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Lecture+C5-3 - More ElectronsSolid Free electron gas model...

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Free electron gas model — Fermi Level Model: 1. Outer most valence electrons become detached and shared throughout the solid 2. Electrons in a 3-d box with infinite walk Box size: *Ignore electron-electron & electron-nucleus interaction. *V=0 when electron is inside the box, V= when outside. More Electrons—Solid lx ly lz ( 29 0 0 ,0 ,0 , , x y z x l y l z l V x y z otherwise < < < < < < = +∞ , , x y z l l l
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Separation of variables Shrodinger`s equation Recall: Solution to infinite potential well ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 2 , , 2 2 2 x y z x y z X x Y y Z z d X d Y d Z E X E Y E Z m dx m dy m dz ψ = - = - = - = h h h ( 29 2 2 2 2 2 2 2 2 x y z E E m m ψ ψ ψ ψ - = - ∇ + ∇ + ∇ = h h ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 sin cos sin cos sin cos x y z x x x x y y y y z z z z E E E E X x A k x B k x Y y A k y B k y Z z A k z B k z = + + = + = + = +
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Boundary conditions ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 0 0 0 x y z X X l Y Y l Z Z l = = = = = = ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 y y y z 0 cos = sin 0 cos = sin y 0 cos = sin y x x x x y y z z B k x A k x B k y A k B k z A k = = = 2 2 2 2 2 2 2 2 2 2 0 , , 1,2,3 , 1,2,3 , 1,2,3 8 sin sin sin 2 2 x y z x y z x y z x x x y y y z z z x y z y x z n n n x y z x y z y x z n n n x y z B B B k l n k l n k l n n n n n n n x y z l l l l l l n n n k E m l l l m π π π π π π ψ π = = = = = = = = = = ÷ ÷ ÷ ÷ = + + = ÷ ÷ L L L h h
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in k space Each lattice point represents a distinct state (different kx, ky, kz) Volume occupied by each state in k space from a 3-d reciprocal lattice in momentum space x y z x y z l l l k spaced apart by k spaced apart by k spaced apart by π π π 2 2 2 2 x y z k k k k = + + 3 3 x y z x y z l l l l l l V π π π π π = = g g x k y k 2 D k space x l π y l π
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If we have N atoms in solid, q free electron per atom, What is the ground state? Volume in k space Volume occupied by each state # of states (↑↓ each state) Octant of a sphere The electron will fill the lower energy (smaller |k|) first Fermi level: 3 3 1 4 8 3 2 F Nq k V π π = ÷ F k 2 2 2 2 2 2 2 2 2 2 2 2 x y z y x z n n n x y z n n n k E m l l l m π = + + = ÷ ÷ h h ( 29 ( 29 1/3 2 2 2 2/3 2 2 3 3 2 2 F F F Nq k free electron density V E k m m π π = = = = l l h h l
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Fermi surface kF: boundary separating occupied and unoccupied states in k space Fermi energy EF: E>EF
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