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Faraday effect and the effective field theory in solid materials
Journal
of Magnetism
and Magnetic
Materials
96 (1991) 155-161
155
North-Holland
Faraday effect and the effective field theory in solid materials Liu Gong-qiang CCAST (World Laboratoryl, P.0. Box 8730, Beijing 100080, China and Department of Applied Physics, Shanghai Jiaotong University, Shanghai 200030, China
Yu Zhi-qiang and Le Zhi-qiang Department of Applied Physics, Shanghai Jiaotong University, Shanghai 200030, China Received
26 May 1989; in revised form 14 September
1990
Having used the classical electromagnetic field theory and combined it with the effective-field conception, in this paper we show that: 1) The real part 19of Faraday rotation $I in any kind of the magneto-optical (MO) material can be expressed in the fundamental form of 0 = VLH,. 2) The differences in the MO properties of the different kinds of the MO media mainly result from both the characteristics and the size of their effective field H, and especially, the different temperature properties in various media are discussed. 3) The relationship between the real part 0 and the imaginary part 4 of + can be described by the effective extinction coefficient n*. 4) The other MO properties are predicted. It is proved that the effective field H,, which is concerned in both the spin-orbital interaction and the exchange interaction (or the indirect exchange interaction) in media, can be expressed in the form of H, = vM (or L:f= ,v,M,). Some theoretical results are verified by experiments.
1. Introduction Magneto-optical (MO) effects result from the interaction between light waves and media with a magnetic moment (containing the induced moment). Thus, to describe the MO effects by the classical theory, the Maxwell equations describing the movement of light waves must be applied. Otherwise, it will be necessary to use equations or physical quantities describing the characteristics of media, such as the Lorentz equation of electron movement [l] or the dielectric permeability tensor ;[2.3]. On the basis of the band theory of metals and the electron wave functions related to the spin-orbit interaction, Argyres [3] once discussed the Faraday and Kerr effects in ferromagnetics by evaluating the polarizability and conductivity tensors. But up to now, since only the external magnetic field H, is considered and various effective fields are not counted in the discussions on the basis of the Lorentz electron movement equa0304~8853/91/$03.50
0 1991 - Elsevier Science Publishers
tion, the different characteristics of the MO effects in the different magnetic media are not explained satisfactorily by the solution resulting from both the Lorentz equation and the Maxwell equations. In the strict sense of the word, the results obtained are not applicable to ferromagnetic, ferrimagnetic, antiferromagnetic and even some paramagnetic media. They cannot explain the behaviour of 9 with temperature, stress and wavelength in the media. Practically, in the ferromagnetic, antiferromagnetic and ferrimagnetic media, besides the Lorentz force acting on the moving electrons due to an external magnetic field, the more important force acting on the electrons is from the effective fields related to the spin-orbital interaction, the exchange interaction (or the indirect exchange interaction), the magnetocrystalline anisotropy, and so on. In some paramagnetic media, the effective field related to the exchange interaction is a factor which can not be negligible. In weak magnetic
B.V. (North-Holland)
156
G.-q LIU er al. / Faraday effect and effective field theor?:
media. especially in the diamagnetic media because the effective fields related to the spin-orbital interaction and the exchange interaction approach zero, the force acting on the electrons from the external magnetic field and the effective field related to the stress anisotropy would be principal. In this paper the effective fields, especially related to the spinorbital interaction, are discussed in detail. At the same time, solving both the Maxwell equations and the Lorentz equation while taking the effective fields into account, we have obtained the definite expression + in relation to M, T and A. Therefore some more significant conclusions have been obtained.
2. The basic magneto-optical
relations
anisotropy through the spin-orbital coupling. Hd is the demagnetizing field, and its magnitude is related closely to the anisotropy on the form of the medium. h is the unit vector in the direction of H,. When eq. (1) is multiplied by Ne/m, using the polarized vector P= Ner, the following equation is obtained:
= (Ne’/m)(
E + P/36,,),
where the damping coefficient y = g/m, N is the electron number per unit volume. Suppose that the incident light is linearly polarized. From eq. (3) combined with the Maxwell equations, we can obtain the following relationships, s*(a’P+ic,,/3PXh)
In media the equation ment is as follows:
of the electron
(3)
=O,
(4a)
moves.H=O,
(4b)
(n/p.,,c)[s~
(aP+i/?P~h)]
=H,
(4c)
-(n/c)(s~H)=a’P+ic,,~Pxh, (1) On the right side of the equation the first term represents the force of the positive charge centre acting on the electron; o0 is the intrinsic frequency of electron movement. The second term signifies the force of the local electric field acting on the electron in the medium. The third term denotes the damping force imposed on the electron in the moving process. The last term stands for the force acting on the electron by the effective field H, H, = He + H,, + Ha + H, + Hd + . . .
(2)
where He is the external field, H, the effective field concerned in both the spin-orbital interaction and the (indirect) exchange interaction, Ha the effective field related to the magnetocrystalline anisotropy, which comes from the anisotropy on the crystal structure, HA the effective field related to the stress anisotropy, which results from the external stress and the inside residual stress in the medium, and its magnitude is concerned in the magnetostriction. The orbit-motion of the electron is indirectly influenced by the (indirect) exchange interaction, the magnetocrystalline and the stress
where s is the unit vector wave vector k, and 7
(Y=
a’ =
in the direction
of the
7.
w; -
cd- -
Ne2/m c&r
(4d)
+
1yw
1 3r,, *
(6)
1,
/3 = p. Hiw/Ne.
(7)
Eqs. (4) are the basic relationship effects and the classical theoretical which all kinds of the MO effects handled.
3. The effective
(5)
for the MO basis upon can be duly
field H,
Generally speaking, the MO effects principally result from the spin-orbital interaction of the excited states and the (indirect) exchange interaction between spins of electrons, and these effects are mainly based on the former in some media, and on the later in the others. Hulme [4] and Argyres once indicated that the spin-orbital interaction can, in a certain approximation, be thought
G. -4. Liu et al. / Faraday effect and effective field theory
of as an effective magnetic field on the motion of the electrons. Now it will be proved that the interaction can be equivalent to the effective field H, = VM (in ferromagnetic medium) or H, = C,v,M, (in ferrimagnetic and antiferromagnetic media) acting on the orbital electrons. The spin-orbital interaction Hamiltonian is
157
Supposing that there are I sublattices in the antiferromagnetic and ferrimagnetic media, we have the expression of the effective field which is concerned in the spin-orbital interaction on every sublattice, + q2M2 + . . . + v,,M,,
H,, = v,,M,
+ . . . + v2,MI,
Hu2 = vz,M, + v,,M, HLS=[‘L*S=SmL*m,,
(8)
where { = (2m*c*/e){‘. Assuming only one electron for every atom contributes to the magnetization, we can consider L, S, mL and m, in eq. (8) respectively as the orbital moment of momentum, the spin moment of momentum, the orbital magnetic moment and the spin magnetic moment of the atom. In ferromagnetic medium,
where vjk( j, k = 1, 2, 3,. . . , I) is the coefficient concerned in the spin-orbital interaction between the jth sublattice and the kth sublattice. The effective field concerned in the spin-orbital interaction at any site in the medium is H, =
Usually the spin-orbital interactions between the atoms can be neglected. Assuming that all atoms have the same interaction coefficient 5, that is, and considering the magnetic moment 5;j = 16i,* of the atom m, = mL, + m,,, we can write eq. (9) as = 5 Cm,,*mi (
- CmL;mL, i
i
1
.
(10)
In most of the ferromagnetic and ferrimagnetic media, the second term in the right side of eq. (10) can be neglected in first approximation. In the case of saturation, m, = M,/N, and M, is the saturated magnetization. Then H Lqsat) = vMs-CmL,9
(11)
i
= vM.xm,,,
K$,,*
+
c
. . . +
K,&
1 ~~v$bf,
+
i=l
c
1 ~,v,@f~
+
. . . +
1=1
=vlM,+v,M,+
c
K,v,,M,
i=l
. ..+v.M,.
04)
It is not difficult to prove that the effective field related to the (indirect) exchange interaction acting on the orbital electrons could be expressed as v’M in ferromagnetic medium and Cl= ,v;‘M, in antiferromagnetic and ferrimagnetic media. But here we should emphasize that v’M or Ci=,v(M, is not the molecular field but the effective field concerned in the molecular field. Based on the formal resemblance, two kinds of the effective fields above can be merged as follows H, = vM (ferromagnetic), H, = i
v;M;
(15)
i=l
where v = l/N. When H, = 0, the magnetic domains in medium are arrayed disorderly, and eq. (10) can be written as HLs = O*C,m,, = 0. Therefore, in general, eq. (10) can be approximately expressed as H,,
+
1 =
H,,
K,&
(9)
HLS = c C S;jm,, - ms,. ’ j
03)
**., HvI = v,,M, + q2M2 + . . . + v,,M,,
(12)
i
where vM is the effective field of the spin-orbital interaction acting on the orbital magnetic moment.
(antiferromagnetic
and ferrimagnetic).
4. The Faraday effect For the Faraday effect s 11h, assuming axis, from eqs. (4~) and (4d), we get
h 11z-
AP, + i BPY = 0, -iBP,
+ APY = 0,
(16)
G. -4. LIU et al. / Faraday effect and effective field them
15x
where,
Using eq. (24), we obtain B = ,8( n*/‘/.&
A = an2/poc2 - a’,
In the condition of the nonzero (16) it can be obtained that
(17)
-q,).
solution
of eqs.
(18)
A = -B,
P,.=
-iPX,
E,.=
it is obtained
from
eqs. (16)
-iE,.
(19)
This is the right circularly polarized light and its index of refraction is n,. From eqs. (4) (17) and (18) it is obtained that 2
n+-
Ne2 3r,m
z z % - w -lyO-
e&G + m
E,,=iE,,
(21)
2 cd” - 0
--Ne*
2
3e,m
-lyw-
ep.,H,w m
’
(22) Let wL = ep,Hi/2m, and when H, < 1/4a X 10”’ A/m (10’ Oe), wL -=Kw. If the damping term is neglected, it may be easily found that z n+ -1 p= n:+2 n’-=
FL0Ne2c2/3 ao’ -&J*+--1
n2+2
Z
+oH,a
[
2m(w:
2 oo-w
2
&-(o-wL)*’ =
ePoHio -~
1+ [
- Ne’/3e,,m)
+1 I
, w2( wi - Ne’/3c,,m) 304
+
2 + ...
8( C& - Ne*/3c,,m)
+ c//h4
+ ...
n+-
n_=
-2d!&_ dw
p.,Ne2c2/3
m
2x0, = -w
Eq. (28) is the well-known Cauchy experimental formula. Substituting eq. (28) into eq. (26). we obtain the Verdet constant in the normal dispersion region, V=(e~Jmc)(h/A2+c/A4+
where h = -h’
19= VL(H,+
dn dX’
(28)
. ..).
(29)
and c = - 2~‘.
In the diamagnetic media, magnetization M is very little, and the magnetocrystalline anisotropy is very faint. For eq. (2) Hi = (H, + H,,), so from eq. (25) it is given that
w~-(~+w~)*’
as the function
(27)
wheren=$(n++n.).Aso-l/X,theaboveexpression can be rewritten as
H,).
(30)
(23) Being considered
+const, I
4.1. In the diumagnetic mediu
poNe2c2/3
m
puNe2c2/3
(26)
1 ,‘2
n = a’ + h’/A’
Ne2c2/m
rl* -I=
(25)
dn/dX.
Ne”c’
x
When A = B, for the same reason, the left circularly polarized light and its index of refraction n- can be given respectively by
VLH,,
is the Verdet constant, h the light wavelength in vacuum, and L the propagated distance in the medium, H, the prcjection of H, in the direction of the light propagation. For the region where w is far from w,), i.e. the normal dispersion region, from eqs. (20) (22) and (24) it could be gotten that
(20)
P,,=iP,,
)=
where
‘=
Ne ‘c’/m
1=
-n
V = (ep,,X/2mc)
A=_tB.
When that
0= $$(n+
4.2. In the paramagnetic
of (w f We), (24)
media
In some paramagnetic media there is a weaker exchange interaction between most neighbouring
G. -4. Liu et al. / Faraday effect and effective field theory
electron
spins;
then H, can be written
as
H, = vM = VXH,.
Here, the indexes of the refraction and left circularly polarized light are (31)
The relation between susceptibility x and temperature T conforms to the Curie-Weiss law x = c/( T - T,), where c and T, are the Curie constant and the Curie temperature respectively. When the influence of H,, HA and Hd can be neglected, eq. (25) can be expressed as 8 = I%( H, + Hv) = v,LH,,
(36)
Let eqs. (20) and (22) be written
(n,++
in,+)‘=
1 + (I&“+‘:;
+l(X+
4.3. In the ferromagnetic, ferrimagnetic media
and
antiferromagnetic
In these media, compared to H,, effective fields in eq. (2), especially neglected. Eq. (25) is also suitable antiferromagnetic and magnetic, media at H, d (4a)-’ x 10” A/m. H,, H,, HA and Hd, 8 is expressed as
the other four H,, cannot be for the ferroferrimagnetic When H, x= approximately
13= Vf vM (ferromagnetic),
e=
(37)
,x~~,r’,2
wz
where W = poNe2c2/m, X = ~0’ - u2 - Ne2/ 3r,m, Y = e/.loHiw/m, Z = yw. According to the definitions of B and 4, and from eqs. (37) it can be obtained that n,B-n,#=
(nL/2h)(A’-
n,# + n,B = (TL/~A)(
B’),
(35)
5. The magnetic circular dichroism The expressions for 8 in section 4 are obtained while damping is neglected, so that 8 is only the real part of the Faraday rotation. Next, the relationship between 8 and # (the imaginary part of Faraday rotation or the magnetic circular dichroism), and their characteristics will be described.
and
+ n:)]
X[n,(A’-B’)+n2(D’-c’)], $J= [L/2X(n:+ni)]
and ferrimagnetic).
(38)
D’ - C’),
where n, = t(n,++ n,_), n, = :(n,++ n,_), then B and $J can be written as follows,
e = [ L/2X(nf (34)
VL C v,~, I=1 (antiferromagnetic
Y)2+z*
(33) Eq.
zZ2
=A’+iC’, (v+%)“=l+
where G = V(vc - Tc)/c, R = l/(vc - T,). (33) has been confirmed by experiment [6].
as follows
wz
(32)
= G(1 + RT),
of the right
n* = n,*+ in,+.
where V, = V(1 + vx) is the Verdet constant in some paramagnetic media. It can be written as v,/x
159
(39
X[q(D’-C’)-n,(A’-B’)]. Generally, N - 1026-1027, w - 1015 s-t, y -=z w. It is supposed that Hi Q (4~)~’ X 10” A/m, and then Z +z A’, B’ and C’, therefore, (D’ - C’) in eqs. (39) can be neglected. From this it can be found that
e/G = -n,/n,
= l/n*.
Here n* is called the effective extinction cient. From eqs. (25) and (40) we obtain $ = vLq*H,.
(40) coeffi-
(41)
G.-q. LIU et (11./ Faraday effectund effectrue field theory
160
0 (deg)
0
100
200
300
-3000 j I5
I1
-
\
12
.
\
i
theoretical
2
Ndf3 -5000[ v/k
experimental
9i\ 6,
I
31
300 1
‘l
0 II_-. 0.6
a..-._,
1.1
1.8
-flOOO.Ly/~
&urn)
Fig. 1. 8-X dispersion curve for (BiTm),(FeGa),O,, the thickness 27.50 pm.
6.
200
100 -‘nnnP
;\
film with
PrF3
Fig. 3. Ratio of the Verdet constant and the magnetic susceptibility as a function of temperature for NdF, and PrF,.
Discussion
From the calculation above, we discuss as follows: (1) In all of the MO media, the real part of the Faraday rotation is 0 = VLH,. In both the normal dispersion region (dn/dX < 0) and the abnormal dispersion region (dn/dX > 0), the signs of 8 are opposite to each other. In the normal dispersion with the region, 8 is satisfied by the relation optical wavelength X as B - (6/X2 + c/A4 + . . ).
I3
(
dr,:
Using the magneto-optical rotation spectrometer [5] (measured accuracy 0.01” ) we have measured /3 - X in the case of the garnet-film sample being magnetically saturated. The sample is a (BiTm),(FeGa),O,, film grown on GGG substrate by the liquid-phase epitaxial method. The experimental results are shown in fig. 1. When H, and M are fixed, from eqs. (29) and (35), 0 = (&/A2 + co/A4 + . . . ). If b, = 1.9 and cg = 1.6, then the theoretical results agree with the experimental results.
e
) O-
(
deg
)
o-
o-o-0
2.0
l
1 .o
-o-
r!a~“~to4ptlcnl
-a-
Elllptlcity
rbtat1on
H,( 0~
I Fig. 2. The real part
B and the imaginary
part of the Faraday rotation vs. field strength thickness 2h = 27.50 pm at X = 0.6328 pm.
)
0
H, for (BiTm)3(FeGa)S0,,
film sample
G. -4. Liu et al. / Faraday effect and effective field theory
‘0.6 - 0.4
- 0.2
0.5
1.0
1.5
2.0
A(m)
Fig. 4. Dispersion relations of 19and 4 for (BiTm),(FeGa)sO,, single crystal film with the thickness 2/z = 11.66 urn.
(2) The imaginary part J/ and the real part 8 of the Faraday rotation have many similar characteristics, i.e. both are directly proportional to the effective field Hi, and are saturated as M tends to saturation as shown in fig. 2. (3) The properties and sizes of the effective field Hi in different magnetic media are different. This is the primary factor that induces different MO characteristics. (4) In some paramagnetic media, 0 = ~~‘,LHi, and V,/x = G(l + RT). This has been verified by the experiments in NdF, (for T > 77 K) and in PrF, (for T > 40 K), and is shown in fig. 3 [6]. It is worth to mention that by eqs. (29) and (33) is shown that the coefficient G is closely related to the light wavelength X or the light frequency w but the coefficient R is not related to X. This has been experimently proved by Leycuras et al. [6] too. (5) In ferromagnetic media, the temperature behaviour of 8 is satisfied with B(T) - vM(T), and in antiferromagnetic and ferrimagnetic media, B(T) - Cf=,vi(T)Mi(T). These have been verified
161
by the experiment in ref. [7]. In antiferromagnetic media, Xi= iM, = 0, and usually Xi= iv,M, # 0 so that the large 0 may exist. (6) 8 would generally vary with the propagation direction of light in the crystal with the form of the medium and with the direction and size of the stress in the medium. In magnetic media the above factors which influence B are much weaker than H, so that it is difficult to measure. In diamagnetic and paramagnetic media the above factors such as the anisotropy in paramagnetic media usually are not negligible. All of these may be measured. (7) The difference between 4 and B can be described by an effective extinction coefficient n*. n* is not only related to the optical loss in medium, but is also a function of the frequency of the incident light. In the general case, the dispersion relationships of \c/ and B are different, but they are similar in the region in which \cI* is approximately not related to w. This has been proven by the experimental results shown in fig. 4. (8) Generally, n* is not closely related to temperature T, so the temperature behaviour of 4 and 8 are roughly similar.
References 111W. Pauli, Pauli Lectures on Physics, vol. 2, Optics and the Theory
of Electrons
(MIT Press, Cambridge, MA, 1973). Properties of Materials, ed. J. Smit (McGraw-Hill, New York, 1971) p. 152. [31 P.N. Argyres, Phys. Rev. 97 (1955) 334. [41 H.R. Hulme, Proc. Roy. Sot. London A 135 (1932) 237. 151G.q. Liu, X.X. Han and H.d. Song, China J. Sci. Instrum. 8 (1987) 360. 161C. Leycuras and H. Le Gall, J. Appl. Phys. 55 (1984) 2161. and K. Witter, Phys. Rev. B 25 [71 P. Hansen, M. Rosenkranz (1982) 4396.
121J.F. Dillon Jr., Magnetic
of Magnetism
and Magnetic
Materials
96 (1991) 155-161
155
North-Holland
Faraday effect and the effective field theory in solid materials Liu Gong-qiang CCAST (World Laboratoryl, P.0. Box 8730, Beijing 100080, China and Department of Applied Physics, Shanghai Jiaotong University, Shanghai 200030, China
Yu Zhi-qiang and Le Zhi-qiang Department of Applied Physics, Shanghai Jiaotong University, Shanghai 200030, China Received
26 May 1989; in revised form 14 September
1990
Having used the classical electromagnetic field theory and combined it with the effective-field conception, in this paper we show that: 1) The real part 19of Faraday rotation $I in any kind of the magneto-optical (MO) material can be expressed in the fundamental form of 0 = VLH,. 2) The differences in the MO properties of the different kinds of the MO media mainly result from both the characteristics and the size of their effective field H, and especially, the different temperature properties in various media are discussed. 3) The relationship between the real part 0 and the imaginary part 4 of + can be described by the effective extinction coefficient n*. 4) The other MO properties are predicted. It is proved that the effective field H,, which is concerned in both the spin-orbital interaction and the exchange interaction (or the indirect exchange interaction) in media, can be expressed in the form of H, = vM (or L:f= ,v,M,). Some theoretical results are verified by experiments.
1. Introduction Magneto-optical (MO) effects result from the interaction between light waves and media with a magnetic moment (containing the induced moment). Thus, to describe the MO effects by the classical theory, the Maxwell equations describing the movement of light waves must be applied. Otherwise, it will be necessary to use equations or physical quantities describing the characteristics of media, such as the Lorentz equation of electron movement [l] or the dielectric permeability tensor ;[2.3]. On the basis of the band theory of metals and the electron wave functions related to the spin-orbit interaction, Argyres [3] once discussed the Faraday and Kerr effects in ferromagnetics by evaluating the polarizability and conductivity tensors. But up to now, since only the external magnetic field H, is considered and various effective fields are not counted in the discussions on the basis of the Lorentz electron movement equa0304~8853/91/$03.50
0 1991 - Elsevier Science Publishers
tion, the different characteristics of the MO effects in the different magnetic media are not explained satisfactorily by the solution resulting from both the Lorentz equation and the Maxwell equations. In the strict sense of the word, the results obtained are not applicable to ferromagnetic, ferrimagnetic, antiferromagnetic and even some paramagnetic media. They cannot explain the behaviour of 9 with temperature, stress and wavelength in the media. Practically, in the ferromagnetic, antiferromagnetic and ferrimagnetic media, besides the Lorentz force acting on the moving electrons due to an external magnetic field, the more important force acting on the electrons is from the effective fields related to the spin-orbital interaction, the exchange interaction (or the indirect exchange interaction), the magnetocrystalline anisotropy, and so on. In some paramagnetic media, the effective field related to the exchange interaction is a factor which can not be negligible. In weak magnetic
B.V. (North-Holland)
156
G.-q LIU er al. / Faraday effect and effective field theor?:
media. especially in the diamagnetic media because the effective fields related to the spin-orbital interaction and the exchange interaction approach zero, the force acting on the electrons from the external magnetic field and the effective field related to the stress anisotropy would be principal. In this paper the effective fields, especially related to the spinorbital interaction, are discussed in detail. At the same time, solving both the Maxwell equations and the Lorentz equation while taking the effective fields into account, we have obtained the definite expression + in relation to M, T and A. Therefore some more significant conclusions have been obtained.
2. The basic magneto-optical
relations
anisotropy through the spin-orbital coupling. Hd is the demagnetizing field, and its magnitude is related closely to the anisotropy on the form of the medium. h is the unit vector in the direction of H,. When eq. (1) is multiplied by Ne/m, using the polarized vector P= Ner, the following equation is obtained:
= (Ne’/m)(
E + P/36,,),
where the damping coefficient y = g/m, N is the electron number per unit volume. Suppose that the incident light is linearly polarized. From eq. (3) combined with the Maxwell equations, we can obtain the following relationships, s*(a’P+ic,,/3PXh)
In media the equation ment is as follows:
of the electron
(3)
=O,
(4a)
moves.H=O,
(4b)
(n/p.,,c)[s~
(aP+i/?P~h)]
=H,
(4c)
-(n/c)(s~H)=a’P+ic,,~Pxh, (1) On the right side of the equation the first term represents the force of the positive charge centre acting on the electron; o0 is the intrinsic frequency of electron movement. The second term signifies the force of the local electric field acting on the electron in the medium. The third term denotes the damping force imposed on the electron in the moving process. The last term stands for the force acting on the electron by the effective field H, H, = He + H,, + Ha + H, + Hd + . . .
(2)
where He is the external field, H, the effective field concerned in both the spin-orbital interaction and the (indirect) exchange interaction, Ha the effective field related to the magnetocrystalline anisotropy, which comes from the anisotropy on the crystal structure, HA the effective field related to the stress anisotropy, which results from the external stress and the inside residual stress in the medium, and its magnitude is concerned in the magnetostriction. The orbit-motion of the electron is indirectly influenced by the (indirect) exchange interaction, the magnetocrystalline and the stress
where s is the unit vector wave vector k, and 7
(Y=
a’ =
in the direction
of the
7.
w; -
cd- -
Ne2/m c&r
(4d)
+
1yw
1 3r,, *
(6)
1,
/3 = p. Hiw/Ne.
(7)
Eqs. (4) are the basic relationship effects and the classical theoretical which all kinds of the MO effects handled.
3. The effective
(5)
for the MO basis upon can be duly
field H,
Generally speaking, the MO effects principally result from the spin-orbital interaction of the excited states and the (indirect) exchange interaction between spins of electrons, and these effects are mainly based on the former in some media, and on the later in the others. Hulme [4] and Argyres once indicated that the spin-orbital interaction can, in a certain approximation, be thought
G. -4. Liu et al. / Faraday effect and effective field theory
of as an effective magnetic field on the motion of the electrons. Now it will be proved that the interaction can be equivalent to the effective field H, = VM (in ferromagnetic medium) or H, = C,v,M, (in ferrimagnetic and antiferromagnetic media) acting on the orbital electrons. The spin-orbital interaction Hamiltonian is
157
Supposing that there are I sublattices in the antiferromagnetic and ferrimagnetic media, we have the expression of the effective field which is concerned in the spin-orbital interaction on every sublattice, + q2M2 + . . . + v,,M,,
H,, = v,,M,
+ . . . + v2,MI,
Hu2 = vz,M, + v,,M, HLS=[‘L*S=SmL*m,,
(8)
where { = (2m*c*/e){‘. Assuming only one electron for every atom contributes to the magnetization, we can consider L, S, mL and m, in eq. (8) respectively as the orbital moment of momentum, the spin moment of momentum, the orbital magnetic moment and the spin magnetic moment of the atom. In ferromagnetic medium,
where vjk( j, k = 1, 2, 3,. . . , I) is the coefficient concerned in the spin-orbital interaction between the jth sublattice and the kth sublattice. The effective field concerned in the spin-orbital interaction at any site in the medium is H, =
Usually the spin-orbital interactions between the atoms can be neglected. Assuming that all atoms have the same interaction coefficient 5, that is, and considering the magnetic moment 5;j = 16i,* of the atom m, = mL, + m,,, we can write eq. (9) as = 5 Cm,,*mi (
- CmL;mL, i
i
1
.
(10)
In most of the ferromagnetic and ferrimagnetic media, the second term in the right side of eq. (10) can be neglected in first approximation. In the case of saturation, m, = M,/N, and M, is the saturated magnetization. Then H Lqsat) = vMs-CmL,9
(11)
i
= vM.xm,,,
K$,,*
+
c
. . . +
K,&
1 ~~v$bf,
+
i=l
c
1 ~,v,@f~
+
. . . +
1=1
=vlM,+v,M,+
c
K,v,,M,
i=l
. ..+v.M,.
04)
It is not difficult to prove that the effective field related to the (indirect) exchange interaction acting on the orbital electrons could be expressed as v’M in ferromagnetic medium and Cl= ,v;‘M, in antiferromagnetic and ferrimagnetic media. But here we should emphasize that v’M or Ci=,v(M, is not the molecular field but the effective field concerned in the molecular field. Based on the formal resemblance, two kinds of the effective fields above can be merged as follows H, = vM (ferromagnetic), H, = i
v;M;
(15)
i=l
where v = l/N. When H, = 0, the magnetic domains in medium are arrayed disorderly, and eq. (10) can be written as HLs = O*C,m,, = 0. Therefore, in general, eq. (10) can be approximately expressed as H,,
+
1 =
H,,
K,&
(9)
HLS = c C S;jm,, - ms,. ’ j
03)
**., HvI = v,,M, + q2M2 + . . . + v,,M,,
(12)
i
where vM is the effective field of the spin-orbital interaction acting on the orbital magnetic moment.
(antiferromagnetic
and ferrimagnetic).
4. The Faraday effect For the Faraday effect s 11h, assuming axis, from eqs. (4~) and (4d), we get
h 11z-
AP, + i BPY = 0, -iBP,
+ APY = 0,
(16)
G. -4. LIU et al. / Faraday effect and effective field them
15x
where,
Using eq. (24), we obtain B = ,8( n*/‘/.&
A = an2/poc2 - a’,
In the condition of the nonzero (16) it can be obtained that
(17)
-q,).
solution
of eqs.
(18)
A = -B,
P,.=
-iPX,
E,.=
it is obtained
from
eqs. (16)
-iE,.
(19)
This is the right circularly polarized light and its index of refraction is n,. From eqs. (4) (17) and (18) it is obtained that 2
n+-
Ne2 3r,m
z z % - w -lyO-
e&G + m
E,,=iE,,
(21)
2 cd” - 0
--Ne*
2
3e,m
-lyw-
ep.,H,w m
’
(22) Let wL = ep,Hi/2m, and when H, < 1/4a X 10”’ A/m (10’ Oe), wL -=Kw. If the damping term is neglected, it may be easily found that z n+ -1 p= n:+2 n’-=
FL0Ne2c2/3 ao’ -&J*+--1
n2+2
Z
+oH,a
[
2m(w:
2 oo-w
2
&-(o-wL)*’ =
ePoHio -~
1+ [
- Ne’/3e,,m)
+1 I
, w2( wi - Ne’/3c,,m) 304
+
2 + ...
8( C& - Ne*/3c,,m)
+ c//h4
+ ...
n+-
n_=
-2d!&_ dw
p.,Ne2c2/3
m
2x0, = -w
Eq. (28) is the well-known Cauchy experimental formula. Substituting eq. (28) into eq. (26). we obtain the Verdet constant in the normal dispersion region, V=(e~Jmc)(h/A2+c/A4+
where h = -h’
19= VL(H,+
dn dX’
(28)
. ..).
(29)
and c = - 2~‘.
In the diamagnetic media, magnetization M is very little, and the magnetocrystalline anisotropy is very faint. For eq. (2) Hi = (H, + H,,), so from eq. (25) it is given that
w~-(~+w~)*’
as the function
(27)
wheren=$(n++n.).Aso-l/X,theaboveexpression can be rewritten as
H,).
(30)
(23) Being considered
+const, I
4.1. In the diumagnetic mediu
poNe2c2/3
m
puNe2c2/3
(26)
1 ,‘2
n = a’ + h’/A’
Ne2c2/m
rl* -I=
(25)
dn/dX.
Ne”c’
x
When A = B, for the same reason, the left circularly polarized light and its index of refraction n- can be given respectively by
VLH,,
is the Verdet constant, h the light wavelength in vacuum, and L the propagated distance in the medium, H, the prcjection of H, in the direction of the light propagation. For the region where w is far from w,), i.e. the normal dispersion region, from eqs. (20) (22) and (24) it could be gotten that
(20)
P,,=iP,,
)=
where
‘=
Ne ‘c’/m
1=
-n
V = (ep,,X/2mc)
A=_tB.
When that
0= $$(n+
4.2. In the paramagnetic
of (w f We), (24)
media
In some paramagnetic media there is a weaker exchange interaction between most neighbouring
G. -4. Liu et al. / Faraday effect and effective field theory
electron
spins;
then H, can be written
as
H, = vM = VXH,.
Here, the indexes of the refraction and left circularly polarized light are (31)
The relation between susceptibility x and temperature T conforms to the Curie-Weiss law x = c/( T - T,), where c and T, are the Curie constant and the Curie temperature respectively. When the influence of H,, HA and Hd can be neglected, eq. (25) can be expressed as 8 = I%( H, + Hv) = v,LH,,
(36)
Let eqs. (20) and (22) be written
(n,++
in,+)‘=
1 + (I&“+‘:;
+l(X+
4.3. In the ferromagnetic, ferrimagnetic media
and
antiferromagnetic
In these media, compared to H,, effective fields in eq. (2), especially neglected. Eq. (25) is also suitable antiferromagnetic and magnetic, media at H, d (4a)-’ x 10” A/m. H,, H,, HA and Hd, 8 is expressed as
the other four H,, cannot be for the ferroferrimagnetic When H, x= approximately
13= Vf vM (ferromagnetic),
e=
(37)
,x~~,r’,2
wz
where W = poNe2c2/m, X = ~0’ - u2 - Ne2/ 3r,m, Y = e/.loHiw/m, Z = yw. According to the definitions of B and 4, and from eqs. (37) it can be obtained that n,B-n,#=
(nL/2h)(A’-
n,# + n,B = (TL/~A)(
B’),
(35)
5. The magnetic circular dichroism The expressions for 8 in section 4 are obtained while damping is neglected, so that 8 is only the real part of the Faraday rotation. Next, the relationship between 8 and # (the imaginary part of Faraday rotation or the magnetic circular dichroism), and their characteristics will be described.
and
+ n:)]
X[n,(A’-B’)+n2(D’-c’)], $J= [L/2X(n:+ni)]
and ferrimagnetic).
(38)
D’ - C’),
where n, = t(n,++ n,_), n, = :(n,++ n,_), then B and $J can be written as follows,
e = [ L/2X(nf (34)
VL C v,~, I=1 (antiferromagnetic
Y)2+z*
(33) Eq.
zZ2
=A’+iC’, (v+%)“=l+
where G = V(vc - Tc)/c, R = l/(vc - T,). (33) has been confirmed by experiment [6].
as follows
wz
(32)
= G(1 + RT),
of the right
n* = n,*+ in,+.
where V, = V(1 + vx) is the Verdet constant in some paramagnetic media. It can be written as v,/x
159
(39
X[q(D’-C’)-n,(A’-B’)]. Generally, N - 1026-1027, w - 1015 s-t, y -=z w. It is supposed that Hi Q (4~)~’ X 10” A/m, and then Z +z A’, B’ and C’, therefore, (D’ - C’) in eqs. (39) can be neglected. From this it can be found that
e/G = -n,/n,
= l/n*.
Here n* is called the effective extinction cient. From eqs. (25) and (40) we obtain $ = vLq*H,.
(40) coeffi-
(41)
G.-q. LIU et (11./ Faraday effectund effectrue field theory
160
0 (deg)
0
100
200
300
-3000 j I5
I1
-
\
12
.
\
i
theoretical
2
Ndf3 -5000[ v/k
experimental
9i\ 6,
I
31
300 1
‘l
0 II_-. 0.6
a..-._,
1.1
1.8
-flOOO.Ly/~
&urn)
Fig. 1. 8-X dispersion curve for (BiTm),(FeGa),O,, the thickness 27.50 pm.
6.
200
100 -‘nnnP
;\
film with
PrF3
Fig. 3. Ratio of the Verdet constant and the magnetic susceptibility as a function of temperature for NdF, and PrF,.
Discussion
From the calculation above, we discuss as follows: (1) In all of the MO media, the real part of the Faraday rotation is 0 = VLH,. In both the normal dispersion region (dn/dX < 0) and the abnormal dispersion region (dn/dX > 0), the signs of 8 are opposite to each other. In the normal dispersion with the region, 8 is satisfied by the relation optical wavelength X as B - (6/X2 + c/A4 + . . ).
I3
(
dr,:
Using the magneto-optical rotation spectrometer [5] (measured accuracy 0.01” ) we have measured /3 - X in the case of the garnet-film sample being magnetically saturated. The sample is a (BiTm),(FeGa),O,, film grown on GGG substrate by the liquid-phase epitaxial method. The experimental results are shown in fig. 1. When H, and M are fixed, from eqs. (29) and (35), 0 = (&/A2 + co/A4 + . . . ). If b, = 1.9 and cg = 1.6, then the theoretical results agree with the experimental results.
e
) O-
(
deg
)
o-
o-o-0
2.0
l
1 .o
-o-
r!a~“~to4ptlcnl
-a-
Elllptlcity
rbtat1on
H,( 0~
I Fig. 2. The real part
B and the imaginary
part of the Faraday rotation vs. field strength thickness 2h = 27.50 pm at X = 0.6328 pm.
)
0
H, for (BiTm)3(FeGa)S0,,
film sample
G. -4. Liu et al. / Faraday effect and effective field theory
‘0.6 - 0.4
- 0.2
0.5
1.0
1.5
2.0
A(m)
Fig. 4. Dispersion relations of 19and 4 for (BiTm),(FeGa)sO,, single crystal film with the thickness 2/z = 11.66 urn.
(2) The imaginary part J/ and the real part 8 of the Faraday rotation have many similar characteristics, i.e. both are directly proportional to the effective field Hi, and are saturated as M tends to saturation as shown in fig. 2. (3) The properties and sizes of the effective field Hi in different magnetic media are different. This is the primary factor that induces different MO characteristics. (4) In some paramagnetic media, 0 = ~~‘,LHi, and V,/x = G(l + RT). This has been verified by the experiments in NdF, (for T > 77 K) and in PrF, (for T > 40 K), and is shown in fig. 3 [6]. It is worth to mention that by eqs. (29) and (33) is shown that the coefficient G is closely related to the light wavelength X or the light frequency w but the coefficient R is not related to X. This has been experimently proved by Leycuras et al. [6] too. (5) In ferromagnetic media, the temperature behaviour of 8 is satisfied with B(T) - vM(T), and in antiferromagnetic and ferrimagnetic media, B(T) - Cf=,vi(T)Mi(T). These have been verified
161
by the experiment in ref. [7]. In antiferromagnetic media, Xi= iM, = 0, and usually Xi= iv,M, # 0 so that the large 0 may exist. (6) 8 would generally vary with the propagation direction of light in the crystal with the form of the medium and with the direction and size of the stress in the medium. In magnetic media the above factors which influence B are much weaker than H, so that it is difficult to measure. In diamagnetic and paramagnetic media the above factors such as the anisotropy in paramagnetic media usually are not negligible. All of these may be measured. (7) The difference between 4 and B can be described by an effective extinction coefficient n*. n* is not only related to the optical loss in medium, but is also a function of the frequency of the incident light. In the general case, the dispersion relationships of \c/ and B are different, but they are similar in the region in which \cI* is approximately not related to w. This has been proven by the experimental results shown in fig. 4. (8) Generally, n* is not closely related to temperature T, so the temperature behaviour of 4 and 8 are roughly similar.
References 111W. Pauli, Pauli Lectures on Physics, vol. 2, Optics and the Theory
of Electrons
(MIT Press, Cambridge, MA, 1973). Properties of Materials, ed. J. Smit (McGraw-Hill, New York, 1971) p. 152. [31 P.N. Argyres, Phys. Rev. 97 (1955) 334. [41 H.R. Hulme, Proc. Roy. Sot. London A 135 (1932) 237. 151G.q. Liu, X.X. Han and H.d. Song, China J. Sci. Instrum. 8 (1987) 360. 161C. Leycuras and H. Le Gall, J. Appl. Phys. 55 (1984) 2161. and K. Witter, Phys. Rev. B 25 [71 P. Hansen, M. Rosenkranz (1982) 4396.
121J.F. Dillon Jr., Magnetic