Transcript 2014.9.17

DEE4521
Semiconductor Device Physics
Lecture 2a:
Density-of-States (DOS), Fermi-Dirac Statistics, and Fermi Level
Prof. Ming-Jer Chen
Department of Electronics Engineering
National Chiao-Tung University
September 18, 2014
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1. To count electrons or holes, it is better
started from the Energy point of view.
2. DOS and Fermi Statistics and Level are
Energy related terminologies.
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Let us focus on those of textbook contents involved:
pp. 71 – 102;
and
start from the (potential) energy band diagram
(Remember: you should have got it from a slide
example and a conductor example).
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What are States?
•
Pauli exclusion principle:
No two electrons in a system can have the same set
of quantum numbers.
•
Here, Quantum Numbers represent States.
• States can be thought of as available seats for electrons
in conduction band, as well as for holes in valence band.
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3-D Carriers (d = 3)
S(E): DOS (density of states) function, the number of states
per unit energy per unit volume.
mdse*: electron DOS effective mass, which carries information
about the DOS in the conduction band
mdsh*: hole DOS effective mass, which carries information
about the DOS in the valence band
*
dse 3 / 2
2
1 2m
S(E)  2 (
)
2 
*
dsh 3 / 2
2
1 2m
S(E)  2 (
)
2 
E  EC
EV  E
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Fermi-Dirac Statistics
Fermi-Dirac distribution function delivers the probability of
filling an existing state at energy E.
f (E) 
1
1 e
( E  E f )/ kBT
1 - f(E): the probability of unfilled state E
Ef: Fermi Level
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Key Concepts:
1. Fermi Level only exists in semiconductors, metals, and around
insulators’ edges, NOT bulk insulators.
2. Fermi-Dirac Statistics (and hence Fermi level) only applies to
mobile electrons, mobile holes, and immobile (valence) electrons,
NOT phonons, plasmons, or (ionized) impurity level.
3. Fermi level must be first constructed in the equilibrium or quasiequilibrium condition and in the neutral or quasi-neutral region.
4. Fermi level is the only direct way with which our EE people can
go into device.
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Fermi level is related to one of laws of Nature:
Conservation of Charge
Positioning of Fermi level can reveal the doping details.
Extrinsic (doping) case
2-13
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Electron concentration

n   S ( E ) f ( E )dE
EC
EV
p   S ( E )(1  f ( E ))dE

Hole concentration
Effective density of states
in the conduction band
C = (Ef – EC)/kBT
n  NC exp(C)
p  NV exp(V)
V= (EV – Ef)/kBT
Case of EV < Ef < EC (Non-degenerate)
NC = 2(mdse*kBT/2ħ2)3/2
NV = 2(mdsh*kBT/2ħ2)3/2
Effective density of states
in the valence band
Note: for EV < Ef < EC, Fermi-Dirac distribution reduces
to Boltzmann distribution.
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Electron distribution function n(E)
Evidence of DOF = 3
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Case of EV < Ef < EC (Non-degenerate)
n  NC exp(C)
C = (Ef – EC)/kBT
p  NV exp(V)
V= (EV – Ef)/kBT
NC = 2(mdse*kBT/2ħ2)3/2
NV = 2(mdsh*kBT/2ħ2)3/2
For intrinsic case where n = p, at least four statements can be drawn:
• Ef is the intrinsic Fermi level Efi
• Efi is a function of temperature T and the ratio of mdse* to mdsh*
• Corresponding ni (= n = p) is the intrinsic concentration
• ni is a function of the band gap (= Ec- Ev)
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DOF (degree of freedom): d
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