# Broken Kramers’ Degeneracy in Altermagnetic MnTe

Suyoung Lee  Center for Correlated Electron Systems, Institute for Basic
Science, Seoul 08826, Korea  Department of Physics and Astronomy, Seoul
National University, Seoul 08826, Korea  Sangjae Lee  The Research Institute
of Basic Sciences, Seoul National University, Seoul 08826, Korea  Saegyeol
Jung  Center for Correlated Electron Systems, Institute for Basic Science,
Seoul 08826, Korea  Department of Physics and Astronomy, Seoul National
University, Seoul 08826, Korea  Jiwon Jung  Center for Artificial Low
Dimensional Electronic Systems, Institute for Basic Science (IBS), Pohang
37673, Korea  Department of Physics, Pohang University of Science and
Technology (POSTECH), Pohang 37673, Republic of Korea  Donghan Kim  Center for
Correlated Electron Systems, Institute for Basic Science, Seoul 08826, Korea
Department of Physics and Astronomy, Seoul National University, Seoul 08826,
Korea  Yeonjae Lee  Center for Correlated Electron Systems, Institute for
Basic Science, Seoul 08826, Korea  Department of Physics and Astronomy, Seoul
National University, Seoul 08826, Korea  Byeongjun Seok  Center for Correlated
Electron Systems, Institute for Basic Science, Seoul 08826, Korea  Department
of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
Jaeyoung Kim  Center for Artificial Low Dimensional Electronic Systems,
Institute for Basic Science (IBS), Pohang 37673, Korea  Byeong Gyu Park
Pohang Accelerator Laboratory, Pohang University of Science and Technology,
Pohang 37673, Korea  Libor Šmejkal  Institut für Physik, Johannes Gutenberg
Universität Mainz, D-55099 Mainz, Germany  Institute of Physics, Czech Academy
of Sciences, Cukrovarnická 10, 162 00 Praha 6 Czech Republic  Chang-Jong Kang
<EMAIL_ADDRESS>Department of Physics, Chungnam National University,
Daejeon 34134, Korea  Changyoung Kim<EMAIL_ADDRESS>Center for
Correlated Electron Systems, Institute for Basic Science, Seoul 08826, Korea
Department of Physics and Astronomy, Seoul National University, Seoul 08826,
Korea

###### Abstract

Altermagnetism is a newly identified fundamental class of magnetism with
vanishing net magnetization and time-reversal symmetry broken electronic
structure. Probing the unusual electronic structure with nonrelativistic spin
splitting would be a direct experimental verification of altermagnetic phase.
By combining high-quality film growth and $in~{}situ$ angle-resolved
photoemission spectroscopy, we report the electronic structure of an
altermagnetic candidate, $\alpha$-MnTe. Temperature dependent study reveals
the lifting of Kramers’ degeneracy accompanied by a magnetic phase transition
at $T_{N}=267\text{ K}$ with spin splitting of up to $370\text{ meV}$,
providing direct spectroscopic evidence for altermagnetism in MnTe.

††preprint: APS/123-QED

Altermagnetism has recently been theoretically identified to be the third
ground state of collinear magnetism, characterized by alternating spin
characters both in real and reciprocal spaces [1, 2]. Although previously
classified as antiferromagnets for their vanishing net magnetization, it has
been shown that altermagnets are fundamentally distinct from antiferromagnets
(AFMs) in the nonrelativistic spin group formalism [3, 4, 5]. The broken time-
reversal symmetry leads to spin split electronic structure even without spin-
orbit interactions [6, 7]. Owing to its unique spin-momentum locked electronic
structure, altermagnet has been proposed to exhibit distinct time-reversal
symmetry breaking phenomena suitable for various spintronics applications [8].

Despite having been vigorously explored in theory [9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21], direct verification of altermagnetic state via
spectroscopic methods $e.g.$ measuring the spin-split bands by angle-resolved
photoemission spectroscopy (ARPES) remains challenging. First of all,
compounds with giant spin splitting are relatively rare so far although
altermagnetic phases are found in a wide range of materials [8]. More
importantly, spin polarization in the experimentally measured band structure
may cancel out due to contributions from mixed domains as detwinning
altermagnetic domains is generally nontrivial. As a result, direct
spectroscopic evidence for spin-split electronic bands with lifted Kramers’
degeneracy remains elusive despite recent report on unusual circular dichroism
[22] texture of an altermagnet in photoemission.

$\alpha$-MnTe is a suitable material for electronic structure study on
altermagnets. As illustrated in Fig. 1(a) and (b), MnTe has a hexagonal NiAs-
type structure (space group $P6_{3}/mmc$) [23]. Below the Néel temperature
($T_{N}=307\text{ K}$ for bulk), it exhibits an A-type AFM ground state where
the spins align ferromagnetically along the $[1\overline{1}00]$ axis on the
$ab$-plane and antiferromagnetically between adjacent layers [24]. The
opposite-spin sublattices, Mn${}_{\text{A}}$ and Mn${}_{\text{B}}$, are not
connected by a simple space inversion or translation but instead by a sixfold
rotation or a mirror operation, which leads to alternating spin polarizations
in the electronic structure as schematically shown in Fig. 1(c) and (d). MnTe
has been predicted to host one of the largest spin splittings among those of
known altermagnetic candidates [1]. Previous study on the transport properties
of MnTe has reported observation of spontaneous anomalous Hall effect arising
from altermagnetic order [25]. However, ARPES measurement on single
crystalline MnTe is challenging due to the 3D crystal structure.

Figure 1:  (a),(b)Crystal structure of MnTe with two opposite-spin sublattices
($\text{Mn}_{\text{A}}$ and $\text{Mn}_{\text{B}}$) and possible sublattice-
transposing transformations containing a real-space mirror or sixfold
rotation. (c) Schematic constant-energy contour on a $k_{x}-k_{y}$ plane with
$k_{z}\neq n\pi/c$ corresponding to a spin group $[C_{2}{\parallel}C_{6z}]$.
(d) Schematic band dispersion plotted along two momenta $k$ and $k^{\prime}$
marked in (c), showing spin-split electronic structure with alternating spin
polarizations. (e),(f) 3D Brillouin zone and spin-degenerate nodal planes
protected by (e) $[C_{2}{\parallel}M_{z}]$ and (f) $[C_{2}{\parallel}C_{6z}]$.
(g),(h) DFT calculated spin-resolved electronic structures of MnTe plotted on
(g) nodal ($k_{z}=0$) and (h) off-nodal ($k_{z}=0.3\pi/c$) plane. Here, the
red and blue colors correspond to spin-up and down, respectively. The spin
splitting is clearly realized along the off-nodal D-U line.

Growing epitaxial films is an alternative way to achieve a clean surface of a
3D crystal, enabling surface-sensitive measurements such as ARPES. In this
Letter, we report successful growth of high quality MnTe films by molecular
beam epitaxy (MBE) and $in\>situ$ ARPES studies on the films. Our temperature-
dependent ARPES data firmly establish breaking of Kramers’ degeneracy
accompanied by the magnetic transition in MnTe, which is consistent with
density functional theory (DFT) calculation results. This work not only
provides the first spectroscopic evidence for spin-split electronic structure
of altermagnets but also helps understand and further engineer intriguing
spin-dependent phenomena in MnTe, including the previously-reported anomalous
Hall effect.

Collinear magnetic phases can be classified based on nonrelativistic spin
groups that involve transformations $[R_{i}{\parallel}R_{j}]$ where $R_{i}$
operates solely in spin space, while $R_{j}$ acts in real space [4, 5].
Altermagnetism is characterized by the interplay of two rotations: a twofold
rotation in spin space combined with a rotation in real space, denoted as
$[C_{2}{\parallel}A$] ($A$: a rotational operator). This distinct symmetry
connects the sublattices with opposite spins through rotation, rather than via
space inversion or translation. As a consequence, the effective time-reversal
symmetry is broken in the electronic structure of an altermagnet [1, 8].

In contrast to the spin-momentum-locked phases arising from broken space-
inversion symmetry and relativistic spin-orbit coupling (SOC), nonrelativistic
spin splitting in an altermagnet is not universally prohibited at time-
reversal invariant momenta [11]. Instead, spin-degenerate nodes are protected
by the symmetry of the spin group. For instance, both space-inversion and
$[C_{2}{\parallel}A$] symmetries preserve spin degeneracy at the $\Gamma$
point [1]. In the case of MnTe, the $[C_{2}{\parallel}M_{z}]$ symmetry,
encompassing a real-space mirror operation with respect to the $ab$-plane,
establishes spin-degenerate nodal planes at $k_{z}=n\pi/c$ (where $n$=integer,
$c$=out-of-plane lattice parameter), which is highlighted in Fig. 1(e).
Similarly, the $[C_{2}{\parallel}C_{6z}t]$ symmetry upholds spin degeneracy
across all $\Gamma$KHA planes (Fig. 1(f)). It is noteworthy that all high-
symmetry lines within a hexagonal Brillouin zone pertain to such nodal planes.
Thus, it is necessary to investigate off-symmetric regions in the Brillouin
zone, such as a constant-$k_{z}$ plane with $k_{z}\neq n\pi/c$, in order to
comprehensively capture the spin-split electronic structure of MnTe.

First, we theoretically confirm the existence of spin-split electronic
structure of altermagnetic MnTe using DFT. Details regarding the methodology
of theoretical calculations are illustrated in the Supplemental Materials (SM)
Sec. I [26]. Fig. 1(g) shows the calculated electronic structure and spin
character along high-symmetry lines $\Gamma$-M-K-$\Gamma$. Although the
inclusion of SOC induces additional band splittings along nodal planes [27],
no net spin polarization is allowed due to symmetry constraints. On the other
hand, spin-split bands are observed along an off-nodal line D-U, which is a
line parallel to $\Gamma$-M, but with $k_{z}=0.3\pi/c$ (Fig. 1(h)).

Figure 2:  Physical properties of MnTe films. (a) Temperature dependent
longitudinal resistivity $\rho_{xx}$ for $T$ between $20$ and $400\text{ K}$.
(b) Magnetization for a magnetic field direction $H//[01\overline{1}0]$
measured at $T=50\text{ K}$ and $300\text{ K}$, showing absence of net
magnetization at $H=0$. Diamagnetic contribution from the substrate is
subtracted. (c),(d) Surface Brillouin zones of $k_{x}-k_{y}$(a) and
$k_{x}-k_{z}$(b) planes. The curve highlighted in (d) corresponds to the
momentum space where ARPES is measured with photon energy $h\nu=21.2\text{
eV}$. (e) Constant energy ARPES map taken at $E-E_{F}=-50\text{ meV}$. Dotted
lines mark the surface Brillouin zone. (f) ARPES $E-k$ cut along the high
symmetry lines in the surface Brillouin zone highlighted in (e).

MBE-grown MnTe film was adopted as a platform to experimentally study the
unique altermagnetic ground state. Basic characterizations of the film can be
found in Supplemental Material Sec. II [26]. The temperature dependent
longitudinal resistivity of a 50 nm-thick MnTe is shown in Fig. 2(a). The Néel
temperature can be determined from the peak in the resistivity curve [28] to
be $T_{N}=267\text{ K}$, which is slightly lower but consistent with the bulk
value [29]. The decrease in resistivity from $T=250\text{ K}$ to $50\text{ K}$
has been attributed to interactions between itinerant and local spins [28].
The magnetization curve (Fig. 2(b)) showing no spontaneous net magnetization
further confirms that the film is in the $\alpha$-MnTe phase.

As shown in the DFT calculation, MnTe has a 3D electronic structure with spin-
degenerate nodal planes at $k_{z}=n\pi/c$. In ARPES, by use of a light source
with a fixed photon energy one can selectively map a $k_{x}-k_{y}$ surface
with a specific $k_{z}$ in the 3D Brillouin zone. Photon-denergy dependent
ARPES reveals that the photon energy of 21.2 eV He-I$\alpha$ corresponds to
$k_{z}=0.3\pi/c$ as shown in Fig. 2(d) (See Supplemental Material Sec.III
[26]). By mapping the bands onto the surface Brillouin zone as illustrated in
Fig. 2(c), we expect to observe spin-polarized bands as predicted in the
calculated electronic structure in Fig. 1(h).

The constant energy contour taken at $E-E_{F}=-50\text{ meV}$ using 21.2 eV
photon is shown in Fig. 2(e). Although MnTe is a semiconductor with a gap of
1.27 eV [30, 31, 32], the observed bands have small spectral weight at the
Fermi energy due to natural $p$-doping by excess Mn, which has been previously
reported in bulk [30] and thin film MnTe [32, 33]. The observed bands show
sixfold rotational symmetry with valence band maximum near the D point, which
is consistent with the crystal symmetry of MnTe and the DFT calculation. The
high-symmetry cut in Fig. 2(f) shows good agreement with the DFT calculation
presented in Fig. 1(h).

Figure 3: Temperature-dependent evolution of MnTe electronic structure. DMFT
results in the (a) paramagnetic ($T=2321\text{ K}$) and (b) altermagnetic
($T=116\text{ K}$) states. Here, the red and blue lines correspond to spin-up
and down states, respectively. ARPES cut (left) and its second derivative
(right) taken along the D-U direction in the (c) paramagnetic and (d)
altermagnetic states taken at $T=300\text{ K}$ and $6\text{ K}$, respectively.

To verify that Kramers’ degeneracy is lifted in the measured bands, we also
performed DFT+dynamical mean field theory (DMFT) calculations and temperature-
dependent ARPES to compare the electronic structures of MnTe in altermagnetic
and paramagnetic states. Fig. 3(a) is the DMFT results along the off-nodal D-U
line for paramagnetic MnTe ($T>T_{N}$). Bands are spin degenerate at all
momenta due to the recovered time reversal symmetry. In the altermagnetic
state ($T<T_{N}$), the number of bands doubles as the initially doubly-
degenerate bands split into spin-polarized bands except for symmetry-protected
nodes (D and U). The antiparallel spin polarizations alternating in the
Brillouin zone shown in Fig. 3(b) are characteristic features of the
altermagnetism.

ARPES spectra were obtained for both paramagnetic and altermagnetic MnTe along
the D-U line at $T=300~{}\text{K}$ and $T=6~{}\text{K}$, respectively.
Notably, the DMFT simulation at $T>T_{N}$ shows precise correspondence to the
ARPES spectra at $T=300~{}\text{K}$ with an energy offset of $\sim$0.23 eV
(see Figs. 3(a) and (c)). This clear alignment strongly supports the validity
of the current DMFT calculations [26]. Both the ARPES and DMFT data reveal a
doubling of the number of bands during the altermagnetic transition,
indicating the lifting of the Kramers’ degeneracy due to broken time-reversal
symmetry. The ARPES and DMFT results suggest substantial spin splittings of a
few hundred meVs between spin-polarized bands along the D-U line.

Figure 4: (a),(b) ARPES cuts taken along the D-U line at temperatures above
(a) and below (b) the $T_{N}$. (c),(d) Detailed temperature-dependent
evolution of the ARPES spectral function taken at (c) D and (d) $0.1\text{
\AA}^{-1}$ away from D. The fitted positions of the peaks are marked in black
circle, red cross and blue diamond, denoting spin-degenerate, spin-up, and
spin-down, respectively. The spin splitting at
$k_{x}=-0.175~{}\text{\AA}^{-1}$ is marked in the black arrow. (e)
Temperature-dependent energy splitting between the spin-up and spin-down
bands. Error bars are obtained from the standard errors of the fitted peak
positions. Energy splittings in the shaded area near the $T_{N}$
($210~{}{\leq}~{}T~{}{\leq}~{}260$ K) could not be reliably determined due to
the uncertainty in the peak fitting at this temperature range.

To analyze the temperature dependence in more detail, we focus on the hole
band centered at D and ascertain the peak positions through the application of
Lorentzian fitting to the ARPES spectra. Due to the D point being a spin-
degenerate node protected by $[C_{2}{\parallel}C_{6z}]$, only one peak is
observed within the energy range of ($-1.2~{}\text{eV}$, $-0.6~{}\text{eV}$)
in both paramagnetic and altermagnetic states. At $T=300~{}\text{K}$, fitting
the peak positions of the energy distribution curves (EDCs) from different
momentum values reveals a single parabolic hole band, as displayed in Fig.
4(a). At $T=10~{}\text{K}$, moving away from D towards U, we find that the
single peak in the EDC taken at D at $-0.9~{}\text{eV}$ splits into two peaks
at off-nodal momenta, forming two hole bands crossing at D, as marked in Fig.
4(b). The splitting between the two bands is largest at
$k_{x}=-0.175~{}\text{\AA}^{-1}$ with splitting energy of $370~{}\pm~{}30$
meV, which is consistent with the magnitude of the splitting found in the DMFT
(Fig. 3(b)). Such momentum-dependent band splitting indicates that the two
bands highlighted with markers in Fig. 4(b) are opposite-spin pair bands
originating from altermagnetism.

To further investigate the temperature-dependent evolution of the electronic
structure, we examine the EDCs at nodal and off-nodal momenta for various
temperatures. At D, as shown in Fig. 4(c), a single peak persists over all
temperatures, exhibiting only slight shifts in energy and thermal broadening
in the spectra. In contrast, at an off-nodal momentum, specifically at
$k_{x}=0.1\text{ \AA}^{-1}$, we observe that a single peak at
$T=300~{}\text{K}$ gradually splits into two spin-split peaks as the system is
cooled down (Fig. 4(d)). The band splitting $E_{\text{splitting}}$ between the
spin-split bands at $k_{x}=0.1\text{ \AA}^{-1}$ is plotted in Fig. 4(e).
$E_{\text{splitting}}$ at $T~{}{\leq}~{}200$ K clearly shows gradual decrease
as the temperature increases, and vanishes above the $T_{N}$ ($T~{}\geq~{}270$
K).

The reduced splitting and thermal broadening at raised temperatures hinder
precise determination of the peak positions at temperature between $T=210$ to
$260$ K (shaded area in Fig. 4(e)). Although exact fitting of
$E_{\text{splitting}}$ to the mean field model $(1-T/T_{N})^{\beta}$ could not
be made due to such uncertainty near the $T_{N}$, the temperature-dependent
evolution of $E_{\text{splitting}}$ at $T{\ll}T_{N}$ and $T>T_{N}$ strongly
suggests continuous lifting of Kramer’ degeneracy accompanied by second order
magnetic phase transition. Notably, the band presenting the energy splitting
exhibits a significant dispersion with a bandwidth of $\sim$1.0 eV, indicative
of an itinerant character. Based on the ARPES and DMFT results, we conclude
that the energy splitting originates from spin separation along the off-nodal
$k$-paths, indicating a breaking of time-reversal symmetry along these paths.
This energy splitting in an altermagnet, to the best of our knowledge, is
measured for the first time, and it directly reflects the itinerant nature of
altermagnetism.

In order to rule out the possibility of band splitting caused by structural
transition inducing band folding or lifting of crystal centrosymmetry, we have
performed temperature dependent low energy electron diffraction (LEED) and
second harmonic generation (SHG). The results are presented in the
Supplemental Material [26]. We have not detected any signatures of structural
reconstruction in the $ab$-plane near the $T_{N}$ in LEED. Furthermore, we
employed the SHG technique which is highly sensitive to breaking of the space-
inversion symmetry ($\mathcal{P}$-symmetry). SHG intensity shows a linear-in
temperature increase across the altermagnetic transition, indicating that the
$\mathcal{P}$-symmetry is preserved. Therefore, the observed band splitting
should be attributed to lifted Kramers’ degeneracy due to broken time-reversal
symmetry.

Ideally, the observation of spin polarizations in the split bands by spin-
resolved ARPES (SARPES) would be the most direct verification of altermagnetic
state. However, it is often challenging to measure spin polarization of time-
reversal symmetry broken states by SARPES due to the formation of multiple
domains which contribute to opposite signs of spin polarizations. With the
beam spot from a He discharge lamp being in the $mm$-scale which is larger
than the typical size of magnetic domains by orders of magnitude, one would
usually obtain collective signals from multiple domains which average out to
zero net polarization. Unlike ferromagnets where a large portion of domains
can be aligned by applying external magnetic field [34, 35, 36], there is no
universal way available at the moment to align antiferromagnetic or
altermagnetic Néel vectors. Several methods including external magnetic field
[37], strain [38] and spin orbit torque [39, 40] are exploited; however, such
methods are highly materials-specific and there is no general solution to
aligning altermagnetic Néel vectors to yield macroscopic spin-polarized
responses.

In the case of MnTe, a sixfold rotational symmetry in the $ab$-plane allows
for 6 domains corresponding to Néel vector
$\mathbf{L}\parallel[1\overline{1}00]$ and five equivalent directions. Such
domains can be reduced to two antiparallel Néel vectors by external magnetic
field [32]; however, one cannot further detwin the two antiparallel domains
that are equal in energy. Therefore, alternative ways to align altermagnetic
domains in MnTe are required to enable spin-resolved spectroscopic study.
Using a ferromagnetic substrate has been suggested as a possible approach
[41]. High-quality MnTe film is an ideal platform to apply various switching
methods, making it a promising workhorse material to explore the electronic
structure and macroscopic properties of altermagnets on a single domain.

In conclusion, by demonstrating momentum-resolved spectroscopic measurements
and the spin split electronic structure in an altermagnetic MnTe for the first
time, our study presented in this Letter lays firm groundwork for further
studies on the novel magnetic ground state and potential future applications
based on its unique properties.

This work was supported by the Institute for Basic Science in Korea (Grant No.
IBS-R009-G2) and the National Research Foundation of Korea (NRF) grant funded
by the Korea government (MSIT). (No. 2022R1A3B1077234). C.-J. K. was supported
by the NRF (Grant No. 2022R1C1C1008200). L. S acknowledges support from JGU
TopDyn initiative.

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