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Semiconductor Device Physics
Lecture 2
Dr.-Ing. Erwin Sitompul
President University
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Chapter 2
Carrier Modeling
Electronic Properties of Si
 Silicon is a semiconductor material.
 Pure Si has a relatively high electrical resistivity at room
temperature.
 There are 2 types of mobile charge-carriers in Si:
 Conduction electrons are negatively charged,
e = –1.602  10–19 C
 Holes are positively charged,
p = +1.602  10–19 C
 The concentration (number of atom/cm3) of conduction
electrons & holes in a semiconductor can be influenced in
several ways:
 Adding special impurity atoms (dopants)
 Applying an electric field
 Changing the temperature
 Irradiation
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Chapter 2
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Bond Model of Electrons and Holes
 2-D Representation
Si
Si
Si
Si
Si
Si
Si
Si
Si
Hole
 When an electron breaks
loose and becomes a
conduction electron, then a
hole is created.
Si
Si
Si
Si
Si
Si
Si
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Si
Si
Conduction
electron
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Chapter 2
Carrier Modeling
What is a Hole?
 A hole is a positive charge associated with a half-filled covalent
bond.
 A hole is treated as a positively charged mobile particle in the
semiconductor.












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

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Chapter 2
Carrier Modeling
Conduction Electron and Hole of Pure Si
• Covalent (shared e–) bonds exists
between Si atoms in a crystal.
• Since the e– are loosely bound,
some will be free at any T,
creating hole-electron pairs.
ni = intrinsic carrier
concentration
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ni ≈ 1010 cm–3 at room temperature
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Chapter 2
Carrier Modeling
Si: From Atom to Crystal
Energy states
(in Si atom)
Energy bands
(in Si crystal)
• The highest mostly-filled
band is the valence band.
• The lowest mostly-empty
band is the conduction band.
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Chapter 2
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Energy Band Diagram
Electron energy
Ec
EG, band gap energy
Ev
• For Silicon at 300 K, EG = 1.12 eV
• 1 eV = 1.6 x 10–19 J
 Simplified version of energy band model, indicating:
 Lowest possible conduction band energy (Ec)
 Highest possible valence band energy (Ev)
 Ec and Ev are separated by the band gap energy EG.
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Chapter 2
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Measuring Band Gap Energy
 EG can be determined from the minimum energy (hn) of photons
that can be absorbed by the semiconductor.
 This amount of energy equals the energy required to move a
single electron from valence band to conduction band.
Electron
Ec
Photon
photon energy: hn = EG
Ev
Hole
Band gap energies
Semiconductor
Band gap (eV)
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Ge
0.66
Si
1.12
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GaAs
1.42
Diamond
6.0
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Chapter 2
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Carriers
 Completely filled or empty bands do not allow current flow,
because no carriers available.
 Broken covalent bonds produce carriers (electrons and holes)
and make current flow possible.
 The excited electron moves from valence band to conduction
band.
 Conduction band is not completely empty anymore.
 Valence band is not completely filled anymore.
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Chapter 2
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Band Gap and Material Classification
Ec
Ev
Ec
EG= ~8 eV
Ec
Ev
SiO2
EG = 1.12 eV
Si
Ec
Ev
Ev
Metal
 Insulators have large band gap EG.
 Semiconductors have relatively small band gap EG.
 Metals have very narrow band gap EG .
 Even, in some cases conduction band is partially filled,
Ev > Ec.
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Chapter 2
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Carrier Numbers in Intrinsic Material
 More new notations are presented now:
 n : number of electrons/cm3
 p : number of holes/cm3
 ni : intrinsic carrier concentration
 In a pure semiconductor, n = p = ni.
 At room temperature,
ni = 2  106 /cm3 in GaAs
ni = 1  1010 /cm3 in Si
ni = 2  1013 /cm3 in Ge
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Chapter 2
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Manipulation of Carrier Numbers – Doping
 By substituting a Si atom with a special impurity atom
(elements from Group III or Group V), a hole or conduction
electron can be created.
Acceptors: B, Ga, In, Al
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Donors: P, As, Sb
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Chapter 2
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Doping Silicon with Acceptors
 Example: Aluminium atom is doped into the Si crystal.
Al– is immobile
 The Al atom accepts an electron from a neighboring Si atom,
resulting in a missing bonding electron, or “hole”.
 The hole is free to roam around the Si lattice, and as a moving
positive charge, the hole carries current.
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Chapter 2
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Doping Silicon with Donors
 Example: Phosphor atom is doped into the Si crystal.
P is immobile
 The loosely bounded fifth valence electron of the P atom can
“break free” easily and becomes a mobile conducting electron.
 This electron contributes in current conduction.
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Chapter 2
Carrier Modeling
Donor / Acceptor Levels (Band Model)
▬
+
ED
Donor Level
Ec
Donor ionization energy
Acceptor ionization energy
▬
Acceptor Level
EA
Ev
+
Ionization energy of selected donors and acceptors in Silicon
Donors
Ionization energy of dopant
EC – ED or EA – EV (meV)
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Sb P
39 45
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Acceptors
As B
54 45
Al In
67 160
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Chapter 2
Carrier Modeling
Dopant Ionization (Band Model)
 Donor atoms
 Acceptor atoms
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Chapter 2
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Carrier-Related Terminology
 Donor: impurity atom that increases n (conducting electron).
Acceptor: impurity atom that increases p (hole).
 n-type material: contains more electrons than holes.
p-type material: contains more holes than electrons.
 Majority carrier: the most abundant carrier.
Minority carrier: the least abundant carrier.
 Intrinsic semiconductor: undoped semiconductor n = p = ni.
Extrinsic semiconductor: doped semiconductor.
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Chapter 2
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Density of States
DE
Ec
Ev
E
Ec
Ev
gc(E)
density of states g(E)
gv(E)
 g(E) is the number of states per cm3 per eV.
 g(E)dE is the number of states per cm3 in the energy range
between E and E+dE).
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Chapter 2
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Density of States
E
DE
Ec
gc(E)
Ec
density of states g(E)
gv(E)
Ev
Ev
 Near the band edges:
g c (E ) 
g v (E ) 
mn* 2mn*  E  Ec 
 h
2
3
mp* 2mp*  Ev  E 
 h
2
3
E  Ec
E  Ev
mn* : effective mass of electron
For Silicon at 300 K,
mn*  1.18mo
mp*  0.81mo
mo  9.1 10 31kg
mo: electron rest mass
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Chapter 2
Carrier Modeling
Fermi Function
 The probability that an available state at an energy E will be
occupied by an electron is specified by the following probability
distribution function:
f (E) 
1
1 e
( E  EF ) / kT
k : Boltzmann constant
T : temperature in Kelvin
 EF is called the Fermi energy or the Fermi level.
If E  EF , f ( E )  0
If E  EF , f ( E )  1
If E  EF ,
f (E)  1 2
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Chapter 2
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Effect of Temperature on f(E)
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Chapter 2
Carrier Modeling
Effect of Temperature on f(E)
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Chapter 2
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Equilibrium Distribution of Carriers
 n(E) is obtained by multiplying gc(E) and f(E),
p(E) is obtained by multiplying gv(E) and 1–f(E).
 Intrinsic semiconductor material
Energy band
diagram
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Density of
states
Probability
of occupancy
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Carrier
distribution
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Chapter 2
Carrier Modeling
Equilibrium Distribution of Carriers
 n-type semiconductor material
Energy band
diagram
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Density of
States
Probability
of occupancy
Erwin Sitompul
Carrier
distribution
SDP 2/24
Chapter 2
Carrier Modeling
Equilibrium Distribution of Carriers
 p-type semiconductor material
Energy band
diagram
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Density of
States
Probability
of occupancy
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Carrier
distribution
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Chapter 2
Carrier Modeling
Important Constants
 Electronic charge, q = 1.610–19 C
 Permittivity of free space, εo = 8.85410–12 F/m
 Boltzmann constant, k = 8.6210–5 eV/K
 Planck constant, h = 4.1410–15 eVs
 Free electron mass, mo = 9.110–31 kg
 Thermal energy, kT = 0.02586 eV (at 300 K)
 Thermal voltage, kT/q = 0.02586 V (at 300 K)
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Chapter 2
Carrier Modeling
Homework 1
Problem 2.5
Develop an expression for the total number of available states/cm3 in the
conduction band between energies Ec and Ec+ γkT, where γ is an arbitrary
constant.
Problem 2.6
(a) Under equilibrium condition at T > 0 K, what is the probability of an
electron state being occupied if it is located at the Fermi level?
(b) If EF is positioned at Ec, determine (numerical answer required) the
probability of finding electrons in states at Ec + kT.
(c) The probability a state is filled at Ec + kT is equal to the probability a
state is empty at Ec + kT. Where is the Fermi level located?
Due date: Thursday, 28.01.10, 8 am.
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