High-field calculations of Landau-like shallow donor states: A finite-difference approach

Abstract

A theoretical method for calculating the energies and wave functions of an electron bound to a shallow donor in a semiconductor, subject to an applied magnetic field, is presented. This approach is particularly useful for describing highly excited Landau-like states, which cannot be dealt with properly using most other theoretical models. First, an adiabatic high-field approximation is used where mixing between different Landau-like states is neglected. Solving the one-electron Schrödinger equation is then reduced to finding solutions to a one-dimensional differential equation for motion along the field axis. We present results in which a finite-difference technique is used to solve this equation numerically. Values for the electron wave function at discrete points along the field axis are then determined. By calculating the discrete Fourier transform of this set of values, an analytical form for the wave function in terms of sines and cosines is obtained. These resultant wave functions are then used to calculate a Hamiltonian matrix in which mixing between different high-field states is included. Diagonalization of this matrix yields improved values for the energies of the impurity states. Where previous results exist, our results are compared with those of other theoretical approaches and from experiments on the donor in GaAs. The advantage of our approach is that it enables both energies and wave functions to be determined without any prior assumptions of the form of the wave function in the field direction. This includes expressions for highly excited states which are difficult to obtain by other means. The results demonstrate the validity of the present method of calculation. ͓S0163-1829͑98͒08315-5͔

Si:P as a laboratory analogue for hydrogen on high magnetic field white dwarf stars

“…NATURE COMMUNICATIONS | DOI: 10.1038/ncomms2466 ARTICLE both free atoms 1,15 and solids 28,29,[33][34][35]37,38 , its characteristic anti-crossings with Dm ¼ 0 and DL ¼ ± 2 have not been observed previously (and neither have the purely linear Zeeman, B>z anisotropy induced, anti-crossings with Dm ¼ ± 1 and DL ¼ 0). The results show that silicon donors provide an excellent laboratory material for studying effects of atomic physics that have previously been beyond the capability of laboratory equipment and available for study only in extreme astrophysical phenomena.…”

Binding energy of shallow donors in a quantum well in the presence of a tilted magnetic field

“…2 shows that for θ = 0 • the difference E b (B = 16T )− E b (B = 0T ) is 0.6 Ry * =8 meV, while for θ = 90 • it is only 0.05 Ry * =0.6 meV, see also Fig. (3). These totaly different values in the two limiting field orientations (case I and case II ) are related to the fact that the QW width is smaller then (or comparable to) the characteristic magnetic length λ c at fields up to 16 T. If the magnetic field is applied along the z-direction (θ = 0 • ), the electron is localized in x-and y-directions by the external magnetic field, as discussed at the outset (the bigger the field, the larger the magnetic localization).…”

“…Most of these theoretical studies have focused on the binding energy of the neutral donor (D 0 ) [1,2,3,4], charged donor [1,5,6], neutral exciton [7,8,9,10], and charged exciton (trion) [9,10,11,12,13] states as a function of magnetic field. The vast majority of these calculations treat the most straightforward case, where the magnetic field is applied parallel to the direction of growth of the QW.…”

Magnetodielectric coupling in nonmagnetic Au/GaAs:Si Schottky barriers

“…The increase in E a with respect to H has been previously studied both theoretically 24,25 and experimentally using optical 26,27 and Hall measurements 22,28 . In the GaAs host, the Bohr radii of the hydrogen-like donor electrons are renormalized upwards by the small effective mass, m GaAs = 0.065m e , and the large relative permittivity, ε GaAs = 13.5.…”