Chem120A+Notes+-+Electronic+Strucutre+of+Solids

Chem120A+Notes+-+Electronic+Strucutre+of+Solids - The...

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The Electronic States in Metal Solids Fermi Level f E Fermi Level dE E Fermi Level Conduction Band For a given number of electrons in the conduction band, where is the Fermi level? The number electron states in an energy interval between E to E +d E. What is the electron distribution at finite temperatures ?
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Free Electron model for Conduction Band The electrons are moving in a 1D wire with length of L. The periodic boundary condition is imposed, . Assume there is no interactions between electrons. The free electron model of a 1D wire L The electronic states are 1/2 ( ) (2 ) exp( ) 2 , 1 ,2 ,3 , k x ikx n kn L ψ π = == ± ± ± L The momentum of electron p k = h The energy 22 2 k k E m = h m The mass of electron (0) ( ) L = () x
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Fermi Level for a 1D Wire n M 4 ± 3 ± 2 ± 1 ± 2 2 16 2 h mL 22 2 2 k hn E mL = 2 2 9 2 h mL 2 2 4 2 h mL 2 2 2 h mL 0 2 2 k p E m = pk = h Electrons occupy the lower energy level first. Each energy level can take up to 4 electrons. Given N electrons in the conduction band, the energy of the highest occupied state (Fermi Energy) is 4 N ⎡⎤ ⎢⎥ ⎣⎦ 2 2 2 24 f hN E = The smallest integer greater than or equal to N / 4 max p p f E Each state takes an interval of Δ p=h/L in the momentum space (The uncertainty principle). The maximum of the momentum satifies The Fermi Energy 2 2 p N p = Δ 2 2 2 2 2 f p E m = ⎛⎞ = ⎜⎟ ⎝⎠ The two results are equivalent when the number of electron N is large.
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The Density of States for a 1D Wire The number of states in the energy interval between E to E + d E 2 EEd E dp n p →+ = Δ 0 2 2 p E m = p p dp −− p p dp + p Ed E + E 2 mm dp dE dE pE == p dE dp m = () / 2 2 E E nd E Lm hE ρ = = The Density of States 0 E E For 1D systems, the density of states decreases with the increase of the energy.
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The Density of States of Nanowire The electronic states 2 222 () 2 xyz E kkk m =+ + h Potential wells in the x and y direction, and Free electron motion in the z direction. y d x d L y x z xy dd L = << ( , , ) sin( )sin( )exp( ) 2 2 2 ,, ; , 1 , 2 , ; 1 , 2 , z y x z xyzx y z kx ky i kz n n n kkkn n n L ψ π === = = ± ± LK The electronic energies E E ρ The electronic states in the x, y directions give the discrete energy levels, and the states in the z direction lead to the band structure.
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Free Electron model for a 2D film x The electronic states are 1 ( ) (2 ) exp( ) 2 2 , ; , 1 ,2 ,3 , kx y y x xyx y xi k x i k y n n kkn n L L ψ π =+ == = ± ± ± L The momentum of electron xy = + pk k hh The energy 22 2 () 2 k kk E m + = h (, ) y
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This note was uploaded on 09/28/2009 for the course CHEM 120A taught by Professor Whaley during the Spring '07 term at University of California, Berkeley.

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Chem120A+Notes+-+Electronic+Strucutre+of+Solids - The...

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