## Calculating the Clogston Chandrasekhar (CC) limit, HP

#### Chuck Agosta - Clark University

A universal property of the FFLO state is that Cooper pairs are broken apart by Pauli paramagnetism as opposed to the formation of vortices. As Clogston and Chadrasakhar 1, 2 showed, the ultimate critical magnetic field for a superconductor should occur when the Zeeman energy is greater than the superconducting energy gap. In the simplest case, the Zeeman energy is $\mu_{B}H$, and the BCS energy gap is $1.746 K_BT_c$. Of course most superconductors of current interest are not accurately described by BCS theory, so we use a semi empirical method to find $H_P$. There are various ways to do this calculation using either the size of te specific heat jump, and the Rutgers formula, or using a more complete dataset of the specific heat jump and its decay as a function of decreasing temperature. Given the difficulty in defining the size of the specific heat jump for broad superconducting transitions, we now prefer to use a temperature sweep of the specific heat from $T_c$ to as low a temperature possible. We discuss this method, the Alpha Model, here.

As we showed in a previous paper 3, following Clogston 1, we can find the critical magnetic field associated with the quenching of superconductivity by estimating the superconducting energy gap by analyzing specific heat data and setting this energy equal to the gain in free energy in a metal with susceptibility $\chi_e$. More specifically, we equate the superconducting condensation energy $$U_c = 1/2 N(E_f) \Delta(0)^2, \label{eq:Uc}$$ where $N(E_f)$ is the density of states at the Fermi energy and $\Delta(0)$ is the superconducting energy gap at zero temperature, with $$\Delta F = 1/2\mu_0\chi_eH_P^2,\label{eq:dF}$$ the magnetic energy of a metal with susceptibility $\chi_e$. The susceptibility $\chi_e$ can be expressed as $1/2(g\mu_B)^2 N(E_f)$, where g is the gyromagnetic ratio, but it is important to notice that $\mu_0H^2$ already has the units of energy density, so $\chi_e$ must be dimensionless. The expression $(g\mu_B)^2 N(E_f)$ has dimensions of $J/T^2m^3$, exactly the inverse of $\mu_0$. Therefore we substitute $\mu_0\chi_e$ into Equation \ref{eq:dF} and after equating $U_c = \Delta F$ and noticing that the density of states cancels out, and setting $g = 2$, we end up with the common result $$B_P = \frac{\sqrt{2}\Delta}{g\mu_B} = \frac{\Delta}{\sqrt{2}\mu_B}.\label{eq:Bp}$$ after using the relation that $B = \mu_0H$ and knowing that B is what we measure in the laboratory. This is the result of a direct comparison of the energy needed to break a Cooper pair with the energy needed to flip a electron spin. Orlando at al. 4 added a correction to formula \ref{eq:Bp} of $1/\sqrt{1+\lambda}$ where $\lambda$ is the electron-phonon interaction parameter, to account for many body effects. This factor was corrected by Schossmann and Carbotte 5 to not have the squareroot in the denominator. McKenzie in Zuo et al.6 adds a practical version of this correction to equation \ref{eq:Bp} defining $g^*$, and shows the enhancement of $g$, namely $g^*/g$, is equivalent to Wilson's ratio. Incorporating $g^*$ into equation \ref{eq:Bp} the result is: $$B_{P}=\frac{\sqrt{2}\Delta}{g^*\mu_{B}}.\label{eq:hp}$$ The ratio g*/g can be found from specific heat and susceptibility measurements, or from spin-splitting of quantum oscillations, a measurement that is common in our laboratory. There is a table with $g^*/g$ found by both methods in McKenzie's paper on the arXiv.7 Despite knowing that $B_P$ is really the more useful parameter in this calculation, we will continue to use $H_P$ as the designation of the Chandrasekhar-Clogston Pauli paramagnetic limit as is common in most articles.

1 Clogston, A.M., "Upper Limit for the Critical Field in Hard Superconductors," Phys. Rev. Lett. 9, 266 (1962).
2 Chandrasekhar, B.S., "A Note on the Maximum Critical Field of High-Field Superconductors," Appl. Phys. Lett., 1, 7, (1962).
3 Charles C. Agosta, "Inhomogeneous Superconductivity in Organic and Related Superconductors," Crystals 8, 285 (2018).
4 Orlando, T.P.; McNiff, E.J., Jr.; Foner, S.; Beasley, M.R. "Critical fields, Pauli paramagnetic limiting, and material parameters of Nb3 Sn and V3 Si," Phys. Rev. B 19, 4545 (1979).
5 Schossmann, M.; Carbotte, J., "Pauli limiting of the upper critical magnetic field," Phys. Rev. B 39, 4210 (1989).
6 F. Zuo, J. S. Brooks, R. H. McKenzie, J. A. Schlueter, J. M. Williams, Paramagnetic limiting of the upper critical field of the layered organic superconductor κ-(BEDT-TTF)$_2$Cu(NCS)$_2$. Phys. Rev. B 61, 750 (2000).
7 R. H. McKenzie, R.H. "Wilson’s ratio and the spin splitting of magnetic oscillations in quasi-two-dimensional metals," arXiv 1999, arXiv.9905044.
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Last updated 8 Oct 2019
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