De Broglie Wavelengths of Electrons

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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 caluculation 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 = (h)²k²/2m
where the wavenumber k is determined by momentum through the relation
p = hk = h/λ
and hence
λ = 2/k
where
k = (2mE)^1/2/h
Thus
λ = 2h/(2mE)^1/2
= h/(2mE)^1/2
= (h/(2m₀)^1/2)(1/((m_eE)^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 = m_em₀ 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/(m_eE_eV)^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.

Material Electron
Energy Effective
Mass Ratio de Broglie
Wavelength 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

Material	Electron Energy	Effective Mass Ratio	de Broglie Wavelength	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

E = (h)2k2/2m

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

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

λ (nm) = 19.65/(meEeV)1/2

E = (h)²k²/2m

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

λ = 2h/(2mE)^1/2
= h/(2mE)^1/2
= (h/(2m₀)^1/2)(1/((m_eE)^1/2)

λ (nm) = 19.65/(m_eE_eV)^1/2