ECE230A_Lecture_slides-Energy_band_theory

ECE230A_Lecture_slides-Energy_band_theory - Section Section...

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Section 2. nergy Band Theory Energy Band Theory 35
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Topics in Energy Band Theory Intuitive reasoning of formation of energy (conduction and valence) bands. Bloch theorem. Kronig-Penny model Nearly free-electron model ight inding approximation (skip due to time limit) Tight-binding approximation (skip due to time limit) k.p approximation (skip due to time limit) Band structures for typical semiconductors (Si, Ge, GaAs, InP. .) oncept of electrons and holes 36 Co cept o e ect o s a d o es Effective mass, crystal momentum, subbands, strain effects.
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Bloch Theorem The Bloch theorem relates the value of the wavefunction within any “unit cell” of a periodic potential to an equivalent point in any other unit ell cell. It allows one to concentrate on a single repetitive nit when seeking a solution to Schrodinger’s unit when seeking a solution to Schrodinger s equation. 37
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Bloch Theorem For a one-dimensional system, the potential of the lattice looks like the following: 38
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Bloch Theorem For a one-dimensional system, the Block theorem is as follows: ) ( ) x a Ux = () += ) ( ) ika a e x = If Then xa ψψ + ikx x eu x ψ = Which implies that ( ) ux a ux + = Where 39
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Bloch Theorem For a three-dimensional system, the Block theorem is as follows: ) ( ) ik a a e r = v v v vv () ra ψψ += v ik r re u r ψ = v v v Where ( ) ur a ur + = v 40
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Bloch Theorem Periodic boundary conditions for N-atom ring with interatomic spacing a: ) ( ) () ikNa x N a e x += xx ψψ ψ =+= Na 1 ikNa e = 2 ; 0, 1, 2,. .., / 2 n kn N Na π = ± ± 41
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Kronig Kronig-Penney Model Penney Model In order to simplify the problem, the potential function is approximated by a rectangular potential using Kronig-Penney Model. 42
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Shroedinger Equation 2 2 riodic function ( ) ( ) () ) () r Vr R r r E r + ∇+ = rr r r r r h periodic function () ( R : lattice vector ( 2 ( ) ( ) Vr m Hr Er ϕϕ =+ = r E: eigenvalue ( ) : eigen function 1-D case: r ϕ r n=- 2 j( n)x a V(x) = Vn e π Since V(x) is a periodic function, it can be represented as a Fourier series. Note that (2 π n/a) is actually the “reciprocal lattice vector” of the 1-D crystal.
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() ( ) : periodic function jkx xu x e ux ϕ = , ( ) jkr kc ux ux na ru r e =+ = rr r ( ) ( ) jkxn a jkna x na u x na e x e + +=+ = 22 wave function squared probability of finding that electron: ( ) xx n a ϕϕ
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Solving the Schrodinger Equation V(x) v=0 o V - b x 1 2 3 4 a b 0 a+b =c 1 In 2 φ (x)=C exp( α x) + Dexp(- α x) 22 In 1, In 2, - - 2m 2m o dd EV E dx dx ϕ ϕϕ ⎛⎞ =+ = ⎜⎟ ⎝⎠ hh ( ) ( ) 2( ) 2 = jx o xA e B e x C e D e mV E mE ββ αα β α −− =
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Boundary Conditions Boundary Condition: (x = 0) cont.
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This note was uploaded on 01/08/2011 for the course ECE ece230a taught by Professor Ece during the Fall '10 term at UCSD.

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ECE230A_Lecture_slides-Energy_band_theory - Section Section...

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