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Amplification of spin waves by moving magnetic flux vortices in magnet-superconductor layered structure
ELSEVIER
J4~ Journalof magnetism , ~ and magnetic ~ materials
Journal of Magnetism and Magnetic Materials 146 (1995) 351-353
Amplification of spin waves by moving magnetic flux vortices in magnet-superconductor layered structure N.I. Polzikova *, A.O. Raevskii Institute of Radioengineering and ElectronicsRussian Academy of Sciences, Moscow 103907, Russian Federation
Abstract A new spin wave amplification mechanism in the structure magnet-type II superconductor is proposed and theoretically investigated. The amplification results from magnetic flux motion due to ac and dc currents in the superconductor and takes place in the region of its electrical stability. The gain may reach 100 dB/cm. The propagation of coherent spin waves (SW) in the magnet-type II superconductor (SC) structure placed in an external magnetic field has been theoretically and experimentally studied [1,2]. Such a field magnetizes the magnet up to saturation and penetrates into the SC in the form of magnetic flux vortices. The ac electromagnetic fields accompanying SW propagation result in vortex motion due to the Lorentz force and SW will undergo the reverse action of this motion. It manifests itself in the change of SW dispersion law and absorption [3,4]. The constant drift current flowing through the SC will also cause vortex motion. When the vortex drift velocity exceeds SW phase velocity, SW amplification will take place. Again the drift current may lead to the N type of SC current-voltage characteristics (CVC). In this case the another mechanism for SW amplification may become available [5,6]. In this paper we consider the possibility for SW amplification at drift velocities less than the SW ~ Although originally accepted for publication in the ICM'94 Proceedings, this paper has not been published therein because it was not presented at the Conference. * Corresponding author. Fax: +7-095-203-8414; email: [email protected].
phase velocities and in the region of positive differential conductivity of the SC. The structure under examination is shown in Fig. 1. In the linear approximation according to the SW amplitude the time and spatial dependence of all variables may be written as e x p [ i ( q x - o~t)], where w is the SW frequency and q is the SW wave number. The SW amplification will take place if the sign change of the SC ac conductivity occurs. The latter should be calculated with allowance for vortex motion due to both ac and dc currents. For this purpose we use the expression for current density I
(E-
(Vxe)
1
(1)
and the vortex motion equation q~0 rlV + Fp = - ~ ( I × B )
(2)
where c is the speed of light, ~ = ic2/4~rA2to is SC conductivity in the absence of vortex motion, A is the London penetration depth, ~7 is the viscous friction coefficient, Fp is pinning force, A 0 is magnetic flux quantum, E = E 0 + e , B = B o + b , I = I 0 + j , V = V 0 + v . Here E 0, B0, I 0, V0 are the dc
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(95)00035-6
N.L Polzikova,A.O. Raeeskii/Journal of Magnetism and MagneticMaterials 146 (1995) 351-353
352
//
1 x
! B°
superconductor magnet
:/
iI
/
q
Fig. 1. The investigated structure and the main vector orientation. (static) values of appropriate quantities, e, b are the electromagnetic fields of the SW, j , v are the ac current density and vortex velocity. Let us consider firstly the case of I o = 0. If oJ << W p - ~ F p / r l, i.e. the pinning force plays the main role. Eqs. (1) and (2) have a solution V = 0, J - - J s = ~ e (Fig. 2). As Re ~ = 0, there is no SW power absorption in this case. In the event of w >> % the pinning force may be omitted as against the viscous friction force. Therefore, the vortices come into motion with the velocity t, = (~o/c~?)j and bunches of magnetic flux vortices arise in accordance with the potential pattern produced by the SW. Then we get from Eqs. (1), (2) j = j + = O'Be, where o"B~c~?/dpoB~<< ] ~ ]. Thus, Re o % > 0 , which means SW power absorption. Let the direct transport current with the density I 0 >> 1c ~-c~o/F p flow along the z-axis in the SC. Then the vortices move in the direction of the SW propagation with the constant speed V0 = loqbo/cr l. e?
(a)
|b)
b
y
lel ~x
~
(d) n ~
. . . . . .
As it follows from Eq. (3), this leads to a substitution o"B ~ O'B(1 -- Vo/Vph), where Vph = w / q is SW phase velocity. This corresponds to the emergence of an additional current density j = - - ( V 0 / V p h ) J + which is out of phase with respect to the current density j÷ (Fig. 2). If the CVC of the SC is nonlinear, i.e. when o"B = o ' B ( E ) = o ' B ( E 0 + e ) , where E 0 = 1 0 / ~ r ~= (Vo/c)B o, an extra linear contribution Jd = ( 0 ° 0 / OEo)Eoe to the total current density arises. As a result, the total current density j = j + + j _ +Ja may be represented as j = cr * e, where ac conductivity of the SC has the form
,,'
=
B(Eo)
2-/+, 3-/_, 4--jd.
-
hVP V° --
+
O°'B(E°) OEo O'B(Eo) E° ) .
(3) It should be noted that the advent of the current densities j_ and Ja favors the vortex bunches to be all the time in phase with ac magnetic induction by (Fig. 2) at any velocity value V0 and at any CVC type. Here lies in the difference between SW amplification by moving vortices in SC with nonlinear CVC and wave amplification by electron flow in a semiconductor with nonlinear CVC [7]. The inequality Re o-* < 0 is a necessary condition for SW instability development. It is important that this condition may be fulfilled also in the case when none of the instability mechanisms considered previously in Refs. [3,6] is carried out. The realization of this condition requires the current density Ja to be out of phase with respect to the current density j+, i.e. -
1 <
< O.
This situation involves in the case of the sublinear CVC. The SW attenuation t o " = Im to is determined from the SW dispersion equation. The latter is found from standard electrodynamic boundary conditions. In the case of thin SC, b << I(V o) = {c2/2'rrtotr .}./2 and in the long-wavelength approximation ql(V 0) << 1, (qd) '/2 << 1 we get
to" = y A H Fig. 2. The phase relationship betwcen electric field (a), current densities (b), magnetic induction (c) and vortex density (d); l-j:
(I
qd
K(O) or • / t r , ( E o )
2 4 + [ K ( 0 ) o " * / e r , ( E 0 ) ] 2 wm
(4)
N.I. Polzikoca, A.O. Raet,skii / Journal of Magnetism and Magnetic Materials 146 (1995) 351-353
where y is the gyromagnetic ratio, ¢,.om ~---4"rryM0, M 0 is magnet saturation magnetization, A H is the half-width of the magnet resonant curve, K ( 0 ) = 2b/ql2(O). To get instability the condition w " < 0 should be satisfied. Let us now consider the nonlinearity mechanism suggested theoretically by Larkin and Ovchinnikov [8]. Such a nonlinearity arises from the nonequilibrium processes in the cores of moving vortices which result in the dependence of the viscous friction coefficient and, hence, the conductivity upon electric field. According to Ref. [8], =
+
E2/E ),
where E, =(BoVF/c){(1- T/Tc)'ro/35} l/z, VF is electron Fermi velocity, T is temperature, Tc is SC transition temperature, ~-p, ~- are the momentum and energy relaxation times respectively. Then o'*
Eo'-----~ OrB( =
(l-x2)
-~'x(l+x
(I q-X2)2
e)
(5)
where x = Eo/E,, ~ = V o ( E , ) / V p h . Let us estimate the effect involved for a high temperature superconductor of Y - B a - C u - O type with Tc = 9 3 K, VF ~ 10 s c m / s e c , (~.p/~.)l/2~ (10- 2 - 1 0 - i) at 77 K, B 0 = 3 × 103 Gs. Then E , ~
353
(10-100) V / c m . Under these parameters values l(0) ~5×10 5 cm. For b = 5 X 1 0 5 cm, d - 10 3 cm, q ~ 102 cm ~ one can obtain that K(0) --- 2. Considering the magnet to be an yttrium-iron garnet ferrite with 4'rrM o (77 K ) = 2 0 0 0 Gs and AH(77 K)_< 1 Oe the total amplification may reach the value up to 10 2 d B / c m at V~)/Vph<_O, 2.
Acknowledgements Work was partly supported by the Russian Scientific Programme "Actual condensed matter problems. Superconductivity" and the Russian Fund of Fundamental Research. [l] A.F. Popkov, Zh, Tekhn. Fiz., 59 (1989) 112. [2] V.B. Anfinogenov, Yu.V. Gulyaev, P.E. Zilberman, I.M. Kotelyanskii, V.B. Kravchenko, N.I. Polzikova, A.A. Sukhanov, Superconductivity: Phys. Chem. Indust. 2 (1989) 7. [3] A.F. Popkov, Zh. Tekhn. Fiz. Pis'ma 15 (1989) 9. [4] N.I. Polzikova and A.O. Raevskii, J. Adv. Sci. 4 11992) 197. [5] N.I, Polzikova and A.O, Raevskii, J. Magn. Magn. Mater, 101 11991) 175. [6] N.I, Polzikova and A.O. Raevskii, Zh. Tckhn. Fiz. Pis'ma 16 1199t)) 73. [7] E.M. Epshtein, Fiz. Tverd. Tela. 8 11966) 274. [8] A.I. Larkin and Yu.N. Ovchinnikov, Soy. Phys. JETP 46 11997) 155.
J4~ Journalof magnetism , ~ and magnetic ~ materials
Journal of Magnetism and Magnetic Materials 146 (1995) 351-353
Amplification of spin waves by moving magnetic flux vortices in magnet-superconductor layered structure N.I. Polzikova *, A.O. Raevskii Institute of Radioengineering and ElectronicsRussian Academy of Sciences, Moscow 103907, Russian Federation
Abstract A new spin wave amplification mechanism in the structure magnet-type II superconductor is proposed and theoretically investigated. The amplification results from magnetic flux motion due to ac and dc currents in the superconductor and takes place in the region of its electrical stability. The gain may reach 100 dB/cm. The propagation of coherent spin waves (SW) in the magnet-type II superconductor (SC) structure placed in an external magnetic field has been theoretically and experimentally studied [1,2]. Such a field magnetizes the magnet up to saturation and penetrates into the SC in the form of magnetic flux vortices. The ac electromagnetic fields accompanying SW propagation result in vortex motion due to the Lorentz force and SW will undergo the reverse action of this motion. It manifests itself in the change of SW dispersion law and absorption [3,4]. The constant drift current flowing through the SC will also cause vortex motion. When the vortex drift velocity exceeds SW phase velocity, SW amplification will take place. Again the drift current may lead to the N type of SC current-voltage characteristics (CVC). In this case the another mechanism for SW amplification may become available [5,6]. In this paper we consider the possibility for SW amplification at drift velocities less than the SW ~ Although originally accepted for publication in the ICM'94 Proceedings, this paper has not been published therein because it was not presented at the Conference. * Corresponding author. Fax: +7-095-203-8414; email: [email protected].
phase velocities and in the region of positive differential conductivity of the SC. The structure under examination is shown in Fig. 1. In the linear approximation according to the SW amplitude the time and spatial dependence of all variables may be written as e x p [ i ( q x - o~t)], where w is the SW frequency and q is the SW wave number. The SW amplification will take place if the sign change of the SC ac conductivity occurs. The latter should be calculated with allowance for vortex motion due to both ac and dc currents. For this purpose we use the expression for current density I
(E-
(Vxe)
1
(1)
and the vortex motion equation q~0 rlV + Fp = - ~ ( I × B )
(2)
where c is the speed of light, ~ = ic2/4~rA2to is SC conductivity in the absence of vortex motion, A is the London penetration depth, ~7 is the viscous friction coefficient, Fp is pinning force, A 0 is magnetic flux quantum, E = E 0 + e , B = B o + b , I = I 0 + j , V = V 0 + v . Here E 0, B0, I 0, V0 are the dc
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(95)00035-6
N.L Polzikova,A.O. Raeeskii/Journal of Magnetism and MagneticMaterials 146 (1995) 351-353
352
//
1 x
! B°
superconductor magnet
:/
iI
/
q
Fig. 1. The investigated structure and the main vector orientation. (static) values of appropriate quantities, e, b are the electromagnetic fields of the SW, j , v are the ac current density and vortex velocity. Let us consider firstly the case of I o = 0. If oJ << W p - ~ F p / r l, i.e. the pinning force plays the main role. Eqs. (1) and (2) have a solution V = 0, J - - J s = ~ e (Fig. 2). As Re ~ = 0, there is no SW power absorption in this case. In the event of w >> % the pinning force may be omitted as against the viscous friction force. Therefore, the vortices come into motion with the velocity t, = (~o/c~?)j and bunches of magnetic flux vortices arise in accordance with the potential pattern produced by the SW. Then we get from Eqs. (1), (2) j = j + = O'Be, where o"B~c~?/dpoB~<< ] ~ ]. Thus, Re o % > 0 , which means SW power absorption. Let the direct transport current with the density I 0 >> 1c ~-c~o/F p flow along the z-axis in the SC. Then the vortices move in the direction of the SW propagation with the constant speed V0 = loqbo/cr l. e?
(a)
|b)
b
y
lel ~x
~
(d) n ~
. . . . . .
As it follows from Eq. (3), this leads to a substitution o"B ~ O'B(1 -- Vo/Vph), where Vph = w / q is SW phase velocity. This corresponds to the emergence of an additional current density j = - - ( V 0 / V p h ) J + which is out of phase with respect to the current density j÷ (Fig. 2). If the CVC of the SC is nonlinear, i.e. when o"B = o ' B ( E ) = o ' B ( E 0 + e ) , where E 0 = 1 0 / ~ r ~= (Vo/c)B o, an extra linear contribution Jd = ( 0 ° 0 / OEo)Eoe to the total current density arises. As a result, the total current density j = j + + j _ +Ja may be represented as j = cr * e, where ac conductivity of the SC has the form
,,'
=
B(Eo)
2-/+, 3-/_, 4--jd.
-
hVP V° --
+
O°'B(E°) OEo O'B(Eo) E° ) .
(3) It should be noted that the advent of the current densities j_ and Ja favors the vortex bunches to be all the time in phase with ac magnetic induction by (Fig. 2) at any velocity value V0 and at any CVC type. Here lies in the difference between SW amplification by moving vortices in SC with nonlinear CVC and wave amplification by electron flow in a semiconductor with nonlinear CVC [7]. The inequality Re o-* < 0 is a necessary condition for SW instability development. It is important that this condition may be fulfilled also in the case when none of the instability mechanisms considered previously in Refs. [3,6] is carried out. The realization of this condition requires the current density Ja to be out of phase with respect to the current density j+, i.e. -
1 <
< O.
This situation involves in the case of the sublinear CVC. The SW attenuation t o " = Im to is determined from the SW dispersion equation. The latter is found from standard electrodynamic boundary conditions. In the case of thin SC, b << I(V o) = {c2/2'rrtotr .}./2 and in the long-wavelength approximation ql(V 0) << 1, (qd) '/2 << 1 we get
to" = y A H Fig. 2. The phase relationship betwcen electric field (a), current densities (b), magnetic induction (c) and vortex density (d); l-j:
(I
qd
K(O) or • / t r , ( E o )
2 4 + [ K ( 0 ) o " * / e r , ( E 0 ) ] 2 wm
(4)
N.I. Polzikoca, A.O. Raet,skii / Journal of Magnetism and Magnetic Materials 146 (1995) 351-353
where y is the gyromagnetic ratio, ¢,.om ~---4"rryM0, M 0 is magnet saturation magnetization, A H is the half-width of the magnet resonant curve, K ( 0 ) = 2b/ql2(O). To get instability the condition w " < 0 should be satisfied. Let us now consider the nonlinearity mechanism suggested theoretically by Larkin and Ovchinnikov [8]. Such a nonlinearity arises from the nonequilibrium processes in the cores of moving vortices which result in the dependence of the viscous friction coefficient and, hence, the conductivity upon electric field. According to Ref. [8], =
+
E2/E ),
where E, =(BoVF/c){(1- T/Tc)'ro/35} l/z, VF is electron Fermi velocity, T is temperature, Tc is SC transition temperature, ~-p, ~- are the momentum and energy relaxation times respectively. Then o'*
Eo'-----~ OrB( =
(l-x2)
-~'x(l+x
(I q-X2)2
e)
(5)
where x = Eo/E,, ~ = V o ( E , ) / V p h . Let us estimate the effect involved for a high temperature superconductor of Y - B a - C u - O type with Tc = 9 3 K, VF ~ 10 s c m / s e c , (~.p/~.)l/2~ (10- 2 - 1 0 - i) at 77 K, B 0 = 3 × 103 Gs. Then E , ~
353
(10-100) V / c m . Under these parameters values l(0) ~5×10 5 cm. For b = 5 X 1 0 5 cm, d - 10 3 cm, q ~ 102 cm ~ one can obtain that K(0) --- 2. Considering the magnet to be an yttrium-iron garnet ferrite with 4'rrM o (77 K ) = 2 0 0 0 Gs and AH(77 K)_< 1 Oe the total amplification may reach the value up to 10 2 d B / c m at V~)/Vph<_O, 2.
Acknowledgements Work was partly supported by the Russian Scientific Programme "Actual condensed matter problems. Superconductivity" and the Russian Fund of Fundamental Research. [l] A.F. Popkov, Zh, Tekhn. Fiz., 59 (1989) 112. [2] V.B. Anfinogenov, Yu.V. Gulyaev, P.E. Zilberman, I.M. Kotelyanskii, V.B. Kravchenko, N.I. Polzikova, A.A. Sukhanov, Superconductivity: Phys. Chem. Indust. 2 (1989) 7. [3] A.F. Popkov, Zh. Tekhn. Fiz. Pis'ma 15 (1989) 9. [4] N.I. Polzikova and A.O. Raevskii, J. Adv. Sci. 4 11992) 197. [5] N.I, Polzikova and A.O, Raevskii, J. Magn. Magn. Mater, 101 11991) 175. [6] N.I, Polzikova and A.O. Raevskii, Zh. Tckhn. Fiz. Pis'ma 16 1199t)) 73. [7] E.M. Epshtein, Fiz. Tverd. Tela. 8 11966) 274. [8] A.I. Larkin and Yu.N. Ovchinnikov, Soy. Phys. JETP 46 11997) 155.