CHAPTER 2: ENERGY BANDS &
CARRIER CONCENTRATION IN
THERMAL EQUILIBRIUM
2.1.1 SEMICONDUCTOR
MATERIALS
The goal is:
 To understand the characteristic of
semiconductor materials – you need PHYSICS.
 Basic Solid-State Physics – materials, may be
grouped into 3 main classes:
(i) Insulators,
(ii) Semiconductors, and
(iii) Conductors
- Electrical conductivity :  =1/.
Figure 2.1. Typical range of conductivities for
insulators, semiconductors, and conductors.
SEMICONDUCTOR’S ELEMENTS


The study of semiconductor materials since 19th
century.
Early 1950s – Ge was the major semiconductor
material, and later in early 1960s, Si has become a
practical substitute with several advantages:
(i) better properties at room temperature,
(ii) can be grown thermally – high quality silicon
oxide,
(iii) Lower cost, and
(iv) Easy to get, silica & silicates comprises 25% of
the Earth’s crust.
COMPOUND SEMICONDUCTORS


Types of compounds:
(i) binary compounds
- combination of two elements.
- i.e GaAs is a III – IV.
(ii) ternary and quaternary compounds
- for special applications purposes.
- ternary compounds, i.e alloy semiconductor AlxGa1-xAs
(III – IV).
- quaternary compounds with the form of AxB1-xCyD1-y , socalled combination of many binary & ternary compounds.
- more complex processes.
GaAs – high speed electronic & photonic applications
2.1.2 BASIC CRYSTAL STRUCTURE


Lattice – the periodic
arrangement of atoms in a
crystal.
Unit Cell – represent the entire
lattice (by repeating unit cell
throughout the crystal)
The Unit Cell
 3-D unit cell shown in Fig. 2.2.
 Relationship between this cell &
the lattice – three vectors: a, b,
and c (not be perpendicular to
each other, not be equal in
length).
 Equivalent lattice point in 3-D:
 m, n, and p - integers.
Fig. 2.2. A generalized
primitive unit cell.
(1)
  
R  ma  nb  pc
UNIT CELL
Large number
of elements
Figure 2.3. Three cubic-crystal unit cells. (a) Simple cubic. (b)
Body-centered cubic. (c) Face-centered cubic.
• simple cubic (sc) – atom at each corner of the cubic lattice, each atoms
has 6 equidistant nearest-neighbor atoms.
• body-centered cubic (bcc) – 8 corner atoms, an atom is located at center
of the cube.
- each atom has 8 nearest-neighbor atoms.
• face-centered cubic (fcc) – 1 atom at each of the 6 cubic faces in addition
to the 8 corner atoms. 12 nearest-neighbor atoms.
THE DIAMOND STRUCTURE
• Si and Ge have a diamond lattice structure shown in Fig. 2.4.
• Fig. 2.4(a) – a corner atom has 1 nearest neighbor in the body diagonal
direction, no neighbor in the reverse direction.
• Most of III-IV compound semiconductor (e.g GaAs) have zincblende lattice
structure.
a/2
a/2
Fig. 2.4. (a) Diamond lattice. (b) Zincblende lattice.
CRYSTAL PLANES & MILLER INDICES
Figure 2.6. Miller indices of some important planes in a cubic crystal.
• Crystal properties along different planes are different – electrical & other devices
characteristics can be dependent on the crystal orientation – use Miller indices.
• Miller indices are obtained using the following steps:
(i) find the interfaces of the plane on the 3 Cartesian coordinates in term of the
lattice constant.
(ii) Take the reciprocal of these numbers and reduce them to smallest 3 integers
having the same ratio.
(iii) Enclose the result in parentheses (hkl) as the Miller indices for a single plane.
BASIC CRYSTAL GROWTH TECHNIQUE
Prof. J. Czochralski,
1885-1953
• Refer to EMT 261.
• 95% electronic industry used Si.
• Steps from SiO2 or quartzite.
• Most common method called Czochralski
technique (CZ).
• Melting point of Si = 1412ºC.
• Choose the suitable orientation <111>
for seed crystal.
Si ingot
Figure 2.8.
Simplified schematic drawing of the Czochralski
puller. Clockwise (CW), counterclockwise (CCW).
VALENCE BONDS
• Fig. 2.11(a) – each atom has 4 ē in
the outer orbit, and share these
valence ē with 4 neighbors.
• Sharing of ē called covalent bonding –
occurred between atoms of same and
different elements respectively.
Figure 2.11. (a) A tetrahedron bond. (b)
Schematic two-dimensional representation of
a tetrahedron bond.
Example:
• GaAs – small ionic contribution that is
an electrostatic interactive forces between
each Ga+ ions and its 4 neighboring As- ions
- means that the paired bonding ē spend more
time in the As atom than in the Ga atom.
Figure 2.12. The basic bond representation
of intrinsic silicon. (a) A broken bond at
Position A, resulting in a conduction electron
and a hole. (b) A broken bond at position B.
ENERGY BANDS
Neils Bohr, 1885-1962
Nobel Prize in physics 1922
• Energy levels for an isolated hydrogen atom are given by the Bohr model*:
m0 q 4
13.6
EH   2 2 2   2 eV
8 0 h n
n
(2)
mo – free electron mass (0.91094 x 10-30kg)
1 eV = 1.6 x 10-19 J
q – electronic charges: 1.6 x 10-19 C
0 – free space permittivity (8.85418 x 10-12 F/m)
h – Planck constant (6.62607 x 10-34 J.s)
n – positive integer called principle quantum number
• For 1st energy state or ground state energy level, n = 1, EH = -13.6eV.
• For the 1st excited energy level, n = 2.
* Find Fundamentals Physics books @ UniMAP Library!!
• For higher principle quantum number (n ≥ 2), energy levels are split.
ENERGY BANDS (cont.)
For two identical atoms:


When they far apart – have same energy.
When they are brought closer:
– split into two energy levels by interaction between the atoms.
- as N isolated atoms to form a solid.
- the orbit of each outer electrons of different atoms overlap &
interact with each other.
- these interactions cause a shift in the energy levels (case of two
interacting atoms).
- when N>>>,  an essentially continuous band of energy. This
band of N level may extend over a few eV depending on the interatomic spacing of the crystal.
ENERGY BANDS (cont.)
Equilibrium inter-atomic
distance of the crystal.
Figure 2.13. The splitting of a
degenerate state into a band of allowed
energies.
Figure 2.14. Schematic presentation of
an isolated silicon atom.
ENERGY BANDS (cont.)




At T = 0K, electrons occupy the lowest energy states:
Thus, all states in the valence band (lower band) will be full, and all
states in the cond. band (upper band) will be empty.
The bottom of cond. band is called EC, and the top of valence band
called EV.
Band gap energy Eg = (EC – EV).
Physically, Eg defined as ‘the energy required to break a bond in
the semiconductor to free an electron to the conduction band
and leave the hole in the valence band’.
(Please remember this important definition). This is one of a
hot-issue in scientific research in the world!!!
The Energy-Momentum Diagram
• The energy of free-electron is given by
p2
E
2m0
(3)
p – momentum, m0 – free-electron mass
• effective mass = mn
d E
mn   2 
 dp 
2
1
(4)
• for the narrower parabola (correspond
to the larger 2nd derivative) – smaller mn
• for holes, mn = mp
Figure 2.16. The parabolic
energy (E) vs. momentum (p)
curve for a free electron.
The Energy-Momentum Diagram
• Fig. at the RHS shows the simplified energymomentum of a special semiconductor with:
- electron effective mass mn = 0.25m0 in cond.
band.
- hole effective mass mp = m0 in the valence
band.
• electron energy is measured upward
• hole energy is measured downward
• the spacing between p = 0 and minimum of
upper parabola is called band gap Eg.
• For the actual case,
i.e Si and GaAs – more complex.
Figure 2.17.
A schematic energy-momentum diagram
for a special semiconductor with mn =
0.25m0 and mp = m0.
GaAs
Si
• Fig. 2.18 is similar to Fig. 2.17.
• For Si, max in the valence band
occurs at p = 0, but min of cond.
band occurs at p = pc (along [100]
direction).
• In Si, when electron makes
transition from max point (valence
band) to min point (cond. band) it
required:
Energy change (≥Eg) + momentum
change (≥pc).
• In GaAs: without a change in
momentum.
• Si – indirect semiconductor.
• GaAs – direct semiconductor.
Band gap
Figure 2.18. Energy band structures of Si and
GaAs. Circles (º) indicate holes in the valence
bands and dots (•) indicate electrons in the
conduction bands.
CONDUCTION
Metals/Conductors
 Very low resistivity.
 Conduction band either is partially filled (i.e Cu) or overlaps in valence
band (i.e Zn, Pb).
 No band gap. Electron are free to move with only a small applied field.
Current conduction can readily occur in conductors.
Insulators
 Valence electrons form strong bonds between neighboring atoms (i.e
SiO2).
 No free electrons to participate in current conduction near room
temperature.
 Large band gap. Electrical conductivity very small – very high
resistivity.
Semiconductors
 Much smaller band gap ~ 1eV.
 At T=0K, no electrons in cond. band. Poor conductors at low
temperatures.
 Eg = 1.12eV (Si) and Eg=1.42eV (GaAs) – at room temp. & normal
atmosphere.
(b)
(b)
(c)
(a)
Figure 2.19. Schematic energy band representations of (a) a conductor with two
possibilities (either the partially filled conduction band shown at the upper portion or the
overlapping bands shown at the lower portion), (b) a semiconductor, and (c) an
insulator.
2.2 INTRINSIC CARRIER
CONCENTRATION





In thermal equilibrium:
- at steady-state condition (at given temp. without any external
energy, i.e light, pressure or electric field).
Intrinsic semiconductor – contains relatively small amounts of
impurities compared with thermally generated electrons and
holes.
Electron density, n – number electrons per unit volume.
To obtain electron density in intrinsic s/c – evaluate the electron
density in an incremental energy range dE.
Thus, n is given by integrating density of state, N(E), energy
range, F(E), and incremental energy range, dE, from bottom of
the cond. band, EC = E = 0 to the top of the cond. band Etop.
• We may write n as
Etop
n
Etop
 n( E )dE   N ( E ) F ( E )dE
0
(5)
0
• Fermi-Dirac distribution function
(probability that ē occupies an electronic
state with energy E)
1
F (E) 
1  exp( E  EF ) / kT
n is in
cm-3,
and N(E) is in
(cm3
(6)
.eV)-1
Figure 2.20. Fermi distribution function
F(E) versus (E – EF) for various
temperatures.
k Boltzmann constant ~ 1.38066 x 10-23 J/K, and T in Kelvin.
EF – energy of Fermi level (is the energy at which the probability of occupation
by electron is exactly one-half)
Enrico Fermi (1901 – 1954)
Nobel Prize in Physics 1938
Paul Adrien Maurice Dirac (1902 – 1984)
Nobel Prize in Physics 1933
“There are two possible outcomes: If the result
confirms the hypothesis, then you've made a
measurement. If the result is contrary to the
hypothesis, then you've made a discovery”
“Mathematics is the tool specially
suited for dealing with abstract
concepts of any kind and there
is no limit to its power in this field”

For energy, 3KT above or below Fermi energy, then
F (E)  e
 ( E  E F ) / kT
F (E)  1  e
for ( E  EF )  3kT
 ( E  E F ) / kT
(7)
for ( E  EF )  3kT
• > 3KT the exponential term larger than 20, and < 3kT – smaller than 0.05.
• At (E – EF) < 3KT – the probability that a hole occupies a state located at
energy E.
(8)
Figure 2.21. Intrinsic semiconductor. (a) Schematic band diagram. (b) Density of states.
(c) Fermi distribution function. (d) Carrier concentration.
• N(E) = (E)1/2 for a given electron effective mass.
• EF located at the middle of band gap.
• Upper-shaded in (d) corresponds to the electron density.

Density of state, N(E) is defined as
 2m 
N ( E )  4  2 n 
 h 
3/ 2
E
(9)
h – Planck constant ~ 6.62607 x 10-34 J.s
• substituting (9) and (8) into (5), thus
n  NC exp ( EF  EC ) / kT , and p  NV exp ( EV  EF ) / kT 
 2mn kT 
NC  X 

2
 h

3/ 2
 2m p kT 

NV  2
2
h


3/ 2
(10)
X  12 for Si, and X  2 for GaAs
• NC and NV effective density of state in conduction band & valence band
respectively. • At room temperature, T = 300K;
Effective density
Si
GaAs
NC
2.86 x 1019 cm-3
4.7 x 1017 cm-3
NV
2.66 x 1019 cm-3
7.0 x 1018 cm-3
• Intrinsic carrier density is obtained by:
np  ni
2
(mass action law)
ni2  N C NV exp(  E g / kT )
(11)
ni  N C NV exp(  E g / 2kT )
Where Eg = EC - EV
Figure 2.22.
Intrinsic carrier densities in Si and
GaAs as a function of the reciprocal of
temperature.
2.3 DONORS & ACCEPTORS




When semiconductor is doped with impurities – become
extrinsic and impurity energy levels are introduced.
Donor : n-type
Acceptor : p-type
Ionization energy:
 0
E D  
 S



2
 mn

 m0

 E H

(12)
• 0 – permittivity in vacuum ~ 8.85418 x 10-14 F/cm
• S – semiconductor permittivity, and EH ~ Bohr’s energy model
Figure 2.23. Schematic bond pictures for (a) n-type Si with donor (arsenic) and (b) ptype Si with acceptor (boron).
NON-DEGENERATE
SEMICONDUCTOR
In previous section, we assumed that Fermi level EF is at least 3KT
above EV and 3KT below EC. Such semiconductor called nondegenerate s/c.
 Complete ionization – conduction at shallow donors in Si and GaAs
where they have enough thermal energy to supply ED to ionize all
donor impurities at room temperature, thus provide the same
number of electrons in the conduction band.
 Under complete ionization conduction, electron density may be
written as
n = ND
(13)
And for shallow acceptors;
p = NA
(14)
Where ND and NA – donor and acceptor concentration respectively.

From electron and hole density (10) and (13) & (14), thus (15)

EC  EF  kT ln( N C / N D )
EF  EV  kT ln( NV / N A )
* Higher donor/acceptor concentration – smaller ∆E.
(15)
Non-degenerate Semiconductor
Figure 2.25. Schematic energy band representation of extrinsic semiconductors with
(a) donor ions and (b) acceptor ions.
NON-DEGENERATE
SEMICONDUCTOR (cont.)
Much closer to conduction band
Figure 2.26. n-Type semiconductor. (a) Schematic band diagram. (b) Density of
states. (c) Fermi distribution function (d) Carrier concentration. Note that np = ni2.
*** p-type semiconductor???
NON-DEGENERATE
SEMICONDUCTOR (cont.)

To express electron & hole densities in term of intrinsic part
(concentration and Fermi level) – used as a reference level
when discussing extrinsic s/c, thus;
n  ni exp ( EF  Ei ) / kT 
p  ni exp ( Ei  EF ) / kT 
(16)
• In extrinsic s/c, Fermi level moves towards either bottom of conduction band
(n-type) or top of valence band (p-type). It depends on the domination of
types carriers.
• Product of the two types of carriers will remains constant at a given temp.
NON-DEGENERATE
SEMICONDUCTOR (cont.)
• The impurity that is present in greater concentration, thus it may determines
the type of conductivity in the s/c.
• Under complete ionization:
n  N A  p  ND
(17)
Solve (17) with mass action law, thus
n
1
N D  N A  B
2
pn  ni2 / nn
1
p p  N A  N D  B 
2
n p  ni2 / p p
with B  ( N D  N A )  4ni2
(18)
(19)
Subscript of n and p refer to n and p-type.
EXAMPLE
Find the energy relationship for a free electron to
the electron mass.
Bonus for each student solve this example till
23rd of July 2008
DEGENERATE SEMICONDUCTOR



For a very heavy doped n-type and p-type s/c, EF will be above EC
or below EV – this refereed to “degenerate semiconductor”.
Approximation of (7) and (8) are no longer use. Electron density
(5) may solved numerically.
Important aspect of high doping ~ band gap narrowing effect
(reduced the band gap), and it given by (at T=300K);
N
Eg  22
meV
18
10
(20)
CONCLUSION REMARKS






The properties of s/c are determined to a large extend by the
crystal structure.
Miller indices are to describe the crystal surfaces & crystal
orientation.
The bonding of atoms & electron energy-momentum
relationship – connection to the electrical properties of
semiconductor.
Energy band diagram is very important to understand using
physics approach why some materials are good and some are
poor in term of conductor of electric current.
Some external/internal changing of s/c (temperature and
impurities) may drastically vary the conductivity of s/c.
The understanding of physics behind the semiconductor
behaviours is very important for Microelectronic Engineer to
handle the problems as well as to produced a high-speed
devices performance.
Motivation
“I was born not knowing and have had
only a little time to change that here
and there”
Richard P. Feynman (1918-1988)
Nobel Prize in Physics 1965