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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

Lundstrom: 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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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?

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( )

“depletion region”

W

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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

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( )

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”

NP junction (equilibrium)

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N P

pp ! NA

ρ ! 0 nn ! ND

ρ ! 0

depletion region xp−xn 0

+

-

EVL > VR

ρ < 0NA

−

ρ > 0ND

+

Voltage vs. position

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V x( )

x

N P

xp−xn

Vbi“built-in potential”

xp−xn

Electron energy vs. position

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E

x

N

P ΔE = qVbi

EC x( )

EV x( )EC

EV

Electric field vs. position

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xN P xp−xn

E

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

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

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

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

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 = ?

Equilibrium built-in potential

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I = 0+Vbi −

Vbi =kBTqln NAND

ni2

⎛⎝⎜

⎞⎠⎟

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.

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