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X-band FMR measurements: establishing of relations between the complex permeability and reflection factor
of ferromagnetic amorphous conductors
Emmanuel Saint-Christophe, René Sardos, Richard Barrue
To cite this version:
Emmanuel Saint-Christophe, René Sardos, Richard Barrue. X-band FMR measurements: establishing of relations between the complex permeability and reflection factor of ferromagnetic amorphous con- ductors. Journal de Physique III, EDP Sciences, 1993, 3 (10), pp.2053-2058. �10.1051/jp3:1993258�.
�jpa-00249063�
Classification Physics Abstracts
75.50 76.50
X-band FMR measurements : establishing of relations between the complex permeability and reflection factor of ferromagnetic
amorphous conductors
Emmanuel Saint-Christophe, Rend Sardos, Richard Barrue (*)
Laboratoire de Physique Exp6rimentale et des Micro-Ondes, 351cours de la Lib6ration,
33405 Talence Cedex, France
(Receii~ed 4 December J992, accepted J3 July J993)
Abstract~ A new calculation for the determination, at ultra-high frequencies, of the complex permeability variations of ferromagnetic amorphous conductors has been developed and gives the possibility of using directly the measurements obtained by ferromagnetic resonance. The equations
set out and reported combine surface impedance relation, boundary conditions and variation studies of the magnetic field dependence of the reflection factor in the case of ultra-high frequencies. The measurements at 9 GHz have been led with an X-band three-wave interierometer of great accuracy with based-cobalt (Vitrovac) and based-iron-boron (Metglas) amorphous
conductor samples.
1. Introduction~
The wide range of applications of amorphous ferromagnetic materials has aroused a growing
interest. Normally~ metals more than a few ~Lm thick are opaque to microwave radiation and since it is the case with amorphous materials, their FMR studies at ultra-high frequencies must be led by reflection. Because the microwave is reflected on the sample, the data recorded (as a
function of an extemal applied magnetic field) show the relative amplitude variations
Ar/r and the phase variations Ap of the complex reflection factor p
= reJ~ (Fig, I).
The aim is then to link those variations with various magnetic factors, particularly the relative magnetic permeability of the material : p
= p' jp ". Here we set out the theoretical calculation that we settled [I] and which links up directly Ar/r and Ap with respectively
AR" and AR '.
(*) Present address L-E-S-I-R-, E-N-S- de Cachan, 61 avenue du Pr6sident Wilson, 94235 Cachan Cedex, France.
2054 JOURNAL DE PHYSIQUE III N° lo
__AY
Fig. 1. Schematic shape of Ar/r and AT variations on FMR reflection measurements of amorphous conductor alloys. Arbitrary units.
2~ Experimental details~
The measurements at 9 GHz of the reflection factor on amorphous conducting materials have
been performed with a three-wave interferometer worked out by Sardos [2] which led to
numerous transmission investigations on other materials [3-5]. We modified this three-wave
interferometer operating in X-band to work by reflection. In addition, a circulator and a rectangular-to-circular wave guide transition introduced before a TE~j rotary joint were
inserted in order to turn the amorphous sample around its axis.
Thus the apparatus enables one to measure separately the weak phase and relative amplitude
variations of p and to conduct their uninterrupted recording with respect to an exterior variable acting on this sample. In our case the exterior variable is a magnetic field. The apparatus is of great accuracy since it enables one to measure values Ar/r m 2.5 x lo ~~ and Ap m 5".
The samples which are investigated are discs 30 ~Lm thick and 24.75 mm in diameter. They
are cut out of amorphous ribbons obtained thanks to the Planar Flow Casting process. The
sample is placed in the xoz plan seeing that the exterior applied static filed H- is taken along
Oz axis and microwave radiation has its direction of propagation along Oy I-e- parallel to the
sample normal. Its electric field e, being along Oz and its magnetic field h~ being along
O-r-
3~ Theory~
Let us consider the definition of the surface impedance of a material given by
le-
Z
=
" (I)
hi
y=0
From Maxwell's equations and the boundary conditions one shows [I] that :
k
~ l~~j~~ l( @1 ~ @~)~~~ ~J (I@1 @ ~)~~~l (2)
Z
= l~° with p* = j«/w p~ = je" (3)
po e
where k is the wave vector of the electromagnetic microwave,
« is the conductivity of the
specimen and e * its complex relative permittivity which is an imaginary value in X-band wave
length range [6, 7]. Moreover po and e~ are respectively the vacuum magnetic permeability
and the vacuum magnetic permittivity.
Z Z~
The reflection factor is given by the following complex relation: p
= ~
where
+ Zo
Zo is the vacuum impedance.
Let us assume that jp = p " + jp'
= p eJ~ and let us consider the equation (3), thus we obtain
~
~ ~~~ ~j ~~ ~~~
~ fi
~~ ~ p j ~~~
~ °~
2 ~ ~ ~ °~~"2
Because p
= r eJ~, the equation (4) will permit us to establish both values of
r and p.
CALCULATION oF
i~. The relation (4) gives for r :
e"+ (p( -2fi~cos)
r~
= (5)
e"+ (p( +2 fi~cos ~
2
Since p cos~ ~
= p + p "), and since e" » p in this kind of study e in X-
2 2 '
band wave length and for a metal [8], one has
r~ fi
~
(6)
One can approximate the relation (6) to the first rate limited development :
r2
=
2 j)
p + p ~~)i/2 (7)
CALCULATION oF q. One multiplies the relation (4) by the conjugate quantity of the
denominator and if one chooses the tangent expression it leads to :
~~ ~
~
/
~"~~~
E /l
having the same considerations as before and noticing that p sin~ ~
= p p ") one
2 2
arrives in consequence at :
tg v~
=
j (I
p p ")~/~ (9)
Yet, the calculation of the reflection factor without any-exterior magnetic field, following
the method we worked out [9], shows that with high reflecting samples, the phase at
2056 JOURNAL DE PHYSIQUE III N° lo
H~ = 0 is very close to gr q~~ ~ gr. Thus, one can write tg q~ = tg (gr + ~') with
~' pratically equal to zero. Therefore :
tg ~ = tg ~~
m w~ (lo)
Considering (lo) and multiplying (7) by (9) one obtains
Substracting (7) from (9) and still having (lo), one obtains :
These two very important relations allow us to directly obtain the values of p' and
p " as a function of r and ~' and then of
r and ~, (~
= ~'+ gr ), of the reflection factor P.
STUDY OF THE VARIATIONS AND DISCUSSION. The reflection measurements, we carried out
at 9 GHz with the X-band three-wave interferometer, have variations as shown in figure I.
Let us consider that
p'(0), p "(0), ro and ~o are respectively the permeability complex, the amplitude and the
phase of the reflection factor in zero exterior magnetic field I-e- H~ = 0; and p'(H),
p "(H), r~ and ~~ when this magnetic field is H worth.
Let us assume that
p "(H p ~~(0
= Ap ~~
p'(H)-p~(0)
= Ap~
r~-ro=Ar and ~H-v~o"V~(-V~i~A~.
Therefore
pi ~,,
=
(I (ro + Ar)~) (~o
gr + A~ ) (13)
4
Pi ~ ( (i (~0 ~ ~)~)~) (~0
" + A~ )~ (14)
"
Considering (13) and (14), we hence can directly obtain the values of p' and p ", for each value of H, as a function of the variation measurements of amplitude and phase of the reflection factor and its measured value in H~ = 0.
In most of the cases, the relative variations of amplitude and the phase variations of
~ are at the rate of 10~ ~ to 10~ ~ [l], so let us see what one gets with the relations (13) and (14).
The relation (13) leads to
~,, ~,,
~~'
4 ~~ ~~° ~ ~~~~~ ~~~ ~ ~~
4 ~~ ~~~ ~~' ~~~~
Since i-o
m I within 10~ ~ (the samples are very reflective) and because ~~ is of the order of i-o
10~~ to 10~~
one thus has
2 ro Ar « I and (Ar )~ « l
Therefore, one is allowed to write AR' ~,,
= (I r() A~ (16)
4 On the other hand, the relation (14) leads to
AR"
=
[ (i (ro
+ Ar)~)2 (v~i + V~)2) [ (i ri)2
~12) (17)
On the measurements we get, A~ is of about 10~~ to 10~~ rad, then (Aq~ )~ is of about
10~ ~ to 10~ ~ rad. Considering the approximation made with q~( already expounded above and
neglecting the terms of the second order for Ar as we did above, one thus has the expression of
AR "
AR " ~,,
~ j (I r~) r0 Ar (18)
To sum up, we get the two following very important relations :
AR'
=
~ (l r() Aq~ (19)
AR " ,,
~
( (r( I
ro Ar (20)
We can also notice that
Ar/ro i A~ ',
= m ~. (21)
Aq~ rj AR
Thus, when we plot the data of Ar/ro as a function of Aq~, we plot within the coefficient
16-j, AR " as a function of AR'. Because we have high reflective materials (ro
~ ) we can
admit that the figures Ar/ro = f(Aq~ ) directly obtained and which have a circular shape, correspond (within a few percent) to those of AR " as a function of AR ' and then allow us to see
very quickly the shape of AR "
=
f (AR'). Because the shapes obtained are circular, one can say that we are in the case of Lorentz type curves. An example is given in figures 2 and 3
/~X /~?", A~~ Ay[
(lo")
, ~ ,
' ~
14,0 '
,
200
o 50
. +
.
» +
.
* so
0
. * 20
h '
~~so
2. Fig.
ig. -
The observed
cobalt alloy oio(Femo
Lorentz type curve and
A~'.
Fig. 3. -The observederromagneticsonance on the same
ased-cobalt alloy
oiolfemo)zmn~(BSi)z~ is shown with the xtreme variations A~[, A~l and A~l. The values of
A~'
2058 JOURNAL DE PHYSIQUE III N° lo
Table I.- p' and p" variations as a function of H with the same based-cobalt alloy
Com(Femo)~Mn5(Bsih~ example. Those data can be dir-ect/y obtained from the FMR measurements w,ith the relations (19) and (20).
m 0 ,1 ~29 10~3 12~ 14.0 .7 18. 25.2
~~, 0 -24.3 -38,1 -57~2 -25.7 73.8 198 186 21.4
~~» 0 8.02 29.4 103 189 231 134 71.4 157
according to measurements we carried out at 9 GHz with a based-cobalt amorphous alloy Com(Femo )~Mn5(BSi)~~ [l]. The values of some variations of AR' and AR" are reported in the table I for the same alloy [1].
4~ Conclusion~
These three relations we have established from the initial conditions of our experiments give
the great advantage of being simple in terms and of linking directly both physic terms
permeability, and the reflection factor of conductor ferromagnetic materials and amorphous in
particular.
Thus it is possible to register directly from FMR measurements with an X-band three wave interferometer, the variations of p " and p' of amorphous conductor ferromagnetic materials and then to know, within an X-band, the complex values of p as a function of H.
References
[1] Saint-Christophe E., Thdse Bordeaux (1991) 62-91.
[2] Sardos R., Rev. Phys. App/. 4 (1969) 29.
[3] Sardos R., Escarmant J. F., Saint-Christophe E., IEEE Trans. Microwave Theoiy Tech. 38 (1990) 330-333.
[4] Ramdani Y., Thkse Bordeaux j1989).
[5] Sardos R.~ Saint-Christophe E., Barrue R., Perron J. C.. Bigot J., 10~Co110queOHD (1989) pp. 357-360.
[6] Born M., Wolf E., Principles of optics (Pergamon Press, 1975).
[7] Mercouroff W., La surface de Fermi des mdtaux (Masson, 1967).
[8] Landau L.. Lifshitz E.. Electrodynamique des milieux continus (MIR, 1969).
[9] Sardos R., Escarrnant J. F., Saint-Christophe E., 10~ Colloque OHD (1989) pp. 175-178.
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