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Surface anisotropy tensors for cubic and hcp ferromagnets
Solid State Communications, Vol. 18, pp. 971—972, 1976.
Pergamon Press.
Printed in Great Britain
SURFACE ANISOTROPY TENSORS FOR CUBIC AND HCP FERROMAGNETS A. Wachniewski Institute of Chemistry and Physics, Medical University of Silesia, 41-200 Sosnowiec, Poland (Received 31 October 1975 by E.F. Bertaut) Point groups of the plane surfaces of cubic and h.c.p. ferromagnets are indicated together with the corresponding forms of the first and second rank surface anisotropy tensors. RECENTLY some papers14 point to the tensorial character of surface anisotropy in ferromagnets. Thus Yu, Turk and Wigen1 established experimentally the rotational symmetry of SWR with regard to the axis perpendicularto the thin film and concluded the tensorial rather than vectorial character of the surface anisotropy. A second rank magnetostatic surface anisotropy tensor was calculated for a few specified surfaces and magnetic lattices on the basis of a modified Lorentz local-field formula.3’4 It is however possible, without discussing the origin of the surface anisotropy, to determine the form of the surface anisotropy tensor of any rank for particular crystal surface. Let us first notice, that the magnetic surface anisotropy can be described by first and second rank tensors corresponding respectively to energy terms which are linear and bilinear in spin components. Higher rank tensors can be considered also, but it seems doubtful if they have any physical significance. As spin is an axial vector it follows immediately that only first rank axial and second rank polar tensors are possible. As in this paper we confine our considerations to ferromagnets, the magnetic properties are formulated with reference to a saturated crystal and the classical point
groups have to be used in simplifying surface anisotropy tensors (see Birss5 cap. 5). To the infinite cubic and h.c.p. crystals correspond m3m and 6/mmm point groups, respectively. On the other hand there are only ten, namely 1, 2, m, mm2, 4, 4mm, 3, 3m, 6, and 6mm, two dimensionals point groups (see e.g. cap. 1 of Kittel6) which can describe the symmetry on a plane crystal surface. It is clear that only those of the ten point groups which are sub-groups of m3m or 6/mmm have to be considered in the case of a cubic or h.c.p. crystal surface. For any such group a corresponding crystal surface can easily be found by considering the group elements. If two groups lead to the same crystal surface then that being a sub-group of the other one has to be rejected. Results of this simple checking are presented in Table 1. With the correspondence between crystal surface and its point group established, the tensor forms can be obtained immediately e.g. from Table 4 of reference 5. For convenience we reproduce in Table 1 the forms of first rank axial and second rank polar tensors. The absence of any tensor component in Table 1 means that this component is zero. The orientation of coordinate axes with respect to the crystallographic directions is indicated if material only. We do not include the group 1
Table 1
Point group
4mm 3m 3m mm2 mm2 m m m
Lattice and surface
Cubic Cubic h.c.p. Cubic h.c.p. Cubic Cubic h.c.p. I
(001) (111) (0001) (011) (1100) (hk0) (hk—k) [0001]
First rank axial tensor
Second rank polar tensor
Orientation of coordinate axes
z
xxyyzzxyyx
x
0 0 0 0 0 z z z
xx xx xx xx xx xx xx xx
[001] [111] [0001] [100] [01—1] [011] [0010] [0001] [1100] [001] [011] [0001]
971
xx xx xx yy yy yy yy yy
zz zz zz zz zz
zz zz zz
0 0 0 0 0 xy xy xy
0 0 0 0 0 yx yx yx
y
z
972
TENSORS FOR CUBIC AND HCP FERROMAGNETS
in Table 1 as this group does not simplify tensors, but it has to be noticed that any crystal surface different from those given in Table I corresponds to this group. Thus the form of surface anisotropy tensors can be read directly from Table 1, the only particularization seems to be the symmetry of second rank tensor.
Vol. 18, No. 8
Finally, we would like to stress that our present results are in general not applicable to a ferromagnetic crystal with a surface stratum of different, e.g. antiferromagnetic, character. We shall analyse such systems in a separate paper.
REFERENCES 1.
YEJ J.T., TURK R.A. & WIGEN P.E., Solid State Commun. 14, 283 (1974).
2.
PUSZKARSKI H., Phys. Status Solidi (b) 63, Ki 15 (1974).
3.
WACHNIEWSKI A., Phys. Status Solidi (b) 66, K59 (1974).
4.
WACHNIEWSKI A. & BIEGALA L., Phys. Stalus Solidi (b) 72, K27 (1975).
5.
BIRSS R.R., Symmetry and Magnetism. North-Holland, Amsterdam (1964).
6.
KITTEL CH., Introduction to Solid State Physics, 2nd ed. Wiley, NY (1956).
Pergamon Press.
Printed in Great Britain
SURFACE ANISOTROPY TENSORS FOR CUBIC AND HCP FERROMAGNETS A. Wachniewski Institute of Chemistry and Physics, Medical University of Silesia, 41-200 Sosnowiec, Poland (Received 31 October 1975 by E.F. Bertaut) Point groups of the plane surfaces of cubic and h.c.p. ferromagnets are indicated together with the corresponding forms of the first and second rank surface anisotropy tensors. RECENTLY some papers14 point to the tensorial character of surface anisotropy in ferromagnets. Thus Yu, Turk and Wigen1 established experimentally the rotational symmetry of SWR with regard to the axis perpendicularto the thin film and concluded the tensorial rather than vectorial character of the surface anisotropy. A second rank magnetostatic surface anisotropy tensor was calculated for a few specified surfaces and magnetic lattices on the basis of a modified Lorentz local-field formula.3’4 It is however possible, without discussing the origin of the surface anisotropy, to determine the form of the surface anisotropy tensor of any rank for particular crystal surface. Let us first notice, that the magnetic surface anisotropy can be described by first and second rank tensors corresponding respectively to energy terms which are linear and bilinear in spin components. Higher rank tensors can be considered also, but it seems doubtful if they have any physical significance. As spin is an axial vector it follows immediately that only first rank axial and second rank polar tensors are possible. As in this paper we confine our considerations to ferromagnets, the magnetic properties are formulated with reference to a saturated crystal and the classical point
groups have to be used in simplifying surface anisotropy tensors (see Birss5 cap. 5). To the infinite cubic and h.c.p. crystals correspond m3m and 6/mmm point groups, respectively. On the other hand there are only ten, namely 1, 2, m, mm2, 4, 4mm, 3, 3m, 6, and 6mm, two dimensionals point groups (see e.g. cap. 1 of Kittel6) which can describe the symmetry on a plane crystal surface. It is clear that only those of the ten point groups which are sub-groups of m3m or 6/mmm have to be considered in the case of a cubic or h.c.p. crystal surface. For any such group a corresponding crystal surface can easily be found by considering the group elements. If two groups lead to the same crystal surface then that being a sub-group of the other one has to be rejected. Results of this simple checking are presented in Table 1. With the correspondence between crystal surface and its point group established, the tensor forms can be obtained immediately e.g. from Table 4 of reference 5. For convenience we reproduce in Table 1 the forms of first rank axial and second rank polar tensors. The absence of any tensor component in Table 1 means that this component is zero. The orientation of coordinate axes with respect to the crystallographic directions is indicated if material only. We do not include the group 1
Table 1
Point group
4mm 3m 3m mm2 mm2 m m m
Lattice and surface
Cubic Cubic h.c.p. Cubic h.c.p. Cubic Cubic h.c.p. I
(001) (111) (0001) (011) (1100) (hk0) (hk—k) [0001]
First rank axial tensor
Second rank polar tensor
Orientation of coordinate axes
z
xxyyzzxyyx
x
0 0 0 0 0 z z z
xx xx xx xx xx xx xx xx
[001] [111] [0001] [100] [01—1] [011] [0010] [0001] [1100] [001] [011] [0001]
971
xx xx xx yy yy yy yy yy
zz zz zz zz zz
zz zz zz
0 0 0 0 0 xy xy xy
0 0 0 0 0 yx yx yx
y
z
972
TENSORS FOR CUBIC AND HCP FERROMAGNETS
in Table 1 as this group does not simplify tensors, but it has to be noticed that any crystal surface different from those given in Table I corresponds to this group. Thus the form of surface anisotropy tensors can be read directly from Table 1, the only particularization seems to be the symmetry of second rank tensor.
Vol. 18, No. 8
Finally, we would like to stress that our present results are in general not applicable to a ferromagnetic crystal with a surface stratum of different, e.g. antiferromagnetic, character. We shall analyse such systems in a separate paper.
REFERENCES 1.
YEJ J.T., TURK R.A. & WIGEN P.E., Solid State Commun. 14, 283 (1974).
2.
PUSZKARSKI H., Phys. Status Solidi (b) 63, Ki 15 (1974).
3.
WACHNIEWSKI A., Phys. Status Solidi (b) 66, K59 (1974).
4.
WACHNIEWSKI A. & BIEGALA L., Phys. Stalus Solidi (b) 72, K27 (1975).
5.
BIRSS R.R., Symmetry and Magnetism. North-Holland, Amsterdam (1964).
6.
KITTEL CH., Introduction to Solid State Physics, 2nd ed. Wiley, NY (1956).