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## The Alpha Model

#### How to find the superconducting energy gap from specific heat measurements

It is possible to estimate the superconducting energy gap $\Delta$ by fitting specific heat data in the superconducting state using the Alpha Model, a semi empirical model loosely based on BCS theory, and created by Padamsee et al 1. Our most recent fits have been done using the more recent version by Johnston 2, which allows direct integration of the specific heat data to find the size of the gap. In this model the ratio $\alpha = \Delta/k_BT_c$, is a free parameter rather than fixing $\alpha$ at the BCS value of 1.764, although the gap in the model has the same temperature dependence as the BCS model. The calculation is based on the universal expression of the entropy, $S = k_B\sum f ln(f)$, where $f$ is the Fermi distribution function. The sum can be turned into an integral over the energy range of the quasiparticles, and for a given $\alpha, T_c$, and $\gamma$, the entropy is calculated as a function of temperature. The calculated specific heat can then be found by taking the derivative of the entropy with respect to the temperature. In the Johnston version the derivative of the entropy integral is done analytically and a new integral results that can be used to find the specific heat directly.

One of the additional complications of fitting the specific heat to the Alpha Model is that the fit is sensitive to symmetry of the superconducting order parameter. The original versions of both Padamsee and Johnston use s-wave pairing. We modified the integral expressions for d-wave symmetry. In most cases we fit the data using both forms of the equation, and pick the best fit to determine $\alpha$, and ideally determine the pairing symmetry, a question which is still not completely settled in the organics. We find d-wave as the best fit for $\kappa$ET-CuNCS as did Taylor et al. 3 although we find s-wave as the best fit for the other organics, which is contrary to some of the other evidence in, for example, the $\kappa$ET-Br material 4. Part of the challenge in organics is that the lattice is soft, so the phonon contribution to the specific heat continues to low temperatures. One way to subtract the phonon contribution is to measure the specific heat at a magnetic field large enough to quench the superconductivity and subtract that curve from the specific heat measured with no magnetic field. Wth this method the phonon contribution is subtracted, but the electronic linear term is also subtracted and mixed with the phonon terms. The linear term determines the constant $\gamma$, which is a measure of the electronic density of states. The ability to determine $\gamma$ is critical to getting a good result for the $\alpha$ fits. It is unclear what our conclusion should be for the pairing symmetry, particularly if we believe that all the organics have the same pairing. We have also noticed that the shape of the fits are not perfect with s or d-wave pairing, which suggests the possibly that the pairing has some other symmetry such as s+d, or anisotropic s-wave, a notion supported by some experiments 5 and recent theory 6.
Material $\alpha$ $g^*/g$ $T_c$(K) $H_p$(T) $\alpha_M$ $H_{orb}^0$(T) $\xi$(Å) $\ell$(Å) r = $\ell/\xi$
$\kappa$-(BEDT-TTF)$_2$Cu(NCS)$_2$ 3.0 7 1.26 8 9.6 21.6 4.9 130 9 13 900 10 13
$\kappa$-(BEDT-TTF)$_2$Cu(NCS)$_2$ 1.75 kbar >3.5 0 1.5 4.5 4.9 >11 363 600 1.7
$\beta$"-(BEDT-TTF)$_2$SF$_5$CH$_2$CF$_2$SO$_3$ 1.94 11 1.0 8 4.5 9.2 3.9 75 9 21 520 13
$\alpha$-(ET)$_2$NH$_4$Hg(SCN)$_4$ 1.76 12 0.86 8 0.96 2.1 5.5 8.1 13 53 681 13 13
$\lambda$-(BETS)$_2$GaCl$_4$ 1.83 14 1.0 8 4.3 8.3 3.9 23.115 31.5 170 16 13
$\kappa$-(ET)$_2$Cu[N(CN)$_2$]Br 2.77 7 1.4 8 11.5 23.8 9.6 161 17 12 260 18 13
CeCoIn$_5$ || 3.0319, 20 0.7319 2.16 9.44 6.5 43.5 21 23 810 13
CeCoIn5 $\bot$ 1.0 1 2.16 4.8 3.7 17 44 1500 34
KFe$_2$As$_2$ 1.75 22 1.3 23 3.14 4.84 2.9 9.9 48 177024 13
LiFeP 1.89 25 1.0 17.6 34.9 2.1 51 26 21 5500 26 13
Table 1. Note that || and $\bot$ These parameters will help determine which materials are good candidates for a first-order $H_{c2}$ transition or the FFLO state. $H_{c2}$ is the extrapolated value from the slope near $T_c$ times 0.7.27 For $\kappa$-(BEDT-TTF)$_2$Cu(NCS)$_2$ or other Pauli limited superconductors the slope near $T_c$ is not related to the coherence length and $H_{c2||}$ is estimated from $H_{c2\bot}$ and the anisotropy of the London penetration depth.28 To be consistent, we always compare $\ell$ and $\xi$ in the parallel direction (in the conducting planes) to calculate r. The mean free path for $\kappa$-(BEDT-TTF)$_2$Cu(NCS)$_2$ under pressure is estimated.

The table above shows the values of $\alpha$ from our fits of specific heat measurements. We have collected other parameters, in particular the Maki parameter $\alpha_M$ (see the Intro to FFLO page) all of which are predictors for inhomogeneous superconductivity.

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