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Magnetic properties of time-reversal breaking superconductors
Journal of Magnet ism-and Magnetic Materials 90 & 91 (1990) 653-654 North-Holland
653
Magnetic properties of time-reversal breaking superconductors M. Sigrist, T.M. Rice
1
and K. Veda
University 0/ Tsukuba, Insitute0/ Materials Science, Tsukuba, Ibaraki 305, Japan Time-reversal breaking superconductors yield local magnetization at inhomogeneities (domain wall, surface. etc.) of the superconducting phase. The magnetization at the surface produces a small paramagnetic response of this superconductor. In such superconductors also vortices with fractional nux quanta are stable on domain walls.
Volovik and Gor'kov noticed that a superconducting state with a multi-component order parameter yields special magnetic properties, if it breaks time-reversal symmetry [1]. This fact has essentially two origins: (1) The supercurrent expression in a multi-component has in general tensorial structure. Therefore, especially in the case of time-reversal breaking superconductors, the spatial variation of the order parameter can produce unconventional currents. (2) Several time-reversal breaking superconducting states possess intrinsic magnetic moments, which are not observable in the bulk region because of the Meissner effect, bur occur at inhomogeneities due to imperfect screening. These properties have been studied on the example of the two-component superconductor with the evenparity order parameter J(k) = iai"kzkx+ vkzky ) in the tetragonal lattice D.4h (II and v are complex functions in space). According to the phase transition shceme of the corresponding Ginzburg-Landau-theory a possible stable state in this system can be of the form (II, v) a (1, ± i), time-reversal breaking and two-fold degenerate. It can be shown that this state has an intrinsic magnetic moment ± UiMUdz(± depending on the relative sign in (1 ± i». In an easy way the magnetic properties of this superconducting phase can be seen at a domain wall (e.g. lying in y-z-plane) between the two states (1, + i) and (I, - i), The continuous variation from one state to the other leads to a variation of the phase of J(k). From the tensorial current expression we obtain a supercurrent which flows along the domain wall (in j-direction, perpendicular to the spatial dependence of the order parameter) inducing a magnetic field B, with opposite sign on both sides of the wall. Of course it must be screened towards the bulk region. Hence it occurs only close to the wall and the net magnetization is vanishing 1
Thcoretlsche Physik. Eidgenossische Technische HochschuleHonggerberg, 8093 ZUrich, Switzerland .
since the magnetic fields on both sides cancel each other. A similar behavior can be found at the surface of the superconductor. where the two components II and v vary differently in space due to different boundary scattering properties. Also in this case the variation of the phase of J(k) produces a current along the surface generating a magnetic field. Differently to the domain wall this field can lead to a finite net magnetization (± M, depending on the relative sign in (1, ± i». A one domain sample could possess a finite magnetization due to the superconducting state. Another unconventional effect is found for multi-domain smaples. Since the surface magnetization has opposite sign in the two types of domains, an external magnetic field leads there .to different free energy densities. By a movement of the domain walls. enlarging the domains with their magnetization parallel to the external field. the total energy can be lowered (fig. 1). This is a small paramagnetic response of this system [2]. This may be a mechanism to explain measurements of paramagnetic response in
()
Hext
()
o
-------.--------Sf
000000
o o o o o o
EBEBEBEBEBEB
EB~
EB EB EB EB EB
OW Fig. 1. Paramagnetic response of the superconductor via a domain wall movement due to different surface magnetization in neighboring domains (DW: domain wall; Sf: surface).
0304-8853/90/503.50 €) 1990 - Elsevier Science Publishers B.V. (North-Holland) and Yamada Science Foundation
654
M. Sigrist et al. / Time-reversal breakingsuperconductors
several unconventional superconductors for external fields which are below the lower critical field (conventional nux creep is excluded) [3]. Finally, it has to be mentioned that also inhomogeneities of the superconducting state around impurities can produce local magnetic moments [4]. The topological structure of vortices in this type of superconductor can be multivaious. In principle each component of the order parameter can form a vortex with fractional nux quantum by winding its phase around a line singularity. However, this is energetically disadvantageous, since the winding of only one component changes the superconducting state in whole sytem by variation of the relative phase. In a discontinuously degenerate system this leads to a huge energy expense. Thus it is more favorable to wind the phases of all components simultaneously creating a vortex with integer standard nux quantum lP = hc/2e. However, the singularity lines of each component need not coincide. They can be separated by a small distance (== ~). Hence vortices are possible which have no normal core [5]. On the other hand it is possible to produce fractional vortices in specially prepared systems, e.g. if a domain wall is present with degenerate structure. Then a line defect, similar to a Bloch line in a ferromagnet, can be created .which corresponds to a phase winding of one
component of the order parameter around this line. This vortex has a nux quantum smaller than the standard nux quantum lP. Its line energy and the corresponding critical field are smaller than that of a bulk vortex. The magnitude of the fractional nux quantum depends not only on the winding component, but also on the structure and especially the geometry of the domain wall. Therefore the lower critical magnetic field may not be so well defined as in conventional superconductors [2]. Time -reversal breaking superconductors can produce a series of unconventional effects which could lead to a clear identification of the superconducting state.
References [I] G.E. Volovik and L.P. Gor'kov, Zh. Eksp, Teor. Fiz. 88 (1985) 1412 [Sov. Phys. JETP 61 (1985) 843]. [2] M. Sigrist, T.M. Rice and K. Veda, Phys. Rev. Lett. 63 (1989) 1727. [3] s .c. Mota, G. Juri , P. Visani and A. Pollini, Physica C 162-164 (1989) 1152. [4] C.H. Choi and P. Muzikar, Phys. Rev. B 39 (1989) 9664. [5] T.A. Tokuyasu, D.W. Hess and J.A. Sauls, (to be published).
653
Magnetic properties of time-reversal breaking superconductors M. Sigrist, T.M. Rice
1
and K. Veda
University 0/ Tsukuba, Insitute0/ Materials Science, Tsukuba, Ibaraki 305, Japan Time-reversal breaking superconductors yield local magnetization at inhomogeneities (domain wall, surface. etc.) of the superconducting phase. The magnetization at the surface produces a small paramagnetic response of this superconductor. In such superconductors also vortices with fractional nux quanta are stable on domain walls.
Volovik and Gor'kov noticed that a superconducting state with a multi-component order parameter yields special magnetic properties, if it breaks time-reversal symmetry [1]. This fact has essentially two origins: (1) The supercurrent expression in a multi-component has in general tensorial structure. Therefore, especially in the case of time-reversal breaking superconductors, the spatial variation of the order parameter can produce unconventional currents. (2) Several time-reversal breaking superconducting states possess intrinsic magnetic moments, which are not observable in the bulk region because of the Meissner effect, bur occur at inhomogeneities due to imperfect screening. These properties have been studied on the example of the two-component superconductor with the evenparity order parameter J(k) = iai"kzkx+ vkzky ) in the tetragonal lattice D.4h (II and v are complex functions in space). According to the phase transition shceme of the corresponding Ginzburg-Landau-theory a possible stable state in this system can be of the form (II, v) a (1, ± i), time-reversal breaking and two-fold degenerate. It can be shown that this state has an intrinsic magnetic moment ± UiMUdz(± depending on the relative sign in (1 ± i». In an easy way the magnetic properties of this superconducting phase can be seen at a domain wall (e.g. lying in y-z-plane) between the two states (1, + i) and (I, - i), The continuous variation from one state to the other leads to a variation of the phase of J(k). From the tensorial current expression we obtain a supercurrent which flows along the domain wall (in j-direction, perpendicular to the spatial dependence of the order parameter) inducing a magnetic field B, with opposite sign on both sides of the wall. Of course it must be screened towards the bulk region. Hence it occurs only close to the wall and the net magnetization is vanishing 1
Thcoretlsche Physik. Eidgenossische Technische HochschuleHonggerberg, 8093 ZUrich, Switzerland .
since the magnetic fields on both sides cancel each other. A similar behavior can be found at the surface of the superconductor. where the two components II and v vary differently in space due to different boundary scattering properties. Also in this case the variation of the phase of J(k) produces a current along the surface generating a magnetic field. Differently to the domain wall this field can lead to a finite net magnetization (± M, depending on the relative sign in (1, ± i». A one domain sample could possess a finite magnetization due to the superconducting state. Another unconventional effect is found for multi-domain smaples. Since the surface magnetization has opposite sign in the two types of domains, an external magnetic field leads there .to different free energy densities. By a movement of the domain walls. enlarging the domains with their magnetization parallel to the external field. the total energy can be lowered (fig. 1). This is a small paramagnetic response of this system [2]. This may be a mechanism to explain measurements of paramagnetic response in
()
Hext
()
o
-------.--------Sf
000000
o o o o o o
EBEBEBEBEBEB
EB~
EB EB EB EB EB
OW Fig. 1. Paramagnetic response of the superconductor via a domain wall movement due to different surface magnetization in neighboring domains (DW: domain wall; Sf: surface).
0304-8853/90/503.50 €) 1990 - Elsevier Science Publishers B.V. (North-Holland) and Yamada Science Foundation
654
M. Sigrist et al. / Time-reversal breakingsuperconductors
several unconventional superconductors for external fields which are below the lower critical field (conventional nux creep is excluded) [3]. Finally, it has to be mentioned that also inhomogeneities of the superconducting state around impurities can produce local magnetic moments [4]. The topological structure of vortices in this type of superconductor can be multivaious. In principle each component of the order parameter can form a vortex with fractional nux quantum by winding its phase around a line singularity. However, this is energetically disadvantageous, since the winding of only one component changes the superconducting state in whole sytem by variation of the relative phase. In a discontinuously degenerate system this leads to a huge energy expense. Thus it is more favorable to wind the phases of all components simultaneously creating a vortex with integer standard nux quantum lP = hc/2e. However, the singularity lines of each component need not coincide. They can be separated by a small distance (== ~). Hence vortices are possible which have no normal core [5]. On the other hand it is possible to produce fractional vortices in specially prepared systems, e.g. if a domain wall is present with degenerate structure. Then a line defect, similar to a Bloch line in a ferromagnet, can be created .which corresponds to a phase winding of one
component of the order parameter around this line. This vortex has a nux quantum smaller than the standard nux quantum lP. Its line energy and the corresponding critical field are smaller than that of a bulk vortex. The magnitude of the fractional nux quantum depends not only on the winding component, but also on the structure and especially the geometry of the domain wall. Therefore the lower critical magnetic field may not be so well defined as in conventional superconductors [2]. Time -reversal breaking superconductors can produce a series of unconventional effects which could lead to a clear identification of the superconducting state.
References [I] G.E. Volovik and L.P. Gor'kov, Zh. Eksp, Teor. Fiz. 88 (1985) 1412 [Sov. Phys. JETP 61 (1985) 843]. [2] M. Sigrist, T.M. Rice and K. Veda, Phys. Rev. Lett. 63 (1989) 1727. [3] s .c. Mota, G. Juri , P. Visani and A. Pollini, Physica C 162-164 (1989) 1152. [4] C.H. Choi and P. Muzikar, Phys. Rev. B 39 (1989) 9664. [5] T.A. Tokuyasu, D.W. Hess and J.A. Sauls, (to be published).