CHAPTER 2: ENERGY BANDS &
CARRIER CONCENTRATION IN
THERMAL EQUILIBRIUM
© M.N.A. Halif & S.N. Sabki
OUTLINE
2.1 INTRODUCTION:
2.1.1 Semiconductor Materials
2.1.2 Basic Crystal Structure
2.1.3 Basic Crystal Growth technique
2.1.4 Valence Bonds
2.1.5 Energy Bands
2.2 Intrinsic Carrier Concentration
2.3 Donors & Acceptors
© M.N.A. Halif & S.N. Sabki
2.1.1 SEMICONDUCTOR MATERIALS
-
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
Refer to your basic electronic devices knowledge (EMT
111/4).
Electrical conductivity σ: Æ ρ=1/σ.
© M.N.A. Halif & S.N. Sabki
Figure 2.1. Typical range of conductivities for
insulators, semiconductors, and conductors.
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SEMICONDUCTOR’S ELEMENTS
The study of semiconductor materials – 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.
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COMPOUND SEMICONDUCTORS
Please refer to the Table 2 (in Sze, pg. 20)
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 , so-called
combination of many binary & ternary compounds.
- more complex processes.
GaAs – high speed electronic & photonic applications
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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 r r
R = ma + nb + pc
© M.N.A. Halif & S.N. Sabki
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.
© M.N.A. Halif & S.N. Sabki
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 Fig. 2.4 (b) structure.
a/2
a/2
Fig. 2.4. (a) Diamond lattice. (b) Zincblende lattice.
© M.N.A. Halif & S.N. Sabki
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 (khl) as the Miller indices for a single plane.
© M.N.A. Halif & S.N. Sabki
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).
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VALENCE BONDS
• Recall your basic electronic devices
EMT 111/4.
• 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.
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ENERGY BANDS
Neils Bohr, 1885-1962
Nobel Prize in physics 1922
• Energy levels for an isolated hydrogen atom are given by the Bohr model*:
m0q 4
13.6
EH = − 2 2 2 = − 2 eV
n
8ε 0 h n
(2)
1 eV = 1.6 x 10-19 J
mo – free electron mass (0.91094 x 10-30kg)
q – electronic charges: 1.6 x 10-19 C
ε0 – 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 @ KUKUM Library!!
• For higher principle quantum number (n ≥ 2), energy levels are split.
© M.N.A. Halif & S.N. Sabki
ENERGY BANDS (cont.)
For two identical atoms:
When they far apart – have same energy.
When they are bought 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.
- this interactions cause a shift in the energy levels (case of two interacting
atoms).
- when N>>>, Æ an essentially continuous band of energy. This band of N
level may extend over a few eV depending on the inter-atomic spacing of
the crystal.
© M.N.A. Halif & S.N. Sabki
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.
© M.N.A. Halif & S.N. Sabki
ENERGY BANDS (cont.)
• Fig. 2.15: schematic diagram of the
formation of a Si crystal from N
isolated Si atoms.
• Inter-atomic distance decreases,
3s and 3p sub shell of N Si atoms will
interact and overlap.
• At equilibrium state, the bands will
split again:
4 quantum states/atom (valence
band)
4 quantum states/atom (cond. band)
Figure 2.15. Formation of energy
bands as a diamond lattice crystal
is formed by bringing isolated
silicon atoms together.
© M.N.A. Halif & S.N. Sabki
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.
Bandgap energy Eg = (EC – EV).
Physically, Eg defined as ‘the energy required to break a bond in the
semiconductor to free an electron to the cond. 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!!!
Bandgap is one of the factors that affect the devices performance.
© M.N.A. Halif & S.N. Sabki
Energy & Momentum
Please recall back your memory in the basic of fundamental
PHYSICS!!
What is the definition of an Energy in PHYSICS???
What is the definition of Momentum in PHYSICS???
© M.N.A. Halif & S.N. Sabki
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
Figure 2.16. The parabolic
energy (E) vs. momentum (p)
curve for a free electron.
• for holes, mn = mp
© M.N.A. Halif & S.N. Sabki
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 bandgap 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.
© M.N.A. Halif & S.N. Sabki
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.
bandgap
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.
© M.N.A. Halif & S.N. Sabki
GaAs - very narrow conduction-band parabola, using (8), we may expected
that GaAs – smaller effective mass (mn = 0.063m0).
Si – wider conduction band parabola, using (8), effective mass mn = 0.19m0.
The different between direct and indirect band structures is very important
for applications in LED and semiconductor laser. This devices require the
types of band structure for efficient generation of photons (you may learn
this applications in Chapter 9 – PHOTONIC DEVICES).
© M.N.A. Halif & S.N. Sabki
CONDUCTION
Metals/Conductors
Semiconductors
Insulators
(Recall you basic knowledge in EMT 111)
Can you imagine their behaviors???
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CONDUCTION
Metals/Conductors
Very low resistivity.
Cond. band either is partially filled (i.e Cu) or overlaps in valence band (i.e
Zn, Pb).
No bandgap. 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 bandgap. Electrical conductivity very small – very high resistivity.
Semiconductors
Much smaller bandgap ~ 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.
© M.N.A. Halif & S.N. Sabki
(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.
© M.N.A. Halif & S.N. Sabki
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 relativity small amounts of
impurities compared with thermally generated electrons and holes.
Electron density, n – number electrons per unit volume.
To obtained this in intrinsic s/c – evaluated the electron density in an
incremental energy range dE.
Thus, n = integrating density of state, N(E), energy range, F(E), and
dE from bottom of the cond. band, EC = E = 0 to the top of the cond.
band Etop.
© M.N.A. Halif & S.N. Sabki
• We may write n as
n=
Etop
Etop
0
0
(5)
∫ n(E)dE = ∫ N(E)F(E)dE
• Fermi-Dirac distribution function
1
(6)
F (E) =
1 + exp( E − E F ) / kT
n is in cm-3, and N(E) is in (cm3 .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)
© M.N.A. Halif & S.N. Sabki
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”
© M.N.A. Halif & S.N. Sabki
(7)
For energy, 3kT above or below Fermi energy, then
F (E ) ≅ e
− ( E − E F ) / kT
F (E ) ≅ 1 − e
for ( E − E F ) > 3kT
− ( E − E F ) / kT
(8)
for ( E − E F ) < 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.
© M.N.A. Halif & S.N. Sabki
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 bandgap.
• Upper-shaded in (d) corresponds to the electron density.
© M.N.A. Halif & S.N. Sabki
From Appendix H in Sze, pg. 540, density of state, N(E) is
defined as
⎛ 2m ⎞
N ( E ) = 4π ⎜ 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 = N C exp [( E F − E C ) / kT ], and p = N V exp [( E V − E F ) / kT
⎛ 2π m n kT ⎞
NC = X ⎜
⎟
2
⎝ h
⎠
3/ 2
NV
⎛ 2 π m p kT ⎞
⎟⎟
= 2 ⎜⎜
2
h
⎝
⎠
]
3/2
(10)
X = 12 for Si, and X = 2 for GaAs
• NC and NV effective density of state in cond. band & valence band respectively.
• At room temperature, T = 300K;
Effetive 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
© M.N.A. Halif & S.N. Sabki
• 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.
© M.N.A. Halif & S.N. Sabki
2.3 DONORS & ACCEPTORS
Recall your basic knowledge in EMT 111 – Electronic Devices.
When semiconductor is doped with impurities – become extrinsic
and impurity energy levels are introduced.
Donor : n-type
Acceptor : p-type
Ionization energy:
⎛ ε0
E D = ⎜⎜
⎝εS
⎞
⎟⎟
⎠
2
⎛ mn
⎜⎜
⎝ m0
⎞
⎟⎟ E H
⎠
(12)
• ε0 – permittivity in vacuum ~ 8.85418 x 10-14 F/cm
• εS – semiconductor permittivity, and EH ~ Bohr’s energy model
© M.N.A. Halif & S.N. Sabki
Figure 2.23. Schematic bond pictures for (a) n-type Si with donor (arsenic) and (b) ptype Si with acceptor (boron).
© M.N.A. Halif & S.N. Sabki
Figure 2.24. Measured ionization energies (in eV) for various impurities in Si and GaAs.
The levels below the gap center are measured from the top of the valence band and are
acceptor levels unless indicated by D for donor level. The levels above the gap center are
measured from the bottom of the conduction band and are donor levels unless indicated
by A for acceptor level.
© M.N.A. Halif & S.N. Sabki
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 non-degenerate s/c.
Complete ionization – cond. 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 cond.
band.
Under complete ionization cond., electron density may be written as
(13)
n = ND
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 − E F = kT ln( N C / N D )
E F − EV = kT ln( NV / N A )
* Higher donor/acceptor concentration – smaller ∆E.
© M.N.A. Halif & S.N. Sabki
Non-degenerate Semiconductor
Figure 2.25. Schematic energy band representation of extrinsic semiconductors
with (a) donor ions and (b) acceptor ions.
© M.N.A. Halif & S.N. Sabki
NON-DEGENERATE SEMICONDUCTOR
Much closer to cond. 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???
© M.N.A. Halif & S.N. Sabki
NON-DEGENERATE SEMICONDUCTOR
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 cond. 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.
© M.N.A. Halif & S.N. Sabki
NON-DEGENERATE SEMICONDUCTOR
• The impurity that is present is 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.
© M.N.A. Halif & S.N. Sabki
EXAMPLE
A Si ingot is doped with 1016 arsenic atom/cm3.
(a) Find carrier concentration and the Fermi level at room
temperature (T = 300K).
(b) Eg = 1.12eV (Si). Sketch the level of energy.
© M.N.A. Halif & S.N. Sabki
Real fab.
> 1015
Figure 2.28. Fermi level for Si and GaAs as a function of temperature and impurity
concentration. The dependence of the bandgap on temperature is shown.
© M.N.A. Halif & S.N. Sabki
Figure 2.29. Electron density as a function of temperature for a Si sample
with a donor concentration of 1015 cm-3.
© M.N.A. Halif & S.N. Sabki
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 ~ bandgap narrowing effect (reduced the
(20)
bandgap), and it given by (at T=300K);
N
∆Eg = 22
meV
18
10
© M.N.A. Halif & S.N. Sabki
CONCLUSION REMARKS
The properties of s/c are determined to a large extend by the crystal
structure.
Miller indices 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 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.
© M.N.A. Halif & S.N. Sabki
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
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Next Lecture:
Carrier Transport Phenomena
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