Kinetics of cation site exchange in mixed garnets

W22-3697/81/050337-14$02.00/O Pergamon Press Ltd.

1. Phys. Chem. Solids Vol. 42. pp. 337-350. 1981 Printed in Great Brilain.

KINETICS OF CATION SITE EXCHANGE IN MIXED GARNETS P. R~SCHMANN Philips GmbH Forschungslaboratorium, Hamburg, D-2000Hamburg 54, Germany (Received 11 July 1980;accepted in revised form 14 November 1980)

Abstract-The redistribution rate of cations between the a- and d-sites in quasibinary mixed yttrium-iron gallium and-iron aluminum garnet single crystals has been investigated for temperatures between 773 and 1523K. The frozen-in cation distributions of quenched and of slow cooled samples were derived from magnetization measurements and are described by a phenomenological kinetic equation. The activation energies derived for the thermally activated process range from 1.3 to 4 eV and are related to the anion vacancy concentration of the crystals. The zero defect intrinsic a-d barrier height is 4.5eV as inferred from the observed linear relation between the frequency factor and the activation energy of the Arrhenius equation. This empirical relation, known as the compensation law, has been further applied to interpret reported intra-sublattice site exchange effects of cations on the c-sites and cation self-diffusion data of YIG. The inferred c-c barrier height of 5.5 eV agrees within the error limits of 20.1 eV with the enthalpy of formation, 5.4eV, and the enthalpy of solution, 5.3 eV, derived with equilibrium models based on the ionic fluid (Temkin) model and using experimental results of the kinetics, the cation distribution and the solubility of garnets. Apparently the c-c bonds form the strongest binding scaffold of the garnet structure, which is manifested by a relation to the melting point. While premelting around the anion vacancies assisted by their thermal fluctuations reduces the effective barrier height for cation migration, the apparently uneffected melting point of the compound itself is reflected by the compensation law.

1. INTRODUCTION Among the numerous metal cations that can be incor-

porated into the octahedral (a) and the tetrahedral (d) sites of the garnet system only Fe3’, Ga’+ and A? are known[l] to form the simple end-member garnets: R3AS0,2. In this formula R represents the cation species on the dodecahedral (c) sites, e.g. yttrium or a rare earth element, and A stands for the cation species on the aand d-sites. The apparent affinity of the Fe, Ga or Al cations to both the a- and d-sites in the garnet system enables a complete substitution of any cation species A by any cation species B among these three cation species. Thus, the somewhat more complex ionic crystals Rs(AI_ZBZ)50,z can be formed. In these mixed crystals of quasibinary character the cations A and B are not statistically distributed over the a- and d-sites but show a more or less pronounced preference for one of the available sites. Fe exhibits an octahedral site preference in the FeGa- and in the FeAl-mixed garnet system, the site stabilizing energy being lower for the latter system. The tetrahedral site preference observed for Ga decreases in going from the FeGa- to the AlGa-mixed garnet system [2,3]. With all cations trivalent, the considered quasibinary mixed garnets are ideal model substances for the investigation of the thermodynamic equilibrium distribution of cations among nonequivalent lattice sites. Furthermore, they provide an interesting possibility to study the kinetics of the cation redistribution process[2,4] and the involved ionic migration steps in a solid. At room temperature the observed actual cation distribution in quasibinary mixed garnets represents a frozen-in high temperature equilibrium distribution PCS Vol. 42, No. 5-A

which depends on the thermal history of the sample and on the kinetics of the cation redistribution process. Measurements of the cation distribution taken at room temperature on samples quenched from high temperatures provide an experimental access to determine the temperature dependence of the equilibrium distribution, and of the redistribution rate as well, if the annealing times are varied. A direct method to determine the cation distribution is the calculation of the site occupancy from the intensities of X-ray or neutron diffraction patterns or from Mossbauer spectra. Indirect methods are based on the measurement of a physical property related to the site occupancy of the cations, e.g. the magnetic moment or the temperature of the compensation of the sublattice magnetizations. The class of substituted yttrium iron garnets, Y,(Fe,-&),O,, with B representing the diamagnetic substituents Ga or Al, are particularly well suited for the derivation of the cation distribution by an indirect method. These crystals remain in the ferrimagnetic ground state up to considerable non-magnetic dilution, Z< 0.35. Since the FMR linewidth in single crystals remains narrow, the saturation magnetization can be determined very accurately from measurements of the resonance frequencies of certain magnetostatic modes. The cation distribution can readily be derived from the magnetization data by using the molecular field theory. In previous investigations this simple and fast experimental technique was applied to determine the temperature dependence of the equilibrium cation distribution[2] and of the redistribution rate [3,4] for the FeGa- and the FeAl-yttrium garnet system. This paper presents a detailed study of the cation redistribution rates in Ga and Al substituted YIG, and of 337

338

P. R~SCHMANN

the rate enhancement due to annealing induced changes of the oxygen vacancy concentration. A phenomenological rate equation is applied to analyze the time dependence of the frozen-in cation distributions of quenched and slow cooled samples. The results of this investigation on the inter-sublattice site exchange effects of cations will be compared with data reported for the intra-sublattice site-exchange of cations on dodecahedral sites, for cation self-diffusion and for the solubility of YIG.

YPLS

+X&M =;

- (2 - XL%B

- Y&A

N3- Y)(hea - h/u) + x(hm - hm) - (2 - x)(hm - hAA) - y(hm - heeN,

The redistribution process towards an equilibrium distribution of the ion species A and B between the nonequivalent a- and d-sites of the garnet structure is describable by a phenomenological rate equation based on a chemical rate theory for the ionic site exchange[M]. Before deriving the specific kinetics equation, the pertaining equilibrium conditions of the cation distribution are discussed. The distribution of the octahedrally and tetrahedrally coordinated cations in a quasibinary mixed garnet system, Rs(AI-ZBZ)5012, is given by:

(1)

(5)

and from (5) one obtains: E

=hBA-hAA_h,.,B-L., AB 2 3

(6a)

E BA =hAB-hBB_hm-hgg 2 3

2. KINETICEQUATIONSFOR SITE EXCHANGE IN GARNETS

{R~}[Az~,Bxl(A,-,B,)O,*,

(3 -

.

(6b)

It follows from eqn (3) that the effective site stabilizing energy describing the cation distribution for a given AB garnet system depends on the level of substitution and on the actual distribution: E%(z, x) = EAB- (2 -

~X)@AB

+

-&A).

(7)

The stabilizing energies for Ga and Al substituted YIG with 0.3 < z < 1.5 have recently been determined along with eqn (3) from experimental cation distribution data for equilibrium temperatures 770 K < LF,< 1600K[3]: EFeCa= 0.272 eV; EFcAl= 0.178 eV E GaFe= - 0.230 eV; EAIFe= - 0.132 eV.

where { } represents a dodecahedral or c-site, [ ] an octahedral or a-site and ( ) a tetrahedral or d-site. The total level of the substitution of cation species A by species B is:

Hence in both garnet systems the Fe3’ cations prefer the octahedral sites and the Ga3+ or A13’ cations prefer the tetrahedral sites. If a mixed AB-garnet sample with a frozen-in dis5z=z=xty. (2) tribution corresponding to T, is heated to a temperature T,# T,, a redistribution of the cations takes place. The It is assumed that the cation species R, which may be transformation of the equilibrium states is describable by yttrium or a rare earth element, occupies exclusively the the “chemical” equation: c-site. Impurities inherently present in actual crystals are neglected. 03) (A,-,) + [&I z M-xl + (4) For a mixed garnet crystal the a- and d-site equilibrium concentration x, and ye depend on the equilibrium temperature T, [3]: Ye@ - XJ

___ = exp [(3 - ye)EAB+X&M - (2 - xe).& x,(3-Ye) - Y,&4l/k~e

(3)

where EAB and EBA are the site stabilizing energies for z+O of the garnet systems R3AS-zBr012 and R$m,A,0,2, respectively, and k is the Boltzmann constant. The equilibrium eqn (3) results from a thermodynamical description of an ionic fluid model recently reported by van Erk[9]. The site stabilizing energies are related to the enthalpies of formation of the four endmember garnets: h]R+L4&1=

LA

h[R3BZA301Z]= hea

(hypothetical)

h[R,A,B,O,,] = hAB (hypothetical) h[R&BdU

- hss,

where k,* and kdo are specific rate constants in the two directions a + d and d + a, the brackets denote the concentrations and sites of the cation species A and B. The rate of change, e.g. of [BX]is given by:

(4)

- $ = k,dx(3 - y) - kday(2- x).

(9)

For t + CQ,equilibrium at r. requires dxldf = 0, and eqn (9) reduces to the equilibrium equation corresponding to eqn (3): ~

- exp E%(z, x,)/kT,,

(10)

where eqn (7) has been used and K, denotes the equilibrium constant. The redistribution process considered in eqns (8) or (9) involves jumps of AB cation pairs into opposite lattice sites. If the barrier between the sites is of that height my EL’,? “” or EC!,?‘. _I resuectivelv, . _ than the probability .

Kinetics of cation site exchange in mixed garnets

stating that the entropies of activation in the two directions are equal which is supported by experimental equilibrium cation distribution data[3]. From conventional rate theory the various parameters, specific to site and cation species, may be identified as follows:

a cation will pass over the barrier is given in terms of an activated jump frequency: P tiB = v$~ exp - EtiB/kT

(11)

where vt.” is a characteristic vibrational frequency of the cation species A,B at the a-sites. The corresponding expression holds for the reverse direction: P ;f

= t&E exp - EiiD/kT.

ko,,=

(114

E:,l/kTe.

vad e

-E,,&T

(W

kdo = vdo exp - [E,, t E%(z, x)]/kT.

E., = f (Et, + E:d)

(14)

and the site stabilizing energy is: Ef,&,

X) =; [E:d + E:c, - E& - E%].

(12)

Table 1. Tentative assignment of the center frequencies S. and Cdof absorption band groups in the far IR region for different garnets

f

7

[al

rIv

(d)

Y3

Ca3 YCa2 Cd3 C=3 Y3 Ca3 Fe32+ Cd3 Y3 Ca3 DY3 Ca3

Fe2

Fe3

Fe2

Fe

Fe2

Ge3

Fe2

Fe

Fe2 Al

Fe

Ge2

Si2

370

0.645

360

0.645

370

0.645

360

0.645

Si3

-400

0.645

Ge3

180/370

0.535/0.64

b)

after from[181

-’

610 6 00/700 700 6 00/870

5

References

R 0.49 0.49/0.3

12.378

11,

12.355

16

0.39

12.32

14

0.49/0.2(

12.19

16

870

0.26

12.01

15

720

0.39

_-

14

Al2

A13

480

0.535

750

0.39

12.002

11,

Al2

Ge

480

0.535

750

0.39

12.12

14

Al2

Si:

480

0.535

920

0.26

11.52

10

Si3

480

0.535

890

0.26

11.824

15

Ga3

410

0.62

640

0.47

12.275

11

A12 Ga2

Si3

420

0.62

880

0.26

11.932

15

Cr2

Ga3

450

0.615

640

0.47

__

13

Cr2

Ge3

450

0.615

720

0.39

__

Ga2

-

a)

a’

‘d cm

YCa2

Shannon

ref.

‘17’

(15)

An estimate of the values of the vibrational terms V& and V& may be obtained as follows. Due to the exponential increase of the Born repulsion forces with decreasing atomic separation characteristic atomic vibrations are expected to be largest for the smallest lattice site, i.e. the d-site. Absorption spectra in the far IR, lOO-lOOOcm_‘,show for various garnet systems two characteristic groups of absorption bands. The center frequencies of these groups,. V, and ;a, depend on the cation species, the respective ionic radius and lattice constant. Table 1 presents a tentative assignment of the groups of absorption bands taken from reported IR

Formally, the equilibrium eqn (12) corresponds to eqns (3) or (10) if it is assumed that the vibrational term on the right side of eqn (12) is equal to one. This amounts to

Lc3

(1%)

Hence, the potential barrier for cation redistribution from the nonpreferred site to the preferred site is:

At equilibrium the time average of the number of jumps from a + d is equal to that from d + a. Since only site reversals of A&pairs are of interest, the influence of the environmental site occupation must be taken into account. In the garnet system the u-site is surrounded by six d-sites and the d-site by four u-sites, disregarding the oxygen (h-sites) and the c-sites. Hence, a [B] cation will have on the average 6( 1- y/3) neighbouring d-sites which are occupied by (A) cations. An (A) cation has 4. (62) neighbouring a-sites being occupied by [B] cations. Multiplication of the jump frequencies (11) and (1la) with the appropriate concentrations and the numbers of available sites gives at equilibrium:

-

339

14 L

16

12

340

P. R~CHMANN

spectra for different mixed and end-member garnet systems[lCl6]. Also given are the ionic radii for the respective oxygen coordination [ 171 and the lattice constants[lQ Neglecting any entropy effects arising from a change of the vibrational spectrum concurrent with the redistribution process leads to v,~ = vda. Further, from the above kinetic_ arguments following eqns (lo)(13) it is assumed that:

7

‘\

z,= b

‘\ ‘\

zz x(K,-1)

‘\ ‘\ ‘. 6

‘\

‘\

‘\

v,d = (v: . ,$)"* (16) v& = (v; USing

,t)"*.

for vf etc. the values of i,,d from Table 1 yields: Ga subst. YIG: v,d =

Vda =

1.5

1o13

S-’

Al subst. YIG: v,d = Vda= 1.6. 1oL3SC'.

(174

(17b)

The basic kinetic equation for the cation redistribution in quasibinary mixed garnets is obtained from the rate eqn (9) and using eqns (2), (7) and (10): dx - - k&{x*+ (3 - z)x - [x2- (2 + z)x t 2z]K,’ exp dt-

-[2(x - xe)(& t EmY%7’J. (18) Equation (18) cannot be solved in a closed form. However, a reasonable approximation can be made for the cases z < 1.5 which are of interest here. Defining for the redistribution process an average equilibrium constant lC at x(t) = (x0 t x,)/2 gives: I?, = K, exp Kxo - x,WAB

+

&,JkT,l

(19)

where x0 = x(t = 0). For Ga and Al substituted YIG with (.EABt EBA)= 0.042 and 0.046 eV, respectively, and for a typical experimental condition: Ix0-x,1 = 0.03 and T, = 12OOK, one obtains !?e = (1 tO.Ol)K,. Thus, the exponent in eqn (18) may be assumed to be one for the analysis of the experimental data x(t) in Section 4(b) and the differential eqn (18) can be solved for this case: x - x,

The application of eqn (22) for analysing the measured frequency separation of magnetostatic modes is straightforward. The changes of the magnetostatic mode frequency separation, Sj, of the samples quenched after different holding times t, is due to a change of the saturation magnetization, A4rM,, resulting from the cation redistribution: A4rMs = c, . Sf

x tz

(23)

(20)

--U=exp-t/7 x t 2, x, -x,

where cf depends on the specific magnetostatic mode pair used [ 191:

where: 1 - = ZK.kod

(214

7

ZK= qz,

Fig. 1. The concentrational parameter z, and the kinetic parameter zK vs the substitution level in quasi-binary mixed garnets R3A5-zBz0,2for different values of the equilibrium constant Kc

t x,) e

2; ” = x,(K, - 1)’

Wb)

(2lc)

Figure 1 presents the concentrational and kinetic parameters, z, and zK, as a function of the substitution level z for different values of K,. Since, up to z = 2 the concentration parameter is z, 9 x,, eqn (20) may be further simplified: x - x, = (x, - x,) em”‘.

(22)

1lO-210-mode cf = 2.679 G/MHz 220-210-mode cf = 1.786G/MHz.

(23a) (23b)

The change of the u-site fraction, Ax, of the diagmagnetic substituent is nearly linearly related to A~T~M, as was found from the molecular field theory[20,21] for Ga or Al substitution levels z 5 1.5: A47rMJz, I”) = m(z, T) . Ax

(24)

where the proportionality factor m(z, T) depends for T = 295 K on z by the empirical formula: m(z, 295 K) = (2650 - 340 z3) G.

(24a)

Kinetics of cation site exchange in mixed garnets 3. EXPERIMENTAL

The investigated substituted YIG single crystals with Ga or Al substitution levels, 0.5
341

samples at T, 2 750°C in order to avoid decomposition of the crystals. The samples treated with this procedure will be referred to as slow cooled samples. (ii) Samples in the as grown state and slow cooled samples with specified pre-anneal conditions were fed directly into the preheated furnace at a temperature T,. After a holding time, t,, the spheres were quickly withdrawn and cooled to room temperature in air or water. Samples from this procedure will be referred to as quenched samples. The majority of the slow cooling and quenching experiments were performed in a temperature and atmosphere controlled horizontal tube furnace. The garnet spheres were placed into an open container made from 0.02 mm platinum foil. The ceramic supports needed to move the sample container into and out of the hot zone of the furnace were made of A&O, parts having a thickness down to 0.3 mm in order to keep the heat capacity small. The initial quench rate upon quick withdrawal of the sample container from the hot zone in the furnace was about SO”C/sec. This rate is sufficient to freeze in the actual cation distribution from r, I 1500K[3]. However, the heating part of the quench procedure in the horizontal tube furnace requires a correction of t, as can be seen from Fig. 2, which presents the normalized change of x as a function of t, for different anneal temperatures observed on as-grown Ga YIG samples with z = 0.93. For T, = 1173 and 1273K the experimental data follow closely the sigmoidal curve calculated from eqn (22). A systematically increasing deviation of the experimental points from the predicted curves is observed in Fig. 2 for T, > 1273K and towards short annealing times. Since after the loading of the platinum container into the hot furnace some time elapses before the samples reach the furnace temperature, a first order correction has been tried in order to find the effective annealing time, (to &), at T,. Using t, = 15 set provides a good fit with the sigmoidal curves as indicated by the correction lines in Fig. 2. This correction applies for repeatedly quenched

1

632

l

1323K

. .

1373K lL23K

A 0 0 V t,-%SeC ”

Fig. 2. Normalized changes of experimentally derived Ga concentrations at the u-sites versus the anneal time at different quench temperatures. The dashed lines represent a 15set correction of 1, accounting for the heating time needed by the samples to reach Tq The quench rate is about SO”C/sec.were calculated for an exponential process.

342

P.

R~SCHMANN

samples as well, an example is also shown in Fig. 2. A considerable reduction of the heating part was accomplished by using a vertical tube furnace. The spheres were filled from the top side of the furnace falling into a platinum container preheated to T,. After the holding time the platinum container was released to fall into a water bath placed beneath the bottom of the furnace. In this way initial quench rates of about j00”C/sec were achieved and the required correction of t, is only t, = 2sec. Figure 3 presents the change Sfin,_llo(t) of the magnetostatic mode separation measured after quenching from the vertical tube furnace at T, = 1523K. A correction on the time scale with nt, provides for the experimental data of all samples a good fit to an exponential curve, where n is the number of quench experiments performed on the sample and t, = 2 sec. The derived time constants will be further discussed in Section 4. 6C

5C

LO

-T;

-z

z

Fig. 4. 4mWS vs the anneal temperature T, for Ga:YIG samples annealed for 10hr in different atmospheres and slow cooled (open symbols), and 49rMSof equilibrated and quenched samples versus the equilibrium temperature T, (filled symbols). The solid curves represent theoretical equilibrium values. The ---- refer to standard anneal pretreatments.

z 30

I

20

10

0

I

5

10 -

15

20

25

A

"

I

@I

toIs

Fig. 3. Measuredchangesof the magnetostaticmode separation between 210- and llO-modes versus the anneal time at 1523K for different Ga: YE spheres. ----represent a 2 set correction of r, accounting for the heating time after loading the samples into a platinum container preheated to 1523K. The numbers n of experiments made on the same sample are indicated. represent fitted exponentials.

The quench experiments in the vertical tube furnace were made for all samples with O2 slowly streaming through the furnace. In the horizontal tube furnace the quench experiments were usually performed in the same atmosphere as used for the pretreatments, as-grown samples were quenched from an O2 ambient and the samples with HZ/N, pretreatments from a Nz ambient. 4. RESULTS

Slow cooled samples Figure 4 presents for two different Ga substituted YIG crystals the 4&& data of slow cooled samples and of equilibrated and subsequently quenched samples vs the anneal temperature T, and the equilibrium temperature T,, respectively. The solid curves in Fig. 4 represent 4~&& at equilibrium calculated from eqn (3) and the molecular field theory. A comparison of the 4nMs(T,) (a)

T, IKI

data of the slow cooled samples with ~QTM~(T,)of the equilibrated samples shows a considerable change in the frozen-in equilibria induced after annealing in different ambient atmospheres. The observed changes in the frozen-in equilibria are directly related to changes in the kinetic properties in the crystals. While annealing in O2 up to 1400K yields an increase of 47&f, as a result of a slowed down cation redistribution rate, in HZ/N1 the 47rMs decreases due to a strong enhancement of the redistribution rate. Complementary to these effects are characteristic changes in the FMR linewidth, AH, of the slow cooled samples which are shown in Fig. 5 as a function of the anneal temperature. Some of this data was already presented and discussed in Ref.[2]. The initial decrease of AH with rising T, results from release of the strain caused by the milling and polishing procedures of the sphere manufacture. The strain release occurs between 900 and 1200K for anneals in Nz or O2 while only about 700K is needed for annealing in the reducing HZ/N2 ambient as indicated by the results for YIG (z = 0) in Fig. 5. The decrease of AH(T,) for annealing in O2 at 1200K < T, < 1400K and the increase of AH( r,) for annealing in O2 at 7’, > 1400K or in N, at T. > 1200K or in Hz/N2 at T, > 700 K is caused by changes in the oxygen vacancy concentration, 8, induced in the samples by annealing in oxydizing, inert or reducing ambient. The charge compensation of the induced oxygen (anion) vacancies is due to the change of valence from Fe3+-+Fe2’, which gives rise to additional FMR

343

Kinetics of cation site exchange in mixed garnets

-

I.

4

3

2 z %I

of the samples used for the investigation of the redistribution rate. The experimentally observed equilibrium data 47rM,(T,) and the annealing induced effects 47rM,(To) of Al substituted YE are shown in Fig. 6. The results are similar to those for Ga substituted YE (z = 0.93) in Fig. 4; however, the corresponding quasi-equilibrium temperatures are higher. This property, as well as the smaller changes of AH( T,), and the higher temperatures required for decomposition effects on the surface altogether, indicate that the annealing induced oxygen vacancy concentration in Al substituted YIG, z > 0.3, is lower[26] than in YIG or in Ga substituted YIG.

1

1 1

0

I

1

1000

500 -

1500 TalKI +

Fig. 5. FMR line width at 9.5 GHz of YE and Ga: YE spheres

grown

slow cooled

vs the anneal temperature. The anneal time for the different atmospheres is 10 and I 4 hr for anneals in H2/N2at ‘f. z 650°C.

losses. Metselaar and Huyberts[25] have shown that the oxygen vacancy concentration S depends on the temperature and on the partial oxygen pressure. Annealing induced changes of S are controlled by the diffusion properties of oxygen in garnets and consequently by the sample size, too. These effects are reflected by the experimentally observed annealing induced changes of 4?rM,( TJ and AH( T.) in Figs. 4 and 5. The properties of the as-grown samples can be included to the present considerations since their cooling rate is identical to that of the slow cooled samples. From Fig. 4 is seen that the frozen-in equilibria of the as-grown samples correspond to different quasi-equilibrium temperatures of T:=970K (z =0.84) and of T:= 1120K (z = 0.93) which were found from a comparison with the quenched-in equilibrium results. The effects of annealing in N2 are also different for the two crystals. The inferred difference of the kinetic properties can be explained by the corresponding FMR linewidth results in Fig. 5. The “as-grown” linewidth is larger for z = 0.84 than for z = 0.93; however, after annealing in O2 the minimum values of AH show the expected order: AH a z. The oxygen vacancy concentration is different for these two asgrown crystals. This assumption is supported by the corresponding effects of annealing in N2 on 47rM$ and AH, see Figs. 4 and 5. Furthermore, it was found that reannealing in one of the atmospheres generally yields reversible effects on 47rM, and AH as long as decomposition at the crystal surface is avoided. Nearly homogeneous distributions of the oxygen vacancies are assumed to be present in the as-grown samples and after the anneals in O2 or N2 at 1100°C for 10 hr and in HZ/N2 at 650°C for 4 hr. These anneal conditions have been chosen as standard pretreatments

(I*

o 50"Clhour quenched . 5aYClS 0 SO~C/S

1SCJZ

Fig. 6. The annealing behaviour of 4&f, in AI:YIG for equilibrated and quenched spheres (filled symbols) and of slow cooled spheres with IOhr annealing in different atmospheres. The solid curve represents theoretical equilibrium values, the dashed curves refer to standard pretreatments.

In Figs. 4 and 6 the experimental 47rM,(T,) of equilibrated and quenched samples decrease for r, > 1600K; the actual turning point depends on the quench rate. Similar effects have been reported for polycrystalline Al substituted samples[27]. This anomaly of the high temperature behaviour will be discussed in Section 4(b). (b) Quenched samples The experimental time constants of the cation redistribution process of as-grown and standard annealed samples with different Ga or Al substitution levels are shown in Figs. 7 and 8, respectively. Figure 7 was already published in Ref.[4] except for the data for z(Ga) = 0.84. Within the range of substitution levels that are accessible by the experimental technique used, a dependence T(Z) could not be resolved from the experimental data. Changes of S during the kinetic experiments are assumed to be negligible. The experimental results T(T.J obey the Arrhenius equation: r = r, exp Eb/kT,.

(25)

The activation energy, Ez, and the pre-exponential

344

P.

R~SCHMANN

m P

o

OL 1373K.lOh

0

as grown

0

N, 1373K. 10h

A

H,IN, 923K. Ih

-$[Kj'

Fig. 7. The time constant for the u-d-site redistribution of cations vs the reciprocal equilibrium temperature of Ga:YIG spheres with different pretreatments. The dashed line represents the inferred intrinsic behaviour for zero anion vacancy concentration.

X00

12M) 1000

803

600

results from an inert atmosphere during the crystal growth which might occur if the pressure release pipe of the crucible[22] is blocked. However, the presence of F’- anion impurity levels or uncompensated divalent cation impurity levels may also lead to differences of the oxygen vacancy concentration[25]. Reported annealing effects on 47rM$ of various rare earth iron garnet materials with Ga substitutions[28-301 show upon different reducing treatments essentially the same defect induced enhancement of the redistribution rate as the Hz/N2 preannealed samples in Fig. 7. These experiments were performed on epitaxial films of about 6pm thickness for bubble memory applications and on platelets of bulk flux grown crystals. The oxygen vacancies invoked for explaining the rate enhancement were created by Si[29] or A1[30]films deposited on the surface of the garnet films or by anneals in forming gas [28,29]. The data, .E: and T-, derived from the experimental results in Figs. 7 and 8 by eqn (25) are transformed to the phenomenological rate constant kod using eqn (21a) and defining analogously to eqn (25) a more general form of eqn (13a): kad = v, exp - E,IkT,. (26) It is noted, that I?, in eqn (26) differs slightly from EZ in eqn (25) because the kinetic parameter ZK depends implicitly on T,, see Fig. 1 and eqns (2la,b,c). The transformed data, Y, and E,, are presented in the semilog plot shown in Fig. 9. The vibrational pre-exponential term and the activation energy are related by:

['Cl

log v, = +

.

no

log V.&

(27)

N,1373K,lOh

rH,/N,1023K,Lh

06

08 -

10 e

12

1L

[K]-'

Fig. 8. The time constant for the a-d-site redistribution of cations vs the reciprocal equilibrium temperature of AI:YIG spheres with different pretreatments. The dashed line represents the inferred intrinsic behaviour for zero anion vacancy concentration.

vibrational term, r,, depend strongly on the thermal history of the samples, as expected from the experimental findings on slow cooled samples. The redistribution rate of the as-grown samples is seen to lie midway between the results for samples with O2 and N2 anneals, which possibly reflects the air ambient during the crystal growth. A different behaviour is found from the data in Fig. 7 for the as-grown Ga:YIG sample with z = 0.84, for which the time constants are nearly identical with the results of N2 preannealed samples. See also Figs. 4 and 5 and the pertaining discussion. This property possibly

V

OO -

I

I

I

1

2

3

,

L

5

E,[eVI

Fig. 9. The logarithm of the preexponential term Y, vs the activation energy for cation redistribution in Ga:YIG and Al: YIG as calculated with T(TJ data from the kinetic equations for garnets. The inferred linear relation between log V, and E. complies with the empirical compensation law for activated processes. ---- indicate the derivation of the intrinsic activation energies for zero defect concentration.

345

Kinetics of cation site exchange in mixed garnets

where v_, and E,, properties which chiometric garnet present. Using for 2 yields:

may be regarded as intrinsic material obviously correspond to a stoicrystal without oxygen vacancies v,,d the estimated values from Section

&,@a) = 4.5 ? 0.1 eV E,,(Al) = 4.6 f 0.1 eV.

-1603

(28)

The difference between these values is of the same order as the estimated error. However, there are some indications justifying the slightly higher value for E.,(Al). The intersection of the log r vs T-’ curves occurs for Al: YIG at a higher temperature close to 1750K, see Fig. 8, than for Ga:YIG near 17OOK,see Fig. 7. Since it is unlikely that vad(Ga) is considerably larger than v,d(Al), it follows from eqns (25)-(27) that &,(A!) must be about 0.1 eV higher. Moreover the difference between the COTresponding site stabilizing energies is about 0.1 eV which leads to the same value of the energy sum in eqn (13b):

e 360"Clhwr

-E,

E,,(B) t E;Qz, x) = 4.75 eV.

(29)

The empirical relation (27) between the frequency factor and the activation energy in the Arrhenius equation is known as the compensation law[31]. Its application to other rate processes in garnets will be discussed in Section 5. Numeric calculations of x(t) from eqn (18) have been made using the data for k& in order to examine the errors arising from the simplified rate eqn (22) for the derivation of T and for the transformation r+&,,d with eqn (21a). The worst case deviation of zKf eqn (21b), remains below f 5% for the investigated samples. Mostly, this value is much lower than the experimental error of T ranging between t 10 and 20%. The frozen-in equilibria resulting from different constant cooling rates have been calculated by a numerical integration of the basic kinetic eqn (18) and using eqn (26) and (27) for k& The time dependence of the temperature is given by: T, = T, - 0 ’ t,

(30)

where To is the temperature at t = 0 corresponding to the isothermal anneal temperature T, of the slow cooling experiments or to T, of the quench experiments. The factor 0 describes the constant cooling rate in units of “C/set. The frozen-in values of the substitution level on the octahedral site, xf, have been calculated for various cooling rates as a function of the activation energy for Ga (z = 0.93) substituted YIG. The results are shown in Fig. 10, the right hand scale represents the quasi-equilibrium temperatures Tf corresponding to the cooling rate determined frozen-in equilibria. Also shown are the experimental values of Tf derived from the slow cooling experiments in Fig. 4 for those standard annealed and as-grown samples which were used for the measurements of the relaxation time. The experimental Tf data for slow cooled samples agree quite well with the calculations based on a chemical rate model and on the

leV1

Fig. 10. Frozen-in values of Ga, on octahedral sites and the corresponding quasi-equilibrium temperature Tf vs the activation energy calculated from the basic kinetic eqn (18) for different equilibrium temperatures T, and cooling rates. In the horizontal region of the curves equilibrium is attained, i.e. xf = x, and T, = T,. Experimental values for slow cooled and quenched samples agree well with the calculations.

derived by a different experimental technique. The influence of the experimentally achievable finite quench rates on the frozen-in equilibria at high temperatures is also displayed by the results presented in Fig. 10. A cooling rate of SO”C/secis sufficient to quenchin the equilibrium value, i.e. xr = x, if E, > 1 eV and T, < 1000K. However, at T, = 1500K the corresponding x, is quenched in only if E, > 3 eV. The experimental 47rM,(T,) data in Fig. 4 of equilibrated samples quenched with SO”C/sec exhibit the predicted deviation for T, > 1500K. The same effects occur for a quench rate of 5tWlsec at T, > 1600K in agreement with the calculations shown in Fig. 10. The decrease of 47rM,(T,) observed in Figs. 4 and 6 at T, > 1650K cannot be explained by the model of a cooling rate determined equilibrium[3]. These high temperatures are close to the magnetite-hematite transition temperature of 1660K in air ambient and close to the eutectic temperature of YIG at 1730K[32]. The different curves log T vs T-l, in Figs. 7 and 8 intersect in this temperature region and the actual kinetic data remain possibly considerably below the intrinsic curve. Indeed, Schultz et al. [33] reported very rapid redistribution rates for laser annealing experiments in the millisecond range on epitaxial Ga substituted YIG films. Possibly the exsolution process of magnetite in a garnet matrix as described by Lava1[34] explains the observed anomalies at high temperatures in terms of a transitional high temperature phase. Further insight into the high temperature kinetic and equilibrium properties of mixed garnets might be obtained by laser annealing techniques kinetic parameters

P. R~SCHMANN 1500

/-

1300

lLC0

1200

1100

35s Gao7

11

a,r

N,

n

.4

. l

Sm,,Ga,,

D

.

E’J, ;Gao,

0 0

oc

host

0,

0

1000

%,

ref

YIGlfl

36 LO. L3. L?

YIGIO

Ll L5. LB

ErlGlfIlbl35.38.LL

07

EuiGlfl

L6

Cao9e%nGeog6 Tbo,,

YlGltl

L2

GdlGlbl

37

D

Gd, Al,,

YIGlb)

39

0

GdosTb,,Euo~Al,,

YIGIbl

39

coo5 l

D

-

1 06

SIJ,

1 07 -

9

I 08 p

3

1K-‘1

Fig. 1I. The relaxationtime for intra-sublattice site exchange of cations at the c-sites in garnets versus the reciprocal anneal temperature. The data were derived from reported annealing effects on the uniaxial growth induced anisotropy constant of epitaxial films cf) and bulk crystal (b) samples. ---- comply with the compensation law using a plausible value for the intrinsic activation energy and assuming 7,, = W3 sec.

due to the much faster heating rates achievable than with conventional annealing techniques.

erence among magnetically inequivalent dodecahedral sites is proposed. Annealing at high temperatures randomizes irreversibly the growth induced ordering of the two or more dodecahedral cation species. Figure 11 presents a semilog plot of the relaxation time vs the reciprocal anneal temperature. The data were taken from reported K,,(T,, t) data of isothermal or isochronal annealing experiments [35-48] of the various epitaxial films cf) and bulk crystals (b) samples with different compositions. The data for different temperatures of the same sample are shown only for the lowest and highest temperature, respectively, and have been connected by solid lines in Fig. 11. Assuming an Arrhenius activation process according to eqn (25), the slopes of the solid lines in Fig. 12 yield: 2.2 eV < Ez < 6.5 eV and 10e4s > 7, > 1o-2os. The values of T, of several orders in magnitude below the typical atomic vibration period of about lo-r3 s have been ascribed by Hagedorn[41] to changes in the kinetic properties caused by the annealing induced increase of the oxygen vacancy concentration towards higher temperatures. Thus, the same effect appears to govern the enhancement of the redistribution rate of the inter- and intra-sublattice site exchange of cations. The value of the intrinsic activation energy of the intra-sublattice redistribution process has been estimated to be E& = 5.5 eV from the data in Fig. 11 by assuming that the compensation law eqn (27) is applicable and defining: vczd +

vcr =

l/T,, = 10” s-l.

(31)

Figure 11 shows some curves calculated from eqns (2% lo6

-

5.IONICMIGRATlONINGARNETS

Besides the considered cation exchange between the a- and d-sites an inter-sublattice ionic site exchange is possible in garnets also between the c- and a-sites, but from ionic size considerations it is unlikely to occur between c- and d-sites. To the knowledge of the author studies of these inter-sublattice site exchange processes involving the dodecahedral cations have not been published yet. However, upon annealing an intra-sublattice ionic site exchange occurs in garnet compounds having two or more cation species at the c-sites. This annealing process causes the irreversible reduction of the growth induced uniaxial anisotropy energy, Ku, in mixed ferrimagnetic garnets. Both the inter- and intra-sublattice ionic site exchange processes may be regarded as basic steps for cation diffusion in garnets. In this section reported c-site intra-sublattice relaxation data and diffusion data for YIG will be analyzed by applying the compensation law.

.

Fe

YIG

0

Be

Be0

19 ev

0

Zn

zno

3 3 ev

Q Ca

coo

3 6 ev

0 Cr

020,

19ev

L 5 ev

J

(a) Intra-sublattice site exchange on c-sites In garnets the growth induced uniaxial anisotropy energy has been found to anneal out in a similar way in bulk and epitaxial film samples at temperatures between 900 and 13SO”C[35-481.Several models have been discussed to explain the physical origin of Ku [49,50]. In these models a short-range pair ordering or a site pref-

1

Fig. 12. The frequency factor for cation self-diffusion in YIG and several binary metal oxides vs the normalized activation energy using the intrinsic activation energies from this work or ED, = ED,,,. Predictions from the compensation law are shown by the -

347

Kinetics of cation site exchange in mixed garnets (27) and (31). It is seen that three data sets agree with the calculated curves, for which is implied, that

DOis often termed the frequency factor and is given by:

S = const. The behaviour of the other experimental data can readily be explained by an increase of S and the resulting decrease of E, towards higher anneal temperatures in agreement with Hagedorn’s model [41]. The spread among the data in Fig. 11 for samples of the same composition as well as for samples with different compositions reflects the large differences of the as-grown oxygen vacancy concentration resulting from different growth conditions. Metselaar et a/.[251 have shown that the oxygen vacancy concentration in YIG is strongly influenced by the concentration of acceptor- or donor-type impurity or dopant levels. A very low value of 6 was inferred for n-type conductivity YIG with a charge imbalance due to Si”+ doping, Gyorgy et al.[Sl] reported that Ga substituted YIG samples with p-type conductivity due to uncompensated Ca’+ additions showed a large value of S and a very rapid oxygen diffusion at temperatures as low as 250°C. Hibiya et al.[52] reported that either type of conductivity may occur in Ca2’-Ge4+ substituted YIG garnet films due to the influence of various growth parameters on the charge imbalance. Consequently, the function S(T,) may be quite different for the various samples of Fig. 11. Changes of S due to oxygen diffusion during the anneal experiments for the inter-subblattice site exchange eh’ects are negliglible because of the short relaxation times, see Figs. 7 and 8. The situation is different for the intra-sublattice site exchange effects in films or even in bulk samples, where the relaxation times are at least an order of magnitude longer than the annealing times needed for attaining the equilibrium of S(T,) by diffusion[49,53].

DO= aa& exp AS, = D*v,

(33)

where (Y is a geometry factor related to the crystal structure, a, is the lattice constant and AS, is the change of entropy of the diffusion event. The activation energy ED of the diffusion process may also be regarded as the potential barrier seen by the diffusing ions and following eqns (25) and (26) it may be assumed: E,cE,=E: ED, A E,,

(34)

From eqns (27) and (31)-(34) the “compensation law” for diffusion is: log 0, = 13EdEfi t log D*.

(35)

A plot of DOvs the activation energy normalized to the earlier derived intrinsic inter- and intra-sublattice energy barriers is shown in Fig. 12 for the cation diffusion data of YIG from Table 2. The self-diffusion data agree reasonably with the predictions from the compensation law eqn (35) as represented by the solid lines for different values of D*. Also shown in Fig. 12 are cation self-diffusion data for a number of simple metal oxides. These data were taken from diffusion data collections given in Refs.[58-61]. The activation energies were normalized to the respective highest value, i.e. assuming arbitrarily E,,,, = E&. In some of these oxides D, varies up to 10 decades and a remarkable agreement is observed with the compensation law. 6. DISCUSSION

(b) Self-diffusion Self-diffusion data published for yttrium[54,55], iron [54,56] and oxygen[53,57] in YIG are compiled in Table 2 in terms of the usual diffusion equation: D = DOexp - EdkT.

(32)

In Section 2 it was shown that the thermodynamic model based on the enthalpies of formation and the kinetic model based on the ad hoc introduced energy barriers between the sites both lead to the same equilibrium equation for the cation distribution, see eqns (3)(5) and (lo)-(15). From these equations a relation be-

Table 2. Self-diffusion data for YE Diffusing

ion cnl2

Y3+

(s) (P) (s)

Fe3+

02-

(s)

single

DCJ set

3.4.105 2.1 10

ED -1

References

eV

5.2

10.53

54

3.6410.18

54

4.6

55

ho.8

(s)

2.3.102 8.6

4.1 a3.38 3.75M.12

54

(P)

0.1 0.4

2.56k0.25 3.2 M.2

54

(PI

10-3 10

2.5 3.7

56

(P)

0.4

2.04

57

(P)

8.4.103

2.9

crystal

(p)

polycrystal

Lo.1

53

348

P. R~SCHMANN

tween E,, and the enthalpies of formation is found:

tion eF. is related to the mole number N of the oxides in the solution at saturation temperature: eFe =

where the sum of the concentrations of species A, B on the [a]- and (d&sites is: [A] t(A) t [B] t (B) = 5.

NWZOXI

barrier height E,, = 4.5 50.1 eV

h YIG = 5.4 t 0.12 eV

(39)

Davies et al.[62] measured the heat of solution of YIG samples in a flux system at 950°C and obtained a value of 94 * 12 Kcal/mole (4.1 + 0.5 eV). Other values derived by these authors with a different experimental procedure agreed within 20%. Recently, van Erk[9] reported an analysis of solubility data for YIG in various oxide flux systems. The molar enthalpy of solution obtained with the ionic-fluid (Temkin) model is AHi = 83 Kcallmole. Similar values have been reported by Tolksdorf and Welz[22]: however, the experimental data for a number of flux systems resulted in nonlinear plots of In L vs T;‘, with T, being the saturation temperature of the solution and L being the solubility product defined by: (40)

In L = - AHi/kTs t In L,

(41)

where C, denotes ionic concentrations of the melt and In L, is an entropy increment. The calculation of AHi from eqns (40) and (41) relies on the assumption that in the course of solubility experiments In L, is not influenced by a change in the fractional molar concentration cG of the garnet forming oxides in the melt. Reasons for differences in lnL, observed for different flux systems have been discussed by van Erk[9]. However, also with a given garnet flux system a change of In L, cannot be excluded at different values of cG and the resulting saturation temperatures. Restricting the discussion to solutions containing only oxides, one finds from the ionic fluid model[9]: l+rZ l+tcc)

1’ 8c;

(42)

where the frequently employed iron excess in the solu-

(43)

The molar concentration of garnet forming oxides in the melt is:

(44 and r2 is the molar ratio of the divalent to trivalent constituents of the solvent. The iron oxide excess in the solution is considered as a part of the solvent, therefore r2 is a constant for a given flux system independent of cG. Thus, an ideal behaviour with In Lo(cG) = const. is expected only for flux systems with rz = 0 and eFe= 0. According to eqns (41) and (42) differences of the AHi and In L, results derived from plots of In L, vs T;’ and also nonlinearities in these plots are expected due to the increase of eFe and the decrease of cG towards lower saturation temperatures of the solubility experiments. From van Erk’s plot of In L vs Ti’ for different YIG flux systems (Fig. l(b) in Ref.[9]) one can infer limiting curves for high and low saturation temperatures. These limiting curves corresponding to high and low values of cG and to low and high values of eFe, respectively, are represented by the parameters: AH, = 5.3 2 0.1 eV In Lo,, = 27.5 5 0.5 (high Ts) In L,, = 31 lr 1(low Ts).

(45)

For the congruent melting YGaG or YALG garnet compounds with cG = 1, r2 = 0 and eoa.*, = 0 eqns (41) and (42) predict at the melting point T,,,, i.e. T, = T,,,: AHi = (In L, t 5.29)kT,,,.

L = Ci[Y”]’ ’ Ci[Fe3+]’

L = 33(5f eFe)5 8*

.

(37)

For the Ga:YIG system it can be shown from eqns (6a) and (6b) that h,, and hABdiffer only by a few per cent if bee exceeds hAa by up to 10%. Thus, for Ga substitution levels I = I and [B] -0.05 typically, the enthalpy of formation of YIG from eqns (36) and (37) is approximated by

Using the intrinsic yields:

3N[Fe,O,] - SN[Y,O,]

(46)

However, since it is not known which kind of particles are present in the melt, the description of the melting process with the Temkin model remains questionable. Assuming for the incongruently melting YIG a fictive melting point at TL = 2000+50 K[32,61] and using AHi(YIG) = 5.3 s 0.1 eV yields from eqn (46): In L,(YIG) = 25.5 * 1.4.

(47)

This value is close to the In Lo,, result in eqn (45). The enthalpy of formation, hvlc = 5.4 eV, from eqn (38) based on kinetic data of solid garnets, and AHi = 5.3 eV, from eqn (45) derived from solubility data, agree well within the error limits of about +O.l eV with the intrinsic barrier height E& = 5.5 eV for the intra-sublattice site exchange between the c-sites. If the agreement between these different quantities is not fortuitous, then the bonds between the dodecahedral sites apparently form the strongest binding scaffold in the garnet structure. The application of the melting formula eqn (46) to other garnet compounds strictly requires to consider the use of specific values of In L,(R3A,0j2). These values,

349

Kinetics of cation site exchange in mixed garnets Table 3. Melting points and calculated enthalpies of homologous garnet series compared with the pertaining ionic radii and the lattice constants

a

0

a’

b)

rVIII

s

8

r-IV

rvl s

Tll

a

K

AHi(eq.46)

Reference!

eV

Y3

A15

012

12.002

1.019

0.535

0.39

2220

5.9

26,

Y3

Ga5

012

12.275

1.019

0.62

0.47

2090

5.55

63

Y3

Fe5

012

12.378

1.019

0.645

0.49

2oooc)

5.3

32,

Lu3

Gag

012

12.183

0.977

0.62

0.47

--

--

Er3

Gag

012

12.255

1.004

0.62

0.47

2040

5.4

Gd3

Gag

012

12.376

1.053

0.62

0.47

1970

5.2

64

Gag

O12

12.433

1.079

0.62

0.47

1890

5.0

64

Nd3

Gag

012

12.508

1.109

0.62

0.47

1790

4.75

64

Pr3

Gag

O12

12.57

1.126

0.62

0.47

--

--

Gd3

SC2A13

012

12.395

1.053

0.7

0.39

2100

5.55

65

ca3

Ga2Ge3

O12

12.251

1.12

0.62

0.39

1650

4.4

66

from

[181

c)

estimated

b) value,

YIG

after

Shannon

melts

incongruently

ref.

reflecting differences of the vibrational entropy increments, are influenced by the ionic radii and atomic masses of the cations. A comparison of the melting points of homologous garnet series[26,62-66] with the

pertaining ionic radii shows that the melting point decreases rapidly towards the largest cation known to form an end-member garnet, see Table 3. The enthalpies calculated with eqn (46) and using In L, = 25.5 indicate that the stability of the garnet structure seems to be related to the differences between the strength of the c-c and a-d site bonds. A quantitative relationship between the effective barrier height Ea for the inter-sublattice site exchange of cations in Ga:YIG and the anion vacancy concentration S has recently been reported [4]: E, = E,,(

1 - e-“-ad3ao),

(48)

where

is the mean distance between oxygen vacancies in terms of the lattice constant and S refers to the garnet formula of the crystallographic unit cell, R,,A,O%,. In the garnet structure an oxygen vacancy is surrounded by four cations, namely one at the a- and d-sites and two at the c-sites. The barrier heights for the considered inter- and intra-sublattice site exchange effects presumably are strongly reduced in the

surrounding of a vacancy. The statistical jumps of the vacancies due to thermal fluctuations at elevated temperatures thus provide a mechanism for the observed enhancement of the cation redistribution rates. It is inferred from this rate enhancement expressed by the

61

64

Sm3

a)

62

1171

reduced barrier height with increasing defect concentration and from the concurrent decrease of the frequency factor in the Arrhenius equation obeying the compensation law that local melting around the defect centres plays an important role in solid state ionic migration. The compensation law relation between the energy parameters of an activated rate process indicates that the free energy of exchange is altered in such a manner that the melting point of the compound itself remains uneffected. A further discussion of the compensation law with reference to thermally activated processes in simple oxides, spinels and garnets and some implications for the defect compensating ability of iron garnets will be presented in a forthcoming paper. Acknowledgemenfs-The author wishes to thank Drs. H. Ddtsch, W. van Erk, W. Schilz and Prof. W. Tolksdorf for stimulating discussions and helpful suggestions. The advice and assistance with the anneal and quench experiments by F. Welz and H. Fricke and the programming by W. Klossner is gratefully acknowledged.

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