# Electronic structure and X-ray magnetic circular dichroism in the MAX phases
T2AlC (T=Ti and Cr) from first principles

L. V. Bekenov  $|$ , S. V. Moklyak, B. F. Zhuravlev, Yu. N. Kucherenko, V. N.
Antonov  $|$

(Received December 20, 2022)

###### Abstract

We study the electronic and magnetic properties of T2AlC (T=Ti and Cr)
compounds in the density-functional theory using the generalized gradient
approximation (GGA) with consideration of strong Coulomb correlations
(GGA+$U$) in the framework of the fully relativistic spin-polarized Dirac
linear muffin-tin orbital (LMTO) band-structure method. The X-ray absorption
spectra and X-ray magnetic circular dichroism (XMCD) at the Cr $L_{2,3}$ and
Cr, Ti, and C $K$ edges were investigated theoretically. The calculated
results are in good agreement with experimental data. The effect of the
electric quadrupole $E_{2}$ and magnetic dipole $M_{1}$ transitions at the Cr
$K$ edge has been investigated.

Key words: electronic structure, X-ray absorption, X-ray magnetic circular
dichroism, MAX phases

###### Abstract

Ми вивчаємо електронні та магнітні властивості сполук T2AlC (T$=$Ti та Cr) в
рамках теорії функціоналу густини з використанням узагальненого градієнтного
наближення (GGA) з урахуванням сильних кулонівських кореляцій (GGA+U),
розраховуючи зонну структуру в формалізмі повністю релятивістського спін-
поляризованого методу діраківських лінійних МТ-орбіталей (LMTO). Теоретично
досліджені рентгенівські спектри поглинання та рентгенівський магнітний
циркулярний дихроїзм (XMCD) на Cr $L_{2,3}$ та Cr, Ti, C $K$ краях поглинання.
Результати розрахунків добре узгоджуються з експериментальними даними.
Досліджено вплив електричного квадрупольного $E_{2}$ та магнітного дипольного
$M_{1}$ переходів на Cr $K$ краю поглинання.

Ключов слова: електронна структура, спектри рентгенівського поглинання,
рентгенівський магнітний циркулярний дихроїзм, МАХ фази

##  1 Introduction

The Mn+1AXn (MAX) phases are layered hexagonal compounds, in which close-
packed layers of M (early transition metals) are interleaved with layers of
group A element (mostly IIIA and IVA), with the X-atoms (C and/or N) filling
the octahedral sites between the M layers. Depending on the stoichiometry, one
gets three different kinds of crystal structures classified as 211 ($n=1$),
312 ($n=2$) and 413 ($n=3$), all being described by the $P6_{3}/mmc$ space
group [1]. These carbides and nitrides have received increasing interest in
recent years due to their important properties, such as low density, high
elastic stiffness, thermal-shock and high-temperature oxidation resistance,
good thermal and electrical conductivities, damage tolerance, and easy
machinability [2, 3, 4, 5, 6, 7]. These unusual properties make these
compounds highly promising candidates for diverse applications in high-
temperature oxidizing environments. Potential applications of the
nanolaminated compounds mentioned to date range from spintronics to
refrigeration, even though the research efforts have so far been focused
solely on the discovery of new magnetic phases and compositions, and
fundamentals of magnetic properties. However, individual properties vary from
phase to phase and no systematic study of MAX phases is available in the
literature. Up till now, more than 70 MAX compounds have been synthesized [8,
9], and the search is still going on. Moreover, an increasing number of
scientists attempt to synthesize magnetic MAX phases [10] because magnetic MAX
phases exhibit a combination of magnetic, metallic, and ceramic-like
properties [11, 12], which are fascinating from the standpoints of both
fundamental science and technological applications. For example, a substantial
progress could happen in magnetic multilayers for magnetic recording and data
storage if strong magnetism can be found in MAX phases with inherently
nanolaminated compounds instead of the presently used artificially grown
sandwich materials.

The ternary carbides have the properties of both metals and ceramics. Like
metals, they are good thermal and electrical conductors. They are relatively
soft with vickers hardness of about 2–5 GPa. Like ceramics, they are
elastically stiff. Some of them (e.g., Ti3SiC2) also exhibit excellent high
temperature mechanical properties. They are resistant to thermal shock and
show unusually good damage tolerance and exhibit excellent corrosion
resistance. Above all, unlike the traditional carbides, they can be machined
by conventional tools without lubricant, which makes them more technologically
important for applications [6, 13]. These excellent properties make the MAX
phases another class of technically important materials. Brushes in electric
motors is another application of these materials. The enormous applications of
this class of materials trigger us to have a better understanding of their
electronic and mechanical properties.

Of special interest to this work are Ti2AlC and Cr2AlC carbides. The Ti2AlC
phase was first synthesized by Nowotny with co-workers [14], but its physical
properties were not characterized until quite recently [2, 14, 15, 16]. The
electronic structure of Ti2AlC was calculated a number of times [17, 18, 19].
The qualitative agreement between the first and last studies is good. The
ternary nanolaminated compound Cr2AlC was discovered in 1963 by Jeitschko et
al. [20]. Since that time it has been the subject of numerous studies, both
experimentally [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32] and
theoretically [9, 10, 11, 12, 13, 22, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42,
43].

In spite of many experimental and theoretical efforts to investigate the
electronic structure and physical properties of the ternary MAX phases, there
are still many contradictions and discrepancies among them. Many questions
still remain with respect to magnetism in the the ternary nanolaminated
compounds (see the review paper by Ingason et al. [1] and references therein).
Schneider et al. [22] calculated the difference in cohesive energy per formula
unit (f.u.) of the ferromagnetic (FM) and antiferromagnetic (AFM)
configurations with reference to the nonmagnetic (NM) configuration. Based on
the small total energy difference between the AFM and NM configurations as
well as the comparatively small local magnetic moment (0.7 $\mu_{\text{B}}$/Cr
atom), the authors speculated that magnetism may be suppressed in Cr2AlC. Sun
et al. [33] also performed total energy calculations for Cr2AlC both at AFM
and NM polarized states. They found that energy difference between the two
states was around 0.3 meV. Therefore, they claimed nonmagnetic ground state in
Cr2AlC. This is in agreement with the first experimental result attempting to
address this point based on nuclear magnetic resonance (NMR) spectroscopy
[25]. In this study, Lue et al. show, from the temperature dependence of the
17Al central lines between room temperature and 77 K, that Cr2AlC is NM in
this temperature range. Ramzan et al. [43] presented the electronic and
structural properties calculated by first principles GGA and GGA+$U$
calculations of Cr2AlC. They show that for the GGA exchange-correlation
potential, the ground state of Cr2AlC is not spin-polarized, with vanishing
magnetic moments on the Cr atoms. In this case, the agreement with experiments
for the equilibrium structure and the bulk modulus is not satisfactory. On the
other hand, using the GGA+$U$ approximation (with Hubbard $U=1.95$ eV and
exchange Hund coupling $J=0.95$ eV), the calculated ground state corresponds
to a FM ordering. In this case, the agreement with experiments was found to be
excellent for the lattice constants and bulk modules. However, for a larger
value of $U$ (2.95 eV), the ground state magnetic order is modified, from FM
(for $U=1.95$ eV) to AFM (for $U=2.95$ eV), and the calculated equilibrium
volume was in strong disagreement with experiments. The magnitude of the
magnetic moments strongly depends on the Hubbard $U$. Dahlqvist et al. [11,
44] state that the magnetic ground state of Cr2AlC is in-plane AF. By
comparing GGA and GGA+$U$ results with experimental data, they found that
using a $U_{\rm eff}=U-J$ value larger than 1 eV results in structural
parameters deviating strongly from experimentally observed values. The spin
magnetic moments at the Cr site $M_{s}^{\rm Cr}$ were found to be equal to 0.7
$\mu_{\text{B}}$ [11] and 1.37 eV for the GGA and GGA+$U$ (for $U_{\rm eff}=1$
eV) [44], respectively. The authors concluded that this class of Cr-based
carbide MAX phases cannot be considered as strongly correlated systems since
both GGA and GGA+$U$ with $U_{\rm eff}\leqslant 1$ eV give the calculated
lattice parameters and bulk modulus close to experimentally reported values,
if low-energy in-plane AFM magnetic states are considered. For larger values
of the $U$ parameter ($U_{\rm eff}>1$ eV), the structural parameters deviate
strongly from experimentally observed values.

Soft X-ray absorption and magnetic circular dichroism in T2AlC were measured
by several authors [37, 27, 32]. Recent X-ray magnetic circular dichroism
experiments performed by Jaouen et al. [32] demonstrate that Cr atoms carry a
net magnetic moment in Cr2AlC ternary phase along the $c$ axis. The spin
magnetic moment at the Cr site was found to be extremely small $M_{s}^{\rm
Cr}=0.05$ $\mu_{\text{B}}$, which is much smaller than the predicted values
for the FM [22, 43] as well as the AFM [11, 44] solutions. SQUID experiments
by Jaouen et al. [45] also produce an extremely small Cr spin magnetic moment
of 0.002 $\mu_{\text{B}}$ in Cr2AlC.

The aim of this paper is the theoretical study from the “first principles” of
the electronic and magnetic structures and the XAS and XMCD in T2AlC (T=Ti and
Cr) carbides. The energy band structure of T2AlC (T=Ti and Cr) carbides is
calculated within the ab initio approach taking into account strong electron
correlations by applying a local spin-density approximation to the density
functional theory supplemented by a Hubbard $U$ term (GGA+$U$) [46]. The paper
is organized as follows. The computational details are presented in section 2.
Section 3 presents the electronic structure, XAS and XMCD spectra of T2AlC
(T=Ti and Cr) carbides at the Cr $L_{2,3}$ and Cr, Ti, and C $K$ edges
calculated in the GGA+$U$ approximation. Theoretical results are compared to
the experimental measurements. Finally, the results are summarized in section
5.

##  2 Crystal structure and computational details

###  2.1 X-ray magnetic circular dichroism

Magneto-optical (MO) effects refer to various changes in the polarization
state of light upon interaction with materials possessing a net magnetic
moment, including rotation of the plane of linearly polarized light (Faraday,
Kerr rotation), and the complementary differential absorption of the left-hand
and right-hand circularly polarized light (circular dichroism). In the near
visible spectral range, these effects result from excitation of electrons in
the conduction band. Near X-ray absorption edges, or resonances, magneto-
optical effects can be enhanced by transitions from well-defined atomic core
levels to transition symmetry selected valence states.

Within the one-particle approximation, the absorption coefficient
$\mu^{\lambda}_{j}(\omega)$ for incident X-ray polarization $\lambda$ and
photon energy $\hbar\omega$ can be determined as the probability of electronic
transitions from initial core states with the total angular momentum $j$ to
final unoccupied Bloch states

$\displaystyle\mu_{j}^{\lambda}(\omega)$ $\displaystyle=$
$\displaystyle\sum_{m_{j}}\sum_{n\bf k}|\langle\Psi_{n\bf
k}|\Pi_{\lambda}|\Psi_{jm_{j}}\rangle|^{2}\delta(E_{n\bf
k}-E_{jm_{j}}-\hbar\omega)\theta(E_{n\bf k}-E_{\rm F}),$ (2.1)

where $\Psi_{jm_{j}}$ and $E_{jm_{j}}$ are the wave function and the energy of
a core state with the projection of the total angular momentum $m_{j}$;
$\Psi_{n\bf k}$ and $E_{n\bf k}$ are the wave function and the energy of a
valence state in the $n$-th band with the wave vector k; $E_{\rm F}$ is the
Fermi energy.

$\Pi_{\lambda}$ is the electron-photon interaction operator in the dipole
approximation

$\Pi_{\lambda}=-e\mbox{\boldmath$\alpha$}\bf{a_{\lambda}},$ (2.2)

where $\bm{\alpha}$ are the Dirac matrices and $\bf{a_{\lambda}}$ is the
$\lambda$ polarization unit vector of the photon vector potential, with
$a_{\pm}=1/\sqrt{2}(1,\pm\mathrm{i},0)$, $a_{\parallel}=(0,0,1)$. Here, “$+$”
and “$-$” denotes, respectively, left-hand and right-hand circular photon
polarizations with respect to the magnetization direction in the solid. Then,
X-ray magnetic circular and linear dichroism are given by $\mu_{+}-\mu_{-}$
and $\mu_{\parallel}-(\mu_{+}+\mu_{-})/2$, respectively. More detailed
expressions of the matrix elements in the electric dipole approximation may be
found in references [47, 48, 49]. The matrix elements due to magnetic dipole
and electric quadrupole corrections are presented in reference [49].

###  2.2 General properties of spin density waves

The magnetic configuration of an incommensurate spin spiral shows the magnetic
moments of certain atomic planes varying in direction. The variation has a
well-defined period determined by a wave vector $q$. When the magnetic moment
is confined to the lattice sites, the magnetization $M$ varies as [50]

$\mbox{\boldmath$M$}(\mbox{\boldmath$r$}_{n})=m_{n}\left[\begin{array}[]{c}\cos(\mbox{\boldmath$qr$}_{n}+\phi_{n})\sin(\theta_{n})\\\
\sin(\mbox{\boldmath$qr$}_{n}+\phi_{n})\sin(\theta_{n})\\\
\cos(\theta_{n})\end{array}\right],$ (2.3)

where the polar coordinates are used and $m_{n}$ is the magnetic moment of
atom $n$ with a phase $\phi_{n}$ at the position $r_{n}$. Here, we consider
only planar spirals, that is, $\theta_{n}=\piup$/2 which also give the minimum
of the total energy. The magnetization of equation (2.3) is not
translationally invariant but transforms as

$\mbox{\boldmath$M(r+R)$}=\mbox{$D$(\boldmath$qR$)\boldmath$M(r)$},$ (2.4)

where $R$ is a lattice translation and $D$ is a rotation around the $z$ axis.
A spin spiral with a magnetization in a general point $r$ in space can be
defined as a magnetic configuration which transforms according to equation
(2.4). Since the spin spiral describes a spatially rotating magnetization, it
can be correlated with a frozen magnon.

Since the spin spiral breaks the translational symmetry, the Bloch theorem is
no longer valid. Computationally, one should use large super-cells to obtain
total-energy of the spin spirals. However, when the spin-orbit interaction is
neglected, spins are decoupled from the lattice and only the relative
orientation of the magnetic moments is important. Then, one can define the
generalized translations which contain translations in the real space and
rotations in the spin space [51]. These generalized translations leave the
magnetic structure invariant and lead to a generalized Bloch theorem.
Therefore, the Bloch spinors can still be characterized by a $k$ vector in the
Brillouin zone, and can be written as

$\psi_{k}(\mbox{\boldmath$r$})=\mathrm{e}^{\mathrm{i}\mbox{\boldmath$kr$}}\left(\begin{array}[]{c}\mathrm{e}^{-\mathrm{i}\mbox{\boldmath$qr$}/2}u_{k}(\mbox{\boldmath$r$})\\\
\mathrm{e}^{+\mathrm{i}\mbox{\boldmath$qr$}/2}d_{k}(\mbox{\boldmath$r$})\end{array}\right).$
(2.5)

The functions $u_{k}(\mbox{\boldmath$r$})$ and $d_{k}(\mbox{\boldmath$r$})$
are invariant with respect to lattice translations having the same role as for
normal Bloch functions. Due to this generalized Bloch theorem, the spin
spirals can be studied within the chemical unit cell and no large super-cells
are needed. Though the chemical unit cell can be used, the presence of the
spin spiral lowers the symmetry of the system. There remain only the space-
group operations that leave invariant the wave vector of the spiral.
Considering the general spin space groups, i.e., taking the spin rotations
into account, the space-group operations which reverse the spiral vector
together with a spin rotation of $\piup$ around the $x$ axis are symmetry
operations [51].

Though the original formulation of the local-spin-density approximation of the
density-functional theory permited the noncollinear magnetic order, first-
principles calculations for this aspect have begun only recently (for a
review, see reference [52]). One application is the study of noncollinear
ground states, for example, in $\gamma$-Fe (references [53, 54, 55]) or in
frustrated antiferromagnets [56, 57]. In addition, the noncollinear
formulation enables the studies of finite-temperature properties of magnetic
materials. Since the dominant magnetic excitations at low temperatures are
spin waves which are noncollinear by nature, it is possible to determine the
magnon spectra and ultimately the Curie temperature from first principles [58,
59, 60, 61, 62]. The noncollinear magnetic configurations were investigated in
the Heusler alloys Ni2MnGa, Ni2MnAl [50], IrMnAl [63], Mn3ZnC [64] and Mn3CuN
[65]. The total energies for different spin spirals were calculated and the
ground-state magnetic structures were identified.

###  2.3 Crystal structure

The M2AX unit cell (the space group is $P6_{3}/mmc$ No. 194) with two formula
units per unit cell is shown in figure 1 with Wyckoff positions: M (4$f$), A
(2$d$), and X (2$a$). The coordination of the M is trigonal prismatic, while
that of X is octahedral. The structure possesses the layer stacking sequence
of M and A atoms along the [0001] direction consisting of M2X slabs and
intercalation of planar packed A-ions. The layered stacking characteristics
can be clearly illustrated in the ($1\bar{2}10$) plane, as displayed in figure
1 (b). The X atoms occupy the interstitial sites of M octahedra [see figure 1
(c)]. The crystal structure has one free internal parameter by $Z_{M}$ that
defines the height of the M atoms above the X sheets. The dimensionless
crystallographic coordinate is $z_{M}=Z_{M}/c$. At the ideal value of
$z_{M}=1/12\sim 0.0833$, the M and A planes are evenly spaced. Parameter
$z_{M}=0.086$ and 0.083 for Cr2AlC and Ti2AlC, respectively.

Figure 1: (Colour online) (a) Crystal structure of the Cr2AlC (the space group
is $P6_{3}/mmc$ No. 194). Blue spheres represent Cr atoms, green and blue
spheres show Al and C atoms, respectively; (b) and (c) present arrangement of
atoms on a ($1\bar{2}10$) and (0001) planes of Cr2AlC, respectively.

###  2.4 Calculation details

The details of the computational method are described in our previous papers
[66, 67, 68, 69] and here we only mention several aspects. Band-structure
calculations were performed using the fully relativistic linear muffin-tin
orbital (LMTO) method [70, 48]. This implementation of the LMTO method uses
four-component basis functions constructed by solving the Dirac equation
inside an atomic sphere [71]. The exchange-correlation functional of a GGA-
type was used in the version of Perdew, Burke and Ernzerhof (PBE) [72].
Brillouin zone (BZ) integration was performed using the improved tetrahedron
method [73]. The basis consisted of Cr and Ti $s$, $p$, $d$, and $f$ and Al
and C $s$, $p$, and $d$ LMTO’s.

We found that the agreement between the theoretically calculated and
experimentally measured XAS and XMCD spectra became much better with taking
into account strong Coulomb correlations. To include the electron-electron
correlations into the consideration, we used the “relativistic” generalization
of the rotationally invariant version of the LSDA+$U$ method [74] which takes
into account SO coupling so that the occupation matrix of localized electrons
becomes non-diagonal in spin indexes. This method is described in detail in
our previous paper [74] including the procedure to calculate the screened
Coulomb $U$ and exchange $J$ integrals, as well as the Slater integrals
$F^{2}$, $F^{4}$, and $F^{6}$.

The screened Coulomb $U$ and exchange Hund coupling $J_{H}$ integrals enter
the LSDA+$U$ energy functional as external parameters and should be determined
independently. These parameters can be determined from supercell LSDA
calculations using Slater’s transition state technique [75, 76], from
constrained LSDA calculations (cLSDA) [77, 76, 78, 79, 80] or from the
constrained random-phase approximation (cRPA) scheme [81]. Subsequently, a
combined cLSDA and cRPA method was also proposed [82]. The cLSDA calculations
produce $J_{H}=0.95$ eV for the Cr in Cr2AlC. It is known that the cRPA method
underestimates the values of $U$ in some cases [83]. On the other hand, the
cLSDA method produces too large values of $U$ [84]. Therefore, in our
calculations we treated the Hubbard $U$ as an external parameter and varied
the effective Hubbard $U_{\rm eff}=U-J_{H}$ from 0 to 3.0 eV. In the case of
$U_{\text{eff}}=U-J=0$, the effect of the GGA+$U$ comes from non-spherical
terms which are determined by $F^{2}$ and $F^{4}$ Slater integrals. This
approach is similar to the orbital polarization (OP) corrections [85, 86, 87,
88, 48]. Therefore, we use the notation GGA+OP throughout the paper for the
$U_{\text{eff}}=U-J=0$ eV approach.

The X-ray absorption and dichroism spectra were calculated taking into account
the exchange splitting of the core levels. The finite lifetime of a core hole
was taken into account by folding the spectra with a Lorentzian. The widths of
core levels $\Gamma_{L_{2,3}}$ for Mn and Ti, and $\Gamma_{K}$ for O were
taken from reference [89]. The finite experimental resolution of the
spectrometer was taken into account by a Gaussian of width 0.6 eV.

##  3 Electronic and magnetic structures

The theoretically calculated electronic band structure of nanolaminated
ternary carbides such as Cr2AlC and Ti2AlC, demonstrated that the valence
bands could be divided into several regions. The lowest lying group of valence
states originates predominantly from the C 2$s$ states. The states located at
higher energy range are hybridization states of Al 3$s-3p$ orbitals. Strong
$pd$ covalent bonding states derived from Ti-C interactions dominate the
adjacent higher energy range. The states just below the Fermi level contain a
relatively weaker $pd$ covalent bonding between Ti and Al. Moreover, the
states near and above the Fermi level are attributed to metal-to-metal $dd$
interactions and antibonding states.

Figure 2: (Colour online) Partial density of states [in states/(atom eV)] of
Cr2AlC in the GGA approximation.

Characteristics of atomic bonding can be vividly illustrated by projected
density of states (PDOS). Figures 2 and 3 show the PDOS of Cr2AlC and Ti2AlC
for comparison. The C 2$s$ states are located between $-14.7$ eV to $-12.1$ eV
and $-12.2$ eV to $-10.4$ eV below the Fermi level in Cr2AlC and Ti2AlC,
respectively. The states, which are approximately located between $-5.8$ and
$-2.8$ eV below the Fermi level ($E_{\rm F}$) in Ti2AlC are C 2$p$ states.
They are well hybridized with Al $sp$ and Ti 3$d$ states. The corresponding C
2$p$ states Cr2AlC are situated lower in energy at the $-8.0$ and $-4.3$ eV
energy interval. Al 3$p$ states occupy a wide energy interval from $-9.0$ eV
to $16.0$ eV and from $-7.7$ eV to 16.0 eV Cr2AlC and Ti2AlC, respectively.
The major peak of the occupied Al 3$p$ states associated with the $pd$
covalent bond is situated at around $-2.5$ eV in Cr2AlC and $-1.0$ eV in
Ti2AlC below $E_{\rm F}$. The 3$d$ orbitals of Cr atoms dominate the states
near the $E_{\rm F}$, and the contribution from Al $p$-derived orbitals is
negligible. In comparison, the states near $E_{\rm F}$ are dominated by Ti
3$d$ orbitals in Ti2AlC, with some contribution from Al $p$ orbitals.

Figure 3: (Colour online) Partial density of states [in states/(atom eV)] of
Ti2AlC in the GGA approximation.

The magnitude of Cr spin and orbital magnetic moments strongly depends on the
Hubbard $U$. The GGA approach produces the spin and orbital moments equal to
0.707 $\mu_{\text{B}}$ and 0.007 $\mu_{\text{B}}$, respectively (see table 1).
The GGA+OP approach gives slightly larger moments: $M_{s}^{\rm Cr}=0.757$
$\mu_{\text{B}}$ and $M_{l}^{\rm Cr}=0.009$ $\mu_{\text{B}}$. For $U_{\rm
eff}=1$ eV, 2 eV and 3 eV, the spin magnetic moments are equal to 0.973
$\mu_{\text{B}}$, 2.097 $\mu_{\text{B}}$, and 2.799 $\mu_{\text{B}}$,
respectively. These values are in good agreement with previous band structure
calculations [43, 11, 44]. The orbital magnetic moments at the Cr site also
strongly increased with increasing the value of Hubbard $U$ and even change
the sign for the $U_{\rm eff}$ = 2 eV and 3 eV (table 1). We found a similar
dependence of the spin and orbital magnetic moments for the AFM solution as
well, although the absolute values of the moments are slightly different from
the FM ordering. The induced spin magnetic moments at the Al and C sites are
much smaller than at the Cr site and have an opposite direction (table 1). It
is interesting to note that the spin magnetic moment at the C site is larger
than the corresponding moment at the Al site. Orbital moments on both the Al
and C sites are quite small.

Table 1: The theoretically calculated spin $M_{s}$, orbital $M_{l}$, and total magnetic moments (in $\mu_{\text{B}}$) of Cr2AlC for the FM solution. Method | Atom | $M_{s}$ | $M_{l}$ | $M_{\rm total}$
---|---|---|---|---
| Cr | 0.707 | 0.007 | 4.134
GGA | Al | $-0.015$ | $-0.001$ | $-0.016$
| C | $-0.064$ | 0.0 | $-0.064$
| Cr | 0.757 | 0.009 | 0.766
GGA+OP | Al | $-0.021$ | $-0.001$ | $-0.022$
| C | $-0.064$ | 0.0 | $-0.064$
| Cr | 0.973 | 0.013 | 0.986
GGA+$U$ | Al | $-0.074$ | $-0.001$ | $-0.075$
$U_{\rm eff}=1$ eV | C | $-0.108$ | 0.0 | $-0.108$
| Cr | 2.097 | $-0.030$ | 2.067
GGA+$U$ | Al | $-0.154$ | $-0.001$ | $-0.155$
$U_{\rm eff}=2$ eV | C | $-0.239$ | $-0.003$ | $-0.242$
| Cr | 2.799 | $-0.030$ | 2.769
GGA+$U$ | Al | $-0.196$ | $-0.001$ | $-0.197$
$U_{\rm eff}=3$ eV | C | $-0.357$ | $-0.002$ | $-0.359$
LSDA reference [22] | Cr | 0.70 | $-$ | $-$
GGA+$U$ reference [43] | Cr | 0.90 | $-$ | $-$
$U_{\rm eff}=1$ eV |  |  |  |
GGA+$U$ reference [43] | Cr | 2.50 | $-$ | $-$
$U_{\rm eff}=2$ eV |  |  |  |
exper.XMCD [32] | Cr | 0.05 | $-$ | $-$
exper.SQIID [45] | Cr | 0.002 | $-$ | $-$

SQUID experiments by Jaouen et al. [45] produce Cr2AlC to be FM with extremely
small Cr spin magnetic moment of 0.002 $\mu_{\text{B}}$. Recent X-ray magnetic
circular dichroism experiments [32] clearly demonstrate that Cr atoms carry a
net magnetic moment of 0.05 $\mu_{\text{B}}$ in Cr2AlC along $c$ axis.
Therefore, these experiments are in strong disagreement with all existing
theoretical calculations concerning the value of the Cr spin magnetic moment
and the exact nature of the magnetic ordering in Cr2AlC ternary phase.

In order to gain an insight into the ground state magnetic configurations of
Cr2AlC, we compared the calculated total energies for different spin ordering
Cr2AlC, namely, NM, FM, and AFM phases as well as possible noncollinear
magnetic (NCM) structures. We found that the total energy of the AFMab state
(AFM ordering in the $ab$ plane) is lower by 25 meV/f.u., 17 meV/f.u., and 12
meV/f.u. than those of the NM, FM, and AFM[0001] (AFM ordering along the $c$
axis) states, respectively, which implies a preferable AFM phase in Cr2AlC
ordered in the $ab$ plane. This conclusion agrees with the previous
estimations by Dahlqvist et al. [11, 44]. However, this AFMab ground state is
still in contradiction with the XMCD experiment by Jaouen et al. [32], because
any AFM ordering produces zero net magnetization in the system. Applying the
noncollinear formalism described in section 2.2 to the Cr2AlC, we found NCM
state which has a lower total energy by 6 meV/f.u. in comparison with the
AFMab state. The NCM state is characterized by a canted AFM spin configuration
(spin magnetic moment $M_{S}^{\rm Cr}=0.768$ $\mu_{\text{B}}$ in the GGA
approach with the polar angles equal to $\theta^{\rm Cr_{1}}=91.7^{\circ}$,
$\phi^{\rm Cr_{1}}=0^{\circ}$ and $\theta^{\rm Cr_{2}}=91.4^{\circ}$,
$\phi^{\rm Cr_{1}}=180^{\circ}$). Such a AFM configuration with the spins
slightly canted out of the ($a,b$) plane produces a small projection of the Cr
spin magnetic moment along the $c$ axis of around 0.047 $\mu_{\text{B}}$,
which is in excellent agreement with the estimation by Jaouen et al. [32] of
0.05 $\mu_{\text{B}}$ using the XMCD measurements and sum rules. Strictly
speaking, we do not have AFM ordering in the ($a,b$) plane but rather
ferrimagnetic (FiM) ordering with a small net magnetization of 0.005
$\mu_{\text{B}}$ in the plane.

We found that Ti2AlC is very close to NM ground state. Though titanium is in
the Ti2+ state ($d^{2}$), the spin magnetic moment at the Ti as well as at Al
sites is less than 10-4 $\mu_{\text{B}}$.

##  4 X-ray absorption and XMCD spectra

Figure 4 presents the X-ray absorption spectra (open circles) at the Cr
$L_{2,3}$ edges (top panel) in Cr2AlC measured at 4.2 K [32] with a 6 T
exteranal magnetic field compared with the theoretically calculated ones (full
blue curve). Since the pure Cr $L_{23}$ edges are structureless, the existence
of these fine structures shows that the chromium is not in a pure metallic
state in Cr2AlC, a well-known fact from previous band structure calculations
related to MAX phases [90]. They all have a mixture of covalent, metallic and
ionic bonds.

Figure 4: (Colour online) Top panel: the X-ray absorption spectra (open
circles) at the Cr $L_{2,3}$ edges in Cr2AlC measured at 4.2 K [32] with a 6 T
magnetic compared with the theoretically calculated ones (full blue curve)
calculated in the GGA approach; Middle panel: the theoretically calculated
(full blue curve) and experimentally measured (open circles) [32] X-ray linear
dichroism spectra; lower panel: the XMCD experimental spectra (open circles)
of Cr2AlC at the Cr $L_{2,3}$ edges and the theoretically calculated one (full
blue line); the onset shows the XMCD spectra at the Gr $L_{2,3}$ edges
separately from the Cr1 and Cr2 sites.

One can observe a well separated double structure at the top of both $L_{3}$
and $L_{2}$ edges. These fine structures are well reproduced by a band
structure calculation, although the theory produces inverse relative
intensities between low and high energy peaks in comparison with the
experimentally observed ones in the $L_{3}$ XAS spectrum. It is interesting to
note that the model multiple scattering calculations performed with the FEFF
[91, 92] code by Jaouen et al. [32] show the same inversion in the relative
intensities as obtained by us.

Figure 4 (middle panel) shows the theoretically calculated (full blue curve)
and experimentally measured (open circles) [32] X-ray linear dichroism
spectra. The theory reproduces the energy position of all the fine structures
quite well, although the negative peaks at 571 eV, 578 eV, and 587 eV are much
lower in theory than in the experiment.

The XMCD experimental spectra (open circles) of Cr2AlC at the Cr $L_{2,3}$
edges and the theoretically calculated spectrum (full blue line) are presented
in the lower panel of figure 4. For the experimental geometry used in
reference [32] it follows that the electric field, E, of the incident X-ray
beam is parallel to the $(a,b)$ plane of the MAX phase. In that case, XAS
measurements mainly probe the unoccupied Cr 3$d_{xy}$, 3$d_{xz}$, 3$d_{yz}$
and 3$d_{x^{2}-y^{2}}$ in-plane orbitals, as well as a much smaller $p\to s$
contribution. The X-ray dichroism at the Cr $L_{2,3}$ edges is quite small due
to cancellation signals from Cr1 and Cr2 sites (see the insert in the lower
panel of figure 4). The agreement between the theory and experiment is quite
good although the intensity of the major positive peak at 578 eV is slightly
overestimated in the theory.

Figure 5: (Colour online) Top panel: the X-ray absorption spectrum (open
circles) at the Cr $K$ edge in Cr2AlC measured at 2.2 K [32] and external
magnetic field of 10 T with the theoretically calculated ones in the GGA
approach taking into account only dipole 1$s$ $\to$ 2$p$ transitions (full
blue curve) and quadrupole 1$s$ $\to$ 3$d$ transitions (dashed red curve)
multiply by factor 10; lower panel: the XMCD experimental spectrum (open
circles) of Cr2AlC at the Cr $K$ edge and the theoretically calculated ones in
dipole approximation (full blue line) and quadrupole transitions (dashed red
curve) multiply by factor 10.

We found that both the GGA and GGA+$U$ with $U_{\rm eff}\leqslant 1$ eV give
the calculated XAS and XMCD spectra close to experimentally measured ones. For
larger values of the $U$-parameter ($U_{\rm eff}>1$ eV), the theoretically
calculated XMCD spectra deviate strongly from experimentally observed spectra.
Therefore, this class of Cr-based carbide MAX phases cannot be considered as
strongly correlated systems. A similar conclusion was also drawn by Dahlqvist
et al. [11, 44] based on the calculations of lattice parameters and bulk
modulus in the Cr2AlC, Cr2GaC and Cr2GeC.

Figure 5 presents the X-ray absorption spectrum (open circles) at the Cr $K$
edge (top panel) in Cr2AlC measured at 2.2 K [32] with magnetic field of 10 T
compared with the theoretically calculated ones (full blue curve). The XAS
spectrum consists of a major peak at 5993.4 eV followed by a local minimum at
5996.0 eV and a low energy shoulder at 5989.3 eV. The theory well reproduces
the energy position and shape of the fine structures. It is worth noticing
that the low energy shoulder at 5989.3 eV is usually attributed to the
quadrupolar $E_{2}$ (1$s\to 3d$) transitions [92, 32]. We investigate the
effect of the electric quadrupole $E_{2}$ and magnetic dipole $M_{1}$
transitions. We found that the $M_{1}$ transitions are extremely small in
comparison with the $E_{2}$ transitions and can be neglected. The $E_{2}$
transitions indeed contribute to the low energy shoulder at 5989.3 eV as well
as to the major peak at 5993.4 eV, although the quadrupolar $E_{2}$
transitions are by two orders of magnitude smaller than the electric dipole
transitions $E_{1}$ (see a red dashed curve at the upper panel of figure 5).
Therefore, the low energy shoulder reflects the energy distribution of C
$N_{p}$ partial DOS (figure 2).

The lower panel shows the XMCD experimental spectrum (open circles) of Cr2AlC
at the Cr $K$ edge and the theoretically calculated ones in the dipole
approximation (full blue line). The dashed-dotted lines show the contribution
from the quadrupole $E_{2}$ (1$s\to 3d$) transitions multiplied by a factor of
10. Again, the contribution of the quadrupolar $E_{2}$ (1$s\to 3d$)
transitions to the Cr $K$ XMCD spectrum is very small. The theory well
reproduces the energy position and the intensity of the major positive peak at
5989.3 eV, while other fine structures are reproduced with less accuracy,
although it is hard to achieve an ideal agreement with the experimental
measurements with such a very weak detected XMCD signal.

Figure 6: (Colour online) Top panel: the X-ray absorption spectrum (open
circles) at the C $K$ edge in Cr2AlC measured at 4.2 K [27] with a 6 T
magnetic compared with the theoretically calculated one (full blue curve) in
the GGA approach; lower panel: the theoretically calculated XMCD spectrum of
Cr2AlC at the C $K$ edge.

Figure 6 presents the X-ray absorption spectrum (open circles) at the C $K$
edge (top panel) in Cr2AlC measured by Lin et al. [27] compared with the
theoretically calculated ones (full blue curve). Our band structure
calculations well reproduce the energy position of all fine structures of the
experimental C $K$ XAS spectrum. Shindo and Oikawa [93] investigated the XAS
spectra of diamond, graphite, and amorphous carbon. They reported that a XAS
peak at 291 eV indicated a strong $\sigma$ bonding state for C. On the other
hand, a $\pi$ bonding state of carbon was distinguished by a $\pi^{*}$ peak at
$\sim 284$ eV in the XAS spectra [27, 93]. In the experimentally measured C
$K$ edge for Cr2AlC, the peak was located at about 290 eV, which implies that
the Cr–C bond in Cr2AlC is a strong $\sigma$ bonding [27].

Figure 6 (bottom panel) presents the theoretically calculated XMCD spectrum at
the C $K$ edge in Cr2AlC. Due to very small spin and orbital magnetic moments
at the C site (see table 1), one would expect a quite small dichroism at this
edge with major negative peak at 291 eV. The experimental measurements of the
XMCD spectrum at the C $K$ edge are highly desirable.

Spin and orbital magnetic moments in Ti2AlC are very small at the Ti and C
sites. Therefore, the XMCD spectra at these edges are not detected yet. The
theoretically calculated XMCD spectra (not shown) are three orders of
magnitude smaller than their XAS spectra.

Figure 7 shows the X-ray absorption spectra (open circles) at the Ti $K$
(upper panel) [37] and C $K$ (lower panel) [37] edges in Ti2AlC compared to
the theoretically calculated spectra (full blue curves). The theory quite well
reproduces the experimental spectra.

Figure 7: (Colour online) top panel: the experimentally measured [37] X-ray
absorption spectrum at Ti $K$ edge in Ti2AlC (magenta circles) and
theoretically calculated (full blue curve) in the GGA approach; lower panel:
the experimentally measured [37] X-ray absorption spectrum at C $K$ edge in
Ti2AlC (magenta circles) and theoretically calculated (full blue curve) in the
GGA approach.

The energy position of a major low energy peak at the C $K$ edge coincides
with the low energy shoulder of Ti $K$ XAS spectrum at 4 eV above the edge
indicating a strong Ti $d$ – C $P$ $\sigma$ bonding in Ti2AlC.

##  5 Summary

The electronic and magnetic structures and X-ray magnetic circular dichroism
of the MAX 211 compounds Cr2AlC and Ti2AlC were investigated theoretically
within GGA and GGA+$U$ approaches in the framework of the fully relativistic
spin-polarized Dirac LMTO band-structure method.

We found the non-collienear magnetic state as a ground state in Cr2AlC which
is characterized by a canted AFM spin configuration (spin magnetic moment
$M_{S}^{\rm Cr}=0.768$ $\mu_{\text{B}}$ in the GGA approach with the polar
angles equal to $\theta^{\rm Cr_{1}}=91.7^{\circ}$, $\phi^{\rm
Cr_{1}}=0^{\circ}$ and $\theta^{\rm Cr_{2}}=91.4^{\circ}$, $\phi^{\rm
Cr_{1}}=180^{\circ}$). Such AFM configuration with spins is slightly canted
out of the ($a,b$) plane and produces a small projection of the Cr spin
magnetic moment along the $c$ axis of around 0.047 $\mu_{\text{B}}$, which is
in excellent agreement with the estimation by Jaouen et al. [32] of 0.05
$\mu_{\text{B}}$ using the XMCD measurements and sum rules. There is a
ferrimagnetic ordering in the ($a,b$) plane with net magnetization of 0.005
$\mu_{\text{B}}$.

We have studied an X-ray magnetic circular dichroism at the Cr $L_{2,3}$ and
Cr, C, and Ti $K$ edges in Cr2AlC and Ti2AlC. The calculations show a good
agreement with the experimental measurements. We cannot validate the
significance for using the LSDA(GGA)+$U$ methods for the study of magnetic MAX
phases since both GGA and GGA+$U$ with $U_{\rm eff}\leqslant 1$ eV give the
calculated XAS and XMCD spectra close to the experimentally measured ones.
Therefore, this class of Cr-based carbide MAX phases cannot be considered as
strongly correlated systems.

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Ukrainian 0

Електронна структура та рентгенівський магнітний циркулярний дихроїзм у МАХ
фазах T2AlC (T$=$Ti або Cr), визначені з перших принципів Л. В. Бекеньов, С.
В. Мокляк, Б. Ф. Журавльов, Ю. Кучеренко, В. М. Антонов

Інститут металофізики ім. Г. В. Курдюмова НАН України, бульвар Академіка
Вернадського, 36, UA-03142 Київ, Україна