Electron spin resonance of impurities in magnetic insulators

ELECTRON SPIN RESONANCE OF IMPURITIES IN MAGNETIC INSULATORS

F. MEHRAN IBM T.J. Watson Research Center, Yorktown Heights, N. Y. 10598, U.S.A. and

K.W.H. STEVENS Physics Department, University of Nottingham, Nottingham, Great Britain

NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM

PHYSICS REPORTS (Review Section of Physics Letters) 85, No. 3 (1982) 123—160. North-Holland Publishing Company

ELECTRON SPIN RESONANCE OF IMPURiTIES IN MAGNETIC INSULATORS F. MEHRAN IBM Ti. Watson Research Center, Yorktown Heights, N.Y. 10598, U.S.A.

and K.W.H. STEVENS Physics Departmen4 University of Noninghwn, Nottingham, Great Britain Received December 1981

Contents: 1. Introduction 1.1. Paramagnetic hosts

125 125

1.2. EPR linewidths 1.3. Electronic g.shifts 1.4. Pseudo crystal field effects (Vanimpurities Vleck paramagnets) 2. paramagnetic EPR spectra hosts of S-state in singlet ground state 2.1. Low temperature spectra 2.2. Illustrative example, PrVO 3~ 2.3. Other examples, HoVO 4:Gd 4, EuVO4, EuAsO4 and Smchalcogenides 2.4. Generalization to high temperatures 3. EPR spectra of S-state impurities in degenerate ground state paramagnetic hosts (cooperative Jahn—Teller compounds)

126 126 127 127 127 128

137 141

3.1. The Jahn—Teller 3~ineffect TmVO 3.2. EPR of Gd

4 and TmAsO4 below the cooperative Jahn—Teller phase transition temperatures 3.3. EPR of Gd~ in TmVO4 and TmAsO4 above the cooperative Jahn—Teller 3~in the phase dilute transition Jahn—Teller temperatures system 3.4. Tm~Yi_~,VOs EPR spectra of Gd 4. Lifetime broadening in EPR spectra of S-state impurities in paramagnetic hosts 5. Conclusions Appendix Acknowledgments References

142 143 144 147 151 155 155 156 157

142

Abstract: It is possible to observe and very useful to study the electron spin resonance of impurities in paramagnetic hosts. Among the interesting phenomena observable and measurable in these studies are: (a) symmetric and antisymmetric exchange interactions between the impurities and the hosts determined from the indirect superhyperfine interactions, pseudo-Zeeman interactions and pseudo crystal field effects, (b) the Jahn—Teller induced random strains effects resulting in the magnetic field dependence of the fine structure linewidths, and (c) the nature of the dynamical interactions of the hosts manifested in the lifetime broadenings of the impurity spectra.

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F Mehran and K WH. Stevens, Electron spin resonance of impurities in magnetic insulators

125

1. Introduction Electron paramagnetic resonance (EPR) studies of impurities are ordinarily carried out in diamagnetic hosts to reduce the magnetic interactions between the impurity and the host ions [35,112, 3, 9J. In this review we are concerned with the spin resonance of impurities in paramagnetic hosts, particularly when the impurity is an S-state ion and the host paramagnetic ions are lanthanide rare earths. The incomplete 4f shells of the rare earth ions are screened from the other ions (and from each other) by their complete 5s and 5p shells [197,75, 211], hence they experience weak crystal fields and the splittings of their ionic levels are small [33, 65, 177]. Thus in addition to first order paramagnetism there can be large second order (Van Vleck) paramagnetism [195],because of the small crystal field splittings. [Van Vleck was responsible for the modern theory of many kinds of para and diamagnetism. However, quite unfairly, only the second order paramagnetism has been named after him.] When the ground state of the ion is non-degenerate, at low temperatures the first order paramagnetism disappears and only the second order paramagnetism remains. In this case the material is called a Van Vleck paramagnet. We will consider only those rare earth host ions which have an even number of 4f electrons. There is then no K.ramers degeneracy resulting from time reversal symmetry [102]. (For an early example of EPR in hosts containing Kramers ions see Bleaney et al. [34].) If the local symmetry of the non-Kramers rare earth ion is orthorhombic or lower, all the crystal field-split levels will be non-degenerate (except for accidental degeneracies). For symmetries higher than orthorhombic, it is possible to have degenerate as well as non-degenerate levels. The highest degree of degeneracy (triplet) occurs only for cubic crystals. In this article we will use illustrative examples from primarily two groups of comp.ounds: the divalent S-state impurities (Eu2~and Mn2~)in the (NaCl) cubic samarium chalcogenides (SmS, SmSe and SmTe) and Gd3~impurities in the tetragonal rare earth zircons. The occupation of the degenerate levels of the rare earth ions leads to first order magnetic, exchange [82, 62, 83, 197, 12, 85, 182] and Jahn—Teller [92, 93, 200, 79, 22].effects. When the degenerate levels are not occupied, it is still possible to have second order effects [193,36, 59, 70, 31, 172]. Depending on the relative strengths of these interactions, one of the effects usually predominates and if the ion—ion interactions are large enough, it would lead to the corresponding cooperative phase transition. The states of the rare earth host ions directly affect the EPR spectra of the impurity ion. Conversely, from the behavior of the EPR spectra of the impurities, one gains information about the various interactions in the host materials. 1.1. Paramagnetic hosts In this review we use as illustrative examples of paramagnetic insulators two classes of compounds: the samarium chalcogenides and the rare earth zircons. The samarium chalcogenides (SmS, SmSe and SmTe) have the (NaC1) cubic structure. The paramagnetic ion is Sm2~with the 4f6, 7F singlet ground 7F t away [141,170]. At low temperatures, 0therefore, these state. The first stateparamagnets. 1 is about 300 cm (semiconducting) compounds [87, 61, 113, 174, 86] have compounds areexcited Van Vleck These been of great interest during the past decade due to their insulator—metal phase transitions under pressure [94, 50], where the samarium ions become mixed-valent [119,68]. Since the samarium ion (in the absence of pressure) is divalent, it is appropriate to use divalent (Eu2~and Mn2~)impurities for spin resonance experiments [120,30]. The tetragonal rare earth zircons have the D~(I4 1/amd)structure. The general formula is (RE) X04 with X as V, As or P and RE as Rare Earth. All rare earths in these compounds are trivalent. These

126

F Mehran and K. WH. Stevens, Electron spin resonance of impurities in magnetic insulators

compounds have been of interest during the past decade for their ideal suitability in the studies of cooperative Jahn—Teller effects [67, 71, 134], enhanced nuclear cooling [7, 13, 15], enhanced nuclear magnetism [14, 37], and the containment of radioactive nuclear wastes [160,4]. We will consider only those trivalent rare earth ions which have even numbers of 4f electrons. These include Tm3~(4f12,3H6 3~(4f’°, ~I8), Th3~(4f8,7F 3~(4f6,7F 2,3H4. Since there free ioniceven ground state), Ho 6), Eu 0) and Pr~(4f are only numbers of electrons involved, only non-Kramers degeneracies remain in a crystal field. The most suitable EPR probe for this series is Gd3~(4f7,8S 712) 1.2. EPR linewidths Before discussing the contribution of the paramagnetic host ions to the impurity linewidth, we describe the various sources of linewidths in the ordinary spectra of S-state ions in diamagnetic hosts: (i) Spin—lattice interaction This interaction between the impurity and the vibrational modes of the lattice [84]takes place through the orbital angular momentum [1041of the impurity and is strongly temperature dependent at high temperatures [201, 144, 117, 180]. Since the S-state impurity ions have basically zero 3~ orbital spectraangular in the momenta, this interaction is negligible for them. For instance the EPR linewidths of Gd diamagnet YVO 4 show negligible temperature dependences in the wide temperature range of 6< T < 550 K [129].This source of line broadening can thus be safely ignored for S-state ions. (ii) Fine, hyperfine and superhyperfine interactions These are respectively the interactions of the electronic spins of the impurities with (a) the crystal field [17], (b) the nuclear spins of their own atoms [152] and (c) the nuclear spins of the ligands [147]. The unresolved splittings due to these interactions are not negligible and should be determined in comparable diamagnets and subtracted from the linewidths of the impurities in the paramagnets under study [129, 130]. (iii) Impurity—impurity interactions This interaction [204,49, 202, 156, 97, 115] which for dilute S-state impurities, is almost exclusively due to the long-range magnetic dipolar interactions, can be drastically reduced by going to highly dilute samples. The remnant magnetic dipolar interactions are negligible in moderately dilute cases [130]. In addition to these sources of line broadenings in diamagnetic hosts, there are three major sources peculiar to paramagnetic hosts. These are: (iv) The static broadening due to the magnetic dipolar and exchange interactions of the impurity with the host ions [204,202, 156, 105]. (v) The life-time broadenings due to the fluctuating fields produced by the relaxation processes of the host ions [204,49, 105, 176, 129]. (vi) The indirect superhyperfine interactions of the electronic states of the impurities with the nuclei of the paramagnetic host ions via the polarization of the host electronic states [128, 130, 131]. 1.3. Electronic g-shifts The normal Zeeman interaction of a magnetic entity in a diamagnetic host is its direct interaction with an external magnetic field (except for negligible diamagnetic corrections [106]). The magnetic moment is the negative of the derivative of this interaction with respect to the applied magnetic field. The g-tensor is the ratio of the magnetic moment to the angular momentum in dimensionless units. When the magnetic center in question is in a magnetically polarizable medium, in addition to the direct

F. Mehran and K.WH. Stevens, Electron spin resonance of impurities in magnetic insulators

127

interaction, there is an indirect interaction with the applied magnetic field via the polarizable medium. As a result, the observed g is shifted from its normal value (in a comparable diamagnet). This shift has been observed in the nuclear magnetic resonance (NMR) in metals (the Knight shift) [99], molecules (the chemical shift) [157]and insulators [6, 8, 189, 44] and in the EPR of impurities in metals [148,149, 153, 154, 169, 207, 145, 161, 183, 188], insulators and semiconductors [88, 90, 120, 127, 129, 130, 30, 73, 21, 165, 164, 140, 116], semiconductors under pressure [205],electron-doped semiconductors [122,206] and mixed-valent semiconductors [101]. 1.4. Pseudo crystal field effects The ordinary crystal field experienced by an impurity in a diamagnet is the static electrical potential produced by the crystal at the impurity [27,28, 151, 198, 60]. In the EPR spectra, this field is manifested in the fine structure of the impurities [17].When the host contains magnetically polarizable ions, the exchange and magnetic dipolar interactions between them and the impurity produce a second order effect which resembles a crystal field [88,90,186, 128, 130]. This effect, like the indirect superhyperfine effect, depends on both the diagonal and the off-diagonal parameters of the impurity-host interactions (see subsection 2.2) and the combination of the two effects, together with the g-shifts, which depend on the diagonal parameters alone, can be used to estimate the magnitude of the off-diagonal as well as the diagonal parameters [130, 131]. 2. EPR spectra of S-state impurities in singlet ground state paramagnetic hosts (Van Vieck paramagnets) 2.1. Low temperature spectra The simplest way to study this case is by beginning to consider a pair of ions, an S-state impurity and a Van Vleck paramagnetic host ion, in an external magnetic field and then generalize to the whole crystal. The impurity (Mn2~,Eu2~or Gd3~)consists of an ion with essentially zero orbital angular momentum (S-state) and a spin angular momentum Sh, some of whose isotopes might also have nuclear angular momenta which would give rise to hyperfine interactions. The Van Vleck ion consists of a free ionic total angular momentum Jh whose (2J + 1) fold degeneracy is reduced in the “weak” crystal field (J remaining a “good” quantum number), with a singlet ground state. It may also have a nuclear angular momentum Ih. The impurity-host ion pair in an applied magnetic field is schematically depicted in fig. 1 (the nuclear angular momentum of the impurity is omitted). In this figure, the direct interactions are represented by dotted lines. (The electronic state of the Van Vleck ion, in its ground state is non-magnetic, hence J has no first order interaction with S, H or I.) The three direct interactions are: (a) the electronic Zeeman interaction between S and H, ~LBgSS H, (b) the nuclear Zeeman interaction between I and H, —/LBgII B and (c) the direct superhyperfine interaction between I and S, i/I,JSJJ. The host ion, though non-magnetic in first order, is highly magnetizable in second order if its excited states are close-by and (some of) the matrix elements ~iIJJ0)of the angular momentum operator J between the ground and excited states are nonzero. The typical second order term in perturbation theory expansion is: .

—

2z

~0I V’IiXiI yb) E

0—E,

1

128

F Mehran and K. W.H. Stevens, Electron spin resonance of impurities in magnetic insulators

V \

/c~

\\\\~///

l~

Fig. 1. Schematic diagram of the various interactions of an S-state impurity (with a spin angular momentum S5) and a Van Vleck host ion (with total electronic angular momentum Ji’t and nuclear angular momentum Ih) in an external magnetic field H.

The potentials V and V’ are respectively the energy operators which take the system from the ground state 0) “up” to the excited state i) and bring it back “down” to the ground state. (They may or may not be the same operator.) Thus, depending on what V and V1 are, the perturbation theory gives rise to six different second order interactions: (i) If V and V’ are both the host electronic Zeeman interaction ~g.,J H, eq. (1) will give rise to the Van Vleck paramagnetism of the host [195]. (ii) If V and V’ are both (the magnetic dipolar and/or exchange) interactions between S and .1, XUSIJ, eq. (1) will produce the pseudo crystal field effect on the impurity [90, 128, 130, 131]. (iii) If V and V’ are both the hyperfine interaction Al J of the host ion, eq. (1) will cause the pseudo-quadrupolar effect on the host nucleus [214,66, 44]. (iv) If V = ILBgJJ H and V’ = xqSJ, eq. (1) will give the pseudo-Zeeman interaction between the impurity and the applied field causing the impurity electronic g-shift [90]. (v) If V = ~ H and V’ = Al J eq. (1) will give the pseudo-Zeeman effect between the nucleus of the host ion and the applied field causing the nuclear g-shift [214,66, 44]. (vi) If V = ~qSiJ~and V’ = Al J, eq. (1) will give the indirect superhyperfine interaction between the impurity electrons and the host nuclei [129, 130, 131]. As all these second order effects occur through the operator J, for convenience we will refer to them collectively as the “J-effects”. .

.

.

2.2. Illustrative example, PrVO

3~ 4: Gd The best material for studying the interactions discussed in the preceding subsection is PrVO 3~ 4: Gd [130]. Figure 2 shows the (tetragonal zircon) crystal structure of this compound. TWe trivalent Ud

F Mehran and K. WH. Stevens, Electron spin resonance of impurities in magnetic insulators

129

~Pr~J~

Pr

a~7.367A Pr V0 4 ~Gd ,D~h

Fig. 2. Crystal structure of PrVO4:Gd [2121.

impurity substitutes for a trivalent Pr at a point of 2, tetragonal D~breaks symmetry, andfive has singlets four nearest Pr 2Jj~)of Pr~ up into and two neighbors. The 9-fold degenerate free ionic state (4f doublets in the crystal field. The energies and states of these levels have been determined by Bleaney et al. [38, 39] and are shown in table 1 and fig. 3. The quantum numbers in the last column of table 1 refer to the “magnetic” quantum number M~of the Pr3~ions with respect to the tetragonal c-axis. The free ionic state (4f7 8~,2) of Gd3~breaks up into four Kramers’ doublets in the crystal field. An applied magnetic field removes the Kramers’ degeneracies and the seven AM 5 = ±1transitions induced by the x-band microwave field between the eight energy levels produce the fine-structure spectra shown in figs. 4a—c. Tables 2 and 3 give some of the results of the low temperature (T = 5 K) comparable EPR experiments on 3~in the diamagnet PrVO4: Gd [130] and, for comparison, the corresponding values for Gd YVO 4 [162,130]. The parameters were obtained by fitting the data to the tetragonal spin Hamiltonian

Energy level i 0 1 2 3 4 5 6

Table 1 3F in PrVO Electronic states and energies of Pr 4, ref. [39] 1) State I i> Degeneracy n~ Energy E1 (cm 1 0 12> (1/V2X~+2)+ 1—2)) 1 35 0.86I4~)+ 0.5110) 2 84 0.971±1)+0.261r3) 1 127 1 276 I?) (1/V~+2)— 1—2)) 1 390 0.5114’)—0.8610) 2 600 0.261±1)—0.971;3)

130

F Mehran and K W.H. Stevens, Electron spin resonance of impurities in magnetic insulators 600 cm~

390 cm~ 2, 3H~

276 cm1

4f

127 cm~ 84 cm~ 35 cm~

\\ \\

o Pr3~FREE ION

TETRAGONAL (D2d) CRYSTAL FIELD (PrVO 4)

Fig. 3. Energy levels

of Pr~in

PrVO4 [39].

[201,155, 177, 179, 95, 89]: Zd

=

~LB[gl(HXSX +

HYSY) + g11H5S~]+ B°2O~ + B°4O~ + B°6On~ + BO~+ ~

(2)

The B~parameters are temperature dependent due to couplings to phonons [173].The O~crystal field operators can be expressed in terms of angular momenta [54, 177]. Only the three parameters g11, g~and B°2are listed in table 2, since the other parameters will not be discussed in this article. (For a review, see Buckmaster and Shing [51].A novel procedure for the diagonalization of the spin Hamiltonian matrix has been devised by Misra [136].)Table 3 gives the EPR fine structure linewidths for two doping levels. Table 2 Low temperature spin Hamiltonian parameters of PrVO4 Gd and YVO4: Gd

~i g1 Bi

PrVO4 : Gd

YVO4 : Gd

1.9939±0.0002 2.0199±0.0002 (—1.435 ±0.001) x 10~cm’

1.9919±0.0002 1.9919±0.0002 (— 1.598 ±0.001) x 10~cm’

Table 3 Low temperature linewidths in gauss of PrVO4: Gd and YVO4: Gd finestructure spectra PrVO4 : Gd

YVO4:Gd

[Gd](ppm) HII[001]

HII[110l

500 1000

33.1±0.6 6.2±0.2 34.5±0.5 7.5±0.3

16.0±0.5 17.9±1.1

HullOOl]

HII[110] 5.4±0.5 6.8±0.9

F Mehran and KWH Stevens, Electron spin resonance of impurities in magnetic insulators

I

I

I

a)

3

I

131

I

4

PrVO4’Gd T 5K

i_AnLool]

I I 3 4 5 MAGNETIC FIELD (kG)

_________________________________________

0

2

I

I

I

I

I

6

7

I

PrVO

4 Gd

A II[I00) I

2

I

I

5

3 4 MAGNETIC FIELD (kG)

c)

14

15

I

I

PrVO4Gd T~5K i~lI[IIo]

I

I I I 3 4 5 MAGNETIC FIELD (kG) 3~at 5 K with the external magnetic fields applied in various directions.

2 Fig. 4. Fine structure spectra of PrVO4: Gd

The slight difference is attributed to the remnant impurity—impurity interactions in the higher doping level. The linewidths in YVO are primarily due157Gd to thewith unresolved the 155Gd and 4: theGd—16% abundant nuclear hyperfine spins 3/2,interactions and much with smaller 15% abundant superhyperfine interactions with the 100% abundant 51V nuclei with nuclear spins 7/2 and still smaller superhyperfine interactions with the 100% abundant ~9Ynuclei with nuclear spins 1/2. Since Y3~is diamagnetic there are no electronic spin—spin interactions between Y3~(or the rest of the ions in YVO 4) 3~.As p 3k is a Van Vleck ion, at low temperatures, there will not be any direct electronic and Gd 1 spin—spin interaction between Pr~and Gd3~either. The hyperfine and direct superhyperfine interactions in PrVO 3~are almost exactly the same as those in YVO 3~except for the negligible 4 : Gd 4: Gd difference between the direct superhyperfine interactions of Gd3~with nsY and ‘41Pr. The lifetime -~

—

132

F Mehran and K WH Stevens, Electron spin resonance of impurities in magnetic insulators

broadenings (spin lattice and fluctuating spins) are negligible since Gd3~is in an S-state and Pr~is in a singlet ground state. The difference between the linewidths of PrVO 4 : with Gd and therefore 3~ 141PrYVO4: [128, Gd 130,are 131]. almost entirely due to the indirect superhyperfine interaction of Gd The low temperature I~amiltonianfor a pair of ions (the impurity Gd3~at the origin and a Pr~ion at x = 3.68, y = 0, z = 1.62 A) [see fig. 2] is: ~°pair = ~Gd

+

Xpr

+

~Od-Pr

(3)

-

has the same form as ~‘Gd in eq. (2). However, the parameters in such as YVO 4. The Pr~Hamiltonian ~W’p~ is: ~‘Gd

=~CF+/-LBgJJ-H+AJJ~I+~~CI.

~‘~d

are for Gd3~in a diamagnet

(4)

The first term in eq. (4), the crystal field (CF) interaction, is the largest term and gives rise to the splittings shown in fig. 3. The second and third terms are respectively the electronic Zeeman interaction with the external field and the magnetic hyperfine interaction between J and I. The last term: =

—/-~BgII

H + P(I~

—

(5)

~I2)

is the nuclear interaction (consisting of Zeeman and electric quadrupolar interactions) in an equivalent diamagnetic host. The most general bilinear form (allowed by symmetry) for the interaction between S and J is [103, 199, 143]: ~Gd-Pr

=

aSJ~+ f3S)Jy + yS~J~ + SS,,J5 + ~

(6)

The first three terms in eq. (6) are the diagonal elements of the symmetric tensor XiI• The last two (off-diagonal) terms may be rewritten as: ~~°olf-diag = ~(8+ 6)(S~J~ + SZJX) + ~(t5 E)(S~J~SzJ~c). —

—

(7)

In eq. (7), the first term is an off-diagonal element of the symmetric interaction tensor (the other off-diagonal elements being zero), and the second term is the nonzero element of the antisymmetric interaction [178,63, 138, 139]. The antisymmetric interaction is the source of the so-called “weak” or “canted” magnetism [139].The five parameters a, f3, y, c5 and s each contain an exchange component and a magnetic dipolar component. The exchange component of the diagonal elements [82, 62] are usually (but not always) larger the off-diagonal parts.a perturbation argument can be used to obtain 3~than occupies its ground state, On the assumption that Pr an effective Hamiltonian which does not contain PI3~electronic variables (see the Appendix): (8) where ~bUd~0i

(9)

F Mehran and KWH Stevens, Electron spin resonance of impurities in magnetic insulators

~

=

(10)

I(~I0)i2

6

133

(11) the summations being taken over the excited states shown in table 1, and (JiB gJHZ + A~I~ + yS~ +

~=

(J.LBgJHX +

=

oS~)2

(12)

A~I~ + aS~+ eS~+(I2ngjHy + A~I~ + flS~)2.

(13)

When the external magnetic field is applied along either the z-axis [001] or the x-axis [100], eq. (8) reduces to: = ~°

— 2~gra~yS~H~ + (AB~)S~2/LB g~A —

1aiiI~H~ + (AP)I~ 2Aj(a1I~s+ ajiI~y)S~ —

=

W° 2~~g1a±aS~H~ + (AB°2)S~2/JBgJAJaIIXHX —

—

+

(AP)I~ 2A~(a±I~a + a11I~5)S~ —

(14a) (14b)

where 2—

~2)

+

a±[~(a2 + ~2)

—

~2]

(15)

AB°2= a11(~ô and =

A~(a~ —a 11).

(16)

In arriving at eqs. (14) from eq. (8), constant and second order (off-diagonal) terms and terms which cancel on taking all four nearest Pr~neighbors into account have been dropped.3~ Inelectrons eqs. (14),and the ‘41Pr first term ~‘° includes all the (direct) first order terms in the interactions of the Gd nuclei with the external magnetic field and the second order electric field effects on Gd3~and 141Pr in an equivalent diamagnet. The second term gives the Gd3~electronic g-shift due to the indirect interaction of Gd3~with the external field through the electronic state of the Pr3~.The third term is the B~-shiftof Gd3~due to the pseudo crystal effect of J on S. The fourth term is the indirect interaction of the nuclear moment I of l4tPr with the external magnetic field through J which gives the nuclear g-shift. The fifth terms are the pseudo quadrupolar effect on I caused by J Finally, the sixth terms are the indirect superhyperfine interactions between Gd3~electronic state and the ‘41Pr nuclear state through .1 If the procedure leading to eqs. (14) is repeated for the other three Pr nearest neighbors and the results are added up, the Gd3~electronic and 141Pr nuclear g-shifts will be given by: Ag~

3~) = —2g~L~ja 1(Gd

3~) = —2g

11

Ag±(Gd 1L±a1 4tPr) = 2g~A.,a Ag11(’ 11 4tPr)= 2gjA.raj Ag±(’

(17)

(18) (19) (20)

134

F Mehran and K.WH. Stevens, Electron spin resonance of impurities in magnetic insulators

where L1~4y

(21)

L1na2(a+/3).

(22)

The L-parameters can be determined from eqs. (17—20): LII L

3~) Ag 141Pr) A ‘ Ag1j(Gd 11( —A Ag~(Gd3~) — ‘ Ag±(1~1Pr)~

(23 )

— — —

24

—

( )

The PrVO

3~g-shifts are calculated with respect to YVO 3~g-values (table 2): 4:Gd 4: Gd Ag 3~)= (20 ±3) x iO~, Ag 3~)= (280±3)x iO~. 1p(Gd 1(Gd The 141Pr nuclear g-shifts with respect to the unshifted value of [109]: g

4 1=9.01X10 are determined by Bleaney et al. [38, 39] Ag

41Pr) = 8.51 x 10~, Ag.L(141Pr) 11(’ The Pr hyperfine constant A., is [3,p. 298]:

=

4.65 x iO~.

A., = (3.65 ±0.03) x 10~cm1. Equations (23) and (24) then give: L

1, L 1. 11 = (—86 ±13) X iO~cm 1 = (—220 ±2) x 10~cm From eqs. (21) and (22) the diagonal parameters in eq. (6) for each neighbor are: y = (—21 ±3)x103cm’,

a + 13

=

(—110±2)x103cnf1.

These parameters consist of both the diagonal exchange and the diagonal magnetic dipolar interactions. The magnetic dipolar contributions from the nearest neighbors to Lt and L.L can be calculated from the classical formula:

~‘dip

~

[s. j—4~.r)(J~r)]

(25)

which contributes +0.022cm1 and —0.011 cm1 to L 11 and L.L respectively. The exchange parts of y and

F Mehran and K WH Stevens, Electron spin resonance of impurities in magnetic insulators

135

a + 13 are therefore —0.027 cm’ and —0.104 cm’ respectively. Since the magnetic dipolar interaction, eq. (25), is long-ranged, the contribution from other (farther) neighbors somewhat modify these values. The difference between the linewidths of PrVO4 : Gd and YVO4: Gd (table 3) can be explained by the indirect superhyperfine interactions (last terms in eqs. (14)). For H along the z-axis the indirect superhyperfine parameter is: 2a~+ y2a~f)”2. (26) A11 = 2A.,(s From table 3, the difference between the low temperature linewidths of PrVO 4: Gd and YVO4: Gd for H along the z-axis is: 9.8 ±0.6 G. [All lines have the same widths in a given spectrum.] The intensity of the individual lines in the superhyperfine pattern caused by n equivalent nuclei of spin I is given by the coefficients of the expansion of: I

~

(‘x’) t41Pr nuclei of spin I = the coefficients are: 4 1, 4, 10, 20, 35, 56, 80, 104, 125, 140, 146, 140 ~,

For n

=

This intensity pattern is shown in fig. 5. When the individual lines are not resolved, the envelope of the intensity distribution will be experimentally observed. It would be interesting to resolve the individual lines by an electron nuclear double resonance (ENDOR) experiment. In the EPR experiments the superhyperfine structures are not resolved and the linewidth is determined by the envelope shown in fig. 5 and is —8A. From the measured width difference between PrVO and YVO4: 141Pr nearest neighbor is 1.2 ±0.1 G. With the measured values of4:y Gd = —0.021 cm1 Gd, and aAu due to each 11 = 0.0145 cm, a~= 0.0791 cm [38, 39], eq. (26) gives the magnitude of the parameter ti: Je~= 0.019±0.OO2cm’. I

I

I

I

I

I— U, I—

I,,

~

I

I

I

I

I

I

I

I

I

/

>-_

w

I

-

/

MAGNETIC FIELD 3~due to the interactions with four nearest neighbor ‘41Pr nuclei of spin 5/2, each Fig. 5.a hyperfine with The intensity interaction of the indirect Al~J superhyperfine pattern of Gd

136

F Mehran and K WH. Stevens, Electron spin resonance of impurities in magnetic insulators

The purely magnetic dipolar contribution to this parameter may be calculated from eq. (25) and is —0.012 cm’. The small difference between the two may thus be attributed to exchange. When H is applied perpendicular to the c-axis, z, the four nearest Pr neighbors of Gd are equivalent only for HII[110]. The Gd3~—141Prindirect superhyperfine parameter for this direction, x’ is: ~ = \/2A 2)a~+ ô2afl1”2 (27) 1[(a2 + 13 The measured difference between the linewidths of PrVO Gd3.5 and YVO4: Gd for HJl[110]the (table 3) is 141Pr neighbor4: of ±0.1 G. Unfortunately, quantity 27.7 ± 0.8 G. This gives an ~ due to one determined from the g-shift is a +13 whereas a2 +132 is involved in ~ However, a2 +132 also appears in the B~-shift(eq. (15)). The only problem is that the difference in the crystal fields of PrVO 4 and YVO4 also causes a B~-shift.There are ways of estimating this latter shift. Urban [194]has found that a plot of B°2vs. c/a [seefig. 2] of various vanadates, arsenates andphosphates an approximately 3~,LuVO :follows Gd3~(both diamagneticstraight hosts) line. and In fig. 6 we have plotted B~ vs. c/a for YVO4: Gd PrVO 3t The B° 3~in the absence of a “J-shift” can be estimated by extrapolating the 2of PrVO4: Gd YVO 4: Gd 3t The 4—LuVO4 line.this Thisand procedure a B°2=2)—is1.803 cm’ for thegives unshifted : Gd difference between the actualgives B~(table 3.67 xx 10~ i0~cm1 which a PrVO4 = 9.2 x iO~cm1 for each Pr neighbor. -

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F Mehran and K WH. Stevens, Electron spin resonance of impurities in magnetic insulators

137

Using this value for AB°2and solving eqs. (15) and (27) simultaneously we find: 2 + ~2 = 0.0026 cm2. 181 = 0.341 cm’, a These values differ from those previously given [130, 131] where the B° 2shift was not corrected for the shift due to crystal field differences between PrVO4 and YVO4. The purely magnetic dipolar contribution to 6 from eq. (25) is —0.012 cm’. The parameter 8 is therefore dominated by exchange. The magnitude of the antisymmetnc exchange parameter can then be estimated:

I~(6 e)I I~8I —

0.17 cm_i

3~—Pr~interaction parameters will be since e~~ modified 6~.All when these the calculated the five Gdnot just the four nearest ones) are taken somewhat effect ofvalues all Pr for neighbors (and into account. 2.3. Other examples, HoVO 4, EuVO4, EuAsO4 and Sm-chalcogenides (i) HoVO4: Gd Although PrVO4: Gd is the best example for illustrating all the effects described in the preceding subsection, it is not the most dramatic example. TheHoVO indirect superhyperfine interaction manifests itself 3~in most vividly in the EPR spectrum of Gd 4 (fig. 7). The enormous nuclear magnetic enhancement in this compound leads to a relatively high nuclear antiferromagnetic phase transition temperature of 4.5 mK 43, 5]. Due largeout “J-effect”, the linesbutin HIIc-axis the fine structure spectra 3~are too wide for[185,42, the experiments to to be the carried in any direction [128].Ho3~ free of Gd ionic state has a 4f’°,~ configuration. In the HoVO 4 crystal field the ground state is a singlet and the first excited state is a doublet at —21 cm’ [24].Bleaney et al. [40,43] have determined: 165Ho)= 0.10 Ag~(

MAGNETIC FIELD (kG) Fig. 7. Fine structure spectrum of HoVO

3~at T= 5 K with the magnetic field along the c-axis. 4 Gd

F Mehran and K. W.H. Stevens, Electron spin resonance of impurities in magnetic insulators

138

and have estimated that Ag11(~Ho)is more than two orders of magnitude smaller. This implies a large 3~)is a1 and a within much smaller a11 [eqs. (19 and 20)]. Mehran et a!. [128] have found that the Ag11(Gd negligible the experimental error: g 11 = 1.997 ±0.005 and that the fine structurealinewidth for Hjlc-axis is —168±5 (comparedjustified to 16 Ginfor PrVO4the : Gd). The 3~)indicates small y [eqs. (17 and 21)]. We are,Gtherefore, ignoring second small Ag11(Gd term in the parentheses in eq. (26) and write: 2~AJsa 1~.

(28) 3~and assume If wethe subtract —~8G (due to other sources line broadening) from the linewidth of Gdsuperhyperfine that four nearest 165Ho neighbors can ofcause the rest of the linewidth by indirect interaction, the parameter A 11 due to this effect for each neighbor would be —15 G. From eqs. (20 and 28): 65Ho).

el

(29)

g.,Aj1IAg~(’

Using: g.,

=

1.2417 [96]

we calculate: Jet =~0.017cm’. The magnetic dipolar contribution to this parameter calculated from eq. (25) is e = —0.020 cm_i. Therefore, the large linewidth can be understood to a good extent by magnetic dipolar interactions alone. The small difference could be due to a (positive) exchange contribution. (ii) EuVO 4, EuAsO4 and Sm-chalcogenides The two ions that Van Vleck [197,p. as prime for his discussion of the second order 3~and Sm2~,245] bothused of which haveexamples the 4f6, 7F paramagnetism were Eu 7F 0 freethis ionic ground states. The first t away. Although energy difference between excited stateand is the 1 which 300 cm to the cases that we have already described in the ground the triplet first excited state isisabout large compared this article, it is by far the smallest for free ions. In the cubic crystal fields of the Sm-chaleogenides the first excited state of Sm2~remains a triplet, whereas in the tetragonal fields of EuVO 4 andshow EuAsO4 the 3~breaks up into a doublet and a singlet [48, 110]. Figures 8—12 the low triplet state of Eu temperature EPR spectra of EuVO 3~,EuAsO 3~,SmS : Eu2~,SmSe : Eu2~and SmTe : Eu2~ 4 : Gd[001] directions. 4:Gd The g-shifts and indirect superhyperfine with the magnetic field applied along broadenings in EuVO 3~g-values 4: Gd and EuAsO4:Gd are small [129]. At low temperatures the Gd are g(EuVO

3~)= 1.9814 ±0.0002 4: Gd g(EuAsO 3~)= 1.9839 ±0.0002 4 : Gd which show small antiferromagnetic interactions between Gd3~and Eu3~in EuVO 4 and EuAsO4.

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The Eu2~g-shifts (from g = 1.992) and indirect superhyperfine linewidths are much larger in the Sm-chalcogenides. At low temperatures the Eu2~g-values are [120,30]: g(SmS : Eu2~)= 2.241 ±0.005 g(SmSe : Eu2~)= 2.092 ±0.005 g(SmTe : Eu2~)= 2.016 ±0.005. Not only the g-shifts of Eu2~in the chalcogenides are much larger in magnitude than those in EuVO 3~and EuAsO 3~,but they are of opposite signs. These differences have been explained 4 : Gd[181]by observing 4: Gdthat the low-lying conduction bands [20, 23, 74] in the Sm-chalcogenide by Stevens semiconductors are important in the Eu2~—Sm2~ exchange. Even more dramatic are the g-shifts (from g = 2) of Mn2~in Sm-chalcogenides [30, 120]: g(SmS : Mn2~)= 1.05 ±0.01 g(SmSe : Mn2~)= 1.567 ±0.005 g(SmTe:Mn2~)=1.885±0.005.

140

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Fig. 10. EPR spectrum of SmS Eu

Fig. 11. EPR spectrum of SmSe Eu2~at 4.2 K with HV[001].

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Fig. 13. g-value of Eu’~in Sm 1...~La~S as a function of x.

2~linewidths due to the indirect superhyperfine interactions with As can be seen in figs. 10—12, the Eu the —15% abundant i47Sm and the —14% abundant 149Sm increase, as expected, in the same direction as the g-shifts in the Sm-chalcogenides. However, due to the complexities caused by the hyperfine interactions with the —48% 151Eu and the —52% i5SEu it is not easy to determine the exact contribution of the superhyperfine interactions to the linewidths. The magnitudes of the g-shifts in semiconductors can be increased by introducing electrons in the conduction band [122, 206]. Mehran et al. [122] have studied the EPR of the mixed system Sm (fig. 2~ 13) and found that the Eu2~—Sm2~ interaction is enhanced by the conduction 1_~La~S:Eu electrons introduced by the La3~ions. The Sm—Sm exchange interaction is also enhanced by the conduction electrons [191, 122, 16]. Similar conclusions have been reached by Walsh et al. [206] for Sm 1_~Gd~S and by Goncharova et a!. [76] for Sm1÷~S.

F Mehran and K. WH Stevens, Electron spin resonance ofimpurities in magnetic insulators

141

2.4. Generalization to high temperatures The calculations performed to determine the g-shifts at low temperatures due to the host ground states can be repeated for each of the excited states if their wave functions and energies are known. Again, the ideal illustrative example is PrVO4: Gd [130] for which Bleaney et al. [39] have determined 3FI all the states and energies arising from the crystal field splitting of the 9-fold degenerate 4 free ionic state of Pr~[table 1]. From eqs. (10 and 11) the values of a11 and a1 for each state can be determined. These values plus the Lii and L1 parameters already determined at low temperatures can be used to find Ag11 and Ag1 for each Pr~level [eqs. (17 and 18)]. The results can then be combined with Boltzmann statistics to give the temperature dependence of the total g-shift: Ag(T) =

Ag, e_~1~T/~ ~t ~

(30)

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Fig. 15. Temperature dependence of Eu’~g-values in SmS and SmSe.

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the theory are expected since, in determining the energy levels and the wave functions of the higher levels, Bleaney et al. [39]have used procedures which are of decreasing accuracy as they proceed from the lower levels, whose values are based on experiments, to the higher levels which are based on calculations. 24 g-shifts in Sm-chalcogenides (fig. 15) [127]and the Gd3~g-shifts The temperature variations of Eu in EuVO 4 and EuAsO4 (fig. 16) [129] and in lanthanide hydroxides [116] have been determined and explained in terms of the effects of the excited states in the hosts. The temperature variation of the linewidths and the crystal field parameters of the impurities at high temperatures are only partially caused by the “J-effects” and are mostly due to other (larger) effects which will be described in section 4.

3. EPR spectra of S-state impurities in degenerate ground state paramagnetic hosts (cooperative Jahn—Teller compounds) 3.1. The Jahn—Teller effect Jahn and Teller [92, 93] have shown that in non-linear molecules, a symmetry which gives rise to a non-Kramers electronic degeneracy is unstable as there is, in all known cases, a nonzero linear electron—nucleon interaction which causes a distortion that reduces the symmetry and removes the degeneracy. This is the static Jahn—Teller effect. However, there are always several equivalent ways that this distortion can occur [200], hence in general the degeneracy of the vibronic (coupled electronnuclear) state is not reduced, although the parameters of the states are affected by this dynamic Jahn—Teller effect [79]. The consequences of the (dynamic) Jahn—Teller effect in EPR were first observed by Bleaney and Ingram [32] in copper fluosilicate and explained by Abragam and Pryce [2]. Similar effects were predicted by Bir [29] and Morgan [137] for shallow impurities in semiconductors and observed by Mehran et al. [121, 123]. The impurity Jahn—TelIer effect has been recently reviewed by Bates [22].

F Mehran and K WH. Stevens, Electron spin resonance of impurities in magnetic insulators

143

In crystals with concentrated Jahn—Teller ions, each ion interacts with the distortion around it and since the distortions are the local manifestations of the lattice vibrations, there is in effect a “virtual phonon exchange” interaction between the Jahn—Teller ions. This effect is similar to the phonon exchange effect in magnetism [184, 114, 146, 18, 19] and gives rise to the cooperative Jahn—Teller phase transitions [71]. In the spinels (with 3d electrons) the Jahn—Teller interactions are large and the transitions occur at high temperatures. In the zircons (with 4f electrons) the interactions are small and the transitions are at low temperatures. The rare earth zircons are the most suitable materials for studying the cooperative Jahn—Teller effects. This is because: (a) the tetragonal symmetry gives the lowest possible degeneracy (doublets) resulting in the simplest theoretical analyses; (b) the transitions are at low temperatures, so there are no complications due to high temperature effects; and (c) they are transparent and may be studied optically. None of these conditions are satisfied in the (cubic) spinels. 3.2. EPR of Gd3’ in Tm V04 and TmAsO4 below the cooperative Jahn—Teller phase transition tern peratures 3 at 1.5 K in the low symmetry (orthorhombic) phase of the Schwab [166]has studied the EPR of Gd cooperative Jahn—Teller systems TmVO 4 [58, 80, 26, 133, 168, 25, 150] with the transition temperature 12, ~H = 2.156 K [41],and TmAsO4 [118,56, 80,seven 25, 108] with TD 6.1 high K. The 13-fold freephase ionic 4f 6of 3~breaks up into three doublets and singlets in =the temperature (D~point Tm symmetry) of TmVO 3~in YVO 4 and TmAsO4. The energy levels of Tm 4 are shown in fig. 17 [100,209].

361 cm~ 335 cm~ 332 cm~

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54 cm~

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0

TETRAGONAL (D2d) CRYSTAL FIELD (YVO 4Tm) 3~ion in YVO

Fig. 17. Energy levels of Tm

4 [1001.

F Mehran and K WH. Stevens, Electron spin resonance of impurities in magnetic insulators .1

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Tm4s04’ Gd TrI.9K

± ~/H~IC

MAGNETIC FIELD (kG) 3~at 1.9 K with the magnetic field applied along the c-axis.

Fig. 18. Fine structure spectrum of TmAsO

4: Gd

The ground doublet strongly couples to the B2g distortion mode of the D~lattice and less strongly to the Big mode [133]. Below the phase transition all Tm3~ions are in the orthorhombic D 2 positions and the electronic degeneracies are removed. The EPR results [166] give the orthorhombic crystal field parameters. The EPR spectrum of TmAsO4: Gd at 1.9 K with the magnetic field applied along the c-axis is shown in fig. 18 [125].Schwab [166]has also studied the low temperature EPR of ThAsO4: Gd which has a cooperative Jahn—Teller phase transition temperature of 27.7 K [210], caused by the near 3~. degeneracy of the lowest four states of Th 3.3. EPR of Gd3~in Tm V0 4 and TmAsO4 above the cooperative Jahn—Teller phase transition tern peratures 3~Jahn—Teller ions theaphase degenerate electronic of the Tm effect [124]. The areAbove split at giventransition time duetemperatures to random the local strains caused by states the Jahn—Teller splittings of the Fe2~Jahn—Teller ions in the (cubic) spinels FeV 2O4 and FeCr2O4 above the cooperative phase transitions were observed in the Mössbauer spectra by Tanaka et al. [187].In the zircon systems, it was originally thought [71, p. 57] that due to zero point motions, the non-uniform strain in the high symmetry phase would not occur. However, since the distortions involved in the zircons are nondegenerate (Big3~ orin B2g), the symmetry possibility phase of distortions above phase transitions cannot be ruled out. The the high of TmVO splitting of Tm 4 [124] and TmAsO4 [125] were observed in the 3~spectra in these hosts. GdAt low temperatures only the ground state doublet of Tm3~is occupied and this state can be represented by a pseudospin o- = ~[201] with a Jahn—Teller coupling eQo-~where 0 represents a local strain variable. Each local Q can be expressed as a linear combination of the amplitudes of the normal modes of the lattice oscillators and the effective Hamiltonian used to describe the Jahn—Teller effect becomes:

F Mehran and K. WH. Stevens, Electron spin resonance of impurities in magnetic insulators

~Tm

=

K[(a

—

145

L)2 + (b — L)2] + e(a — b) ~

+~hw(k)(a~ak+~)+~ exp(ik .R~)~(k)o~(ak+a~k).

(31)

The first term in eq. (31) expresses the requirement that in the absence of a coupling to the doublets the lattice would have a tetragonal unit cell of side L and vibrations would occur about the tetragonal cell positions. When the unit cell is distorted, and has sides a and b, the vibrations take place about the distorted cell positions. The second term describes the coupling of the uniform strain to the doublets, and the third and fourth terms, respectively describe the phonons and their interactions with Tm34 doublets. All pseudospin operators appearing in eq. (31) commute, so that all cr~can simultaneously assume values ±1/2at random and the effective Hamiltonian would only be left with lattice variables. By a displaced oscillator transformation [71],such an expression can be reduced to the form: (32)

where C~is a Jahn—Teller reduction energy associated with a nonzero value for a — b, and C 2 is a Jahn—Teller reduction energy associated with the coupling to the phonons, which now oscillate about the displaced lattice sites. It is important to realize that the lattice displacement is composed of two parts. First, there is that due to the uniform strain a — b, which is the same for each lattice site. The second part, which varies from site to site in the lattice, arises from the pseudospin—phonon coupling. The quantities C~and C2 in eq. (32) are respectively important below and above TD. Below TD, ~,. o~ 0, but ~,, exp(ik . R,,)o~tends to zero, so there is a nonzero uniform strain and diminishing random strain. Above TD, ~,, o~= 0 and ~,. exp(ik R,,) o~~ 0 so there is no uniform strain but nonzero nonuniform strains. A given nonuniform strain pattern in the high temperature phase corresponds to the random distribution of the ±1/2assignments of o~.Since there are many ways that this assignment can be made and the lattice can move from one arrangement to another, the random strains in general will be dynamic. However, the random strain fluctuations slow down with decreasing temperature. 3~is The simplest (~~54 example of afrom cooperative TmVO4. ground doublet of Tm is well separated cm_i) the first Jahn—Teller excited statesystem [100,is209] (fig. The 17). The next best example TmAsO 4 where the first excited (singlet) state is only 13.8 cm_i above the ground doublet [25] and therefore the assumption of o = ~is not justified in this case as it is for TmVO4. An isolated doublet in an axial field will have a g-value of zero for the external field applied perpendicular to the c-axis [78]. This requirement is satisfied in TmVO4 [100,26, 167] where g1 0, g~1= 10.22 but not for TmAsO4 [118, 167] where g1 5.5, gi~ 8.8. 3~impurities in the high temperature phases of TmVO The EPR spectra of Gd 4 and TmAsO4 have been studied by Mehran et al. [124, 125]. The EPR spectrum of TmVO4: Gd at room temperature with the magnetic field applied along the c-axis is shown in fig. 19. To see a resonance at all at room temperature in a magnetic host is fairly unusual. However, in this case the spin-lattice relaxation times of the host are very short The due temperature to transitionsvariation betweenof crystal field(narrowest) split levelslines causing 3~levels. the outer of fig.reduced 19 are broadenings for Gd shown in fig. 20. The lines broaden rapidly with decreasing temperature and are not visible close to the phase transition temperature. .

i~

F Mehran and KWH Stevens, Electron spin resonance ofimpurities in magnetic insulators

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F Mehran and K. WH Stevens, Electron spin resonance of impurities in magnetic insulators

147

The EPR spectra of TmAsO4: Gd at temperatures above and at phase transition are shown in fig. 21. The temperature variation of the outer two lines of fig. 21 are shown in fig. 22. The difference between 3 in TmAsO the linewidths of the high-field and the low-field lines of Gd 4 [fig. 22] at low temperatures indicates that the primary source of line broadening is due 3~ to ions an effect which is induced the applied are split in the randombystrain field of magnetic field. The most logical explanation is that the Tm their Jahn—Teller interactions rendering them non-magnetic in the absence of a magnetic field [197,p. 273] and then the applied magnetic field induces a magnetic moment in them which is an increasing function of the applied field: — lhinduced —

2

33

~cLBgTmH + (/LBgTmH)2

where zl is local random Jahn—Teller splitting of a Tm3~ion. The magnetic field dependence of the fine structure linewidth, of course, is also present in the low symmetry phase where the splitting of Tm3~is caused by uniform strain [figs.18 and 22]. The random strain effects due to the Jahn—Teller effect are in general dynamic. This is especially true when the effects are caused by the Jahn—Teller interactions of the excited states [133, 126, 81, 1]. When the random strains fluctuate very rapidly, their diagonal (secular) effects on the Gd3 EPR linewidth diminish [47, 77, 202, 10, 11, 1051. However the rapid fluctuations contribute to the lifetime non-secular broadening [204, 105, 176] which will be discussed in section 4. 3.4. EPR spectra of Gd3~in the dilute Jahn—Teller system Tm~Y 1_~VO4 3~in the concentrated TmVO The EPR lines of Gd 4 host broaden rapidly below 100 K and cannot be observed near the3~ phase transition. broadening canthis be system reduced the0.10 system with ions. Mehran et a!.The [132]have studied forby x =diluting 0.01, 0.05, and 0.15. non-Jahn—Teller Y Since there is a slight mismatch between the sizes of Tm3~and Y3~,the lines will also be broadened by the ordinary strains [1111.It is an experimental task to distinguish between this kind of strain and those generated by the Jahn—Teller effect. Figure 23 shows the Gd3’ spectrum in Tmo~Yo. 9VO4at 5 K with I

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Fig. 23. Fine structure spectrum of Tm01Y09V04: Gd at 5K with the magnetic field applied along the c-axis.

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the magnetic field along the c-axis. Figures 24 a—d show the temperature variation of the linewidth for various concentrations of Tm3t Since the experiments are performed in the high symmetry phase, the first two terms in eq. (31) as well as the term Ci in eq. (32) are zero and when the pseudospins and the oscillators are “decoupled” by the displaced oscillator transformation, C 2 will be m. C2 =

— n,m,k ~

exp[ik

.

(34)

(R,, — Rm)1ty~iz

In this form the pseudospin—pseudospin interaction resembles the Ising [91]interaction in magnetism. For the EPR experiments it is necessary to apply a magnetic field H and then it is not possible to make an exact displaced oscillator transformation. To a reasonable accuracy the local Hamiltonian for a single Tm3~ion can be described by a spin Hamiltonian: Xiocal

(35)

= /LBgTmHo~x+ /.1~rz

where the two terms are the Zeeman and Jahn—Teller effects and A is a function of the magnetic field H, temperature T, concentration x and time. For a given concentration, the Jahn—Teller splitting A is “switched off” for a magnetic field above a critical value. This behavior is clearly observed at low temperatures in figs. 24a—d. For x = 0.01, the lowest-field line has a width of 18 G, but all the other transitions have the same width of —35 G. At x = 0.05, the lowest field transition again has the narrowest width, the second lowest has the next narrowest and then all the other widths are the same. At x = 0.10, the first three transitions are in sequence of widths, with the next four having identical widths. At x = 0.15, it is the first four in sequence, with the final three the same. For x = 0.01 it would seem that when the external field has reached the value for the first resonance, the average strain splitting is sufficiently large that most of the Tm3~ions are only slightly magnetic, so giving little broadening. By the time H has reached the value for the second resonance, the Zeeman field has overcome most of the strain splitting, the Tm3~moments have been completely “switched on”, and thereafter they remain constant as H is further increased. For the next value of x (0.05), it is not until the field is increased to that of the third transition that the strain is “switched off” and so on. It is difficult to see why this should occur if the main contributions to the random strain come from the yttrium diluents [167,72, 142], but it is consistent with the picture of a Jahn—Teller random strain which is “switched off” as H increases. A detailed picture of how A varies with H, x and T cannot be deduced from the experiments and the theory of spin—phonon interactions [171, 190, 53] is not sufficiently developed to predict what occurs. At lower fields, where the Jahn—Teller splittings of the Tm3~states are not removed by the magnetic field and below a temperature of —10 K (which seems to be rather insensitive to the concentration x), the Gd3~linewidths become temperature independent. This means that, for this region, the random strain fluctuation times are longer than the time scale of the experiment (— iO~sec). The reason for this sharp slowing down of the fluctuations is that the most effective mechanism for the fluctuations (i.e. excitation of Tm3~ions to their first excited state at ~ cm_i) is removed at low temperatures. The EPR studies of Gd3~spectra in the dilute system Tm~Yi_~VO clarify 4 several points: (a) the virtual phonon exchange exists (A increases with concentration), (b) it is long-ranged (it exists in dilute systems) and (c) the random strains produced by it slow down at low temperatures. If the long-range virtual phonon exchange interactions between pseudospins is an oscillatory function -~

F. Mehran and K. WH Stevens, Electron spin resonance of impurities in magnetic insulators

151

of the pseudospin separations, Tm~Yi_~VOwill 4 become the Jahn—Teller analogue of the RKKY [69, 215, 158, 159, 163, 98, 213, 203] spin glasses [148, 149, 52, 64] for it would lead to a Jahn—Teller frustration effect [192]and Jahn—Teller glass. Orbach and Tachiki [146]have shown that if the spins are split in a weak magnetic field, the virtual phonon exchange interaction is indeed oscillatory for real spins. 4. Lifetime broadening in EPR spectra of S-state impurities in paramagnetic hosts When an electronic state of a host ion is degenerate (or nearly degenerate), in addition to the line broadenings described in the previous sections, it causes off-diagonal (non-secular) lifetime broadenings due to the magnetic, exchange or Jahn—Teller fluctuations [204, 105, 176]. The best examples for • studying these effects are EuVO4: Gd and EuAsO4: Gd [129]whose low temperature spectra are shown in figs. 8 and 9 and whose temperature spectra shown in figs. 25 7F and 26. As discussed in 3~ionsroom in these compounds have are singlet ground states subsection 2.3, the Eu 0 and triplet the freetriplet ionic 7F~which are —‘-300 cm_i above the ground states. In the tetragonal zircon structure states splits into a singlet and a doublet. At low temperatures the EPR linewidths of Gd3~ions, caused by the normal sources previously described, are less than about 10 G. The indirect superhyperfine interactions are negligible. As the temperature is increased the doublets of Eu” start getting populated and cause two effects, one of which usually predominates over the other: (a) the dynamic Jahn—Teller effect which produces a fluctuating orthorhombic B~field at the Gd3~sites [124, 125, 126, 129] causing ~ = ±2 transitions among the eight energy levels of Gd3~(b) fluctuating magnetic dipolar and/or exchange effects which produce fluctuating B~fields at the Gd3~sites causing t~M~ = ±1transitions among the Gd3” energy levels [129].The two effects are almost completely mutually exclusive for the dominance of one sharply reduces the effectiveness of the other [129]. For a Gd3* energy level with the magnetic quantum number M~,the lifetime broadening from a dynamic B~field is due to the transition to M~±1 levels. The transition probabilities are proportional to [see, e.g., 107] (MS ±1IS±1M

2

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152

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F Me/iran and K. WH. Stevens, Electron spin resonance ofimpurities in magnetic insulators

The lifetime broadening of the M5 level is therefore:

2 + (Ms — us_tM

~H(M5) = i~H~ + i~H_= K1(J(M5 + 1IS+1M5)1 where K 1 is a constant. The width of the line corresponding to the transition M5 —

1)’ i~H(M5)+ft~H(MS— 1)

2)

(36)

5)!

*—--+

M5

—

1 is [208, 176]:

K2[2S(S+ 1) 2M5(M5

1)1].

(37)

For a fluctuating B~field the lifetime broadenings are due to transitions L~M5= ±2.The transition probabilities are proportional to:

2.

I(M5 ±2IS~IM5)I

The lifetime broadenings of the M 5 level is

2+(M 2). ~‘H(M~) = ~‘H++ ~‘H_ = K3(I(M5 + 2I5~IMs)I 5 — 2IS~IM5)I The width of the line corresponding to the transition M 5 ~—-- M5 — 1 is: 2y2+ 2+ (y + 1)(2x2— 5x + y + 1)] ~‘H(M5’t—--~ M5 — 1)= i~’H(M5)+i~’H(M5— 1) = K4[x where x = S+M 5 andy = S—M5. Table 4 Theoretical linewidths from fluctuating magnetic dipolar or exchange effects (i~H)and dynamical Jahn—Teller effects (~‘H)calculated from eqs. (37) and (39) Transition

AH/K2

±7/2+—-~ ±5/2 13 ±5/2+——~±3/2 23 ±3/2’I—--3’±1/2 29 +1/24—--’—1/2 31

11’H/K4 66 126 186 210

Table 5 Theoretical and experimental Iinewidth ratios for high temperature EuAsO4 Gd, EuVO4:Gd and PrVO4 : Gd spectra Theory

±7/2~——* ±5/2

Experiment

Dipolar or exchange

Jahn— Teller

EuAsO4:Gd

EuVO4 : Gd

PrVO4 : Gd

2.38

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2.82

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2.23 ±0.03

2.3±0.1

1.77

1.91

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1.76±0.05

1.8±0.1

(38)

(39)

F Mehran and K.WH. Stevens, Electron spin resonance of impurities in magnetic insulators

153

Table 4 shows the theoretical linewidths predicted by eqs. (37) and (39). Figure 27 shows the temperature dependence of the fine structure linewidth in EuVO4: Gd and EuAsO4:Gd with the external magnetic field applied along the c-axis. By taking the ratios of the theoretical linewidths, in a given fine structure pattern, the constants K2 and K4 are eliminated and the theoretical ratios can be compared to the experimental ratios, once the low temperature widths are subtracted from the experimental values. These are given in table 5, which shows excellent agreement between the experimental results and the predictions of a fluctuating B~ field. It is not possible, however, to distinguish between exchange and magnetic dipolar effects by this method since they both produce fluctuating B~fields at the Gd positions. On the other hand, since the 90

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154

F Me/iran and K. WH. Stevens, Electron spin resonance of impuritiesin magnetic insulators

lattice constants for EuVO4 and EuAsO4 are comparable, one expects the magnetic dipolar effects to produce comparable fields. The experimental results (fig. 27), however, that the in widths are much 3’4’—Eu3’4 exchangeshow interaction the two comlarger for than for Gd. the TheGd—Gd Gd pounds canEuVO4: be quiteGd different. It isEuAsO4 known :that exchange interactions in GdVO 4 [57]is about a factor of 2 larger than the exchange in GdAsO4 [55]. Since the high-temperature linewidths in EuVO4: Gd are about twice as large as linewidths in EuAsO4 (fig. 27), itexchange appears that the dominant 34’ line broadening in the these compounds is the fluctuating interaction. mechanism for Gd The lifetime broadenings are also dominant in the high temperature spectra of PrVO 4 : Gd (fig. 28) 3’4’ in this compound is shown in [130]. of theatfine structure linewidththeof low Gd temperature widths should be fig. 29.The Fortemperature comparisonvariation with theory high temperatures subtracted. As seen in subsection 2.2, an important contribution to the low temperature widths comes from the indirect superhyperfine interaction which is a temperature dependent effect. The temperature variation of the indirect superhyperfine parameter A 11 is given by [130]: A gj A11 = _L

3’4’)j + [-~2~g ~12 r

~ ~g1(Gd j(r[-i--

34’)j J 12i/2

(40)

.

1j(Gd

The ratios of the linewidths a~high temperatures for HIIC, after the subtraction of non-lifetime broadenings, are shown in table 5 and are in fair agreement with the predictions of a fluctuating B~field theory. The lifetime broadenings of the fine structure spectra of Gd3’4’ impurities have been observed by Bleaney et al. [45] in the cubic salt Cs 2NaHoCl6 and by Maihotra and Buckmaster [116] in the hexagonal lanthanide hydroxides. Bleaney and Wells [41] have seen an electric quadrupole hyperfine structure in the NMR of 5iV in TmVO4 whose linewidths may be explained by the fluctuating effects described in this section.

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F Mehran and K.WH. Stevens, Electron spin resonance of impurities in magnetic insulators

155

5. Conclusions We have seen that by the EPR spectroscopy of magnetic impurities in paramagnetic hosts it is possible to study several interesting effects which are peculiar to these compounds. As examples for hosts we have used Van Vleck paramagnets and cooperative Jahn—Teller crystals. From the indirect superhyperfine interactions and pseudo-Zeeman and pseudo-crystal field effects it is possible to determine the symmetric and antisymmetric exchange interactions between the host and the impurity ions. The antisymmetric exchange is found to be larger than the symmetric exchange in some cases. The random strains causedby the Jahn—Teller effect in the high symmetry phase of the cooperative Jahn—Teller phase of the cooperative Jahn—Teller systems are easily observable by EPR which shows the presence of these effects at very high temperatures. Finally the lifetime broadenings of the EPR spectra of impurities can be used to determine the various dynamical interactions in paramagnetic insulators.

Appendix The Hamiltonian for a Gd3’4’—Pr3’4’ pair in PrVO4: Gd is given by eqs. (3—6), and the requirement is to 3’4’ has spin S = ~ and the Pr34’ is in its ground rewriteThe it as an effective Hamiltonian in which theabsent. Gd state. electronic variables of Pr34’ will then be The procedure which will be followed is an example of a general technique in the perturbation theory of degenerate manifolds [196,46, 135 (p. 719), 175]. The unperturbed ground manifold is described by a projection operator Po. which is the product of the projection operator for Gd3’4, S = ~ manifold and the projection operator for the Pr3~ ground state, the latter being in fact the unit operator. Excited manifolds will have projection operators denoted by F 3~S = ~ manifold and where each a product projection the Gd Hamiltonian, ~1), is the projection1, operator of F, theis excited levelof ofthePr34’ In firstoperator order theforeffective P 34’, 0~CP0,and since J~,J~,and J~all have zero matrix elements34’. within the singlet ground state of Pr = P 0~P0= order H°~d + E0 where E0is:is the ground energy of Pr The second contribution ~(2)

= ~

where e, is the energy associated with P 34’ is in its ground state throughout s~can be 1. Since the Gd e, with E,, the excited state energies of Pr3’4. identified with E 4’), and 0 (the ground energy of Pr3’ of the Hamiltonian in the form: It is convenient to write the state J-containing terms uJx+vJy+wJz

where u=.u 8gjH~+AjI~+aS~+eS~ v= W

+ A1I~+

f3S,,

=/hBgJH~+AJI~+ yS~+5S~.

F Me/iran and K. WH Stevens, Electron spin resonance of impurities in magnetic insulators

156

Then

contains a number of contributions, of which typical ones include the factors:

~(2)

(0IJ5Ii)(iIJ~I0) E~-E, and

(0IJ~Ii)(iIJ,,l0) -

E~-E, where that

to) and I i) are states of

4’. The tetragonal symmetry at the Pr3’4’ site can then be used to show Pr3-

(0~J~i)(i~J,,~0) — 0 -

using the invariance properties of OXO! and ~, ji)(iI. It follows that: ~(2)

+

= ~(2)

~

+

~

where ~,(2)_

~~(2)

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~2V

-

V

2V

~

I(iIJ,,I0)12 E 0-E1

=

1w12 ~

(zIJ~K~

which give the eqs. (8—13) in the text.

Acknowledgments This review is largely based on studies conducted during the past decade with the collaborations of S.E. Blum, W.J. Fitzpatrick, F. Holtzberg, T.N. Morgan, T.S. Plaskett, T.D. Schultz, M.W. Shafer, R.S. Title, J.B. Torrance and the expert technical assistance by R.A. Figat, C.J. Lent, P.G. Lockwood and A.H. Parsons. We also wish to thank R.W. Johnson, J.D. Kuptsis and G. Piguey for the electron microprobe, optical emission and mass spectroscopic analyses, J. Angillelo and J.M. Karansinski for the X-ray orientations, J. Keller for the Hall measurements and A. Moldovan for crystal cuttings. We are indebted to Drs. J.H. Griesmer, R.D. Jenks and D.Y.Y. Yun for the SCRATCHPAD symbolic mathematical programs. Over the years we have greatly benefited from conversations with Drs. B.S. Berry, G. Burns, M. Cardona, J.M.D. Coey, W.P. Dumke, J.R. Fletcher, J.F. Janak, A.G. Konheim, W. Liniger, T.R. McGuire, A.P. Malozemoff, R.L. Melcher, K.A. Muller, A.H. Nethercot, F.M. Odeh, T.

F Me/iran and K WH. Stevens, Electron spin resonance of impurities in magnetic insulators

157

Penney, M. Pomerantz, E. Pytte, J.C. Slonczewski, F. Stern, G.V. Subba-Rao, L.J. Tao, S. von Molnar, P.S. Wolfe and Professor B. Bleaney. We especially appreciate the careful and patient works of Messrs. J.V. Staropoli and D. Carrington for preparing the figures and Mrs. E.P. Cavanaugh for typing the manuscript.

References [1] R.Yu. Abdulsabirov, S.I. Andronenko, L.P. Mezentseva, L.A. Bondar and V.A. Loffe, Soy. Phys. Solid State 23 (1981) 327. [2] A. Abragam and M.H.L. Pryce, Proc. Phys. Soc. London A 63 (1950) 409. [3] A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions (Clarendon Press, Oxford, 1970). [4] M.M. Abraham, L.A. Boatner and M. Rappaz, Phys. Rev. Lett. 45 (1980) 839. [5] AL. Ailsop, B. Bleaney, GJ. Bowden, N. Nambudripad, NJ. Stone and H. Suzuki, Proc. R. Soc. London A 372 (1980) 19. [6] S.A. Al’tshuler and V.N. Yastrebov, Sov. Phys. JETP 20 (1965) 254. [7] S.A. M’tshuler, JETP Lett 3 (1966) 112. [8] S.A. Al’tshuler and MA. Teplov, JE’FP Lett. 5 (1967) 167. [9] S.A. Al’tshuler and B.M. Kozyrev, Electron Paramagnetic Resonance of Compounds of Transition Elements (Izdatel’stvo Nauka, Moscow, 1972) [Ketter Publishing House, Jerusalem, 1974]. [10] P.W. Anderson and P.R. Weiss, Rev. Mod. Phys. 25 (1953) 269. [11] P.W. Anderson, J. Phys. Soc. Japan 9 (1954) 316. [12] P.W. Anderson, Magnetism 1(1963) 25. [13] K. Andres and E. Bucher, Phys. Rev. Lett. 21(1968)1221; J. AppI. Phys. 42 (1971)1522; J. Low Temp. Phys. 9 (1972) 267, [14] K. Andres and E. Bucher, Phys. Rev. Lett. 22 (1969) 600. [15] K. Andres, Cryogenics 18 (1978)437. [16] HJ. Anklam and V.A. Kapustin, Soy. Phys. Solid State 19 (1978)1788. [17] D.M.S. Bagguley and J.H.E. Griffiths, Nature 160 (1947) 532. [18]J.M. Baker and A.E. Mau, Can. J. Phys. 45 (1967)403. [19] J.M. Baker, Rep. Prog. Phys. 34 (1971)109. [20] C. Balseiro, M. Passeggi and B. Alascio, Solid State Comm. 16 (1975) 737. [21] CA. Bates and PH. Wood, Contemp. Phys. 16 (1975) 547. [22]CA. Bates, Phys. Reports 35 (1978) 188. [23] B. Batlogg, E. Kaldis, A. Schlegel and P. Wachter, Phys. Rev. B 14 (1976) 5503. [24]J.E. Battison, A. Kasten, MJ.M. Leask and J.B. Lowry, Phys. Lett. 55A (1975)173; J. Phys. C 10 (1977) 323. [25]J.E. Battison, A. Kasten, MJ.M. Leask, J.B. Lowry and KJ. Maxwell, J. Phys. C9 (1976). [26] P.J. Becker, M.J.M. Leask and R.N. Tyke, J. Phys. C5 (1972) 2027. [27] J. Becquerel, Z. Phys. 58 (1929) 205. [28] H. Bethe, Ann. Phys. 3 (1929) 133. [29] G.L. Bir, Soy. Phys. JETP 24 (1967) 372. [30]RJ. Birgeneau, E. Bucher, L.W. Rupp Jr. and W.M. Walsh Jr., Phys. Rev. B5 (1972) 3412. [31] RJ. Birgeneau, ALP Conf. Proc. 10 (1973) 1664. [32] B. Bleaney and DJ.E. Ingram, Proc. Phys. Soc. London A63 (1950)408. [33] B. Bleaney and H.E.D. Scovil, Proc. Phys. Soc. London A63 (1950)1369; AM (1951) 204; Phil. Mag. 43 (1952) 999. [34] B. Bleaney, R.J. Elliott and H.E.D. Scovil, Proc. Phys. Soc. London AM (1951) 933. [35]B. Bleaney and K.W.H. Stevens, Rep. Prog. Phys. 16 (1953) 108. [36]B. Bleaney, Proc. R. Soc. London A276 (1963)19. [37]B. Bleaney, Physica 69 (1973) 317. [38] B. Bleaney, F.N.H. Robinson, S.H. Smith and M.R. Wells, J. Phys. CiO (1977) L385. [39] B. Bleaney, R.T. Harley, J.F. Ryan, M.R. Wells and M.C.K. Wiltshire, J. Phys. Cii (1978) 3059. [40] B. Bleaney, F.N.H. Robinson and M.R. Wells, Proc. R. Soc. London A362 (1978) 179. [41] B. Bleaney and R.M. Wells, Proc. R. Soc. London A370 (1980) 131. [42]B. Bleaney, Proc. R. Soc. London A370 (1980) 313. [43] B. Bleaney, KY. Loftus and H.M. Rosenberg, Proc. R. Soc. London A372 (1980) 9. [44] B. Bleaney, Bull. Mag. Res. 2 (1981) 7. [45] B. Bleaney, A.G. Stephen, S.H. Choh and M.R. Wells, Proc. R. Soc. London A376 (1981) 253. [46] C. Bloch, NucI. Phys. 6 (1958) 329. [47]N. Bloembergen, E.M. Purcell and R.V. Pound, Phys. Rev. 71(1947) 466; 73 (1948) 679.

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