Download The Kronig-Penney one-dimensional model α and more Lecture notes Chemistry in PDF only on Docsity!
The Kronig-Penney one-dimensional model
Purpose: to demonstrate that in solids, where many atoms stay closely, the interference between atoms will create allowed and forbidden bands of energy for electrons. To simplify the analysis, we only consider a one-dimensional system where atoms are aligned and equally spaced. This constructs a one-dimensional potential function: Where V 0 is the value of potential barrier; a and b are lattice constant, represent distance between atoms. For an electron traveling in the x-direction in free-space, the general solution of the wave equation is,
!( x )=exp ( jkx )
Now, within this periodic potential structure, the solution should be modified,
!( x )= u ( x )exp ( jkx )
Bring this assumed solution back to the Schrodinger equation,
[ ( )] ( ) 0
2 2 2
+ E! V x x =
m
dx
d x
In region I, where V(x) = 0, we have,
1 1 2 2 2 1 2
+ − k − u x =
dx
du x
jk
dx
d u x
V(x) V 0
- (b+a) (^) - b 0 a^ (b+a) x II I II I II (^) I II
where 2 2
α = 2 mE /
In region II, where V(x) = V 0 , we have,
2 2 2 2 2 2 2
+ − k − u x =
dx
du x
jk
dx
d u x
where ( ) 2 (^20) 2 0
mV
E V
m
Equations (1) and (2) are two new equations for envelop u 1 (x) and u 2 (x) in regions I and II, respectively. The general solutions for (1) and (2) is, j k x j k x
u x Ae Be
( ) ( )
1 (^ )
− − +
α α For region I (0 < x < a) j k x j k x
u x Ce De
( ) ( )
2 (^ )
− − +
β β For region II (-b < x <
Boundary conditions:
Field continuity u 1^ (^0 )= u 2 (^0 )
0 0
= =
x dx x
du
dx
du
Periodic structure u 1^ (^ a )= u 2 (− b )
x a dx x b
du
dx
du
= = −
This results in 4 equations for coefficients A, B, C, and D,
A + B − C − D = 0
( α − k ) A −( α+ k ) B −( β− k ) C +( β − k ) D = 0
( ) ( ) ( ) ( )
j − k a − j + k a − j − k b j + k b
Ae Be Ce De
α α β β
Therefore, we have, cos( ) cos( ) sin( ) 2 0 a ka a mV ba a
or, cos( ) cos( ) sin( ) a ka a a M + α = α α (3) where (^2) 0 mV ba M ≡ On the right-hand-side of equation (3),
− 1 <cos( ka )< 1
While on the left-hand-side of equation (3), the value of cos( ) sin( ) a a a M α α α
- (^) is not bounded within ±1. Therefore, in order to have non-trivial solution of equation (3), the parameter 2
α = 2 mE / or ultimately the electron energy E
only has certain allowed values, while other values are forbidden. This gives an explanation of allowed and forbidden energy bands: Allowed energy band: cos( )^1 sin( )
M α
Forbidden energy band: cos( ) 1 sin( )
M α
Although cos( ) cos( ) sin( ) a ka a a M + α = α α can be solved numerically, we only look at two extreme cases: (1) No periodic potential barrier V 0 = 0 or V 0 b = 0 and M=0,
Equation (3) becomes, cos(^ α^ a )^ =cos( ka )
Therefore α = k that is, k mE = 2 or, m k E 2 2 2 = Obviously any E-value is allowed, no restriction. (2) Very high periodic potential barrier V 0 b >> 1 and therefore, M
Equation (3) becomes, cos( ) sin( ) ka a a M =
Since M >>1, the solutions can only be found around sin(^ α^ a^ )=^0 ,
or, a mE n π α ≡ =± 2 with n = 1, 2, 3…. That is, a m n E 2 (^2 ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = π with n = 1, 2, 3…. Obviously, E has only discrete values. k E k E
