Effect of Tilt on Angular Dependence of the Critical Field

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 vecH || ^c. The elliptical cross-section of the ellipsoid corresponds to the case where vecH _|_ ^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 sqrtcos² theta + epsilon² sin² 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 sqrtcos² ø + epsilon² sin² 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 vecH || ^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}/ sqrtcos² ø + epsilon ² sin² 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 sqrtcos² ø + epsilon² sin² 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 / sqrtepsilon² sin² ø + cos ² ø. Plugging this formula for xi(theta) into Eqn. now gives Eqn. :

as desired.