Chapter 3: Electronic effects of doping impurities

1. Chapter 3:Electronic effects of doping impurities

2. Schrödinger Equation for an Impurity Electron Where H0 is the one electron Hamiltonian of the crystal, U is the potential due to the impurity One may expand Y(r) in terms of ynk(r), the Bloch wave functions.

3. Wannier function:

4. Effective Mass Theory Expand the impurity electron wave function in terms of Wannier functions Regarding R as a continuous variable

5. Donor impurity in GaAs

6. Some results

7. Donor impurity in Si and Ge For single valley

8. For each valley, there is a state. Thus, we have six-fold degeneracy. To remove the degeneracy, we treat (valley-obit coupling) Donor impurity in Si and Ge Since this is very similar to the equation for one valley, we can treat the other potential terms by perturbation theory and diagonalize the Hamiltonian. The s state is split into A1(1), E(2), and T2(3).

9. G4(p) G1(s) G4(p) Complications: (i) valence bands are degenerated (ii) the valence bands are warped Acceptor impurity We don’t know how to use the effective mass theory Remembering that from k.p theory where H0=

11. In practice, that equation is still difficult to solve. Only numerical solution is possible.

12. Deep Centers • Shallow impurity levels: wave function extends to many unit cells and can be constructed from one Bloch function index by single wave vector • Deep Centers: wave function localized to small number of unit cells and must be constructed from many Bloch functions from several bands and over different wave vectors