Contents - Max-Planck-Institut für Physik komplexer Systeme

2.16 Magnetically Driven Superconductivity in Heavy Fermion Systems
Unconventional superconductivity has attracted interest
for more than thirty years. Unconventional here
refers to superconducting phases where pairing is not
phonon mediated and which therefore do not conform
to classical BCS theory [1]. In strongly correlated
electron systems unconventional superconductivity
occurs predominantly in close vicinity to magnetism.
This includes superconductivity occuring in
the layered perovskite copper-oxides, the recently discovered
iron-pnictides and the heavy fermions. Particularly
the heavy fermions display a rich variety of
superconducting behavior. In UPd2Al3 e.g. superconductivity
occurs deep inside an antiferromagnetic
phase [2]. PuCoGa5 on the other hand enters a superconducting
phase directly from a paramagnet with
Curie behavior [3]. As a result, antiferromagnetic spin
excitations have been discussed as a possible mechanism
for superconducting pairing in heavy fermions [5]
ever since the seminal discovery of superconductivity
in CeCu2Si2 [4]. Until recently, however, no convincing
verification had emerged that antiferromagnetic quantum
critical fluctuations indeed underlie pairing. Based
on an in-depth study of the magnetic excitation spectrum
in momentum and energy space in the superconducting
and normal states, we were able to conclusively
demonstrate that magnetism drives superconductivity
in the prototypical heavy-fermion compound
CeCu2Si2 [6].
CeCu2Si2 crystallizes in a structure with body-centered
tetragonal symmetry. The phase diagram is shown in
Fig.1(a). So-called A-type crystals (see Fig.1(a)) display
antiferromagnetism at sufficiently low temperatures.
This antiferromagnetic order was found to be an
incommensurate spin-density wave (SDW) with propagation
vector QAF ≈ (0.215 0.215 0.53), which can be
ascribed to a nesting vector of the renormalized Fermi
surface [7]. Approaching the antiferromagnetic quantum
critical point (AF QCP), the thermodynamic and
transport behavior [8] is in line with quantum critical
fluctuations of a three-dimensional SDW; see also
Fig.1(b). We here focus on a crystal located on the paramagnetic
side of the quantum critical point (termed Stype
CeCu2Si2, see Fig.1(a)) for which we are able to
identify the magnetic excitations in the normal and superconducting
state. In both cases, magnetic excitations
are found only in close vicinity of the incommensurate
wavevector QAF . In the superconducting state,
a clear spin excitation gap is observed, see Fig.2(a).
The temperature dependence of the gap ∆(T) follows
a rescaled BCS form (∆(0) ≈ 3.9kBTc) [6]. A linear
dispersion of the spin excitation is visible in both
STEFAN KIRCHNER, OLIVER STOCKERT 3
3 Max Planck Institut für Chemische Physik fester Stoffe, Dresden.
the superconducting and normal state with a mode velocity
v ≈ 4.4 meV ˚A. For the superconducting state
this is shown in Fig.2(b). The value for v is substantially
smaller than the Fermi velocity vF ≈ 57 meV ˚A [9]
(1 meV ˚A = 153 m/s), and indicates a retardation of the
coupling between the quasiparticles and the quantumcritical
spin excitations.
To address, if magnetism indeed drives superconductivity
in CeCu2Si2, we study the energetics across the
transition. The condensation energy ∆EC characterizing
the stability of the superconducting state (S) against
a putative normal state (N) is
∆EC = lim
T →0
GS(T,B = 0) − GN(T,B = 0) (1)
where GS/GN is the Gibbs free energy of the superconducting/normal
state. Eq.(1) implies that ∆EC can be
obtained from the heat capacity. We take the finite field
(B = 2 T> Bc2) data as the putative normal state and
find ∆EC = −η 2.27 · 10 −4 meV/Ce. The factor η > 1
accounts for the fact that only the superconducting volume
fraction contributes to ∆EC.
(a)
T (K)
1.5
1.0
0.5
0
AF
TN
A-type CeCu2Si2
S-type CeCu2Si2
QCP
PM
SC
Tc
g (a. u.)
(b)
(0 0 l) (rlu)
_
CeCu Si hω = 0.2 meV, T = 0.06 K, B = 0
2 2
1.8
1.7
1.6
1.5
1.4
1.3
1.2
0.15 0.2 0.25 0.3
(h h 0) (rlu)
Figure 1: (a) Schematic T −g phase diagram of CeCu2Si2 in the vicinity
of the quantum critical point (QCP) where the antiferromagnetic
phase vanishes as function of the effective coupling constant g. Superconductivity
is observed around the QCP and extends far into the
paramagnetic regime. Composition as well as hydrostatic pressure
can be used to change the coupling constant g and to tune the system
to the QCP. The positions of the A-type and the S-type single
crystals in the phase diagram are marked. (b) Spin fluctuation spectrum
at T = 0.06K, B = 0 and at an energy transfer ω = 0.2meV.
The anisotropy factor between the [110] and the [001] directions is
roughly 1.5.
This has to be contrasted with the change in exchange
energy across the superconducting transition, in order
to ascertain whether the magnetic excitations contribute
significantly to the condensation energy. We
model the exchange interaction between the localized
Ce-moments by including nearest neighbor and nextnearest
neighbor terms appropriate for the tetragonal,
body-centered unit cell: I(q) = I1[cos(qxa)+cos(qya)]+
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Neutron intensity (arb. units)