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Kronig-Penney model
Solutions can be found in region I and region IIMatch boundary conditions
I II
2 22 ( )2
d V x Em dx
Linear differential equations with periodic coefficients
or solutions of the form eikxuk(x)
Have exponentially decaying solutions,
T ( )( ) ( ) ( )ikx ik x a ika ikx ika
k k kTe u x e u x a e e u x e ikae
Kronig-Penney model
Eigenfunctions of the translation operator can be found in terms of any two linearly independent solutions. A convenient choice is:
Solutions can be found that are simultaneous eigenfunctions of the Hamiltonian and the translation operator.
1 21 2(0) 1, (0) 0, (0) 0, (0) 1.d ddx dx
Kronig-Penney model
11 1 21
sin( )( ) cos( ), ( ) k xx k x x k for 0 < x < b
for b < x < a1 2 11 2 1
22 1 2 12
1 2
sin( ( ))sin( )( ) cos( ( ))cos( ) ,cos( ( ))sin( ) sin( ( ))cos( )( ) .
k k x b k bx k x b k b kk x b k b k x b k bx k k
Except for the coefficients, these are the same solutions as we found for light in a layered material.
Kronig-Penney modelat x = a
1 2 11 2 12
2 1 2 121 2
sin( ( ))sin( )( ) cos( ( ))cos( ) ,cos( ( ))sin( ) sin( ( ))cos( )( ) .
k k a b k ba k a b k b kk a b k b k a b k ba k k
1 111 122 221 22
.x a xT Tx a xT T
The translation operator translates the function a distance a.
The elements of the translation operator can be evaluated at x = a.
Kronig-Penney model
111 1
2 222
da ax a xdxx a xda adx
2 1 22 1
2 ( ) 1( ) ( ), ,( ) 2( )ax x xd a adx
The eigen functions and eigen values are
2 4 2 2 11 2 1 2 1
1 2
( )( ) 2cos cos sin sin . d a k ka k a b k b k a b k bdx k k If > 2, the potential acts like a mirror for electrons
Kronig-Penney model1, 2E k k
1, 2 1 2( , , , , )k k V V a b
12
2 4
ikae 211 4tan k a
Kronig-Penney model
(a) The energy-wave number dispersion relation. The dashed line is the Fermi energy. (b) The density of states. (c) The internal energy density (solid line) and Helmholtz free energy density (dashed line). (d) The chemical potential (solid line) and the specific heat (dashed line). All of the plots were drawn for a square wave potential with the parameters: V = 12.5 eV, a = 2 × 10-10 m, b = 5 × 10-11 m, and an electron density of n = 3 electrons/primitive cell.
A separable potential
Y is the product of the solutions to the Kronig-Penney model.
I II
2 2 ( ( ) ( ) ( ))2 V x V y V z Em Y Y Y
( , , ) ( ) ( ) ( )KP KP KPx y z x y z Y
A separable potential I II
http://lampx.tugraz.at/~hadley/ss1/separablecrystals/thermo.html
Band structure in 1-D
Bloch Theorem
( )( ) ( )i k G r ik r iG r ik rk k G k G kG G
r C e e C e e u r
( ) ik rkk
r C e
1
( ) i k G rk GGk Bz
r C e
Eigen function solutions of the Schrödinger equation have Bloch form.
Bloch form 1 2 3 1 2 3( ) ( )
1 2 3( ) ( ) ( )ik r ma na la ik ma na lamnl k k kT r e u r ma na la e r
These k's label the symmetries
( ) ( )ik rk kr e u r
periodic function
Any wave function that satisfies periodic boundary conditions
Bloch waves in 1-D
Plane wave method
( ) ( )iG r ik rMO G kG k
U r U e r C e
2 2 ( )2 MOU r Em
Write U and as Fourier series.
2 22
0 0
1( ) 4iG r
MOj Gj
Ze Ze eU r V Gr r
For the molecular orbital Hamiltonian
volume of a unit cell
Plane wave method
( ) ( )iG r ik rMO G kG k
U r U e r C e
2 2 ( )2 U r Em
2 2 ( )2
ik r i G k r ik rGk k kGk k k
k C e U C e E C em
2 2 02 Gk k GGk E C U Cm
Central equations (one for every k in the first Brillouin zone)
Must hold for each Fourier coefficient. k G k k k G
Plane wave method
2
20
iji j
ZeM V G G
2 22ii iM k Gm
MC EC
Off-diagonal elements:
The central equations can be written as a matrix equation.
Diagonal elements:
Central equations - one dimension
Central equations couple coefficients k to other coefficients that differ by a reciprocal lattice wavevector G.
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
220
2 3 4 522
0 22 3 42 2
2 2 322
03 2 2
220
4 3 2
22
22
22
2
G G G G G
kG G G G G
G G G G G
G G G G G
G G G G G
k G E U U U U Umk G CU E U U U Um
kU U E U U Umk GU U U E U Um
k GU U U U E Um
0
0
0
02
0
G
k G
kk G
k G
CC
CC
2 2 02 k G k GG
k E C U Cm
Central equations - one dimension
Central equations 3d - simple cubic
22
unit cell 0G
ZeU V G
( ) iG rGG
V r U e
Molecular orbital Hamiltonian
2 22 ik Gm
2
2unit cell 0 i j
ZeV G G
2 2 02 Gk k GGk E C U Cm
Central equations:
diagonal elements:
off-diagonal elements:
22
2 2 2 2 2 2 2 2 2 2 2 22 2 2 2 2 2
unit cell 0 unit cell 0 unit cell 0 unit cell 0 unit cell 0 unit cell 02
22 2 2 2
2unit cell 0 unit
2 ˆ2 8 8 4 8 8 16
2 ˆ8 2 8
z
y
k k Ze a Ze a Ze a Ze a Ze a Ze aam V V V V V V
k kZe a Ze aaV m V
2 2 2 2 2 2 2 2
2 2 2 2 2 cell 0 unit cell 0 unit cell 0 unit cell 0 unit cell 0
22
2 2 2 2 2 2 2 22 2 2 2
unit cell 0 unit cell 0 unit cell 0 unit cell 0
4 8 16 82 ˆ
8 8 2 4 16x
Ze a Ze a Ze a Ze aV V V V
k kZe a Ze a Ze a Ze aaV V m V V
2 2 2 22 2
unit cell 0 unit cell 02 2 2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2unit cell 0 unit cell 0 unit cell 0 unit cell 0 unit cell 0 unit cell 0
2 2 2 22
unit cell 0
8 8
4 4 4 2 4 4 4
8
Ze a Ze aV V
Ze a Ze a Ze a k Ze a Ze a Ze aV V V m V V V
Ze a Ze aV
22
2 2 2 2 2 2 2 22 2 2 2 2
unit cell 0 unit cell 0 unit cell 0 unit cell 0 unit cell 0
2 2 2 2 2 2 2 22 2 2
unit cell 0 unit cell 0 unit cell 0 unit cel
2 ˆ8 16 4 2 8 8
8 16 8 4
xk kZe a Ze a Ze a Ze aaV V V m V V
Ze a Ze a Ze a Ze aV V V V
22
2 2 2 22 2 2
l 0 unit cell 0 unit cell 0
22 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2unit cell 0 unit cell 0 unit cell 0 unit cell 0 unit cell 0 unit cell 0
2 ˆ8 2 8
216 2 8 4 8 8
yk kZe a Ze aaV m V
kZe a Ze a Ze a Ze a Ze a Ze aV V V V V V
2ˆ2
zkam
Central equations - simple cubic
Central equations - simple cubic
E [aJ]
G X M RG
Central equations - simple cubic
G X G X
empty lattice
Plane wave method
fcc Z= 0
empty lattice
Plane wave method
fcc hydrogen
Plane wave method
Approximate solution near the Bz boundary2 2 02 k G k G
Gk E C U Cm
For just 2 terms
2 2
222 0
2
kk G
k E U CmCk GU Em
Near the Brillouin zone boundary k ~ G/222
222 2 0
2 2
kk G
G E U CmCGU Em
E
G/222
2 2GE Um
2U
Review: MoleculesStart with the full Hamiltonian
2 22 2 22 2, 0 0 02 2 4 4 4
A A Bi Ai A i A i j A Be A iA ij AB
Z e Z Z eeH m m r r r
Use the Born-Oppenheimer approximation
Neglect the electron-electron interactions. Helec is then a sum of HMO.22 2
10 12 4
AMOAe A
Z eH m r r
The molecular orbital Hamiltonian can be solved numerically or by the Linear Combinations of Atomic Orbitals (LCAO)
2 22 22, 0 0 02 4 4 4
A A Belec ii i A i j A Be iA ij AB
Z e Z Z eeH m r r r