Cleanliness Parameter *Lambda*_{tr}

In Section , the dimensionless
cleanliness parameter *Lambda*_{tr} is
discussed. *Lambda*_{tr} >> 1 indicates
dirty-limit superconducting behavior typical of disordered
materials, while *Lambda*_{tr} << 1 is
indicative of clean-limit superconducting behavior typical of
single-crystals. Below *Lambda*_{tr} is
estimated for C_{4}KHg.

The standard definition of *Lambda*_{tr} is[186]

where *l* is the mean-free path, and

is the BCS coherence length.[252] Since for C_{4}KHg
the only experimental data related to v_{F} is for
in-plane transport, only the in-plane
*Lambda*_{tr} can be calculated.

The Fermi velocity can be estimated for C_{4}KHg by
using the Shubnikov-deHaas data of Timp *et al.*[245] in conjunction with the
rigid-band model for the graphitic *pi* carriers. In
this context, the rigid-band model implies the use of
graphitic band parameters for the shape of the
*pi*-carrier piece of the GIC's Fermi surface. In the
rigid-band picture, only the size of the *pi* piece of
the Fermi surface changes upon intercalation, thus changing
k_{F} and v_{F}. There is ample precedent for
using the rigid-band model to estimate electronic parameters
of the donor GIC's,[245,79] although the approximations
made are not as valid for stage 1 as for higher stages. Since
c-axis transport is believed to be through the intercalant
bands rather than the graphitic *pi* bands,[4,229]
the rigid-band analysis is inapplicable. Therefore, there is
no good way of estimating v_{F} for the c-axis
direction, and no reasonable method to calculate
*xi*_{0} or *Lambda*_{tr}.

Timp and colleagues[245]
measured a maximum in-plane Shubnikov-deHaas frequency of
*nu*_{max} = 2490 T for C_{4}KHg. Since
k_{F} = *sqrt*2e *nu*_{max} /
*hbar* c , this gives k_{F} = 2.75×
10^{7} cm^{-1}.[79] For graphitic *pi* bands,
m^{*} = *hbar*^{2} k_{F} /
p_{0}, where p_{0} = 1.08×
10^{-19} erg-cm.[228] This formula gives
m^{*} = 0.31 m_{e} for the in-plane effective
mass of C_{4}KHg, in excellent agreement with the
m^{*} = 0.32 m_{e} found using
reflectivity.[270] Using
v_{F} = *hbar* k_{F} / m^{*},[228] the result is
v_{F} = 1.0× 10^{8} cm/s. Then
*xi*_{0} = 9000 Å.

The main obstacle in estimating *l*_{a}, the
mean-free-path for in-plane transport, is that the in-plane
resistivity has not been reported below 100 K (see Section
).[72] However,
*rho*_{c} measurements have been published down
to liquid-helium temperature.[85] In order to make further
progress, a reasonable procedure is to estimate

This procedure gives a rough estimate of
*rho*_{a}(4.2K) 0.7 *mu* *Omega*-cm,
which compares well with the measured
*rho*_{a}(4.2K) = 0.6 *mu**Omega*-cm
for C_{8}KHg.[192]

For the mean-free path, the relationship is

For an ellipsoidal Fermi surface, the carrier density
**n** is given by

where I_{c} is the c-axis repeat distance.[79] Then *l* is

Plugging in the numbers gives *l*_{a} 9100
Å at liquid helium temperature.

Finally now it is possible to calculate
*Lambda*_{tr} 0.86. This is indicative of fairly
clean superconductivity for C_{4}KHg for in-plane
transport ( *vec*H || ^c), although not really the
``clean limit''.

All these formulae can be reduced to the expression

where *rho*_{a} is in *Omega*-cm, k
_{F} is in cm^{-1}, T_{c} is in K,
and I_{c} is in Å. Getting k_{F} from
the C_{8}KHg maximum SdH frequency of 1490 T,[245] a similar calculation for
C_{8}KHg shows *Lambda*_{tr} = 0.36. For
C_{8}K, using k_{F} = 4.7×
10^{7} cm^{-1}[234] and using
*rho*_{a} = 0.08 *mu* *Omega*-cm at 4
K (this is actually the value for C_{8}Rb[96]), *Lambda*_{tr}
is 22. The ``dirty'' value of *Lambda*_{tr} for
C_{8}K is a result of its lower T_{c} and
I_{c} that go into the equation above.

alchaiken@gmail.com (Alison Chaiken)

Wed Oct 11 22:59:57 PDT 1995