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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

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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

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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

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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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Band bending in an MOS structure

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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

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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

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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)

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( )

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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( )

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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

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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

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Eref = EF

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Step 2: Draw EC(x)

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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)

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Eref = EF

EC x( )

x

E

EV x( )

EG

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Read the E-band diagram: V(x)

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Eref = EF

EC x( )

x

E

EV x( )

EC x( ) = ECB − qV x( )

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Read the E-band diagram: V(x)

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V x( )

x

E

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Read the E-band diagram: electric field

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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

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x

E

E x( )

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Read the E-band diagram: n(x) and p(x)

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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

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1)  Band bending and the electrostatic potential 2)  Reading an energy band diagram 3)  Drawing energy band diagrams