In 3-D nanomaterials, the three spatial dimensions are all above the nanoscale. Therefore the two aforementioned effects can be neglected. Tuy nhiên, bulk nanocrystalline materials Exhibit a high grain boundary area-to-volume ratio, leading to an electron scattering tăng print. As a consequence, nanosize grains Tend to Reduce the electrical conductivity.
In the case of 2-D nanomaterials with thickness at the nanoscale, quantum confinement along the thickness dimension sẽ occur. Simultaneously, the carrier motion is uninterrupted along the plane of the sheet. In fact, as the thickness is reduced to the nanoscale, the wave functions of electrons are limited to very specific values along the cross-section (see Figure 7:18). This is only electron wavelengths vì có are multiple integers of the thickness Will Be allowed. All other electron wavelengths Will Be absent. In other words, there is a reduction in energy số available for electron conduction states along the thickness direction. The electrons trapped in what is trở gọi a well of width equal potencial to the thickness. In general, the effects of confinement on the energy state for a 2-D nanomaterial with thickness at the nanoscale can be Written as
En = 2πmL n2 222 (7:16)
where h ≡ h / 2π, h is Planck's constant, m is the mass of the electron, L is the width of the potencial well (thickness of 2-D nanomaterial), and n is the principal quantum number. 7:16 Equation assumes infinite-depth security model potencial well. As Mentioned, the carriers are free to move along the plane of the sheet. Therefore the total energy of a carrier has two components, namely a term related to the confinement dimension (Equation 7:16) and a term associated with the unrestricted motion along the two other in-plane dimensions.
To hiểu the energy associated with unrestricted motion, let's the z-direction giả sử to be the thickness direction and x and y, the in-plane directions are chứa delocalized electrons. Under những conditionsEND_SPAN, the unrestricted motion can be Characterized by two wave vectors kx and kx, mà related to the electron's momentum along the x and y directions, respectively, in the form px = py = h-h-kx and ky. The energy electrons delocalized tương ứng to những
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