Crystal electrons in an external magnetic field

In this section we consider the response of electrons in an external magnetic field. This response is particularly dramatic when the electrons are confined in two dimensions and are subjected to a very strong perpendicular magnetic field at 

From the Greek word vaxeprfaig, which means lagging behind. 

To begin, let us explore how electrons can be confined in two dimensions. In certain situations it is possible to form a flat interface between two crystals. We will take x, y to be the coordinate axes on the interface plane and z the direction perpendicular to it the magnetic field will be H = Hi. A particular arrangement of an insulator and doped semiconductor produces a potential well in the direction perpendicular to the interface plane which quantizes the motion of electrons in this direction. This is illustrated in Fig. 7.9 the confining potential Vconf(z) can be approximated as nearly linear on the doped semiconductor side and by a hard wall on the insulator side. The electron wavefunctions V (r) can be factored into two parts  

In the following discussion we will ignore the crystal momentum k and band index normally associated with electronic states in the crystal, because the density of confined electrons in the inversion layer is such that only a very small fraction of one band is occupied. Specifically, the density of confined electrons is of order n = 10 cm , whereas a full band corresponding to a crystal plane can accommodate of order (10 cm ) / = 10 cm states. Therefore, the confined electrons occupy a small fraction of the band near its minimum. It should be noted that the phenomena we will describe below can also be observed for a system of holes at similar density. For both the electron and the hole systems the presence of the crystal interface is crucial in achieving confinement in two dimensions, but is otherwise not important in the behavior of the charge carriers (electrons or holes) under the conditions considered here. 

---

A related feature of ODMR also results in enhanced sensitivity over conventional EPR for measurements on randomly oriented biomolecules that cannot be obtained as single crystals. Conventional EPR measurements are made in an external magnetic field, and because of the magnetic dipole-dipole interaction between the unpaired electrons of the triplet ... 

When a crystal is placed in an external magnetic field, all crystal-field degeneracies are removed. This is called the Zeeman effect and can be studied by different techniques Zeeman spectroscopy, electron paramagnetic resonance (EPR) and magnetic circular dichroism (MCD). 

Let us remark that in crystals consisting of aromatic molecules, to which the theory of Sternlicht and McConnell (26) was applied, the excited triplet states are not three-fold degenerate even when an external magnetic field is absent. Due to the dipole spin-spin interaction between electrons the degeneracy is totally or partially removed, depending on the symmetry of the excited state wavefunction. By a phenomenological description of this splitting the so-called Spin-Hamiltonian is usually applied... 

---