Toward the microscopic understanding of conductance on a quantum mechanical basis

Abstract

When Lew Dawydowitsch Landau¹ published his paper on the diamagnetism of metals [16], he was probably not aware of the importance of his calculation for quantum transport. In this publication he determined the quantized energy levels of free electrons in a homogeneous magnetic field

$$ {E_{n}} = \hbar {\omega _{c}}\left( {n + \frac{1}{2}} \right)$$

, where n ≥ 0 is an integer and ω _c = eB/m is the cyclotron frequency. Similar calculations emphasizing the role of the sample edges were performed by V. Fock² [17], C.G. Darwin [18], and Edward Teller³ [19]. The energy levels that we now call Landau levels played their first important role in experiments on the magnetoresistance of bismuth performed by L. Shubnikov and Wander Johannes de Haas⁴ in 1930 [20]. They found that the measured magnetoresistance oscillated as a function of magnetic field, an effect now called Shubnikov-de Haas effect. A theoretical description of this effect was, for example, given in Ref. [21].