June 17, 2007

Fig. 1: Spatial variation of RKKY susceptibility. |

In most systems, the superconducting and magnetic
ordered states are mutually exclusive. However, materials, including
the quaternary borocarbides, coexistence can occur. The borocarbides,
RT_{2}B_{2}C (R=rare earth, T=transition metal),
crystallize for most members of the rare earth series, and they exhibit
conventional BCS phonon-mediated superconductivity. They have generated
interest in the decade since their discovery for two reasons: the
superconducting transition temperature (T_{c}) is relatively
high for a BCS superconductor, and the onset of magnetism occurs at
temperatures close to T_{c}. The latter reason is particularly
crucial, because it indicates similar energy scales for the two
phenomena, making the borocarbides a good system to study the
competition physics of two different ordering phenomena.

The topic of rare earth magnetism is very rich and complicated, and this section is intended to serve as an introduction. Magnetic ordering in the borocarbides (and other rare earth compounds) is mediated by RKKY (Ruderman-Kittel-Kasuya-Yosida) exchange with the conduction electrons. [1] Consider the spin on an individual atom to be a magnet with a miniscule spatial extent. In the limit where the spatial extent is zero, the local magnetic field becomes a delta function, which can be expressed as fourier integral.

where H_{q}=H. If we assume that the
magnetization M is also a delta function, this implies that the electron
gas responds completely to the perturbation of a magnetic ion: i.e. the
susceptibility is a constant given by χ =
M_{q}/H_{q}. However, the presence of a Fermi surface
makes the electron gas less responsive for spatial frequencies q >
k_{f}, where k_{f} is the Fermi wavevector. Thus, we
must consider the frequency components of the susceptibility as
well.

χ_{q} is the response of an electron gas
to a spatially varying magnetic field. Its value is found by perturbing
the bloch wavefunctions of an electron gas to first order with the
following periodic disturbance:

The result of this calculation is quoted below:

Here, χ_{p} is the pauli susceptibility,
given by:

Performing the integral leads to the following spatially-dependent susceptibility:

This oscillatory susceptibility is plotted in Fig. 1. RKKY interaction allows for the ordering rare earth moments: one moment produces an oscillatory magnetization in its vicinity to which the moments of other atoms couple. The coupling can be ferromagnetic or antiferromagnetic, depending on the atomic distance.

The rare earth elements have the same valence, and
often, as is the case for the nickel borocarbides, compounds will form
for much of the series, providing a systematic variation of the magnetic
properties. As we move to the left along the lanthanide series (Fig.
2), we see a roughly linear increase of T_{N}, the
antiferromagnetic transition temperature, with the de Gennes factor, and
a concurrent suppression of T_{c}. The de Gennes factor (dG) is
related to the Lande g factor and the total angular momentum, J, from
Hund's rules, per

Fig. 2: T_{c} and T_{N} scale roughly
linearly with the de Gennes factor (DG). [2] |

The linear relation between the de Gennes factor and
T_{c} and T_{N} involves two parameters which are
constant across the rare earth series: the strength of the exchange
interaction with the local 4f moments (I) and the density of states at
the Fermi level(N(ε_{f})).

This de Gennes scaling shows that both antiferromagnetism and suppression of superconductivity are governed by the same mechanism: exchange interaction with the 4f electrons. Furthermore, we see a competion between these two ordered states: as magnetism is strengthened, superconductivity is weakened. Next, we shall see how this interplay is evident as the system goes through magnetic phase transitions.

This section will show some key experiments on
ErNi_{2}B_{2}C, which beautifully illustrate the
competition physics of the borocarbides. The crystal structure and the
magnetic ordering is shown in Fig. 3. Although the crystal structure is
the same for other borocarbides, these compounds exibit a variety of
magnetic structures. In the Er compound, superconductivity sets in at
11K, and antiferromagnetism appears at 6K. In this ordered state, the
spins lie in the a-b plane. Along the ordering wavevector, q, each spin
is canted from the axis of its neighbor, creating a "spiral" ordered
state. This ordering is incommensurate: the period of the spiral does
not coincide with an integer number of lattice spacings. At 2.3 K,
however, the periodicity becomes commensurate, and also develops a small
ferromagnetic component.

Fig. 3: ErNi_{2}B_{2}C crystal
structure. All the magnetism in the system comes from Er.
Below 6 K, the Er spins order in an incommensurate spin
density wave [2]. |

Two important parameters for characterizing a
superconductor are the coherence length, ξ, (the "size" of a cooper
pair) and the London penetration depth, λ (the distance which a
DC magnetic field can penetrate into a superconductor). Small angle
neutron scattering measurements showed that both of these quantities
have an anomaly near T_{N} [3], and those results are reproduced
in Fig. 4.

One way of assessing the robustness of a
superconducting state is by how readily it is destroyed by a magnetic
field. There are two fields to consider: the lower critical field
(H_{c1}) where quantized flux first enters a type II
superconductor and the upper critical field (H_{c2}) where
superconductivity is destroyed. These two quantities can be calculated
from x and l using the following relations:

Where φ_{0} is the superconducting flux
quantum and κ = λ/ ξ. These quantities also show a kink
near the magnetic phase transition. The dip in H_{c2}
demonstrates that superconductivity is weakest near T_{N}—a
property that is attributed to fluctuations associated with the phase
transition. However, at lower temperatures, neither of the critical
fields recover to values they would have had in the absence of
long-range antiferromagnetic order, indicating that the magnetism in the
system compromises the robustness of the superconducting state.

The possibility for antiferromagnetism to coexist
with superconductivity is not entirely surprising because the net
magnetism of this order is zero: as long as the coherence length, is
larger than the spacing between the magnetic ions, the cooper pairs do
not see a net magnetic field. However, the transition to weak
ferromagnetism below 2.3K in ErNi_{2}B_{2}C is more
surprising. The London equations predict that a uniform magnetic field
is exponentially suppressed inside a superconductor, and even in type II
superconductors which permit magnetic flux, the field is exponentially
suppressed outside the flux tubes. In general, either the magnetic
phase or the superconducting phase must be altered to permit
coexistence, and in ErNi_{2}B_{2}C, it appears that the
magnetic phase is altered to be more amenable to superconductivity.
Several theories propose an oscillatory magnetic state which coexists
with a uniform superconducting state below 2.3K, with a period smaller
than the London penetration depth, and one such possibility is the
Anderson-Suhl mechanism [2,4]. Since the borocarbides superconductors
have singlet cooper pairs, the susceptibility of the electrons at long
wavelengths drops in the superconducting state (bound pairs with net
spin zero cannot respond to a DC magnetic field). The spin
susceptibility does not recover until q~10 ξ^{-1}, and the
integral for determining the RKKY suscpetibility is altered to suppress
the long wavelength (small q) contributions. The local maximum plotted
in Fig. 1 shifts to a nonzero q, producing an oscillatory magnetization.
Domains of oscillating magnetization were observed below 2.3K using
scanning hall probe microscopy [5].

© 2007 Inna Vishik. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] S. Blundell, *Magnetism in Condensed Matter*
(Oxford, 2001).

[2] K-H Muller and V. N. Narozhnyi. Rep. Prog. Phys.
**64**, 943 (2001).

[3] P.L. Gammel et al., Phys. Rev. Lett. **82**
1756 (1999).

[4] P.W. Anderson and H. Suhl. Phys. Rev. **116**
898 (1959).

[5] H. Bluhm et al., Phys. Rev. B **73**, 014514
(2006).