Show that electrons in states near the conduction band minimum behave as free electrons with an...

Show that electrons in states near the conduction band minimum behave as free electrons with an 'effective mass' m. (Hint: Expand the dispersion relation E(k) locally into a Taylor series)

Solutions

Expert Solution

We expand the energy of the electron in a Taylor series about the extrema of propagation vector

where denotes energy of the conduction or valence band in consideration.

For the bottom or minimum of the conduction band

Neglecting higher order terms

We now have the same quadratic relation in k as for free electron gas, provided we change the definition of mass of the electron as

Hence energy can be written as

Which is equivalent to the dispersion relation for free electron gas.

Hence electrons in conduction band minimum behaves as free electrons with effective mass .

genius_generous answered 1 year ago

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