Fig. 1: A schematic visualization of effective spin 1/2 hole or electron quasiparticles and their Fermi-distribution in a quantum well with and without external magnetic field.

a A visualization of an elementary charge particle with internal quantum motion leading to quantized internal orbital momentum in Plank’s constant \(\hslash\) and associated magnetic momentum characterized by the effective g*-factor, \({{\mbox{g}}}^* \!{\mu }_{B}\), where \({\mu }_{B}=e\hslash /2{m}_{e}\) being the Bohr magneton; Electrons have 1/2 spin which can be either up or down. When a magnetic field is applied to a magnetic moment, then the moment experiences a torque to try to align the moment with the field to minimize energy. b Energy schematic of the quantum well filled with free carriers at zero temperature in zero magnetic field (B = 0, left image) and for B > 0 (right image). In zero field (B = 0) free Fermionic carriers occupy states of the band up to the Fermi energy, \({E}_{F}\). In non-zero field the free carriers’ spectrum is quantized in discrete degenerate Landau levels (LL), each of them is split by the Zeeman energy, \({E}_{Z}\), in two spin sublevels. The two-dimensional hole gas (2DHG) energy spectrum is characterized by the Zeeman and orbital gaps, \({E}_{Z}\) and \({E}_{{orb}}\), respectively. Even or odd number of the discrete levels is occupied, called the filling factor, ν, depending on the magnetic field strength and the 2D carrier areal density controlled by the gate voltage in a field-effect device.