Estimation of the Dimensionless ``Cleanliness'' Parameter

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 C₄KHg.

The standard definition of Lambda_tr is[186]

where l is the mean-free path, and

is the BCS coherence length.[252] Since for C₄KHg 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 C₄KHg 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 xi₀ or Lambda_tr.

Timp and colleagues[245] measured a maximum in-plane Shubnikov-deHaas frequency of nu_max = 2490 T for C₄KHg. Since k_F = sqrt2e nu_max / hbar c , this gives k_F = 2.75× 10⁷ cm^-1.[79] For graphitic pi bands, m^* = hbar² k_F / p₀, where p₀ = 1.08× 10^-19 erg-cm.[228] This formula gives m^* = 0.31 m_e for the in-plane effective mass of C₄KHg, 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× 10⁸ cm/s. Then xi₀ = 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 muOmega-cm for C₈KHg.[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 C₄KHg for in-plane transport ( vecH || ^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 C₈KHg maximum SdH frequency of 1490 T,[245] a similar calculation for C₈KHg shows Lambda_tr = 0.36. For C₈K, using k_F = 4.7× 10⁷ cm^-1[234] and using rho_a = 0.08 mu Omega-cm at 4 K (this is actually the value for C₈Rb[96]), Lambda_tr is 22. The ``dirty'' value of Lambda_tr for C₈K is a result of its lower T_c and I_c that go into the equation above.