**SAN JOSÉ STATE UNIVERSITY
**

*Thayer Watkins*

**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 = ()^{2}k^{2}/2m

where the wavenumber k is determined by momentum through the relation
####
p = k = h/

and hence

= 2/k

where

k = (2mE)^{1/2}/

Thus
#### = 2/(2mE)^{1/2}

= h/(2mE)^{1/2}

= (h/(2m_{0})^{1/2})(1/((m_{e}E)^{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_{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 nanameters the value in
meters must be multiplied by 10^{9}. The final result is that:
#### (nm) = 19.65/(m_{e}E_{eV})^{1/2}

Using this relationship the following examples of de Broglie wavelengths
of electrons versus the lattice constants. All values are for room
temperatures, 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 |

GaAs(2D) | 0.01 eV | 0.067 | 759.09 | 0.5653 |