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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 d
^{2}E/dk^{2}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.

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