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ECE 255: L5
Energy Band Diagrams
Mark Lundstrom School of ECE
Purdue University West Lafayette, IN USA
Spring 2019 Purdue University
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Energy band diagrams
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1) Band bending and the electrostatic potential 2) “Reading” an energy band diagram 3) PN junctions 4) Energy band diagram of a PN junction in equilibrium 5) Forward bias and reverse bias 6) The built-in potential
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Energy band diagrams
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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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Kroemer’s lemma of proven ignorance
“Whenever I teach my semiconductor device physics course, one of the central messages I try to get across early is the importance of energy band diagrams. I often put this in the form of “Kroemer’s lemma of proven ignorance:
If, in discussing a semiconductor problem, you cannot draw an Energy Band Diagram, this shows that you don’t know what you are talking about.”
(Nobel Lecture, 2000)
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Kroemer’s corollary
If you can draw one, but don’t, then your audience won’t know what you are talking about.”
(Nobel Lecture, 2000)
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Band bending in an MOS structure
x
EC
EV
EI
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p = NA
n = ni2 NA
semiconductor
SiO2 oxide
metal gate
+VG V = 0
What happens when we apply a voltage to the gate?
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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 vs. position
x
+VG
V = 0
V x( )
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Electrostatic potential causes band bending
x
V = 0
EPE = EC x( ) = EC +∞( )− qV x( )
VG > 0
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ECB
EVB
EIB
pB = NA
nB = ni2 NA
EC x( )
EI x( )
EV x( )
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Electric field
x
V = 0
EEC x( ) = EC +∞( )− qV x( )
dEC x( )dx
= −qdV x( )dx
= qE
VG > 0
The electric field is proportional to the slope of EC
ECB
EVB
EIB
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
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Electron concentration
x
V = 0
En x( )∝ eqV x( ) kBT
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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)
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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
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EVB
EIB
pB = NA
nB = ni2 NA
ECB
p x( ) = pBe−qV x( ) kBT
qV x( )
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“depletion region”
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Electron and hole concentrations
x
log10 n(x)
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ni2
NA× eqV x=0( ) kBT
nB = ni2 NA
NA × e−qV x=0( ) kBT
NA
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Summary: Band diagrams
x
EC
EV
EI
E
E ∝ dEC x( ) dx
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A band diagram Reading the band diagram
V x( )∝−EC x( )
log p x( )∝ EV x( )− EVB
logn x( )∝ ECB − EC x( )
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Another example: NP junction (equilibrium)
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N P
pp ! NA
ρ ! 0 nn ! ND
ρ ! 0 pn ! ni
2 ND np ! ni
2 NA
“majority carriers”
“minority carriers”
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NP junction (equilibrium)
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N P
pp ! NA
ρ ! 0 nn ! ND
ρ ! 0
depletion region xp−xn 0
+
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EVL > VR
ρ < 0NA
−
ρ > 0ND
+
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Voltage vs. position
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V x( )
x
N P
xp−xn
Vbi“built-in potential”
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xp−xn
Electron energy vs. position
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E
x
N
P ΔE = qVbi
EC x( )
EV x( )EC
EV
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Electric field vs. position
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xN P xp−xn
E
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Carrier densities vs. position
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log10 n x( ), log10 p x( )
xN P xp−xn
pp = NA
pn = ni2 ND
nn = ND
np = ni2 NA
nn
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Equilibrium energy band diagram
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EC
x
W
E
qVbi
I = 0
VA = 0
PN
−xn xp
If we apply a + voltage to the P-side, what happens?
EC
N-side P-side
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Where does the voltage drop?
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R1 R2 R3VA
−V3 +−V2 +−V1 +
R2 >> R1,R2
V2 ≈ ?
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Forward bias à smaller energy barrier
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EC x( )
x
E
−xn xp
q Vbi −VA( )ΔE = qVA > 0
• Electric field decreases in magnitude • Width of depletion region decreases
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Reverse bias à larger energy barrier
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EC
x
W
E
−xn xp
q Vbi −VA( )
• Electric field increases in magnitude • Width of depletion region increases
ΔE = q VA
VA < 0
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Equilibrium built-in potential
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EC
xW
E
−xn xp
EC
N-side P-side
n x( )∝ eqV x( ) kBT nn =ni2
NAeqVbi kBT ND =
ni2
NAeqVbi kBT
nn = ND np = ni2 NA
qVbi = ?
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Equilibrium built-in potential
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I = 0+Vbi −
Vbi =kBTqln NAND
ni2
⎛⎝⎜
⎞⎠⎟
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Summary
Energy band diagrams are a powerful tool for understanding the operation of semiconductor devices.
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To find the electrostatic potential vs. position, turn EC(x) upside down.
To find the electric field vs. position, take the slope of EC(x).
To find the carrier density vs. position, begin where it is know, and then exponentially increase or decrease according to the local electrostatic potential.
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Energy band diagrams
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1) Band bending and the electrostatic potential 2) “Reading” an energy band diagram 3) PN junctions 4) Energy band diagram of a PN junction in equilibrium 5) Forward bias and reverse bias 6) The built-in potential