Microscopic Model of Metal-Semiconductor Contacts and Semiconductor Heterojunctions
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rearrangements and/or chemical reactions at the interface. MODEL OF INTERFACE CHARGE DENSITY
Our model is based on a variational description of the electron charge density at the interface[I-31. The periodic ionic charge density in each component of the interface is replaced by a uniform positive background, and the interface is defined as the abrupt discontinuity of this uniform positive background. The electron charge density embodies a short-range double-layer dipole associated with the valence
electrons interfacial redistribution and a long-range space-charge region arising from
the uncompensated doping density in the semiconductor. Parametric forms are used to describe the electron charge density and consequently the electrostatic Hartree potential generated by this charge density can be evaluated analytically by
integrating Poisson's equation. The numerical values of the parameters specifying the self-consistent charge density are obtained by the simultaneous imposition of
thermodynamic equilibrium boundary conditions and minimization of the interface energy, as described below. The boundary conditions required to determine the Hartree potential at the contact are dictated by equilibrium thermodynamics. Thermal equilibrium requires
equal temperatures on the two sides of the junction; mechanical equilibrium
requires equal pressures: and electron-transfer equilibrium requires the equalization of the bulk Fermi levels[3]. This latter boundary condition implies that the selfconsistent charge density must generate an electrostatic junction potential whose net Mat. Res. Soc. Symp. Proc. Vol. 54. t1986 Materials Research Society
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drop across the contact equals the difference in bulk chemical potentials (separation energy) of the two components of the junction. This electron-transfer equilibrium boundary condition applies to the total potential drop and does not provide any information on the relative magnitude of the long-range and short-range components of the electrostatic potential. These two contributions are obtained independently by minimization of the interface energy written as a local-density functional of the valence electron density[I-4]. Once the space-charge parameters are specified, the Hartree potential is evaluated analytically. The energy functional is then constructed and the interface energy is minimized with respect to the variational parameters. MODEL PREDICTIONS From the model defined in the previous Section, the self-consistent charge densities and Hartree potentials can be evaluated for the interface between any two semiconductors or between a semiconductor and a metal. The equalization of the bulk Fermi levels in each electrode implies that the electrostatic potential drop across the junction must be equal to the difference in bulk chemical potentials of the two components of the contact. In order to rewrite this potential drop in terms of measurable quantities, we must relate the (negative) bulk chemical potential to the (positive) work function for metals or the electron affin
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