Peculiarities of magnetic linear birefringence of rare-earth paramagnetic garnets

Physicn B 17Y (1992)

PHYSICA I

IY-23

North-Holland

Peculiarities paramagnetic

of magnetic garnets

linear

N.P. Kolmakova and A.I. Popov MoscowImtirutr of Electronic 7’echnics. Moscow Received

25 February

hlanuscript

received

The magnetic taking

experimental

of rare-earth

30_339#, Kussiu

lYY1 in final form 25 November

linear birefringence

into account

birefringence

(MLB)

the splittings of excited

data. Agreement

1YYl

for paramagnetic configurations

rare-earth 4fh

(RE)

garnets with RE from Tb to Yh ia analyzed

’M(g).

is found for all RE garnets investigated

Theoretical results are compared with available with RE ions from Th to Tm (Ho included). except

for Yh.

1. Introduction In ref. [l] the observed properties of the MLB were analyzed in the Judd-Ofelt approximation for the heavy RE garnets R,M,O,, (R= Tb, Dy, Ho, Er, Tm, Yb; M = Al, Ga). The field and temperature dependences, the sign and anisotropy of MLB were discussed. The Judd-Ofelt approximation was found to describe the MLB data for terbium. dysprosium, erbium and thulium aluminum and gallium garnets within experimental accuracy. At the same time the sign of the MLB for holmium garnets and the sign and anisotropy of the MLB for ytterbium garnets did not coincide with experimental data. In ref. [l] the assumption was made that the opposite sign of the MLB for holmium garnets may bc ascribed to the dichroism due to the proximity of the absorption band. For ytterbium garnets it was supposed that an extension of the theory beyond the Judd-Ofelt approximation and consideration of splittings of excited configurations should allow one to describe correctly the observed properties of the MLB. In this work we consider the influence of the extension of the theory beyond the Judd-Ofelt approximation on the MLB of RE paramagnetic (gallium and aluminum) garnets with RE from Tb to Yb.

2. Sign of MLB The main contribution of the MLB in RE garnets is given apparently by the parity-allowed electric-dipole transitions between thermally populated levels of the ground 4fN configuration and the The quantitative interpretation of the MLB needs levels of excited 4fjVm’5d and 4fNm’ 5g configurations. information on the energy levels and wave functions of all configurations. The electronic structure of is formed by numerous interactions, many of which are comparable in the 4f”-’ 5d(g) configuration of the magnitude (e.g., the Coulomb interaction of the 5d electron with a 4f”-’ core and the interaction of states 5d electron with crystal field (CF) ( see. e.g., ref. [2]). This hampers greatly the classification Apparently in the first approximation the electronic structure of these for 4f”- ’ Sd(g)-configurations. 0021.4S26/Y2/$OS.O0

@ lYY2 - Elsevier

Science Publishers

B.V. All rights reserved

bv the concept ot tractional configurations can be treated as ;I totality of terms. which cm he obtained parentage [3]. In this case WC have as ;I rule several terms, which differ in the statch ot the CX)I-c.WC consider the wave functions of the terms in the form given in ref. 131

with respect to the pcrmutatlon5 ()I of the core 4 \,, / II, II, ;irc antisymmetric of the Sd(g)-electron (I’ _‘. 4 I: ctcctrons I. 2.. . i ~~1. i + I.. . N: V$ ,,,, J&ris the wave function Y are the Ctebsch-Gordan coefficients. The wave functions of the term\ of the grc~unci 4f’ c ”,,rr,,~ configuration are considered in the form [31 where

the wave functions

function ot lhc /th of fractional parcntagc. (Ii,,,,, :,, ia the b;i\c where GT,l\l,l , arc the coefficients electron. To consider the MLB we determine the contribution to the tensor of potarizability of the KL 1011. caused by electric-dipole transitions from the terms of the ground configuration to the tcrmx of the 3f” ‘5d(g) configurations. Using the irreducible form of the polarizabitity tcnaor 141 WC ha\c

where g and e arc indices Bottzmann factor.

dj:’ denotes

and excited

configurations.

and 0,. is the

I

the 4th component

of the electric-dipole

moment

for the ith electron

From the Wigner-Eckart theorem, formulae for sums of products and for hj-symbols. expression (3) is expressed in the form fk

respectively.

d);‘.

D<, = i

,-

of states of the ground

of the Ctebsch-Gordan

cocfficicnt\

1’’ = N,’ ( Q ),’)( I* ) ) . (J)

x 1Aa,.

‘.,.

where L, L’ and configuration and of the quadruple configuration are

, +

4

‘./.

+

(:6,:., +, i -

L, are the orbital angular momenta of the states of the ground configuration. excited the core of the excited configuration. respectively. ((I:,“(L)) is the thermal average moment in the L-representation. The coefficients A. B and (’ for the -If’ ‘5d(g) as follows:

N. p. K&n&ova. iI.1. Popov I Magneticlinearhirefringence of RE prnets

A(5d)

(L + l)(L,

= -

(2L - l)L(2L

+ L + 2)(-L,

+ L + 3)

L(L + 1) L(L, + L - 2)(L, + L - l)(L,

C(5d) = ~

(2L + l)(L 5(L + l)(L,

A(5g) = -

- L + 2)(L,

+ 1)(X

(9

+ L - 4)(L, + L - 3)(L, - L + 4)(L, - L + 5) + 1)

5(L, + L + 3)(L, + L - 3)(L, - L +4)(-L,

B(5g) = +

- L + 3)

+ 3)

12(2L - l)L(2L

+ L + 4)

12L(L + 1)

~L(L, + L + 3)(L, + L + 6)(L, - L +4)(-L,

C(5g) = -

+ L + 3)

+ 1)

(L, + L + 4)(L, + L - 2)(L, - L +3)(-L,

B(5d) = +

Using garnets

+ L + 3)(L, + L +4)(-L,

21

12(2L + l)(L

(4) we can determine

AS,, = a ,;,

TYTj:‘(

+ 1)(X

the contribution

+ L + 5)

+ 3)

of the RE ions to the dielectric

tensor

of paramagnetic

Qky > ? (6)

Here (Q&)) is the thermal average of the quadrupole moment of 4f electrons for the rth inequivalent site, T is the transformation matrix from the local to laboratory system of coordinates, N,, is the number of RE ions in the unit volume, and n is the average refractive index. The MLB, given by An = (1/2n)(Ae,, - AC_), for different orientations of the magnetic field is determined by the formula

An =

k

(AQ)

,

where (AQ) is the thermal average of the appropriate combination of the induced quadruple moments of the RE ion. To discuss the problem of MLB sign we compare the values of coefficients A, B, C for the 4f”- ‘Sd configuration which are presented in table 1 for all RE ions investigated in ref. [l]. In table 1 signs of

Table 1 Magneto-optical

parameters

Ion

Sign of MLB

Sign of

Th’Dy’Ho” Er” Tm” Yb”

+ t + ~

+ + + + + t

for RE ions. L

L,

A(5d)

B(5d)

C(5d)

A(5g)

B(5g)

C(5g)

3 5 6 6 5 3

0 3 5 6 6 5

-48 -32 -21 -12 -5 0

0 +12 t26 t34 t36 t30

0 0 -0.8 -3 -6 -10

0 PO.6 -2 -5 -8 -13

0 +11 +19 i24 t27 +27

-13 -19 -1s -II -8 -5

(I-1Q)

the

MLB

and of the

quantity

depend on the orientation First

WC should

paramagnctic cluestioncd and give The

(see, the

ground

e.g..

that

the positive

firmly

settled.

ref.

151).

sign

of

diffcrcnces

term

same

is

negative

energy

to the

note

garnets

AE,

(for

;t given

KE

ion

the sign

ot’ (-10

1 cioc4 no1

sign

ot

(Au)

have

tried

for

all

for

bea\>

RI,.

ytterbium

the C‘f- parameters.

m,liich

ions

trom

garnets.

dc~cribc

well

10 )‘I-,’

‘1%

\\herc

the

C’F

alI yectro4copic

do not exist. Ilavc found that \uch parameters ’ cm ’ ) bctwccn the terms of the cscitcd configuration.

II1

may

be

ckil;~

WI L

(-10

are hmall

(- 10’ cm

given

field).

We

to find

(IQ).

of the core.

configuration

arc also

(.IQ)

of the magnetic

in comparison

with

’ ). therefore

wc neglect

of the sign

of the MI-E.

the energy them

differences

in the first

belonging

between

them

~lpproximation.

the

and

From

tahlc

I WC WC that allowed transitions to the Iowa-lying terms. which arc characterid bv the go-ouncl term 01 the core. ~“ivc the correct signs of the MLB for garnets studied except those with i.1, “. C‘onscquc%ntl\ to explain the Ggn of the MLB fol- holmium garnets it is not ncccssary to consider influcncc ot a11 ahsorption band [I]. For ytterbium garnets the consideration of the splitting of the Jf’ ‘5~1 configut ;Ition

does

not

solve

It is known csscntial tions

the problem

that sometimes

contributions

to the

prcscnted

to the two-photon

MLB.

in

ytterbium

garnets

3. Anisotropy In ref.

the KE

/I.

(’ (SCC cqx.

of transitions

garnets

populated

the

anisotropy

the different

changes

structure).

under

This

cncrg!

of the

levels

MLB

in

of quadrupolc

consideration

of the MLB.

the known

had to be modified

For

ytterbium

The

be

MLB

configurations

Anti

not,

is

contribution observed

in

We have thus by the influence configuration tions

w,),

Here

(I, are

parameters. summation

for

‘5f contigur-ation corresponding

the

the 4f’

y\c

cc)ritribu-

‘Sg contiguration

contiguration

\o the problem

however. result

particular

dot\

not

result

drc in the

oU the Ggn of the ML13

ytterbium

of excited

and the sign

calculated

data.

tar

on the excited in comparison

to account value4

is the concept

situatccl

for

of different (e.g..

from

Tb’

of g-tensor\

tc) IX_.

directions

in the incquivalent ions

Gtes

01

) to dc\cribc

the ycctroscopic

\vholc complex

;I

that the main

Does

not

in an essential garnets.

of the

ions

for sonic’ KE

al-c conbicicred

field

data.

of cxpcrimcntal

and anisotropy

of the

MI.13

set of the C’F paramctcr4. contribution

the consideration

contribution

where

configurations

the additional

of the CF

to be small

casts

contiguration

magnetic

use of the C’F knoun

2 single

valid

-If’

(the

of RE

;I little

configuration?

in some for

garnets

making

using

how

of the ground

KE

spectroscopic

bv the ground

of the splittings anisotropy

the

simultaneousI?;

arises.

is given

excited

CF

garnets

described

question

of the

(5)) ‘3g

an opportunity

the anisotropy

ions

co~~lcl not

If’

of the 4t’

analycci

(4).

the

moments

giva

For data.

to

investigation.

quantitatively other

ha\e

of MLR

for

the garnet

,4.

to states

So wc

open.

[I] the thermally

rcsponsihlc induces

for is still

transition\

proccs\a.

and the coefficients

I. Consideration

table

of the MLB

change

the electric-dipole

the

effect

to the :lni\c)troph

of the CF

to the anisotropy i\ the

smallest

so great that it i\ the main

ylittings

(see

one‘

and

ot

of the ML.13’) ref.

[I I).

the

it can

give

the

MLB? contribution -If’ with

to the polarizability

‘5d contiguration. the distance

of the KE

C‘onsideriny

between

ion 6ct 1,

the splitting

the centro

’

GILI\C~

of the excited

ot gravity

of c,ontigur;l-

WC have for Sty-1 [h]

the

Stevens

Calculating over

the

parameters the necessary

values

of indices

for

the

ground

Clcbsch-Gordan .\ = 2. 4;

/I = 0.

multiplet

of the

coefficients t2.

t_4

If’

configuration.

and V)j-symbols

(I/II

.5): I

!I;,

the

and performing

2. 1: 7 = 0.

+ I.

(‘Ithe

a.4

N. P. Kolmakova.

A.I.

( ]T] < t), we obtain expressions for 6a when transforming to the Cartesian Further, we neglect the higher-order very small [7]. The resulting contribution to the

Sn, = where i= lo5 cm-‘), the linear taken as

Popov

I Magnetic

linear hirefringrwcr

of

RE garmvs

23

y’ in the explicit form and determine the corrections for the MLB coordinates and summing over inequivalent sites of RE ions. terms (t > 2) in formula (8) as their contribution is known to be MLB is determined

by the expression

& A,(SQ,) , (OOl), (1 lo), (1 11); a’ is the magneto-optical coefficient (a’ia - 2 x 10mh for hw,, (SQ,) is the thermal average of the appropriate combinations of quadruple components, A, combinations of parameters of the CF acting on the 4f”-‘5d configuration, which can be

(qsd = (qJ4,

(dlr’ld) (flr’lf)

(10)

.

Numerical calculations of the additional contribution 6n, to the MLB from the CF splitting of the excited configuration for all reasonable values of the CF parameters compatible with the available estimates for radial integrals, shielding parameters, etc. [8,9], give the value of 6n, not exceeding 1% of the main quantity An. Consequently, consideration of the CF splitting of the excited configuration results in inessential corrections to the MLB.

4. Conclusion Our analysis has shown that the electric-dipole transitions to the states of the 4f”-‘5d(g) configurations characterizing by the ground terms of the core allows one to account fully for the MLB in all heavy RE paramagnetic garnets (including holmium ones) except for ytterbium garnets. For ytterbium garnets we can find no other possible explanation of the properties of the MLB than the influence of excited configurations ndy4fN” (n = 3, 4), the possible importance of which has been noted, e.g., in refs. [lo, 111.

Acknowledgement We would

like to express

our thanks

to R.Z.

Levitin

for his interest

in this work

References [l] [2] [3] [4] [S] [6] [7] [8] (Y]

N.P. Kolmakova. R.Z. Levitin, A.I. Popov. N.F. Vedernikov, A.K. Zvezdin and V. Nekvasil, Phys. Rev. B 41 (1’390) 6170. M.C. Downer. C.D. Cordero-Montalvo and H. Crosswhite. Phys. Rev. B 28 (1983) 4931. 1.1. Sobel’man, Introduction to the ‘Theory of Atomic Spectra (Pergamon. Oxford, 1972). J.D. Axe, Jr.. Phys. Rev. 136 (1964) A42. V. Nekvasil, Phys. Stat. Sol. (b) 1OY (1982) 67. A.S. Moskvin and V.M. Pleschev, Opt. Spectr. 64 (1988) 721 (in Russian). N.P. Kolmakova, R.Z. Levitin, V.N. Orlov and N.F. Vedernikov. J. Magn. Magn. Mater. 87 (1990) 218. K. Rajnak. J. Chem. Phys. 37 (1962) 2440. C.A. Morrison and R.P. Leavitt. in: Handbook on the Physics and Chemistry of Rare-Earths, Vol. 5. eds. K.A. Gschneidner and L. Eyring (North-Holland, Amsterdam, 1982). 101 B.R. Judd and D.R. Pooler. J. Phys. C 15 (1982) 591. 111 P.J. Becker. Phys. Stat. Sol. (b) 43 (1971) 583.