《固体物理学》课程教学课件（讲稿）第四章 能带理论 04_02 一维周期场中电子运动的近自由电子近似

N(n+l)ce-i(k'-k)xV(x)dx(k'lV(x)|k)=~0-Nanan=0一引入积分变量x=E+nadx =d=5:0→a势场是晶格周期性函数V()= V(=+na)E

  1 ( 1) ( ' ) 0 1 ' ( ) N n a i k k x na n k V x k e V x dx Na        —— 引入积分变量  x    na 势场是晶格周期性函数 V( ) V (  na) dx  d  : 0 a

2-i(k'-k)a-i(k'-k)s(k'lV(x)|k)=()dEeN7-02元k =N-Na2元Zi(k-k)a二1) k'-k=ne2元Nan=0k'=Na-i(k'-k)Na2元VZCe一2) k'-k≠n-i(k'-k)aNN1-ean=0E

1 ( ) ( ) 0 0 1 1 ( ) ( ) N a n i k k i k k a n k V x k e V d e a N                         2 k k n a     [ ] 1 1 1 0 ( ' )       N n i k k a n e N 2 k k n a     1 ( ) ( ) ( - ) 0 1 1 1 [ ] 0 1 N i k k Na i k k a n i k k a n e e N N e              1) 2) 2 2 k l Na k l Na      

2元k'-k=na-i(k'-k)V()d(k'|H|k)=(k|V(x)|k)=qJoV(n)-V(x)第n个傅里叶系数2元k'-k=n(k'|H|k)=V(n)a2元k'-k±n(k'|H'|k)=0a10160

2 ( ) 2 0               k k n k H k V n a k k n k H k a 10 / 60 —— V(x) 第n个傅里叶系数   ( ) 0 1 ( ) a i k k k H k k V x k e V d a             V (n) 2 k  k  n a

h?k?E+V<k'|H"|k)2mER=二级修正项h?k"2E-Ek'o+Vk2m2元k'-k=n(k'|H'|k)=V(n)aVE(2)Z二kRInkk+=2元2maE

二级修正项 2 2 0 2 2 0 2 2 k k k E V m k E V m         2 (2) 0 0 ' ' k k k k k H k E E E      2 (2) 2 2 2 ' 2 2 n k n V E n k k m a                   k H k  V(n) 2 k k n a    

h?k?E+V一电子的能量本征2m值O=0 + E(2)E=E°+E)E(2)=h?nk2k+=2元2maVh?k?+V.ZEkX=2mh?nnkk+-2元2ma4

0 (1) (2) Ek  Ek  Ek  Ek 2 2 0 (1) 2 (2) 2 2 2 2 0 ' 2 2 k k n k n k E V m E V E n k k m a                      —— 电子的能量本征 值 2 2 2 2 2 2 ' 2 2 2 n k n k V E V m n k k m a                     