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Interplay of superconductivity and magnetism in thin films of ferromagnetic superconductors
QD 3
Physica 108B (1981/1043-1044 North-Holland Publishing Company
INTERPLAY OF SUPERCONDUCTIVITY AND MAGNETISM IN THIN FILMS OF FERROMAGNETIC SUPERCONDUCTORS
A. Kotani, ~ S. Takahashi and M. Tachiki
The Research Institute for Iron, Steel and Other Metals Tohoku University, Katahira, Sendai, 980 Japan and H. Matsumoto and H. Umezawa Department of Physics, The University of Alberta Edmonton, Alberta, T6G 2Jl Canada The effect of the electromagnetic interaction between the persistent current and the rare earth magnetic moments is theoretically studied in thin fiJms of ferromagnetic superconductors. Depending on the film thickness and temperature, the ferromagnetic ordering, sinusoidal spin ordering and spiral spin ordering are predicted to occur in coexistence with superconductivity.
i. INTRODUCTION In the interplay of magnetism and superconductivity in ferromagnetic superconductors, ErRh4B 4 and HoMo6S 8, the electromagnetic interaction between the persistent current and the rare earth magnetic moments plays an essentially important role. Through the electromagnetic interaction, the persistent current screens the long range part of the interaction between the magnetic moments. As a result, a uniform ferromagnetic ordering is strongly suppressed, and instead a spin-spiral ordering with finite wave number occurs in the superconducting bulk crystals.[l] However, in thin films of ferromagnetic superconductors, the screening effect due to the persistent current is weakened in the neighborhood of the film surfaces, and we expect that the coexistence of superconductivity and magnetism occurs in a different way from that in bulk systems.[2] 2. FORMULATION We consider a film of thickness 2a located at -a < x < a, where the x-axls is taken in the direction normal to the film surface. All the physical quantities are assumed to be uniform in the yz plane. In the normal state, the interaction between the rare earth magnetic moments ~(x) and ~(x') is written as
-
-
x')~(x'). i r a dx la dx'm(x)y(x, -> 2 -a -a
In the superconducting state, the persistent current
is induced by the magnetic moments, where %L is the London penetration depth and ~(x) is the vector potential. The magnetic field ~(x) induced by ~(x) is described by the Maxwell equation. × ~(x) = 4-~ ~(x).
(2)
C
Substituting (I) into (2) and using the relation ~(x) = ~ x ~(x) = ~(x) + 4 ~ ( x ) [~(x~ being the magnetic induction], we can express h(x) as a functional of m(x). By using the molecular field approximation, m(x) is determined by solving g~B J m(x) = g~BJNBj[-~BT hm(X)]
(3)
under the condition of ~(x) H ~ m ( X ) , ~:here ~m(X) is the molecular field and Bj[X] is the Brillouin function. The molecular field is given by a sum of ~(x) and the unscreened molecular field: ~m(X) = ~(x) +
dx'y(x, x')m(x').
(4)
--a
3. RESULTS We study what types of magnetic ordering appear as the solutions of (3). For simplicity, we assume that the persistent current 3(x) is local and the coupling y(x, x') is the same as that in the bulk system. Then we can put c(x, x') = ~(x - x') and
Tm3/2 j(x)
-c2 4n~ L
dx c(x, x')~(x')
(1)
y(x, x') = 2C--~7~ exp[- (rra)i/2]XD - x'I]'
-a
* Present address: Department of Physics, Faculty of Science, Osaka University, Toyonaka, 560 Japan
03784363/81/0000~000/$0250
© No~h-HollandPublishingCompany
where Tm and C, respectively, are the Curie temperature and the Curie constant of the fictitious bulk system in the normal state, and D is
1043
1044
the magnetic stiffness constant. In Figure 1 we show the magnetic phase diagram for the superconducting thin films in the plane of the temperature T/T m and the film thickness a/% L. The solid line represents the critical temperature of the spontaneous magnetic ordering. For a/% L 0.5, the superconducting ferromagnetic state occurs below the critical temperature. In the thin film of such a small thickness, the coexistence of ferromagnetism and superconductivity becomes possible, because the screening effect by the persistent current, which suppresses their coexistence in the bulk system, is much weakened due to the influence of the film surfaces. For a/~ L ~ 0.5, the screening effect is restored well inside the film, and the oscillation and rotation of the ordered magnetization occur. In the regions between the solid and chain lines, there appear the sinusoidally oscillating spin structures. With increasing film thickness, the number of nodes in the sinusoidal spin pattern increases as shown in the figure. In the region below the chain line, we have tbe rotating spin structure, which is similar to the spin-spiral state in the bulk system. The relative stability between the superconducting and normal states is studied at zero temperature. In Figure 2, we show the stable regions of the superconducting ferromagnetic state, superconducting spiral spin state and normal ferromagnetic state, in the plane of 4~M(O)/ Hc(0) and a/% L. Here, M(0) is the saturation magnetization and Hc(0) is the thermodynamic critical field at 0 K in a fictitious system without the magnetic moments. It is found that even if 4~M(0) is well larger than Hc(O), the superconducting ferromagnetic state can be stabilized when a is chosen to be sufficiently small. From a similar calculation at finite temperatures, it is also found that even if the system is in the normal ferromagnetic state at 0 K, the superconducting sinusoidal and spiral spin states can appear when the temperature is increased. In conclusion, we have predicted that in the thin films of ferromagnetic superconductors the ferromagnetic, sinusoidal and spiral spin orderings are possible to occur in coexistence with superconductivity. An extention of the theory is now in progress by taking account of the nonlocal persistent current and the influence of the surface on y(x, x').
L
parra.
0.8
I
~i...
0.6
[2]
A. Kotani, S. Takahashi, M. Tachiki, H. Matsumoto and H. Umezawa, to be published in Solid State Commun. 37 (1981) 619.
.
/i \'-j /
!
!' II
.
"
/e i/
! spiral
k.
0
0.5
Figure i:
a/XL
1.0
15
Magnetic phase diagram in superconducting thin films.
15 fe rro.
~.10 ,,¢
5
super.
~
REFERENCES H. Matsumoto, H. Umezawa and M. Tachiki, Solid State Commun. 31 (1979) 157: M. Tachiki, A. Kotani, H. Matsumoto and H. Umezawa, Solid State Connnun. 31 (1979) 927.
.
i
0 [i]
.
Figure 2:
ferro.
super. spiral
I II
I
0.5
1.0
a/XL
15
Phase diagram at zero temperature.
Physica 108B (1981/1043-1044 North-Holland Publishing Company
INTERPLAY OF SUPERCONDUCTIVITY AND MAGNETISM IN THIN FILMS OF FERROMAGNETIC SUPERCONDUCTORS
A. Kotani, ~ S. Takahashi and M. Tachiki
The Research Institute for Iron, Steel and Other Metals Tohoku University, Katahira, Sendai, 980 Japan and H. Matsumoto and H. Umezawa Department of Physics, The University of Alberta Edmonton, Alberta, T6G 2Jl Canada The effect of the electromagnetic interaction between the persistent current and the rare earth magnetic moments is theoretically studied in thin fiJms of ferromagnetic superconductors. Depending on the film thickness and temperature, the ferromagnetic ordering, sinusoidal spin ordering and spiral spin ordering are predicted to occur in coexistence with superconductivity.
i. INTRODUCTION In the interplay of magnetism and superconductivity in ferromagnetic superconductors, ErRh4B 4 and HoMo6S 8, the electromagnetic interaction between the persistent current and the rare earth magnetic moments plays an essentially important role. Through the electromagnetic interaction, the persistent current screens the long range part of the interaction between the magnetic moments. As a result, a uniform ferromagnetic ordering is strongly suppressed, and instead a spin-spiral ordering with finite wave number occurs in the superconducting bulk crystals.[l] However, in thin films of ferromagnetic superconductors, the screening effect due to the persistent current is weakened in the neighborhood of the film surfaces, and we expect that the coexistence of superconductivity and magnetism occurs in a different way from that in bulk systems.[2] 2. FORMULATION We consider a film of thickness 2a located at -a < x < a, where the x-axls is taken in the direction normal to the film surface. All the physical quantities are assumed to be uniform in the yz plane. In the normal state, the interaction between the rare earth magnetic moments ~(x) and ~(x') is written as
-
-
x')~(x'). i r a dx la dx'm(x)y(x, -> 2 -a -a
In the superconducting state, the persistent current
is induced by the magnetic moments, where %L is the London penetration depth and ~(x) is the vector potential. The magnetic field ~(x) induced by ~(x) is described by the Maxwell equation. × ~(x) = 4-~ ~(x).
(2)
C
Substituting (I) into (2) and using the relation ~(x) = ~ x ~(x) = ~(x) + 4 ~ ( x ) [~(x~ being the magnetic induction], we can express h(x) as a functional of m(x). By using the molecular field approximation, m(x) is determined by solving g~B J m(x) = g~BJNBj[-~BT hm(X)]
(3)
under the condition of ~(x) H ~ m ( X ) , ~:here ~m(X) is the molecular field and Bj[X] is the Brillouin function. The molecular field is given by a sum of ~(x) and the unscreened molecular field: ~m(X) = ~(x) +
dx'y(x, x')m(x').
(4)
--a
3. RESULTS We study what types of magnetic ordering appear as the solutions of (3). For simplicity, we assume that the persistent current 3(x) is local and the coupling y(x, x') is the same as that in the bulk system. Then we can put c(x, x') = ~(x - x') and
Tm3/2 j(x)
-c2 4n~ L
dx c(x, x')~(x')
(1)
y(x, x') = 2C--~7~ exp[- (rra)i/2]XD - x'I]'
-a
* Present address: Department of Physics, Faculty of Science, Osaka University, Toyonaka, 560 Japan
03784363/81/0000~000/$0250
© No~h-HollandPublishingCompany
where Tm and C, respectively, are the Curie temperature and the Curie constant of the fictitious bulk system in the normal state, and D is
1043
1044
the magnetic stiffness constant. In Figure 1 we show the magnetic phase diagram for the superconducting thin films in the plane of the temperature T/T m and the film thickness a/% L. The solid line represents the critical temperature of the spontaneous magnetic ordering. For a/% L 0.5, the superconducting ferromagnetic state occurs below the critical temperature. In the thin film of such a small thickness, the coexistence of ferromagnetism and superconductivity becomes possible, because the screening effect by the persistent current, which suppresses their coexistence in the bulk system, is much weakened due to the influence of the film surfaces. For a/~ L ~ 0.5, the screening effect is restored well inside the film, and the oscillation and rotation of the ordered magnetization occur. In the regions between the solid and chain lines, there appear the sinusoidally oscillating spin structures. With increasing film thickness, the number of nodes in the sinusoidal spin pattern increases as shown in the figure. In the region below the chain line, we have tbe rotating spin structure, which is similar to the spin-spiral state in the bulk system. The relative stability between the superconducting and normal states is studied at zero temperature. In Figure 2, we show the stable regions of the superconducting ferromagnetic state, superconducting spiral spin state and normal ferromagnetic state, in the plane of 4~M(O)/ Hc(0) and a/% L. Here, M(0) is the saturation magnetization and Hc(0) is the thermodynamic critical field at 0 K in a fictitious system without the magnetic moments. It is found that even if 4~M(0) is well larger than Hc(O), the superconducting ferromagnetic state can be stabilized when a is chosen to be sufficiently small. From a similar calculation at finite temperatures, it is also found that even if the system is in the normal ferromagnetic state at 0 K, the superconducting sinusoidal and spiral spin states can appear when the temperature is increased. In conclusion, we have predicted that in the thin films of ferromagnetic superconductors the ferromagnetic, sinusoidal and spiral spin orderings are possible to occur in coexistence with superconductivity. An extention of the theory is now in progress by taking account of the nonlocal persistent current and the influence of the surface on y(x, x').
L
parra.
0.8
I
~i...
0.6
[2]
A. Kotani, S. Takahashi, M. Tachiki, H. Matsumoto and H. Umezawa, to be published in Solid State Commun. 37 (1981) 619.
.
/i \'-j /
!
!' II
.
"
/e i/
! spiral
k.
0
0.5
Figure i:
a/XL
1.0
15
Magnetic phase diagram in superconducting thin films.
15 fe rro.
~.10 ,,¢
5
super.
~
REFERENCES H. Matsumoto, H. Umezawa and M. Tachiki, Solid State Commun. 31 (1979) 157: M. Tachiki, A. Kotani, H. Matsumoto and H. Umezawa, Solid State Connnun. 31 (1979) 927.
.
i
0 [i]
.
Figure 2:
ferro.
super. spiral
I II
I
0.5
1.0
a/XL
15
Phase diagram at zero temperature.