1 Band Functions and the Bloch Condition

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On the right is the electron dispersion relation (in red) for a square well potential of moderate strength, -10eV. The energies are plotted only for k>0 since E(-k)=E(k). The blue curve is the free electron dispersion relation for comparison, with its origin of energy chosen to be the average of the potential. The Bloch function for the lowest band has been able "to exploit the potential to lower its energy" with respect to that of the pure plane wave.

On the left is a 3 dimensional plot of the wave function of energy -3 eV, with a k-value near the middle of the second band. The real part of the wave function is plotted along the blue axis, the imaginary part along the green axis. As you first see it, the z-axis (in black) is along the line of sight. The function is (nearly) pure real at the "far end" of the cell. The phase at the "near end" is about 240 degrees. Drag the axes with the cursor in this window (left mouse button depressed) to see the 3-D plot in a variety of perspectives. As z moves from one end of the cell to the other you see that the phase of the wave function increases, and has accumulated about 240 degrees of phase moving through the full cell.

Use the show menu over the wave function graph to display the phase of wave function (phase of psi) to confirm your estimate. (The phase is plotted in units of pi. The vertical discontinuities are because the phase is restricted to lie in the range -pi to pi. Hence add (or subtract) 2*pi for each such discontinuity.)

The square of the wave function (psi-squared) is plotted for the third choice in the show menu. Note that the electron density has a minimum in the centers of the wells and is relatively large in the barriers. Explain why the solution seems to give an unfavorable distribution of charge. (Hint: look at the charge density for the state, at about the same k-value, at E = -7.8 eV in the lowest band. These two states must be orthogonal to one another.)

Finally, return to the wave function (psi) plot. Watch how the accumulated phase, from one end of the cell to the other, changes as you choose progressively higher energies, always staying in the same band. (An easy way to get the functions for a given point in the band structure is to double click with the left mouse button with the cursor at the desired energy.) What is the variation in accumulated phase as you move from one end of a band to the other?