Engineering Physics - B20EC2020
Deepak.K
School of applied sciences (Physics)
Ph:8892926677 E-mail- deepak.k@reva.edu.in
UNIT 2: SYLLABUS
1. 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 – (10 Hrs)
2
2. Origin of Energy band formation in solids
Fig a. Splitting of energ levels due to interatomic interaction
Fig b. Energy level vs. inter-atomic distance
3
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, whose 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.
4
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.
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.
5
Indirect band gap semiconductor
In Indirect band gap semiconductor, the minimum energy level of conduction band and maximum energy level of valence
band occur at different values of momentum.
6
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.
Life time of charge carriers is more.
Emission of light has energy equal to energygap Eg=(ℎ𝐶/𝜆) eV
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
7
Intrinsic Semiconductor
Intrinsic silicon crystal at T =0K (a) 2-D representation of silicon crystal
(b) Energy band diagram of intrinsic semiconductor
Silicon crystal at a temperature above 0K (a) Due to thermal energy
breaking of Covalent bonds take place (b) Energy band representation.
8
Concept of hole
Carrier concentration in intrinsic semi-conductors
Fig: Electron- Hole concept
9
Electron concentration in intrinsic semiconductor in conduction band (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.
10
11
n(E) dE = Zc(E) dE۰f(E)
1
WKT 𝑓 𝐸 =
1
𝑓 𝐸 =
𝑒
= 𝑒
E−𝐸𝐹
𝑘𝐵 𝑇
𝑊𝐾𝑇 𝑍𝑐 𝐸 =
3
2
∞
𝐸𝐹
𝑒 𝑘𝐵 𝑇
𝐸 − 𝐸𝑐
1
−𝐸
2 𝑒 𝑘𝐵 𝑇
ⅆ𝐸
𝐸𝐶
to convert integration from 𝐸 to 𝑥,
𝐸𝐹 −𝐸
𝑘𝐵 𝑇
(∵ 𝐸 − 𝐸𝐹 > 𝐾𝐵𝑇 )
let 𝐸 − 𝐸𝑐 = 𝑥 𝑘𝐵 𝑇 → 𝐸 = 𝐸𝑐 + 𝑥 𝑘𝐵 𝑇
𝑜𝑛 ⅆ𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑖𝑛𝑔 ⅆ𝐸 = 𝑘𝐵 𝑇 ⅆ𝑥
3
π
8𝑚𝑒∗
2 ℎ2
𝑍𝑐 𝐸 =
…. 1
E−𝐸𝐹
𝑘𝐵 𝑇
1+𝑒
π
𝑛=
2
8𝑚𝑒∗
ℎ2
2
⋅𝐸
3
8𝑚𝑒∗ 2
ℎ2
π
2
1
2
To change the limits:
… (2)
⋅ 𝐸 − 𝐸𝑐
1
2
. . (3)
Therefore, the total number of electrons per unit volume in CB is
Lower limits
Upper limit
let E-Ec = x KBT,
let E-Ec = x KBT ,
when E=Ec, x=0
when E= ∞, x=∞
∞
𝑛 = ∫ 𝑛 𝐸 ⅆ𝐸 =
𝑓 𝐸 . 𝑧𝐶 𝐸 ⅆ𝐸
π
∴𝑛=
2
𝐸𝐶
π 8𝑚𝑒∗
𝑛=
2 ℎ2
3
2
∞
𝐸 − 𝐸𝑐
1
2
𝑒
𝐸𝐹 −𝐸
𝑘𝐵 𝑇
8𝑚𝑒∗
ℎ2
3
2
𝐸𝐹
𝑒 𝑘𝐵 𝑇
∞
(𝑥 𝑘𝐵
𝑇)1/2
𝑒
−
𝐸𝑐 +𝑥 𝑘𝐵 𝑇
𝑘𝐵 𝑇
𝑘𝐵 𝑇 ⅆ𝑥
0
ⅆ𝐸
𝐸𝐶
12
π
∴𝑛=
2
8𝑚𝑒∗
ℎ2
π 8𝑚𝑒∗
𝑛=
2 ℎ2
3
2
3
2
𝑘𝐵 𝑇
3
2
𝑒
𝐸𝐹 − 𝐸𝑐
𝑘𝐵 𝑇
𝐸𝑐 +𝑥 𝑘𝐵 𝑇
𝑘𝐵 𝑇
𝑘𝐵 𝑇 ⅆ𝑥
∞
𝑥 1/2
0
3
π
𝑛=
2
𝑛=2
2π𝑚𝑒∗ 𝑘𝐵 𝑇
ℎ2
where, 𝑁𝐶 = 2
(𝑥 𝑘𝐵 𝑇)1/2 𝑒
−
0
8𝑚𝑒∗ 𝑘𝐵 𝑇
ℎ2
𝒏 = 𝑵𝑪 𝒆
∞
𝐸𝐹
𝑒 𝑘𝐵 𝑇
2
𝑒
3
𝐸𝐹 − 𝐸𝑐
𝑘𝐵 𝑇
2
𝑒
∞
𝑒 −𝑥 ⅆ𝑥
𝑥 1/2
WKT,
0
𝑒 −𝑥 ⅆ𝑥
𝜋
=
2
𝜋
2
𝐸𝐹 − 𝐸𝑐
𝑘𝐵 𝑇
𝑬𝑭 − 𝑬𝒄
𝒌𝑩 𝑻
2𝜋𝑚𝑒∗
ℎ2
3
𝑘𝐵 𝑇
2
is the pseudo constant for conduction band.
13
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
Let the number of holes per unit volume of the semiconductor having energies E and E+dE in VB is represented by p(E) dE. It is
obtained by multiplying the density of energy states ZV(E) dE [No. of energy states per unit volume] and Fermi-Dirac
distribution function for the Probability of occupation of holes f (E).
P(E) dE = Zv(E) dE۰f(E)
(1)
14
P(E) dE = Zv(E) dE۰f(E)
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 + 𝑒
Divide Nr & Dr by 𝑒
𝐸−𝐸𝐹
𝑘𝐵 𝑇
𝑒
∴1−𝑓 𝐸 =𝑒
=
1 + 𝑒
3
2
𝐸𝑉 − 𝐸
1
2
dE
(3)
Using equation (2) and (3) in (1),
𝐸−𝐸𝐹
𝑘𝐵 𝑇
𝜋 8𝑚𝑝∗
𝑝=
2
−∞ 2 ℎ
𝐸𝑉
3
2
𝐸𝑉 − 𝐸
1
2
𝑒
𝐸−𝐸𝐹
𝑘𝐵 𝑇
ⅆ𝐸
𝑒
𝐸−𝐸𝐹
𝑘𝐵 𝑇
ⅆ𝐸
𝐸
𝑘
2 𝑒 𝐵𝑇
ⅆ𝐸 . . . (4)
then
1
1−𝑓 𝐸 =
𝐸−𝐸𝐹
𝑘𝐵 𝑇
𝑒
𝐸−𝐸𝐹
𝑘𝐵 𝑇
𝜋 8𝑚𝑝∗
𝑍𝑉 𝐸 ⅆ𝐸 =
2 ℎ2
𝐸𝐹 −𝐸
𝑘𝐵 𝑇
𝐸−𝐸𝐹
𝑘𝐵 𝑇
1
=
+1
𝑒
𝐸𝐹 −𝐸
𝑘𝐵 𝑇
∵𝑒
𝐸𝐹 −𝐸
𝑘𝐵 𝑇
2
𝜋 8𝑚𝑝∗
𝑝=
2
−∞ 2 ℎ
𝐸𝑉
>> 1
π 8𝑚𝑒∗
𝑝=
2 ℎ2
3
2
3
2
−𝐸𝐹
𝑒 𝑘𝐵 𝑇
𝐸𝑉 − 𝐸
1
2
𝐸𝑉
𝐸𝑣 − 𝐸
1
−∞
15
π
𝑝=
2
8𝑚𝑒∗
ℎ2
3
2
−𝐸𝐹
𝑒 𝑘𝐵 𝑇
𝐸𝑉
𝐸𝑣 − 𝐸
1
𝐸
2 𝑒 𝑘𝐵 𝑇
ⅆ𝐸 . . (4)
−∞
to convert integration from 𝐸 to 𝑥,
let 𝐸𝑣 − 𝐸 = 𝑥 𝑘𝐵 𝑇 → 𝐸 = 𝐸𝑣 − 𝑥 𝑘𝐵 𝑇
π
𝑝=−
2
8𝑚𝑒∗
ℎ2
π
𝑝=+
2
8𝑚𝑒∗
ℎ2
3
2
𝑘𝐵 𝑇
3
2
𝑘𝐵 𝑇
2
2
𝑒
∞
𝐸𝑣 − 𝐸𝐹
𝑘𝐵 𝑇
𝑊𝐾𝑇
𝜋 8𝑚𝑝∗ 𝐾𝐵 𝑇
𝑝=
2
ℎ2
let Ev-Ec = x KBT ,
−𝐸𝐹
𝑒 𝑘𝐵 𝑇
3
∞
Upper limit
when E= ∞ , x= ∞
π
∴𝑝=
2
𝑥 1/2 𝑒 −𝑥 ⅆ𝑥
𝑥 1/2 𝑒 −𝑥 ⅆ𝑥
𝑥
1
2 𝑒 −𝑥
∙ ⅆ𝑥 =
0
Lower limits
3
𝑒
0
To change the limits:
8𝑚𝑒∗
ℎ2
2
0
𝐸𝑣 − 𝐸𝐹
𝑘𝐵 𝑇
∞
𝑜𝑛 ⅆ𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑖𝑛𝑔 ⅆ𝐸 = − 𝑘𝐵 𝑇 ⅆ𝑥
let Ev-E = x KBT,
3
3
when E=Ev, x=0
0
(𝑥 𝑘𝐵 𝑇)1/2 𝑒
𝐸𝑣 −𝑥 𝑘𝐵 𝑇
𝑘𝐵 𝑇
2𝜋𝑚𝑝∗ 𝐾𝐵 𝑇
𝑝=2
ℎ2
(−𝑘𝐵 𝑇) ⅆ𝑥
∞
𝒑 = 𝑵𝑽 ∙
Where, 𝑁𝑉 = 2
∗𝐾 𝑇
2𝜋𝑚𝑝
𝐵
ℎ2
3
2
∙
3
(𝐸𝑣 −𝐸𝐹 )
𝑒 𝐾𝐵 𝑇
2
∙
∙
𝜋
2
𝜋
2
(𝐸𝑣 −𝐸𝐹 )
𝑒 𝐾𝐵 𝑇
(𝑬𝒗 −𝑬𝑭 )
𝒆 𝑲𝑩 𝑻
2
is pseudo constant for valence band.
16
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
𝑛𝑖 2 = 𝑛 ∙ 𝑝
=2
2𝜋𝑚𝑒∗ 𝑘𝐵 𝑇
ℎ2
2𝜋𝑘𝐵 𝑇
=4
ℎ2
2𝜋𝑘𝐵 𝑇
𝑛𝑖 = 2
ℎ2
2𝜋𝑘𝐵
=2
ℎ2
𝒏𝒊 = 𝑨 ∙ 𝑻
𝟑
3
3
3
2
3
2
𝑒
𝐸𝐹 − 𝐸𝑐
𝑘𝐵 𝑇
3
(𝑚𝑒∗ 𝑚𝑝∗ ) 2
2
∙
3
(𝑚𝑒∗ 𝑚𝑝∗ ) 4
(𝑚∗𝑒 𝑚𝑝∗
−𝑬𝒈
𝟐 ∙ 𝒆𝟐𝑲𝑩 𝑻
3
4
∙2
2𝜋𝑚𝑝∗ 𝐾𝐵 𝑇
ℎ2
3
2
∙
− 𝐸𝐹 −𝐸𝑉
𝑒 𝐾𝐵 𝑇
− 𝐸𝐶 −𝐸𝑉
𝑒 𝐾𝐵 𝑇
∙
∙𝑇
−𝐸𝑔
𝑒 2𝐾𝐵 𝑇
3
2
∙
[𝑠𝑖𝑛𝑐𝑒, 𝐸𝐶 − 𝐸𝑉 = 𝐸𝑔 ]
−𝐸𝑔
𝑒 2𝐾𝐵 𝑇
2𝜋𝑘𝐵
𝑤ℎ𝑒𝑟𝑒, 𝐴 = 2
ℎ2
3
2
(𝑚∗𝑒 𝑚𝑝∗
3
4
17
Extrinsic (or) Impure semiconductor
Extrinsic semiconductors are made by doping impurities to the intrinsic semiconductors. 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.
(a) Representation of n- type silicon at T = 0K
(b) Energy band diagram at T = 0K
P- type semiconductor
P – Type semiconductor is formed by doping with trivalent
impurity atoms (acceptor) like IIIrd group atoms i.e.
Aluminum, Gallium, and Indium etc, to a pure
semiconductor like Ge or Si.
(a) Representation of p- type silicon at T = 0K
(b) Energy band diagram at T = 0K
18
Drift current
The flow of charge carriers due to the applied voltage or electric field is called drift current.
The drift velocity of electrons is given by Vn = µn E
The drift velocity of holes is given by Vp = µp E
The drift current density due to free electrons is given by J n =enµ n E
and the drift current density due to holes is given by J p =epµ p E
Then the total drift current density is J = J n +J p
= enµ n E + epµ p E
J = e (nµ n +pµ p ) E
19
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. The diffusion current density is directly proportional to the
concentration gradient.
The concentration gradient for n-type semiconductor is given by
The concentration gradient for p-type semiconductor is given by
Where Jn & Jp = diffusion current density due to electrons & holes respectively
20
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
The total current density due to holes is the sum of drift and diffusion currents Jp = Drift current + Diffusion current
At equilibrium, the diffusion and drift current cancel each other for both types of charge carriers and hence.
Jn (drift) + Jn (diffusion) = 0 and
Jp (drift) + Jp (diffusion) = 0
21
22
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
23
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”.
24
Hall Effect in n –type Semiconductor
Let us consider an n-type material to which the current is passed 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 -ve y direction.
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.
-ve sign indicates that the Hall field is developed in the -ve y direction.
25
Hall Effect in p-type Semiconductor
Force due to the potential difference = eEH
Force due to magnetic field = Bev
At equilibrium
…... (8)
Equation (11) represents the hall coefficient, and the positive sign indicates that the Hall field is developed in the positive y direction.
26
1. Calculate the intrinsic concentration of charge carriers at 300 K given that m*e = 0.12 m0, m*h = 0.28 m0 and the value of
brand gap = 0.67 eV.
Solution.
Given:
intrinsic concentration is given by
intrinsic concentration is
27
2. The intrinsic carrier density of a semiconductor is n = 2.1 × 1019 m-3. The electron and hole mobilities are 0.4 and 0.2 m/V s
respectively. Calculate the conductivity.
Solution.
Given:
3. Find the conductivity of intrinsic silicon at 300 K. It is given that ni at 300 K in silicon is 1.5 × 1016/m3 and the mobilities of
electrons and holes in silicon are 0.13 m2/V-s and 0.05 m2/V-s, respectively
Solution:
Given:
Intrinsic concentration (ni) = 1.5 ×
1016/m3
Conductivity (σ) = ni e (μe + μh)
Mobility of electrons (μe) = 0.13 m2/V-m
= 1.5 × 1016 × 1.6 × 10−19 (0.13 + 0.05)/Ω-m
Mobility of holes (μh) = 0.05 m2/V-m
= 4.32 × 10−4/Ω-m
28
4. In a P-type germanium, n = 2.1 × 1019 m-3 density of boran 4.5 × 1023 atoms/m. The electron and hole mobility are 0.4 and
0.2 m/v s respectively. What is its conductivity before and after addition of boron atoms.
Solution.
Given:
4. The mobility of electrons (μn) in silicon is 0.21 m2/V-s at 300 K, find the diffusion coefficient of electrons.
29
5. The Hall coefficient (RH) of a semiconductor is 3.22 × 10−4 m3 C−1. Its resistivity is 8.50 × 10−3 Ω-m. Calculate the mobility and
carrier concentration of the carriers.
Sol: Since RH is positive, so the given semiconductor is p-type.
nh = p = hole concentration
nh =
Mobility of holes μp is:
30
6. A pure silicon material has an intrinsic concentration of 1.5 × 1016/m3 at 300 K. If it is doped with donor impurity atoms at
the rate of 1 in 108 atoms of silicon, then calculate its conductivity. Assume that all the impurity atoms are ionized. Given that
the atomic weight of silicon is 28.09, density=2.33×103 kg/m3, electron and hole mobilities are 0.14 m2/V-s and
0.05 m2/Vs, respectively.
Sol: Given data are:
Intrinsic concentration (ni) = 1.5 × 1016/m3
Atomic weight of silicon (A) = 28.09
Density of silicon (D) = 2.33 × 103 kg/m3
Since the doping concentration is 1 in 108 silicon atoms
∴ Electron concentration (n) =
Electron mobility (μe) = 0.14 m2/V-s
Hole mobility (μh) = 0.05 m2/V-s
From law of mass action, hole concentration
= 4.5 × 1011/m3
∴ Conductivity (σ) = e[nμe + pμh] = 1.6 × 10–19[5 × 1020 × 0.14 + 4.5 × 1011 × 0.05]
= 1.6 × 10–19 [70,000 × 1015 + 0.0000225 × 1015] = 1.6 × 10–19 × 70,000.0000225 × 1015
= 11.2/Ω–m
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