Calculating thickness of bismuth nanolayers whose impedance matched with the characteristic impedance of the field of two counter-propagating electromagnetic waves.
Authors:   Kotelnikov Institute of Radio Engineering and Electronics of RAS, Mokhovaya 11, building 7, Moscow 125009, Russia, Pyataikin I.I. 
Published: November 2021

This article determines the thickness dpk of the bismuth layer at which its impedance becomes equal to the characteristic impedance of the field of two electromagnetic waves propagating towards each other in a vacuum. For the first time such a calculation was performed by Kaplan in his article [6]. The impetus for re-examining this problem was the 29-fold discrepancy between the value of the mean free path l0 in the bulk bismuth at room temperature, obtained in the work of Pippard and Chambers [7] and used by Kaplan for his calculation, and the estimate of l0 given in the paper by Cronin et al. [12]. To resolve this contradiction, in the second section of the present article, the mean free path l0 in bismuth has been re-estimated, taking into account the strongly anisotropic energy spectrum of quasiparticles in this material from the very beginning. The set of values of the effective masses and Fermi energies of electrons ϵe and holes ϵh that had been measured in the work of Brandt et al. [15] at liquid helium temperatures was the basis for calculating l0. To determine the Fermi velocities at room temperature, the values of energies ϵe and ϵh were corrected in accordance with the change in the concentration of charge carriers with an increase in temperature from 4.2 K to 300 K. The characteristic electron and hole relaxation times were calculated on the basis of the values of mobilities of charge carriers, measured in the work of Michenaud and Issi [16], and their effective masses, taken from the referred above article by Brandt with coworkers. The values of the Fermi velocities and characteristic relaxation times obtained in this way made it possible to estimate the mean free paths along the main directions of the electron and hole ellipsoids in bismuth. It turned out that at room temperature the characteristic mean free path l0 is about 1800 Å, which is 1.8 times greater than the estimate of l0 given in the article by Cronin et al., and 16 times less than the value of l0 used by Kaplan for calculation of the thickness dpk in his work above. It is clear that such a significant discrepancy is a sufficient basis for re-calculating dpk using the new value of l0 found in the present paper. In the third section of this article, it is discussed how justified the choice of a zero value for the specularity coefficient p, made in Kaplan's work, is. Based on the analysis of the available literature on that issue, a nonzero value of this parameter was substantiated and arguments were given in favor of choosing the specularity coefficient p equal to 0.56, in accordance with the results of the work by Hoffman and Frankl [21]. Calculating the dpk thickness for a given specularity value and the new mean free path l0 which was obtained in the second section of this article yields dpk equal to 177 Å, that is almost five times less than the value of dpk found by Kaplan. It is clear that such a small value of the thickness of the bismuth layer will seriously complicate its separation from the supporting substrate. This makes the prospects for using such layers in the spectroscopic systems mentioned in the Introduction of the present paper rather vague. Unfortunately, this is not the only problem standing in the way of using the unique electrodynamic properties of bismuth layers of the dpk thickness. As mentioned in the fourth section, an uncontrolled change in the specularity coefficient p due to the appearance of a surface electric charge on the layer can also make it difficult to work with this object. Since the problem of producing free-standing bismuth layers of the dpk thickness has not yet been solved, it is difficult to say how serious the problems associated with their uncontrolled electrification will be. As noted in the fourth section of the present paper, much of the significant discrepancy between Kaplan's estimates of the dpk thickness in bismuth and those made in this work is due to the greatly overestimated value of the mean free path l0 borrowed by Kaplan from the Pippard and Chambers article. Therefore, in the final section of the current paper, it is discussed the reasons why these authors obtained such a huge value of l0 in bismuth at room temperature. It is suggested that the large value of l0 obtained in their work is a consequence of the application of the Reuter and Sondheimer theory, which is valid in the case of a spherical Fermi surface, for the analysis of experimental data on the anomalous skin effect in strongly anisotropic bismuth. The final section also discusses one controversial thesis from Kaplan's article concerning the magnitude of dpk in nickel. It is shown that the value of this parameter given in his paper is caused by the use of an erroneous value of the mean free path in this metal.