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Near-surface magnetic structures in iron borate
Journal of Magnetism and Magnetic Materials 86 (1990) 105-114 North-Holland
105
NEAR-SURFACE MAGNETIC S T R U C T U R E S IN IRON B O R A T E
V.E. ZUBOV, G.S. K R I N C H I K Magnetics Chair, Department of Physics, Moscow State University, 119899 Moscow, Lenin Hills, USSR
V.N. SELEZNYOV and M.B. S T R U G A T S K Y Solid-State Physics Chair, Department of Physics, Simferopol State University, 333036 Simferopol, Yaltinskaya st., 4, USSR Received 25 June 1989
Surface magnetism on natural non-basal faces of iron borate monocrystals was found and analysed by means of the magneto-optical and Bitter methods. The surface magnetism here is represented by a macroscopical transition magnetic layer, the result of surface magnetic anisotropy. Anisotropy is caused by the change in surface magnetic ion environment symmetry. The erasure of surface magnetism at a face of the (1014)-type takes place in a field of H c = 1.6 kOe, the same characteristic for faces of the (1120) and (l123)-types is _q<100 Oe. By measuring the temperature-critical field relationship in the temperature range from 77 K upto the Nrel point, it was found that H c is proportional to the crystal's magnetization. Surface anisotropy energy and the structure of transition magnetic layers are calculated for the analyzed types of faces. The developed theory gives a correct description of surface magnetic anisotropy's symmetry, yields the H c field value order which agrees with the experimentally determined one, and also explains the temperature-critical field relationship.
I. Introduction
Earlier studies, conducted by two of this paper's authors resulted in revealing surface magnetism on natural non-basal faces-type (100) of hematite, represented by a microscopic transition layer of a domain boundary type which separates the surface of the crystal from its volume [1]. The existence of this layer results from the presence of surface anisotropy. Surface magnetic anisotropy on nonbasal faces of a-Fe203 is caused by the change in Fe 3+ magnetic ions environment symmetry at the surface of the crystal, compared to its volume. It takes about 20 kOe of external field for saturation in surface magnetisation to occur. Surface magnetism was also observed in rare-earth orthoferrites [2]. Back in 1954 Nrel [3] mentioned the possibility of the occurrence of surface anisotropy in magnets. But because this anisotropy is rather low, experimental observation is made difficult. Weak ferromagnets with magnetic anisotropy of the "ease plane"-type are the case of favourable conditions for observing surface anisotropy. This is due to the small demagnetization field energy level, compared to ordinary ferromagnets, as a
consequence of trifleness of the resulting magnetic moment and also because of practically no magnetic anisotropy in basal plane of the aforementioned crystals. These circumstances add to the role of surface anisotropy in the process of magnetising weak easy-plane ferromagnets. Just that very case applies to hematite above the Morin point, and also to rare-earth orthoferrites near the temperature of spin-reorientation transition. From this point of view some other weak ferromagnets are of interest as well. This paper is dedicated to the study of surface magnetism on natural non-basal faces of iron borate (FeBO3). The phenomenon and first results of its study were published in ref.
[41. 2. Samples and measurement methods The crystal lattice of iron borate may be described by Daro spatial symmetry group of a rhombohedral system. FeBO 3 is classified as a weak ferromagnet with magnetic anisotropy of the "easy plane"-type. The chamfer angle between the two antiparallel sublattices is of 55' magnitude,
0304-8853/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
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V.E. Zubov et al. / Near-surface magnetic structures
Fig, 1. The FeBO3 monocrystals,synthesizedby the gas transport method. Given are the indices of the sample faces. and is responsible for the weak-ferromagnet moment in FeBO 3 crystals. Iron borate samples are traditionally grown by the crystallization from solution in fusion method. This method usually yields thin FeBO 3 plates with a 50-100 ~m thickness. The surface of these crystals is parallel to their basal plane. In this study the sample growing was based on a method of synthesis from gas phase [5], resulting in threedimensional iron borate monocrystals. The samples (see fig. 1) had faces of the following types: (1014), (1120), (1123), (0112) and (0001) *. The analyzed faces with an area of 5-20 mm z had a flat mirror surface. Mirror faces of all the mentioned types but the (0112)-type could be found on the samples, the latter being_the reason why no magneto-optic study of the (0112) face was carried out. The type of a face was determined by the optical goniometry method and was defined more precisely with the help of rSntgenography. The degree of crystals' perfection was judged by the half width of their X-ray wobbling curve for all the faces. This value made up 1 0 - 4 0 " (angular seconds). Magnetic suspension was used for the study of domain structure at the surface of the crystals, and also of its behaviour in the presence of magnetic field. Magneto-optical investigation of FeBO 3 crystals was carried out by measuring equatorial (EKE) * Face indexes are given in the hexagonal system.
and polar (PKE) Kerr effects. The EKE is that the intensity of light, reflected from a ferromagnet, changes during its magnetization. The degree of this effect is proportional to the magnetization component, which is lying in the ferromagnet mirror plane and is perpendicular to the light incidence plane. The PKE is the rotation of the reflected light polarization plane during crystal magnetization; the magnitude of this rotation being proportional to the magnetization component, the one perpendicular to the mirror plane. Since the Kerr effects are proportional to the magnetization and because depth of forming the reflected light is very small (less than 0.1 ~m [6]), magneto-optical effects may be used for measuring magnetization curves of the thin near-surface layer of samples. In this research a dynamic magneto-optical setup with automatic signal registration, analogous to the one mentioned in ref. [7] was used. Measurements of volumetric magnetic properties of crystals were carried out on a compensation set-up, consisting of a solenoid and two instrumentation coils. The two instrumentation coils were connected towards each other and placed inside the solenoid, through which an audio frequency current (90 Hz) was passing. Placing a sample inside one of these two coils produced a signal, which was then measured by a resonant amplifier and registered with the help of an automatic recorder.
3. Experimental investigation of surface magnetism A labyrinth domain structure, analogous to the domain structure of fine-film, bubble-containing materials, was observed on natural non-basal faces of iron borate by use of powder figures method. An example may be found in fig. 2, where photos of the domain structure on the (1120) face, which is perpendicular to the basal plane at different magnitudes of the magnetic field are shown. The field was generated by Helmholtz coils and made perpendicular to the analyzed face. A labyrinth domain structure (fig. 2a) is observed in the absence of the magnetic field. The domain width in this structure is about 30 I~m. Increasing the field transforms this labyrinth structure into a set of
V.E. Zubov et al. / Near-surface magnetic structures
107
lies in the range of 5 0 - 7 0 Oe. Sample thickness has no noticeable effect u p o n the d o m a i n structure. As an example of this, the d o m a i n structure at the surface of this F e B O 3 plates with a thickness of about 150 ~tm which were cut from a massive sample was practically the same as it was before the cut-off. A perpendicular magnetic anisotropy is necessary for the labyrinth d o m a i n structure to appear. Because of the s y m m e t r y conditions, there is no uniaxial anisotropy field in the basal plane of the crystal's volume. Thus, it is natural to assume that there is a surface magnetic anisotropy on non-basal bases of iron borate. Reflecting magneto-optical effects, such as Kerr effects, are an effective means of investigating surface magnetic properties of crystals. Use of various Kerr effects (EKE, PKE, etc.) enables the study of magnetization processes for all three magnetization c o m p o n e n t s at the surface of the samples. Fig. 3 shows the angular relationship of the E K E (curve 1), which was measured at r o o m temperature for the (1014) face of the FeBO3 crystal in a field H = 50 Oe; h~0 = 3.3 eV. The angle X is reckoned in face plane from the intersection line of planes (1014) and (0001). Curve 2 represents the relationship between X and the projection of magnetization on the direction of field in the volume of the sample as it rotates in the (1014) plane. This relationship was also measured in the field H = 50 Oe. One m a y see from this figure that the magnetization curves of the surface differ greatly f r o m those of the volume.
Fig. 2. Photographs of the domain structure at the (1120) face in an external magnetic field which is perpendicular to this face: (a) H = 0 , (b) H = l l Oe, (c) H = 4 5 Oe.
isolated dipole and cylindrical domains. The magnitude of the magnetic field at which these domains disappear depends on the type of face and
4
0.4
m
O.R
0
1i"1=,
~
ll'ii'/ii lrad]
Fig. 3. Angular relationship of the EKE (h~0 ~ 3.3 eV) at the (1014) face, measured in a field H of 50 Oe - curve 1; angular relationship of the projection of the volumetric magnetization on the direction of field Mta at rotation of the sample in the (1054) plane, H = 50 Oe - curve 2; M is the spontaneous magnetization of FeBO3.
V.E. Zubov et at / Near-surface magnetic structures
108
/ 1.0
~
m
I / 4.m
a K
q
~
0.74
1
He!, tt¢ m
D 4
l-It [ kOe] Fig. 4. The (1014)-face surface magnetization curves, measured by means of the EKE along the HA (curve 1) and the EA (curve 2). 3 - the rated curve of the surface magnetization along the HA, derived from formula (11) with H c =1.6 kOe parameter. 4, 5 - the curves of volume magnetization along the HA and the EA, respectively (curves 4 and 5 coincide in the figure). The y-axis is on the left for curves 1, 3 and 4, and on the right for curves 2 and 5. All measurements were carried out at T = 300 K.
The change in the projection of the magnetization in volume on the field direction is connected with the fact that because of the symmetry conditions the weak ferromagnetic moment of FeBO3 always lies in the basal (0001) plane [5]. That is why curve 2 reaches its maximum level at X = 0, where the field is parallel to the (0001) plane. The same curve reaches its minimum level at X = ~r/2, when the angle between the field and the (0001) plane is maximum. In contrast to the results of measurements in volume, the EKE reaches its maximum at X = ~r/2, minimum is reached at X = 0. Therefore, one may come to a conclusion that there is a surface uniaxial magnetic anisotropy with an easy magnetic (EA) axis at the surface of the (1014) plane, with this axis being oriented along the direction which is pe_rpendicular to the intersection line of planes (1014) and (0001). Fig. 4 illustrates the magnetization curves at the surface of the (1014) plane, which were measured by means of the EKE at room temperature for the directions of hard (HA), and easy magnetization (curves 1 and 2, respectively). The projection of the external field on the basal plane (Ht), which determines the processes of spontaneous magnetization rotation in iron borate, is plotted along the x-axis. Surface magnetization is almost completed
at Ht = 300 Oe in the EA direction and at H t = 4 kOe in H A direction. Volumetric magnetization curves for the HA and EA directions (curves 4 and 5, respectively) are practically identical. It should be noted that the curves 4 and 5 are shown in different scales, since the HA and EA of the (1014) face make different angles with the basal plane. The values of these angles are 0 o and 42 °, respectively. Thus the maximum projection of the spontaneous magnetization ( M ) on the EA is cos(42 ° ) M = 0.74M. The results, shown in figs. 3 and 4, allow a conclusion to be drawn that there is a surface uniaxial magnetic anisotropy on the (1014) face of iron borate with a critical magnetization field along HA equal to H k ~ - 1 . 6 kOe, this being defined from theoretical curve 3 (fig. 4). The calculation method of the curve 3 is described in section 5. In the absence of field, face surface magnetization lies in the basal plane, perpendicular to the (1014) and (0001) faces intersection line. The slow approach of curve 1 to saturation may apparently be explained by a gradual change of the form of the magneto-optical signal from the form almost sinusoidal to rectangular with increasing the field magnitude. Measurement of magneto-optical effects were made utilizing the modulation method with the first harmonic of the measured signal being recorded. The magnitude of the first harmonic increases 4/~r fold as the form of the signal changes from sinusoidal to rectangular. Fig. 5 shows the results of investigating the magnetic anisotropy on the (1120) face at room temperature. Represented are the surface magnetization curves of this face for the following directions: in the face plane, parallel to the (0001) and (1120) plane intersection line (curve 1), and also perpendicular to the face (curve 2). These curves were measured by means of EKE and PKE, respectively. Broken lines on this figure show the volumetric magnetization curves in relative units (curves l a and 2a) for the same two directions of field. One may see from the figure that the surface magnetization for both directions takes place at equal fields. Besides, volumetric and surface magnetization curves are almost identical: curves 1, l a when the magnetization is in the face plane, and curves 2, 2a when the magnetization is perpendic-
V.E. Zuboo et al. / Near-surface magnetic structures
6
l
109
,4
a
f~f
~.
//
&
: 0
0
Y0:1
1, 1
--
D
6.6
H IkOel Fig. 5. The E K E (& curve 1) and the P K E (a, curve 2) at h ~ = 3.3 eV on the (1120) face. The (1120) face is perpendicular to the (0001). In the first case, H is parallel to the (1120) and (0001) faces intersection line, in the second perpendicular to the (1120) face. Broken curves are the curves of the volume magnetization of the crystal, given in relative units (curves 1 and la, 2 and 2a correspond to equal directions of the magnetizing field).
ular to the face. Thus, with an accuracy defined by the magnitudes of demagnetization fields ( = 100 Oe, since saturation induction of iron borate is 115 G [8]), we m a y say that there is no surface anisotropy on the (1120) face. A field relationship of the E K E was also analyzed for thin (about 50 ~m) FeBO 3 plates with (0001) faces, obtained by growing from solution in fusion. The crystals with an area of about 5 × 5 m m 2 had distinct growth lines. The magnetization was performed in the basal plane along two directions: parallel to the growth lines (fig. 6, curve 1), and perpendicular to them (curve 2). In this particular case the effect of the demagnetization field may be ignored because of the small thickness of the samples. It is evident from fig. 6 that there are no traces of surface anisotropy on the face of the (0001)-type whatsoever. And it should be mentioned that surface anisotropy on basal faces of hematite associated with growth line orientation was considered in ref. [7]. In order to determine the nature of surface anisotropy it might be useful to analyze its temperature relationship. The (1014) face was chosen for study of field relationship of E K E at different temperatures. Measurements were taken at the spectral m a x i m u m of the effect (ho~ = 3.5 eV). For
0
10
tO
SO H lOe]
Fig. 6. Field relationships of the E K E (h~0 = 3.3 eV) at the (0001) basal plane of a thin plate of FeBO 3, measured at T = 300 K for the two field directions in the (0001) plane: 1 the field is parallel to the growth lines, 2 - the field is perpendicular to these lines.
low-temperature measurements a vacuum optical cryostat was employed. At room temperatures and up to the N6el point ( T N = 348 K) the measurements were conducted inside a scavenged thermostat in a flow of heated air. Temperature was being measured by a c o p p e r / c o n s t a n t a n thermocouple located close to the sample. Sample magnetization was carried out in two basic directions:
1
VO ~La
/
1
0 0 0 0 0 0 0
J BOO
i 1000
H IOel Fig. 7. Field relationships of the E K E (ho~ = 3.5 eV) at the (1014) face along the HA, measured at various temperatures; 1 : 7 7 K, 2 : 3 0 0 K, 3 : 3 1 3 K, 4 : 3 2 3 K, 5 : 3 3 3 K, 6 : 3 3 8 K, 7: 343 K. 80(T ) is a saturation value of the E K E at a corresponding temperature. Broken curves represent the sample surface magnetization curves, calculated with the help of (11).
110
ICE. Zubov et al. / Near-surface magnetic structures
along the intersection line of the analyzed face and the basal plane, and also perpendicular to this line in the (1014) plane. From liquid nitrogen temperature up to the Nrel point, the intersection line of faces (1014) and (0001) is an axis of hard magnetization. Since a sample was placed either inside the cryostat or inside the thermostat, the gap of the employed magnet had to be sufficiently large (24 nun), which yielded a maximum field magnitude in this gap of 1 kOe. This field was usually not enough for magnetization of the crystal to saturation along the HA. That is why in order to normalize the magnetization curves obtained along the HA, the EA magnetization curves were used. Fig. 7 illustrates the normalized EKE curves measured along the HA at different temperatures (solid lines). Broken lines represent the rated face surface magnetization curves and define the values of the critical field H k at different temperatures. Section 5 deals with the question of tracing these curves.
O
~
.
i
zIIC3
i
,/ Fig. 8. A rhombohedron of FeBO 3 crystal with faces of the (10i4) type. The edge length of the rhombohedron is 5.9 A. The used coordinate system is on the fight (the x-axis lies in the face plane of the rhombohedron).
crystal. Calculation yields the following expression for the anisotropy surface energy on the (10i4) face at T = 0 K : O(10i4)
=
--
0.043 sin20 cos2cp + 0.015 sin28 sin 2 q)
- 0.032 cos20 + 0.077 sin 0 cos 8 sin qv [erg/cm2].
4. Surface magnetism theory 4.1. Surface anisotropy energy
Variation in Fe 3+ magnetic ion environment symmetry at the surface of the crystal changes their energy at near-surface layers, compared to the ion energy in the depths of the crystal. Let us now calculate the contribution of long-range magneto-dipole interaction to the magnetic anisotropy of the surface by defining it as the difference between the near-surface and volumetric magneto-dipole energies of magnetic ions. We will analyze the (1014) face in detail; only the key results will be given for other faces. An FeBO 3 rhombohedron with faces of the (1014)-type (fig. 8) was used for the calculation of the surface energy at the (1014) face. The smallest rhombohedron has an edge 5.9 A in length, and a 104.2 ° planar angle at the top of it [5]. The same figure shows the utilized system of coordinates. Here and further, the z-axis coincides with the crystal third-order axis, the x- and y-axis lie in the basal plane with the x-axis along the second-order axis, and the y-axis lying in the symmetry plane of the
(1)
The angles 0 and q0 determine the orientation of Fe 3+ magnetic ion moments. This expression was derived, taking into account the magneto-dipole interaction of Fe 3+ ions inside a rhombohedral, analogous to the one shown in fig. 8, with an edge of 20 × 5.9 A = 118 ,~. The corresponding expressions for (1120), (1123) and (0112) faces surface energies were obtained in a similar way. The following values of 8 and cp minimize the o(10~4) energy, defined by the expression (1): Oo = 2.64,
~0 = ~r/2.
(2)
4.2. Transition magnetic layer in the case of H = 0
The essential part hereafter of FeBO3 crystal thermodynamic potential is given by [9] ep =
½Bin 2 +
~1a2l z +
~(lxmy
-
lymx ) .
(3)
The first term in (3) is the exchange energy, the second characterizes the magnetic uniaxial crystallographic anisotropy, the third the Dzyaloshinsky energy, which is responsible for the existence of
V.E. Zubov et al. / Near-surface magnetic structures
weak ferromagnetism; l i (i = x, y, z) - the components of the reduced antiferromagnetic vector I (/=(I1-I2)/21, I = ] I 1 1= ]121. /1, I 2 - sublattice magnetization) m i - the components of the reduced antiferromagnetic vector m (m = (11 + I2)/2I); B=4IHE, a = 2 I H a, ~ = 2 I H D, H E, H a and H~ are, respectively, the exchange field, effective uniaxial crystallographic anisotropy field and the Dzyaloshinsky field. The parameters mentioned above for iron borate at T = 0 K are: I = 520 cgs units [5], H E = 3 × 103 kOe [10], H a = 3 kOe [11], H D = 100 kOe [12]. The Dzyaloshinsky interaction changes the effective field of uniaxial anisotropy. Therefore, the anisotropy part of the expression (3) may be written as: a
,~2~12
a'
2
~ +-2-91.z = T I_., a ' = 1.5a = 4.7 × 106 e r g / c m 3. The expressions (2) were obtained without considering the magnetic anisotropy in the volume of the crystal. Taking into account the volumetric anisotropy, the equilibrium angles 00 and %, which define the magnetic ion spin orientation at the surface of a face in the absence of a field, are determined by a competition between the surface anisotropy energy and transition magnetic layer energy, the latter being characterized by the variance of 0 from the equilibrium value on the surface to the volumetric equilibrium value. The thermodynamic potential of iron borate may be represented (considering magnetic uniaxial anisotropy and the exchange energy, being the result of spin distribution non-uniformity) as
Az[al'~]2+-~-lz, a' 2 3y l j + 2 ~ 3z1
A1[( 3 / ~ ] 2 + ( 3/~]2] q, = -~-
3x ]
where a = x, y, z; A1, A 2 - exchange parameters; a ' uniaxial magnetic anisotropy constant (taking into account the Dzyaloshinsky interaction). The energy of the near-surface transition layer per unit face area is
Yo = ½fo~ ( A(-a-~ dO " 2 + a' cos20} dS, where A is expressed in terms of A 1, A 2 and an
111
angle between the particular face and the z-axis; S = the distance from the surface of a face deep into the crystal. The following boundary conditions of the transition layer computation problem are available:
818=o=8o,
81s=~=~r/2,
the choice of 00 is arbitrary. The solution of the corresponding Euler equation gives us the particular function O(S), which minimizes the "Y0 energy. This very solution is 70 = Y00(1 - sin 00),
700 = (a'A) 1/2.
Let us now estimate Y0o at T = 0 K. a ' = 4.7 × 10 6 e r g / c m 3. It may be shown, that A ~ ½HEIC 2, C = 3.6 ,~, being the distance between adjacent ions Fe 3+ in FeBO 3 [5]. Using the above-mentioned values for H E and 1 we obtained A ---0.7 X 10 - 6 e r g / c m 2. This gives Y00 -- 1.8 e r g / c m 2. In order to determined the equilibrium magnetic structure of the transition layer near the face of a crystal, one needs to find the minimum of the sum of the surface anisotropy and transition layer energies as a function of 00 and ¢P0. Solution of the equations 3(0 + 7o)/30o = 0 and 30/3~00 = 0 for (1014) gives us the following equilibrium angle magnitudes: 00 = ,~/2,
4~0 = 0.
(4)
Considering uniaxial anisotropy in the volume, the equilibrium value of 00 changes at surface (4), compared with the case (2) (no consideration of anisotropy in volume) as to make the magnetic moments of Fe 3+ ions lie in the basal plane. This is connected with the fact that a typical transition energy (7Oo) is more than ten times of surface anisotropy energy (1). It might be interesting to note that in case the volumetric uniaxial anisotropy is taken into account, the equilibrium value of % (see (4)) changes by ,rr/2 as compared to case (2).
4. 3. Transition layer in magnetic hem Let us now consider the behaviour of the transition magnetic layer in an external field. The thermodynamic potential of an FeBO 3 crystal may
V.E. Zubov et al. / Near-surface magnetic structures
112
be written (taking into account that the fields are relatively small ( H << HD), and that the spins lie in the basal plane (0 o = "~/2)) as
ep = M H t sin(~b - ~ ) , where H t is the projection of the field onto the basal plane of a crystal; q~ is the angle between the direction of H t and the x-axis. The transition layer energy in the presence of external field is given by: = ;~fA(d~]
Y~
2
)
Jo ~ 2 ~,a s ] + MHt sin(~k-ep)
dS.
(5)
The Euler equation is d~ dS =+
sin X 3~o '
(6)
In the limit H t ~ H c, we get qoo ~ r / 2 . case, eq. (8) yields
H,: = 4 a 2 / A M .
(9)
At T = 0 K, the value of a~ will be 0.058 e r g / c m 2 (by using (1)). This results in Hc = 1100 Oe. Now let us define the value of H c at room temperature. H~ is proportional to I, since A - 1 2 , M - I , a~ - 12. According to ref. [8], I(0 K ) / I ( 3 0 0 K) = 1.47, that is why He ( T = 300 K) = 750 Oe. Calculations performed with the (1120), (1123) and (0112) non-basal faces, gave the following values of H c at T = 300 K: 110, 170 and 120 Oe, respectively. The following is an evaluation of the effective width of the transition layer as defined by [1] dS
¢r _ cp0)
where 3,0 = ½A/f~-M~t [cm], X = ¼,~ + ½~ - ½~. The boundary conditions are: X I s=0 = Xo = ¼~r +
½+-
½~0,
x l ~ = ~ = k ~ + ½+ - ½ ( + + . ~ / 2 ) = 0.
In this
¢
=3,o(2-~Po)/sin(4-%)~23,~o.
(10)
Substitution of (6) in (5) results in A f~ Y~ = ~ 3~o Jo
sin2XdS,
dS
3~o dqo sin X
6
R
U p o n integration we obtain y~ = 2Y~o(1 - cos X0), where Y,o = A/3~o = 2A~-~-~tt [erg/cm2]" Ignoring all terms independent of q%, the (1014) face surface anisotropy energy may be written as
~m
O(10T4) a s sin2 cp0, =
where a~ is the surface anisotropy constant. The angle ¢P0 is derived from the equation 3 3q0o (Y~ + °(10T4)) = -Y~o sin X o + a~ sin 2 % = 0.
(7) The magnitude of the critical field, which destroys the transition layer, can be determined from eq. (7). In the case of the (1014) face, the hard magnetization direction coincides with the x-axis (q~ = 0). Bearing this n mind, we obtain -2 A~/-~
sin(¼~r-½qOo)+a ~ sin 2qoo=0.
(8)
O
\,~\\ \\ \ ~ ~ U
10
1B s.lO 6 |cnl]
Fig. 9. The relationships between the angle formed by the M vector and the x-axis (¢Pl = ~°I(S)= ~ 0 ( S ) - ,rr/2), and the distance from the surface inside the crystal, calculated from eqs. (6) and (11) at various magnitudes of Ht for the (10i4) face during magnetization along H A (that is along the x-axis). Curve l: H t = 90 Oe, 2 : 1 8 0 Oe, 3 : 3 0 0 Oe, 4 : 4 6 0 Oe, 5 : 6 7 0 Oe, 6:920 Oe, 7 : 1 2 4 0 Oe. Tangents to these curves, calculated from eq. (10) and represented by broken lines, are used for determining the effective width of the transition layer. The latter is defined as the distance from the origin of the x-axis to the point of intersection of the tangent with the x-axis.
V.E. Zubov et al. / Near-surface magnetic structures
At room temperature 8~0--- 2 / flirt [~tm]. In case H = 100 Oe, 8~0-- 0.2 ~tm. Fig. 9 illustrates the structure of the transition layer at the (1014) face in an external field, directed along the HA. Curves, showing the relationship between the orientation of M, the ferromagnetic vector and the distance from the face surface, were obtained from the eqs. (6) and (11). An experimentally derived value of the critical field H~ = 1.6 kOe was used in the calculations. An approach to calculating the H c field is considered in the following section.
5. Discussion of the results
The eqs. (8) and (9) allow us to determine the relationship of the Fe 3+ ion spin orientation at the surface of the crystal ( % ( H t ) ) at the magnetization of the crystal along the HA, as long as the value of H~ is available: - ~ - t sin(¼-~ - ½¢P0) + ~ H ~ sin 2¢p0 -- 0.
(11)
The same equation may also be used for solving the inverse problem: to derive the value of H c from a given experimental magnetization curve. The surface layer magnetization curve with H~ = 1.6 kOe (curve 3) is in good accordance with the experimentally derived curve 1 in fig. 4 for the (1014) face in its initial and middle parts. The discrepancy between theoretical and experimental curves at H t ~-H~ has already been discussed in section 3. The value of Hc for the (1014) face, calculated from formula (9) with the help of the as = 0.058 e r g / c m 2 surface anisotropy constant, is about 750 Oe at room temperature, being two times less than the experimental value. Experimental and theoretical directions of the easy and hard magnetization coincide. Therefore, inclusion of the surface anisotropy due to magneto-dipole interaction of Fe 3+ ions leads to a qualitative explanation of the observed process and yields the true order of the H c field value. It has already been mentioned in section 3 that no surface anisotropy on faces of the (1120) and (1123) types was found using the magneto-optical
113
method (with the precision, determined by the value of the demagnetization field, that is ~ 100 Oe). Nevertheless, experimental studies of domain structures at the surface of iron borate crystals involving the Bitter method have shown that magnetic domain structures of the bubble type do also exist on those non-basal faces, which showed no evidence of surface anisotropy using the magneto-optical method. Apparently, the powder figure method enables a weaker surface anisotropy to be found. The same value of the anisotropy would be fogged out by the demagnetization fields in case of the analysis by Kerr effects. By suggesting that the surface anisotropy is due to Fe 3+ ion magneto-dipole interaction, we derived that H c --~I (see section 4). The Hc(I) experimental relationship may be obtained, for example, by varying I. The latter could be accomplished, for example, by means of varying the temperature. The low N6el point of iron borate ( T N = 348 K) makes it easy to greatly alter the sublattice magnetization in a comparatively narrow temperature range. The values of Hc for the (1014)-face magnetization curves at different temperatures were obtained from eq. (11) (see fig. 7). At each temperature, the He parameter of eq. (11) was chosen by the least squares method, so that the curve of surface magnetization along the HA, being described by this equation, coincided in the initial and middle parts. The corresponding theoretical curves are represented in fig. 7 by broken lines. The resulting temperature-critical field relationship is shown in fig. 10 by dots. The solid-line curve in the same figure shows the magnetizationtemperature relationship M(T) for the FeBO 3 crystal (or the sublattice magnetization I(T)), adopted from ref. [8]. It is easy to see that the relation H c - I holds true fairly well. Thus, the Hc(I) relationship may be explained by suggesting the magnetodipole mechanism as a cause of the surface anisotropy in the FeBO 3 crystal. Discrepancy of the theoretical and experimental values of //~ field for the (1034) face may be the result of a partial reconstruction of the surface with the Fe 3+ ions shifted from their positions, determined by the crystalline structure in the volume of the crystal. This kind of reconstruction should lead to a noticeable change in the mag-
V.E. Zubov et al. / Near-surface magnetic structures
114
,e
"\
lOO
ROO
|
30o
T [K]
Fig. 10. The t e m p e r a t u r e - H e field (spot) relationship, obtained using the experimental results which are shown in fig. 7. The solid curve represents the M ( T ) relationship in volume according to ref. [8]. The Hc(T ) and M ( T ) curves are normed to the H~ and M values at T = 77 K.
neto-dipole energy of Fe 3+ ions at the surface and near it, since the specific contribution of external Fe 3+ ion layers to surface anisotropy is greater than of the more inward ones. Calculations showed that increasing the distance between the external layer of the Fe 3+ ions and the rest of the crystal by 10% increases the H c field about twofold. Besides, single-ion anisotropy and Dzyaloshinsky interaction of magnetic ions in near-surface regions may also contribute to the surface anisotropy energy.
6. Conclusion
The study of the magnetic properties of iron borate in the near-surface region, carried out with the help of magneto-optical Kerr effects and also by the Bitter method, has shown that there is a surface anisotropy on the non-basal planes of FeBO 3with critical fields H c of: H c = 1.6 kOe at the (1014)-type face and _He < 100 Oe at the faces of the (1120) and the (1123) types. Analysis of the temperature-He relationship showed that the critical field changes proportionally to the sublattice magnetization of the crystal, as the temperature rises.
Including the magneto-dipole contribution to the surface anisotropy energy allows us to describe the symmetry of the observed anisotropy, to find the true order of the H c values for various faces, and also to explain the temperature-critical field relationship. Summarizing the results, it may be said that iron borate supplements the surface magnetism possessing group of weak ferromagnetics. Previously, this group was composed of hematite and rare-earth orthoferrites. The origin of surface magnetism in these crystals is determined by the change in magnetic ion environment symmetry at the surface. The magneto-dipole interaction of magnetic ions makes an essential contribution to the surface anisotropy energy. In the case of hematite and iron borate, this contribution becomes, apparently, the deciding one.
References [1] G.S. Krinchik and V.E. Zubov, ZhETF 69 (1975) 707. [2] E.A. Balykina, E.A. Ganshina and G.S. Krinchik, ZhETF 93 (1987) 1879; Fiz. Tverd. Tela 30 (1988) 570. [3] L. Nrel, J. Phys. Radium 15 (1954) 225. [4] V.E. Zubov, G.S. Krinchik, V.N. Selezneov and M.B. Strugatsky, ZhETF 94 (1988) 290. [5] R. Diehl, W. Jantz, B.I. Nolang and W. Wettling, in: Current Topics in Materials Science, vol. II, ed. E. Kaldis (North-Holland, Amsterdam, 1984) p. 241. [6] G.S. Krinchik, V.E, Zubov and V.A. Lyskov, Opt. Spektrosk. 55 (1983) 204. [7] V.E. Zubov, G.S. Krinchik and V.A. Lyskov, ZhETF 80 (1981) 229. [8] A.M. Kadomtseva, R.Z. Levitin, Yu.F. Popov, V.N. Selezneyov and V.V. Uskov, Fiz. Tverd. Tela 14 (1972) 214. [9] I.E. Dzyaloshinsky, ZhETF 32 (1957) 1547. [10] M. Eibschutz and M.E. Lines, Phys. Rev. B 11 (1973) 4907. [11] L.V. Velikov, A.S. Prokhorov, E.G. Rudashevsky and V.N. Seleznyov, Pis'ma ZhETF 15 (1972) 722. [12] L.V. Velikov, A.S. Prokhorov, E.G. Rudashevsky and V.N. Seleznyov, ZhEFT 66 (1974) 1857.
105
NEAR-SURFACE MAGNETIC S T R U C T U R E S IN IRON B O R A T E
V.E. ZUBOV, G.S. K R I N C H I K Magnetics Chair, Department of Physics, Moscow State University, 119899 Moscow, Lenin Hills, USSR
V.N. SELEZNYOV and M.B. S T R U G A T S K Y Solid-State Physics Chair, Department of Physics, Simferopol State University, 333036 Simferopol, Yaltinskaya st., 4, USSR Received 25 June 1989
Surface magnetism on natural non-basal faces of iron borate monocrystals was found and analysed by means of the magneto-optical and Bitter methods. The surface magnetism here is represented by a macroscopical transition magnetic layer, the result of surface magnetic anisotropy. Anisotropy is caused by the change in surface magnetic ion environment symmetry. The erasure of surface magnetism at a face of the (1014)-type takes place in a field of H c = 1.6 kOe, the same characteristic for faces of the (1120) and (l123)-types is _q<100 Oe. By measuring the temperature-critical field relationship in the temperature range from 77 K upto the Nrel point, it was found that H c is proportional to the crystal's magnetization. Surface anisotropy energy and the structure of transition magnetic layers are calculated for the analyzed types of faces. The developed theory gives a correct description of surface magnetic anisotropy's symmetry, yields the H c field value order which agrees with the experimentally determined one, and also explains the temperature-critical field relationship.
I. Introduction
Earlier studies, conducted by two of this paper's authors resulted in revealing surface magnetism on natural non-basal faces-type (100) of hematite, represented by a microscopic transition layer of a domain boundary type which separates the surface of the crystal from its volume [1]. The existence of this layer results from the presence of surface anisotropy. Surface magnetic anisotropy on nonbasal faces of a-Fe203 is caused by the change in Fe 3+ magnetic ions environment symmetry at the surface of the crystal, compared to its volume. It takes about 20 kOe of external field for saturation in surface magnetisation to occur. Surface magnetism was also observed in rare-earth orthoferrites [2]. Back in 1954 Nrel [3] mentioned the possibility of the occurrence of surface anisotropy in magnets. But because this anisotropy is rather low, experimental observation is made difficult. Weak ferromagnets with magnetic anisotropy of the "ease plane"-type are the case of favourable conditions for observing surface anisotropy. This is due to the small demagnetization field energy level, compared to ordinary ferromagnets, as a
consequence of trifleness of the resulting magnetic moment and also because of practically no magnetic anisotropy in basal plane of the aforementioned crystals. These circumstances add to the role of surface anisotropy in the process of magnetising weak easy-plane ferromagnets. Just that very case applies to hematite above the Morin point, and also to rare-earth orthoferrites near the temperature of spin-reorientation transition. From this point of view some other weak ferromagnets are of interest as well. This paper is dedicated to the study of surface magnetism on natural non-basal faces of iron borate (FeBO3). The phenomenon and first results of its study were published in ref.
[41. 2. Samples and measurement methods The crystal lattice of iron borate may be described by Daro spatial symmetry group of a rhombohedral system. FeBO 3 is classified as a weak ferromagnet with magnetic anisotropy of the "easy plane"-type. The chamfer angle between the two antiparallel sublattices is of 55' magnitude,
0304-8853/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
106
V.E. Zubov et al. / Near-surface magnetic structures
Fig, 1. The FeBO3 monocrystals,synthesizedby the gas transport method. Given are the indices of the sample faces. and is responsible for the weak-ferromagnet moment in FeBO 3 crystals. Iron borate samples are traditionally grown by the crystallization from solution in fusion method. This method usually yields thin FeBO 3 plates with a 50-100 ~m thickness. The surface of these crystals is parallel to their basal plane. In this study the sample growing was based on a method of synthesis from gas phase [5], resulting in threedimensional iron borate monocrystals. The samples (see fig. 1) had faces of the following types: (1014), (1120), (1123), (0112) and (0001) *. The analyzed faces with an area of 5-20 mm z had a flat mirror surface. Mirror faces of all the mentioned types but the (0112)-type could be found on the samples, the latter being_the reason why no magneto-optic study of the (0112) face was carried out. The type of a face was determined by the optical goniometry method and was defined more precisely with the help of rSntgenography. The degree of crystals' perfection was judged by the half width of their X-ray wobbling curve for all the faces. This value made up 1 0 - 4 0 " (angular seconds). Magnetic suspension was used for the study of domain structure at the surface of the crystals, and also of its behaviour in the presence of magnetic field. Magneto-optical investigation of FeBO 3 crystals was carried out by measuring equatorial (EKE) * Face indexes are given in the hexagonal system.
and polar (PKE) Kerr effects. The EKE is that the intensity of light, reflected from a ferromagnet, changes during its magnetization. The degree of this effect is proportional to the magnetization component, which is lying in the ferromagnet mirror plane and is perpendicular to the light incidence plane. The PKE is the rotation of the reflected light polarization plane during crystal magnetization; the magnitude of this rotation being proportional to the magnetization component, the one perpendicular to the mirror plane. Since the Kerr effects are proportional to the magnetization and because depth of forming the reflected light is very small (less than 0.1 ~m [6]), magneto-optical effects may be used for measuring magnetization curves of the thin near-surface layer of samples. In this research a dynamic magneto-optical setup with automatic signal registration, analogous to the one mentioned in ref. [7] was used. Measurements of volumetric magnetic properties of crystals were carried out on a compensation set-up, consisting of a solenoid and two instrumentation coils. The two instrumentation coils were connected towards each other and placed inside the solenoid, through which an audio frequency current (90 Hz) was passing. Placing a sample inside one of these two coils produced a signal, which was then measured by a resonant amplifier and registered with the help of an automatic recorder.
3. Experimental investigation of surface magnetism A labyrinth domain structure, analogous to the domain structure of fine-film, bubble-containing materials, was observed on natural non-basal faces of iron borate by use of powder figures method. An example may be found in fig. 2, where photos of the domain structure on the (1120) face, which is perpendicular to the basal plane at different magnitudes of the magnetic field are shown. The field was generated by Helmholtz coils and made perpendicular to the analyzed face. A labyrinth domain structure (fig. 2a) is observed in the absence of the magnetic field. The domain width in this structure is about 30 I~m. Increasing the field transforms this labyrinth structure into a set of
V.E. Zubov et al. / Near-surface magnetic structures
107
lies in the range of 5 0 - 7 0 Oe. Sample thickness has no noticeable effect u p o n the d o m a i n structure. As an example of this, the d o m a i n structure at the surface of this F e B O 3 plates with a thickness of about 150 ~tm which were cut from a massive sample was practically the same as it was before the cut-off. A perpendicular magnetic anisotropy is necessary for the labyrinth d o m a i n structure to appear. Because of the s y m m e t r y conditions, there is no uniaxial anisotropy field in the basal plane of the crystal's volume. Thus, it is natural to assume that there is a surface magnetic anisotropy on non-basal bases of iron borate. Reflecting magneto-optical effects, such as Kerr effects, are an effective means of investigating surface magnetic properties of crystals. Use of various Kerr effects (EKE, PKE, etc.) enables the study of magnetization processes for all three magnetization c o m p o n e n t s at the surface of the samples. Fig. 3 shows the angular relationship of the E K E (curve 1), which was measured at r o o m temperature for the (1014) face of the FeBO3 crystal in a field H = 50 Oe; h~0 = 3.3 eV. The angle X is reckoned in face plane from the intersection line of planes (1014) and (0001). Curve 2 represents the relationship between X and the projection of magnetization on the direction of field in the volume of the sample as it rotates in the (1014) plane. This relationship was also measured in the field H = 50 Oe. One m a y see from this figure that the magnetization curves of the surface differ greatly f r o m those of the volume.
Fig. 2. Photographs of the domain structure at the (1120) face in an external magnetic field which is perpendicular to this face: (a) H = 0 , (b) H = l l Oe, (c) H = 4 5 Oe.
isolated dipole and cylindrical domains. The magnitude of the magnetic field at which these domains disappear depends on the type of face and
4
0.4
m
O.R
0
1i"1=,
~
ll'ii'/ii lrad]
Fig. 3. Angular relationship of the EKE (h~0 ~ 3.3 eV) at the (1014) face, measured in a field H of 50 Oe - curve 1; angular relationship of the projection of the volumetric magnetization on the direction of field Mta at rotation of the sample in the (1054) plane, H = 50 Oe - curve 2; M is the spontaneous magnetization of FeBO3.
V.E. Zubov et at / Near-surface magnetic structures
108
/ 1.0
~
m
I / 4.m
a K
q
~
0.74
1
He!, tt¢ m
D 4
l-It [ kOe] Fig. 4. The (1014)-face surface magnetization curves, measured by means of the EKE along the HA (curve 1) and the EA (curve 2). 3 - the rated curve of the surface magnetization along the HA, derived from formula (11) with H c =1.6 kOe parameter. 4, 5 - the curves of volume magnetization along the HA and the EA, respectively (curves 4 and 5 coincide in the figure). The y-axis is on the left for curves 1, 3 and 4, and on the right for curves 2 and 5. All measurements were carried out at T = 300 K.
The change in the projection of the magnetization in volume on the field direction is connected with the fact that because of the symmetry conditions the weak ferromagnetic moment of FeBO3 always lies in the basal (0001) plane [5]. That is why curve 2 reaches its maximum level at X = 0, where the field is parallel to the (0001) plane. The same curve reaches its minimum level at X = ~r/2, when the angle between the field and the (0001) plane is maximum. In contrast to the results of measurements in volume, the EKE reaches its maximum at X = ~r/2, minimum is reached at X = 0. Therefore, one may come to a conclusion that there is a surface uniaxial magnetic anisotropy with an easy magnetic (EA) axis at the surface of the (1014) plane, with this axis being oriented along the direction which is pe_rpendicular to the intersection line of planes (1014) and (0001). Fig. 4 illustrates the magnetization curves at the surface of the (1014) plane, which were measured by means of the EKE at room temperature for the directions of hard (HA), and easy magnetization (curves 1 and 2, respectively). The projection of the external field on the basal plane (Ht), which determines the processes of spontaneous magnetization rotation in iron borate, is plotted along the x-axis. Surface magnetization is almost completed
at Ht = 300 Oe in the EA direction and at H t = 4 kOe in H A direction. Volumetric magnetization curves for the HA and EA directions (curves 4 and 5, respectively) are practically identical. It should be noted that the curves 4 and 5 are shown in different scales, since the HA and EA of the (1014) face make different angles with the basal plane. The values of these angles are 0 o and 42 °, respectively. Thus the maximum projection of the spontaneous magnetization ( M ) on the EA is cos(42 ° ) M = 0.74M. The results, shown in figs. 3 and 4, allow a conclusion to be drawn that there is a surface uniaxial magnetic anisotropy on the (1014) face of iron borate with a critical magnetization field along HA equal to H k ~ - 1 . 6 kOe, this being defined from theoretical curve 3 (fig. 4). The calculation method of the curve 3 is described in section 5. In the absence of field, face surface magnetization lies in the basal plane, perpendicular to the (1014) and (0001) faces intersection line. The slow approach of curve 1 to saturation may apparently be explained by a gradual change of the form of the magneto-optical signal from the form almost sinusoidal to rectangular with increasing the field magnitude. Measurement of magneto-optical effects were made utilizing the modulation method with the first harmonic of the measured signal being recorded. The magnitude of the first harmonic increases 4/~r fold as the form of the signal changes from sinusoidal to rectangular. Fig. 5 shows the results of investigating the magnetic anisotropy on the (1120) face at room temperature. Represented are the surface magnetization curves of this face for the following directions: in the face plane, parallel to the (0001) and (1120) plane intersection line (curve 1), and also perpendicular to the face (curve 2). These curves were measured by means of EKE and PKE, respectively. Broken lines on this figure show the volumetric magnetization curves in relative units (curves l a and 2a) for the same two directions of field. One may see from the figure that the surface magnetization for both directions takes place at equal fields. Besides, volumetric and surface magnetization curves are almost identical: curves 1, l a when the magnetization is in the face plane, and curves 2, 2a when the magnetization is perpendic-
V.E. Zuboo et al. / Near-surface magnetic structures
6
l
109
,4
a
f~f
~.
//
&
: 0
0
Y0:1
1, 1
--
D
6.6
H IkOel Fig. 5. The E K E (& curve 1) and the P K E (a, curve 2) at h ~ = 3.3 eV on the (1120) face. The (1120) face is perpendicular to the (0001). In the first case, H is parallel to the (1120) and (0001) faces intersection line, in the second perpendicular to the (1120) face. Broken curves are the curves of the volume magnetization of the crystal, given in relative units (curves 1 and la, 2 and 2a correspond to equal directions of the magnetizing field).
ular to the face. Thus, with an accuracy defined by the magnitudes of demagnetization fields ( = 100 Oe, since saturation induction of iron borate is 115 G [8]), we m a y say that there is no surface anisotropy on the (1120) face. A field relationship of the E K E was also analyzed for thin (about 50 ~m) FeBO 3 plates with (0001) faces, obtained by growing from solution in fusion. The crystals with an area of about 5 × 5 m m 2 had distinct growth lines. The magnetization was performed in the basal plane along two directions: parallel to the growth lines (fig. 6, curve 1), and perpendicular to them (curve 2). In this particular case the effect of the demagnetization field may be ignored because of the small thickness of the samples. It is evident from fig. 6 that there are no traces of surface anisotropy on the face of the (0001)-type whatsoever. And it should be mentioned that surface anisotropy on basal faces of hematite associated with growth line orientation was considered in ref. [7]. In order to determine the nature of surface anisotropy it might be useful to analyze its temperature relationship. The (1014) face was chosen for study of field relationship of E K E at different temperatures. Measurements were taken at the spectral m a x i m u m of the effect (ho~ = 3.5 eV). For
0
10
tO
SO H lOe]
Fig. 6. Field relationships of the E K E (h~0 = 3.3 eV) at the (0001) basal plane of a thin plate of FeBO 3, measured at T = 300 K for the two field directions in the (0001) plane: 1 the field is parallel to the growth lines, 2 - the field is perpendicular to these lines.
low-temperature measurements a vacuum optical cryostat was employed. At room temperatures and up to the N6el point ( T N = 348 K) the measurements were conducted inside a scavenged thermostat in a flow of heated air. Temperature was being measured by a c o p p e r / c o n s t a n t a n thermocouple located close to the sample. Sample magnetization was carried out in two basic directions:
1
VO ~La
/
1
0 0 0 0 0 0 0
J BOO
i 1000
H IOel Fig. 7. Field relationships of the E K E (ho~ = 3.5 eV) at the (1014) face along the HA, measured at various temperatures; 1 : 7 7 K, 2 : 3 0 0 K, 3 : 3 1 3 K, 4 : 3 2 3 K, 5 : 3 3 3 K, 6 : 3 3 8 K, 7: 343 K. 80(T ) is a saturation value of the E K E at a corresponding temperature. Broken curves represent the sample surface magnetization curves, calculated with the help of (11).
110
ICE. Zubov et al. / Near-surface magnetic structures
along the intersection line of the analyzed face and the basal plane, and also perpendicular to this line in the (1014) plane. From liquid nitrogen temperature up to the Nrel point, the intersection line of faces (1014) and (0001) is an axis of hard magnetization. Since a sample was placed either inside the cryostat or inside the thermostat, the gap of the employed magnet had to be sufficiently large (24 nun), which yielded a maximum field magnitude in this gap of 1 kOe. This field was usually not enough for magnetization of the crystal to saturation along the HA. That is why in order to normalize the magnetization curves obtained along the HA, the EA magnetization curves were used. Fig. 7 illustrates the normalized EKE curves measured along the HA at different temperatures (solid lines). Broken lines represent the rated face surface magnetization curves and define the values of the critical field H k at different temperatures. Section 5 deals with the question of tracing these curves.
O
~
.
i
zIIC3
i
,/ Fig. 8. A rhombohedron of FeBO 3 crystal with faces of the (10i4) type. The edge length of the rhombohedron is 5.9 A. The used coordinate system is on the fight (the x-axis lies in the face plane of the rhombohedron).
crystal. Calculation yields the following expression for the anisotropy surface energy on the (10i4) face at T = 0 K : O(10i4)
=
--
0.043 sin20 cos2cp + 0.015 sin28 sin 2 q)
- 0.032 cos20 + 0.077 sin 0 cos 8 sin qv [erg/cm2].
4. Surface magnetism theory 4.1. Surface anisotropy energy
Variation in Fe 3+ magnetic ion environment symmetry at the surface of the crystal changes their energy at near-surface layers, compared to the ion energy in the depths of the crystal. Let us now calculate the contribution of long-range magneto-dipole interaction to the magnetic anisotropy of the surface by defining it as the difference between the near-surface and volumetric magneto-dipole energies of magnetic ions. We will analyze the (1014) face in detail; only the key results will be given for other faces. An FeBO 3 rhombohedron with faces of the (1014)-type (fig. 8) was used for the calculation of the surface energy at the (1014) face. The smallest rhombohedron has an edge 5.9 A in length, and a 104.2 ° planar angle at the top of it [5]. The same figure shows the utilized system of coordinates. Here and further, the z-axis coincides with the crystal third-order axis, the x- and y-axis lie in the basal plane with the x-axis along the second-order axis, and the y-axis lying in the symmetry plane of the
(1)
The angles 0 and q0 determine the orientation of Fe 3+ magnetic ion moments. This expression was derived, taking into account the magneto-dipole interaction of Fe 3+ ions inside a rhombohedral, analogous to the one shown in fig. 8, with an edge of 20 × 5.9 A = 118 ,~. The corresponding expressions for (1120), (1123) and (0112) faces surface energies were obtained in a similar way. The following values of 8 and cp minimize the o(10~4) energy, defined by the expression (1): Oo = 2.64,
~0 = ~r/2.
(2)
4.2. Transition magnetic layer in the case of H = 0
The essential part hereafter of FeBO3 crystal thermodynamic potential is given by [9] ep =
½Bin 2 +
~1a2l z +
~(lxmy
-
lymx ) .
(3)
The first term in (3) is the exchange energy, the second characterizes the magnetic uniaxial crystallographic anisotropy, the third the Dzyaloshinsky energy, which is responsible for the existence of
V.E. Zubov et al. / Near-surface magnetic structures
weak ferromagnetism; l i (i = x, y, z) - the components of the reduced antiferromagnetic vector I (/=(I1-I2)/21, I = ] I 1 1= ]121. /1, I 2 - sublattice magnetization) m i - the components of the reduced antiferromagnetic vector m (m = (11 + I2)/2I); B=4IHE, a = 2 I H a, ~ = 2 I H D, H E, H a and H~ are, respectively, the exchange field, effective uniaxial crystallographic anisotropy field and the Dzyaloshinsky field. The parameters mentioned above for iron borate at T = 0 K are: I = 520 cgs units [5], H E = 3 × 103 kOe [10], H a = 3 kOe [11], H D = 100 kOe [12]. The Dzyaloshinsky interaction changes the effective field of uniaxial anisotropy. Therefore, the anisotropy part of the expression (3) may be written as: a
,~2~12
a'
2
~ +-2-91.z = T I_., a ' = 1.5a = 4.7 × 106 e r g / c m 3. The expressions (2) were obtained without considering the magnetic anisotropy in the volume of the crystal. Taking into account the volumetric anisotropy, the equilibrium angles 00 and %, which define the magnetic ion spin orientation at the surface of a face in the absence of a field, are determined by a competition between the surface anisotropy energy and transition magnetic layer energy, the latter being characterized by the variance of 0 from the equilibrium value on the surface to the volumetric equilibrium value. The thermodynamic potential of iron borate may be represented (considering magnetic uniaxial anisotropy and the exchange energy, being the result of spin distribution non-uniformity) as
Az[al'~]2+-~-lz, a' 2 3y l j + 2 ~ 3z1
A1[( 3 / ~ ] 2 + ( 3/~]2] q, = -~-
3x ]
where a = x, y, z; A1, A 2 - exchange parameters; a ' uniaxial magnetic anisotropy constant (taking into account the Dzyaloshinsky interaction). The energy of the near-surface transition layer per unit face area is
Yo = ½fo~ ( A(-a-~ dO " 2 + a' cos20} dS, where A is expressed in terms of A 1, A 2 and an
111
angle between the particular face and the z-axis; S = the distance from the surface of a face deep into the crystal. The following boundary conditions of the transition layer computation problem are available:
818=o=8o,
81s=~=~r/2,
the choice of 00 is arbitrary. The solution of the corresponding Euler equation gives us the particular function O(S), which minimizes the "Y0 energy. This very solution is 70 = Y00(1 - sin 00),
700 = (a'A) 1/2.
Let us now estimate Y0o at T = 0 K. a ' = 4.7 × 10 6 e r g / c m 3. It may be shown, that A ~ ½HEIC 2, C = 3.6 ,~, being the distance between adjacent ions Fe 3+ in FeBO 3 [5]. Using the above-mentioned values for H E and 1 we obtained A ---0.7 X 10 - 6 e r g / c m 2. This gives Y00 -- 1.8 e r g / c m 2. In order to determined the equilibrium magnetic structure of the transition layer near the face of a crystal, one needs to find the minimum of the sum of the surface anisotropy and transition layer energies as a function of 00 and ¢P0. Solution of the equations 3(0 + 7o)/30o = 0 and 30/3~00 = 0 for (1014) gives us the following equilibrium angle magnitudes: 00 = ,~/2,
4~0 = 0.
(4)
Considering uniaxial anisotropy in the volume, the equilibrium value of 00 changes at surface (4), compared with the case (2) (no consideration of anisotropy in volume) as to make the magnetic moments of Fe 3+ ions lie in the basal plane. This is connected with the fact that a typical transition energy (7Oo) is more than ten times of surface anisotropy energy (1). It might be interesting to note that in case the volumetric uniaxial anisotropy is taken into account, the equilibrium value of % (see (4)) changes by ,rr/2 as compared to case (2).
4. 3. Transition layer in magnetic hem Let us now consider the behaviour of the transition magnetic layer in an external field. The thermodynamic potential of an FeBO 3 crystal may
V.E. Zubov et al. / Near-surface magnetic structures
112
be written (taking into account that the fields are relatively small ( H << HD), and that the spins lie in the basal plane (0 o = "~/2)) as
ep = M H t sin(~b - ~ ) , where H t is the projection of the field onto the basal plane of a crystal; q~ is the angle between the direction of H t and the x-axis. The transition layer energy in the presence of external field is given by: = ;~fA(d~]
Y~
2
)
Jo ~ 2 ~,a s ] + MHt sin(~k-ep)
dS.
(5)
The Euler equation is d~ dS =+
sin X 3~o '
(6)
In the limit H t ~ H c, we get qoo ~ r / 2 . case, eq. (8) yields
H,: = 4 a 2 / A M .
(9)
At T = 0 K, the value of a~ will be 0.058 e r g / c m 2 (by using (1)). This results in Hc = 1100 Oe. Now let us define the value of H c at room temperature. H~ is proportional to I, since A - 1 2 , M - I , a~ - 12. According to ref. [8], I(0 K ) / I ( 3 0 0 K) = 1.47, that is why He ( T = 300 K) = 750 Oe. Calculations performed with the (1120), (1123) and (0112) non-basal faces, gave the following values of H c at T = 300 K: 110, 170 and 120 Oe, respectively. The following is an evaluation of the effective width of the transition layer as defined by [1] dS
¢r _ cp0)
where 3,0 = ½A/f~-M~t [cm], X = ¼,~ + ½~ - ½~. The boundary conditions are: X I s=0 = Xo = ¼~r +
½+-
½~0,
x l ~ = ~ = k ~ + ½+ - ½ ( + + . ~ / 2 ) = 0.
In this
¢
=3,o(2-~Po)/sin(4-%)~23,~o.
(10)
Substitution of (6) in (5) results in A f~ Y~ = ~ 3~o Jo
sin2XdS,
dS
3~o dqo sin X
6
R
U p o n integration we obtain y~ = 2Y~o(1 - cos X0), where Y,o = A/3~o = 2A~-~-~tt [erg/cm2]" Ignoring all terms independent of q%, the (1014) face surface anisotropy energy may be written as
~m
O(10T4) a s sin2 cp0, =
where a~ is the surface anisotropy constant. The angle ¢P0 is derived from the equation 3 3q0o (Y~ + °(10T4)) = -Y~o sin X o + a~ sin 2 % = 0.
(7) The magnitude of the critical field, which destroys the transition layer, can be determined from eq. (7). In the case of the (1014) face, the hard magnetization direction coincides with the x-axis (q~ = 0). Bearing this n mind, we obtain -2 A~/-~
sin(¼~r-½qOo)+a ~ sin 2qoo=0.
(8)
O
\,~\\ \\ \ ~ ~ U
10
1B s.lO 6 |cnl]
Fig. 9. The relationships between the angle formed by the M vector and the x-axis (¢Pl = ~°I(S)= ~ 0 ( S ) - ,rr/2), and the distance from the surface inside the crystal, calculated from eqs. (6) and (11) at various magnitudes of Ht for the (10i4) face during magnetization along H A (that is along the x-axis). Curve l: H t = 90 Oe, 2 : 1 8 0 Oe, 3 : 3 0 0 Oe, 4 : 4 6 0 Oe, 5 : 6 7 0 Oe, 6:920 Oe, 7 : 1 2 4 0 Oe. Tangents to these curves, calculated from eq. (10) and represented by broken lines, are used for determining the effective width of the transition layer. The latter is defined as the distance from the origin of the x-axis to the point of intersection of the tangent with the x-axis.
V.E. Zubov et al. / Near-surface magnetic structures
At room temperature 8~0--- 2 / flirt [~tm]. In case H = 100 Oe, 8~0-- 0.2 ~tm. Fig. 9 illustrates the structure of the transition layer at the (1014) face in an external field, directed along the HA. Curves, showing the relationship between the orientation of M, the ferromagnetic vector and the distance from the face surface, were obtained from the eqs. (6) and (11). An experimentally derived value of the critical field H~ = 1.6 kOe was used in the calculations. An approach to calculating the H c field is considered in the following section.
5. Discussion of the results
The eqs. (8) and (9) allow us to determine the relationship of the Fe 3+ ion spin orientation at the surface of the crystal ( % ( H t ) ) at the magnetization of the crystal along the HA, as long as the value of H~ is available: - ~ - t sin(¼-~ - ½¢P0) + ~ H ~ sin 2¢p0 -- 0.
(11)
The same equation may also be used for solving the inverse problem: to derive the value of H c from a given experimental magnetization curve. The surface layer magnetization curve with H~ = 1.6 kOe (curve 3) is in good accordance with the experimentally derived curve 1 in fig. 4 for the (1014) face in its initial and middle parts. The discrepancy between theoretical and experimental curves at H t ~-H~ has already been discussed in section 3. The value of Hc for the (1014) face, calculated from formula (9) with the help of the as = 0.058 e r g / c m 2 surface anisotropy constant, is about 750 Oe at room temperature, being two times less than the experimental value. Experimental and theoretical directions of the easy and hard magnetization coincide. Therefore, inclusion of the surface anisotropy due to magneto-dipole interaction of Fe 3+ ions leads to a qualitative explanation of the observed process and yields the true order of the H c field value. It has already been mentioned in section 3 that no surface anisotropy on faces of the (1120) and (1123) types was found using the magneto-optical
113
method (with the precision, determined by the value of the demagnetization field, that is ~ 100 Oe). Nevertheless, experimental studies of domain structures at the surface of iron borate crystals involving the Bitter method have shown that magnetic domain structures of the bubble type do also exist on those non-basal faces, which showed no evidence of surface anisotropy using the magneto-optical method. Apparently, the powder figure method enables a weaker surface anisotropy to be found. The same value of the anisotropy would be fogged out by the demagnetization fields in case of the analysis by Kerr effects. By suggesting that the surface anisotropy is due to Fe 3+ ion magneto-dipole interaction, we derived that H c --~I (see section 4). The Hc(I) experimental relationship may be obtained, for example, by varying I. The latter could be accomplished, for example, by means of varying the temperature. The low N6el point of iron borate ( T N = 348 K) makes it easy to greatly alter the sublattice magnetization in a comparatively narrow temperature range. The values of Hc for the (1014)-face magnetization curves at different temperatures were obtained from eq. (11) (see fig. 7). At each temperature, the He parameter of eq. (11) was chosen by the least squares method, so that the curve of surface magnetization along the HA, being described by this equation, coincided in the initial and middle parts. The corresponding theoretical curves are represented in fig. 7 by broken lines. The resulting temperature-critical field relationship is shown in fig. 10 by dots. The solid-line curve in the same figure shows the magnetizationtemperature relationship M(T) for the FeBO 3 crystal (or the sublattice magnetization I(T)), adopted from ref. [8]. It is easy to see that the relation H c - I holds true fairly well. Thus, the Hc(I) relationship may be explained by suggesting the magnetodipole mechanism as a cause of the surface anisotropy in the FeBO 3 crystal. Discrepancy of the theoretical and experimental values of //~ field for the (1034) face may be the result of a partial reconstruction of the surface with the Fe 3+ ions shifted from their positions, determined by the crystalline structure in the volume of the crystal. This kind of reconstruction should lead to a noticeable change in the mag-
V.E. Zubov et al. / Near-surface magnetic structures
114
,e
"\
lOO
ROO
|
30o
T [K]
Fig. 10. The t e m p e r a t u r e - H e field (spot) relationship, obtained using the experimental results which are shown in fig. 7. The solid curve represents the M ( T ) relationship in volume according to ref. [8]. The Hc(T ) and M ( T ) curves are normed to the H~ and M values at T = 77 K.
neto-dipole energy of Fe 3+ ions at the surface and near it, since the specific contribution of external Fe 3+ ion layers to surface anisotropy is greater than of the more inward ones. Calculations showed that increasing the distance between the external layer of the Fe 3+ ions and the rest of the crystal by 10% increases the H c field about twofold. Besides, single-ion anisotropy and Dzyaloshinsky interaction of magnetic ions in near-surface regions may also contribute to the surface anisotropy energy.
6. Conclusion
The study of the magnetic properties of iron borate in the near-surface region, carried out with the help of magneto-optical Kerr effects and also by the Bitter method, has shown that there is a surface anisotropy on the non-basal planes of FeBO 3with critical fields H c of: H c = 1.6 kOe at the (1014)-type face and _He < 100 Oe at the faces of the (1120) and the (1123) types. Analysis of the temperature-He relationship showed that the critical field changes proportionally to the sublattice magnetization of the crystal, as the temperature rises.
Including the magneto-dipole contribution to the surface anisotropy energy allows us to describe the symmetry of the observed anisotropy, to find the true order of the H c values for various faces, and also to explain the temperature-critical field relationship. Summarizing the results, it may be said that iron borate supplements the surface magnetism possessing group of weak ferromagnetics. Previously, this group was composed of hematite and rare-earth orthoferrites. The origin of surface magnetism in these crystals is determined by the change in magnetic ion environment symmetry at the surface. The magneto-dipole interaction of magnetic ions makes an essential contribution to the surface anisotropy energy. In the case of hematite and iron borate, this contribution becomes, apparently, the deciding one.
References [1] G.S. Krinchik and V.E. Zubov, ZhETF 69 (1975) 707. [2] E.A. Balykina, E.A. Ganshina and G.S. Krinchik, ZhETF 93 (1987) 1879; Fiz. Tverd. Tela 30 (1988) 570. [3] L. Nrel, J. Phys. Radium 15 (1954) 225. [4] V.E. Zubov, G.S. Krinchik, V.N. Selezneov and M.B. Strugatsky, ZhETF 94 (1988) 290. [5] R. Diehl, W. Jantz, B.I. Nolang and W. Wettling, in: Current Topics in Materials Science, vol. II, ed. E. Kaldis (North-Holland, Amsterdam, 1984) p. 241. [6] G.S. Krinchik, V.E, Zubov and V.A. Lyskov, Opt. Spektrosk. 55 (1983) 204. [7] V.E. Zubov, G.S. Krinchik and V.A. Lyskov, ZhETF 80 (1981) 229. [8] A.M. Kadomtseva, R.Z. Levitin, Yu.F. Popov, V.N. Selezneyov and V.V. Uskov, Fiz. Tverd. Tela 14 (1972) 214. [9] I.E. Dzyaloshinsky, ZhETF 32 (1957) 1547. [10] M. Eibschutz and M.E. Lines, Phys. Rev. B 11 (1973) 4907. [11] L.V. Velikov, A.S. Prokhorov, E.G. Rudashevsky and V.N. Seleznyov, Pis'ma ZhETF 15 (1972) 722. [12] L.V. Velikov, A.S. Prokhorov, E.G. Rudashevsky and V.N. Seleznyov, ZhEFT 66 (1974) 1857.