04-03-2022

Influence of Thermal Energy

Electronic Materials and Devices
BECE201L
No Electrons in Conduction Bands All Valence Bands are filled up.

Module 2: Semiconductor Fundamentals

Electronic Materials-Energy Bands

 Formation of energy bands,
 Density of States
 Fermi Level, Fermi Distribution

Dr. K. Govardhan,
School of Electronics Engineering,
VIT University

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Band Gap in Intrinsic Semi Conductors Energy Levels for Electrons in a Doped
Semiconductor

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Isolated Atoms Diatomic Molecule

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Six Closely Spaced Atoms

Four Closely Spaced Atoms as fn(R)

conduction band

the level of interest

has the same energy in
each separated atom

valence band

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Solid composed of ~NA

Energy Levels in Solids Na Atoms as fn(R)

Two atoms Six atoms Solid of N atoms

1s22s22p63s1

ref: A.Baski, VCU 01SolidState041.ppt

www.courses.vcu.edu/PHYS661/pdf/01SolidState041.ppt

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Sodium Bands vs Separation Copper Bands vs Separation

Rohlf Fig 14-4 and Slater Phys Rev 45, 794 (1934) Rohlf Fig 14-6 and Kutter Phys Rev 48, 664 (1935)

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Differences down a column in the Periodic Table:

same valence IV-A Elements Conduction and valence bands
config

E It will be important for us to know

how the states are distributed in
mostly states empty energy within the conduction and
conduction “band” valence bands.

Most of the empty states (the

holes) in the valence band are
very near EV.
mostly full states

valence “band” Most of the filled states (the

electrons) in the conduction
band are very near EC.
energy vs. position
Lundstrom: 2018
Sandin

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Energy Band Diagram Energy Band Diagram

intrinisic semiconductor: no=po=ni

Eelectron E(x) E(x)

conduction band conduction band

EC EC
n(E)
S(E) EF=Ei
p(E)
EV EV
valence band valence band

Ehole x x

note: increasing electron energy is ‘up’, but increasing hole energy is ‘down’. where Ei is the intrinsic Fermi level

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Energy Band Diagram Energy Band Diagram

n-type semiconductor: no>po p-type semiconductor: po>no

n0  N C exp[ ( EC  EF ) / kT ] p0  NV exp[ ( EF  EV ) / kT ]
E(x) E(x)

conduction band conduction band

EC EC
n(E) EF n(E)
p(E) p(E) EF
EV EV
valence band valence band

x x

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Density of states
Density of States

The number of states between E and E + dE is D(E)dE,

• The density of electrons (no) can be found precisely
where D(E) is the “density of states” (DOS).
if we know
1. the number of allowed energy states in a small energy
range, dE: S(E)dE
The estimation of DOS is greatly simplified because only the
“the density of states”
region near the band edges are important, and in that region, the
2. the probability that a given energy state will be occupied
by an electron: f(E) bands are nearly parabolic:
“the distribution function”

no =  S(E)f(E)dE
band

valence conduction band

band Lundstrom: 2018

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Density of States in x-direction

States in a finite volume of semiconductor
dk

Finite volume, Ω 0 2 spin

(part of an infinite volume) Lx
 x   uk (x)e ik xx
dk x
# of states   2  N k dk
  Lx Ly Lz Finite number of states 2 Lx 
 0    Lx   e ik L 1 xx

Lx
Nk = = density of states in k-space
Periodic boundary conditions: 
k x Lx  2 j j  1,2,3,...
yˆ
2
kx  j
Lx
x̂
“Brillouin zone”
ẑ
Lundstrom: 2018

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Density-of-states in k-space DOS: k-space vs. energy space

1D:
E but non-uniformly distributed
dk in energy space.
dE
2D: Depends on E(k)
(e.g. different for parabolic
dk x dk y independent of E(k) dk x bands and linear bands)
k
3D:
2 L x
dk x dk y dk z
States are uniformly
distributed in k-space,

Lundstrom: 2018 2 Lundstrom: 2018 2

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Example 1: DOS(E) for 1D nanowire 1D (single subband)

• xˆ E
ẑ ŷ

dE

• Find DOS(E) per unit energy, per unit length, a

• single subband assuming parabolic energy bands.

k
2 L dk
Lundstrom: 2018 2 9
5 Lundstrom: 2018

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1D DOS Don’t forget to multiply by 2

dE

dk dk
k
2 L

Multiply by 2 to
account for the (parabolic
negative k- energy bands)
DOS in subband, n. n = 1, 2, 3... states.
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Lundstrom: 2018 Lundstrom: 2018

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Carrier Concentration Statistics

Density of States (DOS)
Number of electrons and current-voltage
holes available for characteristic of
conduction N(E) – semiconductors
Fabrication and System and Circuit
Processing Design
Pauli Exclusion
Principle
Fermi-Dirac
Statistics

Number of energy states as

Occupation probability of
a function of energy
energy states [distribution
[density of states, g(E)]
function, f(E)]

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