PHYSICAL REVIEW B 106, 085114 (2022)

Anomalously large spin-dependent electron correlation in the nearly half-metallic ferromagnet CoS2
Hirokazu Fujiwara ,1,2,* Kensei Terashima,3 Junya Otsuki ,4 Nayuta Takemori ,4 Harald O. Jeschke ,4 Takanori Wakita,4
Yuko Yano,1 Wataru Hosoda,1 Noriyuki Kataoka ,1 Atsushi Teruya,5 Masashi Kakihana,5 Masato Hedo,6 Takao Nakama,6
Yoshichika Ōnuki,6,7 Koichiro Yaji ,3 Ayumi Harasawa ,2 Kenta Kuroda,2 Shik Shin,2 Koji Horiba ,8Hiroshi Kumigashira ,8 Yuji Muraoka ,1,4 and Takayoshi Yokoya 1,4,†
1

Graduate School of Natural Science and Technology, Okayama University, Okayama 700-8530, Japan
2
Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan
3
National Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0044, Japan
4
Research Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530, Japan
5
Graduate School of Engineering and Science, University of the Ryukyus, Nishihara, Okinawa 903-0213, Japan
6
Faculty of Science, University of the Ryukyus, Nishihara, Okinawa 903-0213, Japan
7
RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan
8
Photon Factory, Institute of Materials Structure Science, High Energy Accelerator Research Organization (KEK), 1-1 Oho,
Tsukuba 305-0801, Japan(Received 7 December 2021; revised 13 June 2022; accepted 19 July 2022; published 9 August 2022)
The spin-dependent band structure of CoS2 , which is a candidate for a half-metallic ferromagnet, was investigated by both spin- and angle-resolved photoemission spectroscopy and theoretical calculations to reappraise
the half-metallicity and electronic correlations. We determined the three-dimensional Fermi surface and thespin-dependent band structure. As a result, we found that a part of the minority spin bands is on the occupiedside in the vicinity of the Fermi level, providing spectroscopic evidence that CoS2 is not a half-metal but very
close. Band calculations using density functional theory with generalized gradient approximation showed good
agreement with the observed majority spin eg bands, while it could not explain the observed band width of theminority-spin eg bands. On the other hand, theoretical calculations using dynamical mean field theory could
better reproduce the strong mass renormalization in the minority-spin eg bands. Our results strongly suggest thepresence of anomalously enhanced spin-dependent electron correlation effects on the electronic structure in thevicinity of the half-metallic state. We also report the temperature dependence of the electronic structure across theCurie temperature and discuss the mechanism of the thermal demagnetization. Our discovery of the anomalously
large spin-dependent electronic correlations not only demonstrates a key factor in understanding the electronic
structure of half-metals but also provides a motivation to improve theoretical calculations on spin-polarizedstrongly correlated systems.
DOI: 10.1103/PhysRevB.106.085114

I. INTRODUCTION

Producing a source of spin-polarized carriers has been achallenging topic of research in solid state physics since theemergence of spintronics which manipulates the spin degreeof freedom of carriers [1,2]. Efforts to realize carriers withlarge spin polarization have been vigorously undertaken instudies on half-metallic ferromagnets (HMFs) [3–7]. HMFs
have metallic electronic structures with an energy gap at the
Fermi level (EF ) for any one electronic spin state in the
ground state, realizing 100% spin polarization at EF [8].
Although many HMF candidates have been predicted fortransition metal oxides and chalcogenides, Heusler alloys,
and two-dimensional materials [9–13], CrO2 has been thesole substance providing ∼100% spin polarization in both
point-contact Andreev reflection (PCAR) and spin-resolved*†

hfujiwara@issp.u-tokyo.ac.jp
yokoya@cc.okayama-u.ac.jp2469-9950/2022/106(8)/085114(13)photoemission spectroscopy experiments [14–18]. Further
discovery and identification of half-metallic materials is necessary to understand characteristic phenomena of HMFs andto realize more thermally stable HMFs.
In CrO2 , there are two claims on electron correlation: a soft x-ray angle-resolved photoemission spectroscopy(SXARPES) study shows that the observed majority spinband is very well explained by dynamical mean-field theory (DMFT) calculations assuming a small effective local
Coulomb repulsion Ueff < 1 eV [19]. On the other hand, in
spin-resolved photoemission spectroscopy reports, the temperature dependence of the minority spin state near EF
provides evidence that strong correlation effects are criticalfor understanding the electronic structure of CrO2 [17,18].
Although these claims seem to be contradictory, they can beinterpreted consistently by assuming the existence of spindependent electron correlation. This spin-dependent electron
correlation is expected to be a universal effect in ferromagnets, but few previous studies have investigated the effect[20–22]. In CrO2 , it is difficult to investigate both majority and085114-1©2022 American Physical Society

HIROKAZU FUJIWARA et al.

PHYSICAL REVIEW B 106, 085114 (2022)

minority spin bands simultaneously because there is no occupied minority spin band at EF at low temperatures.
CoS2 is a ferromagnetic metal whose ordered moment
is reported to be ∼0.9 μB /Co with an easy-axis along the[111] direction [23–25]. The ordered moment is ∼1 μB /Co
6 1
in a low-spin state of Co2+ (configuration: t2geg ), suggestingthat CoS2 has a half-metallic electronic structure including completely spin-polarized eg bands crossing EF . The
half-metallicity has also been supported by theoretical band
structure calculated within generalized gradient approximation (GGA), an experimental study of the de Haas–van Alphen(dHvA) effect, and an optical study measuring reflectivityof CoS2 [10,24,26]. On the other hand, the results of bandcalculations within local spin-density approximation (LSDA),
PCAR, x-ray magnetic circular dichroism, and photoemission
spectroscopy measurements indicated that CoS2 has a nearly
half-metallic electronic structure in which the bottom of thelowest-energy minority spin eg band touches EF or is slightly
occupied near the R point [10,27–32]. This band-structure
picture is also supported by the angle-resolved photoemission spectroscopy (ARPES) study on CoS2 [33] and PCAR
studies on Co1−x Fex S2 . In this system, the spin polarization
at EF is enhanced by hole doping [34,35]. In a polarized
neutron-diffraction study, a non-half-metallic electronic state
was inferred from the observed magnetization distribution, in
which both eg and t2g states contributed to the magnetic moments [36]. This conflicts with the physical picture obtainedby photoemission spectroscopy and density functional theory(DFT) studies based on LSDA [31–33]. So far, there is still
no consensus on the half-metallicity of CoS2 . Therefore, adirect observation of the spin-dependent electronic structure isrequired. If CoS2 has an occupied minority spin band near EF ,we can observe both majority and minority spin bands simultaneously and discuss spin-dependent mass renormalization.
To reveal the spin-resolved electronic structure of CoS2
experimentally, it has been pointed out that measurementswith energy resolution <10 meV are needed. A photoemission
study demonstrated that the peak width of a structure attributed to a minority spin state was 6.5 meV [28]. Clarifying
the fine electronic structure with spin character can provide
the decisive evidence for the half-metallicity of CoS2 .
In this paper, we report on the three-dimensional Fermi
surface (FS) and the band structure of CoS2 single crystals including the spin character determined by high-resolution spinand angle-resolved photoemission spectroscopy (SARPES).
We have found that both majority and minority spin bands
cross EF along the X line, unambiguously demonstratingthat CoS2 does not have a half-metallic electronic structure
down to 20 K. Furthermore, the minority spin bands are
renormalized by electron correlation effects, while the overallcharacter of the majority spin band can be understood by the
one-electron model. This indicates that spin-dependent electron correlation is important for understanding the electronic
structure of CoS2 . Based on the experimentally determinedelectronic structure, we discuss (i) the half-metallicity in
Secs. II A and II B, (ii) the spin-dependent electron correlation effects from comparison with band structures computedin DFT based on GGA and in combination with DMFT(DFT + DMFT) in Secs. II C and II D, (iii) the demagnetization process and the thermal decoherence of near-EF bands

in Sec. II E, and finally, (iv) possible origins of the spindependent electron correlation in CoS2 in Sec. II F.
II. RESULTS AND DISCUSSION
A. Three-dimensional FS and spectral function

The photoemission spectroscopy measurements were performed for a cleaved surface of a CoS2 single crystal. The
low-energy electron diffraction (LEED) pattern is shown in
Fig. 1(a). Figures 1(c)–1(f) show the experimental in-planeFSs for the cut P1 and P2 which are defined in Fig. 1(b). By
determining the Fermi wave vectors (kF ) and using twofold
symmetrization analysis [Figs. 1(e)–1(h)], we found two large
sheets and a pocket around the  point (α, β, and γ , respectively), one pocket around the X  point, and one pocket aroundthe M point in the cut P1.
Figures 2(a) and 2(b) show the observed spectra along theX line. As shown in Fig. 2(a), we observed a peak structure located at E − EF = −1.0 eV, which is accompanied byanother structure at −0.75 eV [Fig. 2(b)] corresponding to
the exchange-split t2g bands. Furthermore, we observed bands
dispersing between −0.5 eV and EF corresponding to the eg
bands. A dispersive band with a bottom at −0.5 eV near the point was observed along the in-plane kx direction [Fig. 2(b)].
This band was also observed in a previous ARPES study [33].
Around the X point, we found that the higher intensity near
EF consists of several bands as indicated by polarizationdependent measurements [Figs. 2(c) and 2(d)]. The band witha bottom around the  point, which forms the FS named α,
crosses EF around X [Figs. 2(c) and 2(d), circularly polarized
and s polarized]. Although the intensity derived from a holeband, which forms the FS named β, is quite small for the circularly polarized light, it becomes clearer for the p-polarized
light, as shown in the right panel in Figs. 2(c) and 2(d),because of an orbital selectivity of photoexcitation [37,38]. In
addition, we found heavy bands located just below EF , which
we call γ and δ. Based on these results, we summarize theband structure in Figs. 2(e) and 2(f). Along the X line, theα, β, and γ bands cross EF and constitute the FSs. Comparing our experimental electronic structure with the publishedtheoretical band structure [27], the β band is assigned to amajority spin hole band forming a hole FS around the  point.
The α band also corresponds to a majority spin band aroundthe  point. The experimental α band, however, crosses EF
although the corresponding theoretical band does not crossEF . Furthermore, there is seemingly no theoretical band in theoccupied states to which we can assign the γ and δ bands. To
clarify the character of the γ and δ bands, we investigated thespin-resolved band structure by high-resolution SARPES, as
described below.
B. Spin-resolved electronic structure

Our high-resolution SARPES results are summarized inFig. 3. Using hν = 6.994 eV, the measured kx line is in the
X MX  plane, as shown in Fig. 3(a). We observed an ellipsoidal FS around the  point and a linear FS along the
ky direction at kx = 0.33 Å−1 , as shown in Fig. 3(b), whichcorresponds to γ and β, respectively, defined in Fig. 1(e).
Figures 3(c)–3(h) show SARPES intensity maps taken along

085114-2

ANOMALOUSLY LARGE SPIN-DEPENDENT ELECTRON …

PHYSICAL REVIEW B 106, 085114 (2022)

FIG. 1. (a) Low-energy electron diffraction (LEED) pattern at an incident electron energy of 185 eV. (b) Brillouin zone of pyrite-type
CoS2 . (c) and (d) Angle-resolved photoemission spectroscopy (ARPES) intensity maps of the Fermi surface (FS) measured at hν = 66 and 87
eV [corresponding to cuts P1 and P2 marked in (b)], respectively, overlaid with the fitted momentum distribution curve (MDC) peak positions
at E = EF showing the kF positions (black circles). (e) and (f) kF plots along P1 and P2, respectively, symmetrized by assuming a twofold
symmetry with respect to the  point. The thick shaded lines are guides for the eye.

the blue line in Fig. 3(b). We observed bands located at
E − EF = −1 eV, −0.5 eV, and −50 meV, just below EF ,and a dispersive hole band corresponding to the β band. From
Fig. 3(i), the bands located at −1 eV split into majority (lower
energy) and minority (higher energy) spin bands, as marked
by black solid lines. These spectral features strongly support
the picture shown by GGA calculations that the t2g bands split
into the majority and minority spin bands [10].
For the eg bands, we obtained the spin-polarized band
structure more clearly. From the correspondence to the band
structure obtained by our ARPES measurements, we identified the eg bands, as shown in Figs. 3(c) and 3(f). At
E − EF = −0.5 eV, the SARPES intensity map for the majority spin character has a clear structure corresponding to
the bottom of the α band shown in Fig. 2(c), although themap for the minority spin bands has no structure at −0.5
eV. In the same way, it is ascertained that the β band has
majority spin character. The result that the spin characterof the α and β bands corresponds to the majority spin isconsistent with the band structures calculated within GGA[10]. On the other hand, the γ and δ bands have minority
spin character, as clearly shown in Fig. 3(e). We discuss detailed characteristics and interpretation of the minority spinbands below.
Here, we discuss the half-metallicity of CoS2 . As bands
forming FSs, we observed not only the majority spin bands but
also the minority spin bands, as shown in Figs. 3(c)–3(h). The
minority spin bands are occupied only down to 100 meV binding energy, providing spectroscopic evidence for near rather
than true half-metallicity of CoS2 . The minority spin bands arepresent down to 20 K, as shown later. This result is consistent
with claims by Andreev reflection and photoemission studies[29,31,32]. The non-half-metallicity is also supported by thefact that the ordered moment μs ∼ 0.9 μB /Co at 2 K [23,24]
is not exactly μs = 1 μB /Co which is the expected orderedmoment when the eg electrons are completely spin polarized.
Our results are also consistent with the dHvA results whichshow a lot of dHvA branches derived from the cross-sectionsof FSs in the kx -ky planes [25].
As seen in Fig. 3(j), the γ band has a very narrow peakjust below EF . Such a very narrow peak just below EF wasobserved by high-resolution spin-integrated photoemission
spectroscopy using the Xe Iα line below TC [32]. Sato et al.
[32] interpreted the peak structure as evidence for the occupied minority spin state around the R point, supported bytheir band-structure calculation results. Our high-resolution
SARPES results, however, show that the peak structure isobserved at kF of the minority spin hole band around the point. Note that the minority spin hole band around the point cannot be attributed to a surface band because the bandshows dispersion with the periodicity of the bulk Brillouinzone along the kz direction, as shown as the intense band near
EF (see Supplemental Material [39]). In addition, a previous
study on the surface of CoS2 (100) has shown that the surface
bands are located between E − EF = −0.7 and −0.2 eV alongthe X direction [40], which supports that the δ band is not to
be interpreted as a surface band. For understanding the minority spin bands around the  point, we analyze a more detailed
comparison of our experimental spectra with the theoreticalband structure, as discussed below.

085114-3

HIROKAZU FUJIWARA et al.

PHYSICAL REVIEW B 106, 085114 (2022)

FIG. 2. (a) Angle-resolved photoemission spectroscopy(ARPES) intensity map as a function of energy up to EF and wave
vector. (b) Second derivative of ARPES intensity map in (a) withrespect to energy. (c) Polarization-dependent ARPES intensity
map around the X point: (left) circularly polarized light, (middle)s-polarized light, and (right) p-polarized light. (d) Second derivativesof ARPES intensity maps in (c) with respect to wave vector. Yellow
filled circles show momentum distribution curve (MDC) peak
positions estimated from the right panel in (c). (e) and (f) ARPES
intensity maps along the MX and the RX  M lines, respectively,
overlaid with plots showing peak positions of energy distribution
curves (EDCs; empty circles) and momentum distribution curves(MDCs; filled circles). Yellow filled circles are the same as in (d)but folded with respect to the X point.

C. Spin-dependent electron correlation

Figure 4(a) shows the theoretical band structure obtained
from GGA with spin-orbit coupling (SOC) calculations withexchange reduced to 74%. The theoretical band structure
without the exchange reduction is shown in the Supplemental
Material [39]. The EF position is determined to conserve thenumber of electrons. For the exchange-reduced band structure, the total moment is calculated to be 0.83 μB /Co, whichis in good agreement with the experimental value [36]. Fromthe correspondence between the theoretical and experimental total moments, it is appropriate to compare our ARPES
and SARPES data with the exchange-reduced band structure.
Note that, even though SOC leads to small additional band
splittings, we discuss orbital character in terms of projection
on l, m, and s quantum numbers which is still useful.
For the majority spin bands, the overall band structure
calculated within GGA + SOC is in line with that obtainedby ARPES shown in Fig. 2. For example, along the Xline, the α and β bands correspond to a band with a bottom
at E − EF ∼ −0.55 eV at the  point and correspond to alight hole band crossing EF , respectively. Furthermore, theorbital character of the α band is the dz2 orbital, while theβ band is mainly characterized by dx2 −y2 orbital character,based on the correspondence with the theoretical band structure. Note that, due to the distortion of the CoS6 octahedra,these two eg orbitals are not degenerate. The difference of theorbital characters causes the polarization dependence of the
ARPES intensity maps obtained around the X point shown in
Figs. 2(c) and 2(d). For the minority spin bands, the γ bandcorresponds to a minority spin band whose bottom touches
EF in the middle of the X line in the exchange-reduced band
structure.
To further discuss the correspondence between the experimental spectra and the theoretical band structures, we
compare the band structures and shapes of the FSs obtainedby ARPES and by the exchange-reduced theoretical band
structure. As shown in Figs. 4(c) and 4(d), the bands α, β, γ ,and δ obtained from ARPES measurements can be assigned
to the bands obtained from the GGA calculations. As shownin Fig. 4(c), we observed five FSs: α, β, γ , a pocket aroundthe X  point, and a pocket around the M point in the cut P1,
as mentioned above. In Fig. 4(d), we show the theoreticalenergy contour lines. The energy contour lines were assigned
to the experimentally observed FSs based on correspondenceof characteristics of the theoretical band structures with thoseof the experimental band structures; for instance, both theexperimental contour line and the theoretical one which areassigned to FS γ are produced by a minority spin hole band.
Comparing Figs. 4(c) with 4(d), squarelike contour lines corresponding to β and trapezoidal contour lines corresponding
to the pocket around the X  point have characteristic shapes
corresponding to those observed by ARPES. This agreementof the theoretical and experimental FSs indicates that theoverall electronic structure can be roughly understood by the
GGA calculation.
However, there is a deviation of our experimental band
structure from our theoretical one in terms of effective mass.
Figures 5(a)–5(d) show comparisons of the ARPES intensity
maps along the X line and the band structure calculated
within GGA. The band widths of the experimentally observed
majority spin bands (α, β) are well explained by the GGA
calculations, while the band width of the γ band observedwith 7 eV laser shown in Fig. 5(c) is much smaller thanthat obtained from the GGA calculations shown in Fig. 5(d).
Estimating the effective mass of the γ band from quadratic
curve fitting, the experimental effective mass was found tobe >5 times larger than that of the GGA calculations. Thissuggests the magnitude of electronic correlation in CoS2 isstrongly dependent on the spin character.
To rigorously discuss the spin dependence of electron correlations, bands with identical orbital character but
different spin characters must be compared because the orbital character-dependent contribution of electron correlations
must be eliminated. We show the ARPES intensity map along
the MR line overlaid with band plots showing the peak positions and show the theoretical bands in Figs. 5(e) and 5(f),respectively. The ε and ζ bands have the same orbital character, mainly dx2 −y2 , but have opposite spin. In the theoreticalband structure, the ε and ζ bands split at the R point due
to SOC, which was not observed by ARPES in which theenergy resolution was 20 meV. Focusing on the effective mass,
the slope at kF of the observed ε band is much larger thanthat of the calculated ε and ε bands. This deviation can be
attributed to mass renormalization of the theoretical ε andε bands due to a correlation effect. On the other hand, the085114-4

ANOMALOUSLY LARGE SPIN-DEPENDENT ELECTRON …

PHYSICAL REVIEW B 106, 085114 (2022)

FIG. 3. (a) Two-dimensional Brillouin zone (gray lines) showing estimated measured k positions for the states near EF (red curves) for
each photon energy. (b) Angle-resolved photoemission spectroscopy (ARPES) intensity map of the Fermi surface (FS) as a function of kx andky measured by 7 eV laser. The FS is obtained with an energy-integration window of ±1meV. (c) ARPES and (d) spin- and angle-resolved
photoemission spectroscopy (SARPES) majority spin bands and (e) SARPES minority spin bands intensity maps as a function of energy to
EF and kx . (f)–(h) Same as (c)–(e) but measured in the vicinity of EF . (i) and (j) Spin-resolved energy distribution curves (EDCs) at k points
corresponding to the blue shaded line cuts in (c) and (f). Black solid makers in (i) indicate peak positions of the structure around −1 eV in the
majority and minority spin spectra.

band width of the ζ band is larger than that of the ζ  andζ  bands. From the point of view of a tight-binding model,
a wider bandwidth indicates a smaller effective mass. Thissuggests that the correlation of the minority spin electrons islarger than that of the majority spin electrons.
Here, we quantitatively compare the magnitude of the spindependent electronic correlation of CoS2 with that of typicalferromagnets. For the electron bands around the  point of
Ni (110), η↑ /η↓ was estimated to be 1.6, where ησ is amass enhancement factor defined by ησ = mσ∗ /mb [index σ
indicates majority (↑) or minority (↓) spin state], where mσ∗
and mb are effective masses given by ARPES and by GGA
calculation, respectively [20]. For the exchange-split electron
bands around the R point of CoS2 (namely, the ε and ζ bands),η↑ /η↓ was estimated to be 1.0/2.6 = 0.38, where we assume
that the effective mass of the ζ band is inversely proportional
to the bandwidth.
These quantitative estimates are summarized in Table I.
From Table I, we make two observations: (i) The mass renor-

malization of the minority spin band in CoS2 is larger thanthat of the majority spin band, while in Ni, the relation of themass renormalizations is inverse. (ii) The ratio of η↑ and η↓ of
CoS2 , (η↑ /η↓ )−1 = 2.6, is much larger than that of Ni(110),1.6. This leads us to suspect that the near-EF electronic structure constituted by unrenormalized majority spin bands andthe large mass renormalization of minority spin bands are
one of the characteristic features of a nearly HMF. A study
on the HMF CrO2 reported a similar result that the majority
spin t2g bands can be well reproduced by LSDA calculation
with Ueff < 1 eV, although minority spin bands cannot be
captured by the calculations because no minority spin t2gband is observed below EF for CrO2 [16,19]. However, it has
been reported that minority spin bands of CrO2 are broadened
by increasing thermal fluctuations due to a correlation effect[18]. This suggests strong correlation for the minority spinelectrons, like the case of CoS2 .
In a previous study of CoS2 using SXARPES, the large
spin-dependent electron correlation as observed in this paper

085114-5

HIROKAZU FUJIWARA et al.

PHYSICAL REVIEW B 106, 085114 (2022)

FIG. 4. Electronic structure of CoS2 in ferromagnetic (FM) state
determined with generalized gradient approximation (GGA) + spinorbit coupling (SOC) exchange correlation functional. (a) Band
structure calculated with exchange splitting reduced to 74%. Bothmajority and minority bands are crossing EF . (b) Corresponding
Fermi surface (FS) with majority and minority FS sheets. (c) Experimentally obtained kF plots, same as Fig. 1(g). The thick shadedlines are guides for the eye. (d) Theoretical contour plots at E −EF = 0 meV (blue; corresponding to FS), E − EF = 10 meV (green),
E − EF = 20 meV (yellow), and E − EF = 30 meV (red), calculated
within GGA.

was not mentioned [33]. Along the X R direction, the kF of theε band is estimated to be ∼0.2π /a away from the R point in
the previous study, while that in this paper is estimated to be
0.5π /a, as shown in Fig. S6 in the Supplemental Material [39].
A reason for the observed smaller kF in SXARPES is due tothe difference in the magnitude of the exchange splitting. In
the SXARPES results, the bottom of the band corresponding
to the ζ band is ∼−0.65 eV, while in our ARPES results,
it is estimated to be −0.6 eV. Since this difference can be
attributed to the difference in the magnitude of the exchange
splitting, the energy of the bottom of the ε band is on the highbinding energy side with an energy scale comparable with thatof the energy difference. Therefore, a larger kF was obtainedin our ARPES than the kF estimated in the SXARPES study.
It is still unclear why this difference in exchange splitting
occurred, but one possibility is that it was due to sample
dependency; calculations by Wu et al. [41] reported that aslight shift in the position of S can change the magnitude ofthe exchange splitting.
Based on our SARPES and band calculation results, theeffective mass of the observed minority spin bands along
the MR line is renormalized significantly, while that of theobserved majority spin bands is rather smaller than that estimated from our GGA calculation. Such mass renormalization
only for the minority spin electrons has been investigated as
spin-dependent electronic correlation in studies on typical fer-

FIG. 5. Comparison of our experimental angle-resolved photoemission spectroscopy (ARPES) results with band structures
calculated within generalized gradient approximation (GGA). (a)ARPES intensity map taken by synchrotron light along the X
direction with peak positions of energy distribution curves (EDCs;
blue empty circles) and momentum distribution curves (MDCs; blue
and yellow filled circles), same as Fig. 1(e). (b) Band structure alongthe X direction obtained from GGA + spin-orbit coupling (SOC)
calculations with exchange splitting reduced to 74%. Red and blue
colors indicate the projected majority and minority spin character,
respectively. (c) Same as (a) but taken by 7 eV laser. (d) Same as(b) but enlarged near EF . (e) Same as (a) but taken along the MR
direction. The light blue dotted line is the curve of the ζ band fittedwith a cosine function to estimate the bandwidth. (f) Same as (b) but
for the MR direction.

romagnetic metals [20–22]. One of the studies implies that theimaginary part of the self-energy of Co 3d bands indicates aspin dependence [22]. There, although the electronic states of
Co obtained by SARPES measurements cannot be explainedby a LSDA + U calculation, which is like our findings, theirspectra calculated by LSDA + DMFT within the one-stepmodel improve the description of the experimental photoemission data. This suggests that the deviation of our experimental
results from our theoretical band structure can be understoodby introducing DMFT.

085114-6

ANOMALOUSLY LARGE SPIN-DEPENDENT ELECTRON …

PHYSICAL REVIEW B 106, 085114 (2022)

TABLE I. Effective mass (m∗ ) compared with the mass given by the GGA calculation (mb ) and the spin-dependent mass renormalization
factor which is defined by the ratio between the effective mass for the majority spin (η↑ ) and that for the minority spin (η↓ ).
ησ ≡ mσ∗ / mb
experiment
CoS2
Ni [20]Majority
Minority
Majority
Minorityη↑ /η↓
1.0/2.6 = 0.38

1.0
2.6
2.2
1.4

ησ ≡ mσ∗ / mb
DFT + DMFT
1.2
1.7

η↑ /η↓
0.70

1.6Note that the spin-dependent electronic correlation is different from an orbital-dependent correlation discussed foriron-based superconductors. For the case of Kx Fe2−y Se2 , themagnitude of electronic correlations depends on the orbital
character of Fe 3d [42]. For the case of the eg bands in CoS2 ,the magnitude of electronic correlations depends on the spinbecause the band shape of the calculated e↑g and e↓g bands isthe same, and splitting them by an exchange interaction does
not change the orbital character.
D. DMFT investigation

We performed DFT + DMFT calculations to investigate
the spin-dependent renormalization beyond GGA. Technicaldetails are summarized in Appendix C. Figure 6 shows thetemperature dependence of the ferromagnetic moment M andthe inverse of the ferromagnetic susceptibility 1/χ for J =1.0 eV and U = 4 eV. From the linear fitting of 1/χ , TC wasdetermined to be 401 K, which is larger than the experimental
value of 122 K. If we consider that DFT + DMFT calculations with density-density interactions tend to overestimate
TC [43], we can consider this parameter set a reasonablechoice. Also, the magnetic moment at a low temperature far
below TC was 0.90 μB /Co which is in good agreement withthe experimental value, 0.93 μB /Co [25]. Numerical resultsfor different parameter sets are provided in the Supplemental
Material [39].
Figure 7 shows the single-particle excitation spectrum calculated within DFT + DMFT for T /TC = 0.58. In Figs. 7(a)and 7(b), the t2g bands split into majority spin bands and

FIG. 6. Magnetic moment M (red squares) and inverse of theferromagnetic susceptibility 1/χ (blue squares) as a function of temperature, obtained dynamical mean-field theory (DMFT) calculations
for U = 4.0 eV, J = 1.0 eV. TC was determined from a linear fit of1/χ by the condition 1/χ = 0.

minority spin bands, although the magnitude of splitting energy is not as large as that calculated within DFT + GGA [10].
The top of the majority spin t2g bands has almost the sameenergy as that of the minority spin t2g bands, while the bottomof the majority spin t2g bands is ∼0.3 eV lower than that of
the minority spin t2g bands. This difference of band width
produces finite spin polarization in the density of states (DOS)of the t2g bands. The spin polarization was observed by our
SARPES measurements as an energy splitting of the majority
spin energy distribution curve (EDC) and minority spin one,as shown in Fig. 3(i). While energy splitting of the t2g bands
estimated by GGA calculations is too large to explain the spin
splitting observed by SARPES, DFT + DMFT calculations
improve the description of the spin splitting of the t2g bands.
As for the eg bands, the correspondence between experimental and theoretical spectral functions has been improved.
Along the X line, two majority eg bands cross EF in thetheoretical spectral functions: a large hole band around the point and a small hole band near the X point. These bands
produce two FSs closed around the  point with the majority
spin character in the X MX  plane, as shown in Fig. 7(c). For
the minority spin character, as shown in Fig. 7(d), unsharp
trapezoidal FSs near the X point are also reproduced. In the
X  MRM plane, the theoretical calculations show that a closed
FS is located around the X  point, as shown in Fig. 7(f). This
FS corresponds to the circular FS observed by ARPES aroundthe X  point shown in Fig. 1(f), although the FS does notappear in GGA calculations [Fig. 4(b)]. For the minority spinstates, the ε band touches EF at the R point in the theorical
spectral function, which produces an electron pocket as shownin Fig. 7(g). In our ARPES intensity maps of the FS shownin Fig. 1(d), intensity at the R point is not so large because
the bottom energy of the ε band is lower than that calculated
within DFT + DMFT.
The DFT + DMFT calculations have improved the theoretical description of spin-dependent electron correlation.
We compare the band dispersions along the MR line calculated within DFT + DMFT with those observed by ARPES,
as shown in Fig. 8. In DFT + DMFT, the bandwidths ofboth ε and ζ bands are reduced compared with those of the
GGA calculations. This can be interpreted as the increase in
the effective mass due to the electronic correlation effects.
Focusing on the magnitude of the bandwidth reduction compared with the GGA calculations, the bandwidth of the ζband is reduced by 20%, while for the ε band, it is reducedby 45%. This spin-dependent reduction of the bandwidth isa result which better explains the spin-dependent electron
correlation effects observed in the spectral function obtainedby ARPES.

085114-7

HIROKAZU FUJIWARA et al.

PHYSICAL REVIEW B 106, 085114 (2022)

FIG. 7. The single-particle excitation spectrum for T /TC = 0.58, U = 4 eV, and J = 1.0 eV for (a) the majority spin and (b) the minority
spin calculated within density functional theory (DFT) + dynamical mean-field theory (DMFT). Momentum dependence of the zero-energy
spectral functions in the X MX  plane in a 20 meV energy window for (c) the majority spin and (d) the minority spin calculated within DFT+ DMFT and (e) obtained by angle-resolved photoemission spectroscopy (ARPES). (f)–(h) Same as (c)–(e) but in the X  MRM plane.

Although the DFT + DMFT yields considerable improvement on the spin-dependent correlations, there arestill quantitative deviations from the ARPES results. Figure 8 indicates that the ε band is in quite good agreement,
while the bandwidth is underestimated by 0.1 eV for theζ band. Additionally, the mass enhancement factor ησ ≡mσ∗ /mb for the minority spin band is estimated to be 1.7 in
DFT + DMFT, while it is 2.6 in the experiment, as summarized in Table I. This deviation indicates that DFT +DMFT underestimates the correlation effects. This implies
that correlations beyond DMFT play a role in the spindependent renormalization in CoS2 , which will be discussed
in Sec. II F.
E. Temperature dependence

In Fig. 9, we examine the temperature evolution of the
ARPES spectra. In Fig. 9(a), the spectral intensity of the δband gets broader rapidly with increasing temperature up to
TC (= 122 K). Concomitantly, the β and γ bands are relatively
enhanced. Except for the broadening, the band structure does
not change significantly with increasing temperature. Moreover, from the EDCs shown in Figs. 9(b) and 9(c), although
the peak structure at EF gets broader rapidly, as seen in the
ARPES intensity maps, the peak structure does not disappear
even above TC . From wide spectra corresponding to the Co 3d(t2g and eg) bands shown in Fig. 9(d), a peak structure corresponding to the α band shifts rapidly by +0.16 eV throughTC . Note that a temperature cycle of the measurements was

conducted to make sure that it is an intrinsic behavior, and not
due to surface aging. Based on our SARPES results which
demonstrate that the α and β (γ and δ) bands are the e↑g(e↓g ) bands, the temperature dependence of the ARPES spectra
demonstrates that the e↑g and e↓g bands are not degenerate even
above TC . This suggests that the origin of the demagnetizationof CoS2 is not Stoner damping but vanishing of the ferromagnetic long-range order by spin fluctuations, as in the case ofiron [44,45].
In a previous spin- and angle-integrated photoemission
study, the broadening of the near-EF peak was interpreted
as a change from the ferromagnetic electronic structure to
the paramagnetic one, that is, as damping of the exchange
splitting [32]. However, by our spin- and angle-resolved measurements, we identified that the exchange splitting remains
above TC , although the magnitude of the splitting reduced
slightly. Therefore, the broadening of the δ band is rather
attributed to a reduction of lifetime of the minority spinelectrons. Figure 9(e) shows the temperature evolution of thewidth of the spectral peak at EF (E ) shown in (c). Here,E shows the T 2 dependence below TC but deviates fromthe T 2 curve above TC . The temperature evolution is nicely inline with that of the resistivity [25]. The fact that a deviationfrom the T 2 dependence of E occurs at TC of the bulk CoS2
sample is consistent with the scenario that the δ band is a bulk
rather than a surface band. The lifetime of the minority spinelectrons (τ ) corresponds to 1/E which rapidly increases below TC , as shown in Fig. 9(f). Because the rapid broadening is085114-8

ANOMALOUSLY LARGE SPIN-DEPENDENT ELECTRON …

PHYSICAL REVIEW B 106, 085114 (2022)

FIG. 8. Comparison of our angle-resolved photoemission spectroscopy (ARPES) results with the spectral function maxima
calculated within density functional theory (DFT) + dynamical
mean-field theory (DMFT). Blue empty (filled) circles show thepeak positions of energy distribution curves (EDCs) [momentum
distribution curves (MDCs)]. Light blue dotted line is the curve of theζ band fitted with a cosine function. Solid green lines show the peak
position of the single-particle excitation spectra calculated within
DFT + DMFT, which are shifted so that the energy at the R point
matches the experimental value.

characterized by TC , it is found that scattering by magnetic
fluctuations is a dominant factor contributing to the broadening of the near-EF peak and the increase of the resistivity.
Furthermore, the rapid broadening is significant for the δband which is highly mass renormalized by correlations. Thissuggests that electron-electron scattering due to Coulomb interaction is also an indispensable factor contributing to thetemperature dependence of the resistivity.
Here, we discuss prospects for realizing a half-metallic
electronic structure at room temperature. The thermal broadening is quite like that predicted by DMFT calculations for
HMFs [13,46–49]. DMFT calculations have demonstrated
that, by thermal fluctuations, the spectrum of the electronic
states for either one of the spin states forming a gap at
EF is broadened and has a finite value at EF . This state
emerging near EF as a result of thermal fluctuations is calleda nonquasiparticle (NQP) state, which has been observed
experimentally as breaking of the half-metallicity in CrO2
and Co2 MnSi [17,18,49]. These facts suggest that, althoughspin-dependent correlations may be a common characteristic of HMFs, as discussed above, electronic correlations
break the half-metallicity at finite temperature, unfortunately.
Furthermore, our results demonstrate that the e↑g bands are
well-described by a one-electron picture despite the 3d nature,
which can be another remarkable feature specific to HMFs
observed in CrO2 [19]. This indicates that realizing weak

FIG. 9. (a) Temperature dependence of the angle-resolved photoemission spectroscopy (ARPES) intensity map along the X line
measured by 7 eV laser. (b) and (c) Energy distribution curves(EDCs) at a kx point marked by the dotted line in (a) and those
divided by the Fermi function at measured temperatures which is
convoluted with a 6 meV Gaussian corresponding to the energy
resolution of the measurement. The EDCs are normalized by theirintensity at −150 meV. (d) EDCs at a kx point marked by the dottedline in (a). The EDCs are normalized by the peak height at 1.0 eV.
Black thick markers show positions of the structure corresponding tothe α band. (e) Temperature evolution of the width of the spectral
peak at EF (E ) shown in (c). The E plots are obtained after
subtracting the energy resolution of our measurement, 6 meV. The
gray dotted line denotes a fit of E = A + BT 2 , where A and B
are fitting parameters, and T is the temperature, to the data in atemperature range of 20–120 K. (f) Temperature evolution of the
lifetime (τ ) of the minority spin electrons occupying the δ band,
estimated by τ = h̄/E.

085114-9

HIROKAZU FUJIWARA et al.

PHYSICAL REVIEW B 106, 085114 (2022)

correlation in the majority spin electronic states might leadto discovery of HMFs. In this paper, we offer insight thatspin-dependent electronic correlations are one of the mostimportant factors that must be understood further for realizing
a half-metallic electronic structure at higher temperatures.

effective mass. DMFT considers only local spin fluctuations,and the feedback from the ferromagnetic magnon is missing.
We conclude that the large spin-dependent electronic correlations originate in the spin-dependence of DOS plus nonlocal
effects such as electron-magnon interactions.

F. Possible origins of the spin-dependent electronic
correlations in CoS2

III. CONCLUSIONSThe large spin-dependent electronic correlations in CoS2
are also observed in half-metal CrO2 , as mentioned above, andmay in fact be a correlation effect specific to HMFs. Therefore, the spin-dependent electronic correlations could originate from the difference in number between majority and minority spinelectrons near EF . The largest electron correlation
term is U α nα↑ nα↓ where the subscripts α = dz2 , dx2 −y2 denote the orbital component. This term only contains electroninteractions between electrons with the same orbital character. In half-metallic electronic states, the magnitude of DOS
near EF varies greatly depending on the spin. Therefore, diagrammatic contributions to the self-energy will reflect theasymmetry in the DOS of majority and minority carriers. Thiseffect might lead to spin-dependent effective masses and could
thus be responsible for a spin dependence of electronic correlations. This model is only relevant for half-metallic electronic
states and might universally apply in HMFs.
Another candidate for the origin of spin-dependent electronic correlations could be the difference of magnitude of the
d-p hybridization. A study of GGA calculations shows thatthe fraction of orbital components that constitute the DOS
near EF varies with the spin orientation [10]. The majority
spin states consist of Co 3d and S 3p orbitals in comparable
proportions, while the minority spin states consist almost entirely of Co 3d orbitals. This difference in magnitude of d-p
hybridization could be observed as intensity difference in our
ARPES measurements: As shown in Fig. 2, the intensity ofthe majority spin band is smaller than that of the minority
spin band near EF . Considering that the cross-section of S
3p electrons is smaller than that of Co 3d at this photon
energy (see Supplemental Material [39]), it can be inferredthat the majority spin eg bands are more strongly hybridized
with the S 3p bands. Also from the valence charge density,
the wave function of the e↑g electrons is extended along the
Co-S direction, while that of the e↓g electrons is more localizedaround the Co nucleus. This originates from the differencein number between majority and minority spin electrons. The
difference of wave-function extent might be another origin of
the spin-dependent electronic correlation in CoS2 .
These two scenarios both attribute the spin-dependent correlations to the spin-dependent DOS as a characteristic of
HMFs. These processes are considered by the DFT + DMFT
calculations, though it is difficult to conclude which process isdominant in CoS2 . On the other hand, quantitative deviations
between theory and experiment discussed in Sec. II D imply
the existence of other factors that are not included in the DFT+ DMFT calculations. One possibility is electron-magnon interaction discussed in the context of the NQP state [13]. It has
been shown by perturbation theory that the electron-magnon
exchange interaction affects the low-energy states of the minority spin band, which could enhance the spin-dependent

We have investigated the electronic structure of pyrite-type
CoS2 by synchrotron ARPES and high-resolution SARPES.
Along the X line, we observed localized bands aroundE − EF = −1 eV and dispersive bands between −0.5 eV and
EF . The overall character of the observed bands is in good
agreement with the Co 3d bands calculated within GGA. In
addition, localized bands were observed near EF . From our
SARPES measurements, we unambiguously identified thatthe near-EF bands had minority spin character, while the GGA
calculations predict that an occupied minority spin band exists
solely around the R point. The deviation of the experimental
spectra from the theoretical band structure is roughly explained by reducing the exchange splitting of the eg band in
GGA calculations, demonstrating that the GGA calculations
overestimated the exchange splitting of CoS2 . However, themodified band structure cannot explain the bandwidth of theminority spin bands, although the bandwidths of the majority
spin bands are in line with the theoretical bandwidths. Thiscan be interpreted because of a spin-dependent electronic
correlation effect. The ratio of the mass enhancement factors(η↑ /η↓ )−1 was estimated to be 2.6, which is much higher thanthat of a typical ferromagnetic metal like Ni where η↑ /η↓ =1.6 We performed DFT + DMFT calculations to unveil theorigin of the large spin-dependent electronic correlations.
DFT + DMFT improves the description of the SARPES
spectra compared with GGA, but a quantitative deviation remains. The large effective mass of the minority spin band wasstill underestimated by the DFT + DMFT calculations. This
result implies the importance of spatial correlations beyond
DMFT for the anomalously large spin-dependent electronic
correlations in half-metallic materials. From our temperaturedependent ARPES experiments, we observed a decoherenceof the mass-renormalized minority spin electrons which ismuch more rapid than that of the majority spin band. The
decoherence of the minority spin band is scaled by TC of
CoS2 , indicating that the decoherence originates from increaseof scattering of the conducting minority spin electrons bymagnetic excitations. From the variation of the eg bands, themajority and minority spin bands are not damped even above
TC , as seen in Fe metal. This indicates that the thermal demagnetization in CoS2 is due to spin fluctuations. Throughour thorough electronic-structure study of nearly HMF CoS2 ,we suggest that spin-dependent electronic correlations are an
important factor which must be considered not only for understanding phenomena appearing in spin-polarized strongly
correlated systems but also for designing half-metallic materials via theoretical calculations.
ACKNOWLEDGMENTS

The laser-based SARPES experiments were conducted
at ISSP with the approval of ISSP (Proposals No. B248,

085114-10

ANOMALOUSLY LARGE SPIN-DEPENDENT ELECTRON …

PHYSICAL REVIEW B 106, 085114 (2022)

No. A182, and No. A177). Part of the ARPES experiments
were performed under the Photon Factory Proposal No.
2016G667. This paper was partially supported by the Programfor Promoting the Enhancement of Research Universities anda Grant-in-Aid for the Japan Society for the Promotion of
Science (JSPS) Fellows (Grant No. 16J03208), and JSPS
KAKENHI (Grants No. JP18K03484, No. 20K20522, No.
20H01853, No. 20K20522, No. 21H01041, No. 21H01003,
No. 21H01041, and No. 21H01003) from the Ministry of
Education, Culture, Sports, Science and Technology of Japan(MEXT). H.F. was supported by a Grant-in-Aid for JSPS
Fellows. Part of the computations were carried out at theSupercomputer Center at the Institute for Solid State Physics,
the University of Tokyo. This paper was supported by MEXT
Leading Initiative for Excellent Young Researchers. We thankA. Ino for providing his ARPES analysis program. We thankM. Shimizu for help with the DFT to DMFT interface. H.F.
thanks T. Taniuchi and K. Okazaki for giving H.F. time toadvance this study.

ing of the samples. The high structural quality of a sample(100) surface was verified by LEED, as shown in Fig. 1(a).
The sample was magnetized along the in-plane [010] direction
by bringing the sample close to a magnet at low temperature
below TC . The approximate magnitude of the magnetic fieldat the sample position was 600 Oe.

APPENDIX A: SAMPLES AND PHOTOEMISSION
MEASUREMENTS

APPENDIX B: DATA PROCESSING

The experimental FS maps in Fig. 1 were obtained withan energy-integration window of ±0.01 eV, and those in
Fig. 3 were done with an energy-integration window of ±1
meV around EF . To obtain absolute values of spin polarization using the VLEED detector, we used the equation
P = (1/M0 Seff )(I+ − I− )/(I+ + I− ), where M0 is the ratio of
the remanent magnetization of the sample to the saturated
magnetization, a calibration factor in terms of magnetic domain, Seff (= 0.25) is the effective Sherman function of theapparatus, and I+(−) is the intensity of the electrons reflectedby the positively (negatively) magnetized target. Then we
obtain the majority (I↑ ) and minority (I↓ ) spin spectra using
I↑(↓) = [1 + (−)P](Itot /2), where Itot = I+ + I− [51,52].

Single crystals of CoS2 were synthesized by the vapor
transport method [25] and were used for ARPES and SARPES
measurements. ARPES experiments were carried out at beamline BL-28A of the Photon Factory, KEK, using synchrotron
radiation as the excitation light source. Circularly polarized light was used, except for the polarization-dependence
ARPES measurements. The total energy and angular resolutions were set to ∼20 meV and 0.3 ° (corresponding to ∼0.02
Å), respectively. Here, EF of the samples was calibrated
by measuring a gold foil that was electrically connected tothe samples. The data were taken at T = 50 K. Laser-based
ARPES and SARPES experiments were performed at theInstitute for Solid State Physics, The University of Tokyo[50]. The p-polarized light with hν = 6.994 eV was used toexcite the photoelectrons. Photoelectrons were analyzed witha combination of a ScientaOmicron DA30L analyzer and avery-low-energy-electron-diffraction (VLEED)-type spin detector. The energy and angular resolutions were set to 6 meV
and 0.3 ° (corresponding to ∼0.005 Å) for spin-integrated
ARPES and 10 meV and 1 ° (corresponding to ∼0.02 Å)
for SARPES, respectively. The base pressure was kept below
1 × 10−8 Pa. Calibration of EF for the samples was achieved
using a gold reference. The data were taken at T = 40 K,
except for the temperature-dependence data in Fig. 7. Clean
surfaces for all measurements were obtained by in situ cleav-

We performed DFT calculations for CoS2 within the
full-potential local orbital (FPLO) [53] basis, using thePerdew-Burke-Ernzerhof parameterization of the GGA exchange correlation functional [54]. We based our calculations
on the pyrite structure of CoS2 given in Ref. [36] with lattice
constant a = 5.539 Å and space group Pa3̄. Here, k meshesof 12 × 12 × 12 in the first Brillouin zone were used. Using
the FPLO projective Wannier functions [55], we constructeda 44-band tight-binding model including Co 3d and S 3p
orbitals (the unit cell includes four formula units).
We performed the DMFT self-consistency calculations using DCORE [56] implemented on the TRIQS library [57,58].
Summations over k and ωn are performed with nk = 83
points and niw = 4096 points (for positive frequencies), respectively. The effective impurity problems were solved
with the hybridization-expansion algorithm [59,60] of thecontinuous-time quantum Monte Carlo method implemented
as CT-HYB-SEGMENT [61] on ALPSCORE library [62]. The
ARPES spectra A(k, ω) = Tr(−1/π )ImĜ(ω + i0) were computed by the analytical continuation from the Matsubara
frequency to the real frequency using the Padé approximation[63]. More information is available in the Supplemental Material [39].

[1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton,
S. von Molnár, M. L. Roukes, A. Y. Chtchelkanova, and D.
M. Treger, Spintronics: a spin-based electronics vision for the
future, Science 294, 1488 (2001).
[2] I. Žutić, J. Fabian, and S. Das Sarma, Spintronics: fundamentals
and applications, Rev. Mod. Phys. 76, 323 (2004).
[3] H. Y. Hwang and S.-W. Cheong, Enhanced intergrain tunnelingmagnetoresistance in half-metallic CrO2 films, Science 278,
1607 (1997).

[4] A. Gupta and J. Z. Sun, Spin-polarized transport and magnetoresistance in magnetic oxides, J. Magn. Magn. Mater. 200, 24(1999).
[5] J. M. D. Coey and M. Venkatesan, Half-metallic ferromagnetism: example of CrO2 , J. Appl. Phys. 91, 8345 (2002).
[6] M. Jourdan, J. Minár, J. Braun, A. Kronenberg, S. Chadov, B.
Balke, A. Gloskovskii, M. Kolbe, H. J. Elmers, G. Schönhense
et al., Direct observation of half-metallicity in the Heusler compound Co2 MnSi, Nat. Commun. 5, 3974 (2014).

APPENDIX C: THEORETICAL CALCULATIONS

085114-11

HIROKAZU FUJIWARA et al.

PHYSICAL REVIEW B 106, 085114 (2022)[7] Y. Ando, Spintronics technology and device development, Jpn.
J. Appl. Phys. 54, 070101 (2015).
[8] R. A. de Groot, F. M. Mueller, P. G. van Engen, and K. H. J.
Buschow, New Class of Materials: Half-Metallic Ferromagnets,
Phys. Rev. Lett. 50, 2024 (1983).
[9] K. Schwarz, CrO2 predicted as a half-metallic ferromagnet, J.
Phys. F: Met. Phys. 16, L211 (1986).
[10] T. Shishidou, A. J. Freeman, and R. Asahi, Effect of GGA on thehalf-metallicity of the itinerant ferromagnet CoS2 , Phys. Rev. B
64, 180401(R) (2001).
[11] S. J. Hashemifar, P. Kratzer, and M. Scheffler, Preserving theHalf-Metallicity at the Heusler Alloy Co2 MnSi(001) Surface: A
Density Functional Theory Study, Phys. Rev. Lett. 94, 096402(2005).
[12] Y.-W. Son, M. L. Cohen, and S. G. Louie, Half-metallic
graphene nanoribbons, Nature (London) 444, 347 (2006).
[13] M. I. Katsnelson, V. Yu. Irkhin, L. Chioncel, A. I. Lichtenstein,
and R. A. de Groot, Half-metallic ferromagnets: from band
structure to many-body effects, Rev. Mod. Phys. 80, 315 (2008).
[14] R. J. Soulen Jr., J. M. Byers, M. S. Osofsky, B. Nadgorny,
T. Ambrose, S. F. Cheng, P. R. Broussard, C. T. Tanaka, J.
Nowak, J. S. Moodera et al., Measuring the spin polarization ofa metal with a superconducting point contact, Science 282, 85(1998).
[15] Y. Ji, G. J. Strijkers, F. Y. Yang, C. L. Chien, J. M. Byers, A.
Anguelouch, G. Xiao, and A. Gupta, Determination of the Spin
Polarization of Half-Metallic CrO2 by Point Contact Andreev
Reflection, Phys. Rev. Lett. 86, 5585 (2001).
[16] H. Fujiwara, M. Sunagawa, K. Terashima, T. Kittaka, T. Wakita,
Y. Muraoka, and T. Yokoya, Intrinsic spin polarized electronic
structure of CrO2 epitaxial film revealed by bulk-sensitive spinresolved photoemission spectroscopy, Appl. Phys. Lett. 106,202404 (2015).
[17] H. Fujiwara, M. Sunagawa, K. Terashima, T. Kittaka, T.
Wakita, Y. Muraok, and T. Yokoya, Observation of intrinsic half-metallic behavior of CrO2 (100) epitaxial films bybulk-sensitive spin-resolved PES, J. Electron Spectrosc. Relat.
Phenom. 220, 46 (2017).
[18] H. Fujiwara, K. Terashima, M. Sunagawa, Y. Yano, T.
Nagayama, T. Fukura, F. Yoshii, Y. Matsuura, M. Ogata, T.
Wakita et al., Origins of Thermal Spin Depolarization in HalfMetallic Ferromagnet CrO2 , Phys. Rev. Lett. 121, 257201(2018).
[19] F. Bisti, V. A. Rogalev, M. Karolak, S. Paul, A. Gupta, T.
Schmitt, G. Güntherodt, V. Eyert, G. Sangiovanni, G. Profeta etal., Weakly-Correlated Nature of Ferromagnetism in Nonsymmorphic CrO2 Revealed by Bulk-Sensitive Soft-X-Ray ARPES,
Phys. Rev. X 7, 041067 (2017).
[20] M. Higashiguchi, K. Shimada, K. Nishiura, X. Cui, H.
Namatame, and M. Taniguchi, Energy band and spin-dependent
many-body interactions in ferromagnetic Ni(110): a highresolution angle-resolved photoemission study, Phys. Rev. B
72, 214438 (2005).
[21] J. Sánchez-Barriga, J. Fink, V. Boni, I. Di Marco, J. Braun, J.
Minár, A. Varykhalov, O. Rader, V. Bellini, F. Manghi et al.,Strength of Correlation Effects in the Electronic Structure ofIron, Phys. Rev. Lett. 103, 267203 (2009).
[22] J. Sánchez-Barriga, J. Braun, J. Minár, I. Di Marco, A.
Varykhalov, O. Rader, V. Boni, V. Bellini, F. Manghi, H. Ebert
et al., Effects of spin-dependent quasiparticle renormalizationin Fe, Co, and Ni photoemission spectra: an experimental andtheoretical study, Phys. Rev. B 85, 205109 (2012).
[23] H. S. Jarrett, W. H. Cloud, R. J. Bouchard, S. R. Butler, C. G.
Frederick, and J. L. Gillson, Evidence for Itinerant d-Electron
Ferromagnetism, Phys. Rev. Lett. 21, 617 (1968).
[24] H. Hiraka and Y. Endoh, Ferromagnetic transition of Heisenberg ferromagnetic metal of CoS2 —Static critical Properties, J.
Phys. Soc. Jpn. 63, 4573 (1994).
[25] A. Teruya, F. Suzuki, D. Aoki, F. Honda, A. Nakamura, M.
Nakashima, Y. Amako, H. Harima, M. Hedo, T. Nakama, andY. Ōnuki, Large cyclotron mass and large ordered moment inferromagnet CoS2 compared with paramagnet CoSe2 , J. Phys.
Soc. Jpn. 85, 064716 (2016).
[26] R. Yamamoto, A. Machida, Y. Moritomo, and A. Nakamura,
Reconstruction of the electronic structure in half-metallic CoS2 ,Phys. Rev. B 59, R7793(R) (1999).
[27] G. L. Zhao, J. Callaway, and M. Hayashibara, Electronic structures of iron and cobalt pyrites, Phys. Rev. B 48, 15781 (1993).
[28] S. K. Kwon, S. J. Youn, and B. I. Min, Itinerant ferromagnetismin half-metallic CoS2 , Phys. Rev. B 62, 357 (2000).
[29] L. Wang, T. Y. Chen, and C. Leighton, Spin-dependent band
structure effects and measurement of the spin polarization in thecandidate half-metal CoS2 , Phys. Rev. B 69, 094412 (2004).
[30] V. N. Antonov, O. V. Andryushchenko, A. P. Shpak, A. N.
Yaresko, and O. Jepsen, Electronic structure, optical spectra,
and x-ray magnetic circular dichroism in CoS2 , Phys. Rev. B
78, 094409 (2008).
[31] T. Takahashi, Y. Naitoh, T. Sato, T. Kamiyama, K. Yamada, H.
Hiraka, Y. Endoh, M. Usuda, and N. Hamada, Para- to ferromagnetic phase transition of CoS2 studied by high-resolution
photoemission spectroscopy, Phys. Rev. B 63, 094415 (2001).
[32] T. Sato, S. Souma, K. Sugawara, K. Nakayama, S. Raj,
H. Hiraka, and T. Takahashi, Xenon-plasma-light low-energy
ultrahigh-resolution photoemission study of Co(S1−x Sex )2 (x =0.075), Phys. Rev. B 76, 113102 (2007).
[33] N. B. M. Schröter, I. Robredo, S. Klemenz, R. J. Kirby, J.
A. Krieger, D. Pei, T. Yu, S. Stolz, T. Schmitt, P. Dudin
et al., Weyl fermions, fermi arcs, and minority-spin carriers inferromagnetic CoS2 , Sci. Adv. 6, eabd5000 (2020).
[34] I. I. Mazin, Robust half metalicity in FexCo1−x S2 , Appl. Phys.
Lett. 77, 3000 (2000).
[35] C. Leighton, M. Manno, A. Cady, J. W. Freeland, L. Wang,
K. Umemoto, R. M. Wentzcovitch, T. Y. Chen, C. L. Chien,
P. L. Kuhns et al., Composition controlled spin polarizationin Co1−x FexS2 alloys, J. Phys.: Condens. Matter 19, 315219(2007).
[36] P. J. Brown, K.-U. Neumann, A. Simon, F. Ueno, and K. R. A.
Ziebeck, Magnetization distribution in CoS2 ; is it a half metallic
ferromagnet? J. Phys.: Condens. Matter 17, 1583 (2005).
[37] Y. Zhang, F. Chen, C. He, B. Zhou, B. P. Xie, C. Fang, W. F.
Tsai, X. H. Chen, H. Hayashi, J. Jiang et al., Orbital charactersof bands in the iron-based superconductor BaFe1.85 Co0.15 As2 ,Phys. Rev. B 83, 054510 (2011).
[38] M. Sunagawa, K. Terashima, T. Hamada, H. Fujiwara, T.
Fukura, A. Takeda, M. Tanaka, H. Takeya, Y. Takano, M. Arita
et al., Observation of a hidden hole-like band approaching the
Fermi level in K-doped iron selenide superconductor, J. Phys.
Soc. Jpn. 85, 073704 (2016).
[39] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevB.106.085114 for additional details of the

085114-12

ANOMALOUSLY LARGE SPIN-DEPENDENT ELECTRON …

PHYSICAL REVIEW B 106, 085114 (2022)

ARPES measurements and their evaluation including additional
spectra, for a description of the DFT and DMFT calculations
with additional results and for details of the determination offerromagnetic ordering temperature and effective mass.
[40] N. Wu, R. F. Sabirianov, W. N. Mei, Ya. B. Losovyj, N. Lozova,
M. Manno, C. Leighton, and P. A. Dowben, The minority spin
surface bands of CoS2 (001), J. Phys.: Condens. Matter 21,
295501 (2009).
[41] N. Wu, Y. B. Losovyj, D. Wisbey, K. Belashchenko, M. Manno,
L. Wang, C. Leighton, and P. A. Dowben, The electronic band
structure of CoS2 , J. Phys.: Condens. Matter 19, 156224 (2007).
[42] M. Yi, Z-K. Liu, Y. Zhang, R. Yu, J.-X. Zhu, J. J. Lee, R. G.
Moore, F. T. Schmitt, W. Li, S. C. Riggs et al., Observationof universal strong orbital-dependent correlation effects in ironchalcogenides, Nat. Commun. 6, 7777 (2015).
[43] A. S. Belozerov, I. Leonov, and V. I. Anisimov, Magnetism ofiron and nickel from rotationally invariant Hirsch-Fye quantum
Monte Carlo calculations, Phys. Rev. B 87, 125138 (2013).
[44] E. Kisker, K. Schröder, W. Gudat, and M. Campagna, Spinpolarized angle-resolved photoemission study of the electronic
structure of Fe(100) as a function of temperature, Phys. Rev. B
31, 329 (1985).
[45] R. Kurzawa, K.-P. Kämper, W. Schmitt, and G. Güntherodt,
Spin-resolved photoemission study of in situ grown epitaxial
Fe layers on W(110), Solid State Commun. 60, 777 (1986).
[46] L. Chioncel, E. Arrigoni, M. I. Katsnelson, and A. I.
Lichtenstein, Electron Correlations and the Minority-Spin BandGap in Half-Metallic Heusler Alloys, Phys. Rev. Lett. 96,
137203 (2006).
[47] V. Yu. Irkhin, M. I. Katsnelson, and A. I. Lichtenstein, Nonquasiparticle effects in half-metallic ferromagnets, J. Phys.:
Condens. Matter 19, 315201 (2007).
[48] L. Chioncel, H. Allmaier, E. Arrigoni, A. Yamasaki, M.
Daghofer, M. I. Katsnelson, and A. I. Lichtenstein, Halfmetallic ferromagnetism and spin polarization in CrO2 , Phys.
Rev. B 75, 140406(R) (2007).
[49] L. Chioncel, Y. Sakuraba, E. Arrigoni, M. I. Katsnelson, M.
Oogane, Y. Ando, T. Miyazaki, E. Burzo, and A. I. Lichtenstein,
Nonquasiparticle States in Co2 MnSi Evidenced Through Magnetic Tunnel Junction Spectroscopy Measurements, Phys. Rev.
Lett. 100, 086402 (2008).
[50] K. Yaji, A. Harasawa, K. Kuroda, S. Toyohisa, M. Nakayama,
Y. Ishida, A. Fukushima, S. Watanabe, C. Chen, F. Komori

et al., High-resolution three-dimensional spin- and angleresolved photoelectron spectrometer using vacuum ultraviolet
laser light, Rev. Sci. Instrum. 87, 053111 (2016).
[51] P. D. Johnson, Spin-polarized photoemission, Rep. Prog. Phys.
60, 1217 (1997).
[52] M. Hoesch, T. Greber, V. N. Petrov, M. Muntwiler, M.
Hengsberger, W. Auwarter, and J. Osterwalder, Spin-polarizedFermi surface mapping, J. Electron Spectrosc. Relat. Phenom.
124, 263 (2002).
[53] K. Koepernik and H. Eschrig, Full-potential nonorthogonal
local-orbital minimum-basis band-structure scheme, Phys. Rev.
B 59, 1743 (1999).
[54] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized GradientApproximation Made Simple, Phys. Rev. Lett. 77, 3865 (1996).
[55] H. Eschrig and K. Koepernik, Tight-binding models for theiron-based superconductors, Phys. Rev. B 80, 104503 (2009).
[56] H. Shinaoka, J. Otsuki, M. Kawamura, N. Takemori, and K.
Yoshimi, DCORE: integrated DMFT software for correlated electrons, SciPost Phys. 10, 117 (2021).
[57] M. Aichhorn, L. Pourovskii, P. Seth, V. Vildosola, M. Zingl, O.
E. Peil, X. Deng, J. Mravlje, G. J. Kraberger, C. Martins, M.
Ferrero, and O. Parcollet, TRIQS/DFTTOOLS: a TRIQS application
for ab initio calculations of correlated materials, Comput. Phys.
Commun. 204, 200 (2016).
[58] O. Parcollet et al., TRIQS: A toolbox for research on interacting
quantum systems, Comput. Phys. Commun. 196, 398 (2015).
[59] P. Werner, A. Comanac, L. de’ Medici, M. Troyer, and A. J.
Millis, Continuous-Time Solver for Quantum Impurity Models,
Phys. Rev. Lett. 97, 076405 (2006).
[60] E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M.
Troyer, and P. Werner, Continuous-time Monte Carlo methods for quantum impurity models, Rev. Mod. Phys. 83, 349(2011).
[61] H. Hafermann, P. Werner, and E. Gull, Efficient implementationof the continuous-time hybridization expansion quantum impurity solver, Comput. Phys. Commun. 184, 1280 (2013).
[62] A. Gaenko, A. E. Antipov, G. Carcassi, T. Chen, X. Chen,
Q. Dong, L. Gamper, J. Gukelberger, R. Igarashi, S. Iskakov
et al., Updated core libraries of the ALPS project, Comput.
Phys. Commun. 213, 235 (2017).
[63] H. J. Vidberg and J. W. Serene, Solving the Eliashberg equations by means of N-point Padé approximants, J. Low Temp.
Phys. 29, 179 (1977).

085114-13