Density functional studies of LaMnO3 under uniaxial strain

Journal of Magnetism and Magnetic Materials 322 (2010) 3653–3657

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Density functional studies of LaMnO3 under uniaxial strain B.R.K. Nanda , S. Satpathy Department of Physics, University of Missouri, Columbia, MO 65211, USA

a r t i c l e in fo

abstract

Article history: Received 9 April 2010 Received in revised form 9 July 2010 Available online 21 July 2010

We study the electronic and magnetic properties of tetragonal LaMnO3 (LMO) under uniaxial strain, appropriate for epitaxially grown LMO heterostructures, from density functional calculations. The optimized tetragonal structure without strain has volume, magnetic order, and Jahn–Teller distortions similar to the bulk LMO, which forms in the orthorhombic structure. Strain affects the relative magnitudes of these distortions and changes the magnetic and conduction properties. While unstrained LMO is a type A antiferromagnet and insulating, we find that it changes to a ferromagnetic metal under tensile strain condition (c-axis stretched). The latter is the result of a diminishing magnitude of the Jahn–Teller distortion with strain, which in turn reduces the splitting of the Mn-eg orbitals, eventually leading to a metallic state, and finally to a ferromagnet due to the double exchange interaction. The calculated Poisson’s ratio from geometry optimization agrees with the experimental values for the bulk. & 2010 Elsevier B.V. All rights reserved.

Keywords: Manganites Jahn–Teller distortions Strain Density-functional

1. Introduction Currently there is a considerable interest on the epitaxially grown oxide interfaces because of potential new physics and promising device applications. One such interface being studied is the epitaxially grown LMO heterostructure on various substrates such as SrTiO3. The heterostructure is often pseudomorphic, assuming the planar lattice constants of the substrate on which it is grown. While bulk LMO forms in the orthorhombic structure, it has the tetragonal structure when grown on a substrate, with the ab plane in registry with the substrate. It also becomes uniaxially strained, with the magnitude of the strain being different for different substrates. A wide range of electronic and magnetic properties varying from metallic ferromagnetic (FM) state to insulating antiferromagnetic (AFM) state or an intermediate canted magnetic state [1–9] has been observed in these heterostructures. In this paper, we study the effect of uniaxial strain on the electronic and magnetic properties of tetragonal LMO by performing density-functional calculations using the linear augmented plane wave method. Our main results can be summarized as follows: (i) The unstrained structure is an insulating type A antiferromagnet with Jahn–Teller (JT) distortions similar to the experimental orthorhombic structure. (ii) With tensile strain, it becomes a ferromagnetic metal, while with compressive strain it becomes a type G antiferromagnetic insulator. (iii) As the tensile strain is increased, the strength of the JT distortions diminishes, which results in a

 Corresponding author.

E-mail address: [email protected] (B.R.K. Nanda). 0304-8853/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2010.07.017

reduced splitting between the Mn-eg states leading to a ferromagnetic metal.

2. Method The density-functional calculations were performed using the linear augmented plane wave (LAPW) method [10] with the general gradient approximation (GGA) [11] for the exchangecorrelation functional. We considered a tetragonal unit cell of LMO and constructed a supercell of four formula units by doubling it both along the ab plane and along the c-axis to accommodate the relevant magnetic ordering, viz., the antiferromagnetic type G, type A, and type C, as well as the ferromagnetic structure. For each strain condition, all four magnetic structures were considered. For each case, the in-plane lattice constant ‘a’ was held fixed and full structural optimization was performed to obtain the out-of-plane lattice constant ‘c’ as well as the internal atomic positions within the cell. For all strain conditions, the atomic sphere radii were taken as 2.5, 1.75, and 1.55 a.u. for La, Mn, and O, respectively. Generally the spin densities are well confined within a radius of about 1.5 a.u. [12]; therefore the resulting magnetic moments as well as the relative magnetic energies do not depend strongly on the variation of the atomic sphere radii. The basis set for solving the Hamiltonian included 4s, 4p, and 3d valence and 3s and 3p semicore functions at the Mn site, 6s, 6p, and 5d valence and 5s and 5p semicore functions for the La site, and 2s and 2p functions for the O sites. Thirty two irreducible k points in the full Brillouin zone were used for the self-consistent calculations.

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3. Results Unstrained crystal structure: As mentioned already, the crystal structure of LMO is orthorhombic in the bulk [13], but assumes a tetragonal structure when grown epitaxially on oxide substrates. In view of this, we first study the equilibrium crystal structure of LMO in the tetragonal structure. For this, we took a series of inplane lattice constants ‘a’ and for each case optimized both the out-of-plane lattice constant ‘c’ and the internal atomic positions. The minimum energies for each lattice constant and for two different magnetic structures are shown in Fig. 1. The global minimum occurs at the lattice constant a0 ¼3.99 A˚ with the ˚ The corresponding volume is 3% corresponding value c0 ¼3.76 A; smaller than the experimental value for the bulk orthorhombic structure, which is reasonable. We considered all four magnetic structures (only two are shown in the figure) and found the type A AFM structure to have the lowest energy in agreement with the experiment. The three important octahedral bond-stretching modes for the MnO6 octahedron are indicated in Fig. 2. For the unstrained tetragonal crystal, consistent with the experiments [13], the calculated ground-state has the type A magnetic configuration. The tetragonal structure allows a checkerboard JT distortion on the ab plane of the type (Q2, Q3) alternating with (  Q2, Q3), while the distortions repeat along the c direction. The strength of the checkerboard JT distortion is related to the lattice constants: pffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffi a ¼ 2=3 Q1 Q3 = 3 and c ¼ a þ 3 Q3 [14]. From the computed atomic positions, we calculated the magnitudes of the JT modes for the global minimum structure, which yielded the values

0.3

Energy (eV)

0.2

A−AFM

0.1

FM

0

3.8

4.0

3.9

˚ respectively, for the in-plane Q20 ¼0.14 A˚ and Q30 ¼ 0.12 A, distortion and the octahedral stretching modes. The ˚ corresponding experimental values are 0.28 and  0.10 A, respectively [13]. Effect of strain on structure: For the strained structure, the in-plane lattice constant was held fixed at a particular value and full structural optimization was made with the four magnetic configurations, viz., type C, type A, type G, and type F, with all but the last one being antiferromagnetic. The type G phase represents the Nee´l order AFM phase, while the type C corresponds to interlayer FM ordering and intra-layer AFM ordering, while type A represents FM ab planes stacked antiferromagnetically. The strain parameters in this paper are measured with respect to the calculated lattice constants a0 and c0 for the global minimum structure, where the two strain parameters exx ¼ ðaa0 Þ=a0 and ezz ¼ ðcc0 Þ=c0 characterize the strain state in the tetragonal structure. The magnitudes of the JT distortions Q1 and Q3 for each strain condition are computed from the expressions for the lattice constant given in the last subsection, while the magnitude of Q2 for each strain condition is obtained from the computed positions of the oxygen atoms from energy optimization. In Fig. 3, we have plotted the total energies for different magnetic structures as a function of the uniaxial strain ezz . From the figure we see that for the unstrained LMO ðezz ¼ 0Þ, the type A AFM phase is the stable configuration, which is consistent with Fig. 1. With tensile strain, LMO becomes ferromagnetic, while our results show that it becomes type G AFM with compressive strain. The reason for these magnetic transitions lies in a complex interplay between the JT interaction and the magnetic double and super exchange interactions, which we will return to in Section 4. From the computed lattice constants, we can estimate Poisson’s ratio. In the tetragonal structure, there are two different Poisson’s ratios, n21 and n31 , where the first index denotes the direction of the longitudinal extension, and the second, the direction of the corresponding lateral contraction [15]. The strain parameters are related by the expression exx ¼ ezz ð2n31 Þ1 ðn21 1Þ. Within the pseudo cubic approximation, the magnitudes of n21 and n31 are taken as equal, yielding a single value n for Poisson’s ratio, which we can determine from the computed optimized lattice constants a and c. In Fig. 4 we have plotted the strain parameters and the calculated Poisson’s ratio, which is about 0.40 in agreement with the measured value of 0.37 [16]. The volume of the lattice is

4.1

a (Å) Fig. 1. Total energy per formula unit as a function of the in-plane lattice constant ‘a’ in the tetragonal structure for the FM and type A AFM magnetic ordering.

2

Energy (eV)

5

0.2

O

C G 0.1 F

4 3

Mn

1

A

z

0

6 Q1

Q2

y x

−0.04

Q3

Fig. 2. Octahedral distortion modes for the pffiffiffiMnO6 octahedron with the eigenvectors: jQ1 S ¼ ðX1 þ X2 Y3 þ Y4 Z5 þp Z6ffiffiffiffiffiffi Þ= 6, jQ2 S ¼ ðX1 þ X2 þ Y3 Y4 Þ=2, and jQ3 S ¼ ðX1 þ X2 Y3 þ Y4 þ 2Z5 2Z6 Þ= 12, where X1 denotes the x coordinate of the first oxygen atom, etc. Of these, Q2 and Q3 are Jahn–Teller active.

G Compressive

0 εzz A

0.04 F Tensile

Fig. 3. Total energies for different magnetic phases of tetragonal LMO as a function of strain. The ground state magnetic ordering is insulating type A AFM, which is predicted to change to a metallic ferromagnet (F) under tensile strain condition.

B.R.K. Nanda, S. Satpathy / Journal of Magnetism and Magnetic Materials 322 (2010) 3653–3657

Mn−d

0 0.4

ν

4

Poisson’s ratio (ν)

0.8 εxx

G

Spin

t2g z2−y2 / z2−x2

0 Spin

eg

x2−1/y2−1

4

−0.04

compressive

0.04

εxx

3655

t2g

0 εzz

0.04

Fig. 4. Strain parameters exx vs. ezz and the computed Poisson’s ratio for the tetragonal LMO.

0.4

t2g

4 A

z2−y2 / z2−x2

0 eg x2−1/y2−1

unstrained

−0.04

DOS (states/eV /Mn)

0

t2g

4 t2g

4 F

eg

0

0 eg

Q3 0.2

4 G

−0.04 Compressive

A 0 εzz

−8

0.04 Tensile

not conserved with strain, for which Poisson’s ratio is needed to be 0.5. The magnitudes of the Q2 and Q3 modes as a function of strain are shown in Fig. 5. Overall, the net strength of the JT distortions increases with compressive strain and diminishes with tensile strain condition. This has an important consequence on the band gap, since the on-site energy splitting of the two Mn-eg orbitals is proportional to the strength of the JT distortion: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

EF

F

Fig. 5. Magnitude of the JT distortions as a function of strain. The ground state magnetic structure (G, A, or F), which changes with strain, is indicated in the figure.

E 7 ¼ 7g Q22 þ Q32 ,

tensile

JT distortion (Å)

Q2 0.2

ð1Þ

where g is the linear JT coupling strength. Effect of strain on the electronic structure: Fig. 6 shows the spinresolved Mn-d densities of states for three different values of strain. The majority t2g states are occupied in all cases, while the eg states are partially occupied, with or without a gap, depending on the strain condition. The gap value is directly linked to the strength of the JT distortion in the crystal as suggested by the expression (1) for the splitting of the on-site energies. Of course, in addition to the splitting, the JT-split eg states are also broadened due to the electronic hopping, which may or may not be enough to close the gap depending on the structure. As seen from Fig. 6, both for the compressive and unstrained cases, the degenerate Mn-eg bands are split sufficiently to open up a gap, while for the case of the tensile strain, there is no gap in the band structure thus forming a metallic state. In the metallic state, the degenerate eg bands in turn mediate a strong Zener double exchange [17] between the t2g core spins both in the plane and out of the plane stabilizing thereby the ferromagnetic ordering. For

−4 0 Energy (eV)

t2g

4

Fig. 6. Mn-d spin-resolved densities-of-states for a specific Mn atom under different strain conditions, viz., ezz ¼ 0:035, 0, and 0.037, respectively, from top to bottom. The spin-m (k) denotes the majority (minority) spin states on the specific Mn atom.

strong compressive strain, the occupied eg bands are well separated from the unoccupied states, thus reducing the ferromagnetic double exchange term, so that the super exchange term between the core spins dominates resulting in a type G AFM structure. For the unstrained lattice, the splitting of the eg states have an intermediate value, resulting in an intermediate strength of the FM double exchange, which in turn stabilizes the type A AFM structure, where the in-plane double exchange overcomes the super exchange, while in the out-of-plane direction, the super exchange wins. Fig. 7 shows the calculated band gap as a function of strain, where we find that the gap abruptly disappears once the ground state becomes ferromagnetic. The estimated value of the band gap for the unstrained condition is 0.35 eV which is much less than the experimental value of 1.8 eV as obtained from the photoemission data [18]. The underestimation of the band gap is a well known deficiency of the band calculations employing local-density functionals and this underestimation is seen for bulk LMO in the orthorhombic structure as well [19,20]. Experimentally, the metallic behavior has been reported both in the bulk LMO under pressure [21] as well as in thin films [22].

4. Model Hamiltonian The results of the density-functional calculations can be understood in terms of a model Hamiltonian that contains the key interactions in the system. Details of this model are discussed elsewhere, [14] but here we outline the basic results of the model to make connection with our density-functional results.

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The model Hamiltonian, restricted to a lattice of Mn atoms, is written as H¼

X1 2

i

þ

KQ 2i þ

X J X~ ~ HiJT þ Si  Sj 2 i /ijS

 X  yij tia,jb cos ciya cjb þ h:c: þ HU , 2 /ijS, ab

ð2Þ

where the first two terms represent the E  e JT coupling [23,24], the next term represents the antiferromagnetic super exchange between the Mn-t2g core spins, followed by the Anderson– Hasegawa hopping term [17] and the Coulomb interaction term, both for the eg electrons. The model Hamiltonian may be solved in the Hartree–Fock approximation, if the Coulomb term is present, else, it can be solved numerically exactly. The results are summarized in Fig. 8, where on the left part, we have shown the energy contours for an isolated Mn octahedron. The case of the isolated octahedron is identical to the type G structure,

1.0

Band gap (eV)

G

0.5 A

F

0 −0.04 Compressive

0 εzz

0.04 Tensile

Fig. 7. Computed band gap as a function of strain. For larger tensile strain values, the system stabilizes in ferromagnetic metallic phase with zero band gap.

Q2 (Å) G−AFM

since the electronic hopping is not present in either cases. The three circular dots in the figure indicate the global minimum energy for the single octahedron, while if the strain is fixed to a certain value, Q3 is fixed, and the minimum energy state is obtained from the minimum along the dashed line, which is shown by the solid squares. There are two equivalent minima, corresponding to the two equivalent orbital orientations x2  1 or y2  1, and with distortions 7 Q2 . As strain is changed, the magnitude of Q2 changes with it, and the system moves along the solid line with arrows, with the orbital orientation eventually becoming z2  1 type. The change of magnitudes of Q2 and Q3 obtained from the model is similar to the density-functional results shown in Fig. 5. Thus Fig. 8(left) indicates that as strain is varied from compressive to tensile, beyond a certain value of the tensile strain, the strength of the Q2 distortion mode quickly goes to zero, pffiffi reducing the overall strength ðQ22 þ Q32 Þ of the JT distortion. The splitting of the eg orbitals, Eq. (1), is in turn proportional to this strength, so that as the splitting diminishes, the Anderson– Hasegawa hopping gain of energy increases, with the gain given by t 2 =D from the second-order perturbation theory, D being the separation between the occupied and the unoccupied eg states. Now, this hopping gain of energy is absent in the type G structure because of the multiplicative Anderson–Hasegawa cosðy=2Þ factor [17] in the hopping term t. However, in the other magnetic configurations, this term is present and make these structures energetically favorable if the JT distortion is not too large. To study this, we have computed the energies of the various magnetic structures, and synthesized the right part of Fig. 8 from the minimum-energy structure for each strain condition. We see that the system changes from G-A-F-Gu as strain changes from compressive to tensile, consistent with the results from our full density-functional calculations shown in Fig. 3. Note that the Gu phase in our model calculation occurs for large tensile strain, not considered in the density-functional calculations. The Gu phase differs from G in its orbital ordering as indicated in Fig. 8. The diminished magnitude of the splitting of the eg orbitals at the same time closes the gap in the band structure, leading to a metallic state for the ferromagnetic structure, as seen from the Fig. 7. The orbital ordering shown in Fig. 8 is also consistent with the density-functional results, indicated in Fig. 6.

Ground state

0.4 y2 − 1

5. Summary

y2 − 1 G

0 z2

−1

A

F

G

/ Q3 (Å)

2

z −1

x2 − 1

x2 − 1 -0.4 -0.4

0

0.4

-0.4

0

0.4 εzz

−0.05

0

0.05 0.10 0.15

−0.05

0

0.05 0.10 0.15

Fig. 8. Energy contours in the Q2  Q3 plane for an isolated octahedron (same for type G phase in our model) (Left). The black squares indicate the minimum-energy structure, if the strain is held fixed at the value indicated by the bold dashed line. The arrows on both figures show how the structure changes as strain is varied in the G-AFM configuration. Such contours were computed for different phases for each strain condition (characterized by ezz or Q3) and their energies were compared. The minimum-energy phase for each strain condition is shown on the Right. The full line with arrows indicates the Q2 values for the minimum energy structure, which changes from G-A-F-Gu as strain is changed from compressive to tensile. The vertical dashed lines in the right figure separate the phase boundaries. The minimum energy structure obtained from the model Hamiltonian follows the same sequence obtained from the full density-functional calculations shown in Fig. 3. The orbital ordering in each strain case is indicated in the figure. The parameters used here are (see Ref. [14] for details): Vdds ¼ 0:5 eV, U ¼ 0, ˚ G ¼1.5 eV/A˚ 2, K¼ 9 eV/A˚ 2, and J ¼ 26 meV. g ¼2.5 eV/A,

In summary, we studied the effect of uniaxial strain on the ground-state structure of tetragonal LaMnO3 from density functional calculations. Our results reveal the existence of various phases under varying strain conditions, where the orbital order, magnetic structure, the JT distortions, as well as the conduction properties change under applied strain. Only the ferromagnetic phase, stable under tensile strain condition, was found to be metallic. The results presented here are relevant to the epitaxially grown LaMnO3 heterostructures, which adopt the tetragonal structure.

Acknowledgment This work was supported by the U.S. Department of Energy through Grant no. DE-FG02-00ER45818.

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