6 p-Wave Superconductivity and d-Vector Representation
197
Here, g
↑,↓
1
∝ V
↑↑,↓↓
∝ χ zz , g
↑,↓
2
∝ V
↑↓,↓↑
∝ (χ xx − χ yy ). Equation (6.59) shows
that, as for any two-band superconductor, T SC should increase if the inter-band coupling is increased. In this case, T SC should increase if the difference between the
transverse susceptibilities increases. This is surprising for two reasons.
The first is that it was believed, since the pioneering work of D. Fay and J. Appel
[21], that Ising anisotropy was most favourable for ferromagnetic superconductors
because transverse fluctuations would be pair-breaking as they would force scattering
from one Fermi sheet to the opposite polarization Fermi sheet. However, this paper
was written before the discovery of superconducting MgB 2 and the following boost
of work on multigap superconductivity: we understand now that these transverse
fluctuations do also induce exchange of Cooper pairs from one Fermi sheet to the
other, which is favourable to superconductivity. And so the prediction is that 2D
anisotropy (rather than uniaxial anisotropy) is the most favourable for ferromagnetic
superconductors (maximizing both χ zz and χ xx − χ yy ).
The second reason is that, experimentally, all the systems where ferromagnetic
superconductivity has been discovered did show a strong uniaxial anisotropy, confirming the prediction from [21]. But making statistics on few elements is always
dangerous. We also found that reducing this uniaxial anisotropy in URhGe, using
stress along the b-axis which increases the χ bb susceptibility without changing χ cc
(for stress below 0.6 GPa), does increase T SC in URhGe: a factor 2 between 0 and
1 GPa [22]. Ising anisotropy is probably not the most favourable, and larger T SC
ferromagnetic superconductors might be awaiting to be discovered.
Coming back to the consequences of (6.58) in terms of order parameter, with
decoupling of the equations for η x and η y , the order parameter should have a line of
node, (k x or k y or k z = 0), a finite S of order (at given k)
S(k) =
δ
η
k
2
i
k
2
i
e z ,
δ =
η
↑
i − η
↓
i
2
;
η =
η
↑
i + η
↓
i
2
.
(6.60)
But it is not chiral (L = 0). The A state could be chiral if η x and η y remain coupled
(in this model, if χ xy = 0), with
L =
(δ y η x + δ x η y )k
2
x + k
2
y
(η 2
x + δ 2
x )k 2
x + (η 2
y + δ 2
y )k 2
y
e z .
(6.61)
Note that in principle too, if such is the case, d is probably not an ESP state any
more, meaning that the 0 component should be non-negligible and all expressions
should be much more complex.
197
Here, g
↑,↓
1
∝ V
↑↑,↓↓
∝ χ zz , g
↑,↓
2
∝ V
↑↓,↓↑
∝ (χ xx − χ yy ). Equation (6.59) shows
that, as for any two-band superconductor, T SC should increase if the inter-band coupling is increased. In this case, T SC should increase if the difference between the
transverse susceptibilities increases. This is surprising for two reasons.
The first is that it was believed, since the pioneering work of D. Fay and J. Appel
[21], that Ising anisotropy was most favourable for ferromagnetic superconductors
because transverse fluctuations would be pair-breaking as they would force scattering
from one Fermi sheet to the opposite polarization Fermi sheet. However, this paper
was written before the discovery of superconducting MgB 2 and the following boost
of work on multigap superconductivity: we understand now that these transverse
fluctuations do also induce exchange of Cooper pairs from one Fermi sheet to the
other, which is favourable to superconductivity. And so the prediction is that 2D
anisotropy (rather than uniaxial anisotropy) is the most favourable for ferromagnetic
superconductors (maximizing both χ zz and χ xx − χ yy ).
The second reason is that, experimentally, all the systems where ferromagnetic
superconductivity has been discovered did show a strong uniaxial anisotropy, confirming the prediction from [21]. But making statistics on few elements is always
dangerous. We also found that reducing this uniaxial anisotropy in URhGe, using
stress along the b-axis which increases the χ bb susceptibility without changing χ cc
(for stress below 0.6 GPa), does increase T SC in URhGe: a factor 2 between 0 and
1 GPa [22]. Ising anisotropy is probably not the most favourable, and larger T SC
ferromagnetic superconductors might be awaiting to be discovered.
Coming back to the consequences of (6.58) in terms of order parameter, with
decoupling of the equations for η x and η y , the order parameter should have a line of
node, (k x or k y or k z = 0), a finite S of order (at given k)
S(k) =
δ
η
k
2
i
k
2
i
e z ,
δ =
η
↑
i − η
↓
i
2
;
η =
η
↑
i + η
↓
i
2
.
(6.60)
But it is not chiral (L = 0). The A state could be chiral if η x and η y remain coupled
(in this model, if χ xy = 0), with
L =
(δ y η x + δ x η y )k
2
x + k
2
y
(η 2
x + δ 2
x )k 2
x + (η 2
y + δ 2
y )k 2
y
e z .
(6.61)
Note that in principle too, if such is the case, d is probably not an ESP state any
more, meaning that the 0 component should be non-negligible and all expressions
should be much more complex.