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Energy Bands and Charge Carriers
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Energy bands and charge carriersin semiconductors
Chapter 3Mr. Harriry (Elec. Eng.)
By: Amir Safaei2006
In the name of God
2
Outlines
3-1. Bonding Forces and Energy Bands in Solids 3-1-1. Bonding Forces in Solids 3-1-2. Energy Bands 3-1-3.
Metals, Semiconductors & Insulators 3-1-4. Direct & Indirect Semiconductors 3-1-5.
Variation of Energy Bands with Alloy Composition
3
Outlines
3-2. Carriers in Semiconductors 3-2-1. Electrons and Holes 3-2-2. Effective Mass 3-2-3. Intrinsic Material 3-2-4. Extrinsic Material 3-2-5. Electrons and Holes in
Quantum Wells
4
Outlines
3-3. Carriers Concentrations 3-3-1. The Fermi Level 3-3-2. Electron and Hole
Concentrations at Equilibrium
5
3-1. Bonding Forces & Energy Bands in Solids In Isolated Atoms In Solid Materials
3rd Band2nd Band
1st Band
Core
6
3-1-1. Bonding Forces in Solids
Na (Z=11) [Ne]3s1
Cl (Z=17) [Ne]3s1 3p5
Na+ Cl_
7
3-1-1. Bonding Forces in Solids
e_
Na+
8
3-1-1. Bonding Forces in Solids
9
3-1-1. Bonding Forces in Solids
Si<100>
10
3-1-2. Energy Bands Pauli Exclusion Principle
C (Z=6) 1s2 2s2 2p2
2 states for 1s level
2 states for 2s level
6 states for 2p level
For N atoms, there will be 2N, 2N, and 6N states of type 1s, 2s, and 2p, respectively.
11
3-1-2. Energy Bands
Atomic separation
Diamond lattice
spacing
En
erg
y
1s
2s
2p
Valence band
Conduction band
2p
2s
2s-2p
4N States
4N States
Eg
1s
12
3-1-3. Metals, Semiconductors & Insulators
For electrons to experience acceleration in an applied electric field, they must be able to move into new energy states. This implies there must be empty states (allowed energy states which are not already occupied by electrons) available to the electrons.
The diamond structure is such that the valence band is completely filled with electrons at 0ºK and the conduction band is empty. There can be no charge transport within the valence band, since no empty states are available into which electrons can move.
13
3-1-3. Metals, Semiconductors & Insulators
The difference bet-ween insulators and semiconductor mat-erials lies in the size of the band gap Eg, which is much small-er in semiconductors than in insulators.
Insulator Semiconductor
Filled
Filled
Empty
Empty
Eg
Eg
14
3-1-3. Metals, Semiconductors & Insulators
Metal
Filled
Partially Filled
Overlap
In metals the bands either overlap or are only partially filled. Thus electrons and empty energy states
Metal are intermixed with-
in the bands so that electrons can move freely under the infl-uence of an electric field.
15
3-1-4. Direct & Indirect Semiconductors A single electron is assumed to travel
through a perfectly periodic lattice. The wave function of the electron is assumed
to be in the form of a plane wave moving.
xjkxk
xexkUx ),()( x : Direction of propagation k : Propagation constant / Wave vector : The space-dependent wave function
for the electron
16
3-1-4. Direct & Indirect Semiconductors
U(kx,x): The function that modulates the
wave function according to the periodically of the lattice.
Since the periodicity of most lattice is different in various directions, the (E,k) diagram must be plotted for the various crystal directions, and the full relationship between E and k is a complex surface which should be visualized in there dimensions.
17
3-1-4. Direct & Indirect Semiconductors
Eg=hνEg Et
k k
EE
Direct IndirectExample 3-1
18
3-1-4. Direct & Indirect Semiconductors
Example 3-1: Assuming that U is constant in
for an essentially free electron, show that the x-component of the electron momentum in the crystal is given by
xx khP Example 3-2
),()( xkUx xk xjkxe
19
3-1-4. Direct & Indirect Semiconductors
x
x
xjkxjk
x
khdxU
dxUkh
dxU
dxexj
heU
P
xx
2
2
2
2
)( Answer:
The result implies that (E,k) diagrams such as shown in previous figure can be considered plots of electron energy vs. momentum, with a scaling factor .
h
20
3-1-4. Direct & Indirect Semiconductors
Si Ge GaAs AlAs Gap
1.11 1350 480 2.5E5 D 5.43
0.67 3900 1900 43 D 5.66
1.43 8500 400 4E8 Z 5.65
2.16 180 0.1 Z 5.66
2.26 300 150 1 Z 5.45
Eg(eV) n p Lattice Å
Properties of semiconductor materials
21
3-1-5. Variation of Energy Bands with Alloy Composition
X
L
1.43eVk
E
0.3eV
AlxGa1-xAs
2.16eV
AlAsGaAsX
E
1.4
2.0
1.8
1.6
2.2
2.4
2.6
2.8
3.0
0 0.2 0.4 0.6 0.8 1
X
L
X
L
22
3-2. Carriers in Semiconductors
Ec
Ev
Eg
0ºK3ºK2ºK4ºK5ºK1ºK6ºK7ºK8ºK9ºK10ºK11ºK12ºK13ºK14ºK300ºK
15ºK16ºK17ºK18ºK19ºK20ºK
Electron Hole PairE H P
23
3-2-1. Electrons and Holes
N
iiVqJ 0)(
k
Ekj-kj
j` j
j
N
ii VqVqJ )()(
0
J jVq) ( jV)q(
24
3-2-2. Effective Mass
The electrons in a crystal are not free, but instead interact with the periodic potential of the lattice.
In applying the usual equations of electrodynamics to charge carriers in a solid, we must use altered values of particle mass. We named it Effective Mass.
25
3-2-2. Effective Mass
Example 3-2: Find the (E,k) relationship for a free electron and relate it to the electron mass.
E
k
26
3-2-2. Effective Mass
khmvp
222
2
22
1
2
1k
m
h
m
pmvE
Answer: From Example 3-1, the electron
momentum is:
m
h
dk
Ed 2
2
2
27
3-2-2. Effective Mass
Answer (Continue): Most energy bands are close to
parabolic at their minima (for conduction bands) or maxima (for valence bands).
EC
EV
28
3-2-2. Effective Mass The effective mass of an electron in a band
with a given (E,k) relationship is given by
2
2
2*
dkEd
hm
X
L
k
E
1.43eV
) ()( or** LXmm
Remember that in GaAs:
29
3-2-2. Effective Mass At k=0, the (E,k) relationship near the
minimum is usually parabolic:
gEkm
hE 2
*
2
2 In a parabolic band, is constant.
So, effective mass is constant.
Effective mass is a tensor quantity.
2
2
dk
Ed
2
2
2*
dkEd
hm
30
3-2-2. Effective Mass
EV
EC
02
2
dk
Ed
02
2
dk
Ed
0* m
0* m2
2
2*
dkEd
hm
Ge Si GaAs
† m0 is the free electron rest mass.
Table 3-1. Effective mass values for Ge, Si and GaAs.
mn
*
mp
*
055.0 m 01.1 m 0067.0 m
037.0 m 056.0 m 048.0 m
31
3-2-3. Intrinsic Material
A perfect semiconductor crystal with
no impurities or lattice defects is
called an Intrinsic semiconductor.
In such material there are no charge
carriers at 0ºK, since the valence
band is filled with electrons and the
conduction band is empty.
32
3-2-3. Intrinsic Material
SiEgh+
e-
n=p=ni
33
3-2-3. Intrinsic Material If we denote the generation rate of EHPs
as and the recombination rate
as equilibrium requires that:
)(Tgi
)( 3scmEHPri
ii gr Each of these rates is temperature depe-
ndent. For example, increases
when the temperature is raised.
)( 3scmEHPgi
iirri gnpnr 200
34
3-2-4. Extrinsic Material
In addition to the intrinsic carriers generated thermally, it is possible to create carriers in semiconductors by purposely introducing impurities into the crystal. This process, called doping, is the most common technique for varying the conductivity of semiconductors.
When a crystal is doped such that the equilibrium carrier concentrations n0 and p0
are different from the intrinsic carrier concentration ni , the material is said to be
extrinsic.
35
3-2-4. Extrinsic Material
0ºK3ºK2ºK4ºK5ºK1ºK6ºK7ºK8ºK9ºK10ºK11ºK12ºK13ºK14ºK50ºK15ºK16ºK17ºK18ºK19ºK20ºK
Ec
Ev
Ed
Donor
V
P
As
Sb
36
3-2-4. Extrinsic Material
0ºK3ºK2ºK4ºK5ºK1ºK6ºK7ºK8ºK9ºK10ºK11ºK12ºK13ºK14ºK50ºK15ºK16ºK17ºK18ºK19ºK20ºK
Ec
Ev
Ea
Acceptor
ш
B
Al
Ga
In
37
3-2-4. Extrinsic Material
h+
Al
e- Sb
Si
38
3-2-4. Extrinsic Material
We can calculate the binding energy by using the Bohr model results, consider-ing the loosely bound electron as ranging about the tightly bound “core” electrons in a hydrogen-like orbit.
rKnhK
mqE 022
4
4, 1;2
39
3-2-4. Extrinsic Material
Example 3-3: Calculate the approximate donor binding energy for Ge(εr=16, mn
*=0.12m0).
40
3-2-4. Extrinsic Material
eVJ
h
qmE
r
n
0064.01002.1
)1063.6()161085.8(8
)106.1)(1011.9(12.0
)(8
21
234212
41931
220
4*
Answer:
Thus the energy to excite the donor electron from n=1 state to the free state (n=∞) is ≈6meV.
41
3-2-4. Extrinsic Material
When a ш-V material is doped with Si or Ge, from column IV, these impurities are called amphoteric.
In Si, the intrinsic carrier concentration ni is about 1010cm-3 at
room tempera-ture. If we dope Si with 1015 Sb Atoms/cm3, the conduction electron concentration changes by five order of magnitude.
42
3-2-5. Electrons and Holes in Quantum Wells
One of most useful applications of MBE or OMVPE growth of multilayer compou-nd semiconductors is the fact that a continuous single crystal can be grown in which adjacent layer have different band gaps.
A consequence of confining electrons and holes in a very thin layer is that
43
3-2-5. Electrons and Holes in Quantum Wells
these particles behave according to the particle in a potential well problem.
GaAs
Al0.3Ga0.7AsAl0.3Ga0.7As
50Å
E1
Eh
1.43eV
1.85eV
0.28eV
0.14eV
1.43eV
44
3-2-5. Electrons and Holes in Quantum Wells
Instead of having the continuum of states
as described by , modified for
effective mass and finite barrier height.
Similarly, the states in the valence band
available for holes are restricted to
discrete levels in the quantum well.
2
222
2mL
hnEn
45
3-2-5. Electrons and Holes in Quantum Wells
An electron on one of the discrete condu-ction band states (E1) can make a transition
to an empty discrete valance band state in the GaAs quantum well (such as Eh), giving
off a photon of energy Eg+E1+Eh, greater
than the GaAs band gap.
46
3-3. Carriers Concentrations In calculating semiconductor electrical
pro-perties and analyzing device behavior, it is often necessary to know the number of charge carriers per cm3 in the material. The majority carrier concentration is usually obvious in heavily doped material, since one majority carrier is obtained for each impurity atom (for the standard doping impurities).
The concentration of minority carriers is not obvious, however, nor is the temperature dependence of the carrier concentration.
47
3-3-1. The Fermi Level Electrons in solids obey Fermi-Dirac statistics. In the development of this type of statistics:
Indistinguishability of the electrons Their wave nature Pauli exclusion principle
must be considered. The distribution of electrons over a range of
these statistical arguments is that the distrib-ution of electrons over a range of allowed energy levels at thermal equilibrium is
48
3-3-1. The Fermi Level
kTfEE
eEf )(
1
1)(
k : Boltzmann’s constant
f(E) : Fermi-Dirac distribution function
Ef : Fermi level
49
3-3-1. The Fermi Level
2
1
11
1
1
1)( )(
kTfEfE
eEf f
Ef
f(E)
1
1/2
E
T=0ºKT1>0ºKT2>T1
50
3-3-1. The Fermi Level
Ev
Ec
Ef
E
f(E)01/21
≈≈
f(Ec
)f(Ec
)
[1-f(Ec)]
Intrinsicn-typep-type
51
3-3-2. Electron and Hole Concentrations at Equilibrium
CE
dEENEfn )()(0
The concentration of electrons in the conduction band is
N(E)dE : is the density of states (cm-3) in the energy range dE.
The result of the integration is the same as that obtained if we repres-ent all of the distributed electron states in the conduction band edge EC. )(0 CC EfNn
52
3-3-2. Electron and Hole Concentrations at Equilibrium
EC
EV
Ef
E
Holes
Electrons
Intrinsicn-typep-type
N(E)[1-f(E)]
N(E)f(E)
53
3-3-2. Electron and Hole Concentrations at Equilibrium
kTFECE
kTFECE
ee
Ef C
)(
)(
1
1)(
kTFECE
eNn C
)(
0
23
) 2
(22
*
h
kTmN nC
54
3-3-2. Electron and Hole Concentrations at Equilibrium
)](1[0 VV EfNp
kTVEFE
kTFEVE
ee
Ef V
)(
)(
1
11)(1
kTVEFE
eNp V
)(
0
23
) 2
(22
*
h
kTmN pV
55
3-3-2. Electron and Hole Concentrations at Equilibrium
kTvEiE
eNp Vi
)(
kTiEcE
eNn Ci
)(
kT
gEkT
vEcE
eNNeNNpn vcvc
)(
00
kTgE
eNNpn vcii
kTgE
eNNn vci2
2
00 inpn
kTFEiE
enp i
)(
0
kTiEFE
enn i
)(
0
56
3-3-2. Electron and Hole Concentrations at Equilibrium
Example 3-4: A Si sample is doped with 1017 As Atom/cm3. What is the equilibrium hole concentra-tion p0 at 300°K? Where is EF relative to Ei?
57
3-3-2. Electron and Hole Concentrations at Equilibrium
3317
20
0
2
0 1025.210
1025.2
cmn
np i
Answer: Since Nd»ni, we can approximate
n0=Nd and
kTiEFE
enn i
)(
0
eVn
nkTEE
iiF 407.0
105.1
10ln0259.0ln
10
170
58
3-3-2. Electron and Hole Concentrations at Equilibrium
Answer (Continue) :
Ev
Ec
EF
Ei1.1eV0.407eV
59
References:
Solid State Electronic Devices Ben G. Streetman, third edition
Modular Series on Solid State Devices, Volume I: Semiconductor Fundamentals
Robert F. Pierret