Contents - Max-Planck-Institut für Physik komplexer Systeme

I2f2(a,c,q) where a = 4.1 ˚A and c = 9.9 ˚A are the
lattice constants for the a- and c-axis respectively, I1
and I2 are the nearest and next-nearest neighbor exchange
interactions, and the precise form of f2 can be
found in [6]. The ratio I1/I2 follows from the distancedependence
of the RKKY interaction: I2 ≈ 0.35I1. The
magnitude of the exchange interaction I1 can be estimated
from the observed paramagnon velocity v, see
inset of Fig. 4(a): I1 ≈ 0.63meV. The difference in exchange
energy between superconducting and normal
state follows as
∆Ex ≡ E S x − E N x = A
g2 µ 2 ∞
B 0
π/a π/a π/c
× dqx
−π/a
× Im
dqy
−π/a
d(ω)
n(ω) + 1
π
dqz I(qx,qy,qz)
−π/c
.
χ S (qx,qy,qz, ω) − χ N (qx,qy,qz, ω)
Crystalline-electric-field effects split the J = 5/2 states
of the Ce 3+ ion up into a ground-state doublet and a
quasi-quartet at high energies (> 30 meV) and result in
g-factors gz ≈ g⊥ ≈ 2, and an almost isotropic spin susceptibility.
The constant A is given by A = η·8·3/(2VB)
with VB being the volume of the Brillouin zone. The
volume fraction η enters again, since the magnetically
ordered regions in S and N yield (essentially) identical
responses for B = 0 and B = 2 T. We finally obtain
∆E x/∆EC = 21.1, so that the gain in exchange
energy is more than an order of magnitude larger than
the condensation energy, thus identifying the build up
of magnetic correlations near the AF QCP as the major
driving force for superconductivity in CeCu2Si2. It
is important to note that the gain in ∆E x comes from
the opening of the spin gap and not the “resonance”like
feature above the spin gap, which tends to reduce
the energy gain (see Fig. 2(c)). This follows from the
fact that Imχ N/S (q, ω) is peaked around the incommensurate
wave vector at which I(q) is positive, i.e.
I(QAF) > 0. Note that a spin resonance as it occurs
e.g. in the cuprate superconductors is not expected for
three-dimensional superconductors [11].
2 /meV f.u.)
ω) ( µ Β
Im χ (Q AF ,h _
14
12
10
8
6
4
2
(a)
(c)
+
−
Im χ N (Q , h AF _
ω)
Im χ S (Q , h AF _
ω)
0 0.2 0.4 0.6 0.8 1
h _
0
ω (meV)
Figure 2: (a) Neutron intensity versus energy transfer ω in S-type
CeCu2Si2 at QAF. The inset in (a) shows the magnetic response at
QAF extending beyond ω = 2meV. (b) Wave vector Q dependence
of the magnetic response around QAF in S-type CeCu2Si2 in the superconducting
state at T = 0.06 K for different energy transfers ω.
Inset: Dispersion of the magnetic excitation around QAF at T=0.06
K. The solid line indicates a fit to the data with a linear dispersion
relation yielding a paramagnon velocity v = (4.44 ± 0.86) meV ˚A
(c) Schematic plot of the imaginay part of the dynamic spin susceptibility
Imχ(QAF, ω) in the normal (N) and superconducting (S)
states. The blue area marked with a ’+’ contributes to the gain in
∆Ex whereas the green area (marked with a ’-’) leads to a reduction
in the overall gain in ∆Ex.
These are striking differences between CeCu2Si2 on the
one hand, and CeCoIn5 [10] and high-Tc cuprate superconductors
on the other. For CeCu2Si2, we can unequivocally
conclude that the build-up of magnetic correlations
near the AF QCP energetically drives the superconductivity.
Evidently, there is a sizeable kinetic energy loss. Superconductivity
in CeCu2Si2 occurs in the spin-singlet
channel. As a result of the opening of the superconducting
gap, the Kondo-singlet formation is weakened
and the spectral weight of the Kondo resonance
is reduced. Since the kinetic energy of the quasiparticles
appears through the Kondo-interaction term
in a Kondo-lattice Hamiltonian, this naturally yields a
large loss to the f-electron kinetic energy.
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2.16. Magnetically Driven Superconductivity in Heavy Fermion Systems 73
(b)