Similar Shubnikov-de Haas oscillations were seen in all pure
*alpha*-phase samples studied. Only very weak
oscillations were seen in mixed-phase *alpha* +
*beta* samples, probably due to a smaller value of
(*Omega*_{c} *tau*) caused by in-plane
disorder. A representative Fourier transform calculated from
the *alpha*-phase SdH data is shown in Figure
a). This scan shows the frequency which was seen
consistently, at (1120 ± 110) T.

In order to verify the authenticity of the highest frequency,
the one used in the subsequent data analysis, various small
portions of the data were Fourier transformed separately, to
see if the 1120 T requency was an artifact of the analysis.
However, it was found that this frequency appeared strongly
no matter what field range of data was used for the Fourier
transform. In addition, in order to get some intuition as to
what the 1120 T frequency should look like experimentally, a
calculated version of the trace was compared to the data
using the fit parameters. The calculation used the accepted
form of the temperature and field dependence of the amplitude
of SdH oscillations:[79]

where A is the amplitude of the oscillations, T_{D}
is the Dingle temperature, and *Omega*_{c} is
the cyclotron frequency, through which A is dependent on the
magnetic field. The Dingle temperature characterizes the
degree of disorder in a solid through:[79]

where *tau*_{col} is the transport collision
time. An unusually large value of the Dingle temperature of
about 5 K had to be used to produce the calculated trace
shown in Figure b). The collision time in
the C_{4}CsBi_{0.6} samples is probably short
due to the scattering from the metal inclusions discussed
previously in Section . Confidence in the
validity of the frequencies obtained from the Fourier
transforms is increased because of the good qualitative
agreement between the calculated curve and the data shown in
Figure b).

**Figure:** a) Fourier transform of the data in
Figure
. Most reproducible frequency is (1120 ± 110) T,
although other frequencies sometimes occur. b) Comparison of
the data from Figure with a simulated trace
calculated using Equation and parameters obtained
from the Fourier transform of the data. The calculated curve
is offset from the data for clarity.

Once the maximum SdH frequency of 1120 T has been determined,
one can use the dilute-limit model as a first approach to
understanding the data. This model assumes that the primary
effect of intercalation is to increase the size of the
graphitic *pi*-derived ellipsoids at the K point of the
Brillouin zone, leaving the shape of the ellipsoids
unmodified. The validity of this assumption for stage I GIC's
could be criticized because of the high intercalant densities
involved. However, the excellent agreement obtained by Timp
*et al.*[245] with
other more direct measurements of Fermi energies and charge
transfers gives some justification for the model's use.
[These values are collected in Table
below.]

Proceeding then cautiously in the spirit of the
Dresselhaus-Leung (DL) model[65], one can relate the compounds'
Fermi energy to its maximal FS cross-section, *cal*
A_{max}^{e}, and thus to the largest observed
SdH frequency, *nu*_{max}. The expressions that
describe this relationship, derived by DL, are:

Here the E's are parameters of the DL model, the
*Gamma*'s are graphite band parameters, and
a_{0} is the graphite in-plane lattice constant.
Equation can be used to solve for
E_{F}, giving a value of 0.93 eV. This value seems
low for a stage 1 donor compound, as Table
shows. Yet, within the accuracy of the approximations
made in the use of Equation , the agreement of this
value with the 1.3 eV obtained by Yang and colleagues from
reflectivity measurements[270] is acceptable. The 1.3 eV
number is itself based on the ``mirror bands'' approximation,
whose quantitative correctness for donor compounds is not
well-established.

With knowledge of the in-plane structure, one can also
C_{4}CsBi_{0.6} also estimate the charge
transfer per carbon atom, f_{C}. f_{C} is
approximately equal to the number of electrons per carbon
atom in a GIC since the carrier density in pristine graphite
is so low.[67] The number of
electrons per carbon atom is the ratio of the number of
electrons per unit cell to the number of carbon atoms per
unit cell. The number of electrons per unit cell is the ratio
of the FS volume to the Brillouin zone (BZ) volume:

e^{-} / C-atom = (e^{-} /unit cell) /
(C-atoms / unit cell)

The volume of a piece of *pi*-electron FS can be
estimated in the spirit of the DL model as the size of a
cylinder with a circular cross-section of area *cal*
A_{max}^{e} and height 2*pi* /
I_{c}:

The total volume of the FS is then the number of cylinders
per BZ (six) times the volume per cylinder. Note that the
number of carbon atoms per unit cell and the volume of the BZ
depend on the in-plane structure of the intercalant. At the
time that Ref. [36] was
published, the authors did not know of any work on the
in-plane structural work on the CsBi-GIC's, and so calculated
f_{C} using the (2X2)R0°, since this structure is
common in donor GIC's.[67] A
new number for f_{C} has been calculated now that a
specific in-plane structure has been observed in these
materials.

Bendriss-Rerhrhaye in her thesis proposes a (3*root*13 X
8)R(15°,0°) rhombic unit cell for
C_{4}CsBi_{0.6}.[17] This proposed structure is
shown in Figure . The BZ associated with
such a unit cell will obviously be much smaller than the
graphitic one, but this is partially compensated for by the
large number of carbon atoms (192) in the unit cell. Using
the 1120 T SdH frequency to calculate *cal*
A_{max}^{e} gives an f_{C} of -0.026
electrons per carbon atom. This value is about two-thirds of
that obtained from reflectivity measurements by Yang *et
al.*[270], just as the
E_{F} obtained from SdH is about two-thirds of that
from reflectivity. [See Table
] The f_{C} number of Yang and colleagues is
obtained directly from their Fermi energy using the
Blinowski-Rigaux (BR) tight-binding model.[27] Therefore it is dependent
on both the mirror-bands approximation (used to get
E_{F}) and the assumptions of the BR model, which,
like the DL model, uses graphitic parameters to describe the
*pi* bands of the intercalation compound. Taking all
approximations into account, the E_{F} and
f_{C} numbers of the two references should be about
equally reliable. The lower value of the Fermi energy
calculated from the SdH data could be due to a high-frequency
oscillation that was not observed. A high-frequency
oscillation appears to have been missed in the compound
C_{4}KH_{x}.[62] A frequency of about 1700 T
would be needed for perfect agreement of the CsBi SdH data
with the data of Yang.

**Figure:** (3*root*13 X
8)R(15°,0°) in-plane unit cell proposed for
C_{4}CsBi_{0.6} by A. Bendriss-Rerhrhaye.[17].

**Table:** Electronic parameters for selected
superconducting GIC's. The stage and phase of each compound
are listed in parentheses. The compounds are listed in order
of increasing superconducting transition temperature.
*Omega*_{p} is the unscreened plasma
frequency.

alchaiken@gmail.com (Alison Chaiken)

Wed Oct 11 22:59:57 PDT 1995