Crystal
Crystal
Kronig-Penney model U(x) 0 0<x<a U(0), a<x<a+b=1 U(x+nl)=U(x),n=0,1,2… -b0 aa+b 2a+b 2(a+b) E>Uo E<Uo http://lamp.tu-graz.ac.at/~hadley/ss1/KronigPenney/KronigPenney.php
Kronig-Penney model a b 0 a b 2a b x U x( ) 2( ) a b U a x a b l x a U x ( 0), 0, 0 ( ) 0 http://lamp.tu-graz.ac.at/~hadley/ss1/KronigPenney/KronigPenney.php U x nl U x n ( ) ( ), 0,1,2 E U 0 E U 0
Kronig-Penney model E>U )= A.eB.e 0<x<a,k=2mE/h C.+D.e -b<x<0,k'=2m(E-U)/h U(x) Wn(x)=ek(aDWj(x-a-b),a<x<2a+b A+B=C+D x=0: ik(A-B)=ik'(C-D) -b0 aa+b 2a+b 2(a+b) Aeika+Be-ika =eik(atb)(Ce-ik'b+Dek'b) x=a: ik(Aeika-Be-ika)=ek(atb(ik')(Ceik-Dekb) cos kacos k'b- 2+2 sin kasink'b cos K(a+b) 2kk
Kronig-Penney model E U 0 . 0, ' 2 / , 0 , 2 / ( ) 0 ' ' C e D e b x k m E U A e B e x a k m E x ik x ik x ikx ikx I sin sin ' cos ( ) 2 ' cos cos ' ' 2 2 k a k b K a b k k k k k a k b a b 0 a b 2a b x U x( ) 2( ) a b I II x e I x a b a x a b iK a b II ( ) ( ), 2 ( ) ( ) '( ) 0 : ik A B ik C D A B C D x ( ) ( ')( ) ( ) : ( ) ' ' ( ) ' ' ika ika iK a b ik b ik b ika ika iK a b ik b ik b ik Ae Be e ik Ce De Ae Be e Ce De x a
Kronig-Penney model E<U using k'→ik' cos kacos(ik- sin kasincos K(b) 2k(ik) oskac 一 2kk sin kashk'b cos K(a+b)
Kronig-Penney model E U0 sin sin ' cos ( ) 2 ' cos cos ' using ' ' ' 2 2 k a ik b K a b k ik k ik k a ik b k ik sin s h ' cos ( ) 2 ' cos ch ' ' 2 2 k a k b K a b k k k k k a k b