Transferred hyperfine interactions for O17: LaMnO3 Текст научной статьи по специальности «Физика»

ISSN 2072-5981

Volume 16, 2014 No. 1, 14101 - 7 pages

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Transferred hyperfine interactions for O17: LaMnO3

O.A. Anikeenok Kazan Federal University, Kremlevskaya 18, Kazan, 420008, Russia

E-mail: [email protected]

(Received January 16, 2014; revised March 16, 2014; accepted March 18, 2014)

Parameters of the transferred hyperfine interactions (THFI) for O17: LaMnO3 have been estimated at room temperature. Satisfactory agreement with experiment has been obtained.

PACS: 76.70.Dx, 76.30.Kg, 71.70.Ch

Keywords: transferred hyperfine interactions, NMR, LaMnO3

1. Introduction

Transferred hyperfine interactions (THFI) on nuclei O17: LaMnO3 were studied in [1] using the nuclear magnetic resonance (NMR) method. The wave function |Mn) was determined from the experimental

data as a superposition of 10) = 13z2 - rand |e) = | x2 - ystates, which are the components of the

doublet eg belonging to the configuration tlgeg of the Mn3+ ion. The wave function was determined

from the ratio of the hyperfine fields of oxygens Oi and O2. Ions of oxygen of Oi lie between the planes of manganese ions. Ions of oxygen of O2 lie in the plane of manganese ions. The covalency parameters were taken proportional to the overlap integrals. The direction from Mn3+ to the oxygen ion O2 determining the largest (long) covalent bond was chosen as the quantization axis. The choice of this coordinate system is dictated by the fact that the nearest environment of the Mn3+ ion is almost orthorhombic. At the same time, it is obvious that similar estimates have most likely the qualitative character for the clarification of mechanisms leading to different values of the local magnetic fields on O1 and O2 nuclei. In this work, the quantitative estimates of these fields at room temperature were obtained.

2. Theory

Processes leading to the appearance of hyperfine fields on nuclei of ligands were considered in [2]. One of these processes describing the transfer of the electron from the ligand to the partly filled shell of the cation and denoted as V1 according to [2] has the following form

V =X q\0\\q| 2 fK\\\)-± + fKf\]{S\ v\), (1)

where v is the operator of the hyperfine interaction, r and ), ^ are quantum numbers of cation and ligand orbitals, respectively, \\0) = \\(I + S) 1 \d), where I is the unit operator, S is the matrix of the overlap of one-electron orbitals. \\q|g) are matrix elements of the operator

1 - N

q = ln(I + S) = S J (I + aS)-1 da « S £(I + atS)-1 Aa , (2)

0 i=1

(I + ai S)-1 is the matrix inverse to the matrix (I + ai S), ai = i / N, Aa = 1/ N, N «106. The

expression to the right of the sign of the approximate equality is the integral sum of the integral under consideration. It exists always, if the basis of one-particle functions is chosen to be linearly independent.

(3)

\\G is the amplitude of the transition of the electron from the ligand to the cation, |a^| is the

energy of the transition of the system from the ground to the excited state.

Ignoring the two-particle corrections and in the case of the Hund term, the operator G of the electron transition can be written in the one-particle form

G = Xa\bc\\G,

where is the orbital of the ligand b, » is the orbital of the cation,

2\\G| 0 = \\\0

s\ 1 + Z\\g (1 - P » (( K) -1) + \\ Hr \

vir\ (1 -P » <^b| W |r R i| %

+i (i - p)m ((11 - i)+^i hlrr-^ti) (4)

% | e |

(1 - P .el | v.) ] + (m + {^))\\h\0-\\T-\]\0

%e F - R e |

+Y\\g (1 - P ^X <?e\| tie)-\\TJnRr]\0 + Z(frb\g (1 - P )%)(I Vb)

% K Ki| Vb ,

where all quantities referring to the cation are marked by the subscript e, and those to the ligand by the subscript b, g is the Coulomb interaction of electrons, P is the operator of permutation of quantum numbers. Summation over rje in the case of the Hund term includes summation over the orbital quantum numbers of the cation in the ground configuration. Summation over rjb includes summation over all orbitals of the ligand in the ground configuration. The quantities s\e 1 and sqq are Hartree-

Fock energies of the electron on the orbital » of the cation and on the orbital of the ligand, respectively, determined for free ions. hk is the operator of the kinetic energy.

\\ HLR » = \\-^ |--7T \\ is the energy of the interaction between the electron on the

n, i Ir -(Rn + ri W

orbital » and the infinite crystal lattice in the ion approximation. The prime at the sign of sum

denotes the exclusion of the interaction between the charge qj and the electron on the orbital ».

Summation over n denotes summation over the unit cells, and summation over i denotes summation over the ions of the unit cell. ne, nb are the numbers of electrons in the ground configuration on the cation and on the ligand, respectively.

The process of the transition of the electron from the oxygen to the valence shell of Mn3+ was considered as the main mechanism of the creation of the hyperfine field on nuclei O2-. The orbitals » are orbitals of the 3d-shell of the Mn3+ ion, and the orbitals are orbitals of the O2- ion.

3. Matrix elements of operators

Further let us present the numerical values of the matrix elements of operators in (1) for the calculation of the hyperfine fields on nuclei O17 in LaMnO3. Let us take the coordinate system for the unit cell according to [3]. Taking into account the nearest environment of two nonequivalent oxygen ions leads to the selection of a complex of five ions. For the temperature T = 298 K we have: 0i(0.07452, 0.48743, 0.25), 02(0.22559, 0.19342, -0.0384), Mni(0.5, 0, 0), Mn2(0, 0.5, 0), Mn3(0, 0.5, 0.5). Values are given in units of crystal lattice constants. The wave functions of 2s, 2p orbitals of oxygen ions were taken from [4]. The wave functions of 3d orbitals of Mn3+ were taken from [5]. Thus, the matrix of the overlap of one-electron orbitals S is the matrix of the 23th order. The isotropy of the shift of the NMR line on nuclei of oxygen ions indicates that the main contribution to the THFI constants is given by the contact part of the hyperfine interaction. The calculations show that the 12s) orbitals of oxygen ions have a rather large overlap with the |z= |2z2 - x2 - yand |xy)

orbitals of manganese ions (all orbitals were determined in the coordinate system of [3]). The amplitudes of the transition to other orbitals are ignored in this work. The matrix elements ff)) ~()||)) ~ 1.005«1. The matrix elements of matrices q, f\o), one-particle operators

hi = |r - Ri\ 1, i = e, b and hk are given in Table 1. The values of the necessary two-center integrals of the Coulomb interaction of electrons are given in Table 2.

Table 1. Matrix elements of matrices q, f\o) and one-particle operators (in a.u.).

Mn2 - O1 Mn2 - O2 Mn1 - O2

a, b z2, 2s xy, 2s z2, 2s xy, 2s z2, 2s xy, 2s

< a||b > -0.0546 0.0007 0.0184 0.0309 0.0306 0.0531

< alq\b > 0.0543 -0.0007 -0.0182 -0.0308 -0.0304 -0.0527

a, b, i z2, 2s, k z2, 2s, e xy, 2s, k xy, 2s, e z2, 2s, k z2, 2s, e

< alhlb > 0.0034 0.0244 0.0021 -0.0132 -0.0032 -0.014

a, b, i z2, 2s, b xy, 2s, b z2, 2s, b xy, 2s, k

< a\h1\b > 0.0370 -0.0189 -0.0214 -0.0057

a, b, i xy, 2s, e xy, 2s, b

< alhlb > -0.0242 -0.0371

The energy ef 1 = efn2+ was calculated on the wave functions of the trivalent Mn3+ ion. This energy can be presented as ef'* = ef113 +f,)\gff in the zero approximation according to the definition of Hartree-Fock energies [2]. According to [5], e^f* =-1.9559. According to the calculations, (z2,z2|g|z2,z2) = 0.9584. According to [4], e^T = -0.6286.

Let us calculate the matrix elements of the operator HLR using the results of [6]. The structural factor G. (g) for LaMnO3 at room temperature has the following form:

Gj (g) = cos(gr.)F1 (g), F2 (g) = 0,

F (g ) = 3 [(-i)"' +(-1)* ][i+(-1)nz ]

+12cos ^x(a1ny + nx /2 + " /2) cos \^n{a2nx + ny /2)

-8cos|^(a3ny + nx / 2 + nz /2)]cos[^(a4nx - ny /2))

-16cos|^(a5ny + nx /2 + nz /2)]cos[^(a6nz)]cos[^(a7nx -ny /2),

where ai = 0.59802, a2 = 0.51562, a3 = 0.47486, a4 = 0.35096, a5 = 0.11316, a6 = 0.42320, a7 = 0.04882.

Let us give further the values of the matrix elements of the operator HLR. For the O1 ion the matrix element (2s|H№ |2s) = - 0.8262. For the O2 ion (2s\Hm |2s) = - 0.8158. For the Mn3+ ion the matrix elements {z2|Hm\z2^ = 1.3542, (xylH^xy) = 1.3495. For the pair Mn2 - O1 the matrix element

(z2k—1—,Iz2) = 0.2621. For the pair Mn1 - O2: (xyl,—1—rIxy) = 0.2615. v 'r - R j| |r - R b\

Table 2. Two-center integrals of the Coulomb interaction of electrons (x10), (ab\g\cd) = (ab|c^ (in a.u.).

< ab\cd > \a > = \z2 > Mn2 - O1

bcd z2, 2s, z2 £, 2s, £ £, £, 2s xz, 2s, xz xz, xz, 2s yz, 2s, yz

< ab\cd > 0.2647 0.2122 -0.0119 0.2470 0.0121 0.2467

bcd yz, yz, 2s 2s, 2s, 2s pz, 2s, pz pz, pz, 2s px, 2s, px px, px, 2s

< ab\cd > 0.0115 0.3350 0.2940 0.1299 0.2709 0.0514

bcd py, 2s, py py, py, 2s

< ab\cd > 0.2679 0.0401

< ab\cd > \a > = \xy > Mn2 - O2

bcd xy, 2s, xy £, 2s, £ £, £, 2s xz, 2s, xz xz, xz, 2s yz, 2s, yz

< ab\cd > -0.1323 -0.1295 -0.0013 -0.1254 -0.0022 -0.1264

bcd yz, yz, 2s 2s, 2s, 2s pz, 2s, pz pz, pz, 2s px, 2s, px px, px, 2s

< ab\cd> -0.0024 -0.1730 -0.1421 -0.0195 -0.1440 -0.0217

bcd py, 2s, py py, py, 2s

< ab\cd> -0.1530 -0.0499

< ab\cd > \a > = \z2 > Mn1 - O2

bcd z2, 2s, z2 £, 2s, £ £, £, 2s xz, 2s, xz xz, xz, 2s yz, 2s, yz

< ab\cd > -0.1180 -0.1528 -0.0228 -0.1260 0.0083 -0.1234

bcd yz, yz, 2s 2s, 2s, 2s pz, 2s, pz pz, pz, 2s px, 2s, px px, px, 2s

< ab|cd > 0.0076 -0.1912 -0.1530 -0.0147 -0.1640 -0.0454

bcd py, 2s, py py, py, 2s

< ab\cd> -0.1601 -0.0368

< ab\cd > \a > = \xy > Mn1 - O2

bcd xy, 2s, xy £, 2s, £ £, £, 2s xz, 2s, xz xz, xz, 2s yz, 2s, yz

< ab\cd > -0.2432 -0.2375 -0.0027 -0.2313 -0.0045 -0.2296

bcd yz, yz, 2s 2s, 2s, 2s pz, 2s, pz pz, pz, 2s px, 2s, px px, px, 2s

< ab|cd > -0.0045 -0.3321 -0.2690 -0.0404 -0.2877 -0.0863

bcd py, 2s, py py, py, 2s

< ab\cd > -0.2716 -0.0415

4. Calculations

Let us find the contributions to the THFI parameters for the considered oxygen ions. Substituting the values of the calculated matrix elements into expression (4), we obtain the following values for the amplitudes of the transition. For the pair Mn2 - Oi we obtain the value l^z2|g|2s) = - 0.09. For the

pair Mn2 - O2: (z2 |g|2s) = 0.026, (xy|G|2s) = 0.044. For the pair Mn1 - O2: [z2 |g|2s) = 0.051,

(xy|G| 2s) = 0.086.

The energy of the transition A ^ can be estimated according to [7, 8]:

I A I = Tmn2+ i To2- + Emn3+ Eo2- CXF + E (6)

\At4\ = -1 +1 + EM - EM -s2s + Eeh , (6)

where 1mn , 1o are the ionization energies of manganese and oxygen ions, in this work E\f , EM have the meaning of the matrix elements of the operator HlR . The energy of the electron-hole interaction for the pair Mn2 - O1: Eeh = -0.2621, and for the pair Mn1 - O2: Eeh = -0.2615. According

to [9], the ionization energy 1mn = 1.234. The O2- ion does not exist in the free state, therefore let us

O2- I I

take the 1 value as zero. The estimates obtained above give the value A3d 2 I «1.3 a.u. We obtain the following values for the covalency parameters. For the pair Mn2 - O1: yz2 2s « 0.069, Mn2 - O2: «-0.020, fxy,2S «-0.033. For the pair Mm - O2: «-0.039, y^ «-0.067.

Let us take the wave functions of the electron belonging to the doublet eg of manganese ions in the form

Mn1: = c1 |xy) - c2| z2), Mn2: |2) = cj xy) + c2| z2). (7)

Wave functions are chosen from the condition of maximum length of the electron density in the direction of the long covalent bond.

Let us introduce the following notations qig=\\ q\ 0), p^0 = \\|0). According to [2], the spin

densities of the considered processes have the form

\ = 4\ -PuY\x + \ \,\' = lxy),|z2),c = \2s), (8)

/¿\,c = 4 - "2 \ -1 y+y, (9)

Transferred hyperfine interactions for O17: LaMnO3

where ij denote the pair Mn3+ - O2-.

Using equalities (7) - (9) we obtain for the isotropic part As (1) of the THFI tensor of the O1 ion

As (1) = 2c22fi^as = 12.75 MHz, (10)

where as = 3.46x 103 MHz is the hyperfine interaction parameter of the oxygen ion determined on the wavefunction |2s). The factor 2 arises owing to the contributions of Mn2 and Mn3 ions. The square of the coefficient c2 = 0.2. Coefficients c1 and c2 are determined from rather good agreement with the experiment. The absence of the coefficient c1 is a consequence of the smallness of the matrix element

(xy||2s).

We obtain for the isotropic part As (2) of the THFI tensor for the O2 ion

As (2) = As (2,Mn1) + As (2,Mn2) = 22.90 MHz, (11)

As(2,Mn1) = (c2 fg> - 2^42^,2s + cfd,)as = 12.57 MHz ,

As(2,Mn2) = (c2fg + + clf^)as = 10.33 MHz ,

where As (2,Mn1) is the contribution from the Mn1ion, As (2,Mn2) is the contribution from the Mn2 ion.

The experimental values can be estimated using the temperature perturbation theory and the experimental temperature dependence of the spectrum. The HN is the Hamiltonian of interaction of the nucleus with magnetic field for the noninteracting spins.

Ht_z.12 <SMIH-ISM ■ITSM1H^., (12)

M, m' kT

where H is magnetic field, Hz is the electron Zeeman interaction, HL = A (SI) is the ligand hyperfine interaction. Magnetic shift As (T) of NMR line is defined experimentally in [1]. The As (T) follows a Curie-Weiss law, but with different temperatures 0 for the O1 and O2. Then for the interacting spins

( ) gPHAS (S +1) ^ 3k (T-0) '

where H is magnetic field value. The experimental values are d(O1) = -15 K, 0(O2) = 23 K. We obtain for the magnetic field of 11.7 T, S = 2 at room temperature

As (O1,T) «1.25 MHz, Asexp (O1,T) = 1MHz, (13)

As(O2,T) «2.56 MHz, Asexp(O2,T) = 2.4 MHz. (14)

It is seen from equalities (13) and (14) that the theory and experiment are in rather good agreement.

The wavefunction |Mn^ = c1|#) + c2where c = 0.995, c2 =-0.10 [1]. It is determined in the

local coordinate system specified above. The wavefunction |Mn^ and the wavefunction Mn1:

H = cj xy^ - c2| zcan be compared if the function Mn1: H = cj xy) - c2| z^ is written in the

coordinate system obtained from the coordinate system of [3] as follows. The origin of the coordinate system should be placed at the Mn1 ion, and the z axis rotated by 90 degrees in the plane passing

through the z axis and the 02 ion forming the largest (long) covalent bond. Then in the new coordinate system the wavefunction Mn1 |l) = c[\6) + c'2, where c[ = 0.998, c2 = 0.06. The coefficients c1 and cl almost coincide. The coefficient c' is determined as the difference of rather large quantities of the same order of magnitude. Its uncertainty is of the order of magnitude within the approximations made under ignoring the overlap of orbitals |xz), |yz), |x2 - y2) with the orbitals of the O2- ion.

Thus, it is obvious that the approximations made in this work should be taken into account in the forthcoming publications.

In Ref. [2] the secondary-quantization representation with a basic of partially nonorthogonal orbitals was applied to impurity center. In accordance with the present paper this method can be applied to concentrated systems.

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