Post on 23-Jan-2021
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Energy Band Diagrams
Mark Lundstrom
School of ECE Purdue University
West Lafayette, IN USA
Lundstrom: Fall 2019
ECE 255: Fall 2019 Purdue University
Goal for this lecture
Lundstrom: Fall 2019 2
VDD
+VD
−
ID VD( ) = ?
R
+ VR −
To set the stage for understanding how PN junction diodes work. i.e. what is ID(VD)?
IR VR( ) =VR R ID
Outline
Lundstrom: Fall 2019 3
1) Band bending and the electrostatic potential 2) Reading an energy band diagram 3) Drawing energy band diagrams
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Energy band diagrams
Lundstrom: Fall 2019
An energy band diagram is a plot of the bottom of the conduction band and the top of the valence band vs. position.
Energy band diagrams are a powerful tool for understanding semiconductor devices because they provide qualitative solutions to the semiconductor equations.
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Energy band diagrams
https://www.pbs.org/wgbh/americanexperience/features/silicon-timeline-silicon/
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Band bending in an MOS structure
xLundstrom: Fall 2019
pB = NA
nB = ni2 NA
semiconductor
SiO2 oxide
metal gate
V = 0+VG
What happens when we apply a voltage to the gate?
EC x→∞( ) = ECB
EVB
Electrostatic potential vs. position
x
+VG
V = 0
V x( )
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Voltage and electron potential energy
E = −qV
+V
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A positive potential lowers the energy of an electron.
-
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Electrostatic potential causes “band bending”
x
V = 0
EPE = EC x( ) = ECB − qV x( )
VG > 0
Lundstrom: Fall 2019
ECB
EVB
pB = NA
nB = ni2 NA
EC x( )
EV x( )
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Electric field
x
V = 0
EEC x( ) = ECB − qV x( )
dEC x( )dx
= −qdV x( )dx
= qE
VG > 0
The electric field is proportional to the slope of EC
ECB
EVB
pB = NA
nB = ni2 NA
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Electric field
x
E
The electric field is proportional to the slope of EC
E = 1
qdEC x( )dx
E
Fe
0
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Electron concentration
x
V = 0
En x( )∝ e−EC kBT ∝ eqV x( ) kBT
Lundstrom: Fall 2019
EVB
EIB
pB = NA
nB = ni2 NA
n x( ) = nBeqV x( ) kBT
qV x( )ECB
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Electron concentration
x
log10 n(x)
Lundstrom: Fall 2019
nB = ni2 NA
n x = 0( ) = ni2 NA( )× eqV x=0( ) kBT
n x( ) = nBeqV x( ) kBT
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Hole concentration
x
V = 0
E p x( )∝ e−qV x( ) kBT
VG > 0
Lundstrom: Fall 2019
EVB
pB = NA
nB = ni2 NA
p x( ) = pBe−qV x( ) kBT
qV x( )ECB
“depletion region”
W
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Electron and hole concentrations (log y-axis)
x
log10 n0 (x), p0 x( )
Lundstrom: Fall 2019
ni2
NA
× eqV x=0( ) kBT
nB = ni2 NA
NA × e−qV x=0( ) kBT
NA
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Electron and hole concentrations (linear y-axis)
x
n0 (x), p0 x( )
Lundstrom: Fall 2019
n0 (x)
p0 (x)
“depletion region”
Reading band diagrams
x
EC
EV
E
E ∝ dEC x( ) dx
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A band diagram Reading the band diagram
V x( )∝−EC x( )
p x( )∝ eEV x( ) kBT ∝ e−qV x( ) kBT
n x( )∝ e−EC x( ) kBT ∝ eqV x( ) kBT
Outline
Lundstrom: Fall 2019 18
1) Band bending and the electrostatic potential 2) Reading an energy band diagram 3) Drawing equilibrium energy band diagrams
Draw the E-band diagram
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N P
pp ! NA
ρ ! 0 nn ! ND
ρ ! 0 pn ! ni
2 ND np ! ni
2 NA
electron concentration high electron concentration low
(hole concentration high) (hole concentration low)
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Step 1: Draw a reference line
Lundstrom: Fall 2019
Eref = EF
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Step 2: Draw EC(x)
Lundstrom: Fall 2019
Eref = EF
EC x( )
x
E
EC is low where the electron concentration is high, and EC is high where the electron concentration is low.
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Step 3: Draw EV(x)
Lundstrom: Fall 2019
Eref = EF
EC x( )
x
E
EV x( )
EG
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Read the E-band diagram: V(x)
Lundstrom: Fall 2019
Eref = EF
EC x( )
x
E
EV x( )
EC x( ) = ECB − qV x( )
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Read the E-band diagram: V(x)
Lundstrom: Fall 2019
V x( )
x
E
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Read the E-band diagram: electric field
Lundstrom: Fall 2019
Eref = EF
EC x( )
x
E
EV x( )
EC x( ) = ECB − qV x( )dEC
dx= q − dV
dx⎛⎝⎜
⎞⎠⎟
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Read the E-band diagram: electric field
Lundstrom: Fall 2019
x
E
E x( )
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Read the E-band diagram: n(x) and p(x)
Lundstrom: Fall 2019
Eref = EF
EC x( )
x
E
EV x( )
n0 x( )∝ e−EC x( ) kBT
p0 x( )∝ e+EV x( ) kBT
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Read the E-band diagram: n(x) and p(x)
Lundstrom: Fall 2019
log10 n x( ), log10 p x( )
xN P xp−xn
pp = NA
pn = ni2 ND
nn = ND
np = ni2 NA
Summary
Energy band diagrams are a powerful tool for understanding the operation of semiconductor devices.
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To draw an E-band diagram:
1) Draw a horizontal line as an energy reference
2) Draw EC(x) low where n(x) is large and EC(x) high where n(x) is small.
3) Draw EV(x) = EC(x) - EG
Summary
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1) Find the electrostatic potential vs. position by turning turn EC(x) upside down.
2) Find the electric field vs. position by taking the slope of EC(x).
3) Find the carrier density vs. position by beginning where it is known, and then exponentially increasing or decreasing it according to the local EC(x).
To read an E-band diagram:
Energy band diagrams
Lundstrom: Fall 2019 31
1) Band bending and the electrostatic potential 2) Reading an energy band diagram 3) Drawing energy band diagrams