Isolated atoms vs. large group of atoms
In an isolated atom:
•electrons are restricted to sets of discrete energy levels within atoms.
•large gaps exist in the energy scale in which no energy states are available.
Electrons in solids are:
•restricted to certain energy ranges and are not allowed at other energies.
The basic difference between the case of an electron in a solid and that of an
electron in an isolated atom:
In the solid the electron has a range, or band, of available
energies.
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Bonding forces in solids
1. Ionic bonding
All electrons are tightly bound to atoms. There are no loosely bound electrons to
participate in current flow–this material is insulator.
Example: NaCl (Sodium Chloride –table salt)
Na (sodium, Z= 11) is 1s2 2s2 2p6 3s1 Cl (chlorine, Z= 17) is 1s2 2s2 2p6 3s2 3p5
In the NaCl lattice each Na atom gives up its outer 3s electron to a Cl atom, so that
the crystal is made up of ions with the electronic structures of the inert atoms Ne and
Ar (Ar has the electronic structure [Ne] 3s23p6).
2. Metallic bonding
The forces holding the lattice together arise from an interaction between the positive
ion cores and the surrounding free electrons –metallic bonding.
Example: Na (sodium) Electronic structure of Na (sodium, Z= 11) is 1s2 2s2 2p6 3s1
The single electron in th eouter orbital is loosely bound and is given up easily in ion
formation –the outer electron of each atom is contributed to the crystal as a whole, so that
the solid is made up of ions with closed shells immersed in a sea of free electrons. These
“free” electrons are free to move about the crystal under the influence of an electric field
providing good electric conductivity.
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Bonding forces in solids
3. Covalent bonding
The bonding forces arise from a quantum mechanical interaction between the shared
electrons :
Each electron pair constitutes a covalent bond
These two electrons are indistinguishable, except that they must have opposite spin to
satisfy the Pauli exclusion principle. Covalent bonding is also found in certain molecules,
such as H2.
Example: Si (silicon)
Electronic structure of Si (Z= 14) is 1s2 2s2 2p6
3s2 3p2
Si has four electron in the outer orbit.
No free electron is available to the lattice in the covalent structure.
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Energy bands
(A General and Simplified Approach)
•When isolated atoms are brought together to form a solid, various interactions occur
between neighboring atoms.
•The forces of attraction and repulsion between atoms will find a balance at the proper
interatomic spacing for the crystal.
•In this process, important changes occur in the electron energy level configurations, and
these changes result in the varied electrical properties of solids.
•When two atoms are completely isolated from each other, they can have identical
electronic structures
•As the spacing between the two atoms becomes smaller, electron wave functions begin
to overlap. The Pauli exclusion principle dictates that no two electrons in a given
interacting system may have the same quantum state; thus there must be a splitting of
the discrete energy levels of the isolated atoms into new levels belonging to the pair
rather than to individual atoms.
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Energy bands -2
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Energy bands -4
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Energy bands -5
Energy levels in Si as a
function of atomic spacing.
The core levels (n=1,2) in Si
are completely filled with
electrons. At the actual
atomic spacing of the crystal,
the 2N electrons in the 3s
subshell and the 2N electrons
in the 3p subshell undergo
sp3 hybridization, and all end
up in the lower 4N states
(valence band) while the
higher lying 4N states
(conduction band) are empty
separated by a band gap.
1s2 ⇨ 2N states
2s2 ⇨ 2N states
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2p6 ⇨ 6N states
3s2 ⇨ 2N states
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3p6 ⇨ 6N states
66
Energy bands -6
In a solid, very large number of atoms is brought together. This
causes that the split energy levels form continuous bands of
energies.
Example:
Formation of energy bands as
a diamond (C) crystal is
formed by bringing together
isolated carbon atoms.
Electronic structure of C (Z=
6)is 1s2 2s2 2p2
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Energy bands -7
Each C atom has available two 1s states, two 2s
states, six 2p states, and higher states.
If we consider N atoms brought together, there
will be 2N states of type 1s, 2N states of type
2s, and 6N states of type 2p.
As the interatomic spacing decreases, energy
levels split into bands.
As the "2s" and "2p" bands grow, they merge
into a single band composed of a mixture of
energy levels. This band of "2s-2p" levels
contains 8N available states.
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Energy bands -8
E
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Energy bands -9
Two atoms
Six atoms
Solid of N atoms
Electrons must occupy different energies due to Pauli Exclusion principle.
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Energy bands -10
As the distance between atoms approaches the equilibrium
interatomic spacing of diamond, the band of "2s-2p" levels
splits into two bands (conduction band and valence band)
separated by an energy gap Eg.
A semiconductor crystal has two bands of available energy
levels separated by an energy gap Eg wide, which contains no
allowed energy levels for electrons to occupy.
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Energy bands
A Formal Approach
A more formal and mathematical approach of derivation of energy bands in solids more
particularly semiconductor crystals involves quantum mechanics, Bloch’s Theorem and
Kronig-Penney Model.
Felix Bloch (October 23, 1905 – September 10, 1983) was a Swiss physicist,
working mainly in the U.S. He received his doctorate in physics in 1928 at
the University of Leipzig under the supervision of Werner Heisenberg. His
doctoral thesis established the quantum theory of solids, using Bloch
waves to describe the electrons.
Ralph Kronig was a
German-American
physicist (March 10, 1904
– November 16, 1995). He
is noted for the discovery
of particle spin and for his
theory of x-ray absorption
spectroscopy.
His theories include the Kronig-Penney
model, the Coster-Kronig transition and
the Kramers–Kronig relation.
© Nezih Pala npala@fiu.edu
William George Penney, Baron
Penney (24 June 1909 – 3 March
1991) was a British physicist who
was responsible for the
development of British nuclear
technology following the World
War II.
A mathematician by training, he became an
expert on wave dynamics. He was tasked with
the development of the British atomic bomb.
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Electrons in Periodic Potential
When we wan to examine the properties of an electron in a periodic lattice, we need to
consider Schrödinger's; equation such that the potential energy term V(r) reflects the fact
that the electron sees a periodic potential.
In a one dimensional lattice, ionized atoms form a periodic potential distribution for
electrons due to Coulomb potential.
U(x)
x
1 ( Zq)(q)
U ( x) 
4 0
| x|
Schrodinger’s equation can be written as
 2 2
  

 
  V (r ) (r )  E(r )
 2m

where the potential energy term is periodic V(r)=V(r+T) and where T is the crystal
translation vector.
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Bloch’s Theorem
Bloch’s theorem applies to waves in periodic structures in general. In a periodic potential
distribution, wave function solutions of Schrodinger’s equation can be written as the
product of a plane wave and a periodic function.

 jk.r
(r )  u(r )e
where k is the wavevector to be determined (called the Bloch wavevector) and where u is
periodic



u (r )  u (r  T )
Thus
 
  jk.(r T )
 jk.r jk.T
 jk.T
(r  T )  u(r  T )e
 u(r )e e
 (r )e
It is important to note that the Bloch theorem shows that the electronic can propagate
through a perfect periodic medium without scattering(i.e. without hitting the atoms).
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Kronig – Penney Model of Band Structure -1
As an approximate model for one dimensional crystal can be given as
a
V=V0
V=V0
V=V0
V=0
-a2
0
V=0
0 xa
0,
V ( x)
V0 , -b  x  0
a1
where a=a1+a2 is the period of the lattice. This is known as Kronig-Penny model.
In the region -a2 ≤ x ≤ 0, the potential V=V0, and the solution of Schrödinger's equation is
I ( x)  Ae jx  Be  jx
where

2m( E  V0 )
2
In the region 0 ≤ x ≤ a1, the potential V=0, and the solution of Schrödinger's equation is
II ( x)  De jx  Fe  jx
© Nezih Pala npala@fiu.edu
where
EEE 6397 – Semiconductor Device Theory

2mE
2
75
Kronig – Penney Model of Band Structure -2
Using Bloch theorem
( x)  u( x)e jkx
and
u ( x)  u ( x  a )
( x  a)  u( x  a)e jk ( xa )  u( x)e jk ( xa )  ( x)e jka
therefore
Using these relationships the wavefunction in the period a1 ≤ x ≤ a1+a can be written as

 Ae

e
( x)  Ae j ( x a )  Be  j ( x a ) e jka ,
j ( x  a )
 Be  j ( x a )
jka
,
a1  x  a
a  x  a1  a
Enforcing the continuity of Ψ and Ψ’ at x=0 and x=a1 leads to the eignevalue equation,
if 0 < E < V0
 2  2
cos ka  cos(a1 ) cosh(a2 ) 
sin(a1 ) sinh(a2 )
2
and if E > V0
2  2
cos ka  cos(a1 ) cos(a2 ) 
sin(a1 ) sin( a2 )
2
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
2m(V0  E )
2
76
Kronig – Penney Model of Band Structure -3
In the preceding equations, the energy E is the only unknown parameter. For a solution to
exist, we must have
1  cos ka  1
Therefore the right hand side of the last two equations denoted as r(E) must obey this
condition. A typical plot of r(E) vs E is as follows:
k 0
k 

a
The figure makes it that there are certain allowed values of energy, called allowed energy
bands and certain unallowed values of energy , called band gaps. That is if E is in an
allowed energy band, Schrodinger’s equation has solution, and if E is in an allowed energy
band there is no solution. Within an allowed band, energy can take any value (i.e. it is not
discretized). Note that as energy increases , the allowed energy bands increase in width,
and so the forbidden bands decrease in width.
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Kronig – Penney Model of Band Structure -4
Since
cos ka  r ( E )
we can generate an important figure called dispersion diagram, which is a diagram of
energy versus wavenumber (E vs. k). To generate the dispersion diagram, start at E=0 and
compute r(E). If |r(E)|=|cos ka|>1, we are at a forbidden energy (i.e. in a bandgap) and we
need to increase E a bit and try again. If |r(E)|≤ 1, we are at an allowed energy (i.e. in an
energy band) and in this case the corresponding wavenumber is
k
1
cos 1 (r ( E ))
a
Since cosine is an even function –k will also be a solution. By increasing E by a small
amount and checking the value of r(E) , we can generate the plot of allowed and unallowed
energy bands. One form of the result will look like as follows. This depiction is known as the
extended zone scheme.
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Kronig – Penney Model of Band Structure -5
The various sections of wavenumber space are divided into what are called Brillouin zones
with the range


a
k

a
Denoting the important first Brillouin zone. The second Brillouin zone is the range

2

k ,
a
a

a
k
2
a
and so on for higher zones.
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Kronig – Penney Model of Band Structure -6
Another depiction arises from noting that the energy bands in the higher Brillouin zones can
be all translated to the first Brillouin zone by shifts of n2π/a . This results in what is called
reduced zone scheme as shown below. This is the most common format for describing the
band structure.
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Kronig – Penney Model of Band Structure -7
We concluded that, due to the periodic potential
associated with the crystalline lattice, there are
allowed and disallowed energy bands. Let us look at
how carrier transport is affected if a band is filled with
electrons or not.
1. If an allowed band is completely empty of
electrons, obviously there are no electrons in the
band to participate in electrical conduction. This
can happen, for example, in a high-energy band
where the energies of the band are above the
energies of the systems electrons.
2. Similarly, and surprisingly, if an allowed band is
completely filled with electrons, those electrons
can not contributive to electric conduction either.
}
}
Conduction
Band
Valence
Band
3. Only electrons in a partially filled energy band can
contribute to conduction.
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Kronig – Penney Model of Band Structure -8
A schematic description of allowed energy levels and energy bands in an atom and
in crystalline materials.
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Kronig – Penney Model of Band Structure -9
The concept of allowed bands of energy separated by bandgaps is central to the
understanding of crystalline materials. Near the band edges it is usually possible to define
the electron E–k relation as
2
2
 (k  k0 )
E
2m *
where k0 is the k-value at the band
edge and m∗ is the effective mass. The
concept of an effective mass is
extremely useful, since it represents
the response of the electron–crystal
system to the outside world.
E vs. k diagram for Si
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Kronig – Penney Model of Band Structure -10
One obtains a series of allowed energy bands separated by bandgaps as shown schematically
in the previous slides. Each band has an E vs. k relation. Examples of such relations
called bandstructure will be shown later in this chapter.
The product of  and the k-vector behaves like an effective momentum for the electron inside
the crystal. The smallest k-values lie in a k-space called the Brillouin zone (see figure below). If
the k-value is chosen beyond the Brillouin zone values, the energy values are simply repeated.
Brillouin zone of the face centered cubic lattice (left) and the hexagonal lattice (right).
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Kronig – Penney Model of Band Structure -11
As shown in the figure for the fcc lattice, the
maximum k-value along the (100) direction is
2π/a(1, 0, 0). This point is called the X-point
and there are five other equivalent points,
due to the cubic symmetry of the lattice.
Similarly, along the (111) direction, the
maximum k-point is π/a(1, 1, 1) and seven
other similar points. This point is called the
L-point.
Thus we commonly display the E-k diagram
with k going from the origin (called the Γpoint) to the X-point and from the origin to
the L-point.
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Effective Mass -1
An electron in empty space has a well defined constant mass. However, sometimes, it is
useful to view mass merely a proportionality constant between force and acceleration,
remembering the Newton’s second law F=ma.
dp d (mv)
Force  Fint  Fext 

dt
dt
where Fint is the collection of internal periodic crystal forces, and Fext is the externally applied
force. It is inefficient to solve this complicated problem involving the periodic crystal
potential (which is obviously different in different semiconductors) every time we try to solve
a semiconductor device problem. It is better to solve the complicated problem of carrier
motion in the periodic crystal potential just once, and encapsulate that information in what
is called the bandstructure, (E, k), whose curvature gives us the effective mass, m*.The
electron then responds to external forces with this new m*. Newton's law is then written as:
d ( m * v)
Fext 
dt
This is clearly an enormous simplification compared to the more detailed problem. The
periodic crystal forces depend on the details of a specific semiconductor; therefore, the
effective mass is different in different materials.
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Effective Mass -2
On the other hand, complete band structure of material does not need to be known, since
only electrons in certain regions of band are of interest. We typically are more interested in
the behavior of electrons near the band edge (conduction band or valence band) since they
contribute to the conduction most. In this case only the local behavior of E-k curve is
important.
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Effective Mass -3
From Newton’s law of motion
dvg
d (k )
F  m*

dt
dt
vg 
And from the group velocity definition
1 d  dE  1 d 2 E d (k )


 2
dt
 dt  dk   dk 2 dt
dvg
Hence
w 1 E

k  k
2
m*  2
d E
dk 2
1 d 2 E d (k ) d (k )
m* 2

2
 dk
dt
dt
In a real three dimensional crystal, effective mass is
  E 

mx , y , z *   
2
 k

x
.
y
,
z


2
1
2
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Effective mass –4
Consider a free electron for simplicity
p  mv  k
1 2 1 p2 2 2
E  mv 

k
2
2 m 2m
Take the second derivative wrt k
Effective mass m* is inversely
proportional to the curvature (second
derivative) of the E - k relationship.
2
2
d E 

2 2
d k
m
2

m  2
d E 2 2
d k
The term k responds to the external forces as if it is the momentum of the electron,
although, as can be seen by comparing the true Newton’s equation of motion, it is clear that
k contains the effects of the internal crystal potentials and is therefore not the true electron
momentum. The quantity k is called the crystal momentum.
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Effective Mass -5
Mathematically, the effective mass is inversely proportional to the curvature of an E vs k plot.
m* is positive near the bottoms of all bands.
m* is negative near the tops of all bands.
A negative effective mass simply means
that, in response to an applied force, the
electron will accelerate in a direction
opposite to that expected from purely
classical considerations.
The negative effective mass helps to describe, conceptually and mathematically, charge
transfer (and therefore conduction) in a partially filled band. We define an empty state left
behind an electron as a positive charge and a positive effective mass and call it hole.
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Effective Mass –6
In any calculation involving the mass of the charge carriers, we must use effective mass
values for the particular material involved.
mn*–the electron effective mass
mp*–the hole effective mass
The n subscript indicates the electron as a negative charge carrier, and the p subscript
indicates the hole as a positive charge carrier.
Material
mn*/m0
mp*/m0
Si
1.18
0.81
Ge
0.55
0.36
GaAs
0.066
0.52
© Nezih Pala npala@fiu.edu
m0–mass of electron in rest
m0= 9.10938188×10-31kg
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Effective mass –7
In crystalline material, we must use altered values of particle mass
(effective mass) to take into account the influence of the periodic
potential on carrier behavior.
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Effective mass –5
E vs. k diagram for Si
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Effective mass –6
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Effective mass –7
In any calculation involving the mass of the charge carriers, we must use effective mass
values for the particular material involved.
mn*–the electron effective mass
mp*–the hole effective mass
The n subscript indicates the electron as a negative charge carrier, and the p subscript
indicates the hole as a positive charge carrier.
Material
mn*/m0
mp*/m0
Si
1.18
0.81
Ge
0.55
0.36
GaAs
0.066
0.52
© Nezih Pala npala@fiu.edu
m0–mass of electron in rest
m0= 9.10938188×10-31kg
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Direct and indirect semiconductors –1
•The real band structure in 3D is calculated with various numerical
methods, plotted as E vs k. Where k is called wave vector.
•p(k) = ħk, where p(k) is momentum.
•For electron transition, both E and p(k) must be conserved.
•A semiconductor is direct if the maximum of the conduction band and
the minimum of the valence band has the same k value.
•A semiconductor is indirect if the maximum of the conduction band and
the minimum of the valence do not have the same k value.
•Direct semiconductors are suitable for making light-emitting devices,
whereas the indirect semiconductors are not.
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Direct and indirect semiconductors –2
A typical bandstructure of a semiconductor near the top of the valence band is shown in the
next slide. We notice the presence of three bands near the valence bandedge. These curves
or bands are labeled I, II, and III in the figure and are called the heavy hole (HH), light hole
(LH), and the split off hole bands.
The bottom of the conduction band in some semiconductors occurs at k = 0. Such
semiconductors are called direct bandgap materials. Semiconductors, such as GaAs, InP, GaN,
InN, etc., are direct bandgap semiconductors. In other semiconductors, the bottom of the
conduction band does not occur at the k = 0 point, but at certain other points. Such
semiconductors are called indirect semiconductors. Examples are Si, Ge, AlAs, etc.
Due to the law of momentum conservation, direct gap materials have a strong interaction
with light. Indirect gap materials have a relatively weak interaction with electrons.
When the bandedges are at k = 0 it is possible to represent the bandstructure by a simple
relation of the form
2 2
E (k )  EC 
 k
2m *
where Ec is the conduction band edge, and the band structure is a simple parabola. The
equation for the E–k relation looks very much like that of an electron in free space as noted
previously.
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Direct and indirect semiconductors –3
 2k 2
E (k )  EC 
2m *
A semiconductor is direct if the maximum of
the valence band and the minimum of the
conduction band do have the same k value.
© Nezih Pala npala@fiu.edu
A semiconductor is indirect if the maximum of
the valence band and the minimum of the
conduction band do not have the same k
value.
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Direct and indirect semiconductors –4
Thus an electron making a smallest-energy transition from the conduction band to the
valence band in GaAs(-direct semiconductor) can do so without a change in k value.
On the other hand, a transition from the minimum point in the Si conduction band to the
maximum point of the valence band requires some change in k.
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Direct and indirect semiconductors –5
Silicon
The most important semiconductor is silicon. Silicon has an indirect bandgap as shown in
the figure. The bottom of the conduction band in Si is at point (∼ (2π/a)(0.85, 0.0); i.e., close
to the X-point. There are six degenerate X-points and, consequently, six conduction band
edge valleys. The near band edge band structure can be represented by ellipsoids of energy
with simple E vs. k relations of the form (for examples for the [100] valley)
2
2
2
 2 k x2  (k y  k y )
E (k ) 

2ml *
2mt *
where we have two masses, the longitudinal and
transverse. The six constant energy surfaces of Si
are ellipsoids and shown in the figure.
The next valley in the conduction band is the Lpoint valley, which is about 1.1 eV above the
bandedge. Above this is the Γ-point edge. Due to
the six-fold degeneracy of the conduction
bandedge, the electron transport in Si is quite
poor because of the very large density of states
near the bandedge, leading to a high scattering
rate in transport.
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Direct and indirect semiconductors –6
GaAs
GaAs is a direct gap material with small electron effective mass. The near bandedge
bandstructure of GaAs is shown in the figure. The bandstructure can be represented by the
relation (referenced to Ec)
2 2
E (k ) 
 k
2m *
with m∗ = 0.067m0. A better relationship is the non-parabolic approximation
E (1  E ) 
 2k 2
with α = 0.67 eV−1.
2m *
For high electric field transport, it is important to note that the valleys
above Γ-point are the L-valleys. There are eight L-points, but, since half of
them are connected by a reciprocal lattice vector, there are four valleys.
The separation ΔEΓL between the Γ- and L- minima is 0.29 eV.
The L-valley has a much larger effective mass than the Γ-valley. For GaAs,
m∗L ∼ 0.25m0. This difference in masses is extremely important for high
electric field transport as will be discussed later.
The valence band of GaAs has the standard HH, LH, and SO bands. Due to
the large spin–orbit splitting, for most purposes the SO band does not
play any role in electronic properties.
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Direct and indirect semiconductors –5
E vs. k diagram for Si and GaAs.
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Direct and indirect semiconductors –6
InN, GaN, and AlN
The III–V nitride family of GaN, InN, and AlN have become quite important due to progress
in the ability to grow the semiconductor. These materials are typically grown with a wurtzite
structure, and have band gaps ranging from ∼1.0 eV to over 6.0 eV. This large bangap is very
useful for short wavelength light emitters and high power electronics. In the figures we show
the band structure for nitrides.
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Metals, semiconductors, and insulators –1
Every solid has its own characteristic energy band structure.
This variation in band structure is responsible for the wide range of
electrical characteristics observed in various materials.
Current flow in solids can occur only if:
1 An electric field is applied
AND
There are empty energy states
2
. available to the electrons to move to
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Metals, semiconductors, and insulators –2
For example:
There can be no charge transport within the valence band of a
semiconductor at T= 0 K, since no empty states are available into
which electrons can move. There are no electrons in the
conduction band, so no charge transport can take place there
either.
Semiconductor materials at 0 K have basically the same structure
as insulators –a filled valence band separated from an empty
conduction band by a band gap containing no allowed energy
states.
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Metals, semiconductors, and insulators –3
The difference in electrical properties of solids lies in the size of
the band gap Eg
Insulator
An insulator typically has
Eg≈10 eV
Diamond: Eg= 5 eV
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Semiconductor
The semiconductor Si has a
band gap Eg= 1.1 eV
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Metal
The bands in metals either
overlap or are only partially
filled
106
Metals, semiconductors, and insulators –4
•Metals have free electrons and partially filled valence bands, therefore
they are highly conductive.
•Semimetals have their highest band filled. This filled band, however,
overlaps with the next higher band, therefore they are conductive but with
slightly higher resistivity than normal metals. Examples: arsenic, bismuth,
and antimony.
•Insulators have filled valence bands and empty conduction bands,
separated by a large band gap Eg (typically > 4 eV), they have high
resistivity.
•Semiconductors have similar band structure as insulators but with a much
smaller band gap. Some electrons can jump to the empty conduction band
by thermal or optical excitation. Eg=1.1 eV for Si, 0.67 eV for Ge and 1.43
eV for GaAs
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Metals, semiconductors, and insulators –5
Band gap energy Eg of semiconductor materials:
Semiconductor
Bandgap Energy Eg (eV)
Carbon (diamond)
5.47
Silicon
1.12
Germanium
0.66
Tin
0.082
Gallium arsenide
1.42
Gallium nitride
3.49
Indium phosphide
1.35
Boron nitride
7.50
Silicon carbide
3.26
Cadmium selenide
1.70
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Electrons and holes –1
As the temperature of a semiconductor
is raised from 0 K or when energy is
transferred from a photon to an
electron in the valence band, some
electrons can receive enough energy
to be excited across the band gap to
the conduction band.
An empty state in the valence band is
referred to as a hole. If the conduction
band electron and the hole are created
by the excitation of a valence band
electron to the conduction band, they
are called an electron-hole pair
(abbreviated EHP).
After excitation to the conduction band, an electron may occupy any of a large number
of empty energy states.
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Electrons and holes –2
Example:
Typical equilibrium number of electron-hole pairs in pure Si at room temperature
(T≈300 K) is about
1010 EHP/cm3
while the Si atom density is
Compare
5×1022 atoms/cm3
Thus the few electrons in the conduction band are free to move about via the many
available empty states.
1010 EHP/cm3 << 5×1022 atoms/cm3
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Electrons and holes –3
In a filled valence band, all available energy states are occupied.
If we even apply an electric field, the net current is zero.
Let’s suppose that there are N electrons/cm3 in the filled valence band. Then, the
current density in a unit volume can be written as
N
J ~ q vi  0
i 1
where (–q) is a charge of an electron; vi–velocity of the i-th electron.
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Electrons and holes –4
Let’s remove now one j-th electron from the valence band.
That creates a hole in it.
The net current density in the valence band is now the sum over all velocities, minus the
contribution of the j-th electron we have removed:
N
J ~ (q) vi  (q)v j
1
i 

0
This is the current density in the valence band when j-th electron is missing.
It is equivalent to that of a positively charged particle with velocity vj, created by the
removed electron: +qvj
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Electrons and holes –5
Of course, the charge transport is actually due to the motion of the
new uncompensated electron (j'). Its current contribution (-q)(-vj) is
equivalent to that of a positively charged particle with velocity +vj.
So, we may treat empty states in the valence band as
charge carriers (holes) with positive charge and positive
mass.
In all the following discussions we shall concentrate on the electrons in
the conduction band and on the holes in the valence band.
The current flow in a semiconductor can be described by the motion of
these two types of charge carriers: electrons and holes.
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Electrons and holes –6
This movement of electrons from right to left is equivalent to travel of the hole  from
left to right.
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Intrinsic materials –1
A perfect semiconductor crystal with no impurities or lattice defects is called an intrinsic
semiconductor.
In such material there are no free charge carriers at T= 0 K, since the valence band is filled
with electrons and the conduction band is empty. At higher temperatures electron-hole
pairs are generated as valence band electrons are excited thermally across the band gap
to the conduction band.
These EHPs are the only charge carriers in intrinsic material.
If one of the Si valence electrons is broken away from its position in the bonding structure
such that it becomes free to move about in the lattice, a conduction electron is created
and a broken bond (hole) is left behind.
The energy required to break the bond is the band gap energy Eg.
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Intrinsic materials –2
Since the electrons and holes are created in pairs, the conduction band electron
concentration n (electrons per cm3) is equal to the concentration of holes in the valence
band p (holes per cm3). Thus for intrinsic material
n= p= ni
where ni is concentration of EHPs in intrinsic material or intrinsic concentration. ni
depends on temperature (!)
Obviously, if a steady state carrier concentration is maintained, there must be
recombination of EHPs at the same rate at which they are generated.
Recombination occurs when an electron in the conduction band makes a transition
(direct or indirect) to an empty state (hole) in the valence band, thus annihilating the
pair.
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Intrinsic materials –3
Band diagram for intrinsic semiconductor
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Extrinsic materials –1
In addition to the intrinsic carriers generated thermally, it is possible to create carriers in
semiconductors by purposely (controllably) introducing impurities into the crystal.
This process is called doping.
By doping, a crystal can be altered so that it has a predominance of either electrons or
holes. There are two types of doped semiconductors:
•p-type (mostly holes)
•n-type (mostly electrons)
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Extrinsic materials –2
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.
Donors
Acceptors
Dopants increasing electron concentration
Dopants increasing electron concentration
P (phosphorus)
B (Boron)
As (Arsenic)
Ga (Gallium)
Sb (Antimony)
In (Indium)
Al (Aluminum)
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Extrinsic materials –3
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Extrinsic materials –4
When impurities or lattice defects are introduced into an otherwise perfect crystal,
additional levels are created in the energy band structure, usually within the band gap.
n-type semiconductors
An impurity from V-column of the periodic table (P, As, and Sb) introduces an energy level
very near the conduction band in Si or Ge. This level is filled with electrons at T= 0 K, and
very little thermal energy is required to excite these electrons to the conduction band.
At T about 50–100 K virtually all of the electrons in the impurity level are "donated" to the
conduction band. Such an impurity level is called a donor level. Thus semiconductors
doped with a significant number of donor atoms will have n0>> ni or p0 at room
temperature. This is n-type material.
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Extrinsic materials –5
n-type
Semiconductor Si (Z= 14): 1s2 2s2 2p6 3s2 3p2
Dopant (donor) As (Z= 33): 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p3
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Extrinsic materials –6
Band diagram for n-type semiconductor
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Extrinsic materials –7
Atoms from III-column (B, Al, Ga, and In) introduce impurity levels in Ge or Si near the
valence band. These levels are empty of electrons at 0 K.
At low temperatures, enough thermal energy is available to excite electrons from the
valence band into the impurity level, leaving behind holes in the valence band.
Since this type of impurity level "accepts" electrons from the valence band, it is called an
acceptor level, and the column III impurities are acceptor impurities in Ge and Si.
Doping with acceptor impurities can create a semiconductor with a hole concentration
p0much greater than the conduction band electron concentration n0 or ni( this type is ptype material)
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Extrinsic materials –8
p-type
Semiconductor Si (Z= 14): 1s2 2s2 2p6 3s2 3p2
Dopant (acceptor) B (Z= 5): 1s2 2s2 2p1
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Extrinsic materials –9
Band diagram for p-type semiconductor
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Extrinsic materials –10
Example:
Calculate the approximate energy required to excite the extra electron of As donor atom
into the conduction band of Si (the donor binding energy).
Solution:
Let’s assume for rough calculations that the As (1s2 2s2 2p2 3s2 3p6 3d10 4s2 4p3) atom has
its four covalent bonding electrons rather tightly bound and the fifth “extra” electron
loosely bound to the atom.
We can approximate this situation by using the Bohr model results, considering the loosely
bound electron as ranging about the tightly bound "core" electrons in a hydrogen-like orbit.
We have to find energy necessary to remove that “extra” electron from As atom.
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Extrinsic materials –11
We can approximate As dopant atom in Si lattice by
using the Bohr model:
the loosely bound “fifth” electron is ranging about
the tightly bound "core" electrons in a hydrogen-like
orbit.
The magnitude of the ground-state energy (n= 1) of
such an electron in the Bohr model is
mq 4
E1 
2K 2 2
Constant K in this case is K=4πεrε0 where εr is
relative dielectric constant of Si
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Approximation of As dopant
atom in Si lattice.
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Extrinsic materials –12
Besides relative dielectric constant of Si, we have to use conductivity effective mass of
electron mn* in Si in the formula for energy E:
mn* q 4
mn* q 4
E

2 2
2(4 0 r ) 
8( 0 r h) 2
1.18(9.1110 31)(1.6 10 19 ) 4
E
8(8.85 10 12 11.8  6.63 10 34 ) 2
1.18  9.11 (1.6) 4  (10 107 )

8  (8.85) 2  (11.8) 2  (6.63) 2 10 92
70.45 10 107
 20


1
.
837

10
(J)
6
92
3.835 10 10
1eV  1.6 10 19 J  E  0.1 eV
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Extrinsic materials –13
Generally, the column-V donor levels lie approximately 0.01 eV below the conduction
band in Ge, and the column-III acceptor levels lie about 0.01 eV above the valence band.
In Si the usual donor and acceptor levels lie about 0.03-0.06 eV from a band edge.
When a semiconductor is doped n-type or p-type, one type of carrier dominates.
For example, when we introduce donors, the number of electrons in conduction band is
much higher than number of the holes in the valence band.
In n-type material:
holes –minority carriers
electrons –majority carriers
© Nezih Pala npala@fiu.edu
In p-type material:
holes –majority carriers
electrons –minority carriers
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