Lect14 - Introduction to the Physics of Solids: Electrical...

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Introduction to the Physics of Solids: Electrical conduction U(r) r a ψ 3 ψ 1 ψ 2 ψ 4 ψ 5 ψ 6
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Overview Overview z Energy levels in a solid z “Bloch wavefunction” z Energy Bands z Electrical conductivity and the free-electron model z Filling the bands with the Pauli Principle z metals z semiconductors
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Today Today Æ Æ filling energy states in solids filling energy states in solids E F +e r n = 3 n = 2 n = 1 Atom Solid Discrete atomic states band of crystal states Fill according to Pauli Principle
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Recall: Molecular Molecular Wavefunctions Wavefunctions and Energies and Energies -- -- H H 2 Atomic ground state: Molecular states: + e r n = 1 ψ A Bonding state Antibonding state r e ) r ( U 2 κ = 0 a / r e ) r ( ψ The wavefunctions for the molecule were (approximately) the sum of 2 atomic wavefunctions. + e r + e ψ even + e r + e ψ odd Now we fill these orbitals with the 2 available electrons (one from each hydrogen atom). Both can go into ‘bonding’, thanks to spin Qmdbw Double well demo
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Simple model of a crystal Simple model of a crystal with covalent bonding with covalent bonding Again start with simple atomic state: + e r n = 1 ψ A Bring N atoms together together forming a 1-d crystal (a periodic lattice). (N atomic states N crystal states): Energy band What do these crystal states look like? -- approximately linear combinations of atomic orbitals.
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Simple model of a crystal Simple model of a crystal with covalent bonding with covalent bonding ψ N Highest energy orbital (N-1 nodes) ψ 1 Lowest energy orbital (zero nodes) Kruse Energy bands
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The “in between" states The “in between" states Real Envelope: e ikx Length of crystal, L Lattice spacing, a ψ n The wavevector k has N possible values from k = π /L to k = π /a. L/a ..... 1,2, n L n k n = = π N = L/a states u(x) depends on the atomic states involved: Bloch Wavefunction for electron in a solid: u(x)e (x) x ik n n = ψ 1s-states 2s-states u(x) x x
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Bloch Bloch Wavefunctions Wavefunctions and the Energy Band and the Energy Band For N = 6 there are six different superpositions of the atomic states that form the crystal states u(x)e (x) x ik n n = ψ ψ 3 ψ 1 ψ 2 ψ 4 ψ 5 ψ 6 Energy Lowest energy wavefunction Highest energy wavefunction Closely spaced energy levels in this “1s-band”
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Bloch Bloch Wavefunctions Wavefunctions and the Free Electron Model and the Free Electron Model Envelope: Re e ikx u(x)e (x) x ik n n = ψ ψ 2 z Bloch wavefunction acts almost like free electron wavefunction: = = = m 2 p m 2 k E Energy Ae (x) 2 2 2 x ik = Free electron: mass" effective " m* * m 2 k E Energy u(x)e (x) 2 n 2 x ik n n = = = = Bloch wave: e - z In a perfectly periodic lattice, an electron moves freely without scattering from the atomic cores !!
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Lect14 - Introduction to the Physics of Solids: Electrical...

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