physics 355. consider the available energies for electrons in the materials. as two atoms are...
TRANSCRIPT
Energy Bands in SolidsEnergy Bands in SolidsPhysics 355Physics 355
Consider the available energies for electrons in the materials.
As two atoms are brought close together, electrons
must occupy different energies due to Pauli Exclusion principle.
Instead of having discrete energies as in the case of free atoms, the available
energy states form bands.
Conductors, Insulators, and Semiconductors
Free Electron Fermi GasFree Electron Fermi Gas
m
kE
2
22
For free electrons, the wavefunctions are plane waves:
)rk()r( ie
Band Gap
zone boundary
“doped”“thermallyexcited”
+
Origin of the Band Gap
To get a standing wave at the boundaries, you can take a linear combination of two plane waves:
axiikx
axiikx
ee
ee/
2
/1
a
xee axiaxi cos2//
21
/ /1 2 2 sini x a i x a x
e e ia
Origin of the Band Gap
Electron Density
Origin of the Band Gap
Bloch Functions
Felix Bloch showed that the actual solutions to the Schrödinger equation for electrons in a periodic potential must have the special form:
where u has the period of the lattice, that is
)rk(kk )r()r(
ieu
)Tr()r( kk
uu
Kronig-Penney Model
(a+b) b 0 a a+b
U(x)
x
U0
The wave equation can be solved when the potential is simple... such as a periodic square well.
Kronig-Penney Model
)(
2 2
22xU
dx
d
m
Region I - where 0 < x < a and U = 0The eigenfunction is a linear combination of plane waves traveling both left and right:
The energy eignevalue is:
iKxiKx BeAe
m
K
2
22
Region II - where b < x < 0 and U = U0
Within the barrier, the eigenfunction looks like this
and
QxQx DeCe
m
QU
2
22
0
Kronig-Penney Model
)(
2 2
22xU
dx
d
m
Kronig-Penney Model
(a+b) b 0 a a+b
U(x)
x
U0
To satisfy Mr. Bloch, the solution in region IIImust also be related to the solution in region II.
III III
)( baikIIIII e
Kronig-Penney Model
A,B,C, and D are chosen so that both the wavefunction and its derivative with respect to x are continuous at
the x = 0 and a.
At x = 0...
At x = a...
)()( DCQBAiK
DCBA
)( baikIIIII e
)()( baikQbQbiKaiKa eDeCeBeAe
)( baikQbQbiKaiKa eDeCeQBeAeiK
Kronig-Penney Model
)(coscoscoshsinsinh2
22bakKaQbKaQb
QK
KQ
Result for E < U0:
To obtain a more convenient form Kronig and Penney considered the case where the potential barrier becomes a delta function, that is, the case where U0 is infinitely large, over an infinitesimal distance b, but the
product U0b remains finite and constant.
02 UQ
and also goes to infinity as U0. Therefore:
222
0)( Lim QKQ
U
Kronig-Penney ModelWhat happens to the product Qb as U0 goes to infinity?
• b becomes infinitesimal as U0 becomes infinite.
• However, since Q is only proportional to the square root of U0, it does not go to infinity as fast as b goes to zero.
• So, the product Qb goes to zero as U0 becomes infinite.
• As a results of all of this...
KQ
Qb
QbQb
1cos
1sin
Kronig-Penney Model
2
2baQP kaKaKa
Ka
Pcoscossin
Kronig-Penney Model
0 2 3ka ka
0
Plot of energy versuswavenumber for theKronig-Penney Potential,with P = 3/2.
Crucial to the conduction process is whether or not there are electrons available for conduction.
Conductors, Insulators, and Semiconductors
Conductors, Insulators, and Semiconductors
Conductors, Insulators, and Semiconductors
Conductors, Insulators, and Semiconductors
“doped”“thermallyexcited”