Discuss effective’s mass of electrons and holes.

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Suresh Chand · Accepted Answer

The electrons in a crystal are not completely free, but instead interact with the periodic potential of the lattice. As a result, their 'wave-particle' motion cannot be expected to be the same as for electrons in free space. Thus, in applying the usual equations of electrodynamics to charge carriers in a solid, we use altered values of particle mass. In doing so, we account for most of the influences of the lattice, so that the electrons and holes can be treated as 'almost free' carriers in most computations. The calculation of effective mass must take into account the shape of the energy bands in three-dimensional k-space, taking appropriate averages over the various energy bands. The effective mass of an electron in a band with a given (E, k) relationship is found in to be (m^* = frac{hbar^{2}}{d^2 E/dk^2}) Thus the curvature of the band determines the electron effective mass. It is clear that the electron effective mass in GaAs is much smaller in the direct  conduction band (strong curvature) than in the L or X minima (weaker curvature, smaller value in the denominator of the m* expression). A particularly interesting feature is that the curvature of d2E/dk2 is positive at the conduction band minima, but is negative at the valence band maxima. Thus, the electrons near the top of the valence band have negative effective mass, according to the above equation. Valence band electrons with negative charge and negative mass move in an electric field in the same direction as holes with positive charge and positive mass. As discussed, we can fully account for charge transport in the valence band by considering hole motion. For a band centered at k = 0 (such as the (Gamma) band in GaAs), the (E, k) relationship near the minimum is usually parabolic (E = frac{hbar^{2}}{2m^{*}} k^2 + E_{c}) Comparing the above two relation indicates that the effective mass m* is constant in a parabolic band. On the other hand, many conduction bands have complex (E, k) relationships that depend on the direction of electron transport with respect to the principal crystal directions. In this case, the effective mass is a tensor quantity. However, we can use appropriate averages over such bands in most calculations.

Discuss effective’s mass of electrons and holes.

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Discuss effective’s mass of electrons and holes.

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