ОПТИКА И СПЕКТРОСКОПИЯ, 2014, том 116, № 5, с. 715-720
XV МЕЖДУНАРОДНЫЙ ФЕОФИЛОВСКИЙ СИМПОЗИУМ
УДК 548.0:535
BROADENING OF PARAMAGNETIC RESONANCE LINES BY CHARGED POINT DEFECTS IN NEODYMIUM-DOPED SCHEELITES
© 2014 г. E. I. Baibekov*, D. G. Zverev*, I. N. Kurkin*, A. A. Rodionov*, B. Z. Malkin*, and B. Barbara**
* Kazan Federal University, 420008 Kazan, Russia ** Institut Néel, CNRS and Université Joseph Fourier, BP166, 38042 Grenoble Cedex 9, France
E-mail: edbaibek@gmail.com Received November 18, 2013
We study paramagnetic resonance linewidth in a series of CaWO4 and CaMoO4 crystals with different concentrations of neodymium ions (0.0031—0.81 at. %). Experimental data are interpreted in the framework of the statistical theory of line broadening by charged point defects. In our calculations, three different contributions are singled out: arising from the local electric fields, electric field gradients and magnetic fields of the nearby point defects. The interaction parameters are determined from the spectroscopic data available for Nd:CaWO4 crystal. Direct calculations of the linewidth are performed for different crystal orientations with respect to external magnetic field. We conclude that major contribution to the broadening comes from the interactions with random electric fields produced by neodymium and charge compensator ions.
DOI: 10.7868/S0030403414050043
1. INTRODUCTION
Calcium tungstate and calcium molybdate single crystals activated with different rare earth ions are well-known for their applications in optics [1, 2] and use in paramagnetic resonance experiments [3, 4]. Quite recently two of them (Er:CaWO4 and Yb:CaWO4) have attracted some attention in view of possible implementations for quantum information processing [5, 6]. Low natural abundance and low magnetic moment of 183W nuclei makes the host matrix almost dia-magnetic and thus provides an opportunity to obtain electron spin coherence times ~3 ^s [5]. In order to reach longer coherence times, it is necessary to understand clearly the role of different interactions that perturb the resonance frequency of the impurity ions. The linewidths and lineshapes of electron paramagnetic resonance in RE:CaWO4 and RE:CaMoO4 crystals (RE stands for the rare earth ion) have been the subject of long and extensive study [7—11]. Three main contributions were singled out: from the electic fields [7], electric field gradients [11] and magnetic fields [9], all produced by the point defects in the host lattice. However, there was no attempt to combine all three mechanisms and perform calculations of the linewidth in the general case of arbitrary direction of external magnetic field B.
The present work aims to correct this deficiency. Section 2 is a brief review of electric field effect and its contribution to the linewidth in the specific case of tetragonal symmetry of the crystal background. In the Sec. 3 and 4, we derive expressions for the broadening induced by random electric field gradients and mag-
netic dipolar interactions generalized for the case of arbitrary B direction, also for tetragonal symmetry. The calculation procedure is based on the statistical theory of line broadening under continuum approximation [12]. Finally in the Sec. 5 we compare our calculations with the experimental data, including our own measurements of Nd:CaWO4 crystal series and literature data for Nd:CaMoO4 crystal [8]. Note that, in contrast to [8], we attempt to calculate the line-widths avoiding fitting procedure of any kind. The interaction parameters are either taken from the direct measurements of electric field shifts [3] or calculated on the basis of available spectroscopic data.
2. ELECTRIC FIELDS
Let us briefly revise the contribution of excessive electric field to the shift of spin resonance frequency in crystals with tetragonal symmetry [3] and the corresponding broadening of paramagnetic resonance lines resulting from randomly distributed charged point defects [7]. The interaction of a given electron spin S (namely, paramagnetic ion in the crystal background) with the electric field E is described most generally by the following Hamiltonian (^g is Bohr magneton):
HE = И BTijkBiSjEk •
(1)
Here and below, the summation over repeating indices is assumed. Tijk is a third-rank tensor with 18 independent coefficients, Tyk = Tjk. Following [3], we adopt
short notation for a pair of indices related to the symmetric part of the tensor:
JT m m m m m
11k = T1k, T 22k = T2k, T33k = T3k,
JT m m m m m
23k = T4k, T13k = T5k, T12k = T6k.
(2)
In scheelites, RE ions occupy calcium sites with £4 point symmetry which is very close to D2d [13]. Symmetry selection rules applied for D2d point group leave only two nonzero components, T41 and T63, out of 18. These can be acquired directly from the measurements of the electric field effect. Regarding that Zeeman interaction of RE ion with magnetic field B is
hz - MBgijBiSj
(3)
where gi} = g^, g1 = g2 = gL, g3 = g|, so that g-tensor components are shifted as
8g(£) = TikEk, one obtains the shift of spin resonance frequency ro:
S®(E) = (®/2 g 2)[(g± + g ,)Tn(E1 sin O +
+ E2 cos O) sin 2© + 2g 1T63E3 sin2 © sin 2O],
where © and O are azimuth and polar angles that determine the direction of the vector B in the local coordinate system of RE ion, g2 = g2 sin2 0 + g cos2 © is the square of effective g-factor. A given point defect at distance r with excessive charge q (in the units of elementary charge e) would produce electric field E =
= qer/Kr2. Here, k is introduced to account for the polarization of the crystal lattice near the charged defect that reduces the field at the site of RE ion. Under continuum approximation, k is close to dielectric constant of the host crystal. Supposing that there are two types of point defects with opposite signs of charge, ±q, and that they can occupy random positions within the lattice, one finally obtains the line shape in the form of Holtsmark distribution [12] (roughly, intermediate between Gaussian and Lorent-zian lineshapes) with the half-width
AE = 3.75C
2/3|q| e® sin 0
Kg (4)
x [(gi + g|)2T42 cos2 0 + g2T623 sin2 0sin2 2®]1/2.
There are two peculiar features of this result. The first is that the half-width is nonlinear with the defect concentration C. The second is specific to the local symmetry of RE ion: there is no broadening in case when the field B is directed along local coordinate axes x, y, z. In ab plane, AE reaches minima at O = 0, n/2, ..., and maxima at O = n/4,3n/4,..., so that
A E min - 0,
A
E max
- 3.75C2/3 \qT63\m/KgL. (5)
3. ELECTRIC FIELD GRADIENTS
Sometimes, interactions with electric field do not cause the shifts of resonance frequency, as in the case when the impurity ions occupy sites with inversion symmetry, or at specific directions of B (see previous section). In such case, one must take into account the next-order terms of interaction, i.e., electric field gradients. Selection rules imply that electron states of a
ground spectral configuration of RE ion, 4 fn, are not affected by the electric field, and the effect is of the next perturbation order involving excited configurations. In contrast, interactions with electric field gradients, though less in magnitude, are not forbidden by the selection rules. In general, one can expect a competition between these two contributions.
By analogy with Eq. (1), we write the Hamiltonian describing the interactions with the electric field gradients 9 kl = -d Ek/ dxl as
HEG = MBVijklBiSjtykl.
Here Vijkl = Vjikl = Vijlk, ^kl = 9lk, and 9 kk = 9 k satisfies Poisson equation, ^^ 9k = 0. Under D2d symmetry, 7 nonzero components out of 36 are left: V11, V12, V13, V31, V33, V44, V66 (the notation (2) is used here and below). The corresponding frequency shift is linear in 9 k :
Sœ(EG) = (ffl/g2)ykl (©, ®)<Pkl - («/g2)Yk (®, ®)<Pk, (6) where
Y1 = g± sin2 0[(Vn - V13) cos2 O + + V12 - V13)sin2 O] + g, cos2 0(V3! - V33), Y2 (®, O) = Y1 (0, n/2 -O), Y3 = 0, Y 4 = (1/2) (g ± + g, ^ sin 20 sin O, Y5 (0, O) = y4 (0, n/2 -O), Y6 = g±V66 sin 0 sin 2O.
A point defect at distance r with excessive charge q produces electric field gradient qkl (r) = qe(3rkrt —
— r 2S kl )/k ' r5. Here the factor k ' = 3k/5 is introduced, where 3/5 stands for the local field correction [12]. In the framework of the statistical theory, the line-shape is given by the distribution
(7)
r(EG)
(ffl) = J dE exp - CA (E)]
where
A (Ç) = fffd 3r{1 - cos[S®(GE) (r) £]}.
The calculation of the above integral represents a rather painful procedure. Omitting unnecessary details, we present only the result:
A © = ^^ (Y,-Y 2 )2 + 3y 3G
37k
Y i -Y 2
Y 3
(9)
Here, yi are the principal values of the symmetric tensor ykl (6), (7). The function 3/n < G(x) < 1 can be replaced by unity (see figure). Note that our result (9) is more general than the one obtained in [14]. The line-shape (8) is Lorentzian
I{EG] (w) = (1 /n)A eg/(w2 + A E) with the half-width
a eg =
4n2qem [Z _ .3 „2 --u— C (y i - Y 2 ) + 3Y 3 •
27 , 2 .........- (10)
27k g
The half-width is linear in the defect concentration. Let us consider two specific cases. When B 11 c axis,
8n2e®
E y
9/3K ' gy y For B in the ab plane,
C\V3i - V3
33
(11)
A^L = -J^L C{[3(Vn - V12)2 + (Vn + Vn - 2V13)2] + W3K (12)
+3[V626 - (V11 - Vi2)2]sin22O}1/2.
On condition that V626 - (V11 - V12)2 > 0, the half-width (12) reaches maxima and minima at the same polar angles O as the one induced by the electric fields (see Eqs. (4) and (5)).
Since it is exceptionally difficult to measure directly the shifts induced by the uniform electric field gradients, no data is available for the interaction coefficients Vk. However, it is possible to calculate these values using the optical spectroscopic data for the particular RE ion. Let us write the interaction of RE ion with the electric field gradients explicitly:
H g =- 319 j«,
i=1
(13)
where r(i i) is the vector connecting RE nucleus to i th el
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