorem Mackintosh and Andersen (1980). Starting from the self-consistent scalar-relativistic potential, the spin-orbit coupling (SOC) is included in a subsequent one-step calculation with spin being along direction $\hat{\bf{n}}$. The MAE is characterized below as $K_{\hat{\bf{n}}}=E_{\hat{\bf{n}}}-E_{001}$, where $E_{001}$ and $E_{\hat{\bf{n}}}$ are the summation of occupied band energies for the magnetization being oriented along the $[001]$ and $\hat{\bf{n}}$ directions, respectively.

Exchange coupling parameters $J_{ij}$ are calculated using a static linear-response approach implemented in a Green’s function (GF) LMTO method, simplified using the atomic sphere approximation (ASA) to the potential and density Ke et al. (2012, 2013). The scalar-relativistic Hamiltonian is used so SOC is not included, although it is a small perturbation on $J_{ij}$’s. In the basis set, $s,p,d$ orbitals are included for $T$ and Al atoms, and $s,p$ orbitals are included for the B atom. A dense $k$-point mesh is used to calculate exchange parameters $J({\bf q})$, e.g., a $32^{3}$ $k$-point mesh for the five-atom cell. The real-space $J({\bf R})$ are obtained by a subsequent Fourier transform. Curie temperatures are estimated in the mean-field approximation (MFA) with $k_{\text{B}}T_{\text{C}}=2/3\sum_{i}J_{0i}$. The coherent potential approximation (CPA) implemented within the LMTO-ASA-GF code is used to address the chemical effects of doping on magnetization and $T_{\text{C}}$. Without using supercell calculations, the CPA provides an elegant and efficient approach to investigate substitutional effects with an arbitrary composition. The details of the methods and applications can be found elsewhere Ke et al. (2013); Ke and Johnson (2016).

Both experimental and theoretically optimized crystal structures are used to investigate the magnetic properties. We fully relax internal atomic positions and lattice constants with the PBE functional using a fast plane-wave method, as implemented within the Vienna ab initio simulation package (VASP) Kresse and Hafner (1993); Kresse and Furthmüller (1996). The nuclei and core electrons are described by the projector augmented-wave potential Kresse and Joubert (1999) and the wave functions of valence electrons are expanded in a plane-wave basis set with a cutoff energy of up to $520\text{\,}\mathrm{e}\mathrm{V}$.

III Results and discussion

III.1 Pure compounds: Exchange coupling, magnetic anisotropy, and spin configurations

Table 1 shows the atomic spin moments $m_{i}$ at each sublattice and magnetization $M$ in Fe2AlB2, which are calculated using the experimental lattice constants and atomic position parameters from Ref. [Cenzual et al., 1991]. Al and B have small moments antiparallel to the Fe sublattice. Within the GGA, a magnetization of $M=$1.36\text{,}\mu_{\mathrm{B}}\mathrm{/}\mathrm{F}\mathrm{e} is obtained using FP. For the sake of comparison, we carry out similar calculation for the parent compound, FeB, and obtain a magnetization of $M=$1.20\text{\,}\mu_{\mathrm{B}}\mathrm{/}\mathrm{F}\mathrm{e}. The smaller Fe moment in FeB is likely due to its smaller Fe-Fe bond length ($2.62\text{\,}\AA$) than in Fe2AlB2. ASA gives a slightly smaller (by $4\text{\,}\mathrm{\char 37\relax}$) magnetization than FP in Fe2AlB2, suggesting that ASA is suitable for this material. The calculated $T_{\text{C}}=$329\text{,}\mathrm{K}$$ is slightly above the upper bound of the reported experimental $T_{\text{C}}$ values. The agreement is fair considering the MFA generally overestimates $T_{\text{C}}$. LDA gives smaller magnetization, especially with ASA, resulting in a smaller $T_{\text{C}}$.

Starting from the FM configuration and using experimental crystal structures, we calculate the exchange coupling $J_{ij}$ in Fe2AlB2 and Mn2AlB2. Figure 2 shows the $J_{ij}$ as a function of the distance $R_{ij}$. In both compounds $J_{ij}$ becomes negligible after $R_{ij}>$6\text{,}\AA$$. The exchange parameters between the first few nearest neighbors are also listed in Table 2. Here, $J_{ij}$ can be treated as stability parameters and a negative $J_{ij}$ indicates that the given spin configuration is not favorable for that particular pair of sites Ke et al. (2013).

For Fe2AlB2, all of the first four nearest exchange parameters are positive. The $J_{ij}$ value increases with distance, reaching maximum at $J_{b}$, and then decrease, which generally agrees with the previous study Cedervall et al. (2016). LDA gives a similar trend but a smaller amplitude of $J_{ij}$ than GGA, which reflects the smaller magnetic moments obtained within LDA.

The magnetic ground state of Mn2AlB2 is not well understood Chai et al. (2015). For simplicity, we start from the FM configuration. The calculated magnetization is $0.42\text{\,}\mu_{\mathrm{B}}\mathrm{/}{Mn}$, which agrees well with a previous FM calculation Chai et al. (2015). Exchange parameters calculated in the FM configuration show a very interesting feature: all dominant $J_{ij}$ are positive for neighbors within the Mn $ab$ plane but negative for neighbors between neighboring Mn layers, namely, $J_{c}$, $J_{abc}$, $J_{ac}$, and $J_{bc}$. This suggests that the FM coupling of Mn atoms is stable within the $ab$ layer but not between neighboring layers. To confirm it, we calculate $J_{ij}$ for the AFM configuration, in which FM Mn $ab$ layers couple antiferromagnetically along the $c$ axis. This AFM configuration gives lower energy than the FM configuration. Moreover, as shown in Fig. 2(b), the dominant exchange parameters become all positive and larger. To better quantify the relative stability of those two spin configurations, we fully relaxed the structure and found that the AFM configuration increases the on-site Mn moment to $0.75\text{\,}\mu_{\mathrm{B}}\mathrm{/}\mathrm{M}\mathrm{n}$ and lowers the total energy by $42\text{\,}\mathrm{m}\mathrm{eV}\mathrm{/}{f.u.}$ Unlike Fe2AlB2, the largest exchange interaction in Mn2AlB2 is $J_{ab}$, which is between the nearest Mn neighbors. The Néel temperature is estimated to be $T_{\text{N}}=$310\text{,}\mathrm{K}$$ within MFA for the AFM configuration with the experimental crystal structure.

Next we consider the SOC effect in Fe2AlB2. Figure  3 shows the energy and the average orbital magnetic moments of each sublattice as functions of spin quantization axis orientation, which rotates from the $c\to a\to b\to c$ axis. The relaxed structure was used for the calculations. The $a$ axis is the easiest axis, which agrees with recent neutron scattering experiments Cedervall et al. (2016). The $c$ axis is the hardest axis, while the anisotropy within the $ab$ plane is very small. Energy changes by $K_{100}=$-0.38\text{,}\mathrm{m}\mathrm{eV}\mathrm{/}{f.u.} ($-1.34\text{\,}\mathrm{M}\mathrm{J}\mathrm{/}\mathrm{m}^{3}$) when the spin quantization axis rotates from the $c$ axis to the $a$ axis. The anisotropy calculated using experimental structure Cenzual *et al.* ([1991](#bib.bib26)) is larger by $\sim10\text{,}\mathrm{\char 37\relax}$, reaching$K_{100}=-1.47\text{\,}\mathrm{M}\mathrm{J}\mathrm{/}\mathrm{m}^{3}$$. B and Al atoms have negligible orbital magnetic moments, as expected for light 2p$and$3p$elements. The orbital moment of Fe is also rather small, and interestingly has the largest value of$0.02\text{,}\mu_{\mathrm{B}}\mathrm{/}{Fe}$when spin is along the hardest$c$axis instead of in the$abplane. Similar behavior has also been found in materials such as FePt Antropov *et al.* ([2014](#bib.bib27)). To elucidate the origin of MAE, we evaluate the anisotropy of the on-site SOC energy with respect to thec$axis and the$ab$plane,$K_{110}^{\text{so}}=\frac{1}{2}\langle V_{\text{so}}\rangle_{110}{-}\frac{1}{2}\langle V_{\text{so}}\rangle_{001}, and resolve it into four spin channels Ke and van Schilfgaarde ([2015](#bib.bib28)); Ke *et al.* ([2016](#bib.bib29)). It is well known that only when the MAE is dominated by the \downarrow\downarrowterm, one may expect an obvious correlation between the orbital moment and MAE, and a larger orbital moment along the easy axis Bruno ([1989](#bib.bib30)); Ke and van Schilfgaarde ([2015](#bib.bib28)). As shown in the inset of Fig. [3](#S3.F3)(b), the amplitudes of four spin components ofK_{\text{so}}^{\sigma\sigma^{\prime}}$are comparable. The$\downarrow\downarrow$term favors the spin to be along the$c$axis, while the other three terms ($\uparrow\uparrow$,$\uparrow\downarrow$, and$\downarrow\uparrow$) favor the spin to lie in the$ab$plane, explaining the absence of the correlation between MAE and orbital moment in Fe*2*AlB*2*. LDA gives a smaller anisotropy ($K_{100}=-0.82\text{\,}\mathrm{M}\mathrm{J}\mathrm{/}\mathrm{m}^{3}$$ using experimental structure) but a similar trend of angular dependence of energy. For Mn*2*AlB*2*, the c$axis is also the hardest axis. Within the$ab$plane, the system has slightly lower energy when the spin is along the$b$ axis. As expected, its anisotropy is much smaller than in Fe2AlB2. The schematic representation of the ground-state spin configurations of Fe2AlB2 and Mn2AlB2 are shown in the insets of Fig. 2.

Figure  4 shows the scalar-relativistic partial density of states (PDOS) projected on individual elements in $T_{2}$AlB2 with $T=\text{Cr}$, Mn, Fe, Co, and Ni. The total density of states (DOS), scaled by $1/2$, is also shown to compare. The total DOS of Fe2AlB2 compares well with previously reported calculations Chai et al. (2015). Al-$3s$ and B-$2s$ states are located between $-12$ and $-7\text{\,}\mathrm{eV}$ below the Fermi level $E_{\rm F}$. Al-$3p$ states hybridize with $T$ atoms at around $-4\text{\,}\mathrm{eV}$ below $E_{\rm F}$. For Mn2AlB2, the Fermi level is located at a pseudogap in the AFM configuration, and the DOS at $E_{\rm F}$ is smaller than in the FM configuration, which again suggests that the AFM configuration is more stable. The calculated hypothetical Co2AlB2 shows a weak magnetic moment of about $0.2\text{\,}\mu_{\mathrm{B}}\mathrm{/}{Co}$. The two peaks at $\pm 0.1\text{\,}\mathrm{eV}$ around the Fermi level, as shown in Fig. 4(d), will be pinched at $E_{\rm F}$ in the non-spin-polarized calculation (not shown). Thus, the small spin polarization decreases the DOS at $E_{\rm F}$ and stabilizes the system. The calculated Cr2AlB2 and hypothetical Ni2AlB2 are nonmagnetic and have small DOS at $E_{\rm F}$. It is worth noting that NixB1-x systems become magnetic only after $x>0.75$ Geest and Kolmogorov (2014).

Table 3 summarizes the lattice parameters, atomic positions, atomic moment of $T$ site, relative total energies, and critical temperatures of $T_{2}$AlB2 with different magnetic configurations. The calculated lattice parameters and atomic positions agree well with experiments. The relaxed lattice parameters of $T_{2}$AlB2 not only vary with element $T$ but also depend on the spin configuration. For Fe2AlB2, lattice parameters $a$ and $b$ decrease, and $c$ increases when the spin configuration changes from $\text{NM}\to\text{AFM}\to\text{FM}$. The calculated lattice parameters using the FM configuration agree the best with experiments. For Mn2AlB2, $b$ and $c$ increase when the spin configuration changes from $\text{NM}\to\text{FM}\to\text{AFM}$. Relative to those of Fe2AlB2, the lattice parameter $a$ of $T_{2}$AlB2 varies within $0.06\text{\,}\AA$ ($2.2\text{\,}\mathrm{\char 37\relax}$) in the sequence of $\text{Mn}<\text{Fe}\approx\text{Cr}<\text{Co}<\text{Ni}$; $b$ varies within $0.30\text{\,}\AA$ ($2.7\text{\,}\mathrm{\char 37\relax}$) in the sequence of $\text{Fe}<\text{Ni}<\text{Cr}\approx\text{Mn}<\text{Co}$; and parameter $c$ varies within $0.16\text{\,}\AA$ ($5.7\text{\,}\mathrm{\char 37\relax}$) in the sequence of $\text{Co}<\text{Mn}<\text{Ni}\approx\text{Fe}<\text{Cr}$. Percentagewise, the largest variation occurs with lattice parameter $c$. As we will show later, Fe2AlB2 has a much stronger magnetoelastic effect along the $c$ axis.

III.2 Alloys: $M$ and $T_{\text{C}}$ in Fe${}_{2-x}T_{x}$AlB2

To investigate how magnetic properties change with the $3d$ substitutions on Fe sites in Fe2AlB2, we first consider the chemical effect by neglecting the structure changes caused by substitution. The LMTO-ASA-CPA method is used to calculate the magnetization and the normalized effective exchange (or MFA estimation of $T_{\text{C}}$ in units of pure Fe2AlB2) in Fe${}_{2-x}T_{x}$AlB2 as functions of doping concentration $x$, with $T=\text{Cr}$, Mn, Co, and Ni. The experimental lattice parameters and atomic positions of Fe2AlB2 are used and the results are shown in Fig. 5. All dopings decrease the magnetization and $T_{\text{C}}$ in Fe2AlB2. The component-resolved atomic spin moments in those alloys are shown in Fig. 6. For Mn doping, we also consider the AFM configuration and show the absolute values of component-resolved moments in Fig. 6(b).

Assuming the FM configuration, Mn has the slightest effect on the decrease of the magnetization and $T_{\text{C}}$. The Fe moment barely changes and even increases with a higher Mn content. The decrease of total magnetization is due to the dilution of Fe moments with smaller Mn moments. With a $25\text{\,}\mathrm{\char 37\relax}$ Mn content, the calculated $T_{\text{C}}$ decreases by $20\text{\,}\mathrm{\char 37\relax}$ while experiments found a larger $\Delta T_{\text{C}}=$-30\text{,}\mathrm{\char 37\relax}$$. The assumption of FM configuration is logical when the Mn amount is small. However, at a higher Mn content, the AFM configuration should also be considered, given that pure Mn2AlB2 is more stable with the AFM configuration. Here we calculate the AFM configuration in CPA, by assuming the spin moments of $3d$ atoms (both Fe and Mn components) are parallel within the $ab$ plane and antiparallel between neighboring planes. As shown in Fig. 6(b), in comparison with the FM configuration, the AFM configuration gives larger Mn moments in the whole doping range and larger Fe moments at $x\geq 0.3$. Within CPA and without considering any lattice relaxation, the AFM configuration becomes more stable than the FM configuration with $x>0.2$. Thus, the larger decrease of $T_{\text{C}}$ observed in experiments is likely caused by the forming of AFM phases in the samples. By systematically investigating solid solutions (Fe1-xMnx)2AlB2, Chai et al. observed both NM and FM Mössbauer spectral components in all Mn-containing samples and attributed them to the clustering of Mn-rich and Fe-rich regions in the samples. Moreover, a spin-glass state has been observed at low temperature in (Fe1-xMnx)2AlB2 with $x=0.25$ and this phenomenon had been interpreted as the result of geometric frustration caused by the triangular configuration of magnetic atoms Du et al. (2015b). Here, we argue that it could be caused by the competition between the FM and AFM configurations along the $c$ axis.

Given that a large DOS lies right above the Fermi level in the minority spin channel as shown in Fig. 4(c), it is not surprising that the electron doping, such as Co or Ni doping, decreases the magnetic moment on Fe sites. With a small amount of Co doping, the magnetization and $T_{\text{C}}$ decrease nearly linearly with Co content. Similar linear dependence of $T_{\text{C}}$ on Co content had been observed in experiments. As shown in Fig. 5(b), calculated $\Delta T_{\text{C}}$ values agree very well with experiments Hirt et al. (2016). With $0.4<x<0.9$, Co atoms in (Fe1-xCox)2AlB2 have a small moment of $\sim 0.2\text{\,}\mu_{\mathrm{B}}\mathrm{/}\mathrm{C}\mathrm{o}$, similar to the Co moment calculated in the fully relaxed structure of Co2AlB2 using FP. However, this small moment becomes unstable in ASA at $x=1$.

In comparison to Co doping, Ni doping has a similar effect on decreasing $T_{\text{C}}$ and an even stronger effect on suppressing the magnetization in Fe2AlB2. As shown in Fig. 6(d), the atomic Ni moment in (Fe1-xNix)2AlB2 is small and coupled antiparallel with the Fe sublattice at small $x$, and negligible for $x>0.3$.

In (Fe1-xCrx)2AlB2, the Cr moment is small and parallel to the Fe sublattice. The maximum Cr moment of $0.2\text{\,}\mu_{\mathrm{B}}\mathrm{/}\mathrm{C}\mathrm{r}$ occurs at $x=0.5$. Like Mn, Cr doping has a smaller effect on decreasing the $T_{\text{C}}$ than Co and Ni. However, Cr doping is not likely to promote the $\text{FM}\to\text{AFM}$ transition, which may compromise the MCE as in the case of Mn doping Du et al. (2015b). Thus, it is worthwhile to investigate Cr doping, which may provide a useful approach to tune the $T_{\text{C}}$ and MCE in Fe2AlB2.

III.3 Effect of lattice distortion

Besides the chemical effect, the volume change caused by substitution may also affect the magnetic properties. As shown in Table 3, the lattice parameters in $T_{2}$AlB2 vary with the element $T$ and spin configuration. To have a rough idea on the magnetoelastic effect in Fe2AlB2, we calculate the magnetization dependence on the three lattice parameters, respectively. Starting from the fully relaxed structure, each of the three parameters is varied while the other two are kept constant. Interestingly, as shown in Fig. 7, the magnetoelastic effect in Fe2AlB2 is very anisotropic. The magnetization has a much stronger dependence on the lattice parameter $c$ than on $a$ or $b$. With $\Delta c=$-6\text{,}\mathrm{\char 37\relax}$$, magnetization decreases by $35\text{\,}\mathrm{\char 37\relax}$ within GGA and $60\text{\,}\mathrm{\char 37\relax}$ within LDA.

This anisotropic magnetoelastic effect can be understood by investigating the electronic structure features near $E_{\rm F}$ and their changes caused by the lattice distortion. The band structures of Fe2AlB2 in the majority and minority channels are shown in Fig. 8(a, c), respectively. The PDOS projected on Fe-$3d$ states and in two spin channels are shown in Fig. 8(b). Particularly of note is a narrow band right above $E_{\rm F}$ in the minority spin channel. This band is found to consist almost entirely of Fe-$d_{xy}$ and $d_{x^{2}-y^{2}}$ ($m=\pm 2$) orbitals. The variations of PDOS with lattice parameter $c$ are shown in Fig. 9. The decreasing of parameter $c$ increases the bandwidth of the $d_{3z^{2}-1}$ state, which has a relatively large density right below $E_{\rm F}$ in the majority spin channel. These antibonding $d_{3z^{2}-1}$ states, located between $-1.3\text{\,}\mathrm{eV}$ and $E_{\rm F}$, shift up toward $E_{\rm F}$ and become less occupied. Correspondingly, the aforementioned peak of $m=\pm 2$ states, located right above $E_{\rm F}$ in the minority spin channel, become more occupied. As a result, the magnetization decreases. The Fe-$d_{yz}$ and $d_{xz}$ states ($m=\pm 1$) have small DOS around $E_{\rm F}$ and contribute less to this magnetization change. With the further decrease of $c$ and then magnetization, the spin splitting becomes smaller, which quickly accelerates the decrease of magnetization as $\Delta c$ approaches to $-6\text{\,}\mathrm{\char 37\relax}$.

III.4 Dopings on B and Al sites

We also consider the substitutions of B and Al atoms with their neighboring elements in the periodic table: Be and C atoms on the B site, and Mg, Si, and Ga atoms on the Al site. The stabilities of those dopings are not well understood, and a careful and complete future investigation is desired. Here we focus on the possible effects of those dopings on the magnetization. Using various configurations of a ten-atom Fe2AlB2 unit cell, we substitute one B or Al atom with a dopant atom and fully relax the structures for Fe2Al(B${}_{0.75}Z_{0.25}$)2 with $Z=\text{Be}$ and C, and Fe2(Al${}_{0.5}Z_{0.5}$)B2 with $Z=\text{Mg}$, Si, and Ga. Their magnetizations calculated with the corresponding lowest energy configuration are denoted in Fig. 7 with respect to the change of lattice parameter $c$.

Only C doping on the B sites noticeably increases the lattice parameter $c$ and magnetization, while most other substitutions decrease the magnetization in Fe2AlB2. Both chemical effect and the magnetoelastic effect contribute to the magnetization enhancement. The lattice parameter $c$ increases by $1.6\text{\,}\mathrm{\char 37\relax}$ and magnetization increases to $1.4\text{\,}\mu_{\mathrm{B}}\mathrm{/}\mathrm{F}\mathrm{e}$ in Fe2Al(B0.75C0.25)2. Unlike B, C has a small moment parallel to the Fe sublattice. Moreover, C doping increases the moments of neighboring Fe atoms by about $0.1\text{\,}\mu_{\mathrm{B}}\mathrm{/}{Fe}$.

For Si doping, without considering lattice relaxation, magnetization decreases by $6\text{\,}\mathrm{\char 37\relax}$ in Fe2(Si0.5Al0.5)B. The relaxation decreases the lattice $c$ by $1.7\text{\,}\mathrm{\char 37\relax}$, and further decreases the magnetization by another $6\text{\,}\mathrm{\char 37\relax}$. Thus, both chemical effect and the magnetoelastic effect contribute to the decreasing of the magnetization. Be and Mg dopings have stronger effects on decreasing the magnetization. With Be and Mg dopings, the DOS peak right below $E_{\rm F}$ in the majority spin shifts toward $E_{\rm F}$ and becomes less occupied and the magnetization decreases to about $1.1\text{\,}\mu_{\mathrm{B}}\mathrm{/}\mathrm{F}\mathrm{e}$. Ga doping has very small effect on the lattice parameters and the magnetization of Fe2AlB2.

IV Conclusion

Using density functional theory, we investigated the intrinsic magnetic properties in $T_{2}$AlB2 and their alloys. For Fe2AlB2, the $a$ axis is the easiest axis, while the $c$ axis is the hardest axis. For Mn2AlB2, we predict that the magnetic ground state is an AFM configuration, with the neighboring Mn layers being antiferromagnetically coupled along the $c$ axis. Co2AlB2 is weakly ferromagnetic, while Cr2AlB2 and Ni2AlB2 are nonmagnetic. All $3d$ substitutions decrease the magnetization and Curie temperature of Fe2AlB2 in the sequence of $\text{Mn}<\text{Cr}<\text{Co}<\text{Ni}$. However, Mn promotes antiferromagnetism when its doping content is larger than $20\text{\,}\mathrm{\char 37\relax}$. The competition between the two configurations at critical compositions may be responsible for the spin-glass states observed in experiments. Unlike Mn, Cr doping is not likely to promote the AFM configuration, and may be useful in tuning $T_{\text{C}}$ in Fe2AlB2. The effects of strain and alloying on magnetic properties are also studied. A very strong anisotropic magnetoelastic effect is found. Magnetization in Fe2AlB2 becomes fragile and quickly decreases with the lattice parameter $c$, while it barely changes with $a$ and $b$. This effect is explained by the displacement of antibonding $d_{3z^{2}-1}$ states right below the Fermi level in the majority spin channel, and the filling of unoccupied $d_{xy}$ and $d_{x^{2}-y^{2}}$ states which have a sharp peak right above the Fermi level in the minority spin channel. Doping or applying pressure to modify the interlayer distance along the $c$ axis may provide an effective way to tune the magnetic properties in Fe2AlB2.

V Acknowledgments

We thank R. W. McCallum, L. Lewis, R. Barua, and B. Jensen for helpful discussions. Work at Ames Laboratory was supported by the U.S. Department of Energy, Advanced Research Projects Agency-Energy (ARPA-E) under Grant No. 1002-2147. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358.