1. Lecture Thursday 6. March 2008
Room 292, 13:15 – 16:00
Topic:
Electrons – Bloch states
(Periodic potentials)
Accompanying texts: Bloch – Fourier notes (in web collection)
Comment: we had a crash of the drawing program; End of lecture somewhat disturbed 2p __ a
G = 2p __ a
G =
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2. Dirac Notation Exposes the structure For examole the operator,
Last Time
Dirac Notation Exposes the structure
For examole the operator,
Or the completeness
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3. Bloch Theorem part 1 – Last lecture
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4. YOU GET BACK THE PLANE WAVES
THIS IS Bloch's Theorem the function u is periodic with the periodicity of the potential
PHYSICS:
YOU GET BACK THE PLANE WAVES
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5. Periodicity in the k-space related to the periodicity in the normal ('configuration' space)
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6. Expanding the wavefunction – all possible k-values; __ a
G =
Brillouin zone
-p/a to p/a
Expanding the wavefunction – all possible k-values;
The distance between k-values – from periodicity on L=N a
(many =N atoms with a between neighbours) Large L -> small k
Expanding the potential – all possible K-values;
The distance between k-values – from periodicity on a
(over 1 atom with a to the neighbour) Small a -> large K (G=2p/a steps)
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7. Text
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8. These pictures can be really understood
most easily from the Fourier Analysis
(English Notes .... )
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9. Rearranging band matrices A band matrix can be transformed
Text
Rearranging band matrices
A band matrix can be transformed
BY REARRANGING THE BASIS
i.e. just changing the order of the basis states
The diagonalization:
the two slosest states are pushed away from each other
a=b case – 'degenerate'
and a,b, different
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10. BAND THEORY
Where are the BANDS?
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11. Text
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12. Text
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13. Text
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