President UniversityErwin SitompulSDP 4/1 Lecture 4 Semiconductor Device Physics Dr.-Ing. Erwin...

President University Erwin Sitompul SDP 4/1

Lecture 4

Semiconductor Device Physics

Dr.-Ing. Erwin SitompulPresident University

http://zitompul.wordpress.com

2 0 1 3


Ev

Ec

Ec

EvGaAs, GaN(direct semiconductors)

Si, Ge(indirect

semiconductors)

PhotonPhoton

Phonon

Direct and Indirect SemiconductorsChapter 3 Carrier Action

E-k Diagrams

• Little change in momentum is required for recombination

• Momentum is conserved by photon (light) emission

• Large change in momentum is required for recombination

• Momentum is conserved by mainly phonon (vibration) emission + photon emission


0nnn

0ppp

, 0n p

Equilibrium valuesDeviation from

equilibrium values

Excess Carrier ConcentrationsChapter 3 Carrier Action

Positive deviation corresponds to a carrier excess, while negative deviation corresponds to a carrier deficit.

Values under arbitrary conditions

pn Charge neutrality condition:

, 0n p


Often, the disturbance from equilibrium is small, such that the majority carrier concentration is not affected significantly.

However, the minority carrier concentration can be significantly affected.

For an n-type material

For a p-type material

0 0, p n n n

0 0, n p p p

“Low-Level Injection”Chapter 3 Carrier Action

This condition is called “low-level injection condition”.The workhorse of the diffusion in low-level injection condition is

the minority carrier (which number increases significantly) while the majority carrier is practically undisturbed.

0p p

0n n


p TR

pc N p

t

G G-equilibrium

p p

t t

NT : number of R–G centers/cm3

Cp : hole capture coefficient

Indirect Recombination RateChapter 3 Carrier Action

Suppose excess carriers are introduced into an n-type Si sample by shining light onto it. At time t = 0, the light is turned off. How does p vary with time t > 0?

Consider the rate of hole recombination:

In the midst of relaxing back to the equilibrium condition, the hole generation rate is small and is taken to be approximately equal to its equilibrium value:

R-equilibrium

p

t

p T 0c N p


R G R G

p p p

t t t

p TR G p

p pc N p

t

n TR G n

n nc N n

t

pp T

1

c N where

where nn T

1

c N

Indirect Recombination RateChapter 3 Carrier Action

The net rate of change in p is therefore:

p T p T 0c N p c N p p T 0c N p p

• For holes in n-type material

• For electrons in p-type material

Similarly,


pp T n T

1 1 nc N c N

Minority Carrier LifetimeChapter 3 Carrier Action

The minority carrier lifetime τ is the average time for excess minority carriers to “survive” in a sea of majority carriers.

The value of τ ranges from 1 ns to 1 ms in Si and depends on the density of metallic impurities and the density of crystalline defects.

The deep traps originated from impurity and defects capture electrons or holes to facilitate recombination and are called recombination-generation centers.


PhotoconductorChapter 3 Carrier Action

Photoconductivity is an optical and electrical phenomenon in which a material becomes more electrically conductive due to the absorption of electro-magnetic radiation such as visible light, ultraviolet light, infrared light, or gamma radiation.

When light is absorbed by a material like semiconductor, the number of free electrons and holes changes and raises the electrical conductivity of the semiconductor.

To cause excitation, the light that strikes the semiconductor must have enough energy to raise electrons across the band gap.


16 30 10 cmp

p n

Example: PhotoconductorChapter 3 Carrier Action

Consider a sample of Si at 300 K doped with 1016 cm–3 Boron, with recombination lifetime 1 μs. It is exposed continuously to light, such that electron-hole pairs are generated throughout the sample at the rate of 1020 per cm3 per second, i.e. the generation rate GL = 1020/cm3/s.

a) What are p0 and n0?

2i

00

nn

p

210

16

10

10 4 310 cm

b) What are Δn and Δp?

LG 20 610 10 14 310 cm• Hint: In steady-state

(equilibrium), generation rate equals recombination rate


Example: PhotoconductorChapter 3 Carrier Action

Consider a sample of Si at 300 K doped with 1016 cm–3 Boron, with recombination lifetime 1 μs. It is exposed continuously to light, such that electron-hole pairs are generated throughout the sample at the rate of 1020 per cm3 per second, i.e. the generation rate GL = 1020/cm3/s.

c) What are p and n?

d) What are np product?

• Note: The np product can be very different from ni

2 in case of perturbed/agitated semiconductor

0p p p 16 1410 10 16 310 cm

0n n n 4 1410 10 14 310 cm

16 1410 10np 30 310 cm 2in


2i

R G R G p 1 n 1( ) ( )

n npp n

t t n n p p

• ET : energy level of R–G center

Net Recombination Rate (General Case)Chapter 3 Carrier Action

For arbitrary injection levels and both carrier types in a non-degenerate semiconductor, the net rate of carrier recombination is:

T i( )1 i E E kTn n e

i T( )1 i

E E kTp n e

where


JN(x) JN(x+dx)

dx

Area A, volume A.dx

N N

1( ) ( )

nAdx x dx x A

t q

J J

Continuity EquationChapter 3 Carrier Action

Consider carrier-flux into / out of an infinitesimal volume:

Flow of current

Flow of electron


NN N

( )( ) ( )

xx dx x dx

x

J

J J

N

thermal other R G processes

( )1

xn n n

t q x t t

J

N ( )1 xn

t q x

J

Continuity EquationChapter 3 Carrier Action

• Taylor’s Series Expansion

P

thermal other R G processes

( )1

xp p p

t q x t t

J

The Continuity Equations


Minority Carrier Diffusion EquationChapter 3 Carrier Action

The minority carrier diffusion equations are derived from the general continuity equations, and are applicable only for minority carriers.

Simplifying assumptions:The electric field is small, such that:

N n N N

n nq n qD qD

x x

EJ

P p P P

p pq p qD qD

x x

J E

• For p-type material

• For n-type material

Equilibrium minority carrier concentration n0 and p0 are independent of x (uniform doping).

Low-level injection conditions prevail.


NL

n

( )1

xn nG

t q x

J

0 0N L

n

( ) ( )1

n n n n nqD G

t q x x

2

N L2n

n n n

D Gt x

2

P L2p

p p p

D Gt x

Minority Carrier Diffusion EquationChapter 3 Carrier Action

Starting with the continuity equation for electrons:

Therefore

Similarly

thermal nR G

n n

t

Lother processes

n

Gt


2p p p

N L2n

n n n

D Gt x

Carrier Concentration NotationChapter 3 Carrier Action

The subscript “n” or “p” is now used to explicitly denote n-type or p-type material.

pn is the hole concentration in n-type materialnp is the electron concentration in p-type material

Thus, the minority carrier diffusion equations are:

2n n n

P L2p

p p p

D Gt x

• Partial Differential Equation (PDE)!

• The so called “Heat Conduction Equation”


p n0, 0n p

t t

2 2p n

N P2 20, 0

n pD D

x x

p n

n p

0, 0n p

L 0G

Simplifications (Special Cases)Chapter 3 Carrier Action

Steady state:

No diffusion current:

No thermal R–G:

No other processes:

• Solutions for these common special-case diffusion equation are provided in the textbook


2n n n

2 2P p P

p p p

x D L

P P pL D

2n n

P 2p

0p p

Dx

n n0 (0)p p

N N nL D Similarly,

Minority Carrier Diffusion LengthChapter 3 Carrier Action

Consider the special case:Constant minority-carrier (hole) injection at x = 0Steady state, no light absorption for x > 0

L 0 for 0G x

The hole diffusion length LP is defined to be:


2n n

2 2P

p p

x L

P P

n ( ) x L x Lp x Ae Be

n ( ) 0p

n n0(0)p p

Pn n0( ) x Lp x p e

Minority Carrier Diffusion LengthChapter 3 Carrier Action

The general solution to the equation is:

A and B are constants determined by boundary conditions:

Therefore, the solution is:

0B

n0 A p

• Physically, LP and LN represent the average distance that a minority carrier can diffuse before it recombines with a majority carrier.


2p 437 cm V s

P p

kTD

q

P P pL D

Example: Minority Carrier Diffusion LengthChapter 3 Carrier Action

Given ND=1016 cm–3, τp = 10–6 s. Calculate LP.

225.86 mV 437 cm V s 211.3cm s

2 611.3cm s 10 s 33.361 10 cm

= 33.61 m

From the plot,


F i i F( ) ( ) i i, E E kT E E kTn n e p n e

N i( )i

F E kTn n e i P( )i

E F kTp n e

P ii

lnp

F E kTn

N i

i

lnn

F E kTn

Quasi-Fermi LevelsChapter 3 Carrier Action

Whenever Δn = Δp ≠ 0 then np ≠ ni2 and we are at non-

equilibrium conditions. In this situation, now we would like to preserve and use the

relations:

On the other hand, both equations imply np = ni2, which does

not apply anymore.The solution is to introduce to quasi-Fermi levels FN and FP

such that:

• The quasi-Fermi levels is useful to describe the carrier concentrations under non-equilibrium conditions


17 30 D 10 cm ,n N

23 3i

00

10 cmn

pn

17 14 17 30 10 +10 10 cmn n n

3 14 14 30 10 +10 10 cmp p p

17 14 31 310 10 =10 cmnp

Example: Quasi-Fermi LevelsChapter 3 Carrier Action

Consider a Si sample at 300 K with ND = 1017 cm–3 and Δn = Δp = 1014 cm–3.

• The sample is an n-typea) What are p and n?

b) What is the np product?


Example: Quasi-Fermi LevelsChapter 3 Carrier Action

Consider a Si sample at 300 K with ND = 1017 cm–3 and Δn = Δp = 1014 cm–3.

c) Find FN and FP?

N i ilnF E kT n n

5 17 10N i 8.62 10 300 ln 10 10F E

0.417 eV

P i ilnF E kT p n

5 14 10i P 8.62 10 300 ln 10 10E F

0.238 eV

N i i P

i iF E kT E F kTnp n e n e

0.417 0.23810 100.02586 0.0258610 10e e

311.000257 10 31 310 cm

Ec

Ev

Ei

FP

0.238 eV

FN

0.417 eV


1.(6.17)A certain semiconductor sample has the following properties:

DN = 25 cm2/s τn0 = 10–6 sDP = 10 cm2/s τp0 = 10–7 s

It is a homogeneous, p-type (NA = 1017 cm–3) material in thermal equilibrium for t ≤ 0. At t = 0, an external light source is turned on which produces excess carriers uniformly at the rate GL = 1020 cm–3 s–1. At t = 2×10–6 s, the external light source is turned off. (a) Derive the expression for the excess-electron concentration as a function of time for 0 ≤ t ≤ ∞.(b) Determine the value of the excess-electron concentration at

(i) t = 0, (ii) t = 2×10–6 s, and (iii) t = ∞.(c) Plot the excess electron concentration as a function of time.

Chapter 3 Carrier Action

Homework 3

2.(4.38)Problem 3.24Pierret’s “Semiconductor Device Fundamentals”.

Due date: 14.10.2013.

President UniversityErwin SitompulSDP 4/1 Lecture 4 Semiconductor Device Physics Dr.-Ing. Erwin...

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