- Email: [email protected]
Theoretical study of unconventional superconducting classes in heavy Fermion compounds
PilYSICA
Physica C 185-189 (1991) 2627-2628 North-Holland
THEORETICAL STUDY OF UNCONVENTIONAL IN H E A V Y F E R M I O N C O M P O U N D S
SUPERCONDUCTING
CLASSES
Kazushige MACHIDA and Masa-aki OZAKI* Department of Physics, Okayama University, Okayama 700, Japan =Niihama National college of Technology, Niihama 792, Japan In order to narrow down possible non-trivial superconducting pairing functions realized in several heavy Fermion materials, a GL theory is developed in connection with the coexisting antiferromagnetism. In UPta an odd-parity state is identified to explain the multiple phase diagram observed. CeCu2Si2 is also analyzed in light of this coexistence. 1. INTRODUCTION
Above the upper critical field several experiments detect
Much attention has been focused on the so-called heavy Fermion materials, in particular, on their unconventional superconductivity. Although precise superconducting pairing classes for the known four compounds; UPta, UBela , URu2Si2 and CeCu2Si~ have not been identified yet, the
some anomalies, signaling the AF transition. Thus in H vs T phase diagram the SC region seems to be covered by the AF. Here we try to narrow down possible pairing classes for these materials, in particular, for UPt3 and CeCu:Si2. 2. THE UPt3 PROBLEM
growing experimental and theoretical evidence undoubtedly indicates that the pairing classes realized are non-trivial, it
The double transition phenomenon was interpreted as
has become clear that all these heavy Fermion superconduc-
arising from splitting the orbital degeneracy of a two-
tors are intimately related to antiferromagnetisln (AF). In
dimensional representation. The pairing function was identi-
the former three compounds the A F occurs at around TN ~-
fied az a vector order parameter (2D-REP scenario 1 ). How-
10.Tc("10.Tc empirical rule") where TN is the Nrel tempera-
ever, according to this scenario the crossing of the two He2
ture. Below the superconducting (SC) transition (To) these
lines occurs only when M Z H
two long-range orders A F and SC coexist microscopically in
which is i~,lco::',radicti(m with the experimental facts where
a sample.
the crossing always appears irrespective of the field direc-
in the basal plane of Doh.
Among these systems UPta is most throughly studied to
tion. In order to explain this fact, we take up an odd-parity
yield definitive evidence for the non-trivial pairing function
state belonging to the one-dimensional representation (1D-
with internal degrees of freedom because the phase diagram
REP scenario 2 ) in the weak spin-orbit coupling. We assume
in the H vs.T plane consists of multiple phases, at least three
that the subla.ttice moment M is rotatable within the basal
phases, where the A F splitts T¢ of an unconventional super-
plane so to keep M . I _ H . We can derive a non-trivial T¢
conducting state with a pairing function which belongs to a
splitting term of the two order parameters M and rh where
degenerate irreducible representation. The magnetism is a
A(k)
driving mechanism to pair electrons in a non-trivial SC state
two transition points Td and T:2; below G l , r/; starts to
, but also plays an impdrtant role to govern the supercon-
grow, followed by % and rl: at 7 ~ ( < T~: ).
= Ei=~,~,~rhl(k)r..
A GL functional gives rise to the
Let us now consider the external field H by ex!endiag
ducting pairing states themselves. In contrast with these three systems in which the in-
the above GL functional. In conlras: with 2D-REP sc
terplay between A F and SC is established experimentally,
the gradient terms do not mix the three components 0, in
However,
1D-REP. Since an odd-parity state has a H 2 term which is
several experiments clearly suggest this proximity: Various
quadratic in 7?. This term is important in giving rise to the
experiments observe the magnetism just at T~=0.6K in the
crossing of the two He2 lines wlfich are otherwise pa~alleI.
absence of an external field H. In (Cei_xTh~:)Cu2Si: a few
Namely, when H / / y ( / / z ) .
percent doping of Th ion induces AF around TN=2--,3K.
ally overcoming r/~ in h:gh fields. Thus we obtain ~he topo-
CeCu2Si2 has been studied less in this respect.
0921-4534/91/$03.50 © 1991 - Elsevier Scicnce Publishers B.V. All rights reserved.
% (rl:) is favored over r]:-men,u-
K Machida, M.-,4. Ozaki / Uncoventional superconducting classes in heavy fermion compounds
2628
logically same phase diagrams for both field directions. Be-
states in general where in the Z-point AF's the moment ro-
cause we assumed that M.I.H in the basal plane, the phase
tates below the lower TN within the basal plane and in the
diagrams are essentially same irrespective of the directions.
X-point AF's the mixing of the AF's which have different Q
We can derive various physical quantities 2 associated with
vectors and spin directions would occur.
these second order transitions such as the specific heat jump or the lower critical field. As for the nodal structure of the 1D-REP with odd-parity the orbital part I(k) could be l(k) = kz or k,(k~+ k~). These states have a line node in the basal plane, which is consistent with various thermodynamic and transport measurements. We emphasize that the 1D-REP with odd-parity in weak spin-orbit coupling (SO) is only the remaining possibility axaong eight possible combinations of odd- vs. even-parity, weak vs. strong spin orbit coupling and 1D- vs. 2D-REP for a pairing function. This case fits to the existing experiments, excluding either singlet 1D-REP states in weak SO case or singlet and triplet 1D-REP states in strong SO case, all of which do not give rise to the splitting term due to AF. 3. THE CeCu2Si2 PROBLEM
4. CONCLUSION We have demonstrated for UPt3 that the one-dimensional pairing function with odd-parity in weak spin-orbit coupling can explain not only the splitting of the transition temperatures at H = 0, but also the isotropy of the phase diagram consisting of multiple phases under various applied field directions, provided that the sublattice moment rotates so as to keep M_I.H within the basal plane. These features were difficult to understand in terms of the 2D pairing function scenario 1. The identified pairing function has the desired properties on the nodal structure and the parity which are revealed experimentally. Theoretically, Choi and Sanls through the analysis of Hc2 and Norman through the pairing mechanism due to spin fluctuations have identified an odd-parity state rather than an even-parity state as favor-
In a similar way, by enumerating group-theoretically the
able pairing symmetry in UPt3. According to our scenario,
AF allowed under the crystalline symmetry in CeCu2Siz the conditions are examined under which a double transitions
even when the AF has disappeared by applied pressure the
of either SC or AF states occurs. We can draw the follow-
applied fields because of t h e / / 2 term . In that case the up-
ing general conclusions for each combination of various AF
per He2 does not exhibit a kink. The phenomenon is quite
and SC states where AF's are characterized by the ordering
similar to the AI - A2 transition in superfluid 3//e. Therefore
vector Q located in the Brillouin zone and SC's belong to a
it is a crucial test to see the phase transition below Td above
2D-REP:
Pc~ ~- 3kbar.
(1) TN>T¢ :
triply spin-degenerate states; r/r , %, r/z should be split under
As for CeCu2Si2 we predict that for almost all simple
(l-a) The X-point AF can cause the Tc splitting where
AF states a splitting of the transition temperature Tc or TN
the moment could be either parallel to the c-a.~ds or within
can he induced by a non-trivial coupling of the two order
the basal plane and the ordering vectors could be either single
paramotor% provided that the existing SC state belgngs to a
Q or double Q.
degenerate representation.
(l-b) The Z-point AF can cause the Tc splitting only when the moment lies in the basal plane and the ordering vector should be single. (2) Tc>TN : (2-a) Among the three possible SC states the SC3 (r/,jTy) = (1,i) never induces the TN splitting for any AF's. (2-b) The remaining SC2 (1,0) and SC3 (1,1) can il, duce the AF double transition for both Z-point and X-point AF
REFERENCES 1. K. Machida and M. Ozaki, J. Phys. Soc. Jpn. 58 (1989) 2244. K. Machida, M. Ozaki and T. Ohmi, ibid. 58 (i989) 4 i i 5 and references therein. 2. K. Machida and M. Ozaki, Phys. Rev. Lett. 66 (1991) 3293. 3. M. Ozaki and K. Machida, J. Phys. Soc. Jpn. 60 (1991) 1452.
Physica C 185-189 (1991) 2627-2628 North-Holland
THEORETICAL STUDY OF UNCONVENTIONAL IN H E A V Y F E R M I O N C O M P O U N D S
SUPERCONDUCTING
CLASSES
Kazushige MACHIDA and Masa-aki OZAKI* Department of Physics, Okayama University, Okayama 700, Japan =Niihama National college of Technology, Niihama 792, Japan In order to narrow down possible non-trivial superconducting pairing functions realized in several heavy Fermion materials, a GL theory is developed in connection with the coexisting antiferromagnetism. In UPta an odd-parity state is identified to explain the multiple phase diagram observed. CeCu2Si2 is also analyzed in light of this coexistence. 1. INTRODUCTION
Above the upper critical field several experiments detect
Much attention has been focused on the so-called heavy Fermion materials, in particular, on their unconventional superconductivity. Although precise superconducting pairing classes for the known four compounds; UPta, UBela , URu2Si2 and CeCu2Si~ have not been identified yet, the
some anomalies, signaling the AF transition. Thus in H vs T phase diagram the SC region seems to be covered by the AF. Here we try to narrow down possible pairing classes for these materials, in particular, for UPt3 and CeCu:Si2. 2. THE UPt3 PROBLEM
growing experimental and theoretical evidence undoubtedly indicates that the pairing classes realized are non-trivial, it
The double transition phenomenon was interpreted as
has become clear that all these heavy Fermion superconduc-
arising from splitting the orbital degeneracy of a two-
tors are intimately related to antiferromagnetisln (AF). In
dimensional representation. The pairing function was identi-
the former three compounds the A F occurs at around TN ~-
fied az a vector order parameter (2D-REP scenario 1 ). How-
10.Tc("10.Tc empirical rule") where TN is the Nrel tempera-
ever, according to this scenario the crossing of the two He2
ture. Below the superconducting (SC) transition (To) these
lines occurs only when M Z H
two long-range orders A F and SC coexist microscopically in
which is i~,lco::',radicti(m with the experimental facts where
a sample.
the crossing always appears irrespective of the field direc-
in the basal plane of Doh.
Among these systems UPta is most throughly studied to
tion. In order to explain this fact, we take up an odd-parity
yield definitive evidence for the non-trivial pairing function
state belonging to the one-dimensional representation (1D-
with internal degrees of freedom because the phase diagram
REP scenario 2 ) in the weak spin-orbit coupling. We assume
in the H vs.T plane consists of multiple phases, at least three
that the subla.ttice moment M is rotatable within the basal
phases, where the A F splitts T¢ of an unconventional super-
plane so to keep M . I _ H . We can derive a non-trivial T¢
conducting state with a pairing function which belongs to a
splitting term of the two order parameters M and rh where
degenerate irreducible representation. The magnetism is a
A(k)
driving mechanism to pair electrons in a non-trivial SC state
two transition points Td and T:2; below G l , r/; starts to
, but also plays an impdrtant role to govern the supercon-
grow, followed by % and rl: at 7 ~ ( < T~: ).
= Ei=~,~,~rhl(k)r..
A GL functional gives rise to the
Let us now consider the external field H by ex!endiag
ducting pairing states themselves. In contrast with these three systems in which the in-
the above GL functional. In conlras: with 2D-REP sc
terplay between A F and SC is established experimentally,
the gradient terms do not mix the three components 0, in
However,
1D-REP. Since an odd-parity state has a H 2 term which is
several experiments clearly suggest this proximity: Various
quadratic in 7?. This term is important in giving rise to the
experiments observe the magnetism just at T~=0.6K in the
crossing of the two He2 lines wlfich are otherwise pa~alleI.
absence of an external field H. In (Cei_xTh~:)Cu2Si: a few
Namely, when H / / y ( / / z ) .
percent doping of Th ion induces AF around TN=2--,3K.
ally overcoming r/~ in h:gh fields. Thus we obtain ~he topo-
CeCu2Si2 has been studied less in this respect.
0921-4534/91/$03.50 © 1991 - Elsevier Scicnce Publishers B.V. All rights reserved.
% (rl:) is favored over r]:-men,u-
K Machida, M.-,4. Ozaki / Uncoventional superconducting classes in heavy fermion compounds
2628
logically same phase diagrams for both field directions. Be-
states in general where in the Z-point AF's the moment ro-
cause we assumed that M.I.H in the basal plane, the phase
tates below the lower TN within the basal plane and in the
diagrams are essentially same irrespective of the directions.
X-point AF's the mixing of the AF's which have different Q
We can derive various physical quantities 2 associated with
vectors and spin directions would occur.
these second order transitions such as the specific heat jump or the lower critical field. As for the nodal structure of the 1D-REP with odd-parity the orbital part I(k) could be l(k) = kz or k,(k~+ k~). These states have a line node in the basal plane, which is consistent with various thermodynamic and transport measurements. We emphasize that the 1D-REP with odd-parity in weak spin-orbit coupling (SO) is only the remaining possibility axaong eight possible combinations of odd- vs. even-parity, weak vs. strong spin orbit coupling and 1D- vs. 2D-REP for a pairing function. This case fits to the existing experiments, excluding either singlet 1D-REP states in weak SO case or singlet and triplet 1D-REP states in strong SO case, all of which do not give rise to the splitting term due to AF. 3. THE CeCu2Si2 PROBLEM
4. CONCLUSION We have demonstrated for UPt3 that the one-dimensional pairing function with odd-parity in weak spin-orbit coupling can explain not only the splitting of the transition temperatures at H = 0, but also the isotropy of the phase diagram consisting of multiple phases under various applied field directions, provided that the sublattice moment rotates so as to keep M_I.H within the basal plane. These features were difficult to understand in terms of the 2D pairing function scenario 1. The identified pairing function has the desired properties on the nodal structure and the parity which are revealed experimentally. Theoretically, Choi and Sanls through the analysis of Hc2 and Norman through the pairing mechanism due to spin fluctuations have identified an odd-parity state rather than an even-parity state as favor-
In a similar way, by enumerating group-theoretically the
able pairing symmetry in UPt3. According to our scenario,
AF allowed under the crystalline symmetry in CeCu2Siz the conditions are examined under which a double transitions
even when the AF has disappeared by applied pressure the
of either SC or AF states occurs. We can draw the follow-
applied fields because of t h e / / 2 term . In that case the up-
ing general conclusions for each combination of various AF
per He2 does not exhibit a kink. The phenomenon is quite
and SC states where AF's are characterized by the ordering
similar to the AI - A2 transition in superfluid 3//e. Therefore
vector Q located in the Brillouin zone and SC's belong to a
it is a crucial test to see the phase transition below Td above
2D-REP:
Pc~ ~- 3kbar.
(1) TN>T¢ :
triply spin-degenerate states; r/r , %, r/z should be split under
As for CeCu2Si2 we predict that for almost all simple
(l-a) The X-point AF can cause the Tc splitting where
AF states a splitting of the transition temperature Tc or TN
the moment could be either parallel to the c-a.~ds or within
can he induced by a non-trivial coupling of the two order
the basal plane and the ordering vectors could be either single
paramotor% provided that the existing SC state belgngs to a
Q or double Q.
degenerate representation.
(l-b) The Z-point AF can cause the Tc splitting only when the moment lies in the basal plane and the ordering vector should be single. (2) Tc>TN : (2-a) Among the three possible SC states the SC3 (r/,jTy) = (1,i) never induces the TN splitting for any AF's. (2-b) The remaining SC2 (1,0) and SC3 (1,1) can il, duce the AF double transition for both Z-point and X-point AF
REFERENCES 1. K. Machida and M. Ozaki, J. Phys. Soc. Jpn. 58 (1989) 2244. K. Machida, M. Ozaki and T. Ohmi, ibid. 58 (i989) 4 i i 5 and references therein. 2. K. Machida and M. Ozaki, Phys. Rev. Lett. 66 (1991) 3293. 3. M. Ozaki and K. Machida, J. Phys. Soc. Jpn. 60 (1991) 1452.