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ECE 535 Notes for Lecture # 3

Class Outline: • Quantum Refresher • Sommerfeld Model– Part 1

M.J. Gilbert ECE 535 – Lecture 3

Quantum Refresher – 1 But…the Drude theory has some problems…

He was nominated 84 times for the Nobel Prize but never won…most ever.

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M.J. Gilbert ECE 535 – Lecture 3

Quantum Refresher – 2

•  To explain the spectrum of photon emissions in hydrogen Bohr proposed the following:

1.  Electrons exist in certain stable orbits. –  This assumption implies that the orbiting

electron does not give off radiation as classical electromagnetics would require of a charge experiencing angular acceleration.

2.  The electron may shift to an orbit of higher or lower energy, thereby gaining or losing energy equal to the energy difference between the two layers.

3.  The angular momentum of the electron in an orbit is always an integral multiple of Planck’s constant divided by 2Π

The Bohr model of Hydrogen…

+q

( ) 220

40 6.13

42 neV

nqm

EH −==!πε

n = 1, 2, 3, …

M.J. Gilbert ECE 535 – Lecture 3

Quantum Refresher – 3 So, we can start to remedy these shortcomings by looking at quantum mechanics…

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M.J. Gilbert ECE 535 – Lecture 3

Quantum Refresher – 4 More Schrodinger equation…

M.J. Gilbert ECE 535 – Lecture 3

Quantum Refresher – 5 Let’s begin thinking about electrons in a metal…

•  In the Sommerfeld model, electrons are confined in a 3D box with zero potential inside and infinite potential outside.

•  The electron states inside the box are given by the Schrodinger equation.

Free electrons (they experience no potential)

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M.J. Gilbert ECE 535 – Lecture 3

Quantum Refresher – 6 So let’s apply the Schrodinger equation to this problem of free electrons in a box…

M.J. Gilbert ECE 535 – Lecture 3

Quantum Refresher – 7 Let’s examine the wavefunction more closely…

•  All electron states can be labeled by a corresponding k-vector.

Problems: •  The “sine” solutions are difficult to work with – we would like more amenable

solutions •  The form of the solutions come from the boundaries – we already know that

most electrons will never see these boundaries.

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M.J. Gilbert ECE 535 – Lecture 3

Quantum Refresher – 8 So let’s solve the problem again, but using different boundaries:

M.J. Gilbert ECE 535 – Lecture 3

Quantum Refresher – 9 Let us now try to label the electron states again, as we did in the case of hardwall (infinite) boundary conditions…

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M.J. Gilbert ECE 535 – Lecture 3

Quantum Refresher – 10 Let’s visualize these states in k-space: •  We can visualize the allowed states on a

3D grid of points in the entire k-space.

M.J. Gilbert ECE 535 – Lecture 3

Quantum Refresher – 11 The last ingredient that we need in order to proceed is the electron spin: •  Electrons have a spin degree of freedom •  The spin can be either “up” or “down”

For right now, let’s assume that the energy does not depend on the spin:

Note: In our discussions, spin will mainly be a property of the electrons that we account for when we determine the number of states but, otherwise, we will forget about its physical properties.

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M.J. Gilbert ECE 535 – Lecture 3

Sommerfeld Model – 1 Now back to the Sommerfeld Model:

•  Let’s say that we have N electrons in the box. •  Let’s also assume that T = 0K and we want to fill the

states to keep the lowest total energy.

The energy of the quantum state is:

Strategy: •  Each grid point can be occupied by two electrons

(spin-up and spin-down)

•  Start filling up the grid points in the spherical region of increasing radii until we have used all electrons.

•  When done, all filled quantum states correspond to grid points within a sphere of radius kF.

M.J. Gilbert ECE 535 – Lecture 3

Sommerfeld Model – 2 Let’s make sure that we are consistent:

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M.J. Gilbert ECE 535 – Lecture 3

Sommerfeld Model – 3 Let’s continue to examine the zero temperature limit… Remember: 1.  All states inside of the “sphere” are filled,

occupied by electrons. 2.  All states outside of the “sphere” are empty.

M.J. Gilbert ECE 535 – Lecture 3

Sommerfeld Model – 4 Zero temperature is boring and not terribly helpful in our goal as materials rarely sit a zero temperature… •  Our simple counting scheme for filling states will no longer work.

•  We need to begin thinking in terms of probabilities.

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M.J. Gilbert ECE 535 – Lecture 3

Sommerfeld Model – 5 Let’s revisit a few things that we already know…