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Magnetic anisotropy in Cr2GeC investigated by X-ray magnetic circular dichroism and ab initio calculations

J. Magn. Magn. Mater. 501, 166470 (2020)
Magnetic anisotropy in Cr2GeC investigated by X-ray magnetic circular dichroism
and ab initio calculations
Martin Magnuson1and Maurizio Mattesini2,3
1Department of Physics, Chemistry and Biology, IFM,
Thin Film Physics Division, Link¨oping University, SE-58183 Link¨oping, Sweden
2Department of Earth’s Physics and Astrophysics,
Complutense University of Madrid, Madrid, E-28040, Spain and
3Instituto de Geociencias (CSIC-UCM), Facultad de CC. F´ısicas, E-28040 Madrid, Spain
(Dated: January 25, 2020)
The magnetism in the inherently nanolaminated ternary MAX-phase Cr2GeC 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 Cr2GeC magnetic properties.
PACS numbers:
Ternary nanolaminated carbides and nitrides, known
as Mn+1AXn-phases, is the sub ject of intense research
[1–4]. Three related crystal structures are classified by
stoichiometry as 211 (n=1), 312 (n=2) and 413 (n=3)
phases, where the letter M denotes an early transition
metal, A is an element in the groups III-V and X is car-
bon or nitrogen. The Mn+1AXn-phases exhibit a techno-
logically important combination of metallic and ceramic
properties [5], that are related to the internal nanolam-
inated crystal structure, the choice of the three con-
stituent elements, as well as the electronic structure and
the chemical bonding between the intercalated atomic
Macroscopic magnetism in MAX-phases was initially
predicted by Luo et al. in hypothetical Fe-containing 211
MAX-phases [6]. Later, more accurate calculations in-
cluding all competing phases have shown that this phase
is thermodynamically unstable [7]. Magnetic macro-
scopic response has also been observed in Cr2xMnxGaC
phases [8] that could be related to strong magnetic per-
ovskite impurities. Recently, a magnetic macroscopic
response has been observed in Mn-doped phase-pure
(Cr,Mn)2GeC at room temperature [9].
Although pure Cr2GeC is macroscopically non-
magnetic at room temperature [10], the Cr contribution
in Cr2GeC has an anti-ferromagnetic (AFM), ferromag-
netic (FM), ferrimagnetic (FiM), paramagnetic (PM) or
non-magnetic (NM) order that depend on the tempera-
ture. Since the magnetic ordering could affect other prop-
erties of the MAX-phases, such as the thermal expan-
sion and the bulk modulus, it is important to know the
atomic magnetic exchange interactions in detail as non-
magnetic calculations yield insufficient agreement with
experiment [10, 11]. Experimentally obtained data of
element-specific magnetic coupling are therefore impor-
tant [12–17]. Moreover, the nature of the correlation ef-
fects of the localized Cr 3dstates make the magnetic
coupling theoretically complicated [18–21].
Previous investigations of Cr2GeC include several the-
oretical studies, where the magnetic coupling in the elec-
tronic structure has been a controversial issue [18–22].
Using standard Density Functional Theory (DFT) within
the PBE scheme, Zhou et al. [22] found that the ground
state at 0 K of Cr2GeC is antiferromagnetic (AFM) while
the ferromagnetic configuration is a metastable state. In
the AFM case, a significant band spin-split of 2 eV
was also predicted at the Fermi level (EF) [22].
However, the Local Density Approximation (LDA)
yields very close (degenerate) energy difference between
FM and AFM ordering in Cr2GeC. LDA+U (Ueff =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 Cr2GeC is a weak AFM material for
different exchange-correlation functionals [18]. Cr2GeC
has similar degenerated magnetic states as Cr2AlC that
has been predicted to have FM ordering [19, 20]. How-
ever, that is opposite to the AFM coupling, predicted by
Zhou et al. [22]. Experimentally, Cr2AlC and Cr2GeC
have been found to be FM at very low temperature (2.2
K) and very high external magnetic fields (10 T) [21].
In this paper, we investigate the magnetism in a high-
quality single-crystal Cr2GeC (0001) thin film sample,
using element-specific, polarization-, and temperature-
dependent magnetic circular dichroism (XMCD)[23]. By
changing the circular polarization from right to left-
handed, and the direction of the applied magnetic field,
the magnetic Cr moments are probed. We provide direct
evidence of predominant ferrimagnetic ordering in the
electronic structure supported by ab initio band struc-
ture calculations. The measurements and calculations in-
dicate a competition between a FM ordering in the basal
ab-plane and an AFM spin distribution along the c-axis
in the hexagonal crystal lattice.
A. Cr2GeC (0001) thin film synthesis
Phase-pure Cr2GeC (0001) thin films were deposited
at 800 oC by dc magnetron sputtering from elemental Cr
(99.95% purity), Ge (99.999%), and graphite (99.999%)
targets in an argon discharge pressure of 4 mTorr on MgO
(111) substrates. The substrates were ultrasonically de-
greased in acetone followed by ethanol for five minutes
and then annealed in vacuum in the deposition chamber
for 1 h at 800 oC prior to deposition. The Ar sputter-
ing gas had a pressure of 0.3 Pa, and the base pressure
was below 5 ×106Pa. The targets (75 mm diameter)
were arranged on a confocal magnetron cluster and lo-
cated at a distance of 18 cm from the substrate, which
was mounted on a rotating sample holder. The Cr and
C targets were run in DC power-control mode, while the
Ge target was RF sputtered. The films were deposited to
a thickness of 190 nm, corresponding to a deposition
time of 30 min. Details on the synthesis process are given
in Refs. [10, 24].
B. X-ray magnetic circular dichroism
The XAS and XMCD spectra were measured in total
electron yield (TEY) mode with 0.15 eV energy reso-
lution at 45oincidence angle on beamline I1011 at the
MAX IV Laboratory [25]. Left and right handed circu-
larly polarized X-rays were provided in the 3rd harmonic
with an elliptically polarizing undulator (EPU) and a col-
limated plane grating monochromator (cPGM) that en-
abled element specific characterization of magnetic prop-
erties of both ferromagnetic and anti-ferromagnetic ma-
terials. Constant magnetic fields of 0.2 and 0.4 T were ap-
plied both in-plane and out-of-plane on the sample with
an octupole magnet both at room temperature and at 44
C. Computational details
1. Electronic ground state and magnetic spin coupling
The electronic structure and magnetic ordering in
Cr2GeC was studied with the wien2k code [30] employ-
ing the density-functional [31, 32] augmented plane wave
plus local orbital (APW+lo) computational scheme. The
Kohn-Sham equations were solved by means of the Wu-
Cohen GGA (GGA W C) [33, 34] for the exchange-
correlation (xc) potential. The well-known shortcom-
ing of the DFT description of the electron-correlation
effects was explicitly treated within a phenomenologi-
cal many-body Hamiltonian, the Hubbard model [39],
where the effective on-site Coulomb interaction (Uef f )
has been calculated [18] for the Cr atom in the hexag-
onal Cr2GeC structure by using the constrained DFT
formalism method [37]. A plane-wave expansion with
RMT ·Kmax =10 was used in the interstitial region, while
the potential and the charge density were Fourier ex-
panded up to Gmax=12. The modified tetrahedron
method [35] was applied to integrate inside the Bril-
louin zone, and a k-point sampling with a 35×35×7
Monkhorst-Pack [36] mesh in the full BZ (corresponding
to 786 irreducible k-points) was considered satisfactory
for the hexagonal Cr2GeC system. Relativistic correc-
tions (e.g., spin-orbit coupling) in the electronic structure
calculation were included in a second-variational proce-
dure using scalar relativistic wave functions [38]. The
magnetic ground-state properties were studied after hav-
ing achieved the relaxed unit cell parameters (a=2.981 ˚
and b=12.044 ˚
The computed GGAW C +Umagnetic spin ordering
in Cr2GeC is generally AFM along the out-of-plane c-
axis with residuals of FM within the ab basal plane. The
details of this magnetic spin coupling were presented in
an earlier work [18], where a Ge-mediated super-exchange
magnetic structure interact between the piled CrC layers
(Fig. 1, bottom panel).
To test the robustness of this magnetic ground-state
scenario, we further applied different theoretical schemes.
This included a pseudopotential calculational method (q-
ESPRESSO [26]) based on a hybrid exchange-correlation
functional carried out on a 2×2×1 supercell (B3LYP[40]),
coupled with a non spin collinear treatment of mag-
netism. The results showed that there is still a general
AFM spin distribution, although a non-vanishing non
collinear magnetic component is evidenced both along
the a- (0.02 µB/cell) and c-axis (0.05 µB/cell).
2. Theoretical XMCD spectra
The absorption cross-section (µ) for incident X-rays
was theoretically determined according to Fermi’s golden
rule by using the one-particle framework and the dipole
approximation by the probability of an electron to be
excited from a core state to a final valence state. Our
dichroic and total absorption spectra were computed
within the wien2k code [30] through the linear com-
binations of the absorption cross-sections (µ+-µand
µ++µ), by using the LAPW basis set formalism of Par-
dini et al. [43]. Convergent theoretical XMCD spectra
were achieved within the same GGA W C +Umethod-
FIG. 1: (Color online) Top panel: Illustration of the hexago-
nal crystal structure of Cr2GeC with atoms of different spins.
The CrC slabs are interleaved by pure layers of Ge. Bottom
panel: Schematic representation of the CrC polyhedra of the
laminate plane.
ology presented earlier in ref. [18]. Reference XMCD
spectra were also computed within the Green’s formal-
ism (i.e., multiple scattering) on a muffin-tin potential
[41] implemented in the FDMNES package [42]. Well-
converged XMCD spectra were obtained by using a clus-
ter’s radius of 8 ˚
A. Such a less precise but computation-
ally faster methodology was systematically applied on a
variety of AFM and FM spin configurations in order to
get a deeper insight on the type of magnetic coupling in
Figure 1 shows the unit cell of Cr2GeC with a hexag-
onal crystal structure containing CrC slabs interleaved
by atomic layers of Ge atoms. Calculated magnetic mo-
ments are indicated by the arrows where the lengths cor-
respond to the relative size of the moments and the di-
rection correspond to the spin orientation obtained from
the results of GGAW C +Ucalculations [18]. The unit
cell shows alternating Cr-C slabs with an in-plane FM
Cr-Cr coupling where the Cr-containing basal ab-planes
Intensity (arb. units)
Photon Energy (eV)
FIG. 2: (Color online) Experimental Cr 2pX-ray absorption
spectra with left and right hand polarization in a magnetic
field of 0.4 T at room temperature. Bottom: XMCD differ-
ence spectra measured with the magnetic field in the basal
ab-plane and out-of-plane along the c-axis.
are piled up into a nanolaminate along the c-axis with an
AFM spin ordering.
Figure 2 (top) shows Cr 2p3/2,1/2X-ray absorption
spectra (XAS) measured at room temperature with left-
and right-hand polarization in a constant magnetic field
of 0.4 T. XMCD difference spectra are shown below for
in-plane and out-of-plane alignment of the magnetic field.
Although the general shapes of the XAS spectra appear
similar for the two orientations of the magnetic field, the
XMCD spectra exhibit significant differences. The ob-
served sub-peak splitting of the 2p3/2and 2p1/2peaks in
XAS is due to the t2gand egsplitting that are salient fea-
tures for the metallic and bonding orbitals, respectively.
The t2g(dxy,dxz ,dyz) orbitals have the lowest energy
while the covalent eg(d3z2
y2) orbitals are lo-
cated at 1.2 eV higher energy in the hexagonal symmetry
surrounding the Cr sites. The t2g-egsplitting at the 2p3/2
and 2p1/2peaks mainly affects the shape of the XAS but
not necessary the shape of the XMCD signal. The lat-
ter depends primary on the difference between left- and
right-hand polarized edges. The 2p3/2/2p1/2branching
ratio of 1.27 is smaller than the statistical ratio of 2:1
and reflects the amount of conductivity and the exchange
and mixed terms between the core-states [27].
The observed intensity oscillations of the XMCD signal
at the 2p3/2-edge of Cr2GeC with negative intensity at
574 eV followed by positive intensity at 579 eV indi-
cate that the magnetism in Cr2GeC consists of a mixture
of FM and AFM coupling of the spins of the Cr 3delec-
Photon Energy (eV)
Intensity (arb. units)
FIG. 3: (Color online) In-plane analysis of experimental Cr 2p
x-ray absorption spectra with left and right hand polarization
in a magnetic field of 0.2 T at 44 K. Bottom: XMCD difference
spectra measured with the magnetic field in the basal ab-
plane. The A1, A2and the A3represent the three integrated
areas needed in the sum-rule analysis.
trons. In the basal ab-plane, the XMCD signal of Cr
show similarities to induced magnetism of Cr 3dstates
in Fe/Cr multilayers [45, 46]. However, along the c-axis,
the spectral shape exhibits a more randomly ordered dis-
tribution of negative and positive intensity that is a sign
of a predominant AFM ordering.
Figure 3 shows XAS and XMCD data measured at
44 K with an external field of 0.2 T in the basal ab-
plane. By applying LHe temperatures, the spectral fea-
tures become slightly sharper than in the room temper-
ature measurements in Fig. 2. In order to quantitatively
estimate the magnetic moments in the basal ab-plane,
the spectra have been analyzed using the sum rules that
relate the integrated XMCD intensities to the element-
specific properties of the 3dorbital angular momentum
morb=-4×nh×A2/3A3and the 3dspin angular momen-
tum mspin=-nh×(6A1-4A2)/A3as originally suggested
by Thole et al. [28]. In this way, the spin and orbital
magnetic moments were determined from the integrated
areas (denoted A1, A2and the A3in Figs. 3 and 4) of the
XMCD signal as experimentally confirmed for Fe and Co
by Chen et al. [29]. As observed in the bottom panel of
Fig. 3, the integrated XMCD curve crosses the zero line
and a possible mixture of FM/AFM is indicated by the
integrated XMCD signal (u++u) that is first negative,
crosses zero and then becomes positive.
From the GGA W C +Ucalculations, we obtained
a total valence charge of 5.55 electrons within the Cr
Intensity (arb. units)
Photon Energy (eV)
FIG. 4: (Color online) Out-of-plane analysis of experimental
Cr 2px-ray absorption spectra with left and right hand polar-
ization in a magnetic field of 0.2 T at 44 K. Bottom: XMCD
difference spectra measured with the magnetic field out-of-
plane along the c-axis. The A1, A2and the A3represent the
three integrated areas needed in the sum-rule analysis.
atomic spheres (n3d=1.60, n3p=2.93 and nspin=1.03,
nh=4.45). The A1, A2and A3notations in Fig. 3 rep-
resent the three integrated areas needed in the sum-rule
analysis, where A3is the integral for the whole 2p3/2,1/2
range that can be precisely determined. The separate A1
and A2integrals of the 2p3/2and 2p1/2ranges depend
on the cutoff energy and the potential orbital overlap be-
tween the edges. The orbital and spin moments were
then determined by applying nh=4.45 and the A1, A2
and the A3values indicated in Fig. 3 and, after correct-
ing for the geometry of the beam and the polarization
rate by 1/cos(45o)/0.95 as, ml=(5.3±0.5)·103µB/atom
and ms=(6.4±1.0)·103µB/atom.
Figure 4 shows XAS and XMCD data measured at 44
K with an external field of 0.2 T applied along the c-axis
(out-of-plane). By applying the sum rules in the same
way as in Fig. 3, we find ml=(5.4±0.5)·104µB/atom
and ms=(3.6±0.7)·104µB/atom. Comparing the
XMCD results in Figs. 3 and 4, the orbital and spin
moments are an order of magnitude smaller out-of-plane
than in-plane. The mland msvalues are similar in-plane
but for out-of-plane, the spin moment msis smaller than
the orbital moment ml.
Figure 5 shows the calculated atom-specific XAS and
XMCD spectra using GGA W C +Uwith relativis-
tic corrections for the four different Cr atoms in the
Cr2GeC unit cell. Note that in this type of calculations,
even though intensities and spectral shapes of the XMCD
Photon Energy (eV)
Intensity (arb. units)
FIG. 5: (Color online) Calculated Cr 2pXAS and XMCD
spectra for the four Cr atoms in Cr2GeC with a spin-collinear
treatment. The shape and amplitude of the curves depends
on the sign and strengths of the Cr moments.
spectra are rather similar, the signs of the magnetic mo-
ments are different. The largest moments were found for
Cr1and Cr4but with opposite signs. The moments of
Cr2and Cr3are 10 times smaller than those of Cr1
and Cr4also with different signs. Such a magnetic cou-
pling scenario, where small spin moment magnitudes are
involved in a sort of entangled competition between FM
and AFM coupling complicates the physical interpreta-
tion of the measured spectra. However, by looking at
the sum of the XMCD spectra of Cr2and Cr4(i.e., the
in-plane atoms coupled FM) we find that the in-plane
XMCD is a non-zero signal, while the sum of Cr1+ Cr2
and Cr3+ Cr4give almost a null XMCD (i.e., an AFM
coupling along the c-axis). Thus, the signal originating
from the sum of Cr2and Cr4will be used (Fig. 6, orange
line) to compare with the experimental XMCD spectrum.
In order to further understand the nature of magnetism
in Cr2GeC, reference XMCD model spectra for frozen
FM and AFM configurations were calculated within the
faster Green’s formalism.
Figure 6 (top panel) shows the experimental XMCD
spectra of Figs. 3 and 4 in comparison to model spectra
with the same spin coupling distribution in the Cr2GeC
as shown in Fig. 1 using both the GGAW C +Uand the
full-multi-scattering methods. Although the Cr2+Cr4
spectral shape is not in perfect quantitative agreement
with the measured spectrum, we note that qualitatively,
this calculation catch up the positive part of the high
Intensity (arb. units)
Energy (eV)
Wien2k (Cr
FIG. 6: (Color online) The upper panel (a) shows the exper-
imental Cr 2pXMCD spectra of Figs. 3 and 4 in comparison
to model spectra the with same FM/AFM configuration as
shown in Fig. 1. The green curve use multi-scattering for-
malism, while the orange curve refers to the Cr2+Cr4spectral
shape using the GGA W C +Umethod. Bottom panel (b)
shows calculated reference XMCD spectra for a pure FM spin
coupling in both Cr2GeC and CrO2.
energy XMCD signals. Fig. 6 (bottom panel), shows
the CrO2reference spectrum in comparison to that of a
pure FM scenario in Cr2GeC. Although the fitting is not
perfect, the multi-scattering method (green line in Fig.
6) shows that a scenario with FM coupling in the basal
ab-plane and AFM along the c-axis of Cr2GeC (i.e., that
of Fig. 1), is clearly in much better agreement than a
pure FM coupling in both directions. In particular, the
first and second peaks around 573.2 eV and 574 eV
in pure FM coupling have opposite signs in comparison
to the experimental results as well as for the calculated
orthogonal FM/AFM coupling in Cr2GeC. This is also
the case for a possible CrO2contribution on the surface,
as the shape of the XMCD spectrum is significantly dif-
ferent from the experimental results and thus unlikely
(bottom panel of Fig. 6). To further test this scenario,
several other 2×2×2 supercells were used to test different
AFM/FM spin couplings within the same Green’s for-
malism and found that only the spin ordering shown in
Fig. 1 provides sufficient agreement with the experimen-
tal data. Furthermore, we note that the full-multi scat-
tering method is generally in better agreement with mea-
surements at the low-energy part of the spectrum, while
at higher energies, the GGA W C +Umethod performs
better. This behavior is in line with the fact that the
more computational demanding GGA W C +Ucalcu-
lations were performed without considering the core-hole
effect, while in the multi-scattering scheme, the Z+1
approximation was used to take this interaction into ac-
count. On the other hand, the Green’s formalism shows a
certain shortcoming description for the energy position of
positive peaks located at higher energies. This behavior
can be addressed to the well-known scissor operator ef-
fect [53] and to the calculated firstprinciples values of
the spin-orbit splitting that are slightly underestimated
for the early transition metals [4].
Table I shows the calculated self-consistent magnetic
moments for Cr2GeC (in µB/cell), and their orbital (ml)
and spin (ms) contributions using GGA W C +Uwith
relativistic corrections. The average spin moments are
<ml>=7.75·105µBand <ms>=1.23·104µB. In gen-
eral, the computed results with small magnetic moments
are in agreement with the experimental AFM coupling al-
though the calculated msvalue is somewhat smaller than
the experimental values obtained from the spin sum rule.
Note that for weak AFM materials such as Cr2GeC, the
orbital moments mlare often more important than the
spin moments msin XMCD. The values of the pairs of or-
bital moments basically have the same magnitudes, and
only a change in sign is observed for Cr1and Cr4as well
as for Cr2and Cr3(Table I). Experimentally, we further
observe that the average spin moments are an order of
magnitude larger in-plane compared to out-of-plane.
In order to shed more light onto the difference be-
tween in-plane vs. out-of-plane spin distributions, we also
carried out hybrid-functional calculations using a non-
collinear spin-treatment. Table II is the result of a differ-
ent calculation scheme based on B3LYP+non-collinear
treatment that yield different results in terms of total
magnetic moments. The results obtained by using the
hybrid-functional methodology shows that each magnetic
moment of Cr, Ge and C atoms in Cr2GeC can be decom-
posed along the laminated ab-plane (mx,my) and along
the c-axis (mz). The ma jor contributions to the spin ap-
pear along the c-axis (mz) of Cr while the small magnetic
moments of C and Ge elements are due to induced po-
larization effects. The experimental observations are in
good agreement with the results achieved within the non-
collinear spin treatment of the Cr2GeC system (Table
II), where the obtained in-plane net magnetic moment
(1.66·102µB) is 2.6 times larger than the out-of-plane
(6.50·103µB). Although a common AFM ordering is
predicted along the c-axis in both cases, the calculations
indicate that electronic correlation effects are critical in
describing the correct magnetic spin ordering in Cr2GeC
that will be further discussed below.
In the XMCD experiments shown in Figs. 3 and 4, the
net AFM coupling observed by the average Cr atoms in-
and out-of-plane is a result of the weighted coupling from
the Cr-Cr, Cr-C and Cr-Ge interactions. Thus, the net
magnetic Cr coupling not only depends on the magni-
tudes and distances between the FM coupled Cr atoms,
but also, on the AFM coupled C atoms that are closer to
the Cr as shown in Fig. 1.
For the quantitative determinations of the average
magnetic moments of Cr, the limitations of the integral
sum rules should be considered [54]. In particular, the
spin moment is sensitive to the number of 3dholes and
the Cr L3/L2branching ratio. This depends on the Cr
3d-band filling, the amount of overlap of the Cr 2p3/2,1/2
edges and the Cr 2p3/2/2p1/2branching ratio. This is
more severe for the earliest transition metals Ti and V
where the 2pspin-orbit splitting is smaller and imply
more mixing of the 2p3/2and 2p1/2core states [15]. More-
over, the spin sum rule <ms>that involves the difference
between the integrated areas at the 2p3/2and 2p1/2edges
is most sensitive to a change of the 2p3/2/2p1/2branch-
ing ratio. Therefore, the <ms>spin moments in the sum
rules are often underestimated while the orbital moments
<ml>usually yield reasonable values.
XMCD measurements on MAX-phases with hexagonal
symmetry are thus complex, not only requiring sample
phase purity but also avoiding surface oxides that quickly
form on fresh surfaces in air. Although Cr2GeC is oxida-
tion resistant and the sample was properly cleaned and
ultrasonically degreased in acetone followed by ethanol
before being measured in ultrahigh vacuum, the forma-
tion of a thin natural nonmagnetic Cr2O3or ferromag-
netic CrO2oxides on the surface cannot be completely
excluded. In addition, the very weak magnetic signals re-
quire extensive measurement times in stable X-ray beams
and magnetic fields. Furthermore, the nature of the cor-
relation effects of localized Cr 3dstates and the com-
peting balance between FM and AFM ordering with a
very small energy difference, makes modeling of the mag-
netic coupling in Cr2GeC rather complicated. If the
non-collinearity of the spins is fully taken into account
in Cr2GeC, the presence of two competing Cr-Cr mag-
netic mechanisms (FM in plane and AFM along the c-
axis), leads to stabilization of a non-perfectly compen-
sated AFM material. Indeed, by employing DFT+U full-
potential calculations with different exchange-correlation
functionals, Mattesini et al. [18] already showed that
Cr2GeC is a weak AFM material. For this purpose,
the Cr2GeC phase can be used as a model test case for
other similar magnetic MAX phases such as, for example,
Cr2AlC [14, 47], and V2GeC [48].
Although the Cr 3dstates theoretically exhibit a topo-
logically FM coupling within the ab-planes, the net mag-
netic moment of Cr2GeC in a unit cell is close to zero
when these planes piles-up anti-ferro-magnetically along
the c-axis. Comparing the results shown in Tables I and
II, they are both in general agreement with an overall
AFM ordering and provide the same magnetic scenario
found experimentally through the XMCD spectra. The
difference is merely on the magnitude of the localized
TABLE I: Calculated (GGA W C +Uwith relativistic corrections) magnetic moments (in µB/cell), and their orbital (ml) and
spin (ms) contributions.
Atom spin mlms
Cr10.01529 0.47522 0.01404
Cr2+0.00177 0.47562 +0.00181
Cr30.00175 +0.47580 0.00238
Cr4+0.01644 +0.47535 +0.01510
TABLE II: Calculated (B3LYP+non-spin collinear) in-plane vector components (mx,my) and out-of-plane (mz) magnetic moments
(in µB/cell) of the Cr, Ge and C atoms of Cr2GeC.
Atom mxmymz
C11.04·1030.0 +0.61
mtot +1.66·1020 -6.50·103
magnetic moments of Cr. Although the hybrid B3LYP
functional coupled to a non-spin collinear treatment pro-
vides Cr magnetic moments that are larger than what
is found within the GGA W C +U[18], both meth-
ods point to the same vertical AFM spin ordering for the
Cr2GeC unit cell.
In addition to the AFM coupling between Cr atoms
of different ab-basal planes, as discussed above, there is,
a small but non-negligible spin-collinear component that
affects the magnetic properties. The non-spin collinear
component influences the stacking of ferromagnetic lay-
ers with spins inside the layers that are not perfectly
vertically aligned, resulting in a non-compensated anti-
ferromagnetic coupling along the c-axis. Thus, it is the
presence of a weak spin-wave effect that makes the entire
system a magnetically strained AFM material, where the
final macroscopic magnetic properties depend on small
changes in the synthesis conditions i.e., temperature and
sputtering flux. We suspect that this is one of the reasons
why the experimental characterization of the magnetic
properties of Cr2GeC has been so far rather controver-
sial [18–22].
The competition between FM and AFM coupling is re-
ferred to as antiferromagnetic spin frustration [11]. We
anticipate that topological constraints from the hexago-
nal crystal lattice should be avoided in order to stabilize
a FM coupled pure MAX-phase. One solution to over-
come this is to break the ordering symmetry and achieve
configuration disorder by alloying and replacing part of
the metal atoms with other magnetic elements such as
Fe and Mn [11]. The observed low Curie temperature for
both Cr2GeC (0 K) [52] and Cr2AlC (73 K) [21], suggest
that magnetic ordering should only be significant below
Tc. However, as shown for (Crx,V1x)2AlC phases in
ref. [11], this is not the case as NM calculations yield
poor results in comparison to experiments. Nonetheless,
it is reasonable to expect that for Cr2GeC the observed
net Cr FM moments are very small at room temperature
We further believe that the relatively small spin mag-
nitudes of the C atoms (C1and C2) shown in Table
II stabilize the ferromagnetic coupling of the Cr atoms
within each layer, while the individual spins of the Ge
atoms (Ge1and Ge2) are essential for establishing a
Ge-mediated super-exchange coupling between the ver-
tically piled layers. This interpretation is consistent with
findings in other two-dimensional materials e.g., metal-
coordinated networks and ferrites [23, 44]. Considering
all possibilities of spin distortions modulated by a 3D
spin-wave distribution, the unit cell is thus not a per-
fect anti-ferromagnet with a null total spin but, instead
exhibits residual magnetic components both along the
ab-basal plane and along the c-axis with a total magne-
tization of 0.02 and 0.05 µB/cell, respectively. Indeed,
in recent GGA W C +Ucalculations of Cr2GeC, Mat-
tesini et al. [18] and Magnuson et al. [10] pointed out
that the magnetic moments on the Cr atoms, although in-
plane FM coupled, are small and largely cancel each other
along the c-axis. The latter vertical coupling could lead
to a perfect AFM ordering or to a ferrimagnetic ground-
state if a non-spin collinear effect is taken into account.
The Heyd-Scuseria-Ernzerh (HSE06) hybrid functional
formalism [49] has been employed for a variety of differ-
ent spin configurations and also provides an AFM ground
state ordering [20, 50]. Thus, in addition to the exist-
ing controversial theoretical interpretations about what
type of magnetic coupling dominates (i.e., in-plane or
out-of-plane), the evident disagreement between theory
(AFM) and experiment (FM) in determining the lower
energy magnetic ordering in Cr2GeC calls for the need
of further experimental and theoretical efforts [18, 20] to
disentangle the temperature dependence and the effect of
For a correct description of magnetic states and cor-
relation effects in MAX-phases, calculations needs to go
beyond the rigid band model. In further studies of poten-
tial magnetic properties in MAX-phases, both different
competing magnetic interactions and temperature effects
should be included and carefully analyzed. The t2g-eg
branching ratio can also be improved by many-body per-
turbation theory by solving the Bethe-Salpether equa-
tion (BSE) implemented in WIEN2kbse [27]. Phonons
are also known to play a role in many of the physi-
cal properties of condensed matter, such as thermal and
electrical conductivity and magnetism [51]. Besides the
phonon-induced electro-structural effects featured earlier
in Cr2GeC [10], electron-phonon interactions can further
influence the competition between AFM and FM cou-
pling in Cr-based MAX-phases. In this sense, the Ge
atoms in Cr2GeC move preferentially within the ab-basal
plane, while Cr and C have preference for the c-axis di-
rection. As shown in previous phonon calculations [10], a
x-, y-displacement of 0.097 ˚
A and a larger z-displacement
of 0.114 ˚
A was found for the Cr atoms. Rapidly moving
atoms is known to change the electronic DOS at EF[10]
and should also affect the magnetic ordering [51]. This
implies that phonons play an important role especially in
strained magnetic systems. Specifically, the movement of
Cr atoms can in different ways affect the spin ordering
along the c-axis vs. the in-plane ordering. It can be ex-
pected that at room temperature experiments where the
phonon amplitudes are larger than at 0 K, the detection
of the in-plane FM coupling becomes more challenging.
On the contrary, the AFM spin coupling along the c-axis
should be less affected by thermally dependent atomic
This work provides a step forward toward understand-
ing the controversial magnetism in an important class
of nanolaminated materials. X-ray magnetic circular
dichroism have been applied to investigate the com-
plicated magnetic coupling mechanism in the Cr2GeC
MAX-phase. The measurements exhibit a predominantly
antiferromagnetic coupling between the Cr layers along
the c-axis that is influenced by a ferromagnetic contribu-
tion in the nanolaminated ab-basal planes. Experimen-
tally, we also find that the net magnetic moments along
the ab-basal plane is ten times larger than along the c-
axis. We showed that this results in an overall residual
spin moment that resembles that of a ferrimagnet. Ab
initio calculations further confirm that Cr2GeC is a mag-
netically strained system that exhibits small but different
resultant magnetic components along the ab-basal plane
and along the c-axis.
We thank the staff at MAX IV Laboratory for ex-
perimental support. We thank P. Eklund, M. Bugnet
and V. Mauchamp for providing the samples and M.
Jaouen for discussions. This work was financially sup-
ported by the Swedish Research Council, Linnaeus Grant
LiLi-NFM and the the Swedish Foundation for Strate-
gic Research (SSF). M. Magnuson acknowledges finan-
cial support from the Swedish Energy Research (no.
43606-1) and the Carl Tryggers Foundation (CTS16:303,
CTS14:310). M. Mattesini acknowledges financial sup-
port by the Spanish Ministry of Economy and Competi-
tiveness (CGL2013-41860-P and CGL2017-86070-R).
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... 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 stability of (Crx,V1−x)2AlC MAX phases, materials of interest for a variety of magnetic as well as high-temperature applications, has been studied using density-functional-theory first-principles calculations. The enthalpy of mixing predicts these alloys to be unstable towards unmixing at 0 K. The calculations also predict, however, that these phases would be thermally stabilized by configurational entropy at temperatures well below the values used for synthesis. The temperature Ts below which they become unstable is found to be quite sensitive to the presence of magnetic moments on Cr ions, as well as to the material's magnetic order, in addition to chemical order and composition. Allowing for magnetism, the value of Ts for (Cr0.5,V0.5)2AlC with chemically disordered Cr and V atoms is estimated to be between 516 and 645 K depending on the level of theory, while if constrained to spin paired, Ts drops to ∼142 K. Antiferromagnetic spin arrangements are found to be favored. The combination of antiferromagnetic frustration and configurational disorder should give rise to interesting spin textures at low temperatures.
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
In this comprehensive yet compact monograph, Michel W. Barsoum, one of the pioneers in the field and the leading figure in MAX phase research, summarizes and explains, from both an experimental and a theoretical viewpoint, all the features that are necessary to understand and apply these new materials. The book covers elastic, electrical, thermal, chemical and mechanical properties in different temperature regimes. By bringing together, in a unifi ed, self-contained manner, all the information on MAX phases hitherto only found scattered in the journal literature, this one-stop resource offers researchers and developers alike an insight into these fascinating materials.
This is a critical review of MAX-phase carbides and nitrides from an electronic-structure and chemical bonding perspective. This large group of nanolaminated materials is of great scientific and technological interest and exhibits a combination of metallic and ceramic features. These properties are related to the special crystal structure and bonding characteristics with alternating strong MC bonds in high-density MC slabs, and relatively weak MA bonds between the slabs. Here, we review the trend and relationship between the chemical bonding, conductivity, elastic and magnetic properties of the MAX phases in comparison to the parent binary MX compounds with the underlying electronic structure probed by polarized X-ray spectroscopy. Spectroscopic studies constitute important tests of the results of state-of-the-art electronic structure density functional theory that is extensively discussed and are generally consistent. By replacing the elements on the M, A, or X-sites in the crystal structure, the corresponding changes in the conductivity, elasticity, magnetism and other material properties make it possible to tailor the characteristics of this class of materials by controlling the strengths of their chemical bonds.
The anisotropy in the electronic structure of the inherently nanolaminated ternary phase Cr$_{2}$GeC is investigated by bulk-sensitive and element selective soft x-ray absorption/emission spectroscopy. The angle-resolved absorption/emission measurements reveal differences between the in-plane and out-of-plane bonding at the (0001) interfaces of Cr$_{2}$GeC. The Cr $L_{2,3}$, C $K$, and Ge $M_{1}$, $M_{2,3}$ emission spectra are interpreted with first-principles density-functional theory (DFT) including core-to-valence dipole transition matrix elements. For the Ge $4s$ states, the x-ray emission measurements reveal two orders of magnitude higher intensity at the Fermi level than DFT within the General Gradient Approximation (GGA) predicts. We provide direct evidence of anisotropy in the electronic structure and the orbital occupation that should affect the thermal expansion coefficient and transport properties. As shown in this work, hybridization and redistribution of intensity from the shallow $3d$ core levels to the $4s$ valence band explain the large Ge density of states at the Fermi level.
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.
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, F[n(r)], independent of v(r), such that the expression E≡∫v(r)n(r)dr+F[n(r)] has as its minimum value the correct ground-state energy associated with v(r). The functional F[n(r)] is then discussed for two situations: (1) n(r)=n0+ñ(r), ñn0≪1, and (2) n(r)=ϕ(rr0) with ϕ 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.
The unpolarized absorption and circular dichroism spectra of the fundamental vibrational transitions of the chiral molecule, 4-methyl-2-oxetanone, are calculated ab initio. Harmonic force fields are obtained using Density Functional Theory (DFT), MP2, and SCF methodologies and a 5S4P2D/3S2P (TZ2P) basis set. DFT calculations use the Local Spin Density Approximation (LSDA), BLYP, and Becke3LYP (B3LYP) density functionals. Mid-IR spectra predicted using LSDA, BLYP, and B3LYP force fields are of significantly different quality, the B3LYP force field yielding spectra in clearly superior, and overall excellent, agreement with experiment. The MP2 force field yields spectra in slightly worse agreement with experiment than the B3LYP force field. The SCF force field yields spectra in poor agreement with experiment.The basis set dependence of B3LYP force fields is also explored: the 6-31G and TZ2P basis sets give very similar results while the 3-21G basis set yields spectra in substantially worse agreements with experiment.