Band Theory 1
-
Upload
achmad-ariebyo -
Category
Documents
-
view
229 -
download
0
Transcript of Band Theory 1
-
7/26/2019 Band Theory 1
1/26
Energy Bands in SolidsEnergy Bands in Solids
Physics 355Physics 355
-
7/26/2019 Band Theory 1
2/26
Consider the available energies for electrons in the materials.
As two atoms are brought
close together, electrons
must occuy different
energies due to Pauli
E!clusion rincile.
"nstead of having discrete
energies as in the case of
free atoms, the available
energy states form bands.
Conductors, "nsulators, and Semiconductors
-
7/26/2019 Band Theory 1
3/26
#ree Electron #ermi $as#ree Electron #ermi $as
-
7/26/2019 Band Theory 1
4/26
-
7/26/2019 Band Theory 1
5/26
-
7/26/2019 Band Theory 1
6/26
==
m
kE
2
22
#or free electrons, the
wavefunctions are lane
waves%
)rk()r(
ie
-
7/26/2019 Band Theory 1
7/26
-
7/26/2019 Band Theory 1
8/26
Band $a
&one boundary
-
7/26/2019 Band Theory 1
9/26
'doed('thermally
e!cited(
-
7/26/2019 Band Theory 1
10/26
)
*rigin of the Band $a
+o get a standing wave at the boundaries, you cantae a linear combination of two lane 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
= = =
-
7/26/2019 Band Theory 1
11/26
*rigin of the Band $a
Electron -ensity
-
7/26/2019 Band Theory 1
12/26
*rigin of the Band $a
-
7/26/2019 Band Theory 1
13/26
Bloch #unctions
#eli! Bloch showed that the actualsolutions to the Schrdinger e/uation for
electrons in a eriodic otential must have
the secial form%
where uhas the eriod of the lattice, that
is
)rk(kk )r()r( = ieu
)Tr()r( kk +=uu
-
7/26/2019 Band Theory 1
14/26
0ronig1Penney 2odel
a+b4 b a a+b
U!4
x
U0
+he wave e/uation can be solved when the otential is
simle... such as a eriodic s/uare well.
-
7/26/2019 Band Theory 1
15/26
0ronig1Penney 2odel
=+ )(2 2
22xU
dx
d
m
Region I - where 0 < x< aand U= 0
The eigenfunction is a linear comination of !lane wa"estra"eling oth left and right#
The energ$ eigne"alue is#
iKxiKx BeAe +=
m
K
2
22
=
-
7/26/2019 Band Theory 1
16/26
Region II - where b< x< 0 and U= U0%ithin the arrier& the eigenfunction looks like this
and
QxQx DeCe +=
m
QU
2
22
0
=
0ronig1Penney 2odel
=+ )(2 2
22xU
dx
d
m
-
7/26/2019 Band Theory 1
17/26
0ronig1Penney 2odel
a+b4 b a a+b
U!4
x
U0
+o satisfy 2r. Bloch, the solution in region """
must also be related to the solution in region "".
III III
)( baik
IIIII e
+
=
-
7/26/2019 Band Theory 1
18/26
0ronig1Penney 2odel
A,B,C, and - are chosen so that both the wavefunctionand its derivative with resect toxare continuous at
thex6 and a.
Atx6 ...
Atx = a...
)()( DCQBAiKDCBA = +=+
)( baikIIIII e
+=
)()( baikQbQbiKaiKa eDeCeBeAe + +=+
( ) ( ) )( baikQbQbiKaiKa eDeCeQBeAeiK + =
-
7/26/2019 Band Theory 1
19/26
0ronig1Penney 2odel
)(coscoscoshsinsinh2
22
bakKaQbKaQbQK
KQ+=+
+7esult for E8 U%
+o obtain a more convenient form 0ronig and Penney considered the
case where the otential barrier becomes a delta function, that is, the
case where Uis infinitely large, over an infinitesimal distance b, but the
roduct U0bremains finite and constant.
02 UQ
and also goes to infinity as U. +herefore%
222
0
)('im QKQ
U
-
7/26/2019 Band Theory 1
20/26
0ronig1Penney 2odel9hat haens to the roduct Qbas Ugoes to infinity:
; bbecomes infinitesimal as Ubecomes infinite.
; >
-
7/26/2019 Band Theory 1
21/26
0ronig1Penney 2odel
=2
2baQP kaKaKaKaP coscossin =+
-
7/26/2019 Band Theory 1
22/26
0ronig1Penney 2odel
= 3 ka ka
Plot of energy versus
wavenumber for the
Kronig-Penney Potential,
with P = !".
-
7/26/2019 Band Theory 1
23/26
Crucial to the
conduction rocess
is whether or notthere are electrons
available for
conduction.
Conductors, "nsulators, and Semiconductors
-
7/26/2019 Band Theory 1
24/26
Conductors, "nsulators, and Semiconductors
-
7/26/2019 Band Theory 1
25/26
Conductors, "nsulators, and Semiconductors
-
7/26/2019 Band Theory 1
26/26
Conductors, "nsulators, and Semiconductors
'doed('thermally
e!cited(