Quantum wells and confined carriers

In Section 2.1. we examined solution to the Schrodinger equation in free space and in a periodic potential and foimd families of wave functions having a simple, quadratic relationship between wave energy and momentum over wide ranges of energy values. Because there are a very large number of such states in a solid, it is normally impossible to distinguish one individual state from another. In this Section, we will examine, briefly, the impact of artificial potential structures on the wave solutions. For a complete discussion of such effects, the reader is referred to books on quantum mechanics. 

Artificial-potential-barrier structures appear in a number of important microelectronic device applications. The best known is probably the laser diode. These ubiquitous devices are deceptive in their outward simplicity - a single small chip of material 

The solutions to Equation 2.53 are proportional to sin(Kx) and cos(kx). Since the wave function is zero outside the well and since the wave function inside and outside must match at the bormdaries, the values of k are forced to be kL = ntt, where n=l,2,3. for the sin(Kx) solution and kL = (n-l/2) c for cos(icx) to produce nodes at 

The two exponential relationships describe the decay of the wave function outside of the well, while the middle relationship in 2.55 describes the wave function within the well. As in the infinite well, the wave function and its first derivative must be continuous across the well boundaries. The boundary conditions yield four equations (two boundaries, two matching conditions) in four unknowns (A,B,C,D), yielding  

Additional equations for C and D can be developed but will not be reproduced here to save space. Furthermore, one can show that  

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