Carrier Concentrations Presented by Chanam Lee August 22nd

Carrier Concentration Carrier Properties State and Carrier Distributions Equilibrium Carrier Concentration Carrier Concentration for the Quantum Well Devices

Carrier Properties Carrier Movement in Free Space Carrier Movement Within the Crystal Intrinsic Carrier Concentration Extrinsic n-Type Semiconductor Extrinsic p-Type Semiconductor

Electronic Materials Two-dimensional representation of an Individual Si atom. Elemental semiconductors Valence

Represents each valence electron

III

IV

V

B

C

Al

Si

P

Ga

Ge

As

Semiconductors When Si (or Ge & GaAs) atoms contact other Si atoms, they form a tetrahederal 2D representation of lattice structure:

Electronic Materials Two dimensions

Three dimensions

Carrier Movement in Free Space

Newton’s second law

dv F = −qE = m0 dt

Carrier Movement Within the Crystal Electrons moving inside a semiconductor crystal will collide with semiconductor atoms ==> behaves as a “wave” due to the quantum mechanical effects The electron “wavelength” is perturbed by the crystals periodic potential

Carrier Movement Within the Crystal m n* (electron effective mass)

m *p (hole effective mass)

dv F = − qE = m dt

dv F = − qE = m dt

* p

* n

Material

mn* / m0

m*p / m0

Si

1.18

0.81

Ge

0.55

0.36

GaAs

0.066

0.52

Density of States Effective Masses at 300 K

Intrinsic Carrier Concentration Contains an insignificant concentration of impurity atoms Under the equilibrium conditions, for every electron is created, a hole is created also n = p = ni As temperature is increased, the number of broken bonds (carriers) increases As the temperature is decreased, electrons do not receive enough energy to break a bond and remain in the valence band.

Extrinsic n-Type Semiconductor Donors (Group V): The 5th in a five valence electrons is readily freed to wander about the lattice at room temperature There is no room in the valence band so the extra electron becomes a carrier in the conduction band Does NOT increase the number of hole concentration

Extrinsic p-Type Semiconductor Acceptors (Group III) : three valence electrons readily accept an electron from a nearby Si-Si bond Completing its own bonding creates a hole that can wander about the lattice Does NOT increase the number of electron concentration

State and Carrier Distribution How the allowed energy states are distributed in energy How many allowable states were to be found at any given energy in the conduction and valence bands? Essential component in determining carrier distributions and concentration

Density of States Fermi Function Dopant States

Density of States:

Number of cm 3

gc (E) =

States

mn* 2mn* ( E − Ec )

π

2

3

,

/

eV

E ≥ Ec

g c ( E )dE represents the number of conduction band states/ cm 3 lying in the energy range between E and E + dE

gv (E) =

m *p 2m *p ( Ev − E )

π

2

3

, E ≤ Ev

g v ( E )dE represents the number of 3 valence band states / cm lying in the energy range between E and E + dE

g c ( Ec ) = g v ( Ev ) = 0

Fermi Function (I) How many of the states at the energy E will be filled with an electron

f ( E) =

1

1 + e ( E −EF ) / kT

f(E), under equilibrium conditions, the probability that an available state at an energy E will be occupied by an electron 1- f(E), under equilibrium conditions, the probability that an available state at an energy E will NOT be occupied by an electron

Fermi Function (II) If E = E F , f ( E F ) = 1 / 2 If E ≥ EF + 3k BT , f ( E ) ≈ exp[( EF − E ) / k BT ] If E ≤ E F − 3k BT , f ( E ) ≈ 1 − exp[( E − EF ) / k BT ] At T=0K (above), No occupation of states above EF and complete occupation of states below EF At T>0K (below), occupation probability is reduced with increasing energy f(E=EF) = 1/2 regardless of temperature. At higher temperatures, higher energy states can be occupied, leaving more lower energy states unoccupied (1-f(E))

Fermi Function (III)

Dopant States n-type: more electrons than Holes Intrinsic: Equal number of electrons and holes p-type: more holes than electrons

Density of States

Occupancy factors

Carrier Distribution

Equilibrium Carrier Concentration Formulas for n and p Degenerate vs. Non-degenerate Semiconductor Alternative Expressions for n and p ni and the np Product Charge Neutrality Relationship Carrier Concentration Calculations Determination of EF

Formulas for n and p n= n=

Etop Ec

mn* 2mn*

π2

Ec

E − Ec kT

E = Ec

when

n=

Etop

3

Letting η =

Let

p=

g c ( E ) f ( E ) dE

ETop * n

m

( E − Ec ) dE ( E − E F ) / kT 1+ e and ηc =

Ebottom

Nc = 2

E F − Ec kT

gv (E)[1− f (E)]dE

2πm kT h * n 2

3/ 2

, Nv = 2

2πm*p kT h2

, η =0

→ ∞ * n 2 3

2m ( kT )

π

Ev

n ≡ Nc 3/ 2

E∞ E0

η

1/ 2

1+ e

(η −η c )

F1 / 2 (η c )

dη

p ≡ Nv

2

π 2

π

F1 / 2 (η c ) F1 / 2 (η v )

3/ 2

Degenerate vs. Non-degenerate Semiconductor E F < E c − 3k B T , f ( E ) =

If

1 ≅ e −(η −η c ) (η −η c ) 1+ e

F1/ 2 (η c ) =

π

2

e ( EF − Ec ) / kT

n = N c e ( E F − Ec ) / k B T , p = N v e ( E v − E F ) / k B T

If

E F > Ev + 3k BT , F1/ 2 (η v ) =

π 2

e ( Ev − EF ) / kT

Alternative Expressions for n and p n0 = N c e

p0 = N v e ( E v − E F ) / k B T

( E F − Ec ) / k B T

When n=ni, EF = Ei, then

ni = N c e( Ei − Ec ) / kBT = N v e( Ev − Ei ) / k BT

N c = ni e

( E c − Ei ) / k B T

, N v = ni e

( Ec −Ei + EF −Ec ) / kBT

n0 = nie

( Ei − E v ) / k B T

( EF −Ei ) / kBT

= ni e

p0 = ni e ( Ei − Ev + Ev − EF ) / k BT = ni e ( Ei − EF ) / k BT

n0 p0 = ni

2

ni and the np Product ni = N c e

( Ei − Ec ) / k BT

n = Nc Nve 2 i

= N ve

− ( Ec − Ev ) / k B T

( E v − Ei ) / k BT

= Nc Nve

ni = N c N v e

− E g / 2 k BT

− E g / k BT

Charge Neutrality Relationship For uniformly doped semiconductor: Charge must be balanced under equilibrium conditions otherwise charge would flow

− A

+ D

qp − qn − qN + qN = 0 Thermally generated + assume ionization of dopant addition all dopant sites

Carrier Concentration Calculations ( p − N A ) + ( N D − n) = 0 ni2 ( − N A ) + ( N D − n) = 0 n n 2 − n( N D − N A ) − ni2 = 0 n=

ND − N A + 2

ND − N A 2

1/ 2

2

+ ni2

,p=

np = n i

2

N A − ND + 2

N A − ND 2

1/ 2

2

+ ni2

+ D

Relationship for N and N N A− =

NA

1+ g A

( E A − E F ) / kT

, N D+ =

− A

ND 1+ gD

( E F − E D ) / kT

The degeneracy factors account for the possibility of electrons with different spin, occupying the same energy level (no electron with the same quantum numbers can occupy the same state) Most semiconductor gD=2 to account for the spin degeneracy at the donor sites gA is 4 due to the above reason combined with the fact that there are actually 2 valence bands in most semiconductors Thus, 2 spins x 2 valance bands makes gA=4

Determination of EF (Intrinsic Material) n = Nce( Ei −Ec ) / kBT = Nve( Ev −Ei ) / kBT = p N c e ( Ei − E c ) / k B T = N v e ( E v − Ei ) / k B T Ei =

Ec + Ev k BT N + ln v 2 2 Nc

2πm kT Nc = 2 h * n 2

3/ 2

, Nv = 2

2πm kT h

* p 2

3/ 2

m *p

Nv = Nc mn*

1/ 2

m*p Ec + Ev 3k BT Ei = + ln * mn 2 4

Determination of EF (Extrinsic Material) n = ni e

( E F − Ei ) / k B T

,

p = ni e

( Ei − E F ) / k B T

n p E F − Ei = kT ln = − kT ln ni ni

or

for ND >>NA and ND >>ni N E F − Ei = kT ln D ni

or

for NA >>ND and NA >>ni E F − Ei = − kT ln

NA ni

Determination of EF (Extrinsic Material)

Fermi level positioning in Si at room temperature as a function of the doping concentration. Solid EF lines were established using Eq. (previous page)

Carrier Concentration for the Quantum Well Devices Density of States 3D vs. 2D Carrier Concentration – 2D Charge Neutrality

Density of States 3D vs. 2D 3D

g (E) = 2D

g=

m 2mE

π

2

3

m

π

2

3

LZ

Energy Dependent

Energy Independent

Carrier Concentration – 2D n=

Etop Ec

g c ( E ) f ( E ) dE

mn* kT n= 2 3 π Lz

p=

i

m*p kT

π

2

3

Lz

[

ln 1 + e ( EFC − Ei ) / kT

[

ln 1 + e i

( E FV − Ei ) / kT

]

]

Charge Neutrality

Phh + Plh = N Γ + N Χ + N L

References Robert F. Pierret, Semiconductor Fundamentals (VOLUME I), Addison-Wesley Publishing Company, 1988, chapter 2 Robert F. Pierret, Advanced Semiconductor Fundamentals (VOLUME VI), Addison-Wesley Publishing Company, 1987, chapter 4 Ben G. Streetman and Sanjay Banerjee, Solid State Electronic Devices, Prentice Hall, Inc., 2000, chapter 3 Peter S. Zory, JR., Quantum Well Lasers, Academic Press, 1993, chapter 1, 7

Effective Mass of Holes - 3D gv (E) =

=

m*p 2m*p ( Ev − E )

π2

3

* * mhh 2mhh ( Ev − E)

π

2 3

+

mlh* 2mlh* ( Ev − E)

π2

3

* 3/ 2 (m *p ) 3 / 2 = (mhh ) + (mlh* ) 3 / 2

[

m = (m ) * p

* 3/ 2 hh

]

* 3/ 2 2 / 3 lh

+ (m )