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d-orbital-frustration-induced ferromagnetic monolayer Cu3O2
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d-orbital-frustration-induced ferromagnetic monolayer Cu3O2 Hongxia Ge a, b, Yuee Xie a, b, Yuanping Chen a, b, * a b
School of Physics and Optoelectronics, Xiangtan University, Xiangtan, 411105, Hunan, China Faculty of Science, Jiangsu University, Zhenjiang, 212013, Jiangsu, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Cu3O2 Ferromagnetism d orbital frustration Flat band
In a frustrated geometry, all types of orbitals are frustrated like spin. d orbital frustration could produce more emergent phenomena because transition elements have strong exchange interactions, spin-orbit coupling and magnetism. However, a real material possessing d orbital frustration is hard to find. Here, we calculate d orbital frustration in a Kagome lattice, which is identified in a new monolayer ferromagnetic Cu3O2. Nine types of d orbitals are considered, and each type generates a group of frustrated states. Three types of d orbitals and corresponding frustration states are found in the monolayer Cu3O2 we proposed. The monolayer structure is similar to a synthesized copper oxide sheet on an Au (111) surface. It is a ferromagnetic nanosheet, whose ferromagnetism is related to d-orbital-frustration-induced flat band. In addition, a possible route to experi mentally synthesize Cu3O2 on SiC surface is discussed.
1. Introduction Some special lattices, such as triangle, Kagome and pyrochlore lat tices, are named as frustrated geometries [1–26]. In these lattice, various degrees of freedom of electrons could be frustrated. Fig. 1(a) shows an example of spin frustration in a Kagome lattice, where three magnetic ions reside on the three lattice sites. Once the first two spins align antiparallel, the third one cannot simultaneously minimize its in teractions with both of them, and thus is frustrated. Spin frustration has been studied extensively because it yields fascinate magnetic phenom ena [1,2,27–30], such as spin ice, spin liquid, spin wave, spin glass and so on. Orbital is another degree of freedom of electrons. All types of orbitals in a frustrated lattice also generate frustrations like spin [4,12,27, 31–35]. Fig. 1(b–d) present examples for frustrations of s, p and d or bitals analogy to Fig. 1(a), respectively, where the orbital at question mark cannot simultaneously favor all of the orbital-orbital interactions with its neighbors. Mathematically, s, p, d and f orbitals can be described by rank-0, 1, 2, 3 tensors, respectively, according to their shapes as well as positive and negative of wavefunctions. p orbital is the closest case with spin (a Rank-1 tensor), and its frustration has been explored in some Kagome lattices, such as Kagome graphene [12] and monolayer P2C3 [14]. These previous studies indicate that orbital frustration pro duces frustrated states and Kagome bands including a flat band and two Dirac bands. The flat band gives rise to a series of many-body
phenomena, such as ferromagnetism [36,37], superconducting [38,39], Wigner crystallization [12,40], and anomalous quantum Hall effect [41]. Comparing with p orbitals, d orbitals have more complicated spatial configurations. The atoms possessing d orbitals are transition elements, which have stronger exchange interactions, spin-orbital coupling and magnetism than other atoms. As such, one can expect that d orbital frustration will lead to more emergent phenomena than p orbital frus tration. For example, a system with a d-orbital-induced flat band could become ferromagnetic spontaneously because the transition-metal atoms have strong intrinsic exchange interactions. For a system with a p-orbital-induced flat band, however, its ferromagnetism can only be stimulated by fractional filling of the flat band [12,14,18]. To capture d-orbital-frustration-induced phenomena, a frustrated lattice made of transition elements should be found firstly. It is known that spin frustration has been extensively studied in some complicated transition-metal compounds where the magnetic transition elements form frustrated lattice. Because spin-spin interaction between magnetic atoms can pass through other nonmagnetic atoms, the nonmagnetic atoms have no effect on the frustrated lattice and thus the frustrated lattice is survived. However, d orbital frustration cannot occur in these compounds because orbital-orbital interactions between the transition elements are hindered by other atoms, and thus the frustrated lattice is concealed. This indicates that d orbital frustration can only be observed in the simpler transition-metal networks. Unfortunately, transition
* Corresponding author. School of Physics and Optoelectronics, Xiangtan University, Xiangtan, 411105, Hunan, China. E-mail address: [email protected] (Y. Chen). https://doi.org/10.1016/j.physb.2019.411826 Received 2 August 2019; Received in revised form 17 October 2019; Accepted 25 October 2019 Available online 30 October 2019 0921-4526/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: Hongxia Ge, Physica B, https://doi.org/10.1016/j.physb.2019.411826
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Fig. 1. Schematic views of spin and orbital frustra tions in a Kagome lattice. (a) Spin frustration, where three magnetic ions reside on the three lattice sites. The two arrows represent two spin directions. (b–d) Frustrations of s, p and d orbitals, respectively, where the orbitals at question marks cannot favor all of the orbital-orbital interactions with its neighbors. The green and purple colors represent positive and nega tive wavefunctions. (For interpretation of the refer ences to color in this figure legend, the reader is referred to the Web version of this article.)
elements are not favorable to form frustrated lattices directly in 2D system. In this paper, we first study nine types of d orbital frustrations in a model of Kagome lattice according to the lattice symmetry. Each type of d orbital generates a group of three states, one singlet and one doublet, which is somewhat similar to the cases of p orbitals. Then, we propose a new Cu3O2 monolayer, as shown in Fig. 3(a), in which the Cu atoms form a Kagome lattice. The new monolayer is selected from a series of monolayers X3O2 (X ¼ transition elements) by analysis of structural stability. It is similar to a synthesized copper oxide sheet on an Au (111) surface [42]. The difference is that the monolayer we proposed is exactly planar while the structure supported by Au substrate is buckling. Elec tron properties of the monolayer Cu3O2 are studied. Three types of d orbitals and their corresponding frustrated states are found. Each group of d orbitals generates a set of spin-polarized Kagome band. The splitting of the Kagome band around the Fermi level induces sponta neous ferromagnetism, and thus Cu3O2 monolayer is intrinsically ferromagnetic. Moreover, our calculations indicate that it is a super conductor with a high critical temperature.
irreducible representations A1g and E2g. We first apply A1g projection operator PA1g to one orbital in the Kagome lattice, say ϕ3 . Each symmetry operation in A1g is applied in turn, and then multiply the result by the character of the A1g representation, and then add the results up:
ψ 1 ¼ PA1g ϕ3 ¼ ϕ3 þ ϕ2 þ ϕ1 þ ϕ1 þ ϕ2 þ ϕ3 þ ϕ3 þ ϕ2 þ ϕ1 þ ϕ3 þ ϕ2 þ ϕ1 þ ϕ3 þ ϕ2 þ ϕ1 þϕ1 þ ϕ2 þ ϕ3 þ ϕ3 þ ϕ2 þ ϕ1 þ ϕ3 þ ϕ2 þ ϕ1 (1)
¼ 8ðϕ1 þ ϕ2 þ ϕ3 Þ: After normalization, one can obtain ψ 1 ¼
p1ffiffi ðϕ 1 3
þ ϕ2 þ ϕ3 Þ, whose
orbital configuration is shown in the left panel in Fig. 2(b). Then, we apply the E2g projection operator PE2g to the orbital, and can obtain 1 6
ψ 2 ¼ pffiffi ð2ϕ3
ϕ1
ϕ2 Þ
(2)
which corresponds to the orbital configuration in the middle panel in Fig. 2(b). The irreducible representation E2g generates two degenerate states. In order to get another degenerate state ψ 3 , we pick a symmetry operation in the D6h group (for example, C3), which will convert ψ 2 into a different state and then make a linear combination of the new state with ψ 2 , such that the result is orthogonal to ψ 2 , rffiffi 3 ðϕ ϕ1 Þ ψ 3 ¼ 2C3 ðψ 2 Þ þ ψ 2 ¼ (3) 2 2
2. d orbital frustion in a Kaomge lattice According to atomic theory, d orbital has five kinds of spatial con figurations [33,34,43], i.e., dxy, dyz, dxz, d2x 2-y, d2z , because of its angular momentum quantum number l ¼ 2. The five kinds of orbitals have different forms in different lattices, according to the symmetry of the lattices. For example, in a Kagome lattice, nine kinds of d orbitals shown in Fig. 2(a) could exist. The space group of Kagome lattice is D6h. Its symmetry axes on the plane are along the lines pointing to the triangle centers or normal to these lines (see the dashed lines in Fig. 2(a)). The nine kinds of d orbitals lie on the mirror planes related to the symmetry axes. Obviously, the orbitals d1 ~ d3 in Fig. 2(a) are analogies to dxy, dyz and dxz, while the orbitals d4 ~ d6 and d7 ~ d9 are analogies to d2x 2-y and d2z , respectively. By applying D6h symmetry operations to the d orbitals in Fig. 2(a), we find that they belong to different irreducible representations, and each kind of d orbital generates three states: one singlet and one doublet. We use the orbital d4 in Fig. 2(a) as an example to explain. In Fig. 2(b), we set the orbitals on atoms I, II and III in a Kagome lattice to be ϕ1 , ϕ2 and ϕ3 , respectively, and all of them are d4 orbital. The orbital satisfies two
After normalization we obtain ψ 3 ¼ p1ffi2ffi ðϕ2
ϕ1 Þ, as shown in the
right panel in Fig. 2(b). The two degenerate states exhibit situations of orbital frustration. The irreducible representations of the other 8 d orbitals are given in Table 1. By using the same process, each d orbital produces a group of three states, one singlet and one doublet. This is similar to the cases of p orbital frustration. Fig. 2(c) and (d) present two groups of three states, corresponding to the orbitals d3 and d8 in Fig. 2(a) respectively. Those states corresponding to other orbitals are given in Fig. S1 in supple mentary information (SI).
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Fig. 2. (a) Nine possible d orbitals in a Kagome lattice. The dashed lines present symmetry lines of the lattice. The orbitals d1 ~ d3 are analogies to dxy, dyz and dxz, while the orbitals d4 ~ d6 and d7 ~ d9 are analogies to d2x 2-y and d2z , respectively. (b) Three quantum states result from orbital frustration of d4 orbital in (a). The left panel is a singlet, while the right two panels are doublet. (c–d) Quantum states result from orbital frustrations of d3 and d8 orbitals in (a), respectively.
force convergence criteria were set to be 10 5 eV and 10 3 eV/Å, respectively. To avoid interaction between adjacent layers, the vacuum distance normal to the layers was kept to 20 Å. A k-point sampling of 7 � 7 � 1 for the primitive cell is used for the Brillouin zone (BZ) inte gration [50]. In the calculation of their phonon spectra, the primitive cell expanded the 2 � 2 � 1 supercell of convention cell. To account for structural thermal stability, we carried out AIMD simulations based on canonical ensemble [51], for which a 4 � 4 supercell containing 80 atoms was used and the AIMD simulations were performed with a Nose-Hoover thermostat from 300 to 1000 K, respectively.
3. Monolayer X3O2 structures and computation method All transition elements are not favorable to directly form Kagome lattice, and thus we can only consider transition metal compounds to indentify d orbital frustration mentioned above. Because there are lots of transition metal oxides exisiting in nature, we propose a simpler lattice made of transition elements X and oxygen, as shown in Fig. 3(a), in which the transition elements X form a Kagome lattice while the O atoms are used to link them. The primitive cell of the lattice, labeled as X3O2, includes three transition-metal atoms and two O atoms. We first access stabilities of the monolayer structures X3O2 (X ¼ transition elements), by using the first-principles calculations within the density functional theory (DFT) formalism as implemented in VASP [44,45]. The Perdew-Burke-Ernzerhof (PBE) functional [46] was employed for the exchange-correlation term according to generalized gradient approximation (GGA) [47,48]. The projector augmented wave (PAW) method [49] was used to represent the ion-electron interaction, and the kinetic energy cutoff of 800 eV was adopted. The energy and
3.1. Electronic and magnetic properties of monolayer Cu3O2 The calculated results indicate that, in all X3O2 (X ¼ 3d transition elements) structures, only the phonon dispersions of Cu3O2 have no soft mode (see Fig. S2 in SI). We note that the monolayer Cu3O2 in Fig. 3(a) is somewhat similar to one of the thin-film copper oxide layers reported by Moller et al. [42] (they observed a variety of thin-film copper oxide 3
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Fig. 3. (a) A Kagome-like structure Cu3O2, which also represents a series of structures X3O2 (X ¼ transition elements). The primitive cell is given in the dashed box. (b–c) Phonon dispersions of monolayer Cu3O2 in the cases of the monolayer is planar and buckling, respectively. The insets in (b–c) present side views of primitive cells of planar and buckling Cu3O2, respec tively. Snapshots of the equilibrium structures of monolayer Cu3O2 at temperatures of (d) 600 and (e) 1000 K after 20 ps AIMD simulations.
1.78 Å. The cohesive energy is 4.05 eV. Its symmetry belongs to the space group P6/MMM (D6h-1). Band structure of monolayer Cu3O2 is shown in Fig. 4(a). It dem onstrates that the structure is a metal because some energy bands cross over the Fermi level. Meanwhile, it is ferromagnetic because the spin-up and spin-down bands split. The magnetic moment of each Cu atom is 0.17 μB. Considering that the PBE functional could underestimate the electron-electron interactions, band structure of monolayer Cu3O2 is recalculated by employing the Hubbard-based PBE þ U (U ¼ 5 eV) corrective scheme, and the result is shown in Fig. 4(b). It is found that spin splitting is strengthened. There only exist spin-down energy bands around the Fermi level, moreover, the contacting point of conduction bands and valence band is a Dirac point. Therefore, the monolayer Cu3O2 in Fig. 3(a) is a half-metal Dirac material. Two mechanisms are usually used to explain origination of ferro magnetism. One is Stoner criterion, the other is flat-band physics. Stoner mechanism for ferromagnetism [52] captures the physics of exchange interaction: spin-polarized electrons maintain a distance from one another due to Fermi statistics to reduce repulsion. However, it does not take into account the correlation effect. Electrons often remain unpo larized even under very strong interactions since they can still avoid one another by developing highly correlated wave functions [12]. Seen from
Table 1 Irreducible representations for the 9 d orbitals in Fig. 2(a). doublet singlet
d1
d2
d3
d4
d5
d6
d7
d8
d9
E2g A2g
E1g B2g
E1g B1g
E2g A1g
E2g A1g
E2g A1g
E2g A1g
E2g A1g
E2g A1g
morphologies on Au (111) surface). The difference is, the O atoms in the experimental samples locate out of the plane (see the inset in Fig. 3(c)), while the monolayer we proposed is exactly planar (see the inset in Fig. 3(b)). We check stability of the Cu3O2 structure reported in the experimental but without Au substrate. The phonon dispersions in Fig. 3 (c) illustrate that it is unstable. This is the reason why the buckling copper oxide sheet forms chemical bonds with Au substrate. We also examine thermal stability of monolayer Cu3O2 we proposed by per forming AIMD simulations in canonical ensemble. After heating up to the targeted temperature of 600 K for 20 ps, we do not observe any structural decomposition (see Fig. 3(d)). The structure reconstruction only occurs at 1000 K during the 20 ps simulation, as show in Fig. 3(e). Therefore, the monolayer Cu3O2 has a rather high thermodynamic sta bility. After structural optimization, lattice constants of monolayer Cu3O2 are a ¼ b ¼ 6.16 Å. The bond length between Cu and O atoms is
Fig. 4. Band structures of monolayer Cu3O2 by adopting (a) PBE functional and (b) Hubbard-based PBE þ U (U ¼ 5 eV) corrective scheme, respectively. The red and blue lines represent spin-up and spin-down energy bands, respectively. (c) Orbital-projected density of states corresponding to band structure in (b). (For inter pretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) 4
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Fig. 5. Orbital-projected band structures and quan tum states corresponding to the bands at Γ point. (a) Projected energy bands of orbitals dxz and dyz, and the crowded bands in the energy range [-2.6, 2.1] eV is amplified in (c) to show the detail. It is noted that the energy bands in (c) eliminate the wide-energy-range Kagome band in (a). (b) Projected energy bands of orbitals d2z . Insets: Quantum states corresponding to the bands at Γi point (i is the index of spin-down bands). These states in (a–c) are similar to those produced by frustrations of orbitals d3, d5 and d2, respectively.
Fig. 4(c), density of states around the Fermi level is mainly attributed by d2x 2-y and dxy orbitals. Although the density of states is high, splitting of the corresponding bands is weak (see Fig. 4(b)). This indicates that Stoner mechanism has small effect here. In general, spin polarization of common bands is typically not favored because the kinetic energy cost is often larger than the exchange energy gain. However, the flat-band physics gives another picture: when electrons fill in the flat band, the kinetic energy penalty of spin polarization does not exist anymore, hence the exchange interaction stabilizes the polarized state. To analyze the origination of ferromagnetism in monolayer Cu3O2, we calculate orbital-projected band structures. Fig. 5(a) shows energy bands attrib uted by dxz and dyz orbitals. There is a set of spin-polarized Kagome bands: the spin-down bands lie around the Fermi level in the energy range [-1.6, 1.2] eV, while the spin-up bands lie in the energy range [-3.0, 0] eV. The splitting between spin-up and spin-down bands is larger than 1.2 eV, which is the largest splitting in all bands. Therefore, the ferromagnetism of monolayer Cu3O2 is originated from flat-band phys ics, i.e., splitting of the flat band in the Kagome bands. For a system with p-orbital-induced flat bands, its ferromagnetism can only be stimulated by fractional filling of the flat band [12,14]. However, the flat band here does not locate around the Fermi level. We think that spontaneous ferromagnetism of monolayer Cu3O2 is a result of collaboration of flat-band physics and strong exchange interactions of Cu atoms. In the insets of Fig. 5(a), we show quantum states in the spin-down (or spin-up) Kagome bands at Γ point. The state at Γ23 is a singlet, while the two states at Γ20 and Γ19 are doublet. One can find that these states are similar to the three states in Fig. 2(c). As mentioned above, they are generated by frustration of d3 orbital in Fig. 2(a). Meanwhile, one can find that there are some energy bands crowding in the energy range [-2.6, 2.1] eV in Fig. 5(a). We amplify them in Fig. 5(c) to show the detail. It is found that they are also a set of spin-polarized Kagome bands. The insets in Fig. 5(c) show the quantum states in the Kagome bands, which are very similar to those in Fig. S1(b) in SI. It means that these Kagome bands are induced by the frustrated states of d2 orbital in Fig. 2(a). Further, in the energy range [-2.6, 2.2] eV, we find another set of spin-polarized Kagome bands, as shown in Fig. 5(b). The states in the inset of Fig. 5(b) illustrate that they correspond to the frustrated states of d5 orbital (see Fig. S1(e) in SI). Therefore, d orbital frustration discussed above can be observed in monolayer Cu3O2. In addition, we further check stabilities of other X3O2 (X ¼ 4d and 5d transition elements) in Fig. 3(a). The results show that only Ag3O2 and Au3O2 are stable (see Fig. S2 in SI). The two structures exhibit similar electronic properties with Cu3O2 (see Fig. S3 in SI). If spin-orbit coupling (SOC) in monolayer X3O2 (X ¼ Cu, Ag, Au) is considered, some band
crossings will be gapped. For example, the original Dirac point on the Fermi level as well as the crossings between flat bands and other bands are gapped, as shown in Fig. S4 in SI. Since copper oxide is regarded as a prototype system for high Tc superconductor, we calculate superconducting properties of monolayer Cu3O2. The approach used to compute the electron-phonon coupling follows the method employing the QUANTUM-ESPRESSO (QE) pro grams by Wierzbowska, Gironcoli and Gianozzi [53,54] with non-spin-polarization. The Cu and O atoms are represented by ultra-soft pseudopotentials and the kinetic energy and charge density cutoffs are chosen to be 100 and 800 Ry, respectively. The results indicate that monolayer Cu3O2 is a superconductor with a high critical temperature Tc ¼ 10.5 K in the case of nonmagnetism. We expect some further experimental studies can be carried out to validate and extend our findings. Our analyses for structural stabilities already demonstrate the high feasibility to synthesize Cu3O2. Considering that some 2D materials have been grown successfully on SiC substrates [55], we explore the possi bility to synthesize Cu3O2 on SiC substrate. According to our computa tions, the lattice constants of a 2 � 2 supercell of SiC (1 1 1) substrate (a0 ¼ b0 ¼ 6.18 Å) are very close to those of the primitive cell of Cu3O2. The mismatch between them is smaller than 0.49%. The adhesion en ergy between Cu3O2 sheet and SiC substrate is 0.95 eV/atom (see Fig. S5 in the SI for more details). We believe the newly predicted Cu3O2 sheet can be synthesized by the pathway similar to those of other 2D materials on the SiC substrate. 4. Conclusions In summary, we first study d orbital frustrations in a Kagome lattice. Nine types of d orbitals and their corresponding frustrated states in a Kagome lattice are discussed. Each type of d orbitals generates a group of three states, one singlet and two degenerate states. Based on a Kagomelike lattice, a new Cu3O2 monolayer, analogy to a structure reported in experimental, is proposed. It is selected from a series of monolayers X3O2 (X ¼ transition elements) by analysis of structural stability. The structure shows good dynamics and thermodynamics stabilities. Several types of d-orbital-frustration states are found in it. Each type of d orbitals generates a set of Kagome band corresponding to three frustrated states. The splitting of flat band and strong exchange interactions of Cu atoms leads to spontaneous ferromagnetism of monolayer Cu3O2. In addition, we proposed a promising approach to prepare the new sheet by epitaxial growth on a SiC substrate, and hope some experiments can be carried out to validate and extend our findings. 5
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Acknowledgments
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Physica B: Physics of Condensed Matter journal homepage: http://www.elsevier.com/locate/physb
d-orbital-frustration-induced ferromagnetic monolayer Cu3O2 Hongxia Ge a, b, Yuee Xie a, b, Yuanping Chen a, b, * a b
School of Physics and Optoelectronics, Xiangtan University, Xiangtan, 411105, Hunan, China Faculty of Science, Jiangsu University, Zhenjiang, 212013, Jiangsu, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Cu3O2 Ferromagnetism d orbital frustration Flat band
In a frustrated geometry, all types of orbitals are frustrated like spin. d orbital frustration could produce more emergent phenomena because transition elements have strong exchange interactions, spin-orbit coupling and magnetism. However, a real material possessing d orbital frustration is hard to find. Here, we calculate d orbital frustration in a Kagome lattice, which is identified in a new monolayer ferromagnetic Cu3O2. Nine types of d orbitals are considered, and each type generates a group of frustrated states. Three types of d orbitals and corresponding frustration states are found in the monolayer Cu3O2 we proposed. The monolayer structure is similar to a synthesized copper oxide sheet on an Au (111) surface. It is a ferromagnetic nanosheet, whose ferromagnetism is related to d-orbital-frustration-induced flat band. In addition, a possible route to experi mentally synthesize Cu3O2 on SiC surface is discussed.
1. Introduction Some special lattices, such as triangle, Kagome and pyrochlore lat tices, are named as frustrated geometries [1–26]. In these lattice, various degrees of freedom of electrons could be frustrated. Fig. 1(a) shows an example of spin frustration in a Kagome lattice, where three magnetic ions reside on the three lattice sites. Once the first two spins align antiparallel, the third one cannot simultaneously minimize its in teractions with both of them, and thus is frustrated. Spin frustration has been studied extensively because it yields fascinate magnetic phenom ena [1,2,27–30], such as spin ice, spin liquid, spin wave, spin glass and so on. Orbital is another degree of freedom of electrons. All types of orbitals in a frustrated lattice also generate frustrations like spin [4,12,27, 31–35]. Fig. 1(b–d) present examples for frustrations of s, p and d or bitals analogy to Fig. 1(a), respectively, where the orbital at question mark cannot simultaneously favor all of the orbital-orbital interactions with its neighbors. Mathematically, s, p, d and f orbitals can be described by rank-0, 1, 2, 3 tensors, respectively, according to their shapes as well as positive and negative of wavefunctions. p orbital is the closest case with spin (a Rank-1 tensor), and its frustration has been explored in some Kagome lattices, such as Kagome graphene [12] and monolayer P2C3 [14]. These previous studies indicate that orbital frustration pro duces frustrated states and Kagome bands including a flat band and two Dirac bands. The flat band gives rise to a series of many-body
phenomena, such as ferromagnetism [36,37], superconducting [38,39], Wigner crystallization [12,40], and anomalous quantum Hall effect [41]. Comparing with p orbitals, d orbitals have more complicated spatial configurations. The atoms possessing d orbitals are transition elements, which have stronger exchange interactions, spin-orbital coupling and magnetism than other atoms. As such, one can expect that d orbital frustration will lead to more emergent phenomena than p orbital frus tration. For example, a system with a d-orbital-induced flat band could become ferromagnetic spontaneously because the transition-metal atoms have strong intrinsic exchange interactions. For a system with a p-orbital-induced flat band, however, its ferromagnetism can only be stimulated by fractional filling of the flat band [12,14,18]. To capture d-orbital-frustration-induced phenomena, a frustrated lattice made of transition elements should be found firstly. It is known that spin frustration has been extensively studied in some complicated transition-metal compounds where the magnetic transition elements form frustrated lattice. Because spin-spin interaction between magnetic atoms can pass through other nonmagnetic atoms, the nonmagnetic atoms have no effect on the frustrated lattice and thus the frustrated lattice is survived. However, d orbital frustration cannot occur in these compounds because orbital-orbital interactions between the transition elements are hindered by other atoms, and thus the frustrated lattice is concealed. This indicates that d orbital frustration can only be observed in the simpler transition-metal networks. Unfortunately, transition
* Corresponding author. School of Physics and Optoelectronics, Xiangtan University, Xiangtan, 411105, Hunan, China. E-mail address: [email protected] (Y. Chen). https://doi.org/10.1016/j.physb.2019.411826 Received 2 August 2019; Received in revised form 17 October 2019; Accepted 25 October 2019 Available online 30 October 2019 0921-4526/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: Hongxia Ge, Physica B, https://doi.org/10.1016/j.physb.2019.411826
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Fig. 1. Schematic views of spin and orbital frustra tions in a Kagome lattice. (a) Spin frustration, where three magnetic ions reside on the three lattice sites. The two arrows represent two spin directions. (b–d) Frustrations of s, p and d orbitals, respectively, where the orbitals at question marks cannot favor all of the orbital-orbital interactions with its neighbors. The green and purple colors represent positive and nega tive wavefunctions. (For interpretation of the refer ences to color in this figure legend, the reader is referred to the Web version of this article.)
elements are not favorable to form frustrated lattices directly in 2D system. In this paper, we first study nine types of d orbital frustrations in a model of Kagome lattice according to the lattice symmetry. Each type of d orbital generates a group of three states, one singlet and one doublet, which is somewhat similar to the cases of p orbitals. Then, we propose a new Cu3O2 monolayer, as shown in Fig. 3(a), in which the Cu atoms form a Kagome lattice. The new monolayer is selected from a series of monolayers X3O2 (X ¼ transition elements) by analysis of structural stability. It is similar to a synthesized copper oxide sheet on an Au (111) surface [42]. The difference is that the monolayer we proposed is exactly planar while the structure supported by Au substrate is buckling. Elec tron properties of the monolayer Cu3O2 are studied. Three types of d orbitals and their corresponding frustrated states are found. Each group of d orbitals generates a set of spin-polarized Kagome band. The splitting of the Kagome band around the Fermi level induces sponta neous ferromagnetism, and thus Cu3O2 monolayer is intrinsically ferromagnetic. Moreover, our calculations indicate that it is a super conductor with a high critical temperature.
irreducible representations A1g and E2g. We first apply A1g projection operator PA1g to one orbital in the Kagome lattice, say ϕ3 . Each symmetry operation in A1g is applied in turn, and then multiply the result by the character of the A1g representation, and then add the results up:
ψ 1 ¼ PA1g ϕ3 ¼ ϕ3 þ ϕ2 þ ϕ1 þ ϕ1 þ ϕ2 þ ϕ3 þ ϕ3 þ ϕ2 þ ϕ1 þ ϕ3 þ ϕ2 þ ϕ1 þ ϕ3 þ ϕ2 þ ϕ1 þϕ1 þ ϕ2 þ ϕ3 þ ϕ3 þ ϕ2 þ ϕ1 þ ϕ3 þ ϕ2 þ ϕ1 (1)
¼ 8ðϕ1 þ ϕ2 þ ϕ3 Þ: After normalization, one can obtain ψ 1 ¼
p1ffiffi ðϕ 1 3
þ ϕ2 þ ϕ3 Þ, whose
orbital configuration is shown in the left panel in Fig. 2(b). Then, we apply the E2g projection operator PE2g to the orbital, and can obtain 1 6
ψ 2 ¼ pffiffi ð2ϕ3
ϕ1
ϕ2 Þ
(2)
which corresponds to the orbital configuration in the middle panel in Fig. 2(b). The irreducible representation E2g generates two degenerate states. In order to get another degenerate state ψ 3 , we pick a symmetry operation in the D6h group (for example, C3), which will convert ψ 2 into a different state and then make a linear combination of the new state with ψ 2 , such that the result is orthogonal to ψ 2 , rffiffi 3 ðϕ ϕ1 Þ ψ 3 ¼ 2C3 ðψ 2 Þ þ ψ 2 ¼ (3) 2 2
2. d orbital frustion in a Kaomge lattice According to atomic theory, d orbital has five kinds of spatial con figurations [33,34,43], i.e., dxy, dyz, dxz, d2x 2-y, d2z , because of its angular momentum quantum number l ¼ 2. The five kinds of orbitals have different forms in different lattices, according to the symmetry of the lattices. For example, in a Kagome lattice, nine kinds of d orbitals shown in Fig. 2(a) could exist. The space group of Kagome lattice is D6h. Its symmetry axes on the plane are along the lines pointing to the triangle centers or normal to these lines (see the dashed lines in Fig. 2(a)). The nine kinds of d orbitals lie on the mirror planes related to the symmetry axes. Obviously, the orbitals d1 ~ d3 in Fig. 2(a) are analogies to dxy, dyz and dxz, while the orbitals d4 ~ d6 and d7 ~ d9 are analogies to d2x 2-y and d2z , respectively. By applying D6h symmetry operations to the d orbitals in Fig. 2(a), we find that they belong to different irreducible representations, and each kind of d orbital generates three states: one singlet and one doublet. We use the orbital d4 in Fig. 2(a) as an example to explain. In Fig. 2(b), we set the orbitals on atoms I, II and III in a Kagome lattice to be ϕ1 , ϕ2 and ϕ3 , respectively, and all of them are d4 orbital. The orbital satisfies two
After normalization we obtain ψ 3 ¼ p1ffi2ffi ðϕ2
ϕ1 Þ, as shown in the
right panel in Fig. 2(b). The two degenerate states exhibit situations of orbital frustration. The irreducible representations of the other 8 d orbitals are given in Table 1. By using the same process, each d orbital produces a group of three states, one singlet and one doublet. This is similar to the cases of p orbital frustration. Fig. 2(c) and (d) present two groups of three states, corresponding to the orbitals d3 and d8 in Fig. 2(a) respectively. Those states corresponding to other orbitals are given in Fig. S1 in supple mentary information (SI).
2
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Fig. 2. (a) Nine possible d orbitals in a Kagome lattice. The dashed lines present symmetry lines of the lattice. The orbitals d1 ~ d3 are analogies to dxy, dyz and dxz, while the orbitals d4 ~ d6 and d7 ~ d9 are analogies to d2x 2-y and d2z , respectively. (b) Three quantum states result from orbital frustration of d4 orbital in (a). The left panel is a singlet, while the right two panels are doublet. (c–d) Quantum states result from orbital frustrations of d3 and d8 orbitals in (a), respectively.
force convergence criteria were set to be 10 5 eV and 10 3 eV/Å, respectively. To avoid interaction between adjacent layers, the vacuum distance normal to the layers was kept to 20 Å. A k-point sampling of 7 � 7 � 1 for the primitive cell is used for the Brillouin zone (BZ) inte gration [50]. In the calculation of their phonon spectra, the primitive cell expanded the 2 � 2 � 1 supercell of convention cell. To account for structural thermal stability, we carried out AIMD simulations based on canonical ensemble [51], for which a 4 � 4 supercell containing 80 atoms was used and the AIMD simulations were performed with a Nose-Hoover thermostat from 300 to 1000 K, respectively.
3. Monolayer X3O2 structures and computation method All transition elements are not favorable to directly form Kagome lattice, and thus we can only consider transition metal compounds to indentify d orbital frustration mentioned above. Because there are lots of transition metal oxides exisiting in nature, we propose a simpler lattice made of transition elements X and oxygen, as shown in Fig. 3(a), in which the transition elements X form a Kagome lattice while the O atoms are used to link them. The primitive cell of the lattice, labeled as X3O2, includes three transition-metal atoms and two O atoms. We first access stabilities of the monolayer structures X3O2 (X ¼ transition elements), by using the first-principles calculations within the density functional theory (DFT) formalism as implemented in VASP [44,45]. The Perdew-Burke-Ernzerhof (PBE) functional [46] was employed for the exchange-correlation term according to generalized gradient approximation (GGA) [47,48]. The projector augmented wave (PAW) method [49] was used to represent the ion-electron interaction, and the kinetic energy cutoff of 800 eV was adopted. The energy and
3.1. Electronic and magnetic properties of monolayer Cu3O2 The calculated results indicate that, in all X3O2 (X ¼ 3d transition elements) structures, only the phonon dispersions of Cu3O2 have no soft mode (see Fig. S2 in SI). We note that the monolayer Cu3O2 in Fig. 3(a) is somewhat similar to one of the thin-film copper oxide layers reported by Moller et al. [42] (they observed a variety of thin-film copper oxide 3
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Fig. 3. (a) A Kagome-like structure Cu3O2, which also represents a series of structures X3O2 (X ¼ transition elements). The primitive cell is given in the dashed box. (b–c) Phonon dispersions of monolayer Cu3O2 in the cases of the monolayer is planar and buckling, respectively. The insets in (b–c) present side views of primitive cells of planar and buckling Cu3O2, respec tively. Snapshots of the equilibrium structures of monolayer Cu3O2 at temperatures of (d) 600 and (e) 1000 K after 20 ps AIMD simulations.
1.78 Å. The cohesive energy is 4.05 eV. Its symmetry belongs to the space group P6/MMM (D6h-1). Band structure of monolayer Cu3O2 is shown in Fig. 4(a). It dem onstrates that the structure is a metal because some energy bands cross over the Fermi level. Meanwhile, it is ferromagnetic because the spin-up and spin-down bands split. The magnetic moment of each Cu atom is 0.17 μB. Considering that the PBE functional could underestimate the electron-electron interactions, band structure of monolayer Cu3O2 is recalculated by employing the Hubbard-based PBE þ U (U ¼ 5 eV) corrective scheme, and the result is shown in Fig. 4(b). It is found that spin splitting is strengthened. There only exist spin-down energy bands around the Fermi level, moreover, the contacting point of conduction bands and valence band is a Dirac point. Therefore, the monolayer Cu3O2 in Fig. 3(a) is a half-metal Dirac material. Two mechanisms are usually used to explain origination of ferro magnetism. One is Stoner criterion, the other is flat-band physics. Stoner mechanism for ferromagnetism [52] captures the physics of exchange interaction: spin-polarized electrons maintain a distance from one another due to Fermi statistics to reduce repulsion. However, it does not take into account the correlation effect. Electrons often remain unpo larized even under very strong interactions since they can still avoid one another by developing highly correlated wave functions [12]. Seen from
Table 1 Irreducible representations for the 9 d orbitals in Fig. 2(a). doublet singlet
d1
d2
d3
d4
d5
d6
d7
d8
d9
E2g A2g
E1g B2g
E1g B1g
E2g A1g
E2g A1g
E2g A1g
E2g A1g
E2g A1g
E2g A1g
morphologies on Au (111) surface). The difference is, the O atoms in the experimental samples locate out of the plane (see the inset in Fig. 3(c)), while the monolayer we proposed is exactly planar (see the inset in Fig. 3(b)). We check stability of the Cu3O2 structure reported in the experimental but without Au substrate. The phonon dispersions in Fig. 3 (c) illustrate that it is unstable. This is the reason why the buckling copper oxide sheet forms chemical bonds with Au substrate. We also examine thermal stability of monolayer Cu3O2 we proposed by per forming AIMD simulations in canonical ensemble. After heating up to the targeted temperature of 600 K for 20 ps, we do not observe any structural decomposition (see Fig. 3(d)). The structure reconstruction only occurs at 1000 K during the 20 ps simulation, as show in Fig. 3(e). Therefore, the monolayer Cu3O2 has a rather high thermodynamic sta bility. After structural optimization, lattice constants of monolayer Cu3O2 are a ¼ b ¼ 6.16 Å. The bond length between Cu and O atoms is
Fig. 4. Band structures of monolayer Cu3O2 by adopting (a) PBE functional and (b) Hubbard-based PBE þ U (U ¼ 5 eV) corrective scheme, respectively. The red and blue lines represent spin-up and spin-down energy bands, respectively. (c) Orbital-projected density of states corresponding to band structure in (b). (For inter pretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) 4
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Fig. 5. Orbital-projected band structures and quan tum states corresponding to the bands at Γ point. (a) Projected energy bands of orbitals dxz and dyz, and the crowded bands in the energy range [-2.6, 2.1] eV is amplified in (c) to show the detail. It is noted that the energy bands in (c) eliminate the wide-energy-range Kagome band in (a). (b) Projected energy bands of orbitals d2z . Insets: Quantum states corresponding to the bands at Γi point (i is the index of spin-down bands). These states in (a–c) are similar to those produced by frustrations of orbitals d3, d5 and d2, respectively.
Fig. 4(c), density of states around the Fermi level is mainly attributed by d2x 2-y and dxy orbitals. Although the density of states is high, splitting of the corresponding bands is weak (see Fig. 4(b)). This indicates that Stoner mechanism has small effect here. In general, spin polarization of common bands is typically not favored because the kinetic energy cost is often larger than the exchange energy gain. However, the flat-band physics gives another picture: when electrons fill in the flat band, the kinetic energy penalty of spin polarization does not exist anymore, hence the exchange interaction stabilizes the polarized state. To analyze the origination of ferromagnetism in monolayer Cu3O2, we calculate orbital-projected band structures. Fig. 5(a) shows energy bands attrib uted by dxz and dyz orbitals. There is a set of spin-polarized Kagome bands: the spin-down bands lie around the Fermi level in the energy range [-1.6, 1.2] eV, while the spin-up bands lie in the energy range [-3.0, 0] eV. The splitting between spin-up and spin-down bands is larger than 1.2 eV, which is the largest splitting in all bands. Therefore, the ferromagnetism of monolayer Cu3O2 is originated from flat-band phys ics, i.e., splitting of the flat band in the Kagome bands. For a system with p-orbital-induced flat bands, its ferromagnetism can only be stimulated by fractional filling of the flat band [12,14]. However, the flat band here does not locate around the Fermi level. We think that spontaneous ferromagnetism of monolayer Cu3O2 is a result of collaboration of flat-band physics and strong exchange interactions of Cu atoms. In the insets of Fig. 5(a), we show quantum states in the spin-down (or spin-up) Kagome bands at Γ point. The state at Γ23 is a singlet, while the two states at Γ20 and Γ19 are doublet. One can find that these states are similar to the three states in Fig. 2(c). As mentioned above, they are generated by frustration of d3 orbital in Fig. 2(a). Meanwhile, one can find that there are some energy bands crowding in the energy range [-2.6, 2.1] eV in Fig. 5(a). We amplify them in Fig. 5(c) to show the detail. It is found that they are also a set of spin-polarized Kagome bands. The insets in Fig. 5(c) show the quantum states in the Kagome bands, which are very similar to those in Fig. S1(b) in SI. It means that these Kagome bands are induced by the frustrated states of d2 orbital in Fig. 2(a). Further, in the energy range [-2.6, 2.2] eV, we find another set of spin-polarized Kagome bands, as shown in Fig. 5(b). The states in the inset of Fig. 5(b) illustrate that they correspond to the frustrated states of d5 orbital (see Fig. S1(e) in SI). Therefore, d orbital frustration discussed above can be observed in monolayer Cu3O2. In addition, we further check stabilities of other X3O2 (X ¼ 4d and 5d transition elements) in Fig. 3(a). The results show that only Ag3O2 and Au3O2 are stable (see Fig. S2 in SI). The two structures exhibit similar electronic properties with Cu3O2 (see Fig. S3 in SI). If spin-orbit coupling (SOC) in monolayer X3O2 (X ¼ Cu, Ag, Au) is considered, some band
crossings will be gapped. For example, the original Dirac point on the Fermi level as well as the crossings between flat bands and other bands are gapped, as shown in Fig. S4 in SI. Since copper oxide is regarded as a prototype system for high Tc superconductor, we calculate superconducting properties of monolayer Cu3O2. The approach used to compute the electron-phonon coupling follows the method employing the QUANTUM-ESPRESSO (QE) pro grams by Wierzbowska, Gironcoli and Gianozzi [53,54] with non-spin-polarization. The Cu and O atoms are represented by ultra-soft pseudopotentials and the kinetic energy and charge density cutoffs are chosen to be 100 and 800 Ry, respectively. The results indicate that monolayer Cu3O2 is a superconductor with a high critical temperature Tc ¼ 10.5 K in the case of nonmagnetism. We expect some further experimental studies can be carried out to validate and extend our findings. Our analyses for structural stabilities already demonstrate the high feasibility to synthesize Cu3O2. Considering that some 2D materials have been grown successfully on SiC substrates [55], we explore the possi bility to synthesize Cu3O2 on SiC substrate. According to our computa tions, the lattice constants of a 2 � 2 supercell of SiC (1 1 1) substrate (a0 ¼ b0 ¼ 6.18 Å) are very close to those of the primitive cell of Cu3O2. The mismatch between them is smaller than 0.49%. The adhesion en ergy between Cu3O2 sheet and SiC substrate is 0.95 eV/atom (see Fig. S5 in the SI for more details). We believe the newly predicted Cu3O2 sheet can be synthesized by the pathway similar to those of other 2D materials on the SiC substrate. 4. Conclusions In summary, we first study d orbital frustrations in a Kagome lattice. Nine types of d orbitals and their corresponding frustrated states in a Kagome lattice are discussed. Each type of d orbitals generates a group of three states, one singlet and two degenerate states. Based on a Kagomelike lattice, a new Cu3O2 monolayer, analogy to a structure reported in experimental, is proposed. It is selected from a series of monolayers X3O2 (X ¼ transition elements) by analysis of structural stability. The structure shows good dynamics and thermodynamics stabilities. Several types of d-orbital-frustration states are found in it. Each type of d orbitals generates a set of Kagome band corresponding to three frustrated states. The splitting of flat band and strong exchange interactions of Cu atoms leads to spontaneous ferromagnetism of monolayer Cu3O2. In addition, we proposed a promising approach to prepare the new sheet by epitaxial growth on a SiC substrate, and hope some experiments can be carried out to validate and extend our findings. 5
H. Ge et al.
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Acknowledgments
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