ECE 340 Lecture 9 : Carrier

Concentrations and the Temperature Dependence

Class Outline: • Intrinsic Carrier Concentrations • Extrinsic Carrier Concentrations •  Thermal Effects Revisited

• What is the charge neutrality relationship? • How are the carrier concentrations

determined for extrinsic material? • What role does temperature play

in carrier concentrations in extrinsic material? • What is compensated material?

M.J. Gilbert ECE 340 – Lecture 9 09/17/12

Things you should know when you leave…

Key Questions

M.J. Gilbert ECE 340 – Lecture 9 09/17/12

Intrinsic Carrier Concentrations

We recall that by using the density of states and the Fermi function…

For electrons:

For holes:

( ) ( )∫=top

c

E

Ec dEEfEgn

( ) ( )[ ]∫ −=v

bottom

E

Ev dEEfEgp 1

We can simplify these integrals to find closed form relations…

23

2

*

23

2

*

22

22

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡=

π

π

TkmN

TkmN

bpv

bnc

23

0

*,

319

,11051.2 ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ ×=mm

cmN pn

vc

3kbT

3kbT ( )

( )TkEE

v

TkEE

c

b

fv

b

cf

eNp

eNn−

−

=

=

M.J. Gilbert ECE 340 – Lecture 9 09/17/12

Intrinsic Carrier Concentrations

Is this the best and easiest??

( ) ηη

η ηη de

Fcc ∫

∞

−+=0

21

2/1 1

( )( ) 3

2

23

2/1 32

32

⎟⎟⎠

⎞⎜⎜⎝

⎛ −==

TkEE

Fb

cfcc ηη

( )( )

TkEE

cb

cf

eF−

=η2/1

But we can use curve fitting (no physics) to cover all of the ranges: ⎥

⎦

⎤⎢⎣

⎡+−=⎥

⎦

⎤⎢⎣

⎡++=

vvbv

ccbcf N

pNpTkE

Nn

NnTkEE

81ln

81ln

Joyce-Dixon Approximation:

M.J. Gilbert ECE 340 – Lecture 9 09/17/12

Intrinsic Carrier Concentrations

Recall that we can also find the dependence on temperature…

For intrinsic semiconductors, we know the following: n = p = ni and Ei = Ef

Then the relations for n and p become:

TkEiE

vi

TkEE

ci

b

v

b

ci

eNn

eNn−

−

=

=

TkEE

iv

TkEE

ic

b

vi

b

ic

enN

enN−

−

=

=

By combining equations, we arrive at two very important and useful results:

TkEE

i

TkEE

i

b

fi

b

if

enp

enn−

−

=

=TkE

vcib

g

eNNn 2−

=

M.J. Gilbert ECE 340 – Lecture 9 09/17/12

Extrinsic Carrier Concentrations

In most situations, we are dealing with extrinsic semiconductors…

Ed

Ea

Ev

Ev

Δx

Donor Doped Semiconductor Band Diagram:

Acceptor Doped Semiconductor Band Diagram:

M.J. Gilbert ECE 340 – Lecture 9 09/17/12

Extrinsic Carrier Concentrations

Let’s consider a small separate parts of a uniformly doped semiconductor…

+

+

+

-

M.J. Gilbert ECE 340 – Lecture 9 09/17/12

Extrinsic Carrier Concentrations

The preceding analysis leads to the charge neutrality relationship…

0

arg3

=−+−=

−+−=

−+

−+

AD

AD

NNnp

qNqNqnqpcm

ech

0=−−− AD NNnpAA

DD

NNNN

=

=−

+

M.J. Gilbert ECE 340 – Lecture 9 09/17/12

Extrinsic Carrier Concentrations

Let’s use the charge neutrality relation…

nnp i2

= 02

=−+− ADi NNnnn

( ) 022 =−−− iAD nNNnn

21

22

22 ⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠

⎞⎜⎝

⎛ −+

−= i

ADAD nNNNNn21

22

22 ⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠

⎞⎜⎝

⎛ −+

−= i

DADA nNNNNp

M.J. Gilbert ECE 340 – Lecture 9 09/17/12

Extrinsic Carrier Concentrations

The previous equations are generic, we can simplify in 4 cases…

NA = ND = 0

n = p = ni

D

i

D

Nnp

Nn2

=

≈

A

i

A

Nnn

Np2

=

≈or

n = p = ni

M.J. Gilbert ECE 340 – Lecture 9 09/17/12

Extrinsic Carrier Concentrations

What about the 4th case? Compensated Semiconductors

+      +      +      +      +      +      +  

-­‐        -­‐        -­‐        -­‐        -­‐        -­‐        -­‐  

Sum of positive charges (holes and ionized donors) must balance sum of negative charges (electrons and ionized acceptors): !

p0 +Nd+ = n0 +Na

-  

M.J. Gilbert ECE 340 – Lecture 9 09/17/12

Extrinsic Carrier Concentrations

Where is the intrinsic level, Ei, really located?

In intrinsic material, we know that n=p. Now use the Boltzmann equations:

TkEE

vTkEE

cb

if

b

fi

eNeN−−

=

But,

23

*

*

⎟⎟⎠

⎞⎜⎜⎝

⎛=

n

p

c

v

mm

NN

So,

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+= *

*

ln43

2 n

pb

vci m

mTkEEE

M.J. Gilbert ECE 340 – Lecture 9 09/17/12

Extrinsic Carrier Concentrations

So where does the Fermi level lie for a doped semiconductor??

Solve for Ef – Ei: TkEE

i

TkEE

i

b

fi

b

if

enp

enn−

−

=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛=−

⎟⎟⎠

⎞⎜⎜⎝

⎛=−

i

Abfi

i

Dbif

nNTkEE

nNTkEE

ln

ln ND >> NA , ND >> ni

NA >> ND , NA >> ni

M.J. Gilbert ECE 340 – Lecture 9 09/17/12

Extrinsic Carrier Concentrations

So where does the Fermi level lie for a doped semiconductor??

Fermi level positioning in Si at 300 K as a function of the doping concentration…

M.J. Gilbert ECE 340 – Lecture 9 09/17/12

Thermal Effects Revisited

What is the role of temperature??