kittel chapter 9 fermi surfaces and metalsw3.phys.nthu.edu.tw/~spin/course/106s/ch 9-1.pdf ·...
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Kittel Chapter 9Fermi Surfaces and Metals
Part 1
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Few people would define a metal as “ a solid with a Fermi surface.”
This may nevertheless be the most meaningful definition of a metal
one can give today; It represents a profound advance in the
understanding of why metals behave as they do. The concept of the
Fermi surface, as developed by quantum physics, provides a precise
explanation of the main physical properties of metals.
A. R. Mackintosh
(1936-1995)Magnetism and neutron scattering; rare-earth metals; solid-state physics
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Periodic boundary conditions
Exp (ikL) = 1
k = ± n 2p / L
Kittel Chapter 6
FREE ELECTRON GAS IN THREE DIMENSIONS
Wave functions satisfying the free particle Schrödinger equation,
and the periodicity condition are of the form of a traveling plane wave:
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At the surface 𝝐f , KfFermi Sphere
Fermi Surface
at the Fermi surface 𝝐F, kf
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Free Electron Model Nearly Free Electron Model
Solutions in 1-D
The Bragg condition (k + G)2 = k2 for diffraction of a wave of wavevector kbecomes in one dimension
k = ±1
2G = ± n𝜋/a, (4)
Where G = 2𝜋n/a is a reciprocal lattice vector and n is an integer. The first reflections and the first energy gap occur at k = ± 𝜋/a. The region in k space between - 𝜋/a and 𝜋/a is the first Brillouin zone of this lattice. Other energy gaps occur for other values of the integer n.
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Standing wave solutions
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Standing waves
cos2px/a sin2px/acharge densityIs a constant
The origin of energy gap
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Felix Bloch
(1905-1983, Swiss)
Awarded the 1952 Nobel Prize for
"their development of new ways
and methods for nuclear magnetic
precision measurements
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Restatement of the Bloch Theorem
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The shape of Fermi surface of copper is deformed by interaction with the lattice.
Fermi surface
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Octahedron
Brillouin zone in reciprocal space:
Truncated Octahedron
fcc reciprocal lattice bcc
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Cu Al
The shape of Fermi surface of copper is deformed due to the interaction with the lattice.
Also refer to page 20, Fig. 8
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Reduced Zone Scheme
k = k’ + G
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1st zone, 2nd zone, 3rd zone
Extended zone scheme
A’ A
C
Reduced zone scheme
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Extended zone
Reduced zone
Periodic zone
-G
-G
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in Fig. 6
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Periodic zone scheme
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See page 37, 38, 39
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electron-likehole-like
gradk
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B
B
KE
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Eq. 13
Shift from (a) by (-1/2 ky, 1/2 kz)
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Ashcroft Mermin
Chapter 9 Electrons in a weak periodic potential
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The Fermi surface always intersects Brillouin zone boundary perpendicularly
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bcc fcc
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like Al
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