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Phase transitions in the mixed state of superconductors with anisotropic pairing
PHYSICA
Physica B 194-196 (1994) 1971-1972 North-Holland
PHASE T R A N S I T I O N S IN THE MIXED STATE OF S U P E R C O N D U C T O R S WITH ANISOTROPIC PAIRING M. E. Zhitomirsky and I. A. Luk'yanchuk L. D. Landau Institute for Theoretical Physics, Moscow, 117940, Russia The symmetry approach is used to classify the superconducting lattices near H~2 and to show that for the unconventional superconductors the phase transition from one-quantum hexagonal lattice to the lattice with double or triplet magnetic flux per unit sell or with reduced number of rotational elements is quite characteristic. In the context of present theory the H - T phase diagram of UPta with two close superconducting phase transitions is analyzed. The characteristic feature of the mixed (vortex) state for the superconductors with unconventional pairing is that the order parameter (superconducting gap) A varies both in k-(position on Fermi surface) and r-(real) spaces simultaneously. The k-dependence is defined by basis functions • i(k) of the irreducible representation (IR) of the crystal which corresponds to the given type of unconventional pairing:
A(k, r ) = ~ yi(r)(I)i(k).
(1)
i
To know the r-dependence, the multicomponent Ginzburg-Landau (GL) functional (like e.g. the GL functional F = A(T-
Tc)rl*rl + a(rfrt) 2 + bo*2r/2+
G = {U(1),T~,L¢,a,a'HR,]}.
+ K1V~rl~Virlj + K2V~l~.Vj~]jq-}-
K
r"7~
• *t-'7
f
*
*
3 V i 7]j V j T]i "4- I~ 4 V z ?]i V z ~]i ,
He2 requires the solution of nonlinear GLequations originated from (2) which is not easily done because of the multicomponent nature of the order parameter. We propose instead the results of the symmetry analysis of the problem presented in [1,2] which predict the qualitatively new feature of the unconventional superconductors - the structural phase transition in the Abrikosov vortex lattice, make it possible to classify such transitions and explain the stability of the kink in Itch(T) dependence in the heavy fermion superconductor UPta. The appearance of the superconducting vortex lattice below Hc2 means the violation of the symmetry of the Fermiliquid state of the normal metal in a magnetic field which is given by the group:
(2)
2e ( i , j = x , y ; V i = Oi - i-~cAi)
for the widespread two-component model) should be minimized. Treatment of the Abrikosov problem of the appearance of the vortex lattice near
(3)
Where: U(1) - gauge group q2 ---+ e i ~ ; Ta - magnetic translations on vector a; Z 6 - rotation about magnetic field; & and &H - the reflections in the perpendicular and parallel to magnetic field planes; R, I - time and space inversions. Considering this transition in terms of Landau theory of second order phase tran-
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0921-4526(93)1617-U
1972
sitions we came [1,2] to the following conclusions: 1. In the most cases near He2 the mixed state of superconductors with unconventional pairing is the hexagonal vortex lattice. 2. The symmetry properties of the lattice are determined by two quantum numbers (c~,N) of the solutions of linearized GL equations (which determine He2). c~ = 4-1 is the parity under &; N - generalizes the Landau level classification which usually appears in the problem of Hc2 in usual superconductors. Note that crystal field anisotropy changes the N-classifications: if magnetic field is directed along n-fold crystal symmetry axis, it reduces to the classification by N over rnod n. 3. There is no one to one correspondence between pairing type at the uniform state at H = 0 and the symmetry of mixed state near He2: for the same pairing type the different quantum numbers ( a , N ) of A(k, r) can be realized. However the pairing type restricts the possible choice of (a,N). (E.g. for the two-component model, if H 1] z, depending on the coefficients K~ in (1), only quantum numbers: ( + , - 1 ) and (+, +1) are possible [1]. 4. The symmetry of the vortex lattice is given by the group H C G:
Trna, e-2iTr(N+I)/6L2r/6, H~,,~={e To, e i~^ iTr
^
^
(4) 5. Depending on the coefficients before the gradient terms in GL functional, it is highly probable to have the another eigensolution (with quantum numbers (a', N')) of the linearized GL equations with critical field slightly below H~2. It leads to the appearance of the new vortex lattice at the background of the old one and therefore to the structural phase transition (at some critical field H*) ill the vortex lattice. If
the vortices of the new lattice appear on the same positions as the vortices of the old lattice, the residual symmetry group is H~,,,N, n H(,,N which implies the distortion of the initial hexagonal lattice. Otherwise, if the new vortices appear between the old ones, the brake of initial hexagonal symmetry occurs with lattice period multiplication and distortion [1]. Such phase transitions are expected in the mixed state of heavy fermion superconductors which seems to have the unconventional pairing type. 6. The complicated H-T phase diagram of the heavy fermion superconductor UPt3 with double phase transition and the intersection of two upper critical fields H~(T) and Hb(T) is usually related with two close phase transitions to the superconducting states (a and b) which transform according to the different IR of the crystal point symmetry group D6. The quantum number classification of the mixed state solution near IIc2(T) = max{H~(T),Hb(T)} is possible if the magnetic field is directed along symmetry crystal axes. The kink in Hc~(T) dependence can exist only if the phases related to a and b have the different quantum numbers. Otherwise it should be absent or at least smeared out. The phase transition lines which correspond to the prolongation of H~(T) and Hb(T) from kink into the region of the mixed state mean the structural phase transitions in the vortex lattice. For the arbitrary directed magnetic field kink should be always smeared out. REFERENCES
1. M. E. Zhitomirsky, I. A. Luk'yanchuk, Zh. Eksp. Teor. Fiz. 101, 1954 (1992) [Soy. P h y s . - J E T P 74, 1046 (1992)]. 2. I. A. Luk'yanchuk, M. E. Zhitomirsky, Physica C 206, 373 (1993).
Physica B 194-196 (1994) 1971-1972 North-Holland
PHASE T R A N S I T I O N S IN THE MIXED STATE OF S U P E R C O N D U C T O R S WITH ANISOTROPIC PAIRING M. E. Zhitomirsky and I. A. Luk'yanchuk L. D. Landau Institute for Theoretical Physics, Moscow, 117940, Russia The symmetry approach is used to classify the superconducting lattices near H~2 and to show that for the unconventional superconductors the phase transition from one-quantum hexagonal lattice to the lattice with double or triplet magnetic flux per unit sell or with reduced number of rotational elements is quite characteristic. In the context of present theory the H - T phase diagram of UPta with two close superconducting phase transitions is analyzed. The characteristic feature of the mixed (vortex) state for the superconductors with unconventional pairing is that the order parameter (superconducting gap) A varies both in k-(position on Fermi surface) and r-(real) spaces simultaneously. The k-dependence is defined by basis functions • i(k) of the irreducible representation (IR) of the crystal which corresponds to the given type of unconventional pairing:
A(k, r ) = ~ yi(r)(I)i(k).
(1)
i
To know the r-dependence, the multicomponent Ginzburg-Landau (GL) functional (like e.g. the GL functional F = A(T-
Tc)rl*rl + a(rfrt) 2 + bo*2r/2+
G = {U(1),T~,L¢,a,a'HR,]}.
+ K1V~rl~Virlj + K2V~l~.Vj~]jq-}-
K
r"7~
• *t-'7
f
*
*
3 V i 7]j V j T]i "4- I~ 4 V z ?]i V z ~]i ,
He2 requires the solution of nonlinear GLequations originated from (2) which is not easily done because of the multicomponent nature of the order parameter. We propose instead the results of the symmetry analysis of the problem presented in [1,2] which predict the qualitatively new feature of the unconventional superconductors - the structural phase transition in the Abrikosov vortex lattice, make it possible to classify such transitions and explain the stability of the kink in Itch(T) dependence in the heavy fermion superconductor UPta. The appearance of the superconducting vortex lattice below Hc2 means the violation of the symmetry of the Fermiliquid state of the normal metal in a magnetic field which is given by the group:
(2)
2e ( i , j = x , y ; V i = Oi - i-~cAi)
for the widespread two-component model) should be minimized. Treatment of the Abrikosov problem of the appearance of the vortex lattice near
(3)
Where: U(1) - gauge group q2 ---+ e i ~ ; Ta - magnetic translations on vector a; Z 6 - rotation about magnetic field; & and &H - the reflections in the perpendicular and parallel to magnetic field planes; R, I - time and space inversions. Considering this transition in terms of Landau theory of second order phase tran-
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0921-4526(93)1617-U
1972
sitions we came [1,2] to the following conclusions: 1. In the most cases near He2 the mixed state of superconductors with unconventional pairing is the hexagonal vortex lattice. 2. The symmetry properties of the lattice are determined by two quantum numbers (c~,N) of the solutions of linearized GL equations (which determine He2). c~ = 4-1 is the parity under &; N - generalizes the Landau level classification which usually appears in the problem of Hc2 in usual superconductors. Note that crystal field anisotropy changes the N-classifications: if magnetic field is directed along n-fold crystal symmetry axis, it reduces to the classification by N over rnod n. 3. There is no one to one correspondence between pairing type at the uniform state at H = 0 and the symmetry of mixed state near He2: for the same pairing type the different quantum numbers ( a , N ) of A(k, r) can be realized. However the pairing type restricts the possible choice of (a,N). (E.g. for the two-component model, if H 1] z, depending on the coefficients K~ in (1), only quantum numbers: ( + , - 1 ) and (+, +1) are possible [1]. 4. The symmetry of the vortex lattice is given by the group H C G:
Trna, e-2iTr(N+I)/6L2r/6, H~,,~={e To, e i~^ iTr
^
^
(4) 5. Depending on the coefficients before the gradient terms in GL functional, it is highly probable to have the another eigensolution (with quantum numbers (a', N')) of the linearized GL equations with critical field slightly below H~2. It leads to the appearance of the new vortex lattice at the background of the old one and therefore to the structural phase transition (at some critical field H*) ill the vortex lattice. If
the vortices of the new lattice appear on the same positions as the vortices of the old lattice, the residual symmetry group is H~,,,N, n H(,,N which implies the distortion of the initial hexagonal lattice. Otherwise, if the new vortices appear between the old ones, the brake of initial hexagonal symmetry occurs with lattice period multiplication and distortion [1]. Such phase transitions are expected in the mixed state of heavy fermion superconductors which seems to have the unconventional pairing type. 6. The complicated H-T phase diagram of the heavy fermion superconductor UPt3 with double phase transition and the intersection of two upper critical fields H~(T) and Hb(T) is usually related with two close phase transitions to the superconducting states (a and b) which transform according to the different IR of the crystal point symmetry group D6. The quantum number classification of the mixed state solution near IIc2(T) = max{H~(T),Hb(T)} is possible if the magnetic field is directed along symmetry crystal axes. The kink in Hc~(T) dependence can exist only if the phases related to a and b have the different quantum numbers. Otherwise it should be absent or at least smeared out. The phase transition lines which correspond to the prolongation of H~(T) and Hb(T) from kink into the region of the mixed state mean the structural phase transitions in the vortex lattice. For the arbitrary directed magnetic field kink should be always smeared out. REFERENCES
1. M. E. Zhitomirsky, I. A. Luk'yanchuk, Zh. Eksp. Teor. Fiz. 101, 1954 (1992) [Soy. P h y s . - J E T P 74, 1046 (1992)]. 2. I. A. Luk'yanchuk, M. E. Zhitomirsky, Physica C 206, 373 (1993).