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Vortex phases of an unconventional superconductor with two components order parameter
Physica B 281&282 (2000) 947}948
Vortex phases of an unconventional superconductor with two components order parameter N. Ogawa!,*, M.E. Zhitomirsky",# !Department of Fundamental Science, Gifu National College of Technology, Sinsei-cyo, Gifu 501-0495, Japan "Theoretische Physik, ETH-Ho( nggerberg, Zu( rich, Switzerland #L.D. Landau Institute for Theoretical Physics, Moscow, Russia
Abstract We examined the single vortex structure of superconductor with the order parameter of two components by numerical calculation. Under certain conditions in the case of time-reversal preserving phase a single vortex dissociates into a pair of half-quantum vortices. We determine the phase diagram for these vortices in the space of parameters of the Ginzburg}Landau functional. A novel characteristic length which is longer than the coherence length m and shorter than the magnetic penetration length j is introduced. ( 2000 Elsevier Science B.V. All rights reserved. L0/$0/ Keywords: Unconventional superconductivity; Vortex state; Phase diagrams
1. Introduction In unconventional superconductors, such as heavy fermion superconductors; UBe and UPt and Sr RuO , 13 3 2 4 the presence of a multi-dimensional order parameter plays an important role in the appearance of complex superconducting phase diagrams [1,2]. Various superconducting states may also be distinguished by the internal structure of their vortices in the mixed state [3}7]. In this study we examined the single-vortex structure of an unconventional superconductor for a model with a twodimensional order parameter. We report the numerical results of the vortex phases and novel characteristic length for the popular model of the Ginzburg}Landau (GL) functional for E irreducible representation of the 1 hexagonal point group D in this paper. 6) 2. Vortex phases of an unconventional superconductor We consider the two-component GL functional F"!a(gH ) g)#b (gH ) g)2#b Dg ) gD2 1 2 # K DHgHD g #K DHgHD g 1 i j i j 2 i i j j * Corresponding author. Fax: #81-58-320-1263. E-mail address: [email protected] (N. Ogawa)
# K DHgHD g #K DHgHD g 3 i j j i 4 z i z i #
h2 h ) H ! , 8p 4p
where the superconducting order parameter is DK (k, r)"g (r)UK (k)#g (r)UK (k) with basis functions x x y y UK (k) transforming according to E representation of the i 1 point group D and D "L !i(2e/+c)A (k"x, y, z). 6) k k k In the presence of strong spin}orbit coupling the formula gives no di!erence between GL functionals for odd and even parity representations. It is invariant under continuous rotations about the z-axis. In the absence of external magnetic "eld H, two kinds of homogeneous superconducting states, complex with broken time reversal symmetry for b '0 and real preserving time-reversal 2 symmetry for b (0, exist. 2 In the time-reversal breaking phase two kinds of vortex states were investigated by Tokuyasu et al. [5], while in this study we report on the numerical calculation results of the vortex structure for a two-component superconductor in the case where the zero-"eld phase does not break time-reversal symmetry. We de"ne new variables c"(K #K )/2K , d"(K !K )/2K , and 2 3 1 2 3 1 p"c/(1#c) for further discussions. Possible types of isolated vortices which appear at H"H are researched c1
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 9 7 7 - 1
948
N. Ogawa, M.E. Zhitomirsky / Physica B 281&282 (2000) 947}948
Fig. 1. Spatial variation of the order parameter. The unit length m is J(K #K )/a¹ where a"a(¹ !¹). (i"10, p"0.2, 0 1 2 c c b /b "!0.3, ¹"0.75¹ ). 2 1 c
Fig. 2. Phase diagram of single vortex states for a two-component unconventional superconductor. The dimension less value p is constructed by the coe$cients of the gradient terms as p"c/(1#c) where c"(K #K )/2K . 2 3 1
for the "eld parallel to the z-axis. Spatial variation of the order parameter for typical molecule vortex (a pair of half-quantum vortices) is shown in Fig. 1. Each of two components of the order parameter in the molecule vortex has its core at separate places. The new phase boundary which distinguishes two vortex states in the p and b plane was determined for 2 di!erent i values in our numerical study (Fig. 2). At the right side of the boundary line the vortex is the molecule vortex, while at left side, both components of the order parameter have its core at the same place. This phase boundary was determined from the following procedure. First, the vortex structure is calculated for various b /b for the "xed p and the size of molecule 2 1 vortex is determined. Then, we extrapolate to the value of b where the distance between the two vortex centers 2 vanishes.
Fig. 3 . i dependence of the distance f between two halfquantum vortices. The distance f is measured by the unit length m . (p"0.2 and b /b "!0.2). 0 2 1
The phase boundary line between the two vortex structures approaches b "!0.5b for p"0 as the 2 1 value of i increases. This result is consistent with the result [7] which is analytically obtained in the limit i<1. The two points with the same b and $p have the 2 same energy, because the sign of p can be simply incorporated in one of the components g $ig . x y The i dependence of the distance between cores corresponding to two elements of the order parameter for the dissociated vortex is calculated. The result for a typical parameter is shown in Fig. 3. f (the distance between cores) seems saturated to a value as i grows. The distance f gives a new characteristic length. The scale of this length is longer than the coherence length m and shorter than the magnetic penetration length j . L0/$0/ Acknowledgements We are grateful to Kazuo Ueda and Manfred Sigrist for useful discussions. This work was performed on the supercomputer at Nagoya University and at University of Kyoto.
References [1] [2] [3] [4]
M. Sigrist, K. Ueda, Rev. Mod. Phys. 63 (1991) 239. H. von LoK hneysen, Physica B 197 (1994) 551. J.A. Sauls, Adv. Phys. 43 (1994) 113. M. Sigrist et al., Proceedings of the Conference Anomalous Complex Superconductors, to be published. [5] T.A. Tokuyasu, D.W. Hess, J.A. Sauls, Phys. Rev. B 41 (1990) 8891. [6] N. Ogawa, M. Sigrist, K. Ueda, Physica B 186}188 (1993) 856. [7] M.E. Zhitomirsky, J. Phys. Soc. Japan 64 (1995) 913.
Vortex phases of an unconventional superconductor with two components order parameter N. Ogawa!,*, M.E. Zhitomirsky",# !Department of Fundamental Science, Gifu National College of Technology, Sinsei-cyo, Gifu 501-0495, Japan "Theoretische Physik, ETH-Ho( nggerberg, Zu( rich, Switzerland #L.D. Landau Institute for Theoretical Physics, Moscow, Russia
Abstract We examined the single vortex structure of superconductor with the order parameter of two components by numerical calculation. Under certain conditions in the case of time-reversal preserving phase a single vortex dissociates into a pair of half-quantum vortices. We determine the phase diagram for these vortices in the space of parameters of the Ginzburg}Landau functional. A novel characteristic length which is longer than the coherence length m and shorter than the magnetic penetration length j is introduced. ( 2000 Elsevier Science B.V. All rights reserved. L0/$0/ Keywords: Unconventional superconductivity; Vortex state; Phase diagrams
1. Introduction In unconventional superconductors, such as heavy fermion superconductors; UBe and UPt and Sr RuO , 13 3 2 4 the presence of a multi-dimensional order parameter plays an important role in the appearance of complex superconducting phase diagrams [1,2]. Various superconducting states may also be distinguished by the internal structure of their vortices in the mixed state [3}7]. In this study we examined the single-vortex structure of an unconventional superconductor for a model with a twodimensional order parameter. We report the numerical results of the vortex phases and novel characteristic length for the popular model of the Ginzburg}Landau (GL) functional for E irreducible representation of the 1 hexagonal point group D in this paper. 6) 2. Vortex phases of an unconventional superconductor We consider the two-component GL functional F"!a(gH ) g)#b (gH ) g)2#b Dg ) gD2 1 2 # K DHgHD g #K DHgHD g 1 i j i j 2 i i j j * Corresponding author. Fax: #81-58-320-1263. E-mail address: [email protected] (N. Ogawa)
# K DHgHD g #K DHgHD g 3 i j j i 4 z i z i #
h2 h ) H ! , 8p 4p
where the superconducting order parameter is DK (k, r)"g (r)UK (k)#g (r)UK (k) with basis functions x x y y UK (k) transforming according to E representation of the i 1 point group D and D "L !i(2e/+c)A (k"x, y, z). 6) k k k In the presence of strong spin}orbit coupling the formula gives no di!erence between GL functionals for odd and even parity representations. It is invariant under continuous rotations about the z-axis. In the absence of external magnetic "eld H, two kinds of homogeneous superconducting states, complex with broken time reversal symmetry for b '0 and real preserving time-reversal 2 symmetry for b (0, exist. 2 In the time-reversal breaking phase two kinds of vortex states were investigated by Tokuyasu et al. [5], while in this study we report on the numerical calculation results of the vortex structure for a two-component superconductor in the case where the zero-"eld phase does not break time-reversal symmetry. We de"ne new variables c"(K #K )/2K , d"(K !K )/2K , and 2 3 1 2 3 1 p"c/(1#c) for further discussions. Possible types of isolated vortices which appear at H"H are researched c1
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 9 7 7 - 1
948
N. Ogawa, M.E. Zhitomirsky / Physica B 281&282 (2000) 947}948
Fig. 1. Spatial variation of the order parameter. The unit length m is J(K #K )/a¹ where a"a(¹ !¹). (i"10, p"0.2, 0 1 2 c c b /b "!0.3, ¹"0.75¹ ). 2 1 c
Fig. 2. Phase diagram of single vortex states for a two-component unconventional superconductor. The dimension less value p is constructed by the coe$cients of the gradient terms as p"c/(1#c) where c"(K #K )/2K . 2 3 1
for the "eld parallel to the z-axis. Spatial variation of the order parameter for typical molecule vortex (a pair of half-quantum vortices) is shown in Fig. 1. Each of two components of the order parameter in the molecule vortex has its core at separate places. The new phase boundary which distinguishes two vortex states in the p and b plane was determined for 2 di!erent i values in our numerical study (Fig. 2). At the right side of the boundary line the vortex is the molecule vortex, while at left side, both components of the order parameter have its core at the same place. This phase boundary was determined from the following procedure. First, the vortex structure is calculated for various b /b for the "xed p and the size of molecule 2 1 vortex is determined. Then, we extrapolate to the value of b where the distance between the two vortex centers 2 vanishes.
Fig. 3 . i dependence of the distance f between two halfquantum vortices. The distance f is measured by the unit length m . (p"0.2 and b /b "!0.2). 0 2 1
The phase boundary line between the two vortex structures approaches b "!0.5b for p"0 as the 2 1 value of i increases. This result is consistent with the result [7] which is analytically obtained in the limit i<1. The two points with the same b and $p have the 2 same energy, because the sign of p can be simply incorporated in one of the components g $ig . x y The i dependence of the distance between cores corresponding to two elements of the order parameter for the dissociated vortex is calculated. The result for a typical parameter is shown in Fig. 3. f (the distance between cores) seems saturated to a value as i grows. The distance f gives a new characteristic length. The scale of this length is longer than the coherence length m and shorter than the magnetic penetration length j . L0/$0/ Acknowledgements We are grateful to Kazuo Ueda and Manfred Sigrist for useful discussions. This work was performed on the supercomputer at Nagoya University and at University of Kyoto.
References [1] [2] [3] [4]
M. Sigrist, K. Ueda, Rev. Mod. Phys. 63 (1991) 239. H. von LoK hneysen, Physica B 197 (1994) 551. J.A. Sauls, Adv. Phys. 43 (1994) 113. M. Sigrist et al., Proceedings of the Conference Anomalous Complex Superconductors, to be published. [5] T.A. Tokuyasu, D.W. Hess, J.A. Sauls, Phys. Rev. B 41 (1990) 8891. [6] N. Ogawa, M. Sigrist, K. Ueda, Physica B 186}188 (1993) 856. [7] M.E. Zhitomirsky, J. Phys. Soc. Japan 64 (1995) 913.