Parallel Critical Field in Thin Niobium Films: Comparison to Theory

Abstract

For the first time, a comparison to the predicted behavior for parallel critical field is carried out for the model of Kogan and the model of Hara and Nagai. In this study, thin niobium films in the moderately dirty regime were considered. Experimental values of the $-C_{2}$ term are seen to be lower than those from the model of Hara and Nagai. A possible reason for this could be not including the non-spherical Fermi surface of niobium into the model. There is clearly disagreement with the model of Kogan as the films get cleaner and thinner, and two films which should be below his critical thickness still show positive values of $-C_{2}$, in disagreement with his theory.

Appendix

Niobium samples used in this study
Sample	Substrate	d (nm)	RRR	$\rho _{\mathrm {10K}}\ (\mathrm {n}\Omega \mathrm {m})$	$T_\mathrm{c0}$ (K)	$D=d/\xi$	$\lambda _{\mathrm {tr}}$	$-C_{2} (\mathrm {Exp.})$	$-C_{2} (\mathrm {HN})$	$-C_{2} (\mathrm {Kogan})$
A	(100) Si	48	4.04	49	8.946	1.3	4.4	0.064	0.071	0.066
B	(100) Si	23	2.68	89	7.777	0.54	10	0.0052(3)	0.0065	0.0054
C	(100) Si	48	2.68	89	8.262	1.2	9.5	0.027(1)	0.036	0.035
D	(100) Si	25	2.51	$1.0{\times }10^{2}$	8.426	0.63	10	0.0072(2)	0.0087	0.0079
E	A plane Sap	25	4.82	39	8.284	0.63	4.2	0.014	0.017	0.0072
F	C plane Sap	52	3.77	54	8.649	1.4	5.5	0.056	0.074	0.066
G	(100) Si	25	3.13	70	8.504	0.65	7.3	0.011	0.012	0.010
H	A plane Sap	22	7.24	24	8.644	0.57	2.4	0.014	0.017	$-$0.017
J	C plane Sap	25	4.07	49	8.477	0.64	5.0	0.012	0.015	0.0094
K	C plane Sap	16	2.53	98	7.615	0.37	11	0.0021	0.0027	0.0019
L	A plane Sap	16	4.12	48	8.021	0.39	5.3	0.0043	0.0049	$-$0.0072
M	C plane Sap	26	2.63	92	8.264	0.64	9.8	0.0077	0.0092	0.0084
N	(100) Si	24	3.01	75	8.600	0.63	7.6	0.0093(1)	0.011	0.0090
P	C plane Sap	20	2.68	89	8.050	0.49	9.7	0.0045	0.0053	0.0043
R	C plane Sap	20	2.59	94	8.060	0.50	10	0.0051(3)	0.0055	0.0044
S	C plane Sap	39	3.27	66	8.520	1.0	6.8	0.024	0.032	0.029

1. All samples except B and C were grown at elevated temperatures, while B and C were grown at room temperature. Here d is the sample thickness, Si and Sap denote silicon and sapphire substrates, respectively, and RRR is the residual resistivity ratio between 290 and 10 K. Note that uncertainties for $T_\mathrm{c0}$ are 3 mK, the uncertainties for $-C_{2}$ experimental are given with the values and for all other values uncertainties are as stated in the text. The last two columns are the theoretical values for $-C_{2}$ for the model of Hara and Nagai (HN) and Kogan