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Mössbauer Studies on Exchange Interactions in CoFe2O4

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2001 Jpn. J. Appl. Phys. 40 4897

(http://iopscience.iop.org/1347-4065/40/8R/4897)

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Jpn. J. Appl. Phys. Vol. 40 (2001) pp. 4897–4902Part 1, No. 8, August 2001c©2001 The Japan Society of Applied Physics

Mossbauer Studies on Exchange Interactions in CoFe2O4

Sam Jin KIM, Seung Wha LEE1 and Chul Sung KIMDepartment of Physics, Kookmin University, Seoul 136-702, Korea1Department of Electronics Engineering, Chungju National University, Chungbuk 380-702, Korea

(Received November 15, 2000; revised manuscript received March 15, 2001; accepted for publication April 23, 2001)

Two polycrystalline samples of CoFe2O4 were prepared by slow cooling and quenching and studied using Mossbauer spec-troscopy and X-ray diffraction. The crystals were found to have a cubic spinel structure with the lattice constants of the slowlycooled sample beinga0 = 8.381A and the quenched sample beinga0 = 8.391A. The temperature dependence of the mag-netic hyperfine field in57Fe nuclei at the tetrahedral (A) and octahedral (B) sites was analyzed based on the Neel theory offerrimagnetism. For the slowly cooled sample, the intersublatticeA–B superexchange interaction and intrasublatticeA–A su-perexchange interaction were antiferromagnetic with a strength ofJA–B = −25.0kB and JA–A = −18.9kB, respectively, whilethe intrasublatticeB–B superexchange interaction was ferromagnetic with a strength ofJB–B = 3.9kB. In the quenched sample,however, their strengths wereJA–B = −22.6kB, JA–A = −17.6kB, andJB–B = 3.9kB, respectively.

KEYWORDS: superexchange interaction, Mossbauer spectroscopy, Neel theory, Debye temperature, cobalt ferrite, magnetichyperfine field

1. Introduction

Cobalt ferrite (CoFe2O4) is a wellknown hard magneticmaterial, which has been studied in detail due to its highcoercivity (5400 Oe) and moderate saturation magnetization(about 80 e.m.u.g−1) as well as its excellent chemical sta-bility.1) It has been widely studied and used as a mag-netic recording material and the importance of its applica-tion is appreciated in various fields.2) Its metallic atoms arein an inverse spinel distribution at room temperature3) asFeA[CoFe]BO4. In this study, the temperature dependence ofthe magnetic hyperfine fields in57Fe nuclei at the tetrahedral(A) and octahedral (B) sites is analyzed based on the Neeltheory of ferrimagnetism.4)

In an attempt to study superexchange interactions in ferritesusing Mossbauer spectroscopy, we prepared CoFe2O4 sam-ples; one was obtained by slow cooling, the other by quench-ing. Two sets of data, namely, magnetic hyperfine field andmagnetic moment, can be used to determine accurate superex-change strengths. However, the total spontaneous magneticmoment near the Neel temperature cannot be correctly deter-mined in a high external magnetic field due to the inducedmagnetic moment. Thus, it would be desirable to obtain twoseries of data unaffected by an external magnetic field.5) Themagnetic hyperfine fields can be measured without the help ofan external magnetic field using the Mossbauer spectroscopy.

Studies on the superexchange interaction of ferrites havebeen reported by a number of authors.6,7) However, in cal-culating the superexchange strength many authors neglectedthe intrasublattice interaction, namely theA–A or B–B in-teraction, as it was weaker than theA–B interaction. Ac-cording to our experimental results on cobalt ferrite, the in-trasublattice superexchange strengthsJA–A and JB–B shouldnot be neglected, as was done by previous authors. To ob-tain a better fitted superexchange parameter to coincide withexperimental results, we applied the Neel theory of ferrimag-netism including the intrasublattice superexchange interac-tion. We had previously briefly reported the superexchangeinteractions in cobalt ferrite based on molecular field theo-ries.8) However in the present article, we report detailed cal-culations and the effects of sample preparation. From an anal-

yses of the Mossbauer spectroscopy data, Debye temperaturesof two magnetic sites on CoFe2O4 are discussed. In this paper,we report our Mossbauer spectroscopy results on CoFe2O4

with particular emphasis on the intersublattice and intrasub-lattice superexchange interactions and Debye temperatures ofmagnetic sites.

2. Experimental Procedure

The two sets of CoFe2O4 samples were prepared using thefollowing ceramic method. A mixture of the proper pro-portions of Fe2O3 and CoO of 99.995% and 99.999% puri-ties, respectively, were ground, then pressed into a pellet at6000 N/cm2, and sealed in an evacuated quartz tube. Thesealed mixture was heated at 1000◦C for 2 days and thenslowly cooled to room temperature at a rate of 10◦C/h. Toobtain a homogenous material, it was necessary to grind thesamples after the first firing and to press the powder into pel-lets before annealing them for a second time in an evacuatedquartz tube. The quenched sample was also obtained accord-ing to the same procedure, but rapidly quenched at a cool-ing rate of 300◦C/s. X-ray diffraction patterns of the sam-ples were obtained using a SCINTAG diffractometer with Cu-Kα radiation in theθ–2θ geometry. The Mossbauer spectrawere recorded using a conventional Mossbauer spectrome-ter of the electromechanical type with a 10-mCi57Co sourcein a Rh matrix. A low temperature was realized using anAPD CS-202 displex closed-cycle refrigeration system with aDMX-20 Mossbauer vacuum shroud interface, while the tem-perature controller was a model DRC-91C manufactured byLake Shore Cryogenics Inc.

3. Results and Discussion

The X-ray diffraction measurements showed that bothslowly cooled and quenched CoFe2O4 samples had the cu-bic spinel of a single phase. A slow scanning speed (0.25◦advance in 2θ per min) was used to optimize the resolution ofclosely spaced reflections. We could not find any other sin-gle diffraction peak of Fe2O3, CoO, and Fe other than that ofCoFe2O4. For the slowly cooled and quenched samples, thelattice constants at room temperature werea0 = 8.381±0.003

4897

4898 Jpn. J. Appl. Phys. Vol. 40 (2001) Pt. 1, No. 8 S. J. KIM et al.

and 8.391±0.003A, respectively, which were determined us-ing the Nelson-Riley function9) and by extrapolating to thebackward diffraction (θ = 90◦).

Mossbauer spectra of CoFe2O4 were taken at various ab-sorber temperatures. Figures 1 and 2 illustrate the represen-tative spectra for the slowly cooled and quenched samples,respectively, which were composed of two six-line hyper-fine patternsA (inner sextet) andB (outer sextet). Using aleast-squares computer program, two sets of six Lorentzianlines were fitted to the Mossbauer spectra below the magneticordering temperature under well-known restraints,10) whichare valid when the quadrupole interaction is much weakerthan the magnetic hyperfine interaction. Table I presents themagnetic hyperfine fields and isomer shifts forA and B pat-terns in the slowly cooled CoFe2O4 sample at some typicaltemperatures, and Table II presents the results of the quenchedCoFe2O4 sample. In the slowly cooled sample, the mag-netic hyperfine fields ofA and B at 13 K are 516± 2 and553± 2 kOe, respectively. The isomer shifts at room tem-perature for theA and B patterns are 0.15 ± 0.01 mm/s and0.25 mm/s relative to the metal iron, respectively, which areconsistent with either a highly covalent high-spin Fe3+ or alow-spin Fe3+.11) However the magnitude of magnetic hyper-fine fields at 13 K mentioned above exclude the latter alterna-tive. Also recalling that the possible existence of Co3+ couldnot be excluded, the possibility of the low-spin Fe2+ was alsoexamined. Generally, the hyperfine fields of the Fe2+ ion areknown to have values less than 300 kOe. Furthermore, the

-12 -8 -4 0 4 8 12

6

4

2

0

A B

S O

R P

T I

O N

( %

)

VELO C ITY (m m / s)

6

4

2

0

750 K

295 K

450 K

13 K

130 K

6

4

2

0

6

4

2

0

4

2

0

Fig. 1. Mossbauer spectra of the slowly cooled CoFe2O4.

electric quadrupole splitting of the Fe2+ is larger than that ofFe3+ ions due to orbital contribution. As shown in Tables Iand II, the electric quadrupole splitting is nearly zero withinexperimental error over the entire temperature range. Hencethe possibility of the Fe2+ ion is also excluded. The smallervalue of the isomer shift at theA site was due to a larger co-valency at theA site. The Fe3+ character of the Fe ions wasalso manifested by the magnitudes of magnetic fields shownin Tables I and II; the hyperfine field values ofA and B pat-terns at room temperature in the slowly cooled CoFe2O4 are498± 2 and 527± 2 kOe, respectively, which are typical val-ues for Fe3+. These assignments are in accord with the resultsof De Grave’s group,12) which were determined in an externalmagnetic field on samples of Ga-doped Co ferrites. From theisomer shift value, electric quadrupole splitting, and magnetichyperfine fields, we conclude that the Fe ions in CoFe2O4 areFe3+. Since Fe3+ ions have neither orbital nor dipolar con-tribution13) to the magnetic hyperfine field, the magnetic hy-perfine field is proportional to the spin of the ferric ion. Thequadrupole shifts are zero within experimental error for theA and B sites below the Neel temperature. This shows thatquadrupole shifts for CoFe2O4 are consistent with those ofthe cubic crystal structure. The Neel temperature,TN, wasmeasured using the thermal scan method. The relative countsduring 10 s as a function of temperature at zero transducer ve-locity was measured, the temperature, at which relative countswere minimized, was determined as the Neel temperature.The determined Neel temperature,TN, was 870± 3 K for the

-12 -8 -4 0 4 8 12

86420

A B

S O

R P

T I

O N

( %

)

VELO C ITY (m m / s)

86420

750 K

295 K

450 K

13 K

130 K

6

4

2

0

4

2

04

2

0

Fig. 2. Mossbauer spectra of the quenched CoFe2O4.

Jpn. J. Appl. Phys. Vol. 40 (2001) Pt. 1, No. 8 S. J. KIM et al. 4899

slowly cooled sample.Figure 3 shows the reduced magnetic hyperfine fields

H(T )/H(0) for the A and B sites of the slowly cooledCoFe2O4 sample as a function of the reduced temperature

Table I. Magnetic hyperfine fieldsH , electric quadrupole shiftsEQ, andisomer shiftsδ for tetrahedral (A) and octahedral (B) sites in slowly cooledCoFe2O4. The value ofδ is relative to that of metallic iron.

T H(A) EQ(A) δ(A) H(B) EQ(B) δ(B)

(K) (kOe) (mm/s) (mm/s) (kOe) (mm/s) (mm/s)

13 516 −0.02 0.25 553 −0.02 0.38

130 514 −0.02 0.24 549 −0.02 0.24

295 498 −0.02 0.16 527 −0.03 0.27

450 467 −0.01 0.06 483 −0.02 0.19

600 393 0.0 −0.02 411 −0.01 0.00

750 276 −0.02 −0.11 303 −0.01 −0.09

±2 ±0.01 ±0.01 ±2 ±0.01 ±0.01

Table II. Magnetic hyperfine fieldsH , electric quadrupole shiftsEQ, andisomer shiftsδ for tetrahedral (A) and octahedral (B) sites in quenchedCoFe2O4. The value ofδ is relative to that of metallic iron.

T H(A) EQ(A) δ(A) H(B) EQ(B) δ(B)

(K) (kOe) (mm/s) (mm/s) (kOe) (mm/s) (mm/s)

13 518 0.01 0.27 555 0.01 0.39

130 513 0.01 0.25 541 0.0 0.36

295 493 0.02 0.15 513 0.02 0.29

450 448 0.01 0.06 466 0.02 0.07

550 402 0.0 −0.01 419 0.0 −0.02

750 301 0.01 −0.09 301 −0.02 −0.19

±2 ±0.01 ±0.01 ±2 ±0.01 ±0.01

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

HB(T

) /

HB(0

)H

A(T

) /

HA(0

)

(T/TN

)

A site

B site

Fig. 3. Reduced magnetic hyperfine fieldsH(T )/H(0) for tetrahedral (A)and octahedral (B) sites in slowly cooled CoFe2O4 as a function of reducedtemperatureT/TN. The solid line represents calculated reduced magneti-zation.

τ = T/TN (TN = 870 K). Also Fig. 4 shows the reducedmagnetic hyperfine fieldsH(T )/H(0) for the A and B sitesof the quenched CoFe2O4 sample as a function of the reducedtemperatureτ = T/TN (TN = 853 K). Since both magneticmoment and magnetic hyperfine field of the ferric ion are pro-portional to its spin, the reduced magnetic hyperfine fieldsshown in Figs. 3 and 4 should be equal to the reduced sublat-tice magnetizationσA andσB . To extract information on thesuperexchange interactions from Figs. 3 and 4, we applied theNeel theory of ferrites14,15) to the two sublattices of CoFe2O4.The molecular field acting on each Fe ion at theA (tetrahe-dral) site is

HA = γαMA − γ MB (1)

and that acting on each Fe or Co ion at theB (octahedral) siteis

HB = −γ MA + γβMB, (2)

where MA and MB are the magnetizations of theA and Bsublattices, respectively. The terms−γ , γα, andγβ are themolecular-field coefficients corresponding toA–B, A–A, andB–B superexchange interactions, respectively. The reducedmagnetizations forA andB sublattices are determined by

σA = BS(x) (3)

and

σB = BS′(y), (4)

respectively. HereBS(x) and BS′(y) denote the Brillouinfunction ofx andy, respectively, defined by

x = gµBS HA

kBT(5)

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

HB(T

) /

HB(0

)H

A(T

) /

HA(0

)

(T/TN

)

A site

B site

Fig. 4. Reduced magnetic hyperfine fieldsH(T )/H(0) for tetrahedral (A)and octahedral (B) sites in quenched CoFe2O4 as a function of reducedtemperatureT/TN. The solid line represents calculated reduced magneti-zation.

4900 Jpn. J. Appl. Phys. Vol. 40 (2001) Pt. 1, No. 8 S. J. KIM et al.

Table III. Lattice constants, Neel temperatures, calculated superexchange parameters, and Debye temperatures of the slowly cooled andquenched CoFe2O4 samples.

a0 TN JA–B JA–A JB–B �A �B

(A) (K) (kB) (kB) (kB) (K) (K)

Slowly Cooled CoFe2O4 8.381 870 −25.0 −18.9 3.9 734 248

Quenched CoFe2O4 8.391 853 −22.6 −17.6 3.9 701 217

±0.003 ±3 ±0.2 ±0.2 ±0.2 ±5 ±5

and

y = g′µBS′ HB

kBT, (6)

whereS is the spin value of anA site Fe3+ andS′ is the aver-age spin value of aB site Co2+ (S = 3/2) or Fe3+ (S = 5/2).Hereg andg′ are Lande or spectroscopic splitting factors ofFe3+ (S = 5/2) in the A and B sites, respectively. Sincethe Fe3+ ion does not have an orbital angular momentum,and assuming quenching of the orbital angular contributionfor Co2+ ions, a value of 2 forg andg′ was used in this cal-culation. The termkB is the Boltzmann constant,T is theabsolute temperature, andµB is the Bohr magneton. It is truethat eqs. (1)–(6) are somewhat crude, because these equationsinclude many pairs of exchange integrals between differentmagnetic ions; indeed it is extremely difficult to obtain ac-curate superexchange integral parameters. To simplify thisproblem, we assumed that each magnetic ion was located in amean-field situation and considered mean exchange parame-ters,JA–B , JA–A, andJB–B .16)

Now, the Neel temperature can be easily determined usingeqs. (1)–(6) to be

TN = 1

2γ [α + βC ′ +

√(αC − βC ′)2 + 4CC ′], (7)

whereC andC ′ are Curie constants for theA and B sublat-tices, respectively. Substituting eqs. (1)–(4) into eqs. (5) and(6), and solving forσA andσB , one can obtain the followingequations

σA = Qτ

(αβ − 1)g2S2(βx + y) (8)

and

σB = Qτ

2(αβ − 1)g′2S′2 (αy + x), (9)

where

Q = 1

6[αp + 2βp′ +

√(αp − 2βp′)2 + 8pp′]. (10)

Here p = g2S(S + 1) and p′ = g′2S′(S′ + 1).For a set of (α, β) values, eqs. (3), (4), (8) and (9) are

simultaneously solved to obtainσA and σB as functions ofthe reduced temperatureτ . Now, one can easily calculatesum of squares of the deviations between the calculated re-duced magnetization and the experimental reduced magne-tization. Varyingα andβ independently at 0.001 intervalsfrom −2.0 to 2.0, we obtained a minimized square sum forboth sites and determined the best parameter. For the slowlycooled sample, a good agreement between the experimentaldata and the theoretical values ofσA and σB was obtainedfor (α, β) = (−0.498, 0.078), as shown by the solid lines in

Fig. 3. The variance defined by

N∑i=1

[σZ (τi )EXP − σZ (τi )]2

N, [Z = A, B] (11)

was also calculated for each fit. Here,σZ (τ )EXP andσZ (τ ) arethe experimental and theoretical values of the reduced sublat-tice magnetizations, respectively. The variances for the bestfits were 1.04× 10−4 for σA and 4.11× 10−5 for σB .

The intersublattice exchange constant between Fe3+ (A)and Fe3+ or Co2+ (B) was calculated to be−25.0 ± 0.2kB

using the following equation:

JA–B = − g2TNkB

12Q. (12)

The exchange constant between the two nearest-neighborFe3+ (A) ions on theA-sublattice is

JA–A = −3α JA–B

2(13)

and that between two nearest-neighbor Fe3+ ions on theB-sublattice is

JB–B = −2β JA–B . (14)

JA–A and JB–B were calculated to be−18.9 ± 0.2kB and3.9 ± 0.2kB, respectively. Using the same procedure forthe quenched sample,σA and σB were also obtained for(α, β) = (−0.513, 0.086). For a set of(α, β) values, thesuperexchange parameters,JA–B , JA–A, and JB–B , were cal-culated to be−22.6±0.2kB, −17.6±0.2kB, and 3.9±0.2kB,respectively. These results ofA–B superexchange strengthwere similar to those of Co-doped magnetite of ref. 17, but theA–A superexchange strength was larger than that in ref. 17 byas much as 8kB. It is noteworthy that the intersublattice su-perexchange interaction was antiferromagnetic and strongerthan the intrasublattice superexchange interaction.

Table III presents the calculated superexchange strengthparameters, lattice constants and measured magnetic order-ing temperatures of the two samples. Comparing the lat-tice constants and magnetic ordering temperatures of the twosamples, for a slowly cooled sample, a small increase inA–B and A–A superexchange strength was interpreted qual-itatively well. According to the integral form of the superex-change, its strength is inversely proportional to the ionic bondlength. For the quenched sample, its lattice constant waslarger than that of the slowly cooled sample. Hence its ionicbond length was greater than that of slowly cooled sample,therefore the superexchange strengths were reduced and a de-crease in magnetic ordering temperature resulted.

Figure 5 shows the temperature dependence of the ratio ofthe absorption area of theA pattern to that ofB pattern for

Jpn. J. Appl. Phys. Vol. 40 (2001) Pt. 1, No. 8 S. J. KIM et al. 4901

both samples over the temperature range up to 400 K. It isnoteworthy in Fig. 5 that at low temperatures the ratio of areasincreases with increasing temperature up to 400 K. Figure 6shows the temperature dependence of the resonant absorptionarea of both samples. The Debye model gives the followingexpression for the recoil-free fraction:18)

f = exp

[− 3ER

2kB�

(1 + 4T 2

�2

∫ �/T

0

xdx

ex − 1

)], (15)

whereER is the recoil energy of57Fe for the 14.4 keV gammaray, and� represents Debye temperatures. Taking the loga-

0 100 200 300 400

0

2

4

T ( K )

I A /

I B

Q uenched

Slow ly cooled

Fig. 5. Temperature dependence of the ratio of the absorption area of theA pattern to that of theB pattern for slowly cooled (solid circles) andquenched (solid squares) CoFe2O4 samples.

0 4 8 12 16-4.0

-3.5

-3.0

-2.5

-2.0

T 2 ( 10,000 K 2 )

ln f

A site

B site

0 4 8 12-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

(a) s low ly cooled C oFe2O

4

(b) quenched C oFe2O

4

A site

B site

T 2 ( 10,000 K 2 )

ln

f

Fig. 6. Natural logarithm of the absorption area,f , vs T 2 for the A andB subspectra of (a) slowly cooled and (b) quenched CoFe2O4. The solidlines were calculated using eq. (16).

rithm of both sides of eq. (15), one obtains

ln f = − 3ER

2kB�

(1 + 4T 2

�2

∫ �/T

0

xdx

ex − 1

). (16)

When ln f is plotted as a function ofT 2, one obtainsa curve which becomes almost linear at low temperatures.Equation (16) with a proper additive constant was fitted to thedata in Fig. 6 using a least-squares computer program to de-termine a Debye temperature.19,20)The estimated Debye tem-peratures of the slowly cooled sample were�A = 734± 5 Kand�B = 248± 5 K. The site distribution12) can be written(Fe3+)A(Co2+Fe3+)BO2−, and recalling that Mossbauer reso-nant absorption areas are proportional to the recoil free frac-tion f , the ratio of absorption areas of theA pattern to that ofB pattern can be written as

IA

IB= f A

fB. (17)

Since �A was much higher than�B , fB decreases morerapidly than f A with increasing temperature according toeq. (15). Therefore, the ratio of the absorption area of theA pattern to that of theB pattern in eq. (17) are expected toincrease with increasing temperature.

4. Conclusions

The temperature dependence of the magnetic hyperfinefield in 57Fe nuclei at tetrahedral (A) and octahedral (B) sitesfor the CoFe2O4 was analyzed by the Neel theory of ferri-magnetism. For a slowly cooled CoFe2O4 sample, the inter-sublatticeA–B superexchange interaction and intrasublatticeA–A superexchange interaction were antiferromagnetic witha strength ofJA–B = −25.0kB and JA–A = −18.9kB, respec-tively, while the intrasublatticeB–B superexchange interac-tion was ferromagnetic with a strength ofJB–B = 3.9kB. Ina quenched sample their strengths wereJA–B = −22.6kB,JA–A = −17.6kB, and JB–B = 3.9kB, respectively. The in-creasing tendency of the ratio of areas in Mossbauer spectrabelow 400 K can be explained in terms of a large differencein Debye temperature between theA andB sites.

Acknowledgements

This study was supported by the Korea Science andEngineering Foundation (97-0702-0401-5) and Brain Korea21 Program.

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