![Page 1: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/1.jpg)
1
Energy Band Diagrams
Mark Lundstrom
School of ECE Purdue University
West Lafayette, IN USA
Lundstrom: Fall 2019
ECE 255: Fall 2019 Purdue University
![Page 2: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/2.jpg)
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
![Page 3: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/3.jpg)
Outline
Lundstrom: Fall 2019 3
1) Band bending and the electrostatic potential 2) Reading an energy band diagram 3) Drawing energy band diagrams
![Page 4: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/4.jpg)
4
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.
![Page 5: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/5.jpg)
5
Energy band diagrams
https://www.pbs.org/wgbh/americanexperience/features/silicon-timeline-silicon/
![Page 6: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/6.jpg)
6
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
![Page 7: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/7.jpg)
Electrostatic potential vs. position
x
+VG
V = 0
V x( )
Lundstrom: Fall 2019 7
![Page 8: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/8.jpg)
Voltage and electron potential energy
E = −qV
+V
Lundstrom: Fall 2019 8
A positive potential lowers the energy of an electron.
-
![Page 9: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/9.jpg)
9
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( )
![Page 10: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/10.jpg)
10
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
![Page 11: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/11.jpg)
11
Electric field
x
E
The electric field is proportional to the slope of EC
E = 1
qdEC x( )dx
E
Fe
0
![Page 12: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/12.jpg)
12
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
![Page 13: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/13.jpg)
13
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
![Page 14: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/14.jpg)
14
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
![Page 15: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/15.jpg)
“depletion region”
W
15
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
![Page 16: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/16.jpg)
16
Electron and hole concentrations (linear y-axis)
x
n0 (x), p0 x( )
Lundstrom: Fall 2019
n0 (x)
p0 (x)
“depletion region”
![Page 17: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/17.jpg)
Reading band diagrams
x
EC
EV
E
E ∝ dEC x( ) dx
17 Lundstrom: Fall 2019
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
![Page 18: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/18.jpg)
Outline
Lundstrom: Fall 2019 18
1) Band bending and the electrostatic potential 2) Reading an energy band diagram 3) Drawing equilibrium energy band diagrams
![Page 19: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/19.jpg)
Draw the E-band diagram
Lundstrom: Fall 2019 19
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)
![Page 20: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/20.jpg)
20
Step 1: Draw a reference line
Lundstrom: Fall 2019
Eref = EF
![Page 21: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/21.jpg)
21
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.
![Page 22: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/22.jpg)
22
Step 3: Draw EV(x)
Lundstrom: Fall 2019
Eref = EF
EC x( )
x
E
EV x( )
EG
![Page 23: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/23.jpg)
23
Read the E-band diagram: V(x)
Lundstrom: Fall 2019
Eref = EF
EC x( )
x
E
EV x( )
EC x( ) = ECB − qV x( )
![Page 24: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/24.jpg)
24
Read the E-band diagram: V(x)
Lundstrom: Fall 2019
V x( )
x
E
![Page 25: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/25.jpg)
25
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⎛⎝⎜
⎞⎠⎟
![Page 26: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/26.jpg)
26
Read the E-band diagram: electric field
Lundstrom: Fall 2019
x
E
E x( )
![Page 27: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/27.jpg)
27
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
![Page 28: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/28.jpg)
28
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
![Page 29: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/29.jpg)
Summary
Energy band diagrams are a powerful tool for understanding the operation of semiconductor devices.
29 Lundstrom: Fall 2019
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
![Page 30: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/30.jpg)
Summary
30
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:
![Page 31: Energy Band Diagrams - nanoHUB · 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](https://reader033.vdocuments.us/reader033/viewer/2022060100/60b0571af0dee866c00aaed1/html5/thumbnails/31.jpg)
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