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ECE 535 Notes for Lecture # 14

Class Outline: • Quantitative Methods for Bandstructure

M.J. Gilbert ECE 535 – Lecture 14

Quantitative Methods for Bandstructure Calculations - 1 Our purpose here is to develop a bag of tricks to understand semiconductor bands from a different perspective…

The Kronig-Penny (KP) model is an exactly solvable model of the quantum mechanics of an electron in a periodic potential.

We will solve this problem in 1D, as usual. To see what the delta functions mean more concretely, consider a 1D closed ring:

Lattice points

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M.J. Gilbert ECE 535 – Lecture 14

Quantitative Methods for Bandstructure Calculations - 2 We have seen this train of delta functions before, so we know the useful identity…

Represent the terms in the exponential as:

Now plug it into the Schrodinger equation:

The wavefunctions are Bloch wavefunctions, which are Fourier expanded…

Note that: Which is just a sum of all of the Bloch coefficients

Plug the wavefunction into the Schrodinger equation:

M.J. Gilbert ECE 535 – Lecture 14

Quantitative Methods for Bandstructure Calculations - 3 Multiply by e-iGx and then integrate over all x…

When doing so, here is a useful identity to use:

Now let’s solve for uG:

Looks like a mess, but we can simplify it by summing both sides over the G’s:

This leaves us with:

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M.J. Gilbert ECE 535 – Lecture 14

Quantitative Methods for Bandstructure Calculations - 4 This seems like a very crazy way of writing unity but note that this is an EXACT form of the solution of Schrodinger’s equation for a periodic potential…

Now we invert the solution: And plot the result…

Our solution plotted for two values of k.

•  Assume a/S > 0 and constant. •  RHS is a complex function of

Ek with a number of poles where it diverges.

The intersections correspond to the only allowed eigenvalues in the problem. There are N distinct eigenvalues:

values of

M.J. Gilbert ECE 535 – Lecture 14

Quantitative Methods for Bandstructure Calculations - 5 If we turn down the strength of the potential by having S approach 0, then the red line goes to +∞ and the intersections are exactly at the poles:

And we have exactly recovered the energy eigenvalues for the nearly free electron model.

If we allow the strength to be negative by setting S < 0…

•  The red line will clearly become negative. •  There is an energy intersection for energy that is negative. •  This energy represents a “bound” state

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M.J. Gilbert ECE 535 – Lecture 14

Quantitative Methods for Bandstructure Calculations - 6 Let’s look at the energy bandstructures for S > 0 and S < 0…

In this plot the energy is in units, F, of:

•  Dashed line is for S = 0 but the electron still obeys lattice periodicity and symmetry.

•  Note that the energies are higher for S > 0 (repulsive) than the NFE except at the zone edges and center.

•  KP model is degenerate w i t h N F E e n e r g y eigenvalues at:

M.J. Gilbert ECE 535 – Lecture 14

Quantitative Methods for Bandstructure Calculations - 7 That the energy eigenvalues for S > 0 are higher or equal to those of the NFE is guaranteed by Hellmann-Feynman (HF) theorem…

The eigenvalues of any Hamiltonian satisfy:

Imagine the KP model as a perturbation to the NFE:

Where: and

Then we must have:

Therefore the perturbed eigenvalue is greater than the unperturbed one. •  True everywhere except at the degeneracies where one

eigenvalue increases and the other remains unchanged. •  Lowest energy is greater than zero but has a narrow band. •  This means the electron has a large effective mass.

For S < 0, HF has opposite conclusions and the eigenvalues are lower than in the NFE.

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M.J. Gilbert ECE 535 – Lecture 14

Quantitative Methods for Bandstructure Calculations - 8 We can see that the tight-binding method arises naturally from the KP approach…

Take a look at the central eigenvalue equation from the KP model:

A few trigonometry identities later and we obtain:

But this is still an exact solution and now the values of Ek that satisfy this equation will form the energy bandstructure for each k. The LHS is limited in terms of value:

But the RHS can exceed unity:

But this restricts the allowed q values for real energy eigenvalues for each k.

when

M.J. Gilbert ECE 535 – Lecture 14

Quantitative Methods for Bandstructure Calculations - 9 Let’s examine the strength term in our transformation…

The zeros of the sinc function occur at x = nπ

•  A band of q values and corresponding energies are allowed near the zeros

•  We can approximate the solution for the first band by expanding the RHS for a large strength near the first zero at n = 1 around qa = π.

strength

Now Taylor expand:

Plug back into our equation:

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M.J. Gilbert ECE 535 – Lecture 14

Quantitative Methods for Bandstructure Calculations - 10 We don’t need to stop at the first root, we can expand around qa = nπ, and retaining only linear terms, we get a better model for the n-th band…

From this point we can start with orbitals localized on the lattice points and make linear combination in Bloch form to obtain the E(k) and eigenfunctions…

Trial wavefunction

Plug back into the Schrodinger equation:

Project back onto our ansatz: To obtain:

M.J. Gilbert ECE 535 – Lecture 14

Quantitative Methods for Bandstructure Calculations - 11 Let’s examine this equation…

Clearly there are N2 terms.

Diagonal terms:

Denominator:

First nearest neighbor terms:

Denominator: Small terms

Eventually we get:

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M.J. Gilbert ECE 535 – Lecture 14

Quantitative Methods for Bandstructure Calculations - 12 Now let’s consider a defect state in the KP model. In other words, at one site there is a strength of S + U0 instead of S.

Question: what is the effect on the energy eigenvalue of the system due to the presence of the defect?

All we have to do is apply the KP method, as outlined earlier in the lecture to obtain:

•  When U0 goes to zero, the LHS blows up and this happens N times in the RHS allowing us to recover the KP model without the defect.

•  We again see the HF theorem as the top of the band splits rapidly but the remainder of the eigenvalues do not change much.

•  We will see later that this corresponds to the formation of an acceptor state at the top of the valence band.

M.J. Gilbert ECE 535 – Lecture 14

Quantitative Methods for Bandstructure Calculations - 13