The determination of the first three magnetic anisotropy constants of NiCr alloys

Solid State Communications, Vol. 17, pp. 59—61, 1975

Printed in Great Britain

Pergamon Press.

THE DETERMINATION OF THE FIRST THREE MAGNETIC ANISOTROPY CONSTANTSOF NiCr ALLOYS K.J. Bowker and M. Heath Department of Physics, University of Nottingham, University Park, Nottingham NG7 2RD, England (Received 15 February 1975 by R. Loudon)

The first three magnetic anisotropy constants, K1 , K2 and K3 of a series of single crystal samples of NiCr alloys have been measured by means of ferron~agneticresonance. Results have been obtained in the temperature range 4 K to room temperature and agree well with what would be expected from an extrapolation of the values for pure Ni. 1 we reported some preIN A RECENT liminary resultspublication obtained by means of ferromagnetic resonance from a series of single crystal samples of NiCr alloys. The compositions were in the range 2—9

above equation gives the following general resonance condition. (12

per cent Cr by weight. Measurements were made at a microwave frequency of 9.40 GH~Zand at temperatures in the range 4°Kto room temperature. The samples consisted of discs of approximately 5 mm diameter and thicknessbetween 0.5 and 1.5mm, the plane of the disc beingsample parallel to the (110) plane.[110] Thus and the plane ofthe contains the [100], [Ill] axes. More experimental details are given in the above publication.’ Further analysis of our data

=

{H cos 6 + (4ir

x {H cos 6

—

—

n)M + f(K

1 , K~, K3, ~)}

nM + g(K1 , K2, K3, ~)}

where n is the demagnetizing factor, K1 2Ø + 3 cos4~) f(K1 ,K2, K3, ~) = 2 + cos K 2 ~(2 + 3 sin2 ~) 2 2 ~ cos —

2MnII

allows us now to present values of the first three anisotropy constants K1, 1(2 and K3. The appropriate resonance conditions can be derived from the general resonance condition of, for 2 example, Smit and 2 1 WijnIfa2F\fa2F\ /a2F\2l ~y)

=

M2sin28

~

2 20 + sin40)(— 2 + cos2 ~ + 3 cos~) +

+ ~(sin2 20 + sin4

=

g(K

1 , K2, K3, ~) x (1—l1cos2.~+12cos4Ø)

(~~) i

0 [~i-) where 0 and ~ are the polar and azimuthal angles, F is the free energy and Oo is the equilibrium orientation of the magnetisation M. The free energy due to magnetocrystalline anisotropy is normally written in the form = Kj(a 1, a2, a3) where a1 are the direction cosines ofM relative to the cube edges, which in their turn can be related to 0 and If FK is restricted to the first three terms and differentiated, substitution into the —

(sin

2O—3l cos40+ 18 cos6Ø)

—~-(— I + l4cos 2M +

(sin 4~+ 2 sin30 cos 2M

=

equilibrium orientation ofM measured from the

[100] axis in the plane of the disc, OH = orientation of H measured as above, 6 = angle between M and H,

~.

59

60

MAGNETIC ANISOTROPY CONSTANTS OF NiCr ALLOYS

Vol. 17, No. 1

T°k 100

— sinô —

-~-~

(zK1+K2sin2o+K3(l+2cos2o_3cos4o)~

/

)

4ffiLJ

2Ø—1).

;/~

,“‘ .-I0

x sin 20(3 cos The technique of the experiment is to rotate the applied field about the sample keeping it in the plane of the disc and note the resonance fields for vanous orientations. The general equation simplifies to the following resonance conditions for the specified axes

_—~-~-

~

300

/

/ //

o2’7C~. X4/,Cr •6°/~Cr

-: erg cm Ni (FranCe)

[100] axis (hv\2 2K

2K

[Hloo+(4lrn)M+H

FIG. 1.K 3 for NiCr alloys.

[Hioo_nM+-~]

Table 1. Values ofK1 and K2 for the range ofalloys

4 K1 x10 (ergcm3)

K 4 2 x i0 (ergcm3)

T°K

Ni

2% Cr

4% Cr

6% Cr

8% Cr

295 195 77 51 20 14 4

—7.0 —25.0 —84.0 —103.0

—1.4 —7.6 —27.1 —30.8

—

—

—0.2 1.5 —8.9 —11.1 12.9

0 —0.2 —2.4 —3.1 —4.1

—

—

—

—4.3

—

—

—4.9

0 —0.170 —0.23 —0.66 —0.70 —0.87

295 195 77 51 20 14 4

—2.8 —10.4 + 7.6 + 21.0

0 —0.07 0.5 —0.3 —2.1 —1.5 +3.5

0 0 —0.47 —0.62 —0.46 —0.96 —0.12

—

119.0

—

—

—

—0.2 —0.7 + 0.4 +2.2 +3.2

—

—

—

—

—

+30.0

[110] axis

—1.1 +0.1 + 5.8 —

[Ill]

(/w~2 =

—

axis

[H 110+ (4ir

x

—

[H11O

—

n)M— ~ M

_nM+~+~+~]

2

M

=

\gI.LBJ

[H111+(4~_n)M__~~_~] X

1LHhh1~_~~ 4K1 4K2

8K3

where n is the demagnetizing factor in the plane of the disc. A further resonance may be obtained with the field perpendicular to the plane of the disc.

Vol. 17, No. 1 ~hv~ 2 =

MAGNETIC ANISOTROPY CONSTANTS OF NiCr ALLOYS

[~t1+ (4ir

{H~+ (41T

—

—

2n1)M— 2

2n1)M + M + ~

61

MM]

compared with the corresponding values for Ni. Figure 1 gives the results for K3 and once again

+

this curve the is the for firstpure timeNisuch is included. data haveAsbeen far as obtained we know by

—

However we have five unknowns, g, M, K1, K2 and K3. A fifth condition is obtained by numerically integrating the experimental curve of resonance field against orientation,this between thenumerical [100] andintegral [110] of axes comparing with the theand general resonance equation for any angle, summed over the same range.

means of ferromagnetic resonance. A few determinations have been made of the first three anisotropy constants of the pure elements by means of torque 4 butmethods, the present for example by Franse and De Vries, method has the advantage that g and M values are obtained at the same time and also the method is possibly simpler. The iterative method of solution of the resonance equations indicates that the absolute

Table I presents data for K 1 and K2 for the alloys, For comparison purposes data is included for pure Ni 3 In our original communication, estimated from Franse. correct allowance had not been made for K 3 having finite values. When this is done consistent values are obtained for K2. As can be seen from the table the variation of K1 and K2 with temperature and composition is very much what one would expect when

values of K2 and K3 are very dependent on the values of g, M and K1.15Thus they On are probably more accurate than per cent. the other no hand the values ofg, M and K1 are changed by a negligible amount if K3 is assumed to be zero, but the corresponding value of K2 must be treated with caution as it will contain contributions from K3 and higher order terms.

REFERENCES 1.

HEATH M. and BOWKER K.J., Solid State Commun. 15,93 (1974).

2.

SMIT J. and WIJN H.P.J.,Adv. Electron. 6,70(1954).

3.

FRANSE J.J.M.,J. Phys. 32, Supp. C. 1, 186 (1971).

4.

FRANSE J.J.M. and DE VRIES G., Physica 39, 477 (1968).