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AEMnSb2 (AE=Sr, Ba): a new class of Dirac materials To cite this article: M Arshad Farhan et al 2014 J. Phys.: Condens. Matter 26 042201
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Related content - Magnetic phase transitions and superconductivity in strained FeTe A Ciechan, M J Winiarski and M SamselCzekaa - Engineering the electronic properties of silicene by tuning the composition of MoX2 and GaX (X = S,Se,Te) chalchogenide templates E Scalise, M Houssa, E Cinquanta et al. - Unexpected buckled structures and tunable electronic properties in arsenic nanosheets: insights from first-principles calculations Yanli Wang and Yi Ding
Recent citations - Electronic structure of the candidate 2D Dirac semimetal SrMnSb2: a combined experimental and theoretical study Shyama Varier Ramankutty et al - Observation of a two-dimensional Fermi surface and Dirac dispersion in YbMnSb2 Robert Kealhofer et al - Two-dimensional transport and strong spin–orbit interaction in SrMnSb2 Jiwei Ling et al
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 042201 (7pp)
doi:10.1088/0953-8984/26/4/042201
Fast Track Communication
AEMnSb2 (AE = Sr, Ba): a new class of Dirac materials M Arshad Farhan1,2 , Geunsik Lee1 and Ji Hoon Shim1,3,4 1
Department of Chemistry, Pohang University of Science and Technology, Pohang 790-784, Korea Physics Division, PINSTECH, PO Nilore, Islamabad 1482, Pakistan 3 Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea 4 Division of Advanced Nuclear Engineering, Pohang University of Science and Technology, Pohang 790-784, Korea 2
E-mail: [email protected] and [email protected] Received 12 November 2013, revised 9 December 2013 Accepted for publication 11 December 2013 Published 6 January 2014 Abstract
The Dirac fermions of Sb square net in AEMnSb2 (AE = Sr, Ba) are investigated by using first-principles calculation. BaMnSb2 contains Sb square net layers with a coincident stacking of Ba atoms, exhibiting Dirac fermion behavior. On the other hand, SrMnSb2 has a staggered stacking of Sr atoms with distorted zig-zag chains of Sb atoms. Application of hydrostatic pressure on the latter induces a structural change from a staggered to a coincident arrangement of AE ions accompanying a transition from insulator to a metal containing Dirac fermions. The structural investigations show that the stacking type of cation and orthorhombic distortion of Sb layers are the main factors to decide the crystal symmetry of the material. We propose that the Dirac fermions can be obtained by controlling the size of cation and the volume of AEMnSb2 compounds. Keywords: first-principles calculation, density functional theory method, Dirac fermions, band structures, metal–insulator transition (Some figures may appear in colour only in the online journal)
1. Introduction
interesting phenomena related to massless Dirac fermions as observed in graphene. Therefore it would be interesting to explore similar compounds with lesser SOC. Such systems include AEMnSb2 (where AE = Sr, Ba) that we deal with in this paper. BaMnSb2 (Ba112), much like SrMnBi2 , consists of periodic repetition of two kinds of layers along the c axis of a tetragonal unit cell, an (MnSb) layer containing edge shared MnSb4 tetrahedrons and a tripartite (BaSb) layer containing Ba atoms, coincident from the c-axis perspective sandwiching a square net of Sb atoms (figures 1(a) and (c)) [6]. On the other hand, SrMnSb2 (Sr112) has an orthorhombic distortion and the Sb square net is slightly distorted to give a zig-zag chain-like structure displacing the otherwise coincident cations to a staggered conformation around the Sb layer (figures 1(b) and (d)) [7, 8].
Recently, the Dirac fermion behavior in SrMnBi2 was theoretically predicted and then observed by quantum oscillation measurement [1–5]. The atomic structure of SrMnBi2 is characterized by an alternate stacking of (SrBi) and (MnBi) layers. Investigation of the electronic structure suggested that electronic states from (MnBi) have energy levels well separated from the Fermi level, so most electronic states near the Fermi level are from (SrBi) layers or more specifically from the Bi square net where a linear energy–momentum dispersion is realized similar to well-known graphene [5]. However, a significantly large spin–orbit coupling (SOC) in the Bi square net opens an energy gap with parabolic dispersion. The Dirac fermions therefore become massive, making it hard to observe 0953-8984/14/042201+07$33.00
1
c 2014 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 26 (2014) 042201
Figure 1. Crystal structures of (a) BaMnSb2 and (b) SrMnSb2 . (c), (d) Top views of tripartite BaSb and SrSb layers, respectively (lines
mark the lateral unit cell). In (c) and (d), coincident (staggered) stacking of two neighboring Ba (Sr) layers is shown, where light (dark) blue Sr ions are on top (bottom) of the Sb layer in (d). The square net (distorted chain) of Sb is also depicted.
In this paper, we examine the crystal and electronic structures of AEMnSb2 (AE = Sr, Ba) and the relationship between the two by using a first-principles method. It will be shown that Ba112 is metallic with Dirac dispersion but Sr112 does not exhibit the Dirac phenomenon. It is claimed that the main origin of the different electronic structure lies in the distortion of the Sb square net. Also we suggest that the presence of Dirac fermions can be controlled by external pressure.
energy cut and k-mesh used in the calculations are 337 eV and 8 × 8 × 2 respectively. The electronic structure calculations are performed by using the full-potential linearized augmented plane wave method [12] embedded in the WIEN2k package [13]. The generalized gradient approximation (GGA) is utilized for the exchange correlation potential [14]. The k-mesh used for sampling the first Brillouin zone is 21 × 21 × 4 with an RK max of 5.50. A comparison of magnetic phases for AEMnSb2 compounds shows that, similar to SrMnBi2 [5], the ground state with antiferromagnetic configuration is more stable than the ferromagnetic phase by 2.9 and 8.0 meV/f.u. for Sr112 and Ba112 respectively. Therefore, all calculations for AEMnSb2 materials have been performed assuming the checkerboard type antiferromagnetic ground state symmetry. Onsite Coulomb interaction for localized Mn d-electrons is also qualitatively treated by a parameter U = 5 eV. The SOC is also taken into account due to Sb atoms. The structural parameters for equilibrium volumes along with experimental values are listed in tables 1 and 2.
2. Calculation methods
As shown in figure 1, the most important structural difference between Ba112 and Sr112 is that Ba (Sr) atoms have the coincident (staggered) conformation across the Sb layer [9, 10]. The total energy of Ba112 and Sr112 is calculated for both types of arrangement for a range of unit cell volumes using the Vienna ab initio simulation package (VASP) [11], a pseudopotential based method, allowing unit cell shape and internal positions of atoms to relax for a fixed volume. The 2
J. Phys.: Condens. Matter 26 (2014) 042201
Figure 2. (a) Calculated total energies of SrMnSb2 (Sr112) by varying the unit cell volume with the coincident geometry (figures 1(a) and
(c)) and the staggered geometry (figures 1(b) and (d)). (b) Sb–Sb bond distances and Sb–Sb–Sb bond angles. Green squares and red triangles correspond to the coincident and staggered types, respectively. The blue circles represent variation of Sb–Sb–Sb angles (◦ ) in the staggered arrangement.
Table 1. Calculated (experimental) lattice parameters of SrMnSb2
one and the difference decreases with compression, as one can see in the inset of figure 2(a). This is because the increased interaction between AE and Sb layers suppresses the distortion. As it is shown in figure 2(b), the intralayer Sb–Sb distances converge toward a common value and the Sb–Sb–Sb angle approaches 90◦ . So a decrease of the total energy for the staggered conformation is mainly from the distortion of the Sb layer. It is known that Bi square net in SrMnBi2 and CaMnBi2 does not exhibit any distortion due to strong SOC; this orthorhombic distortion of the Sb layer into zig-zag chains is known to have its origins in Peierl’s distortion and is sensitive to the vertical distance between AE and Sb layers [15, 16]. Our calculations indicate that this distortion occurs at sufficiently large vertical distances for both types of conformations. As the distance decreases, the interaction between AE and Sb layers suppresses the distortion with gradual transformation of the Sb layer into a square net. The critical distance where the distortion is suppressed completely depends on the AE atom size and the stacking type. From our results at the experimental volume, staggered and coincident conformations have a vertical distance of ˚ and 2.68 A ˚ respectively between Sr and Sb layers, 2.66 A in good agreement with the experimental distance. Also the chain type distortion of the Sb layer does (does not) occur for the staggered (coincident) conformation. So, the total energy of the staggered conformation is lowered, differing only 18 meV/f.u. relative to the coincident one by two origins, the stacking type and the distortion. From figure 2, one can easily conclude that the relative stability of the two conformations in Sr112 does not change under either positive or negative pressure, so that the crystal symmetry at the ambient condition will not be altered by pressure. However, the orthorhombic distortion is suppressed toward the tetragonal symmetry with compression. A similar approach for Ba112 is also used to calculate the relative total energies by varying the unit cell volume for both the conformations as shown in figure 3(a). Near the
(staggered type lattice, figures 1(b) and (d)), space group = Pnma (62). ˚ (23.19 A) ˚ a = 23.0507 A ˚ (4.42 A) ˚ b = 4.4061 A ˚ (4.46 A) ˚ c = 4.4403 A Sr
4c
x, 1/4, z
x = 0.1155 (0.1156) z = 0.7265 (0.7251)
Mn
4c
x, 1/4, z
x = 0.2502 (0.2501) z = 0.2249 (0.2247)
Sb-1
4c
x, 1/4, z
x = 0.0011 (0.0010) z = 0.2256 (0.2190)
Sb-2
4c
x, 1/4, z
x = 0.3198 (0.3229) z = 0.7251 (0.7250)
Table 2. Calculated (experimental) lattice parameters of BaMnSb2 (coincident type lattice, figures 1(a) and (c)), space group = I4/mmm (139).
˚ (4.53 A) ˚ a = 4.4748 A ˚ (24.34 A) ˚ c = 24.7792 A Ba Mn Sb-1 Sb-2
4e 4d 4c 4e
0, 0, z 0, 1/2, 1/4 0, 1/2, 0 0, 0, z
˚ z = 0.1139 (0.1123 A) ˚ z = 0.3138 (0.3179 A)
3. Results and discussion
To investigate the role of the stacking type, we explore the relative stability of the coincident as well as the staggered phases by allowing full relaxation of atomic positions and unit cell shape within a fixed unit cell volume for both Sr112 and Ba112. Figure 2(a) shows the relative total energies of staggered and coincident conformations of Sr112 with different values of the unit cell volume, modeling the hydrostatic pressure effect. It turns out that the staggered one always has a lower energy value than the coincident 3
J. Phys.: Condens. Matter 26 (2014) 042201
Figure 3. (a) Calculated total energies of BaMnSb2 (Ba112) by varying the unit cell volume with the coincident geometry (figures 1(a) and (c)) and the staggered geometry (figures 1(b) and (d)). (b) Sb–Sb bond distances and Sb–Sb–Sb bond angles. Green squares and red triangles correspond to the coincident and staggered types, respectively. The blue circles represent variation of Sb–Sb–Sb angles (◦ ) in the staggered arrangement.
equilibrium volume, the staggered conformation has a lower energy (11 meV/f.u.) than the coincident one, in disagreement with the experimental observation. Such a discrepancy can be understood as the overestimation of the orthorhombic distortion in the calculation of the Sb layer to lower the total energy which in fact is caused by poor prediction of the vertical distance between Ba and Sb layers, where a larger ˚ for coincident conformation at equilibrium distance 2.82 A volume is predicted compared to the experimental value ˚ In other words, a chain type distortion of the Sb (2.73 A). layer for the staggered conformation is obtained after full relaxation, while the coincident one maintains the square shape. The distortion for the staggered conformation alone lowers the total energy significantly, greater than the energy difference from the different stacking type. However, at reduced volumes, enhanced interaction between Ba and Sb layers suppresses the distortion, as one can see in figure 3(b), thus the stability is mainly decided by the stacking type. Therefore there is a crossover of stability between the two conformations, as shown in figure 3(a). The discrepancy of the relative stability in figure 3 is due to inaccurate prediction of Ba height relative to the Sb layer due to inability of current theories. If we obtain a Ba height consistent with experiment, no distortion of the Sb layer is observed at the equilibrium as well as reduced volumes; a direct consequence of which is lowering the coincident symmetry curve of figure 3(a). Thus the stability is always decided by the stacking type upon compression, where the coincident conformation is favored for Ba112. Further analysis for the preferred geometry of cations around the Sb layer was undertaken by calculating the total energies for both stackings assuming a perfect square shape of the Sb layer. It is found that Ba112 and Sr112 favor the coincident and staggered arrangement of AE atoms, respectively, by the total energy difference of about 10 meV/f.u. It is therefore likely that the cation size which acts to induce pressure on the Sb square net may be an important factor in deciding the stacking type. Our calculations for a hypothetical three layered system
show that the complete suppression occurs at the vertical ˚ (2.7 A) ˚ for Sr, and at 2.8 A ˚ (2.9 A) ˚ in the distance of 2.6 A case of Ba for staggered (coincident) stacking. Considering ˚ and 2.73 A ˚ for Sr112 and the experimental distances of 2.68 A Ba112, respectively, one can understand why Sr112 (Ba112) does (does not) display the distortion. Similar to SrMnBi2 , the Sb square net is likely to engender Dirac fermions in AEMnSb2 (AE = Sr, Ba). Therefore, probing the electronic structures for the coincident conformation involving the Sb square net and the staggered conformation involving the distorted Sb layer is the next logical step. In the case of the coincident geometry, shown in figures 4(a) and (b) for Sr112 and Ba112, respectively, band structure and density of state (DOS) calculations reveal that they are metallic. One can see linearly crossing bands along 0 to M near the Fermi level and additional quasi-linear bands with a small energy gap near high symmetry points X and Y. Similar to SrMnBi2 , these bands are caused by the Sb square net from (AESb) tripartite layers with negligible interlayer interaction. Resulting Fermi surfaces are thus highly two dimensional as shown in figure 4(c). In particular, the hole carriers appearing at the middle of 0 to M are known to behave as Dirac fermions [5]. Therefore the anisotropic Dirac fermion behavior observed in SrMnBi2 [1, 2] is also expected in AEMnSb2 (AE = Sr, Ba). Furthermore, the SOC induced gap is as small as 20 meV as compared to 50 meV for SrMnBi2 [5]. The DOS near the Fermi level is mainly contributed by the p-orbitals of the Sb layer which induces the Dirac related band structures as shown in figure 5. Although there is also contribution from AE atoms at and just below the Fermi level, it corresponds to the bands near the 0 point and is not relevant to the Dirac-like features in the band structure. The calculated band structure and corresponding DOS for the staggered geometry are shown in figures 6(a) and (b) for Sr112 and Ba112, respectively. In comparison with the results in figures 4(a) and (b), one can see pronounced band gaps due to the orthorhombic distortion of the Sb layer. Thus, 4
J. Phys.: Condens. Matter 26 (2014) 042201
Figure 5. Calculated partial density of states for coincident and
staggered geometries of Sr112 and Ba112.
oscillating along the y (inter-chain) direction. Similar to the coincident symmetry, DOS is mainly contributed by the Sb layer only and other atoms do not contribute at Fermi level as evident from figure 5. It should be kept in mind that, for Ba112, the experimental structure has coincident geometry while Sr112 exhibits a staggered arrangement, and the electronic structure is calculated for both types of structure for comparison purposes only. The effect of onsite coulomb interaction ‘U’ on the band structure is explained by figure 7 which shows the calculated band structure for Sr112 (staggered) with and without U. For this typical band structure, the Mn bands, although negligible, predominantly manifest at point M to give a small contribution to the Fermi level. However, the inclusion of U in the calculation pushes them away from the Fermi level, diminishing their contribution. Although our calculation uses U = 5 eV, it is evident that any small value of U would also remove the Mn contribution at Fermi level. Our results show that coincident geometry AEMnSb2 behaves as metallic, while Ba112 with staggered arrangement is an insulator. The electronic structure is strongly affected by the Sb square net distortion, which can be suppressed toward the square net by compressing the Sb layer. Therefore it is possible to have an insulator-to-metal transition with pressure. For example, by reducing the staggered conformation unit cell volume of Sr112 by 10% compared to the equilibrium, equivalent to a theoretical hydrostatic pressure of 4.33 GPa, the calculated band structure shown in figure 6(c) shows a great reduction (∼0.8 eV) in the energy gap at the proposed Dirac point relative to that at an ambient pressure
Figure 4. Calculated band structures and density of states for
(a) SrMnSb2 and (b) BaMnSb2 with coincident geometry. (c) Calculated Fermi surface of BaMnSb2 as in (b).
one can expect that, for staggered geometry, Sr112 shows metallic behavior while Ba112 is insulating in the ground state. Interestingly, the linearly crossing bands between 0 and M which are associated with Dirac fermions are moved away from the Fermi level for both materials. This rearrangement is also related to the chain type distortion of the Sb square net. Additionally, a substantial difference can be seen between the magnitude of gap openings near X and Y. This indicates that the wavefunctions of low energy states near Y are highly 5
J. Phys.: Condens. Matter 26 (2014) 042201
Figure 6. Calculated band structures and densities of states for the staggered type atomic structure (figures 1(b) and (d)) of (a) SrMnSb2 and (b) BaMnSb2 . (c) and (d) show the calculated band structure and Fermi surface for the staggered SrMnSb2 with a 10% reduction of the unit cell volume.
Figure 7. Calculated band structure for Sr112 (staggered), (a) with and (b) without Hubbard U. The Mn bands at point M are pushed away
from the Fermi level without changing the remainder of the band structure.
in figure 6(a). The corresponding Fermi surface (figure 6(d)) shows the emergence of anisotropic pocket located at the middle of 0 to M, which can be associated with the Dirac
fermions. It is logical to deduce that in this case more massless Dirac fermions would be realized if higher pressure is exerted toward the Sb square net. One can therefore, in principle, 6
J. Phys.: Condens. Matter 26 (2014) 042201
References
tailor the electronic properties of this family of materials by controlling the size of AE ions and their distance with the Sb layer.
[1] Park J et al 2011 Phys. Rev. Lett. 107 126402 [2] Wang K, Graf D, Lei H, Tozer S W and Petrovic C 2012 Phys. Rev. B 84 220401 [3] Wang K, Graf D, Wang L, Lei H, Tozer S W and Petrovic C 2012 Phys. Rev. B 85 041101 [4] Wang K, Wang L and Petrovic C 2012 Appl. Phys. Lett. 100 112111 [5] Lee G, Farhan M A, Kim J S and Shim J H 2013 Phys. Rev. B 87 245104 [6] Tremel W and Hoffmann R 1987 J. Am. Chem. Soc. 109 124 [7] Burdett J K and Miller G J 1990 Chem. Mater. 2 12 [8] Sologub O, Hiebl K, Rogl P and Bodak O 1995 J. Alloys Compounds 227 40 [9] Brechtel E, Cordier G and Sch¨afer H 1981 J. Less-Common Met. 79 131 [10] Cordier G and Sch¨afer H 1977 Z. Naturforsch. b 32 383 [11] Kresse G and Furthm¨uller J 1996 Phys. Rev. B 54 11169 [12] Weinert M, Wimmer E and Freeman A J 1982 Phys. Rev. B 26 4571 [13] Blaha P, Schwarz K, Madsen G K H, Kavasnicka D and Luitz J 2001 WIEN2k ed K Schwarz (Austria: Technische Universitat Wien) [14] Perdew J P, Burke K and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865 [15] Mozharivskyj Y and Franzen H F 2002 J. Phys. Chem. B 106 9528 [16] Papoian G A and Hoffmann R 2000 Angew. Chem. Int. Edn 39 2408
4. Conclusion
Our investigation of the structural stability of AEMnSb2 materials highlights that the stacking type and the orthorhombic distortion are the two main factors which dictate crystal symmetry. Moreover, our calculations indicate that among the experimental symmetries low-mass Dirac fermions can be observed in the coincident geometry of BaMnSb2 , while the staggered conformation of SrMnSb2 does not manifest these. Since the size and the distance of the cation from the Sb layer are quite important in such different properties, one can engineer the materials for desirable electronic structures by controlling these two parameters. It is further suggested that SrMnSb2 can be transformed into a BaMnSb2 -like structure with gradual emergence of Dirac fermions by application of pressure. Acknowledgments
This work was supported by the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Nos 2013M2B2A9A 03051257, 2011-0010186, 20110030147).
7
FAST TRACK COMMUNICATION
AEMnSb2 (AE=Sr, Ba): a new class of Dirac materials To cite this article: M Arshad Farhan et al 2014 J. Phys.: Condens. Matter 26 042201
View the article online for updates and enhancements.
Related content - Magnetic phase transitions and superconductivity in strained FeTe A Ciechan, M J Winiarski and M SamselCzekaa - Engineering the electronic properties of silicene by tuning the composition of MoX2 and GaX (X = S,Se,Te) chalchogenide templates E Scalise, M Houssa, E Cinquanta et al. - Unexpected buckled structures and tunable electronic properties in arsenic nanosheets: insights from first-principles calculations Yanli Wang and Yi Ding
Recent citations - Electronic structure of the candidate 2D Dirac semimetal SrMnSb2: a combined experimental and theoretical study Shyama Varier Ramankutty et al - Observation of a two-dimensional Fermi surface and Dirac dispersion in YbMnSb2 Robert Kealhofer et al - Two-dimensional transport and strong spin–orbit interaction in SrMnSb2 Jiwei Ling et al
This content was downloaded from IP address 129.78.139.29 on 28/02/2018 at 14:20
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 042201 (7pp)
doi:10.1088/0953-8984/26/4/042201
Fast Track Communication
AEMnSb2 (AE = Sr, Ba): a new class of Dirac materials M Arshad Farhan1,2 , Geunsik Lee1 and Ji Hoon Shim1,3,4 1
Department of Chemistry, Pohang University of Science and Technology, Pohang 790-784, Korea Physics Division, PINSTECH, PO Nilore, Islamabad 1482, Pakistan 3 Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea 4 Division of Advanced Nuclear Engineering, Pohang University of Science and Technology, Pohang 790-784, Korea 2
E-mail: [email protected] and [email protected] Received 12 November 2013, revised 9 December 2013 Accepted for publication 11 December 2013 Published 6 January 2014 Abstract
The Dirac fermions of Sb square net in AEMnSb2 (AE = Sr, Ba) are investigated by using first-principles calculation. BaMnSb2 contains Sb square net layers with a coincident stacking of Ba atoms, exhibiting Dirac fermion behavior. On the other hand, SrMnSb2 has a staggered stacking of Sr atoms with distorted zig-zag chains of Sb atoms. Application of hydrostatic pressure on the latter induces a structural change from a staggered to a coincident arrangement of AE ions accompanying a transition from insulator to a metal containing Dirac fermions. The structural investigations show that the stacking type of cation and orthorhombic distortion of Sb layers are the main factors to decide the crystal symmetry of the material. We propose that the Dirac fermions can be obtained by controlling the size of cation and the volume of AEMnSb2 compounds. Keywords: first-principles calculation, density functional theory method, Dirac fermions, band structures, metal–insulator transition (Some figures may appear in colour only in the online journal)
1. Introduction
interesting phenomena related to massless Dirac fermions as observed in graphene. Therefore it would be interesting to explore similar compounds with lesser SOC. Such systems include AEMnSb2 (where AE = Sr, Ba) that we deal with in this paper. BaMnSb2 (Ba112), much like SrMnBi2 , consists of periodic repetition of two kinds of layers along the c axis of a tetragonal unit cell, an (MnSb) layer containing edge shared MnSb4 tetrahedrons and a tripartite (BaSb) layer containing Ba atoms, coincident from the c-axis perspective sandwiching a square net of Sb atoms (figures 1(a) and (c)) [6]. On the other hand, SrMnSb2 (Sr112) has an orthorhombic distortion and the Sb square net is slightly distorted to give a zig-zag chain-like structure displacing the otherwise coincident cations to a staggered conformation around the Sb layer (figures 1(b) and (d)) [7, 8].
Recently, the Dirac fermion behavior in SrMnBi2 was theoretically predicted and then observed by quantum oscillation measurement [1–5]. The atomic structure of SrMnBi2 is characterized by an alternate stacking of (SrBi) and (MnBi) layers. Investigation of the electronic structure suggested that electronic states from (MnBi) have energy levels well separated from the Fermi level, so most electronic states near the Fermi level are from (SrBi) layers or more specifically from the Bi square net where a linear energy–momentum dispersion is realized similar to well-known graphene [5]. However, a significantly large spin–orbit coupling (SOC) in the Bi square net opens an energy gap with parabolic dispersion. The Dirac fermions therefore become massive, making it hard to observe 0953-8984/14/042201+07$33.00
1
c 2014 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 26 (2014) 042201
Figure 1. Crystal structures of (a) BaMnSb2 and (b) SrMnSb2 . (c), (d) Top views of tripartite BaSb and SrSb layers, respectively (lines
mark the lateral unit cell). In (c) and (d), coincident (staggered) stacking of two neighboring Ba (Sr) layers is shown, where light (dark) blue Sr ions are on top (bottom) of the Sb layer in (d). The square net (distorted chain) of Sb is also depicted.
In this paper, we examine the crystal and electronic structures of AEMnSb2 (AE = Sr, Ba) and the relationship between the two by using a first-principles method. It will be shown that Ba112 is metallic with Dirac dispersion but Sr112 does not exhibit the Dirac phenomenon. It is claimed that the main origin of the different electronic structure lies in the distortion of the Sb square net. Also we suggest that the presence of Dirac fermions can be controlled by external pressure.
energy cut and k-mesh used in the calculations are 337 eV and 8 × 8 × 2 respectively. The electronic structure calculations are performed by using the full-potential linearized augmented plane wave method [12] embedded in the WIEN2k package [13]. The generalized gradient approximation (GGA) is utilized for the exchange correlation potential [14]. The k-mesh used for sampling the first Brillouin zone is 21 × 21 × 4 with an RK max of 5.50. A comparison of magnetic phases for AEMnSb2 compounds shows that, similar to SrMnBi2 [5], the ground state with antiferromagnetic configuration is more stable than the ferromagnetic phase by 2.9 and 8.0 meV/f.u. for Sr112 and Ba112 respectively. Therefore, all calculations for AEMnSb2 materials have been performed assuming the checkerboard type antiferromagnetic ground state symmetry. Onsite Coulomb interaction for localized Mn d-electrons is also qualitatively treated by a parameter U = 5 eV. The SOC is also taken into account due to Sb atoms. The structural parameters for equilibrium volumes along with experimental values are listed in tables 1 and 2.
2. Calculation methods
As shown in figure 1, the most important structural difference between Ba112 and Sr112 is that Ba (Sr) atoms have the coincident (staggered) conformation across the Sb layer [9, 10]. The total energy of Ba112 and Sr112 is calculated for both types of arrangement for a range of unit cell volumes using the Vienna ab initio simulation package (VASP) [11], a pseudopotential based method, allowing unit cell shape and internal positions of atoms to relax for a fixed volume. The 2
J. Phys.: Condens. Matter 26 (2014) 042201
Figure 2. (a) Calculated total energies of SrMnSb2 (Sr112) by varying the unit cell volume with the coincident geometry (figures 1(a) and
(c)) and the staggered geometry (figures 1(b) and (d)). (b) Sb–Sb bond distances and Sb–Sb–Sb bond angles. Green squares and red triangles correspond to the coincident and staggered types, respectively. The blue circles represent variation of Sb–Sb–Sb angles (◦ ) in the staggered arrangement.
Table 1. Calculated (experimental) lattice parameters of SrMnSb2
one and the difference decreases with compression, as one can see in the inset of figure 2(a). This is because the increased interaction between AE and Sb layers suppresses the distortion. As it is shown in figure 2(b), the intralayer Sb–Sb distances converge toward a common value and the Sb–Sb–Sb angle approaches 90◦ . So a decrease of the total energy for the staggered conformation is mainly from the distortion of the Sb layer. It is known that Bi square net in SrMnBi2 and CaMnBi2 does not exhibit any distortion due to strong SOC; this orthorhombic distortion of the Sb layer into zig-zag chains is known to have its origins in Peierl’s distortion and is sensitive to the vertical distance between AE and Sb layers [15, 16]. Our calculations indicate that this distortion occurs at sufficiently large vertical distances for both types of conformations. As the distance decreases, the interaction between AE and Sb layers suppresses the distortion with gradual transformation of the Sb layer into a square net. The critical distance where the distortion is suppressed completely depends on the AE atom size and the stacking type. From our results at the experimental volume, staggered and coincident conformations have a vertical distance of ˚ and 2.68 A ˚ respectively between Sr and Sb layers, 2.66 A in good agreement with the experimental distance. Also the chain type distortion of the Sb layer does (does not) occur for the staggered (coincident) conformation. So, the total energy of the staggered conformation is lowered, differing only 18 meV/f.u. relative to the coincident one by two origins, the stacking type and the distortion. From figure 2, one can easily conclude that the relative stability of the two conformations in Sr112 does not change under either positive or negative pressure, so that the crystal symmetry at the ambient condition will not be altered by pressure. However, the orthorhombic distortion is suppressed toward the tetragonal symmetry with compression. A similar approach for Ba112 is also used to calculate the relative total energies by varying the unit cell volume for both the conformations as shown in figure 3(a). Near the
(staggered type lattice, figures 1(b) and (d)), space group = Pnma (62). ˚ (23.19 A) ˚ a = 23.0507 A ˚ (4.42 A) ˚ b = 4.4061 A ˚ (4.46 A) ˚ c = 4.4403 A Sr
4c
x, 1/4, z
x = 0.1155 (0.1156) z = 0.7265 (0.7251)
Mn
4c
x, 1/4, z
x = 0.2502 (0.2501) z = 0.2249 (0.2247)
Sb-1
4c
x, 1/4, z
x = 0.0011 (0.0010) z = 0.2256 (0.2190)
Sb-2
4c
x, 1/4, z
x = 0.3198 (0.3229) z = 0.7251 (0.7250)
Table 2. Calculated (experimental) lattice parameters of BaMnSb2 (coincident type lattice, figures 1(a) and (c)), space group = I4/mmm (139).
˚ (4.53 A) ˚ a = 4.4748 A ˚ (24.34 A) ˚ c = 24.7792 A Ba Mn Sb-1 Sb-2
4e 4d 4c 4e
0, 0, z 0, 1/2, 1/4 0, 1/2, 0 0, 0, z
˚ z = 0.1139 (0.1123 A) ˚ z = 0.3138 (0.3179 A)
3. Results and discussion
To investigate the role of the stacking type, we explore the relative stability of the coincident as well as the staggered phases by allowing full relaxation of atomic positions and unit cell shape within a fixed unit cell volume for both Sr112 and Ba112. Figure 2(a) shows the relative total energies of staggered and coincident conformations of Sr112 with different values of the unit cell volume, modeling the hydrostatic pressure effect. It turns out that the staggered one always has a lower energy value than the coincident 3
J. Phys.: Condens. Matter 26 (2014) 042201
Figure 3. (a) Calculated total energies of BaMnSb2 (Ba112) by varying the unit cell volume with the coincident geometry (figures 1(a) and (c)) and the staggered geometry (figures 1(b) and (d)). (b) Sb–Sb bond distances and Sb–Sb–Sb bond angles. Green squares and red triangles correspond to the coincident and staggered types, respectively. The blue circles represent variation of Sb–Sb–Sb angles (◦ ) in the staggered arrangement.
equilibrium volume, the staggered conformation has a lower energy (11 meV/f.u.) than the coincident one, in disagreement with the experimental observation. Such a discrepancy can be understood as the overestimation of the orthorhombic distortion in the calculation of the Sb layer to lower the total energy which in fact is caused by poor prediction of the vertical distance between Ba and Sb layers, where a larger ˚ for coincident conformation at equilibrium distance 2.82 A volume is predicted compared to the experimental value ˚ In other words, a chain type distortion of the Sb (2.73 A). layer for the staggered conformation is obtained after full relaxation, while the coincident one maintains the square shape. The distortion for the staggered conformation alone lowers the total energy significantly, greater than the energy difference from the different stacking type. However, at reduced volumes, enhanced interaction between Ba and Sb layers suppresses the distortion, as one can see in figure 3(b), thus the stability is mainly decided by the stacking type. Therefore there is a crossover of stability between the two conformations, as shown in figure 3(a). The discrepancy of the relative stability in figure 3 is due to inaccurate prediction of Ba height relative to the Sb layer due to inability of current theories. If we obtain a Ba height consistent with experiment, no distortion of the Sb layer is observed at the equilibrium as well as reduced volumes; a direct consequence of which is lowering the coincident symmetry curve of figure 3(a). Thus the stability is always decided by the stacking type upon compression, where the coincident conformation is favored for Ba112. Further analysis for the preferred geometry of cations around the Sb layer was undertaken by calculating the total energies for both stackings assuming a perfect square shape of the Sb layer. It is found that Ba112 and Sr112 favor the coincident and staggered arrangement of AE atoms, respectively, by the total energy difference of about 10 meV/f.u. It is therefore likely that the cation size which acts to induce pressure on the Sb square net may be an important factor in deciding the stacking type. Our calculations for a hypothetical three layered system
show that the complete suppression occurs at the vertical ˚ (2.7 A) ˚ for Sr, and at 2.8 A ˚ (2.9 A) ˚ in the distance of 2.6 A case of Ba for staggered (coincident) stacking. Considering ˚ and 2.73 A ˚ for Sr112 and the experimental distances of 2.68 A Ba112, respectively, one can understand why Sr112 (Ba112) does (does not) display the distortion. Similar to SrMnBi2 , the Sb square net is likely to engender Dirac fermions in AEMnSb2 (AE = Sr, Ba). Therefore, probing the electronic structures for the coincident conformation involving the Sb square net and the staggered conformation involving the distorted Sb layer is the next logical step. In the case of the coincident geometry, shown in figures 4(a) and (b) for Sr112 and Ba112, respectively, band structure and density of state (DOS) calculations reveal that they are metallic. One can see linearly crossing bands along 0 to M near the Fermi level and additional quasi-linear bands with a small energy gap near high symmetry points X and Y. Similar to SrMnBi2 , these bands are caused by the Sb square net from (AESb) tripartite layers with negligible interlayer interaction. Resulting Fermi surfaces are thus highly two dimensional as shown in figure 4(c). In particular, the hole carriers appearing at the middle of 0 to M are known to behave as Dirac fermions [5]. Therefore the anisotropic Dirac fermion behavior observed in SrMnBi2 [1, 2] is also expected in AEMnSb2 (AE = Sr, Ba). Furthermore, the SOC induced gap is as small as 20 meV as compared to 50 meV for SrMnBi2 [5]. The DOS near the Fermi level is mainly contributed by the p-orbitals of the Sb layer which induces the Dirac related band structures as shown in figure 5. Although there is also contribution from AE atoms at and just below the Fermi level, it corresponds to the bands near the 0 point and is not relevant to the Dirac-like features in the band structure. The calculated band structure and corresponding DOS for the staggered geometry are shown in figures 6(a) and (b) for Sr112 and Ba112, respectively. In comparison with the results in figures 4(a) and (b), one can see pronounced band gaps due to the orthorhombic distortion of the Sb layer. Thus, 4
J. Phys.: Condens. Matter 26 (2014) 042201
Figure 5. Calculated partial density of states for coincident and
staggered geometries of Sr112 and Ba112.
oscillating along the y (inter-chain) direction. Similar to the coincident symmetry, DOS is mainly contributed by the Sb layer only and other atoms do not contribute at Fermi level as evident from figure 5. It should be kept in mind that, for Ba112, the experimental structure has coincident geometry while Sr112 exhibits a staggered arrangement, and the electronic structure is calculated for both types of structure for comparison purposes only. The effect of onsite coulomb interaction ‘U’ on the band structure is explained by figure 7 which shows the calculated band structure for Sr112 (staggered) with and without U. For this typical band structure, the Mn bands, although negligible, predominantly manifest at point M to give a small contribution to the Fermi level. However, the inclusion of U in the calculation pushes them away from the Fermi level, diminishing their contribution. Although our calculation uses U = 5 eV, it is evident that any small value of U would also remove the Mn contribution at Fermi level. Our results show that coincident geometry AEMnSb2 behaves as metallic, while Ba112 with staggered arrangement is an insulator. The electronic structure is strongly affected by the Sb square net distortion, which can be suppressed toward the square net by compressing the Sb layer. Therefore it is possible to have an insulator-to-metal transition with pressure. For example, by reducing the staggered conformation unit cell volume of Sr112 by 10% compared to the equilibrium, equivalent to a theoretical hydrostatic pressure of 4.33 GPa, the calculated band structure shown in figure 6(c) shows a great reduction (∼0.8 eV) in the energy gap at the proposed Dirac point relative to that at an ambient pressure
Figure 4. Calculated band structures and density of states for
(a) SrMnSb2 and (b) BaMnSb2 with coincident geometry. (c) Calculated Fermi surface of BaMnSb2 as in (b).
one can expect that, for staggered geometry, Sr112 shows metallic behavior while Ba112 is insulating in the ground state. Interestingly, the linearly crossing bands between 0 and M which are associated with Dirac fermions are moved away from the Fermi level for both materials. This rearrangement is also related to the chain type distortion of the Sb square net. Additionally, a substantial difference can be seen between the magnitude of gap openings near X and Y. This indicates that the wavefunctions of low energy states near Y are highly 5
J. Phys.: Condens. Matter 26 (2014) 042201
Figure 6. Calculated band structures and densities of states for the staggered type atomic structure (figures 1(b) and (d)) of (a) SrMnSb2 and (b) BaMnSb2 . (c) and (d) show the calculated band structure and Fermi surface for the staggered SrMnSb2 with a 10% reduction of the unit cell volume.
Figure 7. Calculated band structure for Sr112 (staggered), (a) with and (b) without Hubbard U. The Mn bands at point M are pushed away
from the Fermi level without changing the remainder of the band structure.
in figure 6(a). The corresponding Fermi surface (figure 6(d)) shows the emergence of anisotropic pocket located at the middle of 0 to M, which can be associated with the Dirac
fermions. It is logical to deduce that in this case more massless Dirac fermions would be realized if higher pressure is exerted toward the Sb square net. One can therefore, in principle, 6
J. Phys.: Condens. Matter 26 (2014) 042201
References
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4. Conclusion
Our investigation of the structural stability of AEMnSb2 materials highlights that the stacking type and the orthorhombic distortion are the two main factors which dictate crystal symmetry. Moreover, our calculations indicate that among the experimental symmetries low-mass Dirac fermions can be observed in the coincident geometry of BaMnSb2 , while the staggered conformation of SrMnSb2 does not manifest these. Since the size and the distance of the cation from the Sb layer are quite important in such different properties, one can engineer the materials for desirable electronic structures by controlling these two parameters. It is further suggested that SrMnSb2 can be transformed into a BaMnSb2 -like structure with gradual emergence of Dirac fermions by application of pressure. Acknowledgments
This work was supported by the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Nos 2013M2B2A9A 03051257, 2011-0010186, 20110030147).
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