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Electronic correlation eﬀects in Cr2GeC

Mn+1AXn-phase

Maurizio Mattesini1,2and Martin Magnuson3

1Departamento de F´ısica de la Tierra, Astronom´ıa y Astrof´ısica I, Universidad

Complutense de Madrid, E-28040 Madrid, Spain.

2Instituto de Geociencias (UCM-CSIC), Facultad de Ciencias F´ısicas, Plaza de

Ciencias 1, 28040-Madrid, Spain.

3Department of Physics, Chemistry and Biology (IFM), Link¨oping University,

SE-58183 Link¨oping, Sweden.

E-mail: mmattesi@fis.ucm.es,m.mattesini@igeo.ucm-csic.es

Abstract. The magnetic properties, electronic band structure and Fermi surfaces of

the hexagonal Cr2GeC system have been studied by means of both generalized gradient

approximation (GGA) and the +U corrected method (GGA+U). The eﬀective Uvalue

has been computed within the augmented plane-wave theoretical scheme by following

the constrained density functional theory formalism of Anisimov et al. [1]. On the

basis of our GGA+U calculations, a compensated anti-ferromagnetic spin ordering of

Cr atoms has been found to be the ground state solution for this material, where a

Ge-mediated super-exchange coupling is responsible for an opposite spin distribution

between the ABA stacked in-plane Cr-C networks. Structural properties have also

been tested and found to be in good agreement with the available experimental

data. Topological analysis of Fermi surfaces have been used to qualitatively address

the electronic transport properties of Cr2GeC and found an important asymmetrical

carrier-type distribution within the hexagonal crystal lattice. We conclude that an

appropriate description of the strongly correlated Cr-delectrons is an essential issue

for interpreting the material properties of this unusual Cr-based M AX -phase.

PACS numbers: 71.15.Mb, 71.20.-b, 75.25.-j, 71.18.+y, 72.15.-v

Submitted to: J. Phys.: Condens. Matter

Electronic correlation eﬀects in Cr2GeC Mn+1AXn-phase 2

1. Introduction

The Mn+1AXnor M AX phases are layered hexagonal solids with unusual and sometimes

unique combination of properties [2]. They are made of an early transition metal M,

an A-group element (group III,IV ,V, or V I element), and by an Xelement that is

either C or N. They have attracted much attention due to the peculiar combination of

properties that are normally associated with either metals or ceramics. Just like metals,

they are readily machinable, electrically and thermally conductive, not susceptible to

thermal shock, plastic at high temperature and exceptionally damage tolerant. On the

other hand, they are elastically rigid, lightweight, creep and fatigue resistant as ceramic

materials. It is therefore not surprising that there has been a rapid increase of research

activities on MAX phases by both experimental and theoretical works during recent

years [2, 4, 5, 6]. A magnetic M AX phase that could potentially give rise to functional

materials for spintronics applications [3] has also long been searched. However, to our

knowledge, despite many attempts and eﬀorts, none of the synthesized Mn+1AXnphases

have been found to possess stable magnetic features.

Among the known Mn+1AXnphases, Cr2GeC (Fig 1) is a relatively little studied

member. It has the highest thermal expansion coeﬃcient among all the present known

MAX phases [7, 8, 9], high resistivity, a positive Seebeck coeﬃcient both in- and out-

of-plane [10], and a negative Hall coeﬃcient. The calculated electronic density of states

(DOS) at the Fermi level (EF) is considerably underestimated [11] and there is a large

and anisotropic electron-phonon coupling [12]. In general, Cr-containing Mn+1AXn-

phases have unusually large DOS at the EFstemming from the electronic d-states of

the transition metal. In fact, for Cr2GeC the DOS at EFis by far the highest (22

eV−1·cell−1) measured among the Mn+1AXn-phases and for Cr2AlC (14.6 eV−1·cell−1)

it is the second highest. The carrier mobility is, however, rather limited in Cr-based

MAX phases compared to Ti-containing ones, due to their strongly localized Cr d-

states. The signiﬁcantly correlated nature of the Cr d-electrons also make the magnetic

coupling and ferromagnetic/anti-ferromagnetic ordering largely unknown and rather

complicated to establish [8]. Moreover, there is some disagreement in the literature

about the experimentally determined values for both bulk modulus and equilibrium

volume [13, 14].

For all these reasons, a comprehensive theoretical study is needed to correctly

address the material properties of this unusual M AX phase. Speciﬁcally, the aim

of this work is to study the eﬀect of correlation on the electronic structure and

material properties of the Cr2GeC M AX-phase. Particular attention has been given to

the ground-state magnetic spin ordering, electro-structural correlations and transport

properties. We observe that ferromagnetic Cr layers are anti-ferromagnetically coupled

together via an interleaved Ge-atom, assembling a multilayer material that could, in

principle, be tuned to provide thermodynamic stable magnetic MAX-phases.

Electronic correlation eﬀects in Cr2GeC Mn+1AXn-phase 3

Figure 1. Atomic model of the hexagonal crystal structure of Cr2GeC with space

group P3m1 (156) and 8-atoms per unit-cell. Chromium (Cr), germanium (Ge), and

carbon (C) atoms are depicted in blue (large spheres), green (medium spheres), and

gray (small spheres), respectively. The large red arrows drawn at the Cr1···Cr4atoms

indicate the ground-state magnetic spin conﬁguration inside the unit cell. Small black

arrows represent the fractional magnetic spin moments localized at the Ge and C sites.

The vesta visualization software [15] was used to generate the present ﬁgure.

2. Computational details

2.1. First-principles calculations

The electronic structure of Cr2GeC was computed within the wien2k code [16]

employing the density-functional [17, 18] augmented plane wave plus local orbital

(APW+lo) computational scheme. The APW+lo method expands the Kohn-Sham

orbitals in atomic-like orbitals inside the muﬃn-tin (MT) atomic spheres and plane

waves in the interstitial region. The Kohn-Sham equations were solved by means of

the Wu-Cohen generalized gradient approximation (GGAW C ) [19, 20] for the exchange-

Electronic correlation eﬀects in Cr2GeC Mn+1AXn-phase 4

correlation (xc) potential. For a variety of materials it improves the equilibrium lattice

constants and bulk moduli signiﬁcantly over the local density approximation (LDA) [18],

and performs pretty well for the Cr2GeC material (see results in Table 2). The latter

is the main reason that motivated our choice to adopt the Wu-Cohen approximation in

studying this Cr-based M AX phase.

A plane-wave expansion with RM T ·Kmax=10 was used in the interstitial region,

while the potential and the charge density were Fourier expanded up to Gmax =12.

The modiﬁed tetrahedron method [21] was applied to integrate inside the Brillouin

zone (BZ ), and a k-point sampling with a 35×35×7 Monkhorst-Pack [22] mesh in

the full BZ (corresponding to 786 irreducible k-points) was considered satisfactory for

the hexagonal Cr2GeC system. Magnetic ground-state properties and electronic band

structure features were studied using the relaxed unit cell parameters. All the spin-

polarized calculations were charge converged up to 10−4e.

Convergent and smooth Fermi surfaces (FSs) were achieved by sampling the whole

BZ with 10000 k-points along the 35×35×7 Monkhorst-Pack grid. The presented FS

plots were then generated with the help of the xcrysden graphical user interface code

[23] applying the tricubic spline interpolation with a degree of ﬁve.

2.2. Searching for an eﬀective Hubbard U-value

Density functional theory (DFT) is an upright method for computing ground-state

properties of solids with feeble electronic correlations. However, this method fails to

describe systems with intermediate and strong electron correlations, such as transition-

metal oxides, Kondo systems and rare earths. Such a short-coming description is due

to the spurious self-interaction. Therefore, these materials are very often investigated

by means of a phenomenological many-body Hamiltonian such as the Hubbard model

[24], where the eﬀective on-site Coulomb interaction is an empirical parameter (U) that

permit to reproduce the experimental results of interest. Hence, by using this approach,

the correct determination of Urepresents a critical issue because many properties, such

as magnetism, can vary in MAX phases with the value of U.

When employing the local density approximation for the xc-part, then the LDA+U

method indicates that an orbital-dependent ﬁeld has been introduced to correct for

self-interaction [25]. Particularly, a set of atomic-like orbitals is treated with an

orbital-dependent potential with an associated on-site Coulomb (U) and exchange

(J) interactions. Since in LDA, the electron-electron interactions have already been

considered in a mean-ﬁeld way, one has to identify the parts that occur twice and apply

a double-counting (DC) correction. To overcome such a problem, in the non-spherical

part of potential we used Ueff =U−J[26], setting therefore J=0. Although several

diﬀerent ways of correcting for DC are existing [25, 1, 27, 28], we here use what has

been referred to as the SIC method introduced by Anisimov et al. [25].

The physical meaning of the Uparameter was deﬁned by Anisimov and Gunnarsson

[1], who described it as the Coulombic energy cost of placing two electrons on the same

Electronic correlation eﬀects in Cr2GeC Mn+1AXn-phase 5

Table 1. Calculated on-site Coulomb value (Ueff in eV) for diﬀerent exchange-

correlation functionals.

LDA GGAW C GGAP BE

Ueff (eg) 2.14 2.33 2.08

Ueff (t2g) 2.04 2.09 2.03

site. In an atom, Usimply corresponds to the unscreened Slater-integrals, whereas in

solids the Uef f is much smaller because of screening eﬀects. The Hubbard Udepends

on the type of crystal structure, delectron number, dorbital ﬁlling and most generally

on the degree of electronic localization.

Using the method of Anisimov and Gunnarsson (sometimes called constrained DFT

formalism), the U-value has been calculated for the Cr atom in the hexagonal Cr2GeC

structure. Two kinds of calculations were performed on a 2×2×1 supercell each with

one impurity site forced to have the d-conﬁguration as shown in eq. (1)

Ueff =ε3d↑n+ 1

2,n

2−ε3d↑n+ 1

2,n

2−1+

−EFn+ 1

2,n

2+EFn+ 1

2,n

2−1(1)

where ε3d↑is the spin-up 3deigenvalue and EFthe Fermi energy. The d-character

of the augmented plane waves at the chromium impurity sites was eliminated by setting

the d-linearization energy far above the Fermi level [E(ℓ=2)=20.30 Ry]. Using eq. (1)

we computed an Ueff value of 2.09 eV (t2g) and 2.33 eV (eg) for the chosen GGAW C

xc-functional. Table 1 shows the obtained Hubbard Uparameter for various exchange-

correlation potentials. In agreement with previous studies [29], the eﬀective interaction

between delectrons in egorbitals is larger than that in t2gorbitals.

3. Results

3.1. Magnetic ground-state and equilibrium structural parameters

We have investigated several possible magnetic orders of the moments on the Cr atoms,

either ferromagnetic (FM), antiferromagnetic (AFM), or with no magnetic moments

(NM). As reported earlier, in the case of GGA, the ground state might correspond to

either NM [30] or AFM [8], while for GGA +Uthe antiferromagnetic spin distribution

of Cr atoms along the c-axis turns out to be the most stable solution [30]. In particular,

using the GGAW C functional (present study) we also found small amounts of localized

Cr magnetic moments with an AFM spin ordering inside the in-plane Cr-C networks

(i.e., Cr1(↑)=+0.012 µB, Cr2(↓)=-0.008 µB, Cr3(↓)=-0.012 µB, and Cr4(↑)=+0.008 µB).

However, when using the +Ucorrected functional (GGAW C +U), the ground state

magnetism turns out to be rather diﬀerent, having an alternate FM spin distribution

for the two Cr-C networks (Fig. 1). The computed magnetic moments for the Cr atoms

that belong to the Cr-C network located at nearly half of the c-axis are Cr2(↑)=+0.011 µB

Electronic correlation eﬀects in Cr2GeC Mn+1AXn-phase 6

Figure 2. Polyhedral model for the Cr2GeC phase showing the Cr-C layers (gray)

that are propagating along the a-bcrystal plane and the interleaved Cr atoms (green).

Each polyhedral skeleton consists of three C atoms at the base and a Cr atom at

the vertex. Note the alternating (up/down) vertex distribution within each layer that

produce an ABA stacking order of layers along the c-axis. The Cr atom sitting on

the polyhedral vertices of the middle (bottom-top) Cr-C network is Cr4(Cr1) for the

upward polyhedra and Cr2(Cr3) for its downward counterpart.

and Cr4(↑)=+0.007 µB, whereas for those at the bottom/top of the hexagonal unit cell

amount to Cr1(↓)=-0.007 µBand Cr3(↓)=-0.011 µB. As shown in Fig. 1, the +Ucorrected

functional allows for a Ge-mediated super-exchange magnetic coupling [31] between Cr

atoms belonging to diﬀerent Cr-C networks (Fig. 2). This inter-layer interaction arises

from the mixing of the Cr 3dand the Ge 4pstates, which act along the 103◦angle

bend Cr-Ge-Cr three-point line. The computed unequal values for the Cr magnetic

moments (Cr16=Cr2and Cr36=Cr4) are ascribed to the small amount of Ge1→Cr4and

Ge2→Cr1charge-transfer that is at the base of the super-exchange coupling mechanism

[31]. This generates a very small spin polarization of the Ge atoms that sustains an

antiferromagnetic spin coupling with the Cr atoms that are non directly involved in the

Electronic correlation eﬀects in Cr2GeC Mn+1AXn-phase 7

charge-transfer mechanism (Ge(↑)

1-Cr(↓)

3and Ge(↓)

2-Cr(↑)

2). Also, when introducing the

on-site Coulombic interaction the three C atoms, which constitute the ﬁrst coordination

shell of Cr ions, become slightly spin-polarized, thus stabilizing a FM in-plane Cr spin

distribution (Cr(↓)

1,3-3×C(↑)

2and Cr(↑)

2,4-3×C(↓)

1). It is worth noting that the total magnetic

moment of the unit cell is still zero, as to indicate an almost perfect resulting AFM

spin alignment inside the whole hexagonal lattice. Despite the tiny magnitude of Cr

magnetism computed within the GGAW C +Uscheme, the achieved results are clearly

showing that Cr2GeC has a considerably more complicated magnetic structure than

believed earlier. Relativistic corrections in the electronic structure calculation have also

been included in a second-variational procedure using scalar-relativistic wavefunctions

[32]. Applying the spin-orbit coupling within the atomic spheres along the [0 0 1]

magnetization direction, leads to enhanced Cr magnetic moments while keeping exactly

the same non-relativistic ground-state spin pattern.

Th importance of the super-exchange coupling has been further underlined by

GGAW C +Ucalculations performed on an iso-geometrical Ge-hollow unit cell. In

this case, an AFM material is stabilized with exactly the same in-plane AFM spin

arrangement of Cr atoms found within the GGAW C method. Therefore, only an explicit

description of the strongly correlated nature of the Cr 3delectrons is able to catch the

super-exchange interaction, that maintain an in-plane FM Cr spin ordering. Preliminary

experimental results seem to conﬁrm this important ﬁnding [33], although the presence

of small Cr magnetic moments might lead to a rather weak magnetic energy and therefore

to a low N´eel temperature.

Attention should also be payed to the nature of the interleaved non-magnetic A-

atoms, that might play a crucial role in determining the overall magnetic properties

of this MAX-phase. As a rule of thumb, the smaller the spin polarizability of the

bridging A-atom is, the weaker the inter-layer super-exchange coupling will be [34].

Hence, smaller atoms having tightly bound valence electrons will then tend to weaken

the super-exchange coupling, enhancing the FM behavior of the Cr2GeC crystal phase.

Formulated diﬀerently, this will translate into attempting to tune the Cr-Abonding

type as to reduce its covalent character, thus rising up the observable N´eel temperature.

In table 2, we report the volume, lattice parameters and bulk modulus within

diﬀerent correlation corrected xc-functionals. After including the +Ucorrection,

the equilibrium volume considerably increases giving a good agreement with the

experimental data of Phatak et al.[14], although the bulk modulus gets signiﬁcantly

lower. However, since the GGAW C +Uis correctly treating the correlated nature of the

Cr d-electrons, we believe that most likely this is the right value for the bulk modulus.

3.2. Electronic density of states and band structure

The DOS at the Fermi level is dominated by the Cr transition metal. Fig. 3 illustrates

the calculated Cr-dtotal DOS for both GGAW C and GGAWC +Umethods. A bandwidth

reduction of about 0.75 eV is found when including the on-site Coulombic interaction.

Electronic correlation eﬀects in Cr2GeC Mn+1AXn-phase 8

Table 2. Optimized cell parameters for the ground-state AFM spin conﬁguration.

Property GGAW C GGAWC +U GGAP B E +Uaexp.bexp.c

V(˚

A3) 86.24 92.71 91.21 91.10 92.82

a(˚

A) 2.899 2.981 2.97 2.950 2.958

c(˚

A) 11.875 12.044 12.16 12.086 12.249

c/a 4.097 4.040 4.094 4.097 4.141

Bo(GPa) 254.4 147.6 150 182 169

aUsing U=1.95 eV and J=0.95 eV (Ref.[30]).

bFrom Ref.[13].

cFrom Ref.[14].

The most distinct feature we observe is the shrinking of both valence and conduction

bandwidths. The bottom of the valence band moves to higher energy by 0.25 eV and

the top of the conduction band is reduced towards lower energies by 0.50 eV. The Cr

d-electrons strongly hybridize with the Ge and C p-states and their relative energetic

position determines the degree of hybridization and the width of the valence band. The

Hubbard Ucorrection has therefore a noticeable inﬂuence on the hybridization between

localized and itinerant states. The top of the valence band and the bottom of the

conduction band develop more C 2pand Ge 3pcharacters when including Uand the Cr

states hybridizes accordingly giving rise to smaller bandwidths. A similar behaviour of

Ge states was found on the V2GeC phase [35].

Since Cr2GeC is a metallic system, the DOS at the Fermi level is a key quantity for

stability purposes. The Cr2GeC crystal has its EFpositioned exactly at a local minimum

in the DOS, thus suggesting a higher level of intrinsic stability. As a matter of fact, local

minimum at EFis a good indicator of large structural stability as it represents a barrier

for electrons below the Fermi-level to move into the unoccupied empty states. The

GGAW C +Uslightly increases the number of states at EF(3.95 states/eV) with respect

to GGAW C calculations (3.85 states/eV), while keeping the same topological DOS. As

such, band renormalization does not provide any remarkable eﬀects concerning the total

amount of electronic band ﬁlling of the occupied states.

Figure 4 shows the GGAW C +Ucalculated electronic band structure of Cr2GeC.

The dominant contribution to the electronic density of states at EFderives from metallic

bonding of the Cr d-electron orbitals in the Cr-Ge-C network. Several bands are

formed that cross EF, both electron- and hole-like, thus resulting in a multiband system

dominated by Cr d-character. At the Fermi level, the bands that are crossing the Fermi

energy are 44, 45, 46, 47 and 48, and their numbering follows exactly that of Fig. 4.

Band 44 has the same dxz +dyz orbital character contribution from all the Cr atoms

(Cr1→Cr4), while in band 45 there is an important dz2weight from Cr1and Cr2and

adx2−y2+dxy contribution from Cr3and Cr4. For band 46 all the Cr atoms contribute

with the same amount of both dz2and dx2−y2+dxy characters. The very similar bands

47 and 48 are mainly of dx2−y2+dxy character from all the Cr atoms. Figure 4 also

shows the hole- and electron-like character of the crossing bands. Speciﬁcally, hole-like

Electronic correlation eﬀects in Cr2GeC Mn+1AXn-phase 9

features are positioned at symmetry point M, midway the K-Γ symmetry line, and at

Land H. On the contrary, an electron-like pocket can been seen at symmetry points

Γ, Aand K.

From band structure analysis, important information about electronic transport

properties can be obtained. The Cr-containing M AX carbides are expected to show

the highest resistivity along the series Ti2GeC→V2GeC→Cr2GeC. As a matter of fact,

the reported resistivity for Ti- and V-based M AX phases are in the range 15-30 µΩcm

compared to 53-67 µΩcm for the Cr2GeC material [5]. This is generally due to the

reduced carrier mobilities in Cr-based systems, where strongly localized Cr d-states are

present near the Fermi level [36]. In this regard, it is worth noting that the overall

electronic band structure of Cr2GeC is rather anisotropic, with bands crossing the EF

only along the symmetry lines of the basal xy-plane. Nevertheless, such an intrinsic band

structure anisotropy is similar to that of typical hexagonal-close-packed materials [37],

and therefore cannot be used to quantitatively explain the peculiar transport properties

of Cr-containing MAX phases. In this respect, scattering processes and charge carrier-

phonon coupling should be taken into account [38].

3.3. Fermi surfaces

In metals, the energy states that participate in determining most properties of a material

lie in close proximity to the Fermi energy, that is, the level below which available energy

states are ﬁlled. The Fermi surface thus separates the unﬁlled orbitals from the ﬁlled

ones and represents a surface of constant energy (E=EF) in k-space. The electrical

properties of metals are deﬁned by the shape and size of the Fermi surface, as the

current is due to changes in the occupancy of states, near the Fermi surface. Therefore,

by observing the fermiology of the computed Fermi surfaces one can help addressing,

from a qualitative point of view, the predominant role of each crossing band along either

the zcomponent or inside the xy plane.

Since the velocities of the electrons are perpendicular to the Fermi surface, then

bands 45, 47 and 48 have large components within the basal plane, while bands 44

and 46 play an equally important role along the three kx, ky, and kzaxes. From

band 44 a localized hole-like FS pocket emerges at the M-point (Fig. 6), deﬁned as

the k-vector (1

2,0,0) in the BZ of Fig. 5. The next band (45) has two hole-like band

features, one centered at the symmetry point Mand the other at L(Fig. 7). Band

46 has a mixture of characters, with hole-like features at Mand Land an electron-like

component at K. The Fermi surfaces of bands 47 and 48 have both very similar hole-like

character along the point symmetries Γ-M, K-Γ, A-L and A-H. As with other transition

metal-based materials, the introduced electronic correlation leads to a certain degree of

electron renormalization of the band structure which, however, does not change the

main topology of the Cr2GeC Fermi surfaces.

Electronic correlation eﬀects in Cr2GeC Mn+1AXn-phase 10

-10 -5 0510

Energy, E-EF (eV)

-1.5

-1

-0.5

0

0.5

1

1.5

Density of states (states/spin/eV)

Cr-dup/dn (GGAWC+U)

Cr-dup/dn (GGAWC)

EF

Cr-dup (GGAWC)

Cr-ddn (GGAWC)

Cr-dup (GGAWC+U)

Cr-ddn (GGAWC+U)

Figure 3. Density of states of AFM Cr2GeC for spin-up (upper part) and spin-down

(lower part) Cr-delectrons.

4. Discussion

It has been shown that an appropriate treatment of correlation eﬀects in the Cr2GeC

MAX-phase leads to the discovery of a diﬀerent magnetic spin pattern, where a super-

exchange interaction operates through the non-magnetic Ge ions. The buckled Cr-C

networks that are propagating parallel to the xy-plane of the crystal are thus showing a

FM intra-layer spin distribution and an AFM inter-layer spin ordering. The FM layers

are lined up in a perfect anti-ferromagnetic pattern giving a vanishing total magnetic

moment inside the unit cell. This ﬁnding enables the possibility of tuning the exchange

coupling between ferromagnetically ordered Cr-C layers so as to achieve stable magnetic

MAX phases for electronic and spintronic applications. For instance, the inter-layer

coupling could be varied by changing the inter-layer thickness [39], interface quality [40]

or even by alloying with another M-element [41]. Further studies are being pursued in

Electronic correlation eﬀects in Cr2GeC Mn+1AXn-phase 11

ΓM K ΓA L H A

0.70

0.75

0.80

0.85

0.90

0.95

Energy (Ry)

44

45

46

47 48

band 44

band 45

band 46

band 47

band 48

Figure 4. Energy band structure of Cr2GeC for the spin-up conﬁguration along high

symmetry directions shown in Fig. 5. Only the electronic bands that are crossing the

Fermi level are shown.

Figure 5. Primitive Brillouin zone of the hexagonal unit cell with the used symmetry

points: Γ (0,0,0), M(1

2,0,0), K(2

3,1

3,0), A(0,0,1

2), L(1

2,0,1

2) and H(2

3,1

3,1

2). Reciprocal

lattice vectors are also shown as a∗(kx), b∗(ky), and c∗(kz).

Electronic correlation eﬀects in Cr2GeC Mn+1AXn-phase 12

Figure 6. Constant energy surface for band 44 (spin-up) viewed along the kxand ky

plane of the hexagonal Brillouin zone.

Figure 7. Fermi surface for band 45 (spin-up).

Figure 8. Fermi surface for band 46 (spin-up).

Electronic correlation eﬀects in Cr2GeC Mn+1AXn-phase 13

Figure 9. Fermi surface for band 47 (spin-up).

Figure 10. Fermi surface for band 48 (spin-up).

order to experimentally conﬁrm such a magnetic pattern via a detailed XMCD analysis

[33].

From the calculated Fermi surfaces we have seen that there is only one electron-like

band (band 46) that shows a velocity component along the kzaxis. The other 4 Fermi

surfaces are hole-like with large velocity contributions conﬁned within the basal xy-

plane. Therefore, Cr2GeC appears to be a material characterized by a clear carrier-type

anisotropy, being the positively hole charge carriers responsible for transport properties

within the basal xy-plane, and the negatively electron charge carriers along the vertical

z-axis. This qualitatively explains the reason why the experimentally determined

Seebeck coeﬃcients [33] along the [001] plane (i.e., the in-plane component) are generally

larger than those along the [103] plane (i.e., the out-of-plane direction). Large and

positive Seebeck coeﬃcients indicate that Cr2GeC behaves as a p-type material along

the in-plane direction, having predominantly positive mobile charges (holes). On the

contrary, the much lower magnitude of measured Seebeck coeﬃcients along the [103]

Electronic correlation eﬀects in Cr2GeC Mn+1AXn-phase 14

plane points to an increased negative carrier concentration along the vertical direction

of the hexagonal Cr2GeC crystal. This may also be true in other Cr-based materials

such as Cr2AlC. As shown in this work, determining the super-exchange coupling will

serve as an important method to predict which MAX phases are good candidates with

stable magnetic features.

5. Conclusions

An ad hoc eﬀective Hubbard Uvalue has been computed for various exchange correlation

functionals by using the constrained DFT formalism. We have shown that Cr2GeC

is a weak AFM material with a rather anisotropic electronic band structure. Most

important, by properly accounting for Cr correlation eﬀects we discovered the presence of

a super-exchange coupling between diﬀerent in-plane Cr-C networks of the Cr2GeC unit

cell. Therefore, the magnetic nature of the studied M AX phase is AFM (as proposed

earlier) but with a substantially diﬀerent electro-structural origin. The interleaved

Ge-atoms stabilize the ferromagnetically ordered Cr layers that are exchange coupled

together. If this kind of inter-layer coupling can be tailored, then very attractive layered

magnetic materials can be proposed with a potential use for many electronics and

spintronics applications.

Equilibrium structural parameters were also computed within the GGAW C +Uand

found to be in good agreements with the experimental data of Phatak et al.[14]. The

topology of Fermi surfaces was studied to address the electric transport properties of the

metallic Cr2GeC material. The achieved results indicate that this Cr-containing MAX

phase has a relevant asymmetrical carrier-type structure, where hole carriers dominate

within the basal plane and electrons only contribute to carrier mobility along the z-axis.

Acknowledgments

We thank the Swedish Research Council (VR) for ﬁnancial support.

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