PDF Effective Density of States

E_C

E_V

C

C E E E

N ( ) ∝ −

E E E

N_V( ) ∝ _V −

E_C

E_V

) (

)

( _{C C}

C E N E E

N = δ −

) (

)

( _{V V}

V E N E E

N = δ −

δ- function with magnitude of NC

) (

) ( ) (

/ ) ( 0

C C

kT E E C

C

E f N

e N

dE E f E N n

F C

=

=

=

−

−

∫

∞

) (

) ( )

0 (

C C

C C

E f N

dE E f E E N n

=

−

=

∫

^∞ δ

For intrinsic material,

kT E E C E E C C i

i C i

F N e

E f N

n = ( ) ₌ = ^{−( − )/}

i kT E E V

i N e n

p = ^{−( i− V)/} =

kT E V C i i i

e g

N N n p

n = ² = ^{− /} E kT

V C i

e

g

N N

n =

^{− /2}

∴

and from n_i = p_i N_Ce^{−(EC−Ei)/kT} = N_Ve^{−(Ei−EV)/kT}

g V

n p g

V C

V V

C

i E E

m kT m E

N E kT N E

E

E 2

) 1 4 ln(

3 2

) 1 2 ln(

) 1 2(

1 − + = + + ≅ +

=

∴

Effective Density of States

Mass Action Law

For extrinsic material,

2 / /

) ( / ) (

i kT E V C kT E E kT E E V

CN e e N N e n

N

np = ^{− C− F − F− V} = ^{− g} =

2

n

i

np =

“mass action law” always true for nondegenerate

kT E E kT i

E E C

kT E E C i

i F i

C F C

e n e n

N e N n

n

_{( )/}

/ ) (

/ )

( −

−

−

−

−

=

⇒

=

kT E E kT i

E E V

kT E E V i

i

F i V

i V F

e n e p

N e N p

n

p

_{( )/}

/ ) (

/ )

( −

−

−

−

−

⇒ =

= =

• n decreases as E_Fmoves farther below E_C, and vice versa; p decreases as E_Fmoves farther above E_V.

• When E_Fis about 20 meV (~ kT) from E_C or E_V ( heavily doped semiconductor, > ~ 10¹⁹cm^-3), the Boltzmann approximation is no longer valid.

Degenerate

Degenerated and non-degenerated SC If doping concentration is small

→No interaction between donor electrons (N-type)

→discrete donor energy state

→non-degenerate semiconductors If doping concentration is large

→Donor electrons begin to interact with each other

→discrete donor energy will split into a band of energies

→If N_d~N_c, the band of donor states may overlap the bottom of the conduction band

Location of Fermi level vs. dopant concentration in Si at 300 and 400 K.

General Theory of n and p

• Assumptions: 1) uniformly doped semiconductor and nondegenerate (np = n_i²) 2) full ionization of the dopant atom (shallow impurities)

• From charge neutrality and mass action law, p − n + N_d − N_a = 0 and np = n_i² (n_i² / n) − n + N_d − N_a = 0

n² − (N_d − N_a) n − n_i² = 0 Solve this quadratic equation for the free electron concentration, n, and take only the plus root

2 / 2 1

2

]

2 )

2 [(

ⁱ

a d

a

d

N N N n

n = N − + − +

Similarly for hole concentration, p,

2 / 2 1

2

]

2 )

2 [(

ⁱ

d a

d

a

N N N n

p = N − + − +

1. Intrinsic semiconductor (N_a = 0, N_d = 0) n = n_i and p = n_i

2. N_d – N_a>> n_i (i.e., N-type) n = N_d – N_aand p = n_i²/n

If, furthermore, N_d >> N_a, then n = N_d and p = n_i²/ N_d

3_. N_a – N_d >> n_i (i.e., P-type) p = N_a – N_d and n = n_i²/p

If, furthermore, N_a >> N_d, then p = N_a and n = n_i²/ N_a

4. n_i >> |N_d – N_a|

This can be happened at very high temperature even for doped semiconductor n = p = n_i

All semiconductor become intrinsic at very high temperature.

5. Compensation

Both donors and acceptors are present in a semiconductor and N_d and N_aare comparable and nonzero.

If N_d > N_a (N-type)

+ N_d

N_a

+ + + + + +

If N_d < N_a (P-type)

N_d N_a

+ + + +

If N_d= N_a (exact compensation)

n N N

N_d,eff = _d − _a ≈ N_a,eff = N_a −N_d ≈ p

ni

p n = =

N_d N_a

+ + + +

Temperature Dependence of Carrier Concentrations

n_i

extrinsic

This region should be wide to have good device performance and temperature characteristics.

Wide bandgap material (small n_i) is preferred in general

N_d dominant Example

For donor

n_i>>N_d

◦◦ ◦◦

N_d Very small n~N_d>>n_i

◦

N_d

n~N_d⁺<N_d

◦

N_d

n~0 (at 0 K)

kT E kT

E

kT E v c i

g g

g

e e

KT

e N N T

n

2 / 2

2 / 3

2 /

) (

) (

−

−

−

∝

∝

= Variation of carrier concentration in an N-type

semiconductor over a wide range of temperature

Increasing temperature

Intrinsic Carrier Concentrations in Ge, Si, and GaAs vs. T

• Drift is the motion of charge carriers caused by an electric field

Drift

Electron and Hole Mobilities

• Assume that the mean free time between collisions is τ_mp (τ_mn)

and that the carrier loses its entire drift momentum, m_pv (m_nv) after each collision.

the drift momentum gained between collision is equal to .

q  × τ

_mp

E

Consider the case for hole,

An electric field creates a drift velocity that is superimposed on the thermal velocity

v 

th

due to collision

on average

⇒

^{+ -}

 v

E

sec / 10

3 )

(

^{1/2 7}

cm m

v kT

n

th

= ≈

Thermal velocity

Drude Theory on Mobility

0

0

mp

p p p

v

p mp p mp

F m a m dv q m dv q dt

dt

m dv q m v q

τ

τ τ

= = = ⇒ =

⇒ = ∴ =

∫ ∫

∫

  

  

 

 

E E

E E

mp mp

p p

p p

q q

v m m

τ τ

µ µ

=  =  ∴ =

 E E

• Assume that the carrier loses its entire drift momentum, m_pv, after each collision.

for hole

mn mn

n n

n n

q q

v m m

τ µ µ τ

= −  = −  ∴ =

 E E

for electron

Low field mobility

Mechanisms of Carrier Scattering

,

, ,

, mn mp 1

n p mn mp

n p

q

m amount of scattering

µ = τ ∝ τ ∝ Function of temperature and

doping concentration

• Phonon(or lattice vibration) scattering

Crystal vibration distorts the periodic crystal structure and thus scatters the electron waves.

• Ionized impurity scattering

Ionized impurity scattering becomes dominant at low temperatures. At higher temperature, the

ionized impurity scattering becomes weaker and hence the mobility becomes higher.

What are the imperfections in the crystal that cause carrier collisions or scattering?

,

( _impurity ^impurity , _impurity )

n p

q mean free time of impurity scattering m

µ = τ τ =

3/ 2 1/ 2

1 1 1

ph ph T

amount of scattering phonon density carrier thermal velocity T T

µ

∝

τ

∝ ∝ ∝ ∝ ⁻

× ⋅

,

( _phonon ^ph , _ph )

n p

q mean free time of phonon scattering m

µ = τ τ =

a d

impurity impurity

N N

T impurity

ionized of

Sum

T

scattering of

amount

∝ +

∝

∝

∝

2 / 3 2

/ 3

τ 1 µ

An electron can be scattered by an acceptor ion (a) and a donor ion (b) in a strikingly similar manner, even though the ions carry opposite types of charge.

The same is true for a hole.

Electrical analogy for scattering

1

ph

phonon

R ^∝τ

1

impurity

impurity

R ^∝τ

ph impurity ph

R = R +R ≅ R

impurity

≅ R

: with low doping and high temperature : with heavily doping

and low temperature

R 1

∝

τ

1 1 1

phonon impurity

τ τ

^{= +}

τ

^+

1 1 1

phonon impurity

µ µ µ

∴ = + +

T μ

ionized impurity scattering

phonon scattering

total mobility T⁻3/ 2

∝ T3/ 2

∝ Probability of being scattered by ionized impurity:

Probability of being scattered by phonons:

Total probability of being scattered:

1

i

impurity

P ^∝

τ

1

ph

phonon

P ^∝

τ

...

ph impurity

P = P +P +

P 1

∝

τ

1 1 1

phonon impurity

τ τ

^{= +}

τ

^+

1 1 1

phonon impurity

µ µ µ

∴ = + +

Mobility vs. dopant concentration for Si at 300 K

( )

2

17 0.7

( / ) 420 50

1 [ /1.6 10 ]

p

a d

cm V s

N N

µ ⋅ = +

+ + ×

( )

2

17 0.85

( / ) 1318 92

1 [ /1 10 ]

n

a d

cm V s

N N

µ ⋅ = +

+ + ×

due to free-carrier screening

When the carrier concentration is large, the carriers can distribute themselves to partially screen out the coulomb field of the dopant ions.

Ionized impurity scattering can be neglected.

Mobility decreases with increasing doping.