Oscillator strength and sum rule for intersubband transitions in a superlattice

The oscillator strength for the inter-minisubband optical absorption is calculated for a superlattice. For periodic structures the matrix element 〈n‖x‖n’〉 is ill defined and we show how the definition of the position operator has to be modified in order to obtain the correct result which then leads to the same result as a calculation of the oscillator strength using the matrix element 〈n‖p‖n’〉. A new sum rule for super- lattices is derived, which involves averaging over wave vectors in the first Brillouin zone.