Chap1. The Crystal Structure of Solids - KOCw

Velocity Saturation and Ballistic Transport

Slope=mobility

1

0 1

1 for holes in Si 2 for electrons in Si

d

sat

sat

v v

v

β β

µ µ µ

β β

 →

=     =  → ∞

 +   

   

 

≅

≅



 E E E

E E

Low field region

due to high energy

optical phonon scattering

High field region

2

7

1 40

2

2 / 10 /

sat opt

sat opt

mv E meV

v E m cm s

≈ ≈

≈ ≈

When device size is smaller than the mean free path,

the carriers experience a nearly scattering-free environment called “ballistic

2 mp

p p

p

where qp pq

m

σ = µ = τ : hole conductivity

,

E E , E

E

p drift p p p

p

J qpv qp v q

m

µ σ µ τ

= = = = =

2 3

ampere coulomb electrons cm

cm electron cm s

 = ⋅ ⋅ 

 

 

,

,

n drift n n n

n

J qnv qn v q

m

µ σ µ τ

= − = E = E = − E = E

, ,

( ) ( ) , ,

drift n drift p drift n p n p drift

n p

J J J qn qp J

qn qp

µ µ σ σ σ

σ µ µ

= + = + = + =

= +

E E E

2 mn

n n

n

where qn nq

m

σ = µ = τ : electron conductivity

: conductivity

p

µ v

⇒ =

E

/ 2/

cm V

cm V s s cm

 = ⋅ 

 

 

( )

A S siemens

V cm cm

 = 

 ⋅ 

 

A P-type semiconductor bar of unit area is used

to demonstrate the concept of current density.

1

t t

L L

R = ρ w = w σ ρ 1

= σ : resistivity [ ]

( ) 1

A S siemens V cm cm cm

 = − = Ω ⋅

 ⋅ 

 

Drift and Resistance

Conversion between resistivity and dopant

density of silicon at room temperature.

• The Hall effect is related to the force that acts on a charged particle that moves in a magnetic field [and electric field].

• If a magnetic field is applied perpendicular to the direction in which

holes drift in a P-type semiconductor bar, the path of the holes tends to be deflected. In the x-direction the force is

( E ):

F r = q ur + × v r B Lorentz Force r

Illustration of the Hall effect. (a) The force due to the magnetic field B

deviates the hole trajectory.

(b) Accumulated holes create Hall field E

= -E

that counteracts the force from the magnetic field B

.

y

x z

Hall Effect

• In the x-direction the magnetic force is

• This causes holes to hit the side of the semiconductor bar. The holes accumulating at one side of the bar create an electric field

• This establishes a steady-state hole flow along the bar in the y-direction.

• If the width of the bar is W, a voltage V

can be measured between the opposite sides and it equals to (Hall voltage).

x

v B

− = E

y z

qv B

x

W

− E

E

.

− q

( ) 0

0 0

E

E E

E

x x

y y x y

z z

x x x y z

y y

z z z

F

F q v B F q v v

F B

     

     

= + × ⇒   =    + 

     

     

r ur r r

x

E

F q qv B

⇒ = +

At steady-state, the force along x-axis must be balanced,

E 0 E

x x y z y z

F = q + qv B = ⇒ qv B = − q

• The current density : J

(=qpv

)

• The Hall voltage is then

• Alternatively, R

can be expressed as

• A measurement of the Hall voltage from a known current and magnetic field yields a value for the hole concentration p.

1 1

y

H x y z z z z

s

J I I

V W v B W B W B W B

qp qp t W qp t

= − E = = = =

1

where is known as the Hall coeff cie t i n

z

H z H

H

IB

V I B R

qp t t

R

= =

1 given that

p H

p

p p

R qp

qp µ

σ

σ µ

= =

=

Diffusion

• A process whereby particles tend to spread out or redistribute as a result of their random thermal motion.

⇒ migrating on a macroscopic scale from regions of high particle concentration into region of low particles concentration.

• Within semiconductors, diffusion related carrier (charged particle: electron or hole) transport therefore gives rise to particle currents

Diffusion Current

,

n diffusion n

dn dn

J qD

dx dx

∝ =

dp dp

J qD

dx dx

∝ = −

[Spreading of a pulse of electrons by diffusion]

Particles diffuse from high-concentration locations

toward low-concentration locations.

n(x)

x

electron diffusion J

steady electron injection at surface

uniform semiconductor

x hole diffusion

J

steady hole

injection at surface

Hot point probe measurement

Hot A Cold

p-type

(-) J

energetic hole

diffuse away negative charge build up

Hot Cold

A

n-type

(+) J

energetic electron

diffuse away positive charge build up

p(x)

Diffusion-current equation

x x

Holes Electrons

concentration gradient: + hole flux (-x direction):

hole diffusion current:

|

p diff p

J = − qD ∇ p J n diff | = + qD N ∇ n

The diffusion current density = carrier flux × carrier charge

( ) ( ) ( ) ( )

p p n n

dp x dn x

x D x D

dx dx

φ = − φ = −

concentration gradient: + electron flux (-x direction):

electron diffusion current:

Derivation of Diffusion Current

1

x n(x)

n

2 n

x

: mean free path between collisions



small incremental distance

−  x

: sec

( ):

where A cross tion area A volume



The electron in segment(1) have equal chances of moving left or right, and in a mean free time τ

one-half of them will move into segment (2), vice versa for electrons in (2).

1 2

1 1

( ) ( )

2 n  A − 2 n  A

∴ The net # of electrons passing x

from left to right in one mean free time,

The rate of electron flow in the +x direction/unit area ⇒ electron flux

( electron flow /unit area·sec)

0 1 2

1 1 1

( ) { ( ) ( )}

2 2

n

mn

x n A n A

φ A

= τ  −  (

)

2

n n

= τ  −

+ 

x

Let the mean free path be a small differential length, 

x

x x

n x n n

n

∆

∆ +

= −

−

( ) ( )

1



, where x is taken at the center of segment (1) and . ∆x = 

2 0

( ) ( )

( ) 2

n

mn

n x n x x

x x

φ τ

− + ∆

∴ =

∆



2 2

0 0

( ) ( ) ( )

( ) ( ) lim

2 2

n n

mn x mn

n x n x x dn x

x x

x dx

φ φ

τ

τ

− + ∆ −

= = =

∆

 

dx x D

dn ( )

−

=

→ 0

∆x

from the definition of the derivative

, where D

is called “diffusion coefficient” of electron

Similarly for hole, ( )

p

( )

x D dp x φ = − dx

The diffusion current density becomes

,

( ) ( )

( )

n diffusion n n

dn x dn x

J q D qD

dx dx

= − − =

,

( ) ( )

( )

p diffusion p p

dp x dp x

J = − + q D = − qD

in 3-D

,

n diffusion n

J  = qD ∇ n

,

p diffusion p

J  = − qD ∇ p

Total electron and hole current density

, ,

( ) ˆ ( ) ( )

n n drift n diffusion n n

J J J q n x x qD dn x x

µ dx

= + =  +

  

E : 1-D

( , , ) ( , , ) ( , , )

n n

q n x y z µ x y z qD n x y z

=  + ∇

E : 3-D

, ,

( ) ˆ ( ) ( )

p p drift p diffusion p p

J J J q p x x qD dp x x

µ dx

= + =  −

  

E : 1-D

( , , ) ( , , ) ( , , )

p p

q µ p x y z x y z qD p x y z

=  − ∇

E : 3-D

Total current density flowing in a semiconductor

p

n

J

J

J    +

=

n(x)

+ -

p(x)

 (x) E

,

,

p diffusion p drift

φ φ

→ →

,

,

p diffusion p drift

J J

→ →

,

,

n diffusion n drift

φ φ

→ ←

,

,

n diffusion n drift

J J

← →

Relation between the Energy Diagram and

( ) . ( )

E C x = const − qV x

• E

and E

vary in the opposite direction from the voltage.

• E

and E

are higher where the voltage is lower.

( ) ( )

( ) 1 1

( ) dV x dE

x dE

x

x = − dx = q dx = q dx E

Energy band diagram of a semiconductor

under an applied voltage 0.7 eV is an arbitrary value.

( ), E ( )

V x x

Constancy of the Fermi Level

• It is always satisfied that no discontinuity on gradient can arrive in the equilibrium Fermi level E

, even if nonuniform doping.

Consider two semiconductors attached at x = 0 Material 1

1

( ) D E

)

1

( E f

Material 2

2

( ) D E

)

2

( E f

x x = 0

There is no current flow at equilibrium.

Electron transfer from 1 to 2 must be exactly

balanced by the opposite transfer of electron from 2 to 1.

1 2 1

( ) ( )

( )[1

( )]

r

= D E f E ⋅ D E − f E

2 1 2

( )

( )

( )[1

( )]

r

= D E f E ⋅ D E − f E

1

( ) ( )

( )

( ) ( )

( )

( ) D E f E D E D E f E D E f E

⇒ −

2

( )

( )

( )

( )

( )

( ) ( )

D E f E D E D E f E D E f E

= −

) ( )

(

1

E f E

f =

∴

1 2

( ) / 1 ( ) / 1

[1 e

]

[1 e

]

⇒ + = +

1 2 ^F

0,

F F

E E dE at equilibrium

∴ = ⇒ dx =

A piece of N-type semiconductor in which

the dopant density decreases toward the

right.

Einstein Relationship

0 , 1

n n n

dE

J q n qD dn

dx q dx

µ

⇒ = = E + E =

( )

{ }

( )

/

/

, 0

C F

C F

E E kT C

E E kT

C C

C F

dn d

dx dx N e

N dE

kT e dx

dE dE

n nq

kT dx kT with dx

− −

− −

=

= −

= − = − E =

0

qD

qn qn

µ kT

= E − E

n n

D kT

µ q

∴ =

For electron,

Similarly, for hole,

• Under equilibrium conditions, total current and because electron and hole activity is totally decoupled, must also be independently zero, regardless uniform and nonuniform doping.

0 J =



n p

J and J

 

0

n p

J = J = J =

  

p p

D kT

µ = q

Optical Absorption

• Photons with energies greater than the band gap energy are absorbed



• Photons with energies less than the band gap are transmitted

• The electron and hole created by this absorption process are excess carriers; they must eventually recombine

g

h ν h c E

= λ ≥

Electron-Hole Recombination

where and are excess carrier concentrations for electron and hole, respectively.

n′ p ′

Charge Neutrality

n ′ ≡ p ′

0 0

n n n p p p

≡ + ′

≡ + ′

• Direct recombination: an excess population of electrons and holes

decays by electrons falling from the conduction band to the valence band

 Direct bandgap materials (GaAs, InP……)

 called “radiative recombination” related to the photoemission

• The net rate of change in the conduction band electron concentration is the thermal generation rate α

n

minus the recombination rate

( )

(

( ) ( ) )

dn t n n t p t dt = α −

Direct Recombination

:constant of proportionality or

radiative recombination coefficient

α

( ) ( ) ( )

( ) ( ) ( )

2

0 0

2

0 0

r i r

r

dn t n n n t p p t

dt

n p n t n t

′ = α − α   + ′     + ′  

′ ′

 

= −α  + + 

In the case of low-level injection ( ) and p-type material ( ),

( )

( ) ,

n

dn t n

p n t

dt τ

′ = −α ′ = − ′

( ) (0)

(0)

n t ′ = n ′ e

= n ′ e

The solution is an exponential decay from the initial excess carrier concentration,

: recombination time or carrier lifetime of electron (minority carrier)

0

1

n

r

p τ = α

n ′ = p ′

( 0) n t ′ =

Similarly for n-type material,

0

1

p

r

n

τ = α : recombination time or carrier lifetime of hole (minority carrier)

n

n τ

′ : recombination time rate of electron

p

p τ

′

: recombination time rate of hole

n′ (t), p′ (t) < n

, p

p

> n

More general expression for the carrier lifetime is

(

)

1

r

n p

τ = α +

0

1 , for -type

n

r

p P τ = α

0

1 , for -type

p

r

n N τ = α

1 , for intrinsic

i

r

n

τ = 2α