Effect of Tilt

In this section, a detailed derivation of Eqn. is presented. In order to understand the geometry of the sample tilt, the reader is referred to Figure .

The underlying assumption of the derivation presented here is
that the material in question has uniaxial symmetry. In the
current context, uniaxial symmetry means that the material
has a plane in which H_{c2} is constant. The angle
between the perpendicular to this plane and the magnetic
field is called *theta*.

For graphite-based superconductors, the planes of constant
H_{c2} are those parallel to the graphite layers, and
the perpendicular axis which defines the angle *theta*
is called the c-axis. (The angle *theta* is defined in
Figure .) In-plane isotropy has
not actually been experimentally confirmed in GIC
superconductors because of the difficulty of preparing
single-crystal samples. However, any in-plane anisotropy is
expected to be small because the in-plane band structure
should be relativelty free-electron-like.[112] In addition, the critical
field experiments discussed in this thesis were performed on
HOPG-based samples. Since HOPG is composed of randomly
oriented crystallites in-plane, but has a well-defined c-axis
direction,[67] GIC's prepared
from HOPG should meet the definition of a uniaxial
superconductor even if in-plane anisotropy is present in
single crystals.

The derivation of Eqn. follows the simple
derivation of Eqn. that was published by
Morris, Coleman, and Bhandari.[175] Let's first review their
derivation. These authors started with the idea that a the
coherence length of a uniaxial superconductor in any
direction is equivalent to the length of a vector joining the
center of a biaxial ellipsoid with its edge. The ellipsoid
has a circular cross-section of radius *xi*_{a},
which corresponds to the circular cross-section of a flux
quantum formed when *vec*H || ^c. The elliptical
cross-section of the ellipsoid corresponds to the case where
*vec*H _|_ ^c, where the flux quantum has radii
*xi*_{a} and *epsilon*
*xi*_{a}. Here *epsilon* is the AGL model
anisotropy parameter introduced in Ref. [175]. The biaxial ellipsoid is
shown in Figure a), and its two
cross-sections in Figure a).

For any orientation of the applied field, one of the
coherence lengths in the plane perpendicular to the field
will be *xi*_{a}, the in-plane coherence length.
The other coherence length in this plane will vary
continuously with *theta* from *xi*_{a} to
*epsilon* *xi*_{a}. Thus

using Eqn. . Figure
a) and some simple geometrical manipulations show that
*xi*(*theta*) = *xi*_{a}
*sqrt**cos*^{2} *theta* +
*epsilon*^{2} *sin*^{2}
*theta* . Plugging the *xi* (*theta*)
expression into Eqn. above gives Eqn.
:

as expected. This is the result obained by Morris, Coleman, and Bhandari.[175]

**Figure:** Coherence lengths of a uniaxial
superconductor. As pointed out by Morris *et al.*,[175] the coherence length of
a uniaxial superconductor is the length of a vector from the
center of a biaxial ellipsoid to its edge. a) The case of an
aligned sample, which is described by Eqn. .
(See Figure a).) The ellipsoid has
two radii of length *xi*_{a} and one of length
*epsilon* *xi*_{a}. b) The case where the
sample is tilted by an angle ø, which is described by
Eqn. . Now the ellipsoid is
triaxial, with one of the coherence lengths of size
*xi*_{a} from a) being replaced by one of size
*xi*_{a} *sqrt**cos*^{2}
ø + *epsilon*^{2} *sin*^{2}
ø.

**Figure:** Cross-section of the flux quantum
in a uniaxial superconductor. In all cases, the magnetic
field is directed out of the paper. a) The aligned case. For
*vec*H || ^c, the cross-section of the flux quantum
along the field direction is circular. b) The tilted case.
Now that the ellipsoid that determines
*xi*(*theta*) is triaxial, the cross-section of the
flux quantum is non-circular for all field
orientations.

Now let's turn to the case where the sample is tilted so that
its c-axis is rotated by an angle ø away from the
horizontal plane, which is the plane in which the applied
field is rotated. From examination of Eqn. ,
it is obvious that the measured magnitude of H_{c2, ||
^c} will be H_{c2, || ^c} ^{eff} ==
H_{c2,|| ^c}/ *sqrt**cos*^{2}
ø + *epsilon* ^{2} *sin*^{2}
ø. Note that this gives H_{c2, ||
^c}^{eff} = H_{c2, || ^c} when ø
= 0°. If H_{c2}(*theta*) still is given by
Eqn. , then it follows that
*xi*(0°) = *xi*_{a}
*sqrt**cos*^{2} ø +
*epsilon*^{2} *sin*^{2} ø.

Now that *xi*(*theta*) has been found for
*theta* = 0°, the full form of the function can be
determined by noting that *xi*(90°) still is
*epsilon* *xi*_{a}. The reason is that the
tilt of the c-axis from the horizontal corresponds to a
rotation about an in-plane direction (an ^a direction), so
the in-plane H_{c2} is not affected by the tilt. This
statement is no longer true if H_{c2} is not constant
in the layer plane (non-uniaxial superconductor), although
everything said up to this point holds for that case. Using
standard formulae for an ellipse gives for general
*theta*

where *epsilon*_{eff} = *epsilon* /
*sqrt**epsilon*^{2} *sin*^{2}
ø + *cos* ^{2} ø. Plugging this
formula for *xi*(*theta*) into Eqn.
now gives Eqn. :

as desired.

alchaiken@gmail.com (Alison Chaiken)

Wed Oct 11 22:59:57 PDT 1995