San José State University |
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Thayer Watkins Silicon Valley & Tornado Alley USA |
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The de Broglie Wavelengths of Electrons |
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For nanostructures one of the critical parameters is the lattice constant of the crytal structure relative to the de Broglie wavelength of the electrons in the structure. The purpose of this page is to present the calculation of this wavelength for some relevant semiconductor materials.
The physical characteristics of the electrons in the crystal structure may be approximated by the properties of free electrons adjusted for the effective mass of electrons in the structure. The energy of a free electron is given by the relation:
where the wavenumber k is determined by momentum through the relation
Thus
It only remains to put the relation in convenient units. The effective mass m may be expressed in terms of the effective mass ratio and the rest mass of the electron; i.e., m = m_{e}m_{0} The quantity h/(2m_{0})^{1/2} is 4.9091x10^{-19} in SI units. To get energy in electron-volts the energy in Joules must be divided by 1.602x10^{-19} and thus the coefficient in the equation must be multiplied by its square root. For a wavelength in nanometers the value in meters must be multiplied by 10^{9}. The final result is that:
Using this relationship the following examples of de Broglie wavelengths of electrons versus the lattice constants. All values are for room temperature, 300° K.
Material | Electron Energy | Effective Mass Ratio |
de Broglie Wavelength nm | Lattice Constant | Ratio |
Al | 11.7 eV | 1.0 | 5.7443 | ||
GaAs | 0.050 eV | 0.067 | 339.477 | 0.5653 | 1.67x10^{-3} |
GaAs(2D) | 0.01 eV | 0.067 | 759.09 | 0.5653 | 7.45x10^{-4} |
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