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Electromagnetic excitation and dynamical response of ferromagnetic superconductors
Solid State Communications, Printed in Great Britain.
Vol. 46, No. 11, pp. 81 I-814,
0038-1098183 $3.00 + .OO Pergamon Press Ltd.
1983.
ELECTROMAGNETIC EXCITATION AND DYNAMICAL RESPONSE OF FERROMAGNETIC SUPERCONDUCTORS A. Kotani and K. Okada Department
of Physics, Faculty of Science, Osaka University,
Toyonaka
560, Japan
(Received 22 February 1983 by J. Kanamori)
Electromagnetic excitation modes in ferromagnetic superconductors are studied theoretically as the coupled modes of photon, electric current and magnetic moments, and their dispersion relations are obtained. With the use of these modes, the reflection coefficient is calculated for the electromagnetic wave incident on the surface of a semi-infinite crystal of ferromagnetic superconductor. It is shown that the reflection coefficient decreases rapidly in the neighborhood of the critical temperature where a spontaneous surface magnetization occurs. An anomaly in the surface impedance is also predicted. the magnetization of spins. The electric current is represented by [6]
IN THE INTERPLAY of magnetism and superconductivity in ferromagnetic superconductors [l], ErRh4B4 and HoMob&, the electromagnetic interaction between the persistent current and the rare earth magnetic moments (denoted by spins, hereafter) plays an essentially important role [2-4]. Through the electromagnetic interaction, the persistent current screens the long range part of the interaction between spins. This indicates that there exist the coupled modes (both of static and dynamical modes) of the photon, electric current and spins, and it is important to study the features of the coupled modes to understand the properties of ferromagnetic superconductors. In the static limit, the behavior of the coupled modes has been studied in a previous paper [5], and the penetration of a static magnetic field into a semi-infinite crystal of the ferromagnetic superconductor has been calculated by using the coupled modes. It is the purpose of the present paper to extend the theory to the dynamical case. We first study the behavior of the dynamical coupled modes, and then calculate, by using them, the reflection coefficient of the electromagnetic wave incident on the surface of a semi-infinite magnetic superconductor. The coupled modes are described by the Maxwell equations
1 dr’ dt’ c(r - r’, t - t’)a(r’,
j(r, t) = - -f$-
Vx e(r,t)
=
j(r, t) + 41rV x m(r, t) + + i
-+ih(r,t),
(3)
which gives the persistent current in the static limit. In equation (3), XL is the London penetration depth, c(r, t) is the kernel representing the nonlocal current effect, and a(r, t) is the vector potential. Confining ourselves to the spin paramagnetic phase, we describe the magnetization m(r, t) in the form m(r, t) = J‘ dr’ dt’ x(r - r’, t - t’) h(r’, t’),
(4)
with the dynamical susceptibility x(r, t) of the spin system and the magnetic field h(r, t). In order to obtain the normal mode, we put all the field quantities in the plane wave form; for instance b(r, t) = b(k, w) exp [i(k- r - wt)]. Then, we obtain, from equations equation for b(k, o) as follows:
x k x [k x b(k, o)] V x b(r, t) = f
t’),
L
e(r, t),
(1) (2)
= 0.
(l)-(4),
(5) the wave
(6)
where c(k, w) and x(k, w) are the Fourier components of c(r, w) and x(r, o), respectively. In deriving equation (6) we have also used the relations b(r, t) = V x a(r, t)
(7)
and
where b(r, t) is the magnetic induction, e(r, t) is the electric field, j(r, t) is the electric current and m(r, t) is
b(r, t) = h(r, t) + 4nm(r, 811
t).
(8)
ELECTROMAGNETIC
812
RESPONSE OF FERROMAGNETIC
In the following, we assume that the electric current is local, i.e. c(k, w) = 1, and take the following form for the dynamical spin susceptibility [3] :
x(k a) = x(k)&
SUPERCONDUCTORS
Vol. 46, No. 1%
a. -
(9)
)
where
c
x(k) = T-T,,,+Dk=
r =
B/x(k).
Here T,,, is the Curie temperature in the fictitious normal state, C is the Curie constant, D is the exchange stiffness constant, and B is a constant describing the spin damping F. By assuming b(k, t) as a transverse wave (i.e. k 1 b), we obtain from equations (6) and (9) the relation between k and o of the normal mode: B(T-T;::=)-iwC][&-
(r;i’];O;’
which is the dynamical version of the secular equation (3.1) of [5] _From equation (lo), k2 is expressed as a function of o by (kh,)=
= &[-
(t-1
--i;)-d(1
+
t-
-0”)
1
-4d?(l
t = 0.9
-is=)
2
1 -i+
Fig. 1. Temperature dependence of the wave numbers (k, and k3) and dampings (k2 and k4) of the normal modes for the frequency W = 0.001. The result for the static limit (W = 0) is also shown with the dashed curves.
l/2
-G=) i
I
(11)
where 0 = G.&/C, t = TIT,,, , F = 4nC/T,, d = D/(T,,,hL)
and
(3 = hLBT,/(cC).
Thus we obtain two types of normal modes for each propagation direction k/]k], and write their wave numbers as k, = (k, + ik2)/XL,
kp = (k3 + ik4)/hL.
In Fig. 1 we show, with the solid curves, the temperature dependence of kl - k4 for W = 0.001. The parameters are taken as F = 2.0, d = 0.01 and /3 = 0.01. We also plot, for comparison, the result in the static limit [5] (G = 0) with the dashed curves. As shown in [5], the static modes are simply damping modes for t 2 to = 1.292, and change into the oscillatory damping modes for t < to, where kl = -k3 and k2 = kq. The damping k2 (= k4) vanishes at tp = 0.727, which indicates the onset of the long range magnetic order with a finite wave number k, (= - k3) at tp. When G becomes finite, the degeneracy between kl and -k3, as well as that between k2 and kq, are lifted as shown in Fig. 1. Furthermore, kl and k3 become nonzero even for t>to. The dynamical features of the coupled modes are
Fig. 2. Frequency (6) vs wave number (k, and k3) relation of the normal modes at the temperature t = 0.9, plotted with the solid curves. The dampings (k2 and k4) are shown with the dashed curves. The inset shows the result in the low frequency region, where the effect of spins is essentially important. better represented by the W vs k relation, i.e. the dispersion relation, for a fixed temperature. In Fig. 2, as an example, we plot W as a function of kXL (k = klk4) for t = 0.9. When 0 is larger than 1 .O, i.e. when W is larger than the plasma frequency c/X,, the G-k3 relation (where k4 = 0) represents the dispersion of the transverse plasmon, which is a coupled mode of the photon and the electric current. When G becomes smaller than 1 .O, the plasmon changes into a simply damping mode (where k3 E 0). In the case where G is of the order of 1.0, the spin system cannot follow the plasmon motion, but as W becomes small enough, the coupling between the plasmon and spins becomes important. In the inset of Fig. 2, we show the dispersion
Vol. 46, No. 11 ELECTROMAGNETIC
RESPONSE OF FERROMAGNETIC
relation in such a low frequency region, where ks is found to appear with the finite negative value. This indicates a peculiar situation that the low frequency ks-mode has a phase velocity in the opposite direction to its group velocity, although the damping k4 is also large in this frequency region. Another mode consisting of kI and k2 originates mainly from the spin motion, and is modified by the coupling with the damped plasmon. By using the dynamical coupled modes, we next calculate the reflection coefficient of the electromagnetic wave incident on the surface (at x = 0) of a semiinfinite system (x > 0) of the ferromagnetic superconductor. The incident wave and the reflected wave are, respectively, expressed as hr(x, t) = el(x, t) = n1 exp [iw(x/c -t)],
SUPERCONDUCTORS
8?3
R 1.00 -
0.99 -
0.98 -
(12)
and /2,(x, t) = -ez(x,
t) = n2 exp [-iw(x/c
+ t)].
(13)
Inside the magnetic superconductor, the electromagnetic field is represented by the linear combination of the normal modes with k, = (k, + ikz)/hL and kp = (ks + ikdlb , so that we put the magnetic field in the form ha(x, t) = q, exp [i(k,x --at)]
+ qp exp [i(kpx -or)].
(14) With the use of equations (2), (4) and (8), the magnetization and the electric field are given by m3(x,
f)
C
=
ViX(ki,
a>
exp
[i(k+
-
~f)l,
(15)
i=a,p
e369
f)
=
C
Vi(~/kic)[l
+
4aX(ki,
011
i=cu,p
X
exp [i(kix -tit)].
(16)
By using the Maxwell boundary
conditions
hr(0, t) +
&(O, t) = hs(O, t), ~(0, t) + ez(O, t) = ea(0,0, and by assuming an additional boundary condition [dm,(x, r)/ dx],=e = 0, we can express the amplitudes n2, qor and np as a function of r)r . The additional boundary condition adopted here is a straightforward extension of that in the static limit adopted in [ 51. The reflection coefficient R is obtained as
uk;[k,c--(l =
xpk;[k,c
+4nxa)] + o(1 + 4rrx,)]
-xak;[kpc-u(l -xak:[kpc
Fig. 3. The reflection coefficient of the electromagnetic wave as a function of the temperature with the frequency as a parameter. The temperature r, indicates the critical temperature of the spontaneous surface magnetization. magnetic field he is applied parallel to the surface, the magnetic field h3 and the magnetization m3 inside the ferromagnetic superconductor are expressed as equations (14) and (15) with w = 0. The amplitudes ~,//re and ~/he, which are calculated by using the boundary conditions h3 = ho and dm3/dx = 0 at x = 0, diverge at a temperature r, higher than the critical temperature rp of the spontaneous magnetic order in the bulk system [5] (r, = 0.863 with our parameter values). This means that the spontaneous magnetic order bound near the surface occurs below r,. The screening of the spin-spin interaction by the persistent current is weakened near the surface, so that the critical temperature of the spontaneous surface magnetization is higher than that of the spontaneous bulk magnetization [5, 7,8]. In Fig. 3 we show the temperature dependence of the reflection coefficient R for various values of the
+4nx )] ’ fl 1 > (17) + w(1 + 47qp)]
where we abbreviate ~(k,, a) and x(kp, w) as xol and xp, respectively. Before discussing the behavior of R, we mention briefly the static limit of our problem. When a static
1
frequency G. The deviation of R from 1 .O is due to the energy loss inside the ferromagnetic superconductor, which originates from the motion of the spin system delayed behind the driving electromagnetic wave. When
814
ELECTROMAGNETIC
O_t,
1.0
\ \ \ \ \ -0.05-
1.2
RESPONSE OF FERROMAGNETIC
1.4
t
__---Imp_,__--------
‘~,._,.-=*
t Fig. 4. Temperature dependence pedance for c = 0.001.
J of the surface im-
Gj is sufficiently small (as the case of 3 = O.OOOOl), the spins can follow, except in the extreme neighborhood of fs, the electromagnetic field without time delay, so that we have R z 1 .O. When z is increased, the spin system can still follow the electromagnetic wave at high temperatures, but cannot in the neighborhood of the critical temperature t, because of the critical slowing down of the spin motion. Thus we have a rapid decrease of R near t, for G = 0.0001 - 0.001. For larger ti, the decrease of R near ts becomes gradual, although the deviation of R from 1 .O becomes larger. Our calculation is limited to the temperature range t > ts, since we assume that the spin system is in the paramagnetic phase. When we decrease the temperature below t,, the critical slowing down of the spin motion ceases, so that R is expected to increase toward 1 .O, again. We also calculate the surface impedance 2, and show in Fig. 4 the temperature dependence of Z in the case of i;j = 0.001. It is found that the surface resistivity Re Z increases rapidly in the vicinity oft,, reflecting the energy loss due to the critical slowing down of the spin motion. The occurrence of the spontaneous surface magnetization in ferromagnetic superconductors was predicted theoretically [S, 7,8], but any experimental evidence has not been obtained yet. From the present calculation, it is expected that the critical temperature of the spontaneous surface magnetization can be detected experimentally as a sharp dip of the reflection coefficient of
SUPERCONDUCTORS
Vol. 46, No. t 1
electromagnetic wave or a sharp peak of the surface resistivity. Although our calculation is limited to the case of the local electric current, the essential features are considered to be unchanged by the nonlocal current effect. The frequency range in which the sharp anomaly of the reflection coefficient or the surface resistivity is expected to occur in the close neighborhood of ts will correspond to 10 - 100 GHz. Finally, it should be mentioned that the present theory can be extended to the study of the electromagnetic properties of the film of ferromagnetic superconductors. The reflection coefficient and the transmission coefficient of the electromagnetic wave incident on the film (see [9] for the static limit) are shown to change anomalously as functions of the temperature and the film thickness, and the details will be reported elsewhere. Acknowledgements - The authors would like to express their thanks to Professor J. Kanamori, Professor M. Tachiki, Dr T. Koyama and Mr. S. Takahashi for valuable discussions on the present and related problems. They also thank Mrs A. Egusa for preparing the manuscript. This work was supported partially by a Grantin-Aid for Scientific Research given by the Ministry of Education, Science and Culture of Japan.
REFERENCES 1.
6. 7.
8. 9.
See, for instance, Proc. Znt. Con& Ternary Superconductors, (Edited by G.K. Shenoy, B.D. Dunlap & F.Y. Fradin), North-Holland (1981). H. Matsumoto, H. Umezawa & M. Tachiki, Solid State Commun. 31,157 (1979). M. Tachiki, A. Kotani, H. Matsumoto & H. Umezawa. Solid State Commun. 31.927 (1979). E.I. Blount & C.M. Varma, Phys. Rev. Leit. 42; 1079(1979). -.- \-- .-,A. Kotani, M. Tachiki, H. Matsumoto, H. Umezawa & S. Takahashi, Phys. Rev. B23,5960 (1981). L. Leplae, F. Mancini & H. Umezawa, Phys. Rep. lOC, 151(1974). A. Kotani, S. Takahashi, T. Koyama, M. Tachiki, H. Matsumoto & H. Umezawa, J. Phys. Sot. Japan SO,3254 (1981). T. Koyama, A. Kotani, S. Takahashi, M. Tachiki, H. Matsumoto & H. Umezawa, J. Phys. Sot. Japan 51,3469 (1982). A. Kotani, S. Takahashi, M. Tachiki, H. Matsumoto & H. Umezawa, Solid State Commun. 37, 619 (1981).
Vol. 46, No. 11, pp. 81 I-814,
0038-1098183 $3.00 + .OO Pergamon Press Ltd.
1983.
ELECTROMAGNETIC EXCITATION AND DYNAMICAL RESPONSE OF FERROMAGNETIC SUPERCONDUCTORS A. Kotani and K. Okada Department
of Physics, Faculty of Science, Osaka University,
Toyonaka
560, Japan
(Received 22 February 1983 by J. Kanamori)
Electromagnetic excitation modes in ferromagnetic superconductors are studied theoretically as the coupled modes of photon, electric current and magnetic moments, and their dispersion relations are obtained. With the use of these modes, the reflection coefficient is calculated for the electromagnetic wave incident on the surface of a semi-infinite crystal of ferromagnetic superconductor. It is shown that the reflection coefficient decreases rapidly in the neighborhood of the critical temperature where a spontaneous surface magnetization occurs. An anomaly in the surface impedance is also predicted. the magnetization of spins. The electric current is represented by [6]
IN THE INTERPLAY of magnetism and superconductivity in ferromagnetic superconductors [l], ErRh4B4 and HoMob&, the electromagnetic interaction between the persistent current and the rare earth magnetic moments (denoted by spins, hereafter) plays an essentially important role [2-4]. Through the electromagnetic interaction, the persistent current screens the long range part of the interaction between spins. This indicates that there exist the coupled modes (both of static and dynamical modes) of the photon, electric current and spins, and it is important to study the features of the coupled modes to understand the properties of ferromagnetic superconductors. In the static limit, the behavior of the coupled modes has been studied in a previous paper [5], and the penetration of a static magnetic field into a semi-infinite crystal of the ferromagnetic superconductor has been calculated by using the coupled modes. It is the purpose of the present paper to extend the theory to the dynamical case. We first study the behavior of the dynamical coupled modes, and then calculate, by using them, the reflection coefficient of the electromagnetic wave incident on the surface of a semi-infinite magnetic superconductor. The coupled modes are described by the Maxwell equations
1 dr’ dt’ c(r - r’, t - t’)a(r’,
j(r, t) = - -f$-
Vx e(r,t)
=
j(r, t) + 41rV x m(r, t) + + i
-+ih(r,t),
(3)
which gives the persistent current in the static limit. In equation (3), XL is the London penetration depth, c(r, t) is the kernel representing the nonlocal current effect, and a(r, t) is the vector potential. Confining ourselves to the spin paramagnetic phase, we describe the magnetization m(r, t) in the form m(r, t) = J‘ dr’ dt’ x(r - r’, t - t’) h(r’, t’),
(4)
with the dynamical susceptibility x(r, t) of the spin system and the magnetic field h(r, t). In order to obtain the normal mode, we put all the field quantities in the plane wave form; for instance b(r, t) = b(k, w) exp [i(k- r - wt)]. Then, we obtain, from equations equation for b(k, o) as follows:
x k x [k x b(k, o)] V x b(r, t) = f
t’),
L
e(r, t),
(1) (2)
= 0.
(l)-(4),
(5) the wave
(6)
where c(k, w) and x(k, w) are the Fourier components of c(r, w) and x(r, o), respectively. In deriving equation (6) we have also used the relations b(r, t) = V x a(r, t)
(7)
and
where b(r, t) is the magnetic induction, e(r, t) is the electric field, j(r, t) is the electric current and m(r, t) is
b(r, t) = h(r, t) + 4nm(r, 811
t).
(8)
ELECTROMAGNETIC
812
RESPONSE OF FERROMAGNETIC
In the following, we assume that the electric current is local, i.e. c(k, w) = 1, and take the following form for the dynamical spin susceptibility [3] :
x(k a) = x(k)&
SUPERCONDUCTORS
Vol. 46, No. 1%
a. -
(9)
)
where
c
x(k) = T-T,,,+Dk=
r =
B/x(k).
Here T,,, is the Curie temperature in the fictitious normal state, C is the Curie constant, D is the exchange stiffness constant, and B is a constant describing the spin damping F. By assuming b(k, t) as a transverse wave (i.e. k 1 b), we obtain from equations (6) and (9) the relation between k and o of the normal mode: B(T-T;::=)-iwC][&-
(r;i’];O;’
which is the dynamical version of the secular equation (3.1) of [5] _From equation (lo), k2 is expressed as a function of o by (kh,)=
= &[-
(t-1
--i;)-d(1
+
t-
-0”)
1
-4d?(l
t = 0.9
-is=)
2
1 -i+
Fig. 1. Temperature dependence of the wave numbers (k, and k3) and dampings (k2 and k4) of the normal modes for the frequency W = 0.001. The result for the static limit (W = 0) is also shown with the dashed curves.
l/2
-G=) i
I
(11)
where 0 = G.&/C, t = TIT,,, , F = 4nC/T,, d = D/(T,,,hL)
and
(3 = hLBT,/(cC).
Thus we obtain two types of normal modes for each propagation direction k/]k], and write their wave numbers as k, = (k, + ik2)/XL,
kp = (k3 + ik4)/hL.
In Fig. 1 we show, with the solid curves, the temperature dependence of kl - k4 for W = 0.001. The parameters are taken as F = 2.0, d = 0.01 and /3 = 0.01. We also plot, for comparison, the result in the static limit [5] (G = 0) with the dashed curves. As shown in [5], the static modes are simply damping modes for t 2 to = 1.292, and change into the oscillatory damping modes for t < to, where kl = -k3 and k2 = kq. The damping k2 (= k4) vanishes at tp = 0.727, which indicates the onset of the long range magnetic order with a finite wave number k, (= - k3) at tp. When G becomes finite, the degeneracy between kl and -k3, as well as that between k2 and kq, are lifted as shown in Fig. 1. Furthermore, kl and k3 become nonzero even for t>to. The dynamical features of the coupled modes are
Fig. 2. Frequency (6) vs wave number (k, and k3) relation of the normal modes at the temperature t = 0.9, plotted with the solid curves. The dampings (k2 and k4) are shown with the dashed curves. The inset shows the result in the low frequency region, where the effect of spins is essentially important. better represented by the W vs k relation, i.e. the dispersion relation, for a fixed temperature. In Fig. 2, as an example, we plot W as a function of kXL (k = klk4) for t = 0.9. When 0 is larger than 1 .O, i.e. when W is larger than the plasma frequency c/X,, the G-k3 relation (where k4 = 0) represents the dispersion of the transverse plasmon, which is a coupled mode of the photon and the electric current. When G becomes smaller than 1 .O, the plasmon changes into a simply damping mode (where k3 E 0). In the case where G is of the order of 1.0, the spin system cannot follow the plasmon motion, but as W becomes small enough, the coupling between the plasmon and spins becomes important. In the inset of Fig. 2, we show the dispersion
Vol. 46, No. 11 ELECTROMAGNETIC
RESPONSE OF FERROMAGNETIC
relation in such a low frequency region, where ks is found to appear with the finite negative value. This indicates a peculiar situation that the low frequency ks-mode has a phase velocity in the opposite direction to its group velocity, although the damping k4 is also large in this frequency region. Another mode consisting of kI and k2 originates mainly from the spin motion, and is modified by the coupling with the damped plasmon. By using the dynamical coupled modes, we next calculate the reflection coefficient of the electromagnetic wave incident on the surface (at x = 0) of a semiinfinite system (x > 0) of the ferromagnetic superconductor. The incident wave and the reflected wave are, respectively, expressed as hr(x, t) = el(x, t) = n1 exp [iw(x/c -t)],
SUPERCONDUCTORS
8?3
R 1.00 -
0.99 -
0.98 -
(12)
and /2,(x, t) = -ez(x,
t) = n2 exp [-iw(x/c
+ t)].
(13)
Inside the magnetic superconductor, the electromagnetic field is represented by the linear combination of the normal modes with k, = (k, + ikz)/hL and kp = (ks + ikdlb , so that we put the magnetic field in the form ha(x, t) = q, exp [i(k,x --at)]
+ qp exp [i(kpx -or)].
(14) With the use of equations (2), (4) and (8), the magnetization and the electric field are given by m3(x,
f)
C
=
ViX(ki,
a>
exp
[i(k+
-
~f)l,
(15)
i=a,p
e369
f)
=
C
Vi(~/kic)[l
+
4aX(ki,
011
i=cu,p
X
exp [i(kix -tit)].
(16)
By using the Maxwell boundary
conditions
hr(0, t) +
&(O, t) = hs(O, t), ~(0, t) + ez(O, t) = ea(0,0, and by assuming an additional boundary condition [dm,(x, r)/ dx],=e = 0, we can express the amplitudes n2, qor and np as a function of r)r . The additional boundary condition adopted here is a straightforward extension of that in the static limit adopted in [ 51. The reflection coefficient R is obtained as
uk;[k,c--(l =
xpk;[k,c
+4nxa)] + o(1 + 4rrx,)]
-xak;[kpc-u(l -xak:[kpc
Fig. 3. The reflection coefficient of the electromagnetic wave as a function of the temperature with the frequency as a parameter. The temperature r, indicates the critical temperature of the spontaneous surface magnetization. magnetic field he is applied parallel to the surface, the magnetic field h3 and the magnetization m3 inside the ferromagnetic superconductor are expressed as equations (14) and (15) with w = 0. The amplitudes ~,//re and ~/he, which are calculated by using the boundary conditions h3 = ho and dm3/dx = 0 at x = 0, diverge at a temperature r, higher than the critical temperature rp of the spontaneous magnetic order in the bulk system [5] (r, = 0.863 with our parameter values). This means that the spontaneous magnetic order bound near the surface occurs below r,. The screening of the spin-spin interaction by the persistent current is weakened near the surface, so that the critical temperature of the spontaneous surface magnetization is higher than that of the spontaneous bulk magnetization [5, 7,8]. In Fig. 3 we show the temperature dependence of the reflection coefficient R for various values of the
+4nx )] ’ fl 1 > (17) + w(1 + 47qp)]
where we abbreviate ~(k,, a) and x(kp, w) as xol and xp, respectively. Before discussing the behavior of R, we mention briefly the static limit of our problem. When a static
1
frequency G. The deviation of R from 1 .O is due to the energy loss inside the ferromagnetic superconductor, which originates from the motion of the spin system delayed behind the driving electromagnetic wave. When
814
ELECTROMAGNETIC
O_t,
1.0
\ \ \ \ \ -0.05-
1.2
RESPONSE OF FERROMAGNETIC
1.4
t
__---Imp_,__--------
‘~,._,.-=*
t Fig. 4. Temperature dependence pedance for c = 0.001.
J of the surface im-
Gj is sufficiently small (as the case of 3 = O.OOOOl), the spins can follow, except in the extreme neighborhood of fs, the electromagnetic field without time delay, so that we have R z 1 .O. When z is increased, the spin system can still follow the electromagnetic wave at high temperatures, but cannot in the neighborhood of the critical temperature t, because of the critical slowing down of the spin motion. Thus we have a rapid decrease of R near t, for G = 0.0001 - 0.001. For larger ti, the decrease of R near ts becomes gradual, although the deviation of R from 1 .O becomes larger. Our calculation is limited to the temperature range t > ts, since we assume that the spin system is in the paramagnetic phase. When we decrease the temperature below t,, the critical slowing down of the spin motion ceases, so that R is expected to increase toward 1 .O, again. We also calculate the surface impedance 2, and show in Fig. 4 the temperature dependence of Z in the case of i;j = 0.001. It is found that the surface resistivity Re Z increases rapidly in the vicinity oft,, reflecting the energy loss due to the critical slowing down of the spin motion. The occurrence of the spontaneous surface magnetization in ferromagnetic superconductors was predicted theoretically [S, 7,8], but any experimental evidence has not been obtained yet. From the present calculation, it is expected that the critical temperature of the spontaneous surface magnetization can be detected experimentally as a sharp dip of the reflection coefficient of
SUPERCONDUCTORS
Vol. 46, No. t 1
electromagnetic wave or a sharp peak of the surface resistivity. Although our calculation is limited to the case of the local electric current, the essential features are considered to be unchanged by the nonlocal current effect. The frequency range in which the sharp anomaly of the reflection coefficient or the surface resistivity is expected to occur in the close neighborhood of ts will correspond to 10 - 100 GHz. Finally, it should be mentioned that the present theory can be extended to the study of the electromagnetic properties of the film of ferromagnetic superconductors. The reflection coefficient and the transmission coefficient of the electromagnetic wave incident on the film (see [9] for the static limit) are shown to change anomalously as functions of the temperature and the film thickness, and the details will be reported elsewhere. Acknowledgements - The authors would like to express their thanks to Professor J. Kanamori, Professor M. Tachiki, Dr T. Koyama and Mr. S. Takahashi for valuable discussions on the present and related problems. They also thank Mrs A. Egusa for preparing the manuscript. This work was supported partially by a Grantin-Aid for Scientific Research given by the Ministry of Education, Science and Culture of Japan.
REFERENCES 1.
6. 7.
8. 9.
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