Exact Fermi energy, susceptibilities, and their singularities for an electron gas in a uniform magnetic field at T = 0 °K

✍ Scribed by Y.B Suh

• Publisher: Elsevier Science
• Year: 1975
• Tongue: English
• Weight: 614 KB
• Volume: 94
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1016/0003-4916(75)90168-2

✦ Synopsis

Exact analysis is presented to derive the magnetic response functions and their singularities of free-electron gas in a uniform magnetic field of arbitrary strength at T = 0 'Ii. The newly defined functions, A,(s) = XVI (S -n)p of p =-= -$, 4, 8, are _ _ employed to obtain the Fermi energy, magnetization, and susceptibility as functions of B. 1-t is revealed that the spin susceptibility is composed of two parts, xSI and ,I~?. where xSC is purely oscillatory diamagnetic. A graphical method of finding the Fermi energy Q(B) as a function of B has been obtained. The system is shown to become totally one-dimensional electron gas in the field B greater than BL = (2vn)?i3 and the total energy satisfies E, = &&3)N. The obvious extension of the present theory to the Bloch electrons on the ellipsoidal constant energy surface is also discussed.

1. Introduction

It is well known that a free-electron gas in a uniform magnetic field shows singularities in its susceptibility and other properties as well[l, 21, especially at T = 0°K. The inherent singularities in the density of states cause difficulties in the subsequent analysis of the exact form. One often uses the well-known EM (Euler-Maclaurin) formula; however, this smooths out all the singularities of the main feature and leads to incorrect results at T = 0°K. One is better justified in using this formula or the Poisson summation formula (see Section 5 for their definitions) for the system at nonzero temperatures [3], in which the latter separates the magnetic response functions into two parts, stationary and oscillatory.

However, it should be kept in mind that the oscillatory part has no singularities at these temperatures. For this reason, the system at T = 0°K requires a more careful analysis. The use of the EM formula is less justified, since it amounts to replacing a singular function with a continuous one, which in turn leads to incorrect physical (consequences. Here we carry the formulations in the exact form as far as we can, #and later some remarks will be made concerning the use of the abovementione:d formula.