PHYSICAL REVIEW B 100, 201102(R) (2019)
Rapid Communications Editors’ Suggestion
Ideal Weyl semimetal induced by magnetic exchange
J.-R. Soh ,
1
F. de Juan,
1,2,3
M. G. Vergniory,
2,3
N. B. M. Schröter,
4
M. C. Rahn,
1,*
D. Y. Yan,
5
J. Jiang,
1,6,7
M. Bristow,
1
P. A. Reiss,
1
J. N. Blandy,
8
Y. F. Guo,
6,9
Y. G. Shi,
5
T. K. Kim,
10
A. McCollam,
11
S. H. Simon,
1
Y. Chen,
1,6
A. I. Coldea,
1
and A. T. Boothroyd
1,†
1
Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom
2
Donostia International Physics Center, 20018 Donostia-San Sebastian, Spain
3
IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013 Bilbao, Spain
4
Paul Scherrer Institute, WSLA/202, 5232 Villigen PSI, Switzerland
5
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
6
School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China
7
Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
8
Department of Chemistry, University of Oxford, Inorganic Chemistry Laboratory, Oxford OX1 3QR, United Kingdom
9
CAS Center for Excellence in Superconducting Electronics (CENSE), Shanghai 200050, China
10
Diamond Light Source, Harwell Campus, Didcot OX11 0DE, United K ingdom
11
High Field Magnet Laboratory (HFML-EMFL), Radboud University, 6525 ED Nijmegen, Nijmegen, The Netherlands
(Received 27 February 2019; published 13 November 2019)
We report theoretical and experimental evidence that EuCd
2
As
2
in magnetic fields greater than 1.6 T
applied along the c axis is a Weyl semimetal with a single pair of Weyl nodes. Ab initio electronic structure
calculations, verified at zero field by angle-resolved photoemission spectra, predict Weyl nodes with wave vectors
k = (0, 0, ±0.03) × 2π/c at the Fermi level when the Eu spins are fully aligned along the c axis. Shubnikov–de
Haas oscillations measured in fields parallel to c reveal a cyclotron effective mass of m
∗
c
= 0.08m
e
and a Fermi
surface of extremal area A
ext
= 0.24 nm
−2
, corresponding to 0.1% of the area of the Brillouin zone. The small
values of m
∗
c
and A
ext
are consistent with quasiparticles near a Weyl node. The identification of EuCd
2
As
2
as a
model Weyl semimetal opens the door to fundamental tests of Weyl physics.
DOI: 10.1103/PhysRevB.100.201102
Weyl semimetals (WSMs) exhibit exceptional quantum
electronic transport due to the presence of topologically pro-
tected band crossings called Weyl nodes [1,2]. The nodes
come in pairs with opposite chirality, but their number and
location in momentum space is otherwise material specific.
Weyl nodes are distinct from other topological features
of electron band structures in several respects, including the
following: (1) the bulk bands that cross at a Weyl node
are nondegenerate, (2) the associated Weyl fermions have a
definite chirality, and (3) the Weyl nodes are protected against
perturbations that do not couple the nodes [1–3]. Moreover,
the individual nodes within a pair act as a source and a sink
of Berry curvature, a topological property of the electronic
wave functions which relates directly to several anomalous
transport phenomena [1,4].
Weyl semimetal phases in crystals require either broken
spatial inversion symmetry, or broken time-reversal symmetry
(TRS), or both. There are a number of experimental real-
izations of the first type (with broken inversion symmetry
only), especially in the TaAs structural family [5–9], but
magnetic WSMs (with broken TRS) are still rare. The few
known candidates are complicated by multiple pairs of Weyl
*
Present Address: Institute of Solid State and Materials Physics,
TU-Dresden, 01062 Dresden, Germany.
†
andrew.boothroyd@physics.ox.ac.uk
nodes and/or by extra (nontopological) Fermi surface pockets
which shroud the Weyl nodes [3,10–14]. Magnetic WSMs are
important for fundamental studies of Weyl fermions because it
is possible for such systems to have only a single pair of Weyl
nodes which, due to inversion symmetry, are guaranteed to be
at the same energy and so have a vanishing density of states.
By contrast, WSMs formed by breaking inversion symmetry
(but with TRS) have a minimum of four nodes which are in
general separated in energy.
Following the initial discoveries [5–7] there is now a need
for better material realizations of WSMs, ideally comprising a
single pair of Weyl nodes located at or very close to the Fermi
level and in an energy window free from other overlapping
bands. Here we propose the layered intermetallic EuCd
2
As
2
[15,16] to be such a system. We show that Weyl nodes in
EuCd
2
As
2
are magnetically induced via exchange coupling,
emerging when the Eu spins are aligned by a small external
magnetic field applied along the c axis.
Bulk single crystals of EuCd
2
As
2
(ECA) were grown by
aNaCl/KCl flux method [17]. The magnetic, transport, and
crystallographic properties of the crystals were fully consis-
tent with earlier reports [15–17] (see Supplemental Material
[18]). Magnetotransport measurements were performed on a
Quantum Design physical properties measurement system for
fields B < 16 T, and at the High Field Magnet Laboratory,
Nijmegen, and the National High Magnetic Field Labora-
tory, Tallahassee, for fields up to 38 and 45 T, respectively.
2469-9950/2019/100(20)/201102(6) 201102-1 ©2019 American Physical Society
J.-R. SOH et al. PHYSICAL REVIEW B 100, 201102(R) (2019)
(a)
a
b
c
(c)
(b)
B
Eu
Cd
As
FIG. 1. Crystal structure of EuCd
2
As
2
and location of Weyl
nodes in the Brillouin zone. (a) The trigonal unit cell for the
space group P
¯
3m1 (No. 164). The Weyl fermions predominantly
occupy orbitals in the double-corrugated Cd
2
As
2
layers, which are
sandwiched between the Eu layers [15]. (b) The Weyl nodes lie
along the A--A high-symmetry line and are separated by k
z
(not
shown to scale). (c) In the fully polarized state, singly degenerate
conduction and valence bands meet at a pair of Weyl nodes (shown
here schematically). The nodes act as a source and sink of Berry
curvature (indicated by arrows).
Angle-resolved photoemission spectroscopy (ARPES) mea-
surements were performed at the high-resolution branch line
of the beamline I05 at the Diamond Light Source, UK [19]. A
Scienta R4000 analyzer was used to select a photon energy
of 130 eV, which approximately corresponds to the k
z
= 0
measurement plane. The sample temperature was T ∼ 5K.
To calculate the band structure we employed density func-
tional theory (DFT) as implemented in the Vienna Ab-initio
Simulation Package (
VASP)[20,21]. The exchange-correlation
term is described according to the Perdew-Burke-Ernzerhof
prescription together with projected augmented-wave pseu-
dopotentials [22]. For the self-consistent calculations we used
a10× 10 × 5 k-points mesh. The kinetic energy cutoff was
set to 550 eV. The spin-polarized band structures are calcu-
lated within GGA + U , with the value of U chosen to be
5 eV to match the position of the Eu 4 f bands in the ARPES
spectrum. The lattice parameters used in the calculations were
a = b = 0.443 nm and c = 0.729 nm (see Supplemental Ma-
terial [18]).
The trigonal crystal structure of ECA, shown in
Fig. 1(a), contains alternating layers of Eu
2+
and [Cd
2
As
2
]
2−
(Ref. [23]). The Eu ions carry a localized magnetic moment
with spin S = 7/2 and essentially zero orbital angular mo-
mentum. Below T
N
= 9.5 K the spins order in an A-type
antiferromagnetic (AFM) structure in which the spins form
ferromagnetic layers which stack antiferromagnetically along
the c axis [Fig. 2(a)][15]. A relatively small magnetic field
(B
c
= 1.6TatT = 2KforB c) can be used to coerce the
Eu spins into a fully aligned state [15]. ECA is metallic at
temperatures down to T ∼ 80 K, but for lower temperatures
the resistivity increases to a sharp maximum at T
N
before
(a)
(b)
(d)(c)
B = 0 B > B
c
E
F
FIG. 2. Exchange-induced Weyl nodes in EuCd
2
As
2
.(a),(c)In
zero field, the Eu spins order in an A-type antiferromagnetic structure
at T < T
N
, with the spins lying in the ab plane. The corresponding
band structure is gapped at , and every band is doubly degenerate
due to the combination of inversion and time-reversal symmetries.
(b), (d) The Eu moments can be fully polarized along the c axis in
a small coercive field (B > B
c
), lifting the double degeneracy of the
bands and creating a pair of Weyl nodes along -A,atk
z
= 0.03 r.l.u.
falling again at lower temperatures. This behavior has been
interpreted as due to scattering of conduction electrons by
fluctuations of localized Eu magnetic moments which are
exchange coupled to the Cd and As orbitals [15,16].
In previous ab initio electronic structure calculations,
where spins in the AFM state were found to be aligned with
the c axis, ECA was predicted to host a band inversion of
doubly degenerate As 4p and Cd 5s states near the Fermi
level (E
F
), producing a crossing along the -A line protected
by C
3
symmetry [15,24]. Experimentally [15] (and in more
recent calculations [25]) the spins are in fact found to point
perpendicular to the c axis [Fig. 2(a)] breaking C
3
symmetry,
so an avoided crossing at finite momentum would be expected.
We performed new ab initio calculations of the electronic
structure of ECA with the Eu spins configured (1) in the
AFM state [Fig. 2(a)], and (2) in the ferromagnetic state with
Eu spins fully aligned along the c axis [Fig. 2(b)]. In the
AFM state, the combination of time-reversal and inversion
symmetry requires every band to be doubly degenerate. For
a wide range of parameters, the AFM state displays a small
direct gap at [Fig. 2(c) and Supplemental Material [18]]
which is mostly insensitive to the orientation of the spins.
When the Eu spins are fully spin polarized along the c axis
the double degeneracy is lifted, and a single pair of Weyl
nodes appears at E
F
with no other Fermi surface pockets
in the Brillouin zone [Fig. 2(d) and Supplemental Material
[18]]. These Weyl nodes lie along the -A high-symmetry
line [see Fig. 1(b)] at wave vectors k = (0, 0, ±k
0
) with k
0
0.03 × 2π/c = 0.26 nm
−1
. ECA in a small magnetic field
applied along the c axis, therefore, is predicted to be a Weyl
semimetal with a single pair of Weyl nodes located at E
F
.
201102-2
IDEAL WEYL SEMIMETAL INDUCED BY MAGNETIC … PHYSICAL REVIEW B 100, 201102(R) (2019)
(f)
(a)
(c)
(b)
(e) (g)
(d)
FIG. 3. ARPES and high-field magnetotransport of EuCd
2
As
2
. (a), (c) ARPES spectrum as a function of wave vector along the M--M
high-symmetry line measured at T 5 K with incident photon energy of 130 eV. The data shown here are a sum of two measurements taken
with linear-vertical and linear-horizontal polarization, respectively, to compensate the effect of selection rules. Nonlinear scaling was applied
to the intensity to enhance the visibility of bands with a small photoemission cross section. (b), (d) Electronic bands calculated by DFT (in
red). (e) The second derivative of the longitudinal resistivity ρ
xx
(B) as a function of field applied along the c axis. (f) Plot of 1/B
int
at the
minima and maxima in ρ
xx
against Landau level index, with integers assigned to the minima, where B
int
= B + μ
0
M is the internal field. The
Shubnikov–de Haas (SdH) frequency F is obtained from the gradient of the linear fit shown. (g) Temperature dependence of the amplitude in
the SdH oscillation, measured at B = 10 T. The line is a fit to the Lifshitz-Kosevich formula, from which the cyclotron effective mass m
∗
c
of
the charge carriers is estimated. The quoted error in m
∗
c
derives from the least-squares fit, but the uncertainty in the measurement is expected
to be larger because of the long period of the oscillations and the relatively narrow field range.
The band splitting in the saturated phase is found to be
∼100 meV, which is two orders of magnitude larger than the
Zeeman splitting in the saturation field B
c
. From this we can
conclude, first, that the calculations, which include exchange
but no Zeeman interaction, are a good representation of the
experimental situation in which a small magnetic field is
used to align the Eu spins, and second, that the existence
of the Weyl nodes is driven by exchange coupling to the Eu
spins.
In order to validate the ab initio predictions we carried
outARPES and quantum oscillation measurements. ARPES
data on ECA for T < T
N
and zero applied magnetic field
are presented in Figs. 3(a) and 3(c). These k-E plots are for
k along the M--M path [see Fig. 1(b)] and show steeply
dispersing bands approaching E
F
. The spectra are in good
agreement with the ab initio band structure [Figs. 3(b) and
3(d)] calculated for the observed AFM state with spins lying
in the plane. The agreement is best when E
F
is shifted slightly
downward by about 50 meV, which indicates that the sample
is very slightly hole doped.
Our quantum oscillation measurements are summarized in
Figs. 3(e)–3(g). Figure 3(e) shows the second derivative of
the in-plane longitudinal resistance measured at T = 1.4K
in magnetic fields applied parallel to the c axis (B c)upto
37 T, well above the coercive field (see Supplemental Material
[18] for details of the data treatment). For B < B
c
there is
a very strong variation in magnetoresistance associated with
the progressive canting of the spins towards the c axis, shown
also in Fig. 4(a), but at higher fields the curve displays clear
Shubnikov–de Haas (SdH) oscillations. Only a single SdH
oscillation frequency could be resolved, consistent with a
B
x
y || b
z || c
J
x
(a) (b)
FIG. 4. Magnetotransport of EuCd
2
As
2
.(a)ρ
xx
and σ
yx
as a
function of field. (b) Definition of the xyz axes relative to the crystal
orientation and directions of the current and applied field used in the
measurement.
201102-3
J.-R. SOH et al. PHYSICAL REVIEW B 100, 201102(R) (2019)
single band. Moreover, we do not find any evidence for a spin
splitting of the Landau levels, in contrast to the SdH data on
the structurally related Dirac semimetal Cd
3
As
2
(Ref. [26]). A
lack of spin splitting would be consistent with the prediction
that for B > B
c
the bands are already split by a constant
exchange field, implying that the observed SdH oscillations
correspond to the small pockets derived from the Weyl points
when E
F
is shifted downwards slightly, as suggested by the
ARPES measurements.
The maxima and minima of the oscillations are plotted
on a Landau level index plot in Fig. 3(f), with minima in
ρ
xx
assigned to the integers [27]. The SdH frequency ob-
tained from the gradient is F = 25 ± 3 T, which converts
via the Onsager relation F = (¯h/2π e)A
ext
to an extremal
area of the Fermi surface perpendicular to the c axis of
A
ext
= 0.24 nm
−2
,ork
F
= 0.28 nm
−1
assuming a circular
cross section. This A
ext
value represents approximately 0.1%
of the cross-sectional area of the Brillouin zone perpendicular
to the c axis. Measurements at higher fields up to 45 T did not
find any additional oscillation frequencies (see Supplemental
Material [18]), and confirmed that the maximum centered
on 30 T in Fig. 3(d) corresponds to the quantum limit (first
Landau level).
Figure 3(f) shows the temperature dependence of
the SdH oscillation amplitude up to 40 K. By fit-
ting the data to the Lifshitz-Kosevich formula [ampli-
tude ∼X/ sinh X , where X = 2π
2
k
B
Tm
∗
c
/e¯hB and m
∗
c
=
(¯h
2
/2π )dA
ext
/dE] we obtain a cyclotron effective mass
of about m
∗
c
= 0.08m
e
. The observation that m
∗
c
/m
e
1
is consistent with quasiparticles near a Dirac or Weyl
node. A small effective mass is also found in our
ab initio calculations. Assuming k
F
= 0.28 nm
−1
, as deter-
mined from the SdH data, and a circular Fermi surface cross
section, we obtain m
∗
c
= 0.18m
e
from the calculated band
structure. We caution, however, that the measured and cal-
culated m
∗
c
are not directly comparable because the Lifshitz-
Kosevich formula assumes that the Landau levels are equally
spaced and that many levels are filled, neither of which applies
here.
Our quantum oscillations and ARPES results, as well as
previous optical reflectivity measurements which found clear
evidence for a very low carrier density [16], all point to a
very small Fermi surface, and support the prediction that in
the spin-polarized state of ECA there is a single pair of Weyl
nodes located close to along -A, in a small window of
energy free from other bands. The small effective mass and
Fermi surface area from the SdH data, together with p-type
Hall transport [15], indicate that the crystals used in this study
are slightly hole doped. From the SdH measurements and
ab initio in-plane dispersion we estimate that E
F
is located
approximately 52 meV below the Weyl node (see Supplemen-
tal Material [18]), which is consistent with the shift applied to
the DFT bands in order to match the ARPES data.
In recent years, there has been a great deal of interest
in measuring anomalous transport effects caused by Berry
curvature in topological semimetals, especially the anomalous
Hall effect (AHE) [4,28]. In a Weyl semimetal, the Berry
curvature is associated with the separation k of the Weyl
nodes in k space, as illustrated in Fig. 1(c), and for a single
pair of nodes at E
F
the anomalous Hall conductivity has the
universal form [4]
σ
AHE
yx
=
e
2
2πh
k. (1)
In experiments, k is typically field dependent due to the
effect of field on the band splitting. This makes it difficult
to separate the anomalous and semiclassical contributions to
the Hall effect, as the latter is also field dependent. In ECA,
however, k is almost constant for fields above the saturation
field B
c
= 1.6 T, which makes it straightforward to isolate the
anomalous part of the Hall resistivity. In principle, therefore,
ECA is an ideal system for studying the AHE experimentally.
Figure 4(a) presents measurements of the longitudinal
magnetoresistance ρ
xx
and the anomalous part of the trans-
verse (Hall) conductivity σ
AHE
yx
at T = 2 K as a function
of field applied parallel to the c axis [the experimental ge-
ometry is shown in Fig. 4(b), and the procedure to obtain
the anomalous part of σ
yx
is described in the Supplemental
Material [18]]. There are rapid changes in ρ
xx
at low field
due to the reorientation of the Eu spins in the magnetic field,
as noted earlier, but above the saturation field B
c
= 1.6T
ρ
xx
decreases monotonically with field. The field range in
Fig. 4(a) is below that where quantum oscillations become
observable [see Fig. 3(e)]. The σ
AHE
yx
(B) curve is an odd func-
tion of field, increasing rapidly for 0 < B < B
c
and remaining
constant for B > B
c
, consistent with a nonzero anomalous
Hall conductivity.
Assuming k 0.52 nm
−1
from our ab initio results,
Eq. (1) predicts the anomalous Hall conductance for ECA to
be σ
AHE
yx
30
−1
cm
−1
, which is significantly larger than
observed. This prediction, however, applies only when the
Weyl nodes lie exactly at E
F
. In the samples used here the
nodes are slightly shifted from E
F
, and in this situation other
factors are expected to affect the Berry phase [28]. One such
factor is disorder. We have found the AHE in ECA to be
reduced by the polishing process used to shape the Hall bar
samples. Although it has been argued that disorder-induced
contributions to the AHE are absent when E
F
is near the
nodes [4], the presence of a tilt in the dispersion makes these
contributions possible in the form of skew scattering [29,30].
The significant tilt predicted in our ab initio calculations
[Fig. 2(d)] might explain why the AHE is so reduced.
The simple structure of the Weyl nodes in ECA makes it an
ideal material with which to study the different contributions
to the AHE. This could be achieved by tuning the position of
E
F
relative to the Weyl nodes by doping or application of pres-
sure, or by controlling the level of defects by irradiation. The
degrading effects of polishing could be avoided by studying
transport phenomena with thin-film samples.
More generally, ECA could provide the means to test
predictions of other exotic physics in Weyl semimetals, such
as the anomalous Nernst and thermal Hall effects [14,31],
nonreciprocal effects in light propagation [32], the repulsive
Casimir effect [33], or to probe the effects of the chiral
anomaly in the optical absorption [34] and nonlocal transport
[35].
Note added. Very recently, Ma et al. [36] reported ARPES
results on EuCd
2
As
2
which provide evidence for the existence
of dynamically fluctuating Weyl points due to ferromagnetic
201102-4
IDEAL WEYL SEMIMETAL INDUCED BY MAGNETIC … PHYSICAL REVIEW B 100, 201102(R) (2019)
correlations present in zero field at T > T
N
. The results in
Ref. [36] provide experimental confirmation that the DFT
calculations correctly predict the effects of ferromagnetic
polarization of the Eu spins on the band structure.
The work at Oxford was supported by the UK En-
gineering and Physical Sciences Research Council (EP-
SRC) (Grants No. EP/I004475/1, No. EP/N034872/1, No.
EP/M020517/1, and No. EP/N01930X/1), the John Fell
Fund, and the Oxford Centre for Applied Superconductivity.
J.-R.S. acknowledges support from the Singapore National
Science Scholarship, Agency for Science Technology and
Research. F.d.J. acknowledges funding from the European
Union’s Horizon 2020 research and innovation programme
under the Marie Sklodowska Curie Grant Agreement No.
705968. M.C.R. was supported by the Oxford University
Clarendon Fund, the Los Alamos National Laboratory Direc-
tor’s Fund, and the Alexander von Humboldt-Stiftung. Crystal
growth was carried out with support from the National Key
Research and Development Program of China (Grants No.
2017YFA0302901 and No. 2016YFA0300604) and the Strate-
gic Priority Research Program (B) of the Chinese Academy
of Sciences (Grant No. XDB07020100). We acknowledge the
Diamond Light Source for time on beamlines I11 (proposal
EE18786) and I05 (Proposal SI19234). Work at the HFML, a
member of the European Magnetic Field Laboratory (EMFL),
was supported in part by the EPSRC via its membership of
the EMFL (Grant No. EP/N01085X/1). Work at the NHMFL
was sponsored by the National Science Foundation under
Cooperative Agreement No. DMR-1157490, and by the State
of Florida. We thank D. Graph for technical support at the
NHMFL.
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![FIG. 1. Crystal structure of EuCd2As2 and location of Weyl nodes in the Brillouin zone. (a) The trigonal unit cell for the space group P3̄m1 (No. 164). The Weyl fermions predominantly occupy orbitals in the double-corrugated Cd2As2 layers, which are sandwiched between the Eu layers [15]. (b) The Weyl nodes lie along the A- -A high-symmetry line and are separated by kz (not shown to scale). (c) In the fully polarized state, singly degenerate conduction and valence bands meet at a pair of Weyl nodes (shown here schematically). The nodes act as a source and sink of Berry curvature (indicated by arrows).](/figures/fig-1-crystal-structure-of-eucd2as2-and-location-of-weyl-4uzhr22n.webp)