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The de Broglie Wavelengths of Electrons

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:

E = h2k2/2m

where the wavenumber k is determined by momentum through the relation

p = hk = h/λ
and hence
λ = 2π/k
where
k = (2mE)1/2/h

Thus

λ = 2πh/(2mE)1/2
= h/(2mE)1/2
= (h/(2m0)1/2)(1/((meE)1/2)

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 = mem0 The quantity h/(2m0)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 109. The final result is that:

λ (nm) = 19.65/(meEeV)1/2

Using this relationship the following examples of de Broglie wavelengths of electrons versus the lattice constants. All values are for room temperature, 300° K.

MaterialElectron
Energy
Effective
Mass Ratio
de Broglie
Wavelength
nm
Lattice
Constant
Ratio
Al11.7 eV1.05.7443  
GaAs0.050 eV0.067339.4770.56531.67x10-3
GaAs(2D)0.01 eV0.067759.090.56537.45x10-4

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