Before working Exercise, read the Guidelines in the Computational Exercises of Chapter 5,...

Before working Exercise, read the Guidelines in the Computational Exercises of Chapter 5, which discuss the Schrödinger equation adapted to numerical solution, with m and ii both defined as 1. Choosing only ψ(0) and ψ(Δx),the following relationship gives ψ(2Δx), then, by reapplication, ψ at every multiple of Δx.

ψ(2Δx) = 2ψ(Δx) - ψ(0) + 2Δx2(U(Δx) - E)ψ(Δx)

Here we consider only U(x) that are symmetric about x = 0, so that ψ(x) must be either an odd or even function of x. Our potential energy will model a one-dimensional crystal as an array of finite wells. A flexible (though not terribly elegant) way to define this U(x) starts by defining a "single well function" that is I everywhere except in a unit-width region centered on X, where it is B, standing for bottom.

sw(.x, X, B) - (B + l)/2 + sign[l/4 - (X - x)^2]*(B – 1)/2

If x differs from X by less than 1/2, this function is B, and if by more than 1/2, it is 1. Now we can define a U(x) that is U0 everywhere but in wells centered on x = 0, X1 X2, and X3, where it is 0.

U(x) = U0 sw(x, 0,0) sw(x, X,, 0) sw(x, X2,0) sw(x, X3, 0)

The assumption of symmetric U(x) allows us to plot \b for positive x only (we know it is odd or even), but it also means that, in effect, our crystal has three more wells at negative x, so it has seven atoms. (Note: Many further studies are possible within the framework provided. Exercise 80, for example, can investigate various atomic spacings, showing the effect on band width. The number of atoms can also easily be varied.)

Donor and Acceptor Levels: Define U(x) with X1 X2, and X3 set to 1.2, 2.4, and 3.6, respectively, which gives seven equally spaced wells separated by walls of width 0.2. For U0 use 20, and for Δx, use 0.001. (a) (Note: This part can be skipped if also working Exercise 80.) Following the Chapter 5 guidelines on choosing ψ(0) and ψ(Δx), test both odd and even functions at different trial values of E by finding ψ at all multiples of Δx and plotting the results from x = 0 to x = 5. Find 14 allowed energies. Note that the indicator of having passed an allowed energy is the flip of the diverging large-x tail. The lowest energy or two will take the most work. Except for these, there is no need to exceed three significant figures, Afterward, make a scatter plot of E versus n, where n goes from 1 to 14. (b) Now replace two atoms with impurity atoms as follows: For either the well at 1.2 or the well at 2.4, change B from 0 to 0.1. (Note: Changing one well automatically changes the corresponding well at negative x. We avoid changing the outermost wells simply because it doesn't work as nicely.) Tins puts the bottom of the altered well at 0,1 U0, or 2 units. Again find 14 energies, and make another scatter plot, (c) Repeat part (b), but choosing -0.1 for B in the "impurity" atom, putting its bottom at -2 units, (d) Discuss how the impurities added in parts (b) and (c) correspond to atoms whose valence differs from that of the intrinsic atoms, (e) If each intrinsic atom comes with two electrons, and the impurities come with one and three, respectively, which states would be filled in parts (b) and (c)? Remember that there are two spin states, (f) Discuss the overall result of adding the impurities.