Chapter 4 - study notes - solid state electronic devices, Lecture notes of Electronics

Download Chapter 4 - study notes - solid state electronic devices and more Lecture notes Electronics in PDF only on Docsity!

EE

Introduction to solid state electronics

Fall 2017

Chapter 4

Excess Carriers in Semiconductors

Optical Absorption

 Incident Photons:

E > E

G

absorbed , E < E

G

transmitted (out)

 Step a: Absorption of photon

(high probability when E = hf > E

G

 electron transition VB to CB

 Excess carrier

 Step b: Electron relaxes to

lowest energy in CB

 Step c: Recombination

 electron transition CB to VB

(with radiation?)

Incident photons transmitted photons Semiconductors

Optical Absorption

Experiment

Probability of absorption in dx is identical at any dx  Degradation of the intensity

- d I (x)/dx = a I (x)  I (x) = I o e - a x

 I

t

= I

o e

- al

a = absorption coefficient (cm

- )

a high for E > E

G

a small for E < E

G

At E = EG = hf  Absorption edge

l Monochromatic Source dx Io x It Detector

Optical Absorption

Luminescence

E = hf = hc/ l = 1.24/ l (in m m)

 In visible range:

1) E

vis

> E

G

silicon

 Si absorbs visible (opaque)

2

2) E

vis

< E

G

ZnS

 ZnS transparent

3) E

G

GaP within visible range

 GaP semi transparent

 Luminescence = Light radiated from material due to

carrier recombination (transition of carrier from CB to VB)

 Highly probable in direct bandgap semiconductor

Carrier Lifetime and Photoconductivity

Direct Recombination of Electrons and holes

1. Recombination = annihilation of electron hole pair  **Transition of a free electron from the CB to the VB

2. Energy lost as a photon (radiation) 3.** Rate of change (decay) of electrons at any time t proportional to n(t) and p(t)

dn(t)/dt = arni

2

- arn(t)p(t)

( a

r n i 2 = thermal generation (Ch. 3)) n(t) = no + dn(t) p(t) = po + dp(t)

ddn(t)/dt = a

r n i 2

- a r [(n o

+ dn(t))(p

o

+ dp(t))]

= arni

2

- arni 2 - ar[(no+ po) dn(t)) + dn 2

(t)] ( dn(t) = dp(t))

ddn(t)/dt = - ar[(no+ po) dn(t)) + dn

2 (t)]

At low level injection dn(t) << no, po  d dn(t)/dt = - ar[(no+ po) dn(t))]

Carrier Lifetime and Photoconductivity

Direct Recombination of Electrons and holes

In p-type material po >> no

 Decay rate d dn(t)/dt = - ar po dn(t)

 dn(t) = Dn exp (- a

r p o

t) = Dn exp (-t/ t

n

t

n

= 1/( a

r p o ) = Electron (minority carrier) recombination lifetime In n-type material no >> po  dp(t) = Dp exp (-t/ tp)

t p = 1/( a r no ) = Hole (minority carrier)

recombination lifetime In general t n = 1/[ a r ( n o + p o)]

Plot ln( dn) versus t  ln( dn) = ln Dn – t/ t

Slope = 1/ t

Carrier Lifetime and Photoconductivity

Indirect Recombination; trapping

 Shallow states (close VB / CB) act as TRAPS not recombination centers

 Excess carrier increases conductivity from s

o

to s

ph

 Recombination of excess carriers reduce s

ph

to s

o

 Lifetime t = Characteristic time for Photoconductivity decay

s(t) = q (n(t) mn + p(t) mp)

Sample resistance R(t) = V/I monitored versus t

 G(t)  s (t)  dn(t)

Carrier Lifetime and Photoconductivity

Steady state carrier generation

 Carrier generation rate g = arnp

 At thermal equilibrium no = po  gth = gth(T) = arnopo

 At steady state non-equilibrium a ll generation phenomena add up (thermal + optical +..) and p ≠ po and n ≠ n o

g = gth(T)+ gop= a r np = ar(no + dn)(po + dp) = arnopo+ ar[(no+po) dn+ dn

2 ]

At low level injection gop = ar(no + po) dn = dn/ tn

 dn = g

op

t

n

= dp

 tn ≠ tp if trapping is not negligible

Carrier Lifetime and Photoconductivity

Photoconduction – Photoconductive devices

 dn = g

op

t

n

, dp = g

op

t

p

 Conductivity s = qn m

n

+ qpm

p

= q (n

o

+dn) m

n

+ q (p

o

+dp) m

p

= s

o

+ Ds

 Conductivity change due to light = Photoconductivity

Ds = s

ph

= qdn m

n

+ qdpm

p

= qg

op

( t

n

m

n

+ t

p

m

p

 In some materials (e.g amorphous Si) s

ph

>> s

o

 Photoconductivity is used in optical sensing

Diffusion of carriers

Diffusion = result of random motion in a gradient

concentration distance Motion by diffusion Concentration gradient

 Carriers in a semiconductor

diffuse in a carrier concentration

gradient by random thermal

motion and scattering

n(x) t = 0 t 1 t 2 t 3 x Pulse of electrons spread by diffusion 0 l n(x) xo- l xo xo+l (1) (2) n 1 n 2 n(x) Divide range into segments wide = mean free path between collisions l l

Diffusion of carriers

Diffusion

 f

n

(x

o

) = (n

1

- n 2

) = limit n(x) – n(x+ Dx)

= -^

dn(x)

=

• Dn dn(x)

D

n

= diffusion constant of electrons

 Similarly f

p

(x) = -D

p

dp(x)/dx

 The diffusion current density:

J

n

= -q f

n

(x) = q D

n

dn(x)/dx

Jp = q fp(x) = -q Dp dp(x)/dx

Dx

l

2 t’

2 D x  0

l

2 t’

l

2 t’

2

dx dx

n(x) p(x) Jn f n Jp f p

Diffusion of carriers

Diffusion and drift currents

 In the presence of an electric field (drift)

J = J

drift

+ J

diffusion

Jn = q n mn e + q Dn dn(x)/dx

J

p

= q p m

p

e - q D

p

dp(x)/dx

 Major Contribution of majority carrier to drift component

 Minority carrier can have large diffusion component

 The total current density

J (x) = J

n

(x) + J

p

(x)

Diffusion of carriers

Einstein Relation

At equilibrium the current J = 0, J

n

= 0, J

p

J

p

(x) = qp(x) m

p

e(x) – qD

p

dp(x)/dx = 0

But from 

 Substitute in e(x) above 

 D

p

/ m

p

= kT/q = D

n

/ m

n

dx

dE

dx q

dp x

p x

D

x

i p

p ( )^1

m

e

Ei EF kT i

p x n e

( )/



i i F

n

p x

E E kT

= + ln

dx dp x p x kT dx dEi ( ) ( ) = 0 +

dx

dp x

p x

kT

dx q

dp x

p x

D

x

p

p ( )

m

e

Einstein Relation

Diffusion of carriers

Example

A semiconductor sample is doped with N

d

=N

o

e

-ax

Find built-in field at equilibrium

e (x) = -kT/q. 1/n

o

(x) dn

o

(x)/dx ,

n

o

(x) ≈ N

d

(x)  dn

o

(x)/dx = -aN

o

e

-ax

 e (x) = -kT/q. (1/N

o

e

-ax

). (-aN

o

e

-ax

) = a kT/q = constant +ive

Nd(x)=no(x) x EF x xj Na E Ec Ev xj n-type p-type

**1. At equilibrium EF = constant

2. Constant E slope**

 Constant e