Measurement of band offsets and interface charges by the C–V matching method

The present article describes a novel application of capacitance–voltage measurements to determine simultaneously the band discontinuities ${\mathrm{(\Delta E}}_V,$ ${\mathrm{\Delta E}}_C)$ and interface charge density $\mathrm{(\sigma )}$ of heterojunctions. The method, which we refer to as $\mathrm{C–V}$ matching, complements the most versatile $\mathrm{C–V}$ profiling technique proposed by Kroemer and successfully applied by others. In contrast to the $\mathrm{C–V}$ profiling which is limited to isotype heterojunctions, the new method is applicable to $\mathrm{p-n}$ heterojunctions as well. The methodology is based on three cardinal equations which are not controversial—the lineup of the bands relative to the common Fermi level (at equilibrium) or the quasi-Fermi levels (when voltage is applied), the charge neutrality and the expression for the total capacitance of the heterostructure. The three equations are formulated for equilibrium as well as nonequilibrium conditions, using quasi-Fermi levels and the quasi-equilibrium approximation. The three cardinal equations are defined by the two constant (albeit unknown) interface parameters ${\mathrm{(\Delta E}}_V\mathrm{,\sigma )}$ which are assumed to be independent of the voltage and two variables ${\mathrm{(\phi }}_{s_1}$ , ${\phi }_{s_2})$ , which describe the total band bending on each side of the heterointerface and vary with the applied voltage. The actual interface parameters ${\mathrm{\Delta E}}_V,$ $\sigma$ are determined by $\mathrm{C–V}$ matching between the calculated and the measured curve. The metric for the optimal match between calculated and measured capacitance vectors is discussed. The methodology presented in this study is general and can be applied to semiconductor-semiconductor and semimetal-semiconductor heterojunctions. It is illustrated here for the HgTe-CdTe semimetal-semiconductor heterojunction, which cannot be evaluated by the $\mathrm{C–V}$ profiling. The significance of the simultaneous determination of the band discontinuities and interface charges of heterojunctions is also discussed. In addition, the methodology presented in this article models the behavior of biased heterojunctions under nonequilibrium conditions, taking into consideration the values of band offset and interface charge density of an actual heterointerface.