FMR properties of epitaxial Fe3O4 films on MgO(100)

Thin Solid Films 307 (1997) 250-259

ELSEVIER

FMR properties of epitaxial Fe304 films on MgO(100) B. A k t a s

*

Department of Physics, Gebze Institate of Technology, 41410 Cayiroca-Gebze, Kocaeli, Turkey

Received 4 February 1997: accepted 13 May I997

Abstract

The magnetic properties of epitaxial F%O 4 films on MgO(100) substrate have been investigated. Spin wave resonance (SWR) technique has been used to study the magnetic anisotropy as a function of temperature between 4-300 K. The ferromagnetic resonance (FMR) spectra exhibit unusual unidirectional character upon cooling the sample in an external field H c. At higher temperatures, generally both the bulk and the surface modes appear as long as the external field is applied within a small angular interval with respect to the film normal. The bulk and the surface FMR modes are well separated from each other when the field, Hap, is parallel to the cooling field, H c, which is normal to the film plane. However these FMR modes surprisingly overlap and become only a single and broader absorption line when the applied field is antiparallel to the cooling field. Using a field-induced unidirectional exchange surface and bulk anisotropy, with the usual magneto-crystalline anisotropy energy, we have simulated the experimental spectra and deduced anisotropy and magnetic stiffness parameters as a function of temperature. © 1997 Elsevier Science S.A. Keywords: Anisotropy;Iron oxide; Magnetic properties and measurements: Surface and interface states

1. Introduction

The surface effects on magnetic properties of a material are becoming the subject of intense current researches, as the trend to higher magnetic recording densities creates a need for smaller magnetic particles having larger surfaceto-volume ratios. If the surface magnetic properties differ from those of the bulk, they can dominate the overall magnetic behavior of small particles or very thin films. In fact, Vassiliou et al. [1] showed that the magnetic anisotropy for the particles in diameter of about 30 A is two orders of magnitude larger than that of the bulk ferrite ( T - F e 3 Q ) crystals. The ferrites have been used as magnetic recording media for a long time [2,3]. Scientists spent great efforts to explain their magnetic properties [4-8]. There are two kinds of sites for Fe ++ and Fe +++ ions having different magnetic moments at room temperature. Site A has the tetrahedral oxygen configuration, while site B has the octahedral configuration. A remarkable feature of the spinels is that all exchange integrals J ~ , JBB, and JAB are negative and favour 'antiparallel' alignment of the spins connected by the interaction [6,7]. However, since

* Corresponding author. 0040-6090/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S0040-6090(97)00311-8

the interactions between nonidentical pairs ( A - B ) are much stronger than those between identical pairs ( A - A or B-B), the resulting alignment is a ferrimagnetic order, that is, the A spins are parallel to each other and the B spins are parallel to each other so that A spins may be antiparallel to B spins (even the ground state correspond to a spiral configuration [8] at lower temperatures). As the temperature decreases, the Fe30 4 shows a phase transition from cubic to orthorhombic crystalline structure, and its magnetization is reduced [9]. Since the exchange interactions (super exchange) are mediated by oxygen ions, a small distortion in the octahedra or tetrahedra may have considerable effects on the magnetic properties of ferrites. Recently, researchers have been applying various surface treatment techniques [10-14] to enhance and control the magnetic behavior of magnetic media. It is obvious that the surface anisotropy can be studied more easily on single crystal thin films than on fine particles. The information gathered from thin films might be applied to fine particles to get a reasonable model for small magnetic particles. Ferromagnetic resonance (FMR) has proven to be one of the most sensitive and useful techniques [15-24] to study the surface anisotropy, as well as other magnetic properties of magnetic thin films. With this motivation we used FMR technique to study the magnetic properties of

B. Aktas / Thin Solid Fihns 307 (1997) 250-259

epitaxial F % Q thin films on MgO(100). Our study revealed that the magnetic properties of epitaxial single crystal Fe304 are remarkably changed from those of the bulk samples. Some films also showed a significant surface anisotropy at high temperature while this anisotropy gains a directional character upon cooling the sample in an external field.

251

185L,,,1

.......

184

, ........

f'" \'/

, W

182

/

The sample preparation was given in an independent work [25] and it can be briefly sunmlarized here as follows: Fe304 thin films were deposited on MgO(100) substrates using reactive dc sputtering technique and characterized by X-ray diffraction, vibrating sample magnetometer, SQUID magnetometer, torque magnetometer and Mossbauer spectroscopy. X-ray diffraction studies of the 400 and 800 reflections indicated single crystal Fe304 films under a tensile stress. This single crystalline structure is consistent with torque measurements of the in-plane anisotropy of the films with thickness of 2000-3000 A. The cubic anisotropy constant (K~) obtained from torque measurement at high fields shows satisfactory agreement with the value for bulk I=e304 over a wide temperature range. However, these fihns exhibit several magnetic anomalies [25]. The magnetization shows a rapid increase with the field up to about 50 mT, and then it continues to increase linearly and very slowly even up to 7 T which is maximum experimental field value. For example, the value of magnetization for some samples is only 85% of the values for bulk Fe304 at 2 T. Also, Mossbauer spectroscopy shows that a very small fraction of the moments are nearly randomly distributed in three dimensions and not in-plane as dictated by film shape anisotropy. Lastly, magnetization measured in low applied fields gives a very small value for cubic anisotropy constant, K~, compared to that for bulk samples. The standard electron spin resonance (ESR) techniques have been used for FMR measurements. A Varian associates reflection type spectrometer operating ~ 9.15 GHz (Z; 0

H H

sl

I

/

(y, b)

(X, a) Fig. 1. Relative orientations of the applied magnetic field vector, H, equilibrium magnetization vector, M, and experimental references axes system that coincide with the crystalline abe cubic axes system.

150 r,, 0

~

/

/

" ....

'

'

'"t

?'

//. Ii

I

181

2. Experimental procedures

,/

"

i

/'.

.

.

,j .... ~ . . . . . . . . . I .... i .... f .... 100 200 300

.2-

r,:,,t 400

o(. (degrees)

F]g. 2. The resonancefield values as a function of angle of applied magnetic field, H, in the sample plane. The filled circles are the experimental values while the continuous line represents the theoretical values obtained from the general resonance equation (Eq. (3)) by using the fitted anisotropyparameters.

employs magnetic field modulation with phase-sensitive detection so that the detected signal is proportional to the field-derivative of the absorbed power. Static component of the magnetic field is oriented along the horizontal plane, while the microwave component is vertical. A sample holder has been used to rotate the thin film sample around the vertical axis so that the relative orientation of dc field can be varied in-plane a n d / o r out-of-plane measurements. The sample geometry and relative orientation of the applied field, H and the magnetization vector, M, with respect to the reference axes system (x, 3', z) are illustrated in Fig. 1. A typical FMR sample is 2 X_3 mgn___and_ aligned via cleavage face of MgO(100) substrates.

3. Experimental results When the applied dc magnetic field direction isclose enough to that of the film normal, the FMR spectra consist of two well-resolved peaks at room temperature. However, as the field is rotated away from the film normal, these two lines come close to each other and overlap, giving a relatively broader single absorption line. Fig. 2 shows in-plane angular variation of the resonance fields for F e 3 0 4 thin film of 3012 A in thickness. The angle of the applied magnetic field is measured from one of the (100) direction. As seen from this figure, Fe304 film on MgO(100) exhibits a fourfold symmetry with a period of 90 ° in the film plane, suggesting a cubic crystalline structure for the film. However, the maximum amplitude of the resonance field variation with the angle is about 3.5 mT. This value is at least one order of magnitude smaller than that one can theoretically predict from the cubic anisotropy parameter 2 K J M (46 mT [26] for the bulk sample of the same compound) as will be explained in Section 4 of this paper. On the other hand, from dc magnetization, we know that the FMR resonance field value is small to reach exact

252

B. Aktas / Thin Solid Fibns 307 (1997) 250-259 !

I

P~

I

;

;

ZFC

:~# 1 1

"E =

;-

i .~r-ZFC

I

J/

=

t-t

1

/

Backgraound ]

I

i- #

"'"

I

; iI. t

n-ZFC

-1

i

0.0

0.1

±

i

0.2 H (T)

i

I

0.3

i

0.4

Fig. 3. FMR spectra of FeaO4 thin film. The open triangles represent the spectrum that was recorded by applying the measurement field parallel to the remnant field of the electromagnet, after cooling the sample down to 4.2 K (n-FC case). The dashed line corresponds to the spectrum that was plotted after the sample was rotated by 180° relative to remnant field direction (r-FC case). Finally, the sample was rotated to its original direction, applying a field of 0.4 T, the sample is magnetized along remanent field direction, the field was decreased down to zero value, the sample was rotated by 180° again and then the spectrum (continuous line) was recorded the second time for r-ZFC case.

cooling field direction that is opposite to the applied field for r-FC case at low temperature. For r - Z F C case, the magnetization is directed in opposite direction of small magnetic field due to strong unidirectional anisotropy field induced along the cooling field (remanent field of the electromagnet) direction. Therefore, the resonance (continues lines) starts again nearly at zero field and traces the negative field part of the resonance line for the n-ZFC case. As the applied field is increased further, the magnetization vector rotates towards the applied field direction and a second resonance absorption takes place at about - 0.2 T in according to the dispersion relation (Eq. (2)). In fact, this resonance absorption corresponds to the main resonance peak for r-ZFC case. One can obtain the rigid component of the unidirectional anisotropy field H A ( = A H / 2 ) from the separation, ( A H ) , between the main peaks for r-ZFC and n-ZFC cases. Low field losses in soft ferromagnetic samples can usually give rise to a large increase in rf absorption due to domain resonance in multidomain phase of the sample. However, even at low applied field, there is a strong unidirectional-induced anisotropy field as evidenced from a large amount of shift of the resonance field for r-FC ease relative to that for n-FC case. Therefore, the resonance

I

saturation o f the magnetization (see Fig. 3 in Ref. [25]'). Also the torque measurement gives very small value for K l at lower field, whereas one obtains a value close to that of the bulk sample at 2 T (see Fig. 7 in Ref. [25]). This indicates that, although it is structurally single crystalline [25] at lower field, the film is magnetically disordered due to some strong and randomly distributed microscopic anisotropy fields. The temperature behavior of the F M R spectra are shown in Figs. 3 - 5 . The sample was cooled down to 4.2 K in the remanent field (5 roT) of the electromagnet, and then the spectra were recorded by sweeping the external field either parallel ( n - Z F C case) or antiparallel ( r - Z F C case) to the remanent field in the film plane. Since the power supply is not bipolar, the sample has been rotated by 180 ° by a goniometer to sweep the external field in reverse direction relative to the cooling field, H c, for r-ZFC case. Therefore, the spectra for both r-ZFC and n-ZFC cases appear at the positive side of the field in the figures. As shown in Fig. 3, there is a large hysteresis effect on the magnetic state at low temperatures. The resonance field for n-ZFC case starts even at negative applied field region, which is much lower than what we would expect for such a system having relatively small demagnetizing field due to the geometry ( 4 w M m 0.5 T). This indicates a strong anisotropy field induced along the cooling field direction, and this induced anisotropy field is rigidly coupled to the lattice so strongly that it tries to make magnetization parallel to the

"~" 2¢-

~ !

4

I

I

~1 r'FC

['~ n~fC ~

-o

I

E

0

E

''

I ['!"

0.0

,,

i~ / 0.1

I

0.2 H (T)

0.3

0.4

Fig. 4. FMR spectra of Fe304 fihn at 42 K for FC case. The sample was cooled down to 4,2 K in a cooling field of 0.4 T in the sample plane and applying the measurement field parallel to the cooling field the first trace (open triangles) was plotted (n-FC case). Then the applied field was decreased down to zero value and the sample was rotated by 180° relative to the cooling field, and the spectrum was recorded again (r-FC ease) by increasing the applied field up to 0.4 T (dashed curve) and decreasing the field down to zero value (heavy triangles). Successive increase and decrease of the field in opposite direction of the cooling field do not change the spectra much for r-FC cases as indicated by double arrows. Finally, the ~amplewas rotated to its original direction and the continuous line was plotted by increasing the field along the cooting field direction the second time as indicated by double arrows.

B. Aktas

Thin Solid Films 307 (1997) 2 5 0 - 2 5 9

2.5

Sc zo

px

-Fc;

t ,Y t

c'-'

-dJ 1.0r-,

E

0.5-~ T=70K 0.00.2

0.3

0.4 H (T)

I0.5

0.6

Fig. 5. FMR spectra of Fe304 film for perpendicular field cooling case at some selected temperatures as indicated above each spectrum. The filled triangle represents the spectrum recorded by applying the external field paralIel to the cooling field direction (n-FC) while the open triangles correspond to r-FC case in which the sample was rotated by 180 ° relative

to the cooling field in zero field after n-FC spectra. The continuous lines are the corresponding theoretical curves that were calculated by using both surface and bulk directional anisotropy in addition to usual crystalline anisotropy as described in the text. It should be noted that the spectrum is substantially changed about 70 K, suggesting a phase transition.

absorption appeared at lower external field takes place at strong effective (internal) field. In other words, these peaks do not belong to the domain resonance. Low field losses for usual ferromagnetic samples should be the same for either direction of the external field (parallel or antiparalleI to the cooling field). So this unidirectional character is a clear sign for the field-induced exchange unidirectional anisotropy. There is another field-induced anisotropy for bulk ferrite. But this anisotropy is originated from a transition from cubic to orthorhombic structure. The easy axis (c-axis) becomes parallel to the external field during the cooling. This anisotropy field is bidirectional, that is, positive or negative direction of the c-axis is the same. However, in our case the induced anisotropy include both axial (bidirectional) and unidirectional components, and the unidirectional component is induced along the cooling field direction. The resonance field values and the line shape are affected by successive sweep of the field as seen in Fig. 3. Fig. 4 illustrates the spectra that were recorded after cooling the sample in the presence of an external field of about 0.4 T in the plane of the sanaple (FC case). When the sweep field is applied parallel to the cooling field (n-FC case, open triangles) the resonance starts again below the zero field. After this first sweep, we rotated the sample by 180 ° relative to the cooling field and repeated to plot the

253

FMR line (r-FC case, dashed line). Now the remanent magnetization is directed opposite to the applied field at the beginning of this sweep (at lower fields). As the field increases in negative direction relative to remanent magnetization, the remaining parts of the resonance line for n-FC case at negative field side appear at zero field region, but the phase of the FMR signal for the same peak is changed because the phase of the modulation field of the spectrometer with respect to the internal field (and magnetization) is reversed. As the applied field increases further in the r-FC case, magnetization first starts to change sign and then a second resonance peak takes place at relatively higher field that corresponds to the main FMR mode for r-FC case. After this trace, we continued to record the line by decreasing the field from - 0 . 4 T. The magnetization and the internal field are now parallel to each other and relatively sharper and more intense: only one FMR peak is observed in this case (heavy triangles). The successive increase and decrease of the applied field in negative direction did not change the line shape much as indicated by double arrows. The amount of the separation of the resonance line for n-FC and r-FC cases is two times of rind component of the exchange anisotropy field (unidirectional anisotropy), and it is very close to that obtained fromlZFC case~i However, the spectra for both n-FC and r-FC cases (cooling field is 0.4 T) shift as a whole towards negative fieId side much more than those for ZFC case. This extra shift is m a n l y originated from a phase transition from cubic to orthorhombic crystal structure in which the easy axis, c, becomes parallel to the external cooling field as will be explained below. Lastly, the sample was rotated to its original position, and the spectrum for n-FC case was recorded the second time as indicated by double arrows (continuous line). As seen, there appeared a weaker and dramatically distorted line in this case: that is, there is a very big hysteresis effect on the magnetic state of the film at low temperature for parallel geometry (when the external magnetic fi_e!d lies .!n the sample plane). The most substantial effects on the spectra were observed when the cooling field (0.4 T) is perpendicular to the film plane (perpendicular geometry). Fig. 5 shows the spectra taken by sweeping the applied field-oriented paraliel a n d / o r antiparallel to the cooling field (which is perpendicular to the film plane) at some selected temperatures. At 4 K, the relatively narrower and well-resolved four resonance peaks (heavy triangles) were observed for n-FC case. However, these peaks surprisingly overlap and become a much broader single peak (open triangles) for ~r-FC case. As well "known, the multipeak resonance line generally arises from the pinning of the spins at the surface of the ferromagnetic films and their relative intensities of the higher order bulk modes progressively decrease with mode number except that for surface mode. The simulated spinwave resonance spectra obtained fi'om the model in the theoretical section are in good agreement with the experimental spectra for perpendicular geometry. Also, the angu-

254

B. Aktas / Thin Solid Films 307 (1997) 250-259

0.6

S

0.5

-£ I

correlation between the line width and the resonance field values. As the applied field is rotated away from the film normal, the line gets more and more broad and takes maximum value when the field lies in the plane of the sample. This implies relatively strong random anisotropy field in the plane of the sample.

J

0.3

J

,

x

) ~ m 3

)~"

0.2 L

/

~"

0.1 ;-

4. T h e o r e t i c a l

model

l] / / '

' , **~._~_~ ,¢~,'

A

0.0"-"*"'' ~''' 0 40

AH

/"

",

~,

~''' ~''' ~ ...... 80 120 160 200

a n g l e (degrees) Fig. 6. R e s o n a n c e field values for various F M R modes and the peak-top e a k line width as a function o f angle of the external field relaUve to

cooling field direction (film normal) in the bc-plane.Continuouslines are fitted values for correspondingFMR modes that they were calculated by using Eq. (12) with the values for magneticparametersgiven in Table 1.

lar behavior of the experimental spectra show a satisfactory agreement with the spin wave theory. Therefore, this asymmetric behavior of the FMR spectra with respect to the cooling field direction is an obvious sign for an unidirectional surface anisotropy induced upon cooling the sample in an external field. The distance between the main resonance peaks (most intense ones) for n-FC and r-FC cases is about 0.2 T and corresponds to an exchange unidirectional bulk anisotropy field H A of about 0.1 T, which is slightly higher than that for parallel field cooling case. This might imply that there is a correlation between the exchange interaction and the stress anisotropy by means of modulated exchange due to the change of the crystalline parameters by stress at the substrate surface. As the temperature increases, the single resonance peak for r-FC cases splits progressively and takes the same structure as that for n-FC cases in every respects. Accordingly, the separations of the resonance lines for r-FC from corresponding modes for n-FC case progressively decrease and vanishes at about 70 K. Moreover, the resonance peaks get more and more broad with increasing temperature at this temperature regime. Fig. 6 illustrates the angular dependencies of both the experimental (symbols) and the theoretical resonance values of various modes for FC cases for perpendicular geometry at 4 K. The angle was measured from the cooling field direction (perpendicular to the film), and the external field was rotated toward one of the cubic axis in the sample plane. The experimental resonance values were taken as an intersection of the resonance line and the base line. Also, the peak-to-peak line width was plotted in the same figure. It should be noted that there is a close

The theoretical resonance fields in Fig. 2 are calculated by using the following expression: "~ ~

~

~

2

2

"~

2

E = - M • H + K i ( a i ' a 2 + o~'a~" + or2 c~3 ) 4- Ke-ffo~3

(1) for the magnetic energy density, E, for a cubic single crystal film in an external field well-above Verwey transition temperature. Here, the first term represents Zeeman energy density, the second term accounts for cubic magneto-crystalline energy density with anisotropy parameter, K~, and, ai's are the direction cosines of M, and in the last term Keff(=Ku-2~M'-) represents the effective uniaxial anisotropy parameter including both the shape anisotropy ( 2 ~ M z) and any perpendicular anisotropy K~ arising from magneto-elastic coupling due to lattice mismatch [25]. For a general direction of the magnetic field, H, the basic dispersion relation for FMR is given [27,28] by =

×

Mssin20 0q~2

1

M s 00 ~

OzE )2

M~sin 0 0q~00

(2)

where y is the gyro-magnetic ratio, ~o is the microwave frequency, 0 and q~ are the usual spherical polar angles for M, D is the exchange stiffness parameter and k n is the spin wave vector for nth mode. If the external dc field is applied in the film plane (0 n = 90 °) and makes an angle q~n with the x-axis (one of the cubic axis) then, using the energy expression Eq. (1) in Eq. (2), one can get the following resonance condition (when MH > > K 1, K U) (y)2=(-~-Lc°s(4~n)+H)×( 2K,

+ --if-

K1 . , ?

2K1M + 4 q r M )

(3)

for uniform mode (k = 0). Using this expression, we obtained best fit (continuous line in Fig. 2) to the experimental values (filled circles) with the parameters 2 K 1/M = 2.2 mT, 2K1/M + 47rM + 2 K J M = 380 mT, (~o/y) = 315 mT and g = 2 . 1 4 . The cubic anisotropy parameter, 2 K1/M, is more than one order of magnitude smaller than a value of 46 mT Oe [26] for the bulk sample of the same

B. AI, tas / Thin Solid Fihns 307 (1997) 2 5 0 - 2 5 9

compound. As mentioned above, the resonance field value in FMR experiment is too small to reach exact saturation of the magnetization. That is, at lower field the film is magnetically disordered (unsaturated), although it is structurally single crystalline. Therefore, each individual spin sees different anisotropy field according to its own orientation in such a tow syrmnetric crystal. To explain the undersaturation effects on the measured anisotropy, we assumed a Gaussian angular distribution for the spin's orientation relative to the applied magnetic field. The mean value for the angles of randomly distributed spins can be taken to be zero. Therefore, the number of spins directed within a solid angle g2 and S2 + d/2 around H is nd~Q = C exp

sin0 dO d~

(4)

where C is a normalization constant giving the total number of spins N in the unit volume, /3 is the angle of the spins with respect to the applied field H, /3o is a field-dependent parameter representing angular distribution width. In Eq. (4), 0 and q~ are the spherical polar angles of the spins for which they are measured relative to Cartesian crystallographic axes with z (or c) directed along the film normal. Applying the external field at the angle 9~ in the sample plane, the relationship among the /3, the spherical polar angles, 0, p and 9H becomes cos/3 = sin0 cos( q~n - 9)

(5)

If the moment of each spin is /x, then the average magneti-zation along the applied field H is simply written as MH = fs~/.LCexp -- ~

cos/3 sin0 dO d~

(6)

The z-component of the torque, ~ , acting upon the spins oriented at the solid angle f2 is

r~ =

(7)

where,

is the crystalline anisotropy energy of each individual spin having a direction cosines a~, a 2, a3 with respect to the cubic crystallographic axes a, b and c, respectively. The z-component of the average torque acting on an individual spin by the crystalline anisotropy is obtained by using the energy Eq. (8) in the torque Eq. (7) as IKIC

(/3)2

~= 2 N faexp --~0 sina0sin(4~)d0d~

(9)

This is the expression that has been used to calculate the torque corresponding to dc measurements for parallel geometry. The values for /3 is inserted from Eq. (5). For perfect alignment of the spins along the applied field, M

255

has its saturation values, M s, and the torque expression reduces to 1

= -~ Kisin(4~ )

(10)

which would be obtained from well-known cubic anisotropy energy

E=KI(afa 2 + afol 2 + a;a;

/

Using this cubic energy in the dispersion relation Eq. (2), one can get magneto-crystalline anisotropy field, H~ = 2 KI/M = 46 mT just as for the bulk samples. - - - ~ e low temperature FMR spectra were analyzed using the following magnetic energy density for single crystalline ferrite film which is cubic at room temperature '~

t

4

E= - M . H + M. (HAD +HAg ) + ( Kaa a + K a a a ~ 4 r 4 +K ba~"~ + K~o G + K ca 2"~ + K c% ) (1t) The first term is the Zeeman energy, the second term represents the unidirectional exchange anisotropy energy including both rigid (HAR along the cooling field) and dynamic component (HAD, that is, an isotropic anisotropy field [29-31] that is always directed along the direction of the applied field). The third term in Eq. (11) represents the magneto-crystalline anisotropy for a rhombic Fe304 crystal at low temperatures with the easy axis c along the direction of the cooling field that was taken along one of the -cubic axes at higher temperature for perpendicular FC case. Axes a and b are rotated by 45 ° relative to the remaining two cubic axes at the same film plane [32]. However, the rotation takes place either direction in equal probability on the various regions of the film plane (i.e., positive or negative 45 ° rotation). Taking average over a and b axes, the magneto-crystalline anisotropy term can be approximated to an uniaxial form having c direction as symmetry axis. Therefore, Eq. (11) can be replaced by

E = - M . H } - M . ( H A D + H A R ) q - K 3 a 3q-K;IR'¢ (12) Here, the t e r m K3o~~ includes both the ftrst-order effective uniaxial magnetic anisotropy and the shape (demagnetization) anisotropy energy. For perpendicular field cooling case, both the uniaxial and unidirectional axes are induced along the film normal. Thus, using Eq. (12) in general resonance condition (Eq. (2)), one can obtain the following expression for the resonance field for nth FMR mode,

( ~ ) 2 = [ (H+HaD)sinOHsinO +Dk:] x

1( H

+ H ~ ) c o s ( 0n - 0) + HaRcos 0

2K3 cos(20) - 3K; sin~-(20) M

4K;

M c°s40+Dk~

M

] (13)

256

B. Aktas / 77~in Solid Fihns 307 (1997) 250-259

50

for a general direction of the measurement field after cooling the sample in the field along film normal. Here, 0 should be determined by the static equilibrium condition

M

4K;

MNs / .873 [

I--

MH s i n ( P - 0n) +MHARsinO- K3 sin(20) - --cos30 M

v

+__:82

E D..

sin0 = 0

e'-'

[ -25

(16)

can be modelted for temperature dependence of both of the surface and the bulk unidirectional exchange anisotropy field. Here K(0) is the zero temperature value of either surface or bulk anisotropy and Tsg is spin glass transition temperature.

5. Theoretical results and discussions

For room temperature analysis, using various values for 130 in Eqs. (6) and (9), we calculated the magnetization and the torque on M, as a function of angle, ~p~ of the applied field H in the plane of the film. Fig. 7 exhibits the calculated results corresponding to the effective anisotropy field which would be obtained from both the FMR and the dc torque measurements. As can be seen from this figure, the effective anisotropy field comes close to its expected value (2 Kt/M--46 mT) as the magnetization rises to its saturation values, M s, which corresponds to a perfect alignment of the spins along H. However, the randomness of the spins affects the apparent anisotropy field dramatically. Only about 10% decrease in magnetization due to random distribution causes a nearly 80% decrease in the apparent anisotropy field. Therefore, when the applied field is not strong enough to saturate magnetization, both FMR and dc torque measurements techniques inevitably give smaller values to the effective anisotropy field. A skeptical reader may take this undersaturation effects as an evidence that the resonance condition (Eq. (3)) is incorrect: but in view of its successes on the interpretation of the experimental data below, we wish to retain it. In fact the magnetization is not very far from its saturation value here. Landau and Lifshitz dynamic equation of motion for average macroscopic magnetization gives also the same expression for resonance condition. A small undersatura-

y

F r

(15)

/'

-5oi

This includes both unidirectional (first term) and uniaxial surface anisotropy energy. Also the following expression [33]

K(T)=D(O) 1 -

IX "ff~

£

(14)

for magnetization. Considering the directional character of the FMR modes, we employed a surface anisotropy energy to determine the spin wave vector k as: E, = Kecos 0 + KsaCOS20

I

0

P

I

50 100 150 angle (degrees)

I 200

Fig. 7. The calculated effective anisotropy field by using the bulk anisotropy vatue for cubic anisotropy parameter, 2 K u / M (46 mT) with a Gaussian distribution for spin orientation for various values of the distribution w~dth (or average magnetization). It should be noted that the model give almost the bulk value for perfect alignment of the spins corresponding to the saturation magnetization M S.

tion effect manifests itself on the distortion on the FMR line shape. The low temperature theoretical FMR spectra are calculated using the anisotropy energy given by Eqs. (t2) and (15) in the calculation procedure in Ref. [21]. The results are plotted in Fig. 5 with the corresponding experimental ones for perpendicular geometry at some selected temperatures. As shown in this figure, simulated FMR spectra reproduce the experimental ones quite well. Some deviations, especially in line width, from experimental spectra come from some field training effects that we could not include in our model. Also, the angular variations of the calculated resonance field values for various spin wave modes are shown in Fig. 6 together with corresponding experimental results at 4.2 K. Relatively small deviation from the theory can be tolerated by considering large line broadening with the angle, which makes it difficult to determine the resonance field more accurately. The fitted parameters are given in Table 1. Table 1 Magnetic parameters deduced from the simulated spectra at different temperatures of Fe304 film Parameters

4K

30 K

65 K

M (10 -3 T) HAR (10 -4 T) HAD (10 . 4 T) K 3 (102 J / m 3) K~ (10 ~"J / m 3) K~a (10 . 4 J / m z) D (10- ]0 J / ( m T ) A H ~ (10 .3 T) A H ~ (t0 .3 T)

416 850 300 -324 -76 2 4 9 25

414 525 250 -330 -78 3 5 30 30

423 50 150 -79 -79 -3.5 6.6 50 50

t3. Ak.tas / T h i n Solid Fibns 307 (19971 2 5 0 - 2 5 9

100 ,, ................................. 4 25

"~ 40

~v

201 0 . . . . . . . . . . . . . . . . . . . . . . '*"!; 0 20 40 60 "800 T (K) Fig. 8. The rigid components of both the unidirectional bulk (heavy triangles) and surface (open triangles) anisotropies as a function of temperature. The dotted and continuous lines are the calculated ones by assuming a linear temperaturedependencegiven in the text.

The g-value for Fe-based compounds is generally 2.12. The magnetization was taken from dc magnetization measurements on the same sample [25]. From this analysis, we observed that the hard direction for surface spins is the film normal making easy plane surface anisotropy over a wide temperature range 4-300 K. From the angular variation of the FMR spectra at higher temperature, the peak at higher field side was identified as a surface mode, which appears for perpendicular direction of measurement field. The drastic change in this surface mode is due to surfaceinduced directional anisotropy field. The temperature dependence of both the bulk and the surface directional anisotropies are shown in Fig. 8, which were deduced by using the simulated spectra. The dotted and the continuous lines are the corresponding fitted curves that were deduced from Eq. (16). There is a small deviation from the experimental results at lower temperature region. But the fitted values of the parameter, T~g, is almost the same for both surface and bulk directional anisotropies. This means tha{ the physical origin of the bulk and surface exchange anisotropies is the same, and the unidirectional anisotropy in the bulk reflects itself on the surface, but the strength of this anisotropy at the surface varies due to lower syn~netry at the surface [34]. Regarding the magnetic energy parameters in Table 1: the value of the magnetization below 70 K is smaller than bulk value at room temperature. This is a natural result of Verwey transition of Fe304 from cubic to orthorhombic structure at lower temperatures. The rigid component, H ~ , of unidirectional anisotropy at 4 K is relatively larger compared to most of the spin-glass materials in the literature. That is, the system is magnetically highly disordered. The value of HAR for field cooling case decreases linearly and more rapidly than the dynamic component, HaD, with increasing temperature up t6 70 K. There is a strong correlation between rigid and dynamic component of the unidirectional anisotropy. The bulk anisotropy parameters are practically constant below Verwey transition temperature, T,.. Both demagne-

257

tizing field-corrected uniaxial ( K 3) and biaxial (K;) components of the magneto-crystalline bulk anisotropy parameters in Eq. (12) are negative for field-cooled case. The easy direction for magnetization is perpendicular to the film surface. These findings are consistent with well-known properties of the bulk Fe304 ferrite. As explained above, when Fe304 single crystal is cooled down below T, in the presence of an external field, the cubic Fe30 a transforms into orthorhombic crystalline structure in such a way that the easy axis (c-axis) becomes parallel to the external field. It should be remembered that the magnetic parameters in Table 1 were deduced from the spectra for perpendicular field cooling case. Therefore, the easy axis was induced along the direction perpendicular to the film plane: The axial component (Ks~) of the surface anisotropy in Table 1 is smaller than the unidirectional component (K d) in Fig. 7 at 4 K. However, Ksa increases while K d decreases as the temperature rises. Perhaps the rigid component of exchange anisotropy gains rotational character with the magnetization at relatively higher temperatures. The effect of this rotation manifest itself in the increment of Ks~. On the other hand, effective exchange interaction parameter, D, progressively decreases as the directional anisotropy grows at lower temperatures. This should be due to the transition of the sample from ferromagnetic to disordered magnetic structure. Lastly, peak-to-peak line widths A H " and A H r for n-FC and r-FC cases, respectively, increase with the increasing temperature. However, it is much smaller for n-FC case than that for r-FC case at 4 K. A similar behavior has been observed in literature [35] for reentrant NiMn alloys. This effect has been attributed to the elastic rotation of internal unidirectional anisotropy that was induced along the initial magnetization during the cooling of the sample. At lower temperature, the anisotropy becomes much more rigid, resulting in a longer relaxation time (smaller line width). The increment in line width at higher temperatures in the FC case should be originated from rapid relaxation of both the bulk and the surface unidirectional anisotr0pies.

6. Conclusion The epitaxial Fe304 film on MgO(100) shows some differences in magnetic properties as compared to those in bulk single crystal F%O 4. The FMR spectra of F%O 4 thin films include well-resolved surface induced modes in addition to the bulk FMR modes. Moreover, both of this surface and bulk modes have unidirectional character at low temperatures, upon cooling the sample in an external field. This is a clear sign for an exchange anisotropy. The exchange anisotropy value at the surface differs from that of the bulk of the same film. For thin film, strong exchange anisotropy field is induced upon cooling the sample in the presence of an external magnetic field. As

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B. Aktas / Thin Solid Fihns 307 (1997) 250-259

well-known, the exchange anisotropy is generally originated from a competing ferromagnetic and anti-ferromagnetic interactions in a magnetic system. However, all kinds of interactions in the bulk F e 3 0 4 magnetite are anti-ferromagnetic as explained above. Possible exchange interactions mediated by oxygen ions (super exchange) between A-A, A-B, and B - B pairs are different in magnitude. Normally, A - B interaction is much larger than A - A and B - B interactions. As a result, Fe304 has ferromagnetic order (to be more precise, a spiral spin configuration) that gives rise to a relatively sharp hysteresis loop for magnetization. It should be noted that there have already been competing interactions between A - B and B - B (or A-A) sites to orient the spins in site B (or A) in opposite direction of any other nearest neighboring spin. None of the spins likes another spin to be oriented along its own direction. Such kind of interaction naturally causes a frustration for spin alignments, but in the stoichiometric F e 3 0 4 or Fe:O 3 magnetite they result in a good ordered spiral spin configuration. Therefore, one might expect that a small change of the crystalline parameters of epitaxiaI F e 3 Q on MgO(100) due to a little misfit at the interface, modifies the relative exchange interactions among the different pairs. This in turn might cause a disorder in the spin alignments and a big exchange anisotropy field in F%O 4 films. But Krebs et al. [36] showed that a single crystal Fe304 grown on MgO by Molecular Beam Epitaxy (MBE) method behaves essentially like a bulk sample. So the exchange anisotropy seems to be originated from some other effects in our samples. On the other hand, Aragon et al. [37] studied nonstoichiometric Fe3t~_ ~O 4 (0 < ~ < 0.006) magnetite and found that a slight amount deviation from exact composition rate has profound effects on physical properties. For instance, Verwey transition, remnant magnetization, cubic magnetic anisotropy constant K 1 and exchange parameter(JAB, dominant interaction between A and B sites) are reduced with increasing nonstoichiometry. Even some Fe304 samples from our different runs give different magnetic cubic and exchange anisotropy values depending on a very small change of preparation conditions (partial oxygen pressure and temperature). Comparing the values for exchange parameter JAB from this work (the value for D in Table 1 corresponds to a value of 30 K for JAB) and from Aragon's study for various Fe concentrations, one can conclude that our sample is very close to exact Fe304 composition rate. A fast decrease in resonance field at the temperatures below 90 K implies a decrease in Verwey transition in our case. We also could make some Fe304 films giving very small exchange anisotropies and magneto-crystalline cubic anisotropy that are very close to those of bulk sample even at relatively small applied field. The results will be published in a separate paper. All findings above suggest a very small Fe ion deficiency (which, one might naturally expect, a tendency from F%O 4 to Fe203 composition rate) to be responsible for these unexpected magnetic behaviors.

Finally, unidirectional surface anisotropy field was phenomenologically introduced by Puszkarski [24] for surface spin pinning. However, the observed angular behavior of most of the FMR spectra in literature [20] show generally higher symmetry than unidirectional surface field. As far as we are aware, ours is the first [25] clear observation of unidirectional bulk and surface anisotropy in FMR lines of F e 3 Q thin films. Further investigation in this direction is under consideration. It should also be noted that, recently, Avgin and Huber [38] calculated the spin wave stiffness for Ca-doped YIG where uncompensated Ca causes a strong random ferromagnetic interactions between Fe ions occupying different sublattices. A similar interaction picture due to a slight defect may also occur in our system. We will do further investigation along that direction.

Acknowledgements The author would like to thank Dr. David Margulies for his kindness to supply the characterized samples. This work has been supported by the Scientific and Technical Research Council of Turkey and Islamic Development Bank.

References [1] J.K. Vassiliou, V. Mehrotra, M.W. Russell, E.P. Giannelis, R.D. McMichael, R.D. Shull, R.F. Ziolo, J. Appl. Phys. 73 (10) (1993) 5109. [2] F. Jorgensen, The Complete Handbook of Magnetic Recording, TAB Books, 1980, p. 191. [3] A.E. Berkowitz, IEEE Trans. Magn. MAG-22 (1986) 466, review. [4] L. Neel, Ann. Phys. 3 (1948) 137. [5] E.W. Goner, J.A. Schulkes, Phys. Rev. 90 (1953) 487. [6] Y. Yafet. C. Kittel, Phys. Rev. 87 (t952) 290. [7] C. Kittel, An Introduction to Solid State Physics, Seventh edn., 1996, p. 459. [8] T.A. Kaplan, K. Dwight, D. Lyons, N. Menyuk, J. Appl. Phys. 32 (1961) 13S. [9] E.J.W. Verwey, Nature 144 (1939) 327. [I0] F.J. Parker, A.E, Berkowitz, IEEE Trans. Magn. MAG-22 (t986) 466. [11] F. Itoh, M. Satou, Jpn. I. Appl. Phys. 14 (I975) 2091. [12] F. Itoh, M. Satou, Y. Yamazaki, IEEE Trans. Magn. MAG-13 (1977) 1385. [13] F.E. Spada, A.E. Berkowitz, N.J. Prokey, J. Appl. Phys. 69 (8) (1991) 4475. [14] A.E. Berkowitz, F.E. Parker, E.L. Hall, G. Podolsky, IEEE Trans. Magn. MAG-24 (1988) 2871. [I5] B. Heinrich, K.B. Urquhant, A.S. Arrott, J.F. Cochran, K. Myrtle, S.T. Purcell, Phys. Rev. Lett. 59 (1987) 1756. [16] G.A. Pfinz, G.T. Rado, J.J. Krebs, J. Appl. Phys. 53 (1982) 2087. [17] Z. Zhang, P.E. Wigen, S.S.P. Parkin, J. Appl. Phys. 69 (1991) 5649. [18] C. K,ittel, Phys. Rev. 110 (1958) 1295. [19] A. Layadi, J.O. Artmam B.O. Hail, R.A. Hoffman, C.L. Jensen, D.J. Chakrabarti, D.A. Saunders, J. Appl. Phys. 64 (1988) 5760. [20] P.E. Wigen, Thin Solid Films 114 (1984) 135. [21] B. Aktas, M. Clzdemir, Physica B 193 (1994) 125,

B, Aktas / Thin Solid Films 307 (1997) 250-259

[22] B. A,ktas, Solid State Commun. 87 (1993) 1067. [23] B. Aktas, Y. 0net, H.Z. Durusoy, J. Magn. Magn. Mater. 119 (1993) 339. [24] H. Puszkarski, Progr. Surf. 9 (1979) 191, and references therein. [25] D.T. Margulies, F.T. Pm'ker, F.E. Spada, R.S. Goldman, J. Li, R. Sinclair, A.E. Berkowitz, Phys. Rev. B 53 (i996) 9175, and the references therein. [26] B.A. Calhoun, Phys. Rev. 94 (1954) 1577. [27] L.J. Maksymowicz. D. Sendorek, J. Magn. Magn. Mater. 37 (1983) I77. [28] V.S. Speriosu, M.M. Chen, T. Suzuki, IEEE 25 (i969) 3875. [29] M.A. Manheimer, S.M. Bhagat, DJ, Woif, J. Appl. Phys. 5i (i985) 347.

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[30]P. Jamet, J.C. D umais~ J. Seiden, K~Knor~ J. Magn. IVlagn. Mater. i5 (1980) i97. [31] D. Youm, S. Schultz, Phys. Rex'. B 34 (i986) 7958, [32] H.J. Williams, R.M. Bozorth, M. Goertz, Phys. Rev. 9i (1953) i107. [33] D. Youm, S. Schultz, Phys. Rex,. B 34 (1986) 7958. [34] J.F. Cochran~ B. Heimich, A.S. Arrott~ Phys. Re,,'. B 34 (!988) 7788. [35] B. Aktas, Y. Oner, E.A. Harris, Phys. Rev. B 39 (i988) 528. [36] J.J. Krebs, D.M. Lind, S.D. Berry, J. Appl. Phys. 73 (1993) 6457. [37] R. Aragon, Phys. Rev, B 4-6 (199_2) 5328-5334, [38] I. Avginl O.L. Huber, J. Appl. Phys. 75 (i994) 5518.