ArticlePDF Available

Abstract and Figures

The magnetic properties, electronic band structure and Fermi surfaces of the hexagonal Cr(2)GeC system have been studied by means of both generalized gradient approximation (GGA) and the +U corrected method (GGA + U). The effective U value has been computed within the augmented plane wave theoretical scheme by following the constrained density functional theory formalism of Anisimov and Gunnarsson (1991 Phys. Rev. B 45 7570-74). On the basis of our GGA + U calculations, a compensated antiferromagnetic 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 has been used to qualitatively address the electronic transport properties of Cr(2)GeC, and found an important asymmetrical carrier-type distribution within the hexagonal crystal lattice. We conclude that an appropriate description of the strongly correlated Cr d electrons is an essential issue for interpreting the material properties of this unusual Cr-based MAX phase.
Content may be subject to copyright.
Electronic correlation effects in Cr2GeC
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.
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
eV1·cell1) measured among the Mn+1AXn-phases and for Cr2AlC (14.6 eV1·cell1)
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 104e.
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 =UJ[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.
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 =ε3dn+ 1
2ε3dn+ 1
EFn+ 1
2+EFn+ 1
where ε3dis 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(=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 103angle
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 Ge1Cr4and
Ge2Cr1charge-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()
3and Ge()
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()
2and Cr()
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
A3) 86.24 92.71 91.21 91.10 92.82
A) 2.899 2.981 2.97 2.950 2.958
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
(Cr1Cr4), while in band 45 there is an important dz2weight from Cr1and Cr2and
adx2y2+dxy contribution from Cr3and Cr4. For band 46 all the Cr atoms contribute
with the same amount of both dz2and dx2y2+dxy characters. The very similar bands
47 and 48 are mainly of dx2y2+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 Ti2GeCV2GeCCr2GeC. 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)
Density of states (states/spin/eV)
Cr-dup/dn (GGAWC+U)
Cr-dup/dn (GGAWC)
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
Energy (Ry)
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
3,0), A(0,0,1
2), L(1
2) and H(2
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
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.
We thank the Swedish Research Council (VR) for financial support.
[1] Anisimov V I and Gunnarsson O 1991 Density-functional calculation of effective Coulomb
interactions in metals Phys. Rev. B 45 7570-7574.
[2] Barsoum M W 2000 The M(N+1)AX(N) phases: A new class of solids; Thermodynamically stable
nanolaminates Prog. Solid State Chem. 28 201-281.
[3] Wolf S A, Awschalom D D, Buhrman R A, Daughton J M, von Molnar S, Roukes M L,
Chtchelkanova A Y, and Treger D M 2001 Spintronics: A spin-based electronics vision for
the future Science 294 1488-1495.
[4] Barsoum M W and Radovic M 2011 Elastic and Mechanical Properties of the MAX Phases
Annual Review of Materials Research 41 195-227.
[5] Eklund P, Beckers M, Jansson U, Hoegberg H, and Hultman L 2010 The M(n+1)AX(n) phases:
Materials science and thin-film processing Thin Solid Films 518 1851-1878.
Electronic correlation effects in Cr2GeC Mn+1AXn-phase 15
[6] Wang J and Zhou Y 2009 Recent Progress in Theoretical Prediction, Preparation, and
Characterization of Layered Ternary Transition-Metal Carbides Annual Review of Materials
Research 39 415-443.
[7] Bouhemadou A 2009 Calculated structural, electronic and elastic properties of M2GeC (M=Ti,
V, Cr, Zr, Nb, Mo, Hf, Ta and W) Appl. Phys. A 96 959-967.
[8] Zhou W, Liu L, and Wu P 2009 First-principles study of structural, thermodynamic, elastic, and
magnetic properties of Cr2GeC under pressure and temperature J. Appl. Phys. 106 033501-
[9] Scabarozi T H, Amini S, Leaffer O, Ganguly A, Gupta S, Tambussi W, Clipper S, Spanier J E,
Barsoum M W, Hettinger J D and Lofland S E 2009 Thermal expansion of select M >n+1 >
AX >n (M=early transition metal, A=A group element, X=C or N) phases measured by high
temperature x-ray diffraction and dilatometry J. Appl. Phys. 105 013543-013551.
[10] Barsoum M, Scabarozi T H, Amini S, Hettinger J D and Lofland S E 2011 Electrical and Thermal
Properties of Cr2GeC J. Am. Ceram. Soc. 94 4123-4126.
[11] Drulis M K, Drulis H, Hackemer A E, Leaffer O, Spanier J, Amini S, Barsoum M W, Guilbert
T and El-Raghy T 2008 On the heat capacities of Ta2AlC, Ti2SC, and Cr2GeC J. Appl. Phys.
104 023526-023533.
[12] Eklund P, Bugnet M, Mauchamp V, Dubois S, Tromas C, Jensen J, Piraux L, Gence L, Jaouen
M and Cabioch T 2011 Epitaxial growth and electrical transport properties of Cr2GeC thin
films Phys. Rev. B 84 075424-075433.
[13] Manoun B, Amini S, Gupta S, Saxena S K and Barsoum M W 2007 On the compression behavior
of Cr2GeC and V2GeC up to quasi-hydrostatic pressures of 50 GPa Phys. Condens. Matter.
19 456218-456225.
[14] Phatak N A, Kulkarni S R, Drozd V, Saxena S K, Deng L, Fei Y, Hu J and Ahuja R 2088
Synthesis and compressive behavior of Cr2GeC up to 48 GPa J. Alloy. Comp. 463 220-225.
[15] Momma K and Izumi F 2008 VESTA: a three-dimensional visualization system for electronic and
structural analysis J. Appl. Crystallogr. 41 653-658.
[16] Blaha P, Schwarz K, Madsen G K H, Kvasnicka D and Luitz J 2001 An Augmented Plane Wave
+ Local Orbitals Program for for Calculating Crystal Properties (Karlheinz Schwarz, Techn.
Universit¨at Wien, Austria), ISBN 3-9501031-1-2.
[17] Hohenberg P and Kohn W 1964 Inhomogeneous Electron Gas Phys. Rev. 136 B864-B871.
[18] Kohn W and Sham L J 1965 Self-Consistent Equations Including Exchange and Correlation
Effects Phys. Rev. 140 A1133-A1138.
[19] Wu Z and Cohen R 2006 More accurate generalized gradient approximation for solids Phys. Rev.
B73 235116-235122.
[20] Tran F, Laskowski R, Blaha P and Schwarz K 2007 Performance on molecules, surfaces, and solids
of the Wu-Cohen GGA exchange-correlation energy functional Phys. Rev. B 75 115131-115145.
[21] Bl¨ochl P E, Jepsen O and Andersen O K 1994 Improved tetrahedron method for Brillouin-zone
integrations Phys. Rev. B 49 16223-16233.
[22] Monkhorst H J and Pack J D 1976 Special points for Brillouin-zone integrations Phys. Rev. B 13
[23] Kokalj A 2003 Computer graphics and graphical user interfaces as tools in simulations of matter
at the atomic scale Comp. Mater. Sci. 28 155-168.
[24] Hubbard J 1963 Electron correlations in narrow energy bands Proc. R. Soc. London Ser. A 276
[25] Anisimov V I, Solovyev I V, Korotin M A, Czy˙zyk M T and Sawatzky G A 1993 Density-functional
theory and NiO photoemission spectra Phys. Rev. B 48 16929-16934.
[26] Dudarev S L, Botton G A, Savrasov S Y, Humphreys C J and Sutton A P 1998 Electron-energy-
loss spectra and the structural stability of nickel oxide:An LSDA+U study Phys. Rev. B 57
[27] Czy˙zyk M T and Sawatzky G A 1994 Local-density functional and on-site correlations: The
Electronic correlation effects in Cr2GeC Mn+1AXn-phase 16
electronic structure of La2CuO4and LaCuO3Phys. Rev. B 49 14211-14228.
[28] Petukhov A G, Mazin I I, Chioncel L and Lichtenstein A I 2003 Correlated metals and the
LDA+U method Phys. Rev. B 67 153106-153110.
[29] S¸sioˇglu E, Friedrich C and Bl¨ugel S 2011 Effective Coulomb interaction in transition metals from
constrained random-phase approximation Phys. Rev. B 83 121101(R)-121105(R).
[30] Ramzan M, Leb´egue S and Ahuja R 2012 Electronic and mechanical properties of Cr2GeC with
hybrid functional and correlation effects Solid State Comm. 152 11471149.
[31] Anderson P W 1950 Antiferromagnetism. Theory of Superexchange Interaction Phys. Rev. 79
[32] Singh D 1994 Plane waves, pseudopotentials and the LAPW method, Kluwer Academic, Boston.
[33] Magnuson M, Mattesini M, Bugnet M, Mauchamp V, Cabioch T, Hultman L and Eklund P 2012
in manuscript.
[34] Sherman D M 1985 The electronic-structures of Fe-3+ coordination sites in iron-oxides-
applications to spectra, bonding, and magnetism Phys. Chem. Minerals 12 161-175.
[35] Magnuson M, Wilhelmsson O, Mattesini M, Li S, Ahuja R, Eriksson O, H¨ogberg H, Hultman L,
and Jansson U 2008 Anisotropy in the electronic structure of V2GeC investigated by soft x-ray
emission spectroscopy and first-principles theory Phys. Rev. B 78 035117-035126.
[36] Hettinger J D, Lofland S E, Finkel P, Meehan T, Palma J, Harrel K, Gupta S, Ganguly A,
El-Raghy T, and Barsoum M W 2005 Electrical transport, thermal transport, and elastic
properties of M2AlC (M=Ti, Cr, Nb, and V) Phys. Rev. B 72 115120-115126.
[37] Sanborn B A, Allen P B and Papaconstantopoulus D A 1989 Empirical electron-phonon coupling
constants and anisotropic electrical resistivity in hcp metals Phys. Rev. B 40 6037-6044.
[38] Magnuson M, Mattesini M, Nong N V, Eklund P and Hultman L 2012 Electronic-structure
origin of the anisotropic thermopower of nanolaminated Ti3SiC2determined by polarized x-
ray spectroscopy and Seebeck measurements Phys. Rev. B 85 195134-195142.
[39] Gr¨unberg P, Schreiber R, Pang Y, Brodsky M B and Sowers H 1986 Layered Magnetic Structures:
Evidence for Antiferromagnetic Coupling of Fe Layers across Cr Interlayers Phys. Rev. Lett. 57
[40] Ikeda S, Hayakawa J, Ashizawa Y, Lee Y M, Miura K, Hasegawa H, Tsunoda M, Matsukura F
and Ohno H 2008 Tunnel magnetoresistance of 604% at 300 K by suppression of Ta diffusion
in CoFeB/MgO/CoFeB pseudo-spin-valves annealed at high temperature Appl. Phys. Lett. 93
[41] Dahlqvist M, Alling B, Abrikosov I A and Rosen J 2011 Magnetic nanoscale laminates with
tunable exchange coupling from first principles Phys. Rev. B 84 220403(R)-220408(R).

Supplementary resource (1)

... Experimentally obtained data of element-specific magnetic coupling are therefore important [12][13][14][15][16][17]. Moreover, the nature of the correlation effects of the localized Cr 3d states make the magnetic coupling theoretically complicated [18][19][20][21]. ...
... Previous investigations of Cr 2 GeC include several theoretical studies, where the magnetic coupling in the electronic structure has been a controversial issue [18][19][20][21][22]. Using standard Density Functional Theory (DFT) within the PBE scheme, Zhou et al. [22] found that the ground state at 0 K of Cr 2 GeC is antiferromagnetic (AFM) while the ferromagnetic configuration is a metastable state. ...
... LDA+U (U ef f =2.04 eV) points to a FM ground state and the importance of electron correlation effects. Using the Hubbard-corrected Generalized Gradient Approximation (GGA+U), it has been shown that Cr 2 GeC is a weak AFM material for different exchange-correlation functionals [18]. Cr 2 GeC has similar degenerated magnetic states as Cr 2 AlC that has been predicted to have FM ordering [19,20]. ...
... Experimentally obtained data of element-specific magnetic coupling are therefore important [12][13][14][15][16][17]. Moreover, the nature of the correlation effects of the localized Cr 3d states make the magnetic coupling theoretically complicated [18][19][20][21]. ...
... Previous investigations of Cr 2 GeC include several theoretical studies, where the magnetic coupling in the electronic structure has been a controversial issue [18][19][20][21][22]. Using standard Density Functional Theory (DFT) within the PBE scheme, Zhou et al. [22] found that the ground state at 0 K of Cr 2 GeC is antiferromagnetic (AFM) while the ferromagnetic configuration is a metastable state. ...
... LDA+U (U ef f =2.04 eV) points to a FM ground state and the importance of electron correlation effects. Using the Hubbard-corrected Generalized Gradient Approximation (GGA+U), it has been shown that Cr 2 GeC is a weak AFM material for different exchange-correlation functionals [18]. Cr 2 GeC has similar degenerated magnetic states as Cr 2 AlC that has been predicted to have FM ordering [19,20]. ...
Full-text available
The magnetism in the inherently nanolaminated ternary MAX-phase Cr$_{2}$GeC is investigated by element-selective, polarization and temperature-dependent, soft X-ray absorption spectroscopy and X-ray magnetic circular dichroism. The measurements indicate an antiferro-magnetic Cr-Cr coupling along the $c$-axis of the hexagonal structure modulated by a ferromagnetic ordering in the nanolaminated $ab$-basal planes. The weak chromium magnetic moments are an order of magnitude stronger in the nanolaminated planes than along the vertical axis. Theoretically, a small but notable, non-spin-collinear component explains the existence of a non-perfect spin compensation along the $c$-axis. As shown in this work, this spin distortion generates an overall residual spin moment inside the unit cell resembling that of a ferri-magnet. Due to the different competing magnetic interactions, electron correlations and temperature effects both need to be considered to achieve a correct theoretical description of the Cr$_{2}$GeC magnetic properties.
... It is known that GGA-PBE does not sufficiently describe the electron-electron correlation. The incorporation of electron-electron correlation effect leads to more accurate magnetic properties including magnetic orderings of MAX-phases containing the elements Cr and Mn [41,42]. Therefore, we employed GGA with onsite electron correlation (U eff = U-J, where U and J are onsite Coulomb and exchange interactions) in these calculations for Mn-3d states [43]. ...
Magnetic MAX phases including their surfaces exhibit promising functional properties for magneto-electronic devices and self-monitoring smart coatings. By employing an integrated ab-initio approach, here we investigate a new member of the MAX phase, Mn2AuN. This compound satisfies the chemical, mechanical and dynamical stability criteria, leading to a possibility of its synthesis, and exhibits electronic and elastic anisotropy. The identified ferromagnetic configuration (with high Curie temperature) is the most energetically favorable among the five different spin configurations i.e., non-magnetic, ferromagnetic, and three anti-ferromagnetic. From the coatings perspective, the surface properties of Mn2AuN(001) terminations are investigated considering the four possible surface termination models. Remarkably, the magnetism does not get vanished even after cleaving the bulk unit cell into the (001)-surfaces. While evaluating the surface energetics in different chemical potentials, the N-001 terminated surface comes out to be the most stable termination contrasting with carbide MAX phases and is the only termination that exhibits the magnetic characteristics.
... The need for using DFT + U methods for studying magnetic MAX phases have been debated, see Ref. 30 for further details. The first studies motivate the use of + U to get a better correlation between measured and calculated bulk modulus for Cr 2 AC (A = Al, Ga, Ge) [32][33][34][35][36][37] . We have later shown that a better match can be achieved without any + U but through extended unit cells to describe non-trivial magnetic configurations 38,39 . ...
Full-text available
The atomically laminated Mn2GaC has previously been synthesized as a heteroepitaxial thin film and found to be magnetic with structural changes linked to the magnetic anisotropy. Related theoretical studies only considered bulk conditions and thus neglected the influence from possible strain linked to the choice of substrate. Here we employ first principles calculations considering different exchange–correlation functionals (PBE, PW91, PBEsol, AM05, LDA) and effect from use of + U methods (or not) combined with a magnetic ground-state search using Heisenberg Monte Carlo simulations, to study influence from biaxial in-plane strain and external pressure on the magnetic and crystal structure of Mn2GaC. We find that PBE and PBE + U, with Ueff ≤ 0.25 eV, gives both structural and magnetic properties in quantitative agreement with available experimental data. Our results also indicate that strain related to choice of substrate or applied pressure is a route for accessing different spin configurations, including a ferromagnetic state. Moreover, the easy axis is parallel to the atomic planes and the magnetocrystalline anisotropy energy can be increased through strain engineering by expanding the in-plane lattice parameter a. Altogether, we show that a quantitative description of the structural and magnetic properties of Mn2GaC is possible using PBE, which opens the way for further computational studies of these and related materials.
MXenes (transition metal carbides and nitrides/carbonitrides) are a novel type of two-dimensional (2D) materials that are derived from MAX-phase precursors. Their structural and compositional diversity, outstanding electrical conductivity, and tunable surface properties have made MXenes one of the best and most versatile families of 2D materials, with rapidly growing fields of application. Since their inception, MXenes have attracted enormous research interest in various research fields, including physics, chemistry, materials science, nanotechnology, nanomedicine, and environmental science. This chapter briefly introduces fundamental aspects of MXenes such as MAX-phase precursors, MXene derivatives, and methods of synthesis and processing. Various MXene properties such as electronic, optical, magnetic, and mechanical properties are then briefly discussed. The next section briefly describes the current advances of MXene-based composites in various application fields. Finally, the chapter highlights the current challenges and research prospects for the development of MXene-based advanced composite materials for specific applications.
In this work, we investigate the magnetic stability of Cr2SiC and electronic properties using the linear enhanced plane wave method (FP-LAPW) at full potential in generalized gradient approximation (GGA) and (GGA + U). An anti-ferromagnetic spin order of Cr atoms is shown to be the ground state of this compound. Based on its electron band structures and state density (DOS), Cr2SiC has a metallic character and from the analysis of the projected site and of the moment densities, it follows that the bond is obtained by hybridization of Cr-3d with C -2p and Cr-3d with Si-3p states. Also note that the Cr–C bond is more covalent than Cr–Si. In the ferromagnetic phase (FM), the calculation of the polarized spin indicates that the total magnetic moment of Cr2SiC increases from 1.99 to 2.34 μB. Finally, we expect that the theoretical study of this material will give an idea for experimenters in the future.
The temperature dependence of the electrical conductivity of Pt nanotubes (NTs) with different thicknesses synthesized by a wetting method using an Al2O3 membrane was studied. Pt NTs exhibited circular pores with an average diameter of ∼200 nm. From XRD, the prepared Pt NTs displayed a cubic crystal structure. Pt metal was identified based on the binding energy peak at 71 eV via XPS analysis. Pt NTs with thicknesses of 5 and 12 nm behaved like a semimetal, whereas Pt NTs with thicknesses of 25 and 29 nm showed normal metallic electrical conduction characteristics. This metal-to-semimetal transition was induced as the thickness and grain sizes of the Pt NTs were decreased. The critical metal-to-semimetal transition temperature of Pt NTs with average tube wall thicknesses of ∼5 nm was measured at ∼37 °C. However, the critical temperature could not be measured for NTs with a thickness of 12 nm. It is assumed that the critical temperature would be far below 0 °C. This transition behavior resulted from both a discontinuity in the density of states due to the quantum confinement effect and the increased energy barrier for conduction of electrons accompanied by the increased density of grain boundaries. These results presented here signify a vital step in the direction of realizing high-performance nanoelectronic devices.
Full-text available
The family of 2D materials has expanded quite rapidly, especially with the addition of transition metal carbides and nitrides called MXenes, in the last decade. Since their discovery in 2011, about 30 different MXenes have been synthesized, and the structure and properties of several dozens have been predicted by first-principles approaches. Given the outstanding advances in the MXene field, it is thus appropriate to review the most relevant properties of these MXenes and point out their potential applications. In this article, the structural, transport, magnetic, vibrational, mechanical, and electrochemical properties of MXenes are overviewed. The goal is to illustrate how the chemical versatility in the intrinsic composition and surface terminations combined with the potential addition of a fourth element enable to tune MXenes properties to meet the targeted applications.
We combine predictive ab-initio calculations with experimental verification of bulk materials synthesis for exploration of new and potentially magnetic atomically laminated i-MAX phases. Two such phases are discovered: (Cr2/3Sc1/3)2GaC and (Mn2/3Sc1/3)2GaC, where the latter compound displays a two-fold increase in Mn content compared to previously reported bulk MAX phases. Both new compounds exhibit the characteristic in-plane chemical order of Cr(Mn) and Sc, and crystallize in an orthorhombic structure, space group Cmcm, as confirmed by scanning transmission electron microscopy (STEM). From density functional theory (DFT) calculations of the magnetic ground state, including the electron-interaction parameter U, we suggest an antiferromagnetic ground state, close to degenerate with the ferromagnetic state.
Recent research has highlighted the potential of MAX phase properties including its machinability, good mechanical behaviour, high electric and thermal conductivity as well as good oxidation and erosion resistance to a range of applications including electrical contact coatings, high-temperature heating elements, and barrier- and protection coatings in gas-turbine engine. Successful engineering of MAX phases requires an understanding of how the manufacturing conditions and the deposition parameters influence the composition, crystallographic, electronic structure and morphology. It is, in turn, important to know how these determine the anisotropy of physical, mechanical and technological properties of the material. A focus point of this part of the review is the systemic study of the regular trends in the preparation parameter–structure–properties relationships of MAX phase coatings. In particular, the Cr–Al–C (Y) coating synthesised by the HIPIMS technique will be considered and compared with other synthesised MAX phase bulk ceramics and coatings.
With its extreme accuracy and reasonable computational efficiency, the linearized augmented planewave (LAPW) method has emerged as the standard by which density functional calculations for transition metal and rare-earth containing materials are judged. This volume presents a thorough and self-conta
From a theory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous system of interacting electrons are developed. These methods are exact for systems of slowly varying or high density. For the ground state, they lead to self-consistent equations analogous to the Hartree and Hartree-Fock equations, respectively. In these equations the exchange and correlation portions of the chemical potential of a uniform electron gas appear as additional effective potentials. (The exchange portion of our effective potential differs from that due to Slater by a factor of 23.) Electronic systems at finite temperatures and in magnetic fields are also treated by similar methods. An appendix deals with a further correction for systems with short-wavelength density oscillations.
A method is given for generating sets of special points in the Brillouin zone which provides an efficient means of integrating periodic functions of the wave vector. The integration can be over the entire Brillouin zone or over specified portions thereof. This method also has applications in spectral and density-of-state calculations. The relationships to the Chadi-Cohen and Gilat-Raubenheimer methods are indicated.
State-of-the-art electronic-structure calculations based on the local-density approximation (LDA) to the density functional fail to reproduce the insulating antiferromagnetic ground state in the parent compounds of the high-temperature oxide superconductors. Similar problems have been observed earlier in classical transition-metal oxides such as FeO, CoO, and NiO. In this work we present the method which delivers the correct insulating antiferromagnet ground state in the correlated oxides preserving other properties as well as the efficiency of the standard LDA. The method embeds the relevant (for a given system of electrons) part of the Hubbard Hamiltonian into the Kohn-Sham LDA equation. The resulting Hamiltonian attempts to fix two intrinsic problems of the LDA: the deficiency in forming localized (atomiclike) moments and the lack of discontinuity of the effective one-particle potential when going from occupied to unoccupied states. We present the detailed study of La2CuO4 and LaCuO3. In the case of La2CuO4 the energy gap and the value of the localized magnetic moment in the stable insulating antiferromagnetic solution are in good agreement with experiment. We compare our results with standard local spin density approximation calculation and multiband Hubbard model calculations, as well as with results of spectroscopy: inverse photoemission, valence photoemission, and x-ray absorption at the K edge of oxygen. In the case of LaCuO3 such an extensive comparison is limited due to the limited data available for this compound. We discuss, however, the electric and magnetic properties and the insulator-metal-insulator transitions upon increase of oxygen deficiency.
Mn+1AXn compounds have gathered huge momentum because of its exciting properties. In this paper we report the synthesis of ternary layered ceramic Cr2GeC, a 211 Mn+1AXn compound by hot-pressing. Scanning electron microscopy and X-ray diffraction have been employed to characterize the new synthesized phase. High-pressure compressibility of Cr2GeC were measured using diamond anvil cell and synchrotron radiation at room temperature up to 48GPa. No phase transformation was observed in the experimental pressure range. The bulk modulus of Cr2GeC calculated using the Birch–Murnaghan equation of state is 169±3GPa, with K′=3.05±0.15.
Density functional calculations have been performed to obtain the structural and electronic properties as well as elastic constants of cubic SrCl2. The compound is found to be an insulator with an indirect band gap of 5.18 eV. The band gap initially increases and then decreases as a function of pressure. Calculated equilibrium properties such as lattice constant, bulk modulus and elastic constants are in good agreement with experimental studies. The calculated C44 value agrees well with experiment only when the chlorine atoms are allowed to displace under strain, indicating the importance of inner strain relaxation. From the elastic constants theoretical values of the Young’s modulus, shear modulus, Poisson’s ratio, sound velocities, and Debye temperature of SrCl2 are obtained.
This paper deals with the ground state of an interacting electron gas in an external potential v(r). It is proved that there exists a universal functional of the density, Fn(r), independent of v(r), such that the expression Ev(r)n(r)dr+Fn(r) has as its minimum value the correct ground-state energy associated with v(r). The functional Fn(r) is then discussed for two situations: (1) n(r)=n0+n(r), n/n01, and (2) n(r)= (r/r0) with arbitrary and r0. In both cases F can be expressed entirely in terms of the correlation energy and linear and higher order electronic polarizabilities of a uniform electron gas. This approach also sheds some light on generalized Thomas-Fermi methods and their limitations. Some new extensions of these methods are presented.
A method is given for generating sets of special points in the Brillouin zone which provides an efficient means of integrating periodic functions of the wave vector. The integration can be over the entire Brillouin zone or over specified portions thereof. This method also has applications in spectral and density-of-state calculations. The relationships to the Chadi-Cohen and Gilat-Raubenheimer methods are indicated.