# Discuss effective’s mass of electrons and holes.

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• 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.