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Chapter 4

The Semiconductor in Equilibrium

Chapter 4 In this chapter, we will apply the concepts of quantum mechanicsto a semiconductor material.In particular, we wish to determine the concentration of electronsand holes in the conduction and valence bands.

Section 4.1.1 Equilibrium Distribution of electrons and holesThe distribution (with respect to energy) of electrons in theconduction band is given by the density of allowed quantum statestimes the probability that an electron occupies a state.where f

(E) is the Fermi-Dirac probability function and gF

(E) is thec

density of quantum states in the conduction band.

) ( ) ( ) (^

E f E g E n^

F

c

=

Equilibrium Distribution of electrons and holesThe distribution (with respect to energy) of holes in the valenceband is given by the density of allowed quantum states in thevalence band times the probability that a state is

not

occupied by

an electron. To find the thermal-equilibrium electron and hole concentrations,we need to determine the position of the Fermi energy E

withF

respect to the bottom of the conduction-band energy E

and the topc

of the valence-band energy E

[^ .v

]) ( (^1) ) ( ) (^

E f E g Ep

F

v^

−

=

Intrinsic SemiconductorThe following figure shows a possible density of states for an E

nearF

the midgap energy. The area under the curves in the figure is then

the total density of electrons in the conduction band andthe total density of holes in the valence band.

Intrinsic Semiconductor^ It was assumed in the figure that the effectivemasses of the electron and hole were equal.When this is not the case, the density of states function will not besymmetrical about the midgap energy.

The n

and p 0

Equations 0

After some mathematics, we can write the electron concentration as:Define a parameter called effective density of states function in theconduction band N

.c

3

(^

)

*^

2

0

2 2 2

c^

F E^

E kT

m kTn

n^

e

π^ h

−^

− ⎡^

⎤

⎢^

⎥

⎣^

⎦

⎛^

⎞

=^

⎜^

⎟

⎝^

⎠

3 *^

2 2 2 2

n

c

m kT

N^

π h ⎛^

⎞

=^

⎜^

⎟

⎝^

⎠

The n

and p 0

Equations 0

Then the thermal-equilibrium electron concentration can be written asN^ c^

is found for T = 300 K on page 713 in your text.

⎤ ⎥⎦

⎡^ ⎢⎣

− −

=^

EE kT

c

F c e N n

) (

0

The n

and p 0

Equations 0

Again, define a parameter called effective density of states function inthe valence band N

.v

Then the thermal-equilibrium hole concentration can be written as

3 *^

2 2 2 2

p

v

m kT

N^

π h ⎛^

⎞

=^

⎜^

⎟

⎜^

⎟

⎝^

⎠

⎤ ⎥ ⎦

⎡^ ⎢ ⎣

− −

=^

kT

E E v

v F e N p

)

(

0

The magnitude of N

is on the order of 10v

19 cm

-3^ at T = 300 K for most

semiconductors.

Exercise 4.1 on page 110Calculate the thermal equilibrium electron and hole concentration in siliconat T = 300K.Egap = 1.12 eV from table B.4 on page 713.

⎤ ⎥⎦ ⎡^ ⎢⎣

−−

=^

EE kT c

Fc eN n

) ( 0

What is N

? Back to table B.4 on page 713.c

N^ c

= 2.8 x 10

19 cm

-^.

3

(^ (

))

19

1

0

(2.

10

)

E^ E^ c^ c

eVeV

cm

n^

−^ e −^ − ⎡^

⎤

⎢^

⎥

⎣^

⎦

=^

×^

3

19

1

(2.

10

)

eVeV cm^

−⎡ (^) e

⎤ ⎢^

⎥ ⎣^

⎦

=^

×

[^

] 3

19 1 (2.

10

−) e (^) cm

=^

×^

3 19

4

1

(2.

10

)(2.

10

)

cm

−

=^

×^

×

3 15 1

10

thermal equilibrium electron concentration cm

=^

×

where the Fermi energy level is 0.22 eV below the conduction band energy E

.c

E= EF^

• 0.22 eVc

thermal

T

kT

Let V

V^

e =^

=^

eV

Section 4.1.3 The Intrinsic Carrier ConcentrationFor an intrinsic semiconductor, the concentration of electrons in theconduction band is equal to the concentration of holes in the valenceband. n^ i^

= electron concentrationp= hole concentrationi where n

= pi^

as stated earlier for an intrinsic semiconductor.i

The Fermi energy level for the intrinsic semiconductor is called theintrinsic Fermi energy E

= EF

.Fi

⎤ ⎥⎦

⎡^ ⎢⎣

− − = =^

EE kT

c i

Fi c e N n n

) (

0

The Intrinsic Carrier Concentrationand for the hole concentrationWe take the product of the last two equations, we find the square of theelectron/hole concentration.

⎡^ ⎢⎣

= =^

EE kT

v

i

v Fi e N p p

⎤ ⎥⎦

⎡^ ⎢⎣

− − ⎤ ⎥⎦

⎡^ ⎢⎣

− −

=^

EE kT

EE kT

v c i

v Fi

Fi c

e

e

N

N

n

) ( ) (

2

⎤ ⎥ ⎦ ⎡ −⎢ ⎣

⎤ ⎥⎦ ⎡^ ⎢⎣

− −

⋅

⋅ =^

E kT vc

EE kT vc i

gap

vc

e NN

e NN n

) (

2

The Intrinsic Carrier ConcentrationNote that the N

and Nc^

values given in the text are for T = 300 K.v

From eqn 4.10 on page 107 and eqn 4.18 on page 109, we see that N

v

and N

have a strong dependence on temperature.c^

So for temperatures other than 300 K, we must adjust N

and Nv

by thec

following:“Temperature Twiddle factor” =

3 2

300

K) (^300) n

(other tha

⎞ ⎟ ⎠

⎛ ⎜ ⎝

K

T

The thermal voltage V

also changes! Recalculate VT

= kt/e for theT

different temperature.

Exercise 4.3 on page 113Find the intrinsic carrier concentration in silicon ata) T = 200K.

kT^

⎛ eV

=^

⎜^

⎝^

⎠^

eV

23 19

J K

or kT

K

− C

⎛^

×

= ⎜^

×

⎝^

eV

3

2

1

1

E^ g kT

i^

v^ c

T

n^

N N

e^

where T

Temperaturein K

−⎛ ⎞ ⎜ ⎟⎜ ⎟⎝ ⎠

⎛^

=^

⎜^

⎝^

N= 2.8 x 10c^

19 cm

-3^ , N

= 1.04 x 10v

19 cm

-3^ , and E

= 1.12 eV.gap

3

2

19

19

)^

n^ i

−⎛ e

⎞ ⎜^

⎟ ⎝^

⎠

⎛^

=^

×^

×^

⎜^

⎝^

⎠^

9 16

cm

=^

×

4 13

i^

cm

n^ =

×