- Email: [email protected]
(SrTiO3)n superlattices
Author’s Accepted Manuscript Effects of electron correlations application to Ti atoms on physical properties of (LaMnO3)m /(SrTiO3)n superlattices A. Aezami, M. Abolhassani, M. Elahi www.elsevier.com/locate/jmmm
PII: DOI: Reference:
S0304-8853(15)30930-6 http://dx.doi.org/10.1016/j.jmmm.2015.12.066 MAGMA60990
To appear in: Journal of Magnetism and Magnetic Materials Received date: 16 September 2014 Revised date: 15 July 2015 Accepted date: 20 December 2015 Cite this article as: A. Aezami, M. Abolhassani and M. Elahi, Effects of electron correlations application to Ti atoms on physical properties of (LaMnO 3)m /(SrTiO3)n superlattices, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2015.12.066 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Effects of electron correlations application to Ti atoms on physical properties of (LaMnO3)m /(SrTiO3)n superlattices
A. Aezami1a), M. Abolhassani1, M. Elahi1 1
Department of physics, Science and Research Branch, Islamic Azad University, Tehran, Iran
a)
E-mail: [email protected]
Abstract Magnetic structures and Curie temperatures of the (LaMnO3)m/(SrTiO3)n superlattices (SLm-n) with m=1, 2, 3 and n=1, 2, 3, 8 were investigared, using density functional theory implemented in Quantum-Espresso open source code. By applying on-site coulomb interaction (Hubbard term U) to Ti atoms for all of these superlattices, using StonerWolfarth model, it was found that the magnetic order of interfacial atoms of these superlattices changed to ferromagnetic by implying U=5 eV on Ti atoms. The inclusion of electron-electron correlation with U=5 eV on the Ti atoms for all of the superlattices made the two dimensional electron gas (2DEG) formed at the interfaces, halfmetallic. The obtained values of Curie temperature, calculated within mean field approximation with U=5 eV on the Ti atoms, are in good agreement with the experimental results. Keywords: Superlattice supercell, (LaMnO3) m/(SrTiO3)n SuperLattices (SL), DFT, Stoner-Wolfarth model, MFA, Curie temperature, 2DEG
I.
Introduction
Superlattices prepared by atomic layer epitaxy using molecular beam epitaxy or pulsed laser deposition exhibit novel magnetic properties and electronic structures which have not been observed in their respective bulk constituents. In LaMnO3/SrTiO3 superlattice, atomic layers of LaMnO3 and SrTiO3 are combined which are of current interest, because of their various electronic and magnetic phases. Recently, it was experimentally demonstrated that the interface between the non-magnetic SrTiO3 and antiferromagnetic ordering (A-type) LaMnO3 became ferromagnetic. It has been reported that depending on the strain state of the superlattice which is controlled by the ratio of LaMnO3 to SrTiO3 thicknesses, the sign of the spin-spin coupling at the interface can be reversed [1]. Liu et al [2] have calculated magnetic properties of the LaMnO3/SrTiO3 superlattices using GGA+U approximation with UMn=3 eV and UTi=0 eV. They have found that the LaMnO3 layer in these superlattices favor the ferromagnetic metallic state 1
rather than A-type antiferromagnetic insulator and it is revealed that tiny magnetic moment has been induced to Ti ions of the SrTiO3 layer near the interface. In this work, we have investigated the effects of the inclusion of electron-electron correlations on the Ti atoms in LaMnO3/SrTiO3 superlattices with m=1, 2, 3 and n=1, 2, 3, 8 unit cell using Density Functional Theory (DFT) calculations [3-5]. II.
Structural and computational details
In this investigation, we have chosen the cubic perovskite structure of LaMnO3 and SrTiO3 in order to study their electronic and magnetic properties using the plane-wave pseudopotential method implemented in Quantum-Espresso open-Source code. We have calculated Hubbard U parameter based on linear response approach [6-7] for Mn and Ti atoms in LaMnO3 and SrTiO3, respectively. The obtained values are 3.618 eV and 4.831 eV for Mn and Ti atoms in LaMnO3 and SrTiO3, respectively, which are comparable to other reported values of 3.5 eV and 5 eV respectively [8-11]. Our calculations revealed that the bulk SrTiO3 has the same perovskite structure as bulk LaMnO3 but are 0
electronically very different. The lattice parameters for LaMnO3 and SrTiO3 are calculated to be 3.874 A , 0
3.911 A , respectively. In bulk LaMnO3, electronic configuration of Mn+3 is t2g3 eg1 while in bulk SrTiO3, the conduction bonds correspond to the bands composed of mainly Ti t2g and eg states [13] that has an empty t2g
valence band. At the ground state, bulk LaMnO3 is an antiferromagnetic (A-type) Mott-
insulator and bulk SrTiO3 is a paramagnetic band-insulator [14].
Fig. 1. Schematic structure of the repetition of atomic layers in the SL1-2 superlattice along z axiz.
2
. Fig. 1 schematically show a prototype of (LaMnO3)m / (SrTiO3)n superlattice with m=1 and n=2. In each superlattice, Mn0 and Ti0 atoms are interior atoms inside the LaMnO3 and SrTiO3 parts respectively and Mn1 and Ti1 atoms are interfacial atoms surrounded by SrO and LaO layers. LaMnO3/SrTiO3 superlattices are grown on SrTiO3 substrate so we have taken lattice parameter a (in xy-plane) in coincidence with the experimental lattice parameter of the substrate (SrTiO3) and the lattice parameter c 0
0
(in z or out of plane axis) are taken to be 3.919 A and 3.787 A for SrTiO3 and LaMnO3, respectively, such that our superlattices to preserve their bulk volumes [15]. From now on, we have labeled (LaMnO3)m/(SrTiO3)n SuperLattices with SLm-n, for example the (LaMnO3)1/(SrTiO3)2 superlattice is denoted by SL1-2. III.
Result and discussions
1. Magnetic ordering of interfacial atoms In this section, we use the Stoner criterion to identify the magnetic ordering of interfacial atoms (ferro or antiferro-magnetic). Stoner model is based on a criterion for magnetism that is ascribed to the localized d electrons and is directly applicable to transition metals. By minimizing the energy, a condition for magnetism analogous to the stoner criterion arises which is U ( E f ) 1 . Where ( E f ) is the density of states of a paramagnetic state at the Fermi energy. So, the spin↑ and spin↓ bands are splitted by an energy gap which is proportional to the magnetization. This so called exchange splitting (∆) is an important concept in theories of band ferromagnetism [16]. Note that, in absence of on-site electron interaction (i.e. U=0), the formation of spontaneous magnetic moment changes and leads to the gain in the exchange 1 energy by E x Im , where I is the Stoner parameter when U=0. Thus, for the appearance of 2
ferromagnetism, the Stoner criterion, I ( E f ) 1 , must be satisfied. Generally, if the Stoner criterion is satisfied, the ferromagnetic state is favorable [11, 16]. By the assumption of ( E f ) being symmetric around Ef in energy interval ∆, it is easy to find that by minimizing the total energy, the Stoner parameter is related to the exchange splitting by the relation I
. The obtained values of ∆ and ( E f ) using first m
principles calculations are gathered in Table 1 for (LaMnO3)m/(SrTiO3)n superlattices with U=0 eV and U=5 eV for Ti interfacial atoms and U=3.5 eV for Mn interfacial atoms.
3
Table 1. Exchange splitting (∆) and the parameters related to Stoner criterion on the interface, for Ti atoms with U=0 eV and U=5 eV and for Mn atoms with U=3.5 eV calculated for the SL1-1, SL1-2, SL1-3, SL1-8, SL3-2, SL38 and SL2-8 superlattices.
∆ (eV)
Iρ (Ef)
∆ (eV)
Uρ (Ef)
∆ (eV)
Uρ (Ef)
superlattice
UTi=0
UTi=0
UMn=3.5 eV
UMn=3.5 eV
UTi=5 eV
UTi=5 eV
SL1-1
0.14
0.86
1.89
5.18
0.44
1.05
SL1-2
0.21
0.90
2.03
8.93
0.21
1.45
SL1-3
0.23
0.88
2.12
8.79
0.12
1.65
SL1-8
0.19
1.00
2.54
10.54
0.09
2.30
SL3-2
0.43
3.85
2.59
10.01
0.43
3.15
SL3-8
0.41
1.34
2.51
15.01
0.23
2.00
SL2-8
0.38
1.05
2.46
11.31
0.18
1.55
As the third column of Table 1 illustrates when UTi=0, for Ti1 (interfacial atom) the Stoner criterion is not satisfied for the SL1-1, SL1-2 and SL1-3 superlattices but is satisfied for the SL1-8, SL3-2, SL3-8 and SL2-8 superlattices. This justifies the antiferromagnetic ordering for Ti1 atoms of SL1-1, SL1-2 and SL1-3 superlattices and the ferromagnetic ordering of Ti1 atoms for SL1-8, SL3-2, SL3-8 and SL2-8 superlattices in the interfacial layers. For SL1-8 for which I ( E f ) equals to one, the exchange interaction energy calculations gathered in Table 3 shows that this superlattice has ferromagnetic order. Furthermore, the fifth column of Table 1 shows that for Mn1 (interfacial atom) the Stoner criterion is satisfied for all of the superlattices, so the ferromagnetic ordering for Mn1 atom in the interfacial layers is justified. Then we applied an on-site Coulomb repulsion U to the Ti atoms in our superlattices. As the seventh column of Table 1 shows that in this situation, the stoner criterion is also satisfied for Ti1 atoms in the interfacial layers for all superlattices. By investigating the density of states of majority and minority spins for all our superlattices, we observed that the inclusion of the on-site coulomb interaction with value of U=5 eV led to half-metallicity as it can be seen in Figures 2(a) and 2(b) and it also enhanced the stabilities of the interface magnetizations in all of superlattices as are presented in Table 2.
4
Fig. 2. Total Dos of majority and minority spin for the SL1-1, SL1-2, SL1-3 and SL1-8 superlattices (a), SL3-2, SL3-8 and SL2-8 superlattices (b) for UTi=0 eV and UTi=5 eV. The Fermi energy is denoted by the vertical dotted line. The densities of states for spin up have shown by “↑” and the densities of states for spin down have shown by “↓”.
5
Table 2. Calculated total magnetic moments of superlattices. The results indicate that the total magnetic moment increases with the inclusion of the on-site coulomb interaction with value of U=5 eV on the Ti atoms. Magnetic moment (μB)
Magnetic moment (μB)
Superlattice
With UTi=0
With UTi=5 eV
SL1-1
3.89
4.001
SL1-2
3.72
4.000
SL1-3
3.60
4.002
SL1-8
3.18
4.001
Then, to illustrate the magnetic configuration, exchange interaction energies for nearest neighbor atoms of all our superlattices were calculated. The obtained results are shown in Table 3. The exchange interaction energy J is defined as the energy difference between the ferromagnetic alignment and antiferromagnetic alignment of two neighboring atoms and a negative sign of J corresponds to ferromagnetic interaction, while a positive one corresponds to an antiferromagnetic interaction. As it can be seen in Table 3, the exchange interaction energy for neighboring Ti atoms with UTi=5 eV are more larger than the one with UTi=0 eV. In addition, we calculated the exchange interaction energy for all nearest neighboring atoms (i.e. Mn and Ti atoms). The results show that by taking into account electronelectron correlations on Ti atoms, the ferromagnetic ordering is induced in all of our superlattices at interfacial layers.
6
Table 3. Exchange interaction energies calculated for Ti0-Ti0 atoms ( J 1 ), Ti1-Ti1 atoms ( J 2 ) and Ti0-Ti1 atoms ( J 3 ) using LDA+U approximation for both values of UTi=0 eV and UTi=5 eV. A negative sign of J corresponds to ferromagnetic interaction, while a positive one corresponds to an antiferromagnetic interaction.
J1
superlattice
J2
(eV)
J3
(eV)
(eV)
SL1-1
UTi=0 eV -
UTi=5 eV -
UTi=0 eV -0.19×10-6
UTi=5 eV -1.4×10-3
UTi=0 eV -
UTi=5 eV -
SL1-2
-
-
-0.13×10-6
-1.01×10-3
0.08×10-6
0.84×10-3
SL1-3
-3.8×10-6
-3.1×10-3
0.09×10-6
-0.96×10-3
-0.6×10-6
-0.65×10-3
SL1-8
0.8×10-6
3.2×10-3
0.05×10-6
-0.75×10-3
2.5×10-6
0.38×10-3
SL3-2
-
-
-
-
-0.1×10-6
-0.86×10-3
SL3-8
-0.9×10-6
-3.8×10-3
-
-
-3.9×10-6
-0.74×10-3
SL2-8
-0.7×10-6
-2.5×10-3
-
-
-4.3×10-6
-0.65×10-3
2. Curie Temperatues Curie temperatures for all of our superlattices were estimated based on statistical mechanics approaches to Heisenberg Hamiltonian, with the exchange interaction energies calculated by Density Functional Theory. Among different approaches for estimation of the Curie temperature, mean field approximation (MFA) gives a qualitative description of the magnetization in whole temperatures [17]. In this article, we used this approximation to calculate the Curie temperatures for superlattices under our investigation. The MFA estimate of the Curie temperature is given by the following equation [18, 19]: k BTcMFA
2 J 0i 3 i 0
(1)
Where J 0i s are the exchange interaction energies of all neighboring atoms. Using Eq. 1, the Curie temperature of each of our superlattices is given by k BTcMFA
2 ,Ti J 0Tii ,Ti J Mn,Mn J 0Mn i 3 i 0 0i
(2)
7
First, Curie temperatures (Tc) were calculated for the cases without the inclusion of on-site Coulomb interaction U on Ti atoms in our superlattices, which the results are gathered in the second column of Table 4. In these calculations the exchange interaction energies between the Mn atoms (consisting of the exchange interaction energy of Mn0-Mn0, Mn0-Mn1, Mn1-Mn1 pair atoms) were only considered because the values of the other pair-wise exchange interaction energies such as Mn-Ti and Ti-Ti are very smaller than that of Mn atoms in all superlattices. For instance, our calculations showed tha in SL1-1 superlattice the exchange interaction energies of Mn1-Mn1 pair is -8 meV in contrast to the exchange interaction energies of Ti1-Ti1 pair which is of -0.00019 meV, similarly in SL3-8 superlattice the exchange interaction energy of Mn0-Mn0 pair is of -3 meV while the exchange interaction energies of Ti0-Ti0 and Mn0-Ti1 pairs are of -0.0009 meV and -0.0003 meV, respectively.
Table 4. Curie temperatures calculated with U=0 and U=5 eV applied to Ti atoms for all superlattices under our investigation, using mean field approximation (MFA). Tc (K)
Tc (K)
Superlattice
[with UTi=0 eV]
[with UTi=5 eV]
SL1-1
82.77
109.92
SL1-2
68.07
85.95
SL1-3
58.79
76.27
SL1-8
40.23
53.94
SL3-2
44.87
91.67
SL3-8
31.72
71.79
SL2-8
27.85
62.51
Then, Cuie temperatures were calculated by inclusion of on-site Coulomb interaction on the Ti atoms with moderate values of U=5 eV. Our calculations showed that in this situation, the exchange interaction energies of Mn-Ti and Ti-Ti pairs are comparable to the exchange interaction energy of Mn-Mn pair. For instance, in SL1-1 superlattice, the exchange interaction energy of Mn1-Mn1 pair is of -4.9 meV compared to the exchange interaction energies of Ti1-Ti1 pair which is of -1.4 meV, and similarly in SL3-8 superlattices, the exchange interaction energy of Mn0-Mn0 pair is of -3.1 meV and the exchange interaction energies of Ti0-Ti0 and Mn0-Ti1 pairs are of -3.8 meV and -0.94 meV, respectively. So Tc calculations were performed using the exchange interaction energies of all Mn-Mn, Mn-Ti and Ti-Ti
8
pairs, consisting of Mn1-Mn1, Mn1-Mn0, Mn1-Ti1, Mn1-Ti0, Mn0-Mn0, Mn0-Ti1, Ti1-Ti1, Ti0-Ti1 and Ti0-Ti0 pairs. The calculated results are gathered in the third column of Table 4. The Curie temperature values calculated with the inclusion of moderate U=5 eV on the Ti atoms are in good agreement with the experimental results [20].
3. Conclusion In summary, we have studied the exchange interaction and magnetic configuration of the (LaMnO3)m/(SrTiO3)n superlattices with m=1, 2, 3 and n=1, 2, 3, 8 in the presence of on-site Coulomb interaction on Ti atom. We calculated the exchange interaction energies for neighboring strong moment atoms. To make sure of the magnetic configurations of atoms located at the interface, we used the StonerWolfarth model. The half metallic 2DEG were observed in these superlattices by applying of electronelectron correlations with U=5 eV to Ti atoms. Furthermore, by taking into account the electron-electron correlation on the Ti atoms with UTi=5 eV, we found that the Curie temperatures of all superlattices calculated by mean field approximation, are in good agreement with the experimental results.
Acknowledgements We thank Dr. I. Abdolhosseini Sarsari from Isfahan University of Technology for support and helpful discussions. References [1] J. G. Barriocanal, et al, Nature. Commun. 1 (2010) 82. [2] H. M. Liu, et al, J. Appl. Phys. 113 (2013) 17D902. [3] G. Parr, and W. Yang., Density-Functional Theory of Atoms & Molecules (Oxford University Press. Newyork (1989). [4] N. D. Mermin., Phys. Rev. A. 137 (1965) 1441. [5] A. I. Liechtenstein, et al., Phys. Rev. B. 52 (1995) R5467. [6] V. I. Anisimov, et al., J. Phys.: Condens. Matter. 9 (1997) 767. [7] I. V. Solovyev, et al., Phys. Rev. B. 50 (1994) 16861. [8] B. R. Nada, and S. Satpathy, Phys. Rev. Lett. 101 (2008) 127201. [9] E. Dagotto, T. Hotta, and A. Moreo, phys. Reports. 344 (2001) 1-153.
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[10] J. Kanamori, J. Appl. Phys. 31 (1960) 45-235. [11] K. Janicka, et al., J. Appl. Phys. 103 (2008) 07B508. [12] R. Pencheva, et al., Phys. Rev. B. 74 (2006) 035112. [13] K. V. Benthem and C. Elsasser, J. Appl. Phys. 90 (2001) 12. [14] J. G. Barriocanal, et al, Adv. Mater. 22 (2010) 627-632. [15] R. B. K. Nada and S. Satpathy, Phys. Rev. B. 79 (2009) 054428. [16] P. Fazekas, Lecture Notes on Electron Correlation and magnetism (world Scientific, Singapore) (1999). [17] http://nbn-resolving.de/urn/resolver.pl?urn=nbn%3Ade%3Agbv%3A3-000011020. [18] L. Sheng, et al, Phys. Rev. Lett. 79 (2010) 1710. [19] Y. Liu, et al, Phys. Rev. B. 82 (2010) 094435. [20] W. S. Choi, et al, Phys. Rev. B. 83 (2011) 195113.
Highlights
Calculated the magnetic structure and Curie temperature of the (LaMnO 3)m/(SrTiO3)n superlattices (SLm-n) with m=1, 2, 3 and n=1, 2, 3, 8 by mean field approximation.
Using Stoner-Wolfarth model for all of these superlattices before and after adding U term on the Ti atoms.
By implying U=5 eV on the Ti atoms, the magnetic order of interfacial atoms of these superlattices has changed to ferromagnetic.
The 2DEG formed at the interface half-metallic have made in these superlattices by the inclusion of electron-electron correlation with U=5 eV on the Ti atoms for all of the superlattices.
10
PII: DOI: Reference:
S0304-8853(15)30930-6 http://dx.doi.org/10.1016/j.jmmm.2015.12.066 MAGMA60990
To appear in: Journal of Magnetism and Magnetic Materials Received date: 16 September 2014 Revised date: 15 July 2015 Accepted date: 20 December 2015 Cite this article as: A. Aezami, M. Abolhassani and M. Elahi, Effects of electron correlations application to Ti atoms on physical properties of (LaMnO 3)m /(SrTiO3)n superlattices, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2015.12.066 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Effects of electron correlations application to Ti atoms on physical properties of (LaMnO3)m /(SrTiO3)n superlattices
A. Aezami1a), M. Abolhassani1, M. Elahi1 1
Department of physics, Science and Research Branch, Islamic Azad University, Tehran, Iran
a)
E-mail: [email protected]
Abstract Magnetic structures and Curie temperatures of the (LaMnO3)m/(SrTiO3)n superlattices (SLm-n) with m=1, 2, 3 and n=1, 2, 3, 8 were investigared, using density functional theory implemented in Quantum-Espresso open source code. By applying on-site coulomb interaction (Hubbard term U) to Ti atoms for all of these superlattices, using StonerWolfarth model, it was found that the magnetic order of interfacial atoms of these superlattices changed to ferromagnetic by implying U=5 eV on Ti atoms. The inclusion of electron-electron correlation with U=5 eV on the Ti atoms for all of the superlattices made the two dimensional electron gas (2DEG) formed at the interfaces, halfmetallic. The obtained values of Curie temperature, calculated within mean field approximation with U=5 eV on the Ti atoms, are in good agreement with the experimental results. Keywords: Superlattice supercell, (LaMnO3) m/(SrTiO3)n SuperLattices (SL), DFT, Stoner-Wolfarth model, MFA, Curie temperature, 2DEG
I.
Introduction
Superlattices prepared by atomic layer epitaxy using molecular beam epitaxy or pulsed laser deposition exhibit novel magnetic properties and electronic structures which have not been observed in their respective bulk constituents. In LaMnO3/SrTiO3 superlattice, atomic layers of LaMnO3 and SrTiO3 are combined which are of current interest, because of their various electronic and magnetic phases. Recently, it was experimentally demonstrated that the interface between the non-magnetic SrTiO3 and antiferromagnetic ordering (A-type) LaMnO3 became ferromagnetic. It has been reported that depending on the strain state of the superlattice which is controlled by the ratio of LaMnO3 to SrTiO3 thicknesses, the sign of the spin-spin coupling at the interface can be reversed [1]. Liu et al [2] have calculated magnetic properties of the LaMnO3/SrTiO3 superlattices using GGA+U approximation with UMn=3 eV and UTi=0 eV. They have found that the LaMnO3 layer in these superlattices favor the ferromagnetic metallic state 1
rather than A-type antiferromagnetic insulator and it is revealed that tiny magnetic moment has been induced to Ti ions of the SrTiO3 layer near the interface. In this work, we have investigated the effects of the inclusion of electron-electron correlations on the Ti atoms in LaMnO3/SrTiO3 superlattices with m=1, 2, 3 and n=1, 2, 3, 8 unit cell using Density Functional Theory (DFT) calculations [3-5]. II.
Structural and computational details
In this investigation, we have chosen the cubic perovskite structure of LaMnO3 and SrTiO3 in order to study their electronic and magnetic properties using the plane-wave pseudopotential method implemented in Quantum-Espresso open-Source code. We have calculated Hubbard U parameter based on linear response approach [6-7] for Mn and Ti atoms in LaMnO3 and SrTiO3, respectively. The obtained values are 3.618 eV and 4.831 eV for Mn and Ti atoms in LaMnO3 and SrTiO3, respectively, which are comparable to other reported values of 3.5 eV and 5 eV respectively [8-11]. Our calculations revealed that the bulk SrTiO3 has the same perovskite structure as bulk LaMnO3 but are 0
electronically very different. The lattice parameters for LaMnO3 and SrTiO3 are calculated to be 3.874 A , 0
3.911 A , respectively. In bulk LaMnO3, electronic configuration of Mn+3 is t2g3 eg1 while in bulk SrTiO3, the conduction bonds correspond to the bands composed of mainly Ti t2g and eg states [13] that has an empty t2g
valence band. At the ground state, bulk LaMnO3 is an antiferromagnetic (A-type) Mott-
insulator and bulk SrTiO3 is a paramagnetic band-insulator [14].
Fig. 1. Schematic structure of the repetition of atomic layers in the SL1-2 superlattice along z axiz.
2
. Fig. 1 schematically show a prototype of (LaMnO3)m / (SrTiO3)n superlattice with m=1 and n=2. In each superlattice, Mn0 and Ti0 atoms are interior atoms inside the LaMnO3 and SrTiO3 parts respectively and Mn1 and Ti1 atoms are interfacial atoms surrounded by SrO and LaO layers. LaMnO3/SrTiO3 superlattices are grown on SrTiO3 substrate so we have taken lattice parameter a (in xy-plane) in coincidence with the experimental lattice parameter of the substrate (SrTiO3) and the lattice parameter c 0
0
(in z or out of plane axis) are taken to be 3.919 A and 3.787 A for SrTiO3 and LaMnO3, respectively, such that our superlattices to preserve their bulk volumes [15]. From now on, we have labeled (LaMnO3)m/(SrTiO3)n SuperLattices with SLm-n, for example the (LaMnO3)1/(SrTiO3)2 superlattice is denoted by SL1-2. III.
Result and discussions
1. Magnetic ordering of interfacial atoms In this section, we use the Stoner criterion to identify the magnetic ordering of interfacial atoms (ferro or antiferro-magnetic). Stoner model is based on a criterion for magnetism that is ascribed to the localized d electrons and is directly applicable to transition metals. By minimizing the energy, a condition for magnetism analogous to the stoner criterion arises which is U ( E f ) 1 . Where ( E f ) is the density of states of a paramagnetic state at the Fermi energy. So, the spin↑ and spin↓ bands are splitted by an energy gap which is proportional to the magnetization. This so called exchange splitting (∆) is an important concept in theories of band ferromagnetism [16]. Note that, in absence of on-site electron interaction (i.e. U=0), the formation of spontaneous magnetic moment changes and leads to the gain in the exchange 1 energy by E x Im , where I is the Stoner parameter when U=0. Thus, for the appearance of 2
ferromagnetism, the Stoner criterion, I ( E f ) 1 , must be satisfied. Generally, if the Stoner criterion is satisfied, the ferromagnetic state is favorable [11, 16]. By the assumption of ( E f ) being symmetric around Ef in energy interval ∆, it is easy to find that by minimizing the total energy, the Stoner parameter is related to the exchange splitting by the relation I
. The obtained values of ∆ and ( E f ) using first m
principles calculations are gathered in Table 1 for (LaMnO3)m/(SrTiO3)n superlattices with U=0 eV and U=5 eV for Ti interfacial atoms and U=3.5 eV for Mn interfacial atoms.
3
Table 1. Exchange splitting (∆) and the parameters related to Stoner criterion on the interface, for Ti atoms with U=0 eV and U=5 eV and for Mn atoms with U=3.5 eV calculated for the SL1-1, SL1-2, SL1-3, SL1-8, SL3-2, SL38 and SL2-8 superlattices.
∆ (eV)
Iρ (Ef)
∆ (eV)
Uρ (Ef)
∆ (eV)
Uρ (Ef)
superlattice
UTi=0
UTi=0
UMn=3.5 eV
UMn=3.5 eV
UTi=5 eV
UTi=5 eV
SL1-1
0.14
0.86
1.89
5.18
0.44
1.05
SL1-2
0.21
0.90
2.03
8.93
0.21
1.45
SL1-3
0.23
0.88
2.12
8.79
0.12
1.65
SL1-8
0.19
1.00
2.54
10.54
0.09
2.30
SL3-2
0.43
3.85
2.59
10.01
0.43
3.15
SL3-8
0.41
1.34
2.51
15.01
0.23
2.00
SL2-8
0.38
1.05
2.46
11.31
0.18
1.55
As the third column of Table 1 illustrates when UTi=0, for Ti1 (interfacial atom) the Stoner criterion is not satisfied for the SL1-1, SL1-2 and SL1-3 superlattices but is satisfied for the SL1-8, SL3-2, SL3-8 and SL2-8 superlattices. This justifies the antiferromagnetic ordering for Ti1 atoms of SL1-1, SL1-2 and SL1-3 superlattices and the ferromagnetic ordering of Ti1 atoms for SL1-8, SL3-2, SL3-8 and SL2-8 superlattices in the interfacial layers. For SL1-8 for which I ( E f ) equals to one, the exchange interaction energy calculations gathered in Table 3 shows that this superlattice has ferromagnetic order. Furthermore, the fifth column of Table 1 shows that for Mn1 (interfacial atom) the Stoner criterion is satisfied for all of the superlattices, so the ferromagnetic ordering for Mn1 atom in the interfacial layers is justified. Then we applied an on-site Coulomb repulsion U to the Ti atoms in our superlattices. As the seventh column of Table 1 shows that in this situation, the stoner criterion is also satisfied for Ti1 atoms in the interfacial layers for all superlattices. By investigating the density of states of majority and minority spins for all our superlattices, we observed that the inclusion of the on-site coulomb interaction with value of U=5 eV led to half-metallicity as it can be seen in Figures 2(a) and 2(b) and it also enhanced the stabilities of the interface magnetizations in all of superlattices as are presented in Table 2.
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Fig. 2. Total Dos of majority and minority spin for the SL1-1, SL1-2, SL1-3 and SL1-8 superlattices (a), SL3-2, SL3-8 and SL2-8 superlattices (b) for UTi=0 eV and UTi=5 eV. The Fermi energy is denoted by the vertical dotted line. The densities of states for spin up have shown by “↑” and the densities of states for spin down have shown by “↓”.
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Table 2. Calculated total magnetic moments of superlattices. The results indicate that the total magnetic moment increases with the inclusion of the on-site coulomb interaction with value of U=5 eV on the Ti atoms. Magnetic moment (μB)
Magnetic moment (μB)
Superlattice
With UTi=0
With UTi=5 eV
SL1-1
3.89
4.001
SL1-2
3.72
4.000
SL1-3
3.60
4.002
SL1-8
3.18
4.001
Then, to illustrate the magnetic configuration, exchange interaction energies for nearest neighbor atoms of all our superlattices were calculated. The obtained results are shown in Table 3. The exchange interaction energy J is defined as the energy difference between the ferromagnetic alignment and antiferromagnetic alignment of two neighboring atoms and a negative sign of J corresponds to ferromagnetic interaction, while a positive one corresponds to an antiferromagnetic interaction. As it can be seen in Table 3, the exchange interaction energy for neighboring Ti atoms with UTi=5 eV are more larger than the one with UTi=0 eV. In addition, we calculated the exchange interaction energy for all nearest neighboring atoms (i.e. Mn and Ti atoms). The results show that by taking into account electronelectron correlations on Ti atoms, the ferromagnetic ordering is induced in all of our superlattices at interfacial layers.
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Table 3. Exchange interaction energies calculated for Ti0-Ti0 atoms ( J 1 ), Ti1-Ti1 atoms ( J 2 ) and Ti0-Ti1 atoms ( J 3 ) using LDA+U approximation for both values of UTi=0 eV and UTi=5 eV. A negative sign of J corresponds to ferromagnetic interaction, while a positive one corresponds to an antiferromagnetic interaction.
J1
superlattice
J2
(eV)
J3
(eV)
(eV)
SL1-1
UTi=0 eV -
UTi=5 eV -
UTi=0 eV -0.19×10-6
UTi=5 eV -1.4×10-3
UTi=0 eV -
UTi=5 eV -
SL1-2
-
-
-0.13×10-6
-1.01×10-3
0.08×10-6
0.84×10-3
SL1-3
-3.8×10-6
-3.1×10-3
0.09×10-6
-0.96×10-3
-0.6×10-6
-0.65×10-3
SL1-8
0.8×10-6
3.2×10-3
0.05×10-6
-0.75×10-3
2.5×10-6
0.38×10-3
SL3-2
-
-
-
-
-0.1×10-6
-0.86×10-3
SL3-8
-0.9×10-6
-3.8×10-3
-
-
-3.9×10-6
-0.74×10-3
SL2-8
-0.7×10-6
-2.5×10-3
-
-
-4.3×10-6
-0.65×10-3
2. Curie Temperatues Curie temperatures for all of our superlattices were estimated based on statistical mechanics approaches to Heisenberg Hamiltonian, with the exchange interaction energies calculated by Density Functional Theory. Among different approaches for estimation of the Curie temperature, mean field approximation (MFA) gives a qualitative description of the magnetization in whole temperatures [17]. In this article, we used this approximation to calculate the Curie temperatures for superlattices under our investigation. The MFA estimate of the Curie temperature is given by the following equation [18, 19]: k BTcMFA
2 J 0i 3 i 0
(1)
Where J 0i s are the exchange interaction energies of all neighboring atoms. Using Eq. 1, the Curie temperature of each of our superlattices is given by k BTcMFA
2 ,Ti J 0Tii ,Ti J Mn,Mn J 0Mn i 3 i 0 0i
(2)
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First, Curie temperatures (Tc) were calculated for the cases without the inclusion of on-site Coulomb interaction U on Ti atoms in our superlattices, which the results are gathered in the second column of Table 4. In these calculations the exchange interaction energies between the Mn atoms (consisting of the exchange interaction energy of Mn0-Mn0, Mn0-Mn1, Mn1-Mn1 pair atoms) were only considered because the values of the other pair-wise exchange interaction energies such as Mn-Ti and Ti-Ti are very smaller than that of Mn atoms in all superlattices. For instance, our calculations showed tha in SL1-1 superlattice the exchange interaction energies of Mn1-Mn1 pair is -8 meV in contrast to the exchange interaction energies of Ti1-Ti1 pair which is of -0.00019 meV, similarly in SL3-8 superlattice the exchange interaction energy of Mn0-Mn0 pair is of -3 meV while the exchange interaction energies of Ti0-Ti0 and Mn0-Ti1 pairs are of -0.0009 meV and -0.0003 meV, respectively.
Table 4. Curie temperatures calculated with U=0 and U=5 eV applied to Ti atoms for all superlattices under our investigation, using mean field approximation (MFA). Tc (K)
Tc (K)
Superlattice
[with UTi=0 eV]
[with UTi=5 eV]
SL1-1
82.77
109.92
SL1-2
68.07
85.95
SL1-3
58.79
76.27
SL1-8
40.23
53.94
SL3-2
44.87
91.67
SL3-8
31.72
71.79
SL2-8
27.85
62.51
Then, Cuie temperatures were calculated by inclusion of on-site Coulomb interaction on the Ti atoms with moderate values of U=5 eV. Our calculations showed that in this situation, the exchange interaction energies of Mn-Ti and Ti-Ti pairs are comparable to the exchange interaction energy of Mn-Mn pair. For instance, in SL1-1 superlattice, the exchange interaction energy of Mn1-Mn1 pair is of -4.9 meV compared to the exchange interaction energies of Ti1-Ti1 pair which is of -1.4 meV, and similarly in SL3-8 superlattices, the exchange interaction energy of Mn0-Mn0 pair is of -3.1 meV and the exchange interaction energies of Ti0-Ti0 and Mn0-Ti1 pairs are of -3.8 meV and -0.94 meV, respectively. So Tc calculations were performed using the exchange interaction energies of all Mn-Mn, Mn-Ti and Ti-Ti
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pairs, consisting of Mn1-Mn1, Mn1-Mn0, Mn1-Ti1, Mn1-Ti0, Mn0-Mn0, Mn0-Ti1, Ti1-Ti1, Ti0-Ti1 and Ti0-Ti0 pairs. The calculated results are gathered in the third column of Table 4. The Curie temperature values calculated with the inclusion of moderate U=5 eV on the Ti atoms are in good agreement with the experimental results [20].
3. Conclusion In summary, we have studied the exchange interaction and magnetic configuration of the (LaMnO3)m/(SrTiO3)n superlattices with m=1, 2, 3 and n=1, 2, 3, 8 in the presence of on-site Coulomb interaction on Ti atom. We calculated the exchange interaction energies for neighboring strong moment atoms. To make sure of the magnetic configurations of atoms located at the interface, we used the StonerWolfarth model. The half metallic 2DEG were observed in these superlattices by applying of electronelectron correlations with U=5 eV to Ti atoms. Furthermore, by taking into account the electron-electron correlation on the Ti atoms with UTi=5 eV, we found that the Curie temperatures of all superlattices calculated by mean field approximation, are in good agreement with the experimental results.
Acknowledgements We thank Dr. I. Abdolhosseini Sarsari from Isfahan University of Technology for support and helpful discussions. References [1] J. G. Barriocanal, et al, Nature. Commun. 1 (2010) 82. [2] H. M. Liu, et al, J. Appl. Phys. 113 (2013) 17D902. [3] G. Parr, and W. Yang., Density-Functional Theory of Atoms & Molecules (Oxford University Press. Newyork (1989). [4] N. D. Mermin., Phys. Rev. A. 137 (1965) 1441. [5] A. I. Liechtenstein, et al., Phys. Rev. B. 52 (1995) R5467. [6] V. I. Anisimov, et al., J. Phys.: Condens. Matter. 9 (1997) 767. [7] I. V. Solovyev, et al., Phys. Rev. B. 50 (1994) 16861. [8] B. R. Nada, and S. Satpathy, Phys. Rev. Lett. 101 (2008) 127201. [9] E. Dagotto, T. Hotta, and A. Moreo, phys. Reports. 344 (2001) 1-153.
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[10] J. Kanamori, J. Appl. Phys. 31 (1960) 45-235. [11] K. Janicka, et al., J. Appl. Phys. 103 (2008) 07B508. [12] R. Pencheva, et al., Phys. Rev. B. 74 (2006) 035112. [13] K. V. Benthem and C. Elsasser, J. Appl. Phys. 90 (2001) 12. [14] J. G. Barriocanal, et al, Adv. Mater. 22 (2010) 627-632. [15] R. B. K. Nada and S. Satpathy, Phys. Rev. B. 79 (2009) 054428. [16] P. Fazekas, Lecture Notes on Electron Correlation and magnetism (world Scientific, Singapore) (1999). [17] http://nbn-resolving.de/urn/resolver.pl?urn=nbn%3Ade%3Agbv%3A3-000011020. [18] L. Sheng, et al, Phys. Rev. Lett. 79 (2010) 1710. [19] Y. Liu, et al, Phys. Rev. B. 82 (2010) 094435. [20] W. S. Choi, et al, Phys. Rev. B. 83 (2011) 195113.
Highlights
Calculated the magnetic structure and Curie temperature of the (LaMnO 3)m/(SrTiO3)n superlattices (SLm-n) with m=1, 2, 3 and n=1, 2, 3, 8 by mean field approximation.
Using Stoner-Wolfarth model for all of these superlattices before and after adding U term on the Ti atoms.
By implying U=5 eV on the Ti atoms, the magnetic order of interfacial atoms of these superlattices has changed to ferromagnetic.
The 2DEG formed at the interface half-metallic have made in these superlattices by the inclusion of electron-electron correlation with U=5 eV on the Ti atoms for all of the superlattices.
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