Unit 2 : SEMICONDUCTOR PHYSICS Intrinsic Semiconductors – Energy band diagram – concept
of hole-direct and indirect semiconductors – Carrier concentration in intrinsic
semiconductors – extrinsic semiconductors – Carrier concentration in N-type & P-type
semiconductors – Carrier transport: Velocity-electric field relations – drift and diffusion
transport – Einstein’s relation – Hall effect and devices.
Contents
1.
Introduction: ................................................................................................................................... 2
2.
Origin of Energy band formation in solids ...................................................................................... 3
3.
Direct and Indirect band gap semiconductors: ............................................................................... 4
4.
concept of hole ............................................................................................................................... 6
5.
Intrinsic Semiconductor .................................................................................................................. 6
6.
Carrier concentration in intrinsic semi-conductors ........................................................................ 7
7.
Electron concentration in intrinsic semiconductor in conduction band (n) ................................... 8
8.
Hole concentration in the valance band of intrinsic semiconductor(p) ....................................... 10
9.
Intrinsic carrier concentration (ni) ................................................................................................ 11
10.
Extrinsic (or) Impure semiconductor ........................................................................................ 12
11.
P- type semiconductor .............................................................................................................. 12
12.
Drift current .............................................................................................................................. 13
13.
Diffusion current ....................................................................................................................... 14
14.
Einstein relation ........................................................................................................................ 15
15.
Hall Effect .................................................................................................................................. 16
16.
Hall Effect in n –type Semiconductor........................................................................................ 16
17.
Hall Effect in p-type Semiconductor ......................................................................................... 17
18.
Hall Coefficient Interms of Hall Voltage .................................................................................... 18
1
Semiconductors:
1.
Introduction:
In this chapter, most of our discussions and examples will be based on Si, the ideas are
applicable to Ge and to the compound semiconductors such as GaAs, InP, and others.
Intrinsic Si means an ideal perfect crystal of Si that has no impurities or crystal defects such
as dislocations and grain boundaries. The crystal thus consists of Si atoms perfectly bonded
to each other in the diamond structure. At temperatures above absolute zero, we know that
the Si atoms in the crystal lattice will be vibrating with a distribution of energies. Even
though the average energy of the vibrations is at most 3kT and incapable of breaking the SiSi bond, a few of the lattice vibrations in certain crystal regions may however be sufficiently
energetic to “rupture” a Si-Si bond. When a Si-Si bond is broken, a “free” electron is created
that can wander around the crystal and also contribute to electrical conduction in the
presence of an applied field. The broken bond has a missing electron that causes this region
to be positively charged. The vacancy left behind by the missing electron in the bonding
orbital is called a hole. An electron in a neighbouring bond can readily tunnel into this
broken bond and fill it, thereby effectively causing the hole to be displaced to the original
position of the tunnelling electron. By electron tunnelling from a neighbouring bond, holes
are therefore also free to wander around the crystal and also contribute to electrical
conduction in the presence of an applied field. In an intrinsic semiconductor, the number of
thermally generated electrons is equal to the number of holes (broken bonds). In an
extrinsic semiconductor, impurities are added to the semiconductor that can contribute
either excess electrons or excess holes. For example, when an impurity such as arsenic (As)
is added to Si, each As atom acts as a donor and contributes a free electron to the crystal.
Since these electrons do not come from broken bonds, the numbers of electrons and holes
are not equal in an extrinsic semiconductor, and the As-doped Si in this example will have
excess electrons. It will be an n-type Si since electrical conduction will be mainly due to the
motion of electrons. It is also possible to obtain a p-type Si crystal in which hole
concentration is in excess of the electron concentration due to, for example, boron doping.
Semiconductors: Semiconductors are the class of materials whose conductivity lies between
good conductors (metals) and insulators. They have a narrow bandgap, because of which a
significant number of thermally excited electrons are available for conduction at room
temperature.
Eg: Si, Ge, As, Ga, etc.
Types of semiconductors:
Intrinsic semiconductors: They are pure semiconductors in which number of electrons are
equal to number of holes and their conductivity is due to only thermal excitation.
Extrinsic semiconductors: They contain very small, intentionally added impurities (dopants),
by which, they possess higher electric conductivity.
Types; p-type and n-type semiconductors.
2
Electrons and hole charge carriers: In an intrinsic semiconductor, for every electron freed from the bond, there will be one hole
created in the crystal. Therefore, the number of conduction electrons is equal to the
number of holes at any given temperature.
In an extrinsic semiconductor, the type of dopant added to the crystal decides whether the
charge carriers are electrons or holes. Further, due to thermal excitation, electron-hole pair
is created by breaking of bonds, which become additional charge carriers. In n-type
semiconductors, electrons are majority carriers and holes minority carriers, In p-type its
vice-versa.
2.
Origin of Energy band formation in solids
The isolated atoms have sharply defined energy levels (quantized), the electrons are
tightly bound and have discrete, sharp energy levels. As the atoms are far apart their
orbitals do not overlap. In particular, if each atom is in its ground state, the electrons in each
atom occupy identical quantum states. When two identical atoms are brought closer, the
outermost orbits of these atoms overlap and interact. When the wave functions of the
electrons of the different atoms begin to overlap considerably, the energy levels split into
two (energy levels spread out into bands of allowed energies). For a solid of N atoms, each
of the energy levels of an atom splits into N levels of energy as in figure (a). The electrons of
different atoms cannot remain in the same state because of Pauli Exclusion Principle (a
particular state can at most accommodate two electrons of opposite spins). Thus, when
atoms are brought together, the levels must split to accommodate electrons in different
states. Though they appear continuous, a band is actually a very large number of closely
spaced discrete levels. The width of this band depends on the degree of overlap of the
electrons of adjacent atoms and is largest for the outermost atomic electrons. In solid, many
atoms are brought together that the split energy levels form a set of energy bands of very
closely spaced levels with forbidden energy gaps between them as in figure b.
The band corresponding to the outermost orbit is called conduction band and the next band
is called valence band. The gap between these two allowed bands is called forbidden energy
gap or bandgap. According to the width of the gap between the bands and band occupation
by electrons all solids can be classified broadly into three groups namely, conductors,
semiconductors and insulators.
Fig a. Splitting of energy levels due to interatomic interaction
3
Fig b. Energy level vs. inter-atomic distance
3.
Direct and Indirect band gap semiconductors:
According to the shape of the band gap as a function of the momentum, semiconductors are
classified as
1. Direct band gap semiconductors
2. Indirect band gap semiconductors.
(1) Direct band gap semiconductors: In direct band gap semiconductors, the electrons at
the bottom of the conduction band and the holes at the top of the valence band on the
either side of the forbidden energy gap have the same value of the crystal momentum. The
direct recombination between the electrons in the conduction band and holes in the
valence band takes place. When the electron hole recombination takes place, the
momentum of electrons and holes, remains the same and band gap energy is emitted as
light. Fig shows the diagram of electron energy versus momentum for direct band gap
semiconductor.
(2) Indirect band gap semiconductor
In Indirect band gap semiconductor, the conduction band minimum energy level and
valence band maximum energy level occur at different values of momentum. When an
electron recombines with a hole, the electron must lose some momentum so that it has the
same momentum corresponding to the energy maximum of the valence band. The
conservation of momentum during, the recombination process requires the emission of a
third particle known as phonon with momentum zK where zSK is the difference in
4
momentum between the minimum energy level in conduction band and maximum energy
level in valence band.
Direct band gap semiconductor
Indirect band gap semiconductor
These are impure or Extrinsic or compound
semiconductors
Examples : InP, GaAs, GaAsP etc
These are pure or intrinsic or elemental
Semiconductors
Examples : Ge, Si
Here an electron from CB can recombine
with a hole in VB directly by emitting light of
photon of energy ‘h𝜗’.
Here an electron from CB cannot recombine
directly with holes in VB. But can recombine
through traps by emitting light without
emission of photon or light.
They are used to fabricate LEDs, Laser
Diodes etc.
They are used to amplify the signals in
electronic devices like rectifiers, transistors,
amplifiers etc.
Life time (recombination rate) of charge
carriers is less.
Emission of light has energy equal to energy
gap Eg=(ℎ𝐶/𝜆) eV
Life time of charge carriers is more.
No emission of light. It conducts only
Electricity.
The minimum energy of Conduction band
(CB) and maximum energy of valence band
(VB) have the same value of wave vector, i.e.
k1 = k2
The minimum energy of Conduction band
(CB) and maximum energy of valence band
(VB) have the different values of wave
vector, i.e. k1 ≠ k2
Figure: E-K curve
Figure: E-K curve
5
4.
Concept of hole
Semiconductors are different from metals and insulators because they contain "almostempty" conduction band and "almost-full" valence band. This implies the transport of
carriers must be seen from both bands. To make such arguments, we introduce holes in
"almost-full" valence band. These are not present in the semiconductor but simply, they
are missing electrons spaces during the electron transport. The fact is that the electrons are
the only charge carriers exists in semiconductors. Holes behave as particles with the same
properties as the electrons would have same occupying states, but they carry a positive
charge. When certain energy is given to an electron in almost filled valence band, the
departure of electron creates a hole, giving chance for another electron to occupy. It means
that the energy of the hole is associated with the rise of electron energy. The electron and
hole concept is schematically shown in the figure. Now it is clear here that the electron and
hole numbers are equal (in intrinsic semiconductors). But if we make either one of them
more by doping.
Fig: Electron- Hole concept
5.
Intrinsic Semiconductor
Pure germanium or silicon called an intrinsic semiconductor. Each atom possesses four
valence electrons in outer most orbits. At T = 0K a 2-D representation of the crystal of silicon
& band diagram is shown in the figure.
Intrinsic silicon crystal at T =0K (a) 2-D representation of silicon crystal
(b) Energy band diagram of intrinsic semiconductor
6
Explanation: At 0K, all the valence electrons of silicon atoms are in covalent bonds and their
energies constitute a band of energies called valance band (VB). So at 0K, VB is completely
filled & conduction band (CB) is empty. If we rise temperature (T>0K), some of the electrons
which are in covalent bonds break the bonds become free and move from VB to CB. The
energy required should be greater than the energy gap of a semiconductor (E>Eg). The
electron vacancy or deficiency created in VB is called holes. This is shown in the figure.
Silicon crystal at a temperature above 0K (a) Due to thermal energy breaking of
Covalent bonds take place (b) Energy band representation.
6.
Carrier concentration in intrinsic semi-conductors
In a semiconductor both electrons and holes are charge carriers (know as carrier
concentration) whereas in metals only electrons. A semiconductor in which holes and
electrons are created by thermal excitation across the energy gap is called an intrinsic
semiconductor. In an intrinsic semiconductor the number of holes is equal to the number of
free electrons.
At T = 0K, valence band is completely filled, and conduction band is completely
empty. Thus, the intrinsic semiconductor behaves as a perfect insulator.
At T > 0K, the electron from the valence band move to conduction band across the
band gap. Thus, there are number of free electrons and holes in intrinsic semiconductor.
Fermi level lies in midway between conduction band and valance band in intrinsic
semiconductors.
7
7.
Electron concentration in intrinsic semiconductor in conduction band (n)
Definition: The number of free electrons per unit volume of the conduction band of a given
intrinsic semiconductor is called electron concentration, represented by “n”.
(a) Energy band diagram of silicon at T = 0K
(b) Energy band diagram of silicon at T > 0K
Derivation: Let the number of free electrons per unit volume of the semiconductor having
energies E and E+dE in CB is represented by n(E) dE. It is obtained by multiplying the density
of energy states ZC(E) d(E) [No. of energy states per unit volume] and Fermi-Dirac
distribution function for the Probability of occupation of electrons fc(E) Therefore,
n(E)dE = Zc(E) dE۰f(E)
where, f(E) is the Fermi distribution function and Zc (E) is the density of states factor.
𝑓(𝐸) =
1
(1)
𝐸−𝐸𝐹
⁄𝐾 𝑇
𝐵
1+𝑒
Since the fermi level for an intrinsic semiconductor must lie well within the forbidden
energy region and since KBT >> Eg, at all temperature, (E-EF) >> KBT. Therefore equation 1 can
be written as,
(𝐸𝐹 −𝐸)
1
𝑘𝐵 𝑇
𝑓(𝐸) =
=
𝑒
𝐸−𝐸𝐹
⁄𝐾 𝑇
𝐵
1+𝑒
π 8𝑚𝑒∗
𝑍𝑐 (𝐸) = [ 2 ]
2 ℎ
3⁄
2
⋅𝐸
1⁄
2
(2)
In equation (2), E is the KE of electrons and starts from bottom of the CB, hence,
π 8𝑚𝑒∗
𝑍𝑐 (𝐸) = [ 2 ]
2 ℎ
3⁄
2
⋅ (𝐸 − 𝐸𝑐 )
1⁄
2
𝑚𝑒∗ is the effective mass of electrons.
Therefore, the total number of electrons per unit volume in CB is
∞
𝑛 = ∫ 𝑛(𝐸) ⅆ𝐸 = ∫ 𝑧𝐶 (𝐸) ⋅ 𝑓(𝐸) ⅆ𝐸
𝐸𝐶
8
(3)
π 8𝑚𝑒∗
𝑛= [ 2 ]
2 ℎ
3⁄
2
∞
∫ (𝐸 −
1
𝐸𝑐 ) ⁄2
(𝐸𝐹 −𝐸)
𝑒 𝑘𝐵 𝑇 ⅆ𝐸
(4)
𝐸𝐶
π 8𝑚𝑒∗
= [ 2 ]
2 ℎ
3⁄
2
∞
𝐸𝐹
𝑒 𝑘𝐵 𝑇
∫ (𝐸 − 𝐸𝑐 )
1⁄
2
−𝐸
𝑒 𝑘𝐵 𝑇 ⅆ𝐸
𝐸𝐶
to convert integration from 𝐸 to 𝑥, let 𝐸 − 𝐸𝑐 = 𝑥 𝑘𝐵 𝑇 → 𝐸 = 𝐸𝑐 + 𝑥 𝑘𝐵 𝑇
𝑜𝑛 ⅆ𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑖𝑛𝑔 ⅆ𝐸 = 𝑘𝐵 𝑇 ⅆ𝑥
To change the limits: Lower limits
let E-Ec = x KBT,
let E-Ec = x KBT ,
when E=Ec, x=0
when E= ∞, x=∞
π 8𝑚𝑒∗
∴𝑛= [ 2 ]
2 ℎ
π 8𝑚𝑒∗
𝑛= [ 2 ]
2 ℎ
∞
WKT, ∫ 𝑥1/2 𝑒 −𝑥 ⅆ𝑥 =
0
3⁄
2
𝐸𝐹
𝑒 𝑘𝐵 𝑇
3⁄
2
2𝜋𝑚𝑒∗
ℎ2
∫ (𝑥 𝑘𝐵 𝑇)1/2 𝑒
(𝑘𝐵 𝑇)
𝐸 +𝑥 𝑘𝐵 𝑇
−[ 𝑐
]
𝑘𝐵 𝑇
𝑘𝐵 𝑇 ⅆ𝑥
3⁄
2
𝑒
𝐸 − 𝐸𝑐
[ 𝐹
]
𝑘𝐵 𝑇
∞
∫ 𝑥1/2 𝑒 −𝑥 ⅆ𝑥
0
√𝜋
2
3⁄
2
2π𝑚𝑒∗ 𝑘𝐵 𝑇
𝑛 = 2[
]
ℎ2
𝒏 = 𝑵𝑪 𝒆
∞
0
π 8𝑚𝑒∗ 𝑘𝐵 𝑇
𝑛= [
]
2
ℎ2
where, 𝑁𝐶 = 2 [
Upper limit
𝑒
𝐸 − 𝐸𝑐
[ 𝐹
]
𝑘𝐵 𝑇
3⁄
2
𝑒
√𝜋
2
𝐸 − 𝐸𝑐
[ 𝐹
]
𝑘𝐵 𝑇
𝑬 − 𝑬𝒄
[ 𝑭
]
𝒌𝑩 𝑻
𝑘𝐵 𝑇]
3⁄
2
is the pseudo constant for conduction band.
9
8.
Hole concentration in the valance band of intrinsic semiconductor(p)
Definition: The number of holes per unit volume of the valance band of a given intrinsic
semiconductor is called hole concentration, represented by “p”.
(a) Energy band diagram of silicon at T = 0K
(b) Energy band diagram of silicon at T > 0K
Derivation:
As the presence of holes regarded as the absence of an electron, the Fermi function in the
valence band (VB) is equal to 1 – f(E)
∴ expression for density (concentration) of holes in VB is
𝐸𝑉
𝑝 = ∫ 𝑝(𝐸) ⅆ𝐸 = ∫ 𝑍𝑉 (𝐸) [1 − 𝑓(𝐸)] ⅆ𝐸
(1)
−∞
1
Where 1 − 𝑓(𝐸) = 1 −
1+
1
𝑊𝐾𝑇, 𝑓(𝐸) =
1+
∴ 1 − 𝑓(𝐸) = 𝑒
=
𝐸−𝐸𝐹
⁄𝐾 𝑇
𝐵
𝑒
−1
𝐸−𝐸𝐹
⁄𝐾 𝑇
𝐵
𝑒
𝐸−𝐸𝐹
⁄𝐾 𝑇
𝐵
𝐸−𝐸𝐹
⁄𝐾 𝑇
𝐵
𝜋 8𝑚𝑝∗
(𝐸)ⅆ𝐸
𝑍𝑉
= [ 2 ]
2 ℎ
𝐸−𝐸𝐹
⁄𝐾 𝑇
𝐵
1+
𝑏𝑢𝑡 𝑒
𝐸−𝐸𝐹
⁄𝐾 𝑇
𝐵
𝑒
1+𝑒
≫1
(2)
3⁄
2
(𝐸𝑉 − 𝐸)
1⁄
2
dE
(3)
Using equation (2) and (3) in (1),
𝜋 8𝑚𝑝∗
𝑝=∫
[ 2 ]
−∞ 2 ℎ
𝐸𝑉
π 8𝑚𝑒∗
𝑝= [ 2 ]
2 ℎ
3⁄
2
3⁄
2
(𝐸𝑉 − 𝐸)
1⁄
2
𝑒
𝐸𝑉
−𝐸𝐹
𝑒 𝑘𝐵 𝑇 ∫ (𝐸𝑣 − 𝐸)
(𝐸−𝐸𝐹)
⁄
𝐾𝐵 𝑇
𝐸
1⁄
2 𝑒 𝑘𝐵 𝑇
ⅆ𝐸
ⅆ𝐸
(4)
−∞
to convert integration from 𝐸 to 𝑥, let 𝐸𝑣 − 𝐸 = 𝑥 𝑘𝐵 𝑇 → 𝐸 = 𝐸𝑣 − 𝑥 𝑘𝐵 𝑇
𝑜𝑛 ⅆ𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑖𝑛𝑔 ⅆ𝐸 = −𝑘𝐵 𝑇 ⅆ𝑥
10
To change the limits: Lower limits
Upper limit
let 𝐸𝑣 − 𝐸 = 𝑥 𝑘𝐵 𝑇,
let 𝐸𝑣 − 𝐸 = 𝑥 𝑘𝐵 𝑇,
when 𝐸 = ∞, 𝑥 = ∞
when 𝐸 = 𝐸𝑣, 𝑥 = 0
π 8𝑚𝑒∗
∴𝑝= [ 2 ]
2 ℎ
3⁄
2
−𝐸𝐹
𝑒 𝑘𝐵 𝑇
0
1/2
∫ (𝑥 𝑘𝐵 𝑇)
𝐸 −𝑥 𝑘𝐵 𝑇
[ 𝑣
]
𝑒 𝑘𝐵 𝑇
(−𝑘𝐵 𝑇) ⅆ𝑥
∞
π 8𝑚𝑒∗
𝑝=− [ 2 ]
2 ℎ
3⁄
2
π 8𝑚𝑒∗
𝑝=+ [ 2 ]
2 ℎ
3⁄
2
(𝑘𝐵 𝑇)
3⁄
2
𝑒
0
𝐸 − 𝐸𝐹
[ 𝑣
]
𝑘𝐵 𝑇
∫ 𝑥1/2 𝑒 −𝑥 ⅆ𝑥
∞
(𝑘𝐵 𝑇)
3⁄
2
𝑒
∞
𝐸 − 𝐸𝐹
[ 𝑣
]
𝑘𝐵 𝑇
∫ 𝑥1/2 𝑒 −𝑥 ⅆ𝑥
0
∞
𝑊𝐾𝑇 ∫ 𝑥
1⁄
2 𝑒 −𝑥
∙ ⅆ𝑥 =
0
𝜋 8𝑚𝑝∗ 𝐾𝐵 𝑇
𝑝= [
]
2
ℎ2
3⁄
2
∙𝑒
2𝜋𝑚𝑝∗ 𝐾𝐵 𝑇
𝑝=2 [
]
ℎ2
𝒑 = 𝑵𝑽 ∙ 𝒆
Where, 𝑁𝑉 = 2 [
9.
3⁄
2
∙𝑒
∙
√𝜋
2
(𝐸𝑣 −𝐸𝐹 )
𝐾𝐵 𝑇
(𝑬𝒗 −𝑬𝑭 )
𝑲𝑩 𝑻
∗𝐾 𝑇
2𝜋𝑚𝑝
𝐵
ℎ2
(𝐸𝑣 −𝐸𝐹 )
𝐾𝐵 𝑇
√𝜋
2
]
3⁄
2
is pseudo constant for valence band.
Intrinsic carrier concentration (ni)
Definition: The no. of free electrons and holes per unit volume of the intrinsic
semiconductor is called intrinsic carrier concentration (ni) remains constant.
For an intrinsic semiconductor, density of electrons is equal to density of holes.
i.e., 𝑛 = 𝑝 = 𝑛𝑖 = 𝑖𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛
2
𝑛𝑖 = 𝑛 ∙ 𝑝
2𝜋𝑚𝑒∗ 𝑘𝐵 𝑇
= 2[
]
ℎ2
3⁄
2
𝐸 − 𝐸𝑐
[ 𝐹
]
𝑒 𝑘𝐵 𝑇
2𝜋𝑚𝑝∗ 𝐾𝐵 𝑇
∙2 [
]
ℎ2
3⁄
2
∙
−(𝐸𝐹 −𝐸𝑉 )
𝑒 𝐾𝐵 𝑇
−(𝐸𝐶 −𝐸𝑉 )
2𝜋𝑘𝐵 𝑇 3 ∗ ∗ 3⁄
𝐾𝐵 𝑇
2
= 4[
]
(𝑚
𝑚
)
∙
𝑒
𝑒 𝑝
ℎ2
= 2[
2𝜋𝑘𝐵 𝑇
ℎ2
3⁄
2
]
2𝜋𝑘𝐵
= 2[ 2 ]
ℎ
𝑛𝑖 = 𝐴 ∙
3
𝑇 ⁄2
∙
3⁄
2
(𝑚𝑒∗ 𝑚𝑝∗ )
3⁄
4
3
(𝑚𝑒∗ 𝑚𝑝∗ ) ⁄4
−𝐸𝑔
∙ 𝑒 2𝐾𝐵 𝑇
∙
3
𝑇 ⁄2
∙
[𝑠𝑖𝑛𝑐𝑒, 𝐸𝐶 − 𝐸𝑉 = 𝐸𝑔 ]
−𝐸𝑔
2𝐾
𝑒 𝐵𝑇
−𝐸𝑔
2𝐾
𝑒 𝐵𝑇
2𝜋𝑘𝐵
𝑤ℎ𝑒𝑟𝑒, 𝐴 = 2 [ 2 ]
ℎ
3⁄
2
(𝑚𝑒∗ 𝑚𝑝∗ )
11
3⁄
4
10. Extrinsic (or) Impure semiconductor
Introduction: The conductivity of an intrinsic semiconductor can be increased by adding
small amounts of impurity atoms, such as III rd or Vth group atoms. The conductivity of silica
is increased by 1000 times on adding 10 parts of boron per million part of silicon. The
process of adding impurities is called doping and the impurity added is called dopant.
N – Type semiconductor
In a pure (intrinsic) semiconductor, when pentavalent an impurity like Phosphorous atom
consisting of five valance electrons is doped, and then concentration of electrons increases
than holes. Hence the given semiconductor formed is called N – type semiconductor. This is
shown in the figure 7a below. By adding donor impurities, the free electrons generated or
donated, form an energy level called as “Donor energy level” i.e. ED is shown in the figure
below.
(a) Representation of n- type silicon at T = 0K (b) Energy band diagram at T = 0K
In the figure (b) EF is Fermi energy level is in between Ec & Ed at T = 0K. Hence EF=(EC +ED)/2.
The donor level “𝐸d” is near to EF consisting of free electrons. But CB is empty.
11. P- type semiconductor
P – Type semiconductor is formed by doping with trivalent impurity atoms (acceptor) like III
rd group atoms i.e. Aluminum, Gallium, and Indium etc, to a pure semiconductor like Ge or
Si. As the acceptor trivalent atoms has only three valance electrons & Germanium, Silicon
has four valence electrons; holes or vacancy is created for each acceptor dopant atom.
Hence holes are majority and electrons are minority. It is shown in the figure a below. Also
an acceptor energy level „EA‟ is formed near VB consisting of holes, as shown in the figure
below.
12
(a) Representation of p- type silicon at T = 0K (b) Energy band diagram at T =0K
As temperature increases (T>0K) the electrons in VB which are in covalent bonds break the
bonds become free and move from VB to acceptor energy level EA.
12. Drift current
The flow of charge carriers due to the applied voltage or electric field is called drift
current. When the voltage is applied to a semiconductor, the free electrons move towards
the positive terminal of a battery and holes move towards the negative terminal of a
battery. The applied voltage causes the electrons to drift towards the positive terminal.
Electrons continuously collide with the atoms and change the direction each time. The
average velocity that an electron or hole achieved due to the applied voltage or electric field
is called drift velocity.
The drift velocity of electrons is given by Vn = µn E
The drift velocity of holes is given by Vp = µp E
Where Vn = drift velocity of electrons,
Vp = drift velocity of holes
µn = mobility of electrons
µp = mobility of holes
E = applied electric field
The drift current density due to free electrons is given by J n = e n µ n E
and the drift current density due to holes is given by J p = e p µ p E
Then the total drift current density is J = J n + J p = e n µ n E + e p µ p E
J=e(nµn+pµp)E
13
13. Diffusion current
The process in which, charge carriers (electrons or holes) in a semiconductor moves
from a region of higher concentration to a region of lower concentration is called diffusion.
The current due to diffusion of charges is diffusion current. The diffusion continues till to
reach the uniform concentration of electrons.
Both drift and diffusion current occur in semiconductor devices whereas only drift current
occur in a conductor. Diffusion current occurs without an external voltage or electric field.
The direction of diffusion current is same or opposite to that of the drift current.
Concentration gradient
The diffusion current density is directly proportional to the concentration gradient.
Concentration gradient is the difference in concentration of electrons or holes in a given
area. If the concentration gradient is high, then the diffusion current density is also high.
Similarly, if the concentration gradient is low, then the diffusion current density is also low.
The concentration gradient for n-type semiconductor is given by
The concentration gradient for p-type semiconductor is given by
Where Jn = diffusion current density due to electrons
Jp = diffusion current density due to holes
Diffusion current density
The diffusion current density due to electrons is given by
Where Dn is the electron diffusion coefficient [cm2/s]
The diffusion current density due to holes is given by
Where Dp is the hole diffusion coefficient [cm2/s].
Total current density
The total current density due to electrons is the sum of drift and diffusion currents.
Jn = Drift current + Diffusion current
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The total current density due to holes is the sum of drift and diffusion currents.
Jp = Drift current + Diffusion current
The total current density due to electrons and holes is given by
J = J n + Jp
At equilibrium, the diffusion and drift current cancel each other for both types of charge
carriers and hence.
Jn (drift) + Jn (diffusion) = 0
Jp (drift) + Jp (diffusion) = 0
Thus, for non-uniform doping in equilibrium, we have:
•Ef is constant and No net currents
•Carrier Concentration gradients that result in a diffusion current component.
•A “Built in” electric field that result in a drift current component.
•BOTH electron and hole components must sum to zero. i.e. Jn=Jp=0
14. Einstein relation
Einstein relation gives direct relation between the diffusion coefficient and mobility of charge
carrier.
At equilibrium, the drift current balances the diffusion current and are opposite to each
other.
Jn (drift) + Jn (diffusion) = 0
= 0 --- (1)
Einstein compare the moment of charge carries to moment of gas molecules. According to
Boltzmann statistics, the concentration of gas molecules is given by
−𝑒𝐸𝑥⁄ )
𝐾𝑇
𝑛 = 𝐶 𝑒(
--- (2)
differentiate w r t ‘x’ partially
ⅆ𝑛
− 𝑒𝐸
−𝑒𝐸
= 𝐶 𝑒 ( ⁄𝐾𝑇) (
)
ⅆ𝑥
𝐾𝑇
ⅆ𝑛
−𝑒𝐸
= 𝑛(
)
− − − (3) (𝑓𝑟𝑜𝑚 (2))
ⅆ𝑥
𝐾𝑇
Substitute Eqn (3) in Eqn (2)
𝑒𝑛𝜇𝑛 𝐸 + 𝑒𝐷𝑛 𝑛 (
−𝑒𝐸
) =0
𝐾𝑇
𝑒𝐷𝑛
𝐾𝑇
𝐷𝑛 𝐾𝑇
=
einstein relation for electrons
𝜇𝑛
𝑒
𝜇𝑛 =
Similarly for holes
𝐷𝑛 𝐾𝑇
=
𝜇𝑛
𝑒
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15. Hall Effect
If a piece of semiconductor carrying a current is placed in a transverse (or
perpendicular) magnetic field, an electric field is produced inside the conductor in a
direction normal (or perpendicular) to both the current and the magnetic field. This
phenomenon is known as “Hall effect” and the generated voltage is called “Hall voltage”.
I-Current, B-Magnetic field, EH-Electric field
16. Hall Effect in n –type Semiconductor
Let us consider an n-type material to which the current is allowed to pass along xdirection from left to right (electrons move from right to left) and the magnetic field is
applied in z-directions, as a result Hall voltage is produced in y direction.
Since the direction of current is from left to right the electrons moves from right to
left in x-direction as shown in Figure.
Now due to the magnetic field applied the electrons move towards downward
direction with the velocity ‘v’ and cause the negative charge to accumulate at face (1) of the
material as shown Figure.
Therefore a potential difference is established between face (2) and face (1) of the specimen
which gives rise to field EH in the negative y direction.
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All 3 quantities 𝐸𝐻 , 𝐽𝑥 𝑎𝑛ⅆ 𝐵 can be measured and so Hall coefficient and current density
can be found out.
17. Hall Effect in p-type Semiconductor
Let us consider a p-type material for which the current is passed along x-direction from left
to right and magnetic field is applied along z-direction as shown in Figure Since the direction
of current is from left to right, the holes will also move in the same direction.
Now due to the magnetic field applied, the holes move towards the downward
direction with velocity ‘v’ and accumulate at the face (1) as shown in Figure.
A potential difference is established between face (1) and (2) in the positive y direction.
Force due to the potential difference = eEH
.. (8)
[Since hole is considered to be an electron with same mass but positive charge
negative sign is not included].
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At equilibrium eqn. (7) = eqn . (8)
Equation (11) represents the hall coefficient and the positive sign indicates that the Hall field
is developed in the positive y direction.
18. Hall Coefficient Interms of Hall Voltage
Half coefficient (RH) is defined as the Hall field developed per unit current density per
unit applied magnetic field.
If the thickness of the sample is‘t’ and the voltage developed is ‘VH’ then Hall voltage
18
Applications of Hall effect :a) Determination of semiconductor type:
For an n-type semiconductor the Hall coefficient is -ve whereas for p-type semiconductor it
is +ve. Thus, the sign of Hall coefficient can be used to determine whether a given
semiconductor is n-type or p-type.
b) Calculation of carrier concentration:
The Hall voltage 𝑉𝐻 is measured as usual by placing 2 probes at the center of the top and
bottom faces of the sample. If the magnetic flux density is 𝐵 Wb/m2, then
1
𝑛=
𝑒𝑅𝐻
c) Determination of mobility :
If the conduction is due to one type of carriers, for eg; electrons, we have
𝜎 = 𝑛𝑒𝜇𝑛
𝜎
𝜇𝑛 =
= 𝜎𝑅𝐻
𝑛𝑒
𝑉𝑥 𝑏
𝜇𝑛 = 𝜎 ( )
𝐼𝑥 𝐵
Knowing 𝜎, we can determine the mobility 𝜇𝑛 .
d) Measurement of magnetic flux density :
Since Hall voltage 𝑉𝐻 is proportional to magnetic flux density 𝐵 for a given current 𝐼𝑥 thro a
sample, the Hall effect can be used as a basics for design of magnetic flux density meter.
e) Measurement of power in an electromagnetic wave :
In an electromagnetic wave in a magnetic field 𝐻 and the electric field 𝐸 are at right angles.
Thus, if a semiconductor sample is placed parallel to 𝐸 it will derive a current 𝐼 in the
semiconductor. The semiconductor is subjected simultaneously to a transverse magnetic
field 𝐻 producing a Hall voltage across the sample. The Hall voltage will be proportional to
the product of 𝐸 and 𝐻 i.e., to the magnitude of Poynting vector of electromagnetic wave.
Thus, the Hall effect can be used to determine the power flow in an electromagnetic wave.
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