De Broglie wavelength and Fermi wavelength

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As we know that the properties of nanoscale material are quite diffrent from the bulk's. But what's the threshold of the size, or what the upper limmit where significant difference can be observed? The De Broglie wavelength or the Fermi wavelength of the electron in material? How to understand that?
 

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Simon Bridge
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The threshold where one approximation takes over from another one is fractal in nature - it depends on the situation being studied and what you want to know about it.
 
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As we know that the properties of nanoscale material are quite diffrent from the bulk's. But what's the threshold of the size, or what the upper limmit where significant difference can be observed? The De Broglie wavelength or the Fermi wavelength of the electron in material? How to understand that?

The Fermi wavelength is nothing but the de Broglie wavelength of electrons near the Fermi energy. Most of the properties of solids, involving electronic transport, are described by the dynamics of the electrons near the Fermi energy. While describing the transport of these particular electrons, one has to account for their scattering by a potential. This potential could be a result of disorder or lattice vibrations (phonons). If the variations in this potential are comparable to the de Broglie (or Fermi) wavelength then quantum effects become more prominent. If, however, this is not the case, i.e. the potential is slowly varying, then one can use the semi-classical description of the system; one example is the Boltzmann transport equation. In this regime, we have used "semi" because we still use band theory, which is quantum, but ignore the wave nature of the electrons in these bands.

Another example where one encounters a rapidly varying potential is in a semiconductors quantum well. Here is one example:

http://www.intenseco.com/images/graphics/technology/figure-1-quantum_well_intermixing.png [Broken]

The width of the quantum well typically ranges from a couple of nanometers to a couple of tens of nanometers. There is, however, one major difference between this well and the 1D quantum well one encounters in standard quantum mechanics. Say that the above figure represents the potential profile along the z direction; the potential in the xy plane, however, is flat. For electrons whose energies lie inside the well, there are a macroscopic number of states which can travel in the xy plane, while still being confined in the z direction. In a way, you could consider this structure as "bulk" in the xy plane and "nanoscale" in the z direction. This distinction comes from the fact that there are macroscopic numbers of "modes" in the xy plane, while there are only a few modes in the z direction. By "modes" I simply mean traveling states. The modes in the z direction are analogous to the discrete eigenstates of the 1D quantum well like:

http://britneyspears.ac/physics/fbarr/images/qwdiag.gif

As you increase the width of the well more modes will start getting squeezed into the well. When the well is sufficiently wide you will have a macroscopic number of modes. Consequently, you now have a macroscopic number of modes in all three directions, which is what happens in a bulk solid. Now, instead of only having a potential well in the z direction you could have decided to have it in two directions. In that case you will end up with the quantum wire. Alternatively, if you choose to have a potential well in all three directions, then you would end up with a quantum dot. You can read more about this here:

www.phys.ttu.edu/~cmyles/Phys5335/Lectures/12-2-10.ppt

OR

http://users.ece.gatech.edu/~alan/E...tures/King_Notes_Density_of_States_2D1D0D.pdf
 
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