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Lecture 13 - EEE 434/591Quantum Mechanics L13:1 David K...

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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L13:1 EEE 434/591—Quantum Mechanics David K. Ferry Regents’ Professor Arizona State University
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L13:2 Periodic Potentials Leon Brillouin Felix Bloch A periodic potential is repetitive through the structure of interest, much like the atomic potentials in a crystal. The potential has a lattice constant d , which describes this periodicity. This gives a periodicity in momentum space (Brillouin zone) and the use of special waves (Bloch waves).
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L13:3 Quantum wells Barriers The quantum wells produce bound states. The barriers must be sufficiently thin that the wave function tunnels through the barrier. Then, the bound states broaden into bands.
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L13:4 http://www.Nanohub.org
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L13:5 The periodicity of the potential enforces the same periodicity upon the wave function, within a phase factor. Hence Since we must have
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L13:6 The general periodic function in a periodic lattice has a specific form known as a Bloch function . The periodicity in real space also creates a periodicity in the momentum vector k. This part of the Bloch function is cell periodic , in that it has the exact periodicity of the potential. We will demonstrate this periodicity in one dimension, in order to be sure that we understand the implications of this new periodicity.
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L13:7 0 1 2 N -1 N We now enforce periodic boundary conditions, as The maximum value for k is when n = N , or
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L13:8 If we shift the k vector by some multiple of K , say RK , then Thus There exists a comparable lattice in the Fourier transform space (where k lives). This lattice is known as the “reciprocal lattice” and has its own set of basis vectors. The reciprocal lattice of the FCC lattice is a BCC lattice, but the basic structure is truncated in the (111) directions.
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L13:9
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L13:10 Electron Motion in a Periodic Potential If we write φ = kd , then n =1, 2, …, N There are N independent values of k , spaced by Δ k = 2 π / L. We refer to the First Brillouin Zone as the range of k where The First Brillouin Zone holds N values of k . In a metal, each atom gives up one free electron, but each state can hold two (± spin) electrons so that the metal has a ½ -filled highest band. In silicon, each unit cell has two atoms (basis) so that there are 2N states per band, which accommodate 4N electrons—exactly the number per atom—silicon is an insulator as the highest band is full.
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L13:11 Electron Motion in a Periodic Potential Hence, our wave function satisfies two equations:
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