Exploring novel two-dimensional (2D) materials with intrinsic magnetism or topological band features is a focus of current research. Here, based on first-principles calculations, we study a 2D structure of MnAl, which, in the bulk form, is a well-known permanent magnet. We show that in 2D, MnAl can stabilize in a square lattice with single-atom thickness. The ground state is an antiferromagnet (AFM) with checkerboard type magnetic ordering and an estimated Néel temperature of 60 K. The state has large magnetic moment (∼4 μB per Mn) and sizable anisotropy (∼0.27 meV/f.u.), analogous to bulk MnAl. In the electronic band structure, the state exhibits a single type-I AFM nodal loop at the Fermi level, which is protected by mirror symmetry in the absence of spin–orbit coupling. Spin–orbit coupling opens only a small gap at the loop, preserving the band inversion feature. Furthermore, we show that a small strain (∼1%) can drive a magnetic phase transition from the checkerboard AFM to a stripe-type AFM state, accompanied by a significant change in the band structure. Our result offers an intriguing platform for exploring the interplay among magnetism, topology, and phase transitions in low dimensions.

1.
K. S.
Novoselov
,
A. K.
Geim
,
S. V.
Morozov
,
D.
Jiang
,
Y.
Zhang
,
S. V.
Dubonos
,
I. V.
Grigorieva
, and
A. A.
Firsov
, “
Electric field effect in atomically thin carbon films
,”
Science
306
,
666
669
(
2004
).
2.
K. S.
Novoselov
,
A.
Mishchenko
,
A.
Carvalho
, and
A. H.
Castro Neto
, “
2D materials and van der Waals heterostructures
,”
Science
353
,
aac9439
(
2016
).
3.
R.
Bian
,
C.
Li
,
Q.
Liu
,
G.
Cao
,
Q.
Fu
,
P.
Meng
,
J.
Zhou
,
F.
Liu
, and
Z.
Liu
, “
Recent progress in the synthesis of novel two-dimensional van der Waals materials
,”
Natl. Sci. Rev.
9
,
nwab164
(
2022
).
4.
M.
Gibertini
,
M.
Koperski
,
A. F.
Morpurgo
, and
K. S.
Novoselov
, “
Magnetic 2D materials and heterostructures
,”
Nat. Nanotechnol.
14
,
408
419
(
2019
).
5.
Q. H.
Wang
,
A.
Bedoya-Pinto
,
M.
Blei
,
A. H.
Dismukes
,
A.
Hamo
,
S.
Jenkins
,
M.
Koperski
,
Y.
Liu
,
Q. C.
Sun
,
E. J.
Telford
,
H. H.
Kim
,
M.
Augustin
,
U.
Vool
,
J. X.
Yin
,
L. H.
Li
,
A.
Falin
,
C. R.
Dean
,
F.
Casanova
,
R. F. L.
Evans
,
M.
Chshiev
,
A.
Mishchenko
,
C.
Petrovic
,
R.
He
,
L.
Zhao
,
A. W.
Tsen
,
B. D.
Gerardot
,
M.
Brotons-Gisbert
,
Z.
Guguchia
,
X.
Roy
,
S.
Tongay
,
Z.
Wang
,
M. Z.
Hasan
,
J.
Wrachtrup
,
A.
Yacoby
,
A.
Fert
,
S.
Parkin
,
K. S.
Novoselov
,
P.
Dai
,
L.
Balicas
, and
E. J. G.
Santos
, “
The magnetic genome of two-dimensional van der Waals materials
,”
ACS Nano
16
,
6960
(
2022
).
6.
B.
Huang
,
G.
Clark
,
E.
Navarro-Moratalla
,
D. R.
Klein
,
R.
Cheng
,
K. L.
Seyler
,
D.
Zhong
,
E.
Schmidgall
,
M. A.
McGuire
,
D. H.
Cobden
,
W.
Yao
,
D.
Xiao
,
P.
Jarillo-Herrero
, and
X.
Xu
, “
Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit
,”
Nature
546
,
270
273
(
2017
).
7.
C.
Gong
,
L.
Li
,
Z.
Li
,
H.
Ji
,
A.
Stern
,
Y.
Xia
,
T.
Cao
,
W.
Bao
,
C.
Wang
,
Y.
Wang
,
Z. Q.
Qiu
,
R. J.
Cava
,
S. G.
Louie
,
J.
Xia
, and
X.
Zhang
, “
Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals
,”
Nature
546
,
265
269
(
2017
).
8.
L.
Kou
,
Y.
Ma
,
Z.
Sun
,
T.
Heine
, and
C.
Chen
, “
Two-dimensional topological insulators: Progress and prospects
,”
J. Phys. Chem. Lett.
8
,
1905
1919
(
2017
).
9.
Y.
Hou
,
T.
Zhang
,
J.
Sun
,
L.
Liu
,
Y.
Yao
, and
Y.
Wang
, “
Progress on 2D topological insulators and potential applications in electronic devices
,”
Chin. Phys. B
29
,
097304
(
2020
).
10.
X.
Feng
,
J.
Zhu
,
W.
Wu
, and
S. A.
Yang
, “
Two-dimensional topological semimetals
,”
Chin. Phys. B
30
,
107304
(
2021
).
11.
Y. J.
Jin
,
R.
Wang
,
J. Z.
Zhao
,
Y. P.
Du
,
C. D.
Zheng
,
L. Y.
Gan
,
J. F.
Liu
,
H.
Xu
, and
S. Y.
Tong
, “
The prediction of a family group of two-dimensional node-line semimetals
,”
Nanoscale
9
,
13112
13118
(
2017
).
12.
B.
Feng
,
B.
Fu
,
S.
Kasamatsu
,
S.
Ito
,
P.
Cheng
,
C.-C.
Liu
,
Y.
Feng
,
S.
Wu
,
S. K.
Mahatha
,
P.
Sheverdyaeva
,
P.
Moras
,
M.
Arita
,
O.
Sugino
,
T.-C.
Chiang
,
K.
Shimada
,
K.
Miyamoto
,
T.
Okuda
,
K.
Wu
,
L.
Chen
,
Y.
Yao
, and
I.
Matsuda
, “
Experimental realization of two-dimensional Dirac nodal line fermions in monolayer Cu2Si
,”
Nat. Commun.
8
,
1007
(
2017
).
13.
C.
Niu
,
P. M.
Buhl
,
G.
Bihlmayer
,
D.
Wortmann
,
Y.
Dai
,
S.
Blügel
, and
Y.
Mokrousov
, “
Two-dimensional topological nodal line semimetal in layered X2Y (X=Ca, Sr, and Ba; Y=As, Sb, and Bi)
,”
Phys. Rev. B
95
,
235138
(
2017
).
14.
S.
Li
,
Y.
Liu
,
S.-S.
Wang
,
Z.-M.
Yu
,
S.
Guan
,
X.-L.
Sheng
,
Y.
Yao
, and
S. A.
Yang
, “
Nonsymmorphic-symmetry-protected hourglass Dirac loop, nodal line, and Dirac point in bulk and monolayer X3 SiTe6 (X= Ta, Nb)
,”
Phys. Rev. B
97
,
045131
(
2018
).
15.
L.
Gao
,
J. T.
Sun
,
J. C.
Lu
,
H.
Li
,
K.
Qian
,
S.
Zhang
,
Y. Y.
Zhang
,
T.
Qian
,
H.
Ding
,
X.
Lin
,
S.
Du
, and
H. J.
Gao
, “
Epitaxial growth of honeycomb monolayer cuse with Dirac nodal line fermions
,”
Adv. Mater.
30
,
e1707055
(
2018
).
16.
P.
Zhou
,
Z. S.
Ma
, and
L. Z.
Sun
, “
Coexistence of open and closed type nodal line topological semimetals in two dimensional B2C
,”
J. Mater. Chem. C
6
,
1206
1214
(
2018
).
17.
R.-W.
Zhang
,
C.-C.
Liu
,
D.-S.
Ma
, and
Y.
Yao
, “
From node-line semimetals to large-gap quantum spin Hall states in a family of pentagonal group-IVA chalcogenide
,”
Phys. Rev. B
97
,
125312
(
2018
).
18.
K.-H.
Kim
,
B. J.
Yang
, and
H.-W.
Lee
, “
Spin-orbit-stable type-II nodal line band crossing in n-doped monolayer MoX2 (X=S, Se, Te)
,”
Phys. Rev. B
98
,
245422
(
2018
).
19.
S.-G.
Xu
,
B.
Zheng
,
H.
Xu
, and
X.-B.
Yang
, “
Ideal nodal line semimetal in a two-dimensional boron bilayer
,”
J. Phys. Chem. C
123
,
4977
4983
(
2019
).
20.
C.
Zhong
,
W.
Wu
,
J.
He
,
G.
Ding
,
Y.
Liu
,
D.
Li
,
S. A.
Yang
, and
G.
Zhang
, “
Two-dimensional honeycomb borophene oxide: Strong anisotropy and nodal loop transformation
,”
Nanoscale
11
,
2468
2475
(
2019
).
21.
W.
Wu
,
Y.
Jiao
,
S.
Li
,
X.-L.
Sheng
,
Z.-M.
Yu
, and
S. A.
Yang
, “
Hourglass Weyl loops in two dimensions: Theory and material realization in monolayer GaTeI family
,”
Phys. Rev. Mater.
3
,
054203
(
2019
).
22.
Z. F.
Wang
,
B.
Liu
, and
W.
Zhu
, “
Hourglass fermion in two-dimensional material
,”
Phys. Rev. Lett.
123
,
126403
(
2019
).
23.
C.
Liu
,
B.
Fu
,
H.
Yin
,
G.
Zhang
, and
C.
Dong
, “
Strain-tunable magnetism and nodal loops in monolayer MnB
,”
Appl. Phys. Lett.
117
,
103101
(
2020
).
24.
H.
Guo
,
J.
Zhao
,
C.
Chen
,
S.
Li
,
W.
Jiang
,
H.
Fan
,
X.
Tian
, and
S. A.
Yang
, “
Nonsymmorphic nodal-line metals in the two-dimensional rare earth monochalcogenides MX
,”
J. Mater. Sci.
55
,
14883
14892
(
2020
).
25.
J.-W.
Rhim
and
Y. B.
Kim
, “
Landau level quantization and almost flat modes in three-dimensional semimetals with nodal ring spectra
,”
Phys. Rev. B
92
,
045126
(
2015
).
26.
Y.
Liu
,
S. A.
Yang
, and
F.
Zhang
, “
Circular dichroism and radial Hall effects in topological materials
,”
Phys. Rev. B
97
,
035153
(
2018
).
27.
J.-W.
Rhim
and
Y. B.
Kim
, “
Anisotropic density fluctuations, plasmons, and friedel oscillations in nodal line semimetal
,”
New J. Phys.
18
,
043010
(
2016
).
28.
Z.
Yan
,
P.-W.
Huang
, and
Z.
Wang
, “
Collective modes in nodal line semimetals
,”
Phys. Rev. B
93
,
085138
(
2016
).
29.
J.
Li
,
Y.
Li
,
S.
Du
,
Z.
Wang
,
B.-L.
Gu
,
S.-C.
Zhang
,
K.
He
,
W.
Duan
, and
Y.
Xu
, “
Intrinsic magnetic topological insulators in van der Waals layered MnBi2Te4-family materials
,”
Sci. Adv.
5
,
eaaw5685
(
2019
).
30.
Y.
Gong
,
J.
Guo
,
J.
Li
,
K.
Zhu
,
M.
Liao
,
X.
Liu
,
Q.
Zhang
,
L.
Gu
,
L.
Tang
,
X.
Feng
,
D.
Zhang
,
W.
Li
,
C.
Song
,
L.
Wang
,
P.
Yu
,
X.
Chen
,
Y.
Wang
,
H.
Yao
,
W.
Duan
,
Y.
Xu
,
S.-C.
Zhang
,
X.
Ma
,
Q.-K.
Xue
, and
K.
He
, “
Experimental realization of an intrinsic magnetic topological insulator
,”
Chin. Phys. Lett.
36
,
076801
(
2019
).
31.
M. M.
Otrokov
,
I. P.
Rusinov
,
M.
Blanco-Rey
,
M.
Hoffmann
,
A. Y.
Vyazovskaya
,
S. V.
Eremeev
,
A.
Ernst
,
P. M.
Echenique
,
A.
Arnau
, and
E. V.
Chulkov
, “
Unique thickness-dependent properties of the van der Waals interlayer antiferromagnet MnBi2Te4 films
,”
Phys. Rev. Lett.
122
,
107202
(
2019
).
32.
D.
Zhang
,
M.
Shi
,
T.
Zhu
,
D.
Xing
,
H.
Zhang
, and
J.
Wang
, “
Topological axion states in the magnetic insulator MnBi2Te4 with the quantized magnetoelectric effect
,”
Phys. Rev. Lett.
122
,
206401
(
2019
).
33.
M. M.
Otrokov
,
I. I.
Klimovskikh
,
H.
Bentmann
,
D.
Estyunin
,
A.
Zeugner
,
Z. S.
Aliev
,
S.
Gaß
,
A. U. B.
Wolter
,
A. V.
Koroleva
,
A. M.
Shikin
,
M.
Blanco-Rey
,
M.
Hoffmann
,
I. P.
Rusinov
,
A. Y.
Vyazovskaya
,
S. V.
Eremeev
,
Y. M.
Koroteev
,
V. M.
Kuznetsov
,
F.
Freyse
,
J.
Sánchez-Barriga
,
I. R.
Amiraslanov
,
M. B.
Babanly
,
N. T.
Mamedov
,
N. A.
Abdullayev
,
V. N.
Zverev
,
A.
Alfonsov
,
V.
Kataev
,
B.
Büchner
,
E. F.
Schwier
,
S.
Kumar
,
A.
Kimura
,
L.
Petaccia
,
G. D.
Santo
,
R. C.
Vidal
,
S.
Schatz
,
K.
Kißner
,
M.
Ünzelmann
,
C. H.
Min
,
S.
Moser
,
T. R. F.
Peixoto
,
F.
Reinert
,
A.
Ernst
,
P. M.
Echenique
,
A.
Isaeva
, and
E. V.
Chulkov
, “
Prediction and observation of an antiferromagnetic topological insulator
,”
Nature
576
,
416
422
(
2019
).
34.
Y.
Deng
,
Y.
Yu
,
M. Z.
Shi
,
Z.
Guo
,
Z.
Xu
,
J.
Wang
,
X. H.
Chen
, and
Y.
Zhang
, “
Quantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi2Te4
,”
Science
367
,
895
900
(
2020
).
35.
C.
Liu
,
Y.
Wang
,
H.
Li
,
Y.
Wu
,
Y.
Li
,
J.
Li
,
K.
He
,
Y.
Xu
,
J.
Zhang
, and
Y.
Wang
, “
Robust axion insulator and chern insulator phases in a two-dimensional antiferromagnetic topological insulator
,”
Nat. Mater.
19
,
522
527
(
2020
).
36.
H.
Kōno
, “
On the ferromagnetic phase in manganese-aluminum system
,”
J. Phys. Soc. Jpn.
13
,
1444
1451
(
1958
).
37.
A. J. J.
Koch
,
P.
Hokkeling
,
M. G. v. D.
Steeg
, and
K. J.
de Vos
, “
New material for permanent magnets on a base of Mn and Al
,”
J. Appl. Phys.
31
,
S75
S77
(
1960
).
38.
A.
Sakuma
, “
Electronic structure and magnetocrystalline anisotropy energy of MnAl
,”
J. Phys. Soc. Jpn.
63
,
1422
1428
(
1994
).
39.
L.
Pareti
,
F.
Bolzoni
,
F.
Leccabue
, and
A. E.
Ermakov
, “
Magnetic anisotropy of MnAl and MnAlC permanent magnet materials
,”
J. Appl. Phys.
59
,
3824
3828
(
1986
).
40.
J. H.
Park
,
Y. K.
Hong
,
S.
Bae
,
J. J.
Lee
,
J.
Jalli
,
G. S.
Abo
,
N.
Neveu
,
S. G.
Kim
,
C. J.
Choi
, and
J. G.
Lee
, “
Saturation magnetization and crystalline anisotropy calculations for MnAl permanent magnet
,”
J. Appl. Phys.
107
,
09A731
(
2010
).
41.
S. H.
Nie
,
L. J.
Zhu
,
J.
Lu
,
D.
Pan
,
H. L.
Wang
,
X. Z.
Yu
,
J. X.
Xiao
, and
J. H.
Zhao
, “
Perpendicularly magnetized MnAl thin films epitaxied on GaAs
,”
Appl. Phys. Lett.
102
,
152405
(
2013
).
42.
P. E.
Blochl
, “
Projector augmented-wave method
,”
Phys. Rev. B
50
,
17953
17979
(
1994
).
43.
G.
Kresse
and
J.
Furthmüller
, “
Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set
,”
Phys. Rev. B
54
,
11169
11186
(
1996
).
44.
G.
Kresse
and
J.
Furthmüller
, “
Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set
,”
Comput. Mater. Sci.
6
,
15
50
(
1996
).
45.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
, “
Generalized gradient approximation made simple
,”
Phys. Rev. Lett.
77
,
3865
3868
(
1996
).
46.
S. L.
Dudarev
,
G. A.
Botton
,
S. Y.
Savrasov
,
C. J.
Humphreys
, and
A. P.
Sutton
, “
Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study
,”
Phys. Rev. B
57
,
1505
1509
(
1998
).
47.
A.
Togo
,
F.
Oba
, and
I.
Tanaka
, “
First-principles calculations of the ferroelastic transition between rutile-type and CaCl2-type SiO2 at high pressures
,”
Phys. Rev. B
78
,
134106
(
2008
).
48.
R. F.
Evans
,
W. J.
Fan
,
P.
Chureemart
,
T. A.
Ostler
,
M. O.
Ellis
, and
R. W.
Chantrell
, “
Atomistic spin model simulations of magnetic nanomaterials
,”
J. Phys. Condens. Matter.
26
,
103202
(
2014
).
49.
J.
Gao
,
Q.
Wu
,
C.
Persson
, and
Z.
Wang
, “
Irvsp: To obtain irreducible representations of electronic states in the VASP
,”
Comput. Phys. Commun.
261
,
107760
(
2021
).
50.
K. B.
Wiberg
and
P. R.
Rablen
, “
Comparison of atomic charges derived via different procedures
,”
J. Comput. Chem.
14
,
1504
1518
(
1993
).
51.
J. G.
Ángyán
,
G.
Jansen
,
M.
Loos
,
C.
Hättig
, and
B. A.
Hess
, “
Distributed polarizabilities using the topological theory of atoms in molecules
,”
Chem. Phys. Lett.
219
,
267
273
(
1994
).
52.
F.
de Proft
,
C.
van Alsenoy
,
A.
Peeters
,
W.
Langenaeker
, and
P.
Geerlings
, “
Atomic charges, dipole moments, and fukui functions using the hirshfeld partitioning of the electron density
,”
J. Comput. Chem.
23
,
1198
1209
(
2002
).
53.
K.
Kim
,
S. Y.
Lim
,
J. U.
Lee
,
S.
Lee
,
T. Y.
Kim
,
K.
Park
,
G. S.
Jeon
,
C. H.
Park
,
J. G.
Park
, and
H.
Cheong
, “
Suppression of magnetic ordering in XXZ-type antiferromagnetic monolayer NiPS3
,”
Nat. Commun.
10
,
345
(
2019
).
54.
Q.
Liu
,
J.
Xing
,
Z.
Jiang
,
X.
Jiang
,
Y.
Wang
, and
J.
Zhao
, “
2D tetragonal transition-metal phosphides: An ideal platform to screen metal shrouded crystals for multifunctional applications
,”
Nanoscale
12
,
6776
6784
(
2020
).
55.
B.
Feng
,
R. W.
Zhang
,
Y.
Feng
,
B.
Fu
,
S.
Wu
,
K.
Miyamoto
,
S.
He
,
L.
Chen
,
K.
Wu
,
K.
Shimada
,
T.
Okuda
, and
Y.
Yao
, “
Discovery of Weyl nodal lines in a single-layer ferromagnet
,”
Phys. Rev. Lett.
123
,
116401
(
2019
).
56.
C. L.
Kane
and
E. J.
Mele
, “
Z2 topological order and the quantum spin Hall effect
,”
Phys. Rev. Lett.
95
,
146802
(
2005
).
57.
Y. X.
Zhao
and
Y.
Lu
, “
PT-symmetric real Dirac fermions and semimetals
,”
Phys. Rev. Lett.
118
,
056401
(
2017
).
58.
K.
Wang
,
J. X.
Dai
,
L. B.
Shao
,
S. A.
Yang
, and
Y. X.
Zhao
, “
Boundary criticality of PT-invariant topology and second-order nodal-line semimetals
,”
Phys. Rev. Lett.
125
,
126403
(
2020
).
59.
K. S.
Kim
,
Y.
Zhao
,
H.
Jang
,
S. Y.
Lee
,
J. M.
Kim
,
K. S.
Kim
,
J. H.
Ahn
,
P.
Kim
,
J. Y.
Choi
, and
B. H.
Hong
, “
Large-scale pattern growth of graphene films for stretchable transparent electrodes
,”
Nature
457
,
706
710
(
2009
).

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