band theory of soids5!1!13
TRANSCRIPT
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IntroductionBloch stated this theory in 1928. According to this theory, thefree electrons moves in a periodic field provided by the lattice.
The Energy band theory of solids is the basic principle of
semiconductor physics and it is used to explain the differences
in electrical properties between Metals, Insulators and
Semiconductors.
Band Theory of Solids
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Electron in a periodic potential
( Bloch Theorem )
A crystalline solid consists of a lattice which is composed of a
large number ofpositive ion cores at regular intervals and the
conduction electrons move freely throughout the lattice.
The variation of potential inside the metallic crystal with the
periodicity of the lattice is explained by Blochtheorem.
The periodic potential V (x) may be defined by means of the
lattice constant a as V (x) = V ( x + a )
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One dimensional periodic potential in crystal.
a+++ +
0V
+ +++++
+ +++++
Periodic positive ion cores Inside metallic crystals.
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electrontheofmotiontheofstatetherepresentskwherea)(xU(x)U
lattice.crystalaofyperiodicitwithperiodicais(x)UWhere
)(x)exp(ikxU(x)
equationrSchrodingetheofsolution
ldimensionaonetheshown thathasBloch
0a)]V(x[Eh
m8
dx
d
0V][Eh
m8
dx
d
equationr waveSchrodingeFrom
kk
k
k
2
2
2
2
2
2
2
2
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KroningPenney Model
According to Kroning - Penney model the electrons move in a
periodic potential field provided by the lattice.
The potential of the solid varies periodically with the periodicity of
space lattice.
Consider Schrdinger equation for this case, we can find the
existence of the energy gap between the allowed values of energy
of electron.
For one dimensional periodic potential field.
0)]([8
2
2
2
2
xVEh
m
dx
d
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+++++
V0
0 a-b
12
0][8
)1(
2
2
2
2
Eh
m
dx
dregionfor
0][8
)2(
02
2
2
2
VEh
m
dx
dregionfor
)2.....(0
)1.....(0
2
2
2
2
2
2
dx
d
dx
d2
2
2
2
2
02
8
8[ ]
where
mE
h
mV E
h
0,)(
0,0)(
0
xbvxv
axxv
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.....(a)(x)eU(x)ikx
k
)(x).....(bUa)(xUkk
.....(c)(x)eNa)(xikNa
kk
According to Bloch, the solution of a Schrodinger equation
Where Uk(x) is the periodicity of the lattice i.e,.
According to Bloch theorem
By using above a, b, and c Bloch conditions, the solutions of
equations (1) & (2) becomes
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2mEh
2
strengthBarriercalledisbV
barrierpotentialtheofpowerscatteringtheisp
bV
h
ma4where..P
acosa
asinPcoska
0
02
2
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+1
-1
+1
-1
a
ac osa
asineP
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Conclusions
1. The motion of electrons in a periodic lattice is characterizedby the bands of allowed energy separated by forbiddenregions.
2. As the value ofa increases, the width of allowed energy
bands also increases and the width of the forbidden bandsdecreases. i.e., the first term of equation deceases on theaverage with increasing a .
3. Let us now consider the effect of varying barrier strength P. if
V0b is large ,i.e. if p is large ,the function described by the lefthand side of the equation crosses +1 and -1 region as shownin figure. thus the allowed bands are narrower and theforbidden bands are wider.
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a0
p
If P tends to infinite the allowed band reduces to one
single energy level
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0p
a
4. If P tends to zero
No energy levels exist:
all energies are allowed to the electrons.
22
2
22
2
2
2
2
2
22
2
22
22
2
1
2)
2(
1
)2(
)2
)(8(
)2(
2
coscos
mvm
p
h
p
m
hE
m
h
E
m
hE
kmE
mEk
k
k
kaa
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Brillouin Zones
The Brillouin zone is a representation of permissive
values of k of the electrons in one, two or three
dimensions.
Thus the energy spectrum of an electron moving in
the presence of a periodic potential fields is divided
into allowed zones and forbidden zones.
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Allowedbands
Energy gap
First
Brillouin zone
E
k
Energy gap
a
a
2
a
3
a
a
2
a
3
E-k Diagram
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Origin of Energy band formation in Solids
When we consider isolated atom, the electrons are tightly
bound and have discrete, sharp energy levels.
When two identical atoms are brought closer the outer mostorbits of these atoms overlap and interact.
If more atoms are brought together more levels are formed andfor a solid of N atoms , each of the energy levels of an atomsplits into N levels of energy.
The levels are so close together that they form an almostcontinuous band.
The width of this band depends on the degree of overlap ofelectrons of adjacent atoms and is largest for outer most
atomic electrons.
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N energy levels
N atoms
E
E1
E2
E3
E2
E1
E1
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The energy bands in solids are important indetermining many of physical properties of solids.
The allowed energy bands
(1) Valance band
(2) Conduction band
The band corresponding to the outer most orbit iscalled conduction band and the next inner band iscalled valence band. The gap between these twoallowed bands is called forbidden energy gap.
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Classifications of materials into Conductors,Semiconductors & Insulators:-
On the basis of magnitude of forbidden band thesolids are classified into insulators, semiconductors
and conductors.
Insulators:
In case of insulators, the forbidden energy band is
very wide as shown in figure.
Due to this fact the electrons cannot jump from
valance band to conduction band.
In insulators at 00k and the energy gap between
valance band and conduction band is of the order.
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Forbidden gap
Valance band
Conduction band
INSULATORSForbidden gap
Valance band
Conduction band
SEMI CONDUCTORS
Valance band
Conduction band
CONDUCTORS
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SEMI CONDUCTORS
In semi conductors the forbidden energy ( band ) gap is very smallas shown in a figure.
Ge and Si are the best examples of semiconductors.
Forbidden ( band ) is of the order of 0.7ev & 1.1ev.
CONDUCTOS:
In conductors there is no forbidden gap. Valence and conduction
bands overlap each other as shown in figure above.The electrons from valance band freely enter into conduction band.