AdvSemi_lec2_2013-03-05

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Advanced

Semiconductor

Devices

Lecture 2

Advanced

Semiconductor

Devices

Lecture 2

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2 © V. Ariel 2013 Advanced Semiconductor Devices Lecture 2

Lecture outline

Review of energy band structure

Simplified band structure and electron concentration

Intrinsic semiconductor

Extrinsic semiconductors in equilibrium

Semiconductor current

Einstein relationship

Non-equilibrium semiconductors

Continuity equation

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Electron in one-dimensional atomic lattice

22

20

d

dx

0

2

2m E U

0 E U

Shrodinger wave equation – Kronig-Penney model

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Graphical solution

The solution can be compared to the free electron solution

Fold all energy solutions into the first Brillouin zone

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Electrons and holes

Effective mass is positive for electrons in band 3 – conduction band

Particle mass is negative at the top of band 2 – valence band

Model empty electron states at the top of the valence band as positive

electric charges with positive mass - holes

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Energy bands in real semiconductors

Energy band structure in Si as a function of the wave-vector k

Note that the minimum of the conduction band is not directly

above the maximum of the valence band. Si – indirect gap

semiconductor

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4 - band energy model

Simplified energy band structure for the direct gap semiconductor

One conduction band

3 valence bands with different effective masses of holes

Effective mass defined for all carriers as

2

/

k m

E k =

¶ ¶

h

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Very simple energy band structure

Simple 2 – band energy structure

Assume that the top of the valence band and the bottom of the

conduction bands are parabolic and isotropic using density of

states effective masses 22

*

22

*

2

2

C

n

V

p

E E k m

E E k m

-

-

h;

h;

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Fermi energy and carrier density

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Maxwell – Boltzmann approximation

Can obtain a simple approximation for the Fermi-Dirac integral

Where we define the effective densities of states in the

conduction and valence bands

3/2*

2

3/2*

2

2 2

22

n

C

p

V

m kT

N

m kT N

p

p

æ ö

º ç ÷è ø

æ öº ç ÷ç ÷

è ø

h

h

( )

( )

/

/

3

3

F C

V F

E E kT

C C F

E E kT

V F V

n N e E E kT

p N e E E kT

-

-

= - ³

= - ³

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Intrinsic semiconductors

Charge neutrality without impurities

in p n= =Write for intrinsic carrier concentration

using Maxwell – Boltzmann approximation

( ) ( )/ /i C V i

E E kT E E kT

i C V n N e N e

- -= =

Obtain intrinsic carrier concentration

( )/ /2 V C G E E kT E kT

i C V C V n N N e N N e- -= =

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Intrinsic semiconductors

Obtain intrinsic carrier concentration

/2G

E kT

i C V n N N e-=

*

*3 ln2 4

pC V i

n

m E E E kT

m

æ ö+= + ç ÷ç ÷

è ø

3/2*

*ln ,

2 2

pC V V V i

C C n

m E E N N kT E

N N m

æ öæ ö+= + = ç ÷ç ÷ ç ÷

è ø è ø

Derive the intrinsic Fermi level

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Extrinsic carrier concentration

General carrier concentration in terms of

intrinsic parameters

( )

( )

/

/

F i

i F

E E kT

i

E E kT

i

n n e

p n e

-

-

=

=

Obtain equilibrium charge balance2

in p n× = 0 D A p n N N + -- + - =

Ionized impurity concentrations

( )

( )

1 exp /

1 exp /

D D

D F D

A A

A A F

N N g E E kT

N N

g E E kT

+

-

= é ù+ -ë û

=é ù+ -ë û

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Fermi energy in extrinsic materials

Given carrier concentrations, can determine the Fermi level

For n- or p-type materials

( )

( )

/

/

ln ln

F i

i F

E E kT

i

E E kT

i

F i

i i

n n e

p n e

n p E E kT kT

n n

-

-

=

=

æ ö æ ö- = = -ç ÷ ç ÷

è ø è ø

ln ,

ln ,

D F i D i

i

Ai F A i

i

N E E kT N n

n N

E E kT N nn

++

--

æ ö- ç ÷

è øæ ö

- ç ÷è ø

; ?

; ?

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Alternative expressions

Carrier concentration and the Fermi level


( ) ( )/ /,

ln ln

F C V F E E kT E E kT

C V

F C V

C V

n N e p N e

n p E E kT E kT

N N

- -= =

æ ö æ ö= + = -ç ÷ ç ÷

è ø è ø

Dn N +» A p N -»

ln ln ,

ln ln ,

D F C D i

C C

AV F A i

V V

n N E E kT kT N n

N N

p N E E kT kT N n N N

++

--

æ ö æ ö- = ç ÷ ç ÷

è ø è ø

æ ö æ ö- ç ÷ ç ÷è ø è ø

; ?

; ; ?

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Extrinsic carrier concentration

In general

Obtain quadratic equation for n

Which we solve for n and p

0 D A p n N N + -- + - =2

in p

n=

2

0i D A

nn N N

n

+ -- + - = ( )2 2 0 D A in n N N n+ -- - - =

2

2

2

2

2 2

2 2

D A D Ai

A D A Di

N N N N n n

N N N N p n

+ - + -

- + - +

æ ö- -= + +ç ÷

è ø

æ ö- -= + +ç ÷

è ø

Typical values for Si10 31.5 10in cm-» ´

14 3 19 3, 10 10 A D N N cm cm- --:

For n- or p-type materials Dn N +» A p N -»

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Energy band bending

1 1C V

dV

V dx

dE dE

q dx q dx

= -Ñ = -

= =

E

E

Potential energy is a function of

position and can be linked to

electrostatic potential

Express for the electric field

( ) ( )

( )1

pot

n

C ref

E x qV x

V E E q

= -

= - -

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Electron drift current

Electric field causes drift of

carriers resulting in

drift current

Write for electron current

density

Here t 0

is the transit time

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Electron drift current

Assuming constant electric filed

We get apparent violation of

Ohm's law!!!

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Total drift current

Similarly for holes

Obtain total drift current

Note that carrier velocities are

dr

p p J qpm = E *

p

p

p

q

m

t m =

dr dr dr

n p n p J J J qn qpm m = + = +E E

n nm = - E V p pm = E V

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Semiconductor resistivity

Define semiconductor conductivity and resistivity from Ohm's law


1dr J s r

= =E E

1n pqn qps m m

r = = +

1

D nqN

r

m

;i Dn n n N ? ;

1

A pqN r

m ;

i A p n p N ? ;

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Diffusion Current

Diffusion is due to redistribution of carriers caused by random

thermal motion

Assume, that equal number of particles moves in both + x and - x

directions in each cross-section

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Diffusion Current

Assume that all carriers move with the same average velocity

Assume that all carriers move with the same average velocity V

The average distance between two electron collisions is l

Average life-time between collisions is

l t =

V

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Total current

Determine current

2

pdiff

p P P

l dp J qD D

dx= - º

V

Can express total current as a sum of drift and diffusion currents

dr diff

n n n n N

dr diff p p p p P

qn qD n

qp qD p

m

m

= + = + Ñ

= + = - Ñ

J J J

J J J

E

E

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Equilibrium Fermi level

Fermi level is equivalent to chemical potential by definition

Chemical potential determines the flow of particles between parts of

the system

Therefore, Fermi level inside a material is constant as a function of

position under thermodynamic equilibrium (total current is 0)

const (x) F E =

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Writing for electrons from Boltzmann approximation

C E

V E

const (x) F E =G

E

F C E E

kT C n N e

-

= ×

F C C E E dE dn d nn

dx dx kT kT dx

-æ ö= = -ç ÷

è ø

Under thermodynamic equilibrium const (x) F E =

E

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C E

V E

const (x) F E =G

E

Total current under thermodynamic equilibrium is 0

0 C C n n n n n

dE dE dn n J qn qD n qD

dx dx kT dxm m = = + = -E

Einstein relationship follows for electrons (and similarly for holes)

n n p p

kT kT D D

q qm m = =

E

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Non-equilibrium: Quasi-Fermi levels

In non-equilibrium, can not describe semiconductor properties witha single constant Fermi level

Introduce quasi-Fermi levels E FN

and E FP

for electrons and holes

FN C V FP E E E E

kT kT

C V n N e p N e

- -

= × = ×

C E

G E FN E

FP E

V E

0

0

n n n

p p p

= + D

= + D

Excess carrier concentrations: Δn and Δ p

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Non-equilibrium total current

In general, for electrons in 3-dimensions

C E

G E FN E

FP E

V E

Similarly, for holes

Total current

dr diff

n n n n n n FN qn qD n n E m m = + = + Ñ = ÑJ J J E

dr diff

p p p p p p FP qp qD p p E m m = + = - Ñ = ÑJ J J E

n p n FN p FP n E p E m m = + = Ñ + ÑJ J J

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Generation -recombination of carriers

Assume low-level injection

Minority carrier recombination rate can be approximated as

0 0, p p n nD D= =

,n p

n p R R

t t

D D; ;

Band-to-band R-G in indirect band semiconductors is complex and

requires involvement of phonons for momentum conservation

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Continuity equation

Δ x

( )n

J x ( )n

J x x+ D

n

G

n

R

Consider electron current in a semiconductor bar

Assume electron generation and recombination

Let Δx approach zero to derive the continuity equation for electrons

( )( ) ( )n n

n n

J x J x xdn x G R x

dt q q

+ DD = - + - D

- -

Similarly for holes

( )

1 n

n n

dJ dn

G Rdt q dx= + -

( )1 p

p p

dJ dpG R

dt q dx= - + -

Mi it i diff i ti

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Minority carrier diffusion equation

No electric field

Low level injection in p

-type semiconductor The equilibrium minority carriers

Approximate minority carrier current

Obtain minority carrier diffusion equation

0=E

n n n n

dn dn

J qn qD qDdx dxm = + ;E

2

2

1 pnn

d ndJ D

q dx dx

D; ( )

1 p nn n

d n dJ G R

dt q dx

D= + -

0 p p pn n n= + D

0 p pn pD =

0 const( ) pn x=

( )2

2

p p

n n n

d n d n D G R

dt dx

D D= + -

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AdvSemi_lec2_2013-03-05