ch1 2 carrier_action

8/11/2019 Ch1 2 Carrier_action

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Carrier Action

Drift

Definition, drift current, mobility, resistivity

DiffusionDefinition, diffusion current, Einstein relationship

Recombination-Generation

Definition, R-G statistics, minority carrier lifetimes

Equation of States

Continuity equations, minority carrier diffusion equations

Supplemental Concepts

Diffusion lengths, quasi-Fermi levels



Drift: Definition and physical picture

Definition: Drift is charged-particle motion in response to an applied electric field

Physical picture

Microscope view

Charged particles get energy from E-field and move

Collide with other particles (atom, etc), change direction, and at

same time lose energy

Again get energy from E-field and move, then repeat

Macroscope view

Viewed as a charged particle under electrical field

+

+ _

_

+ +

+

vd



Drift Current

Consider the p-type semiconductor bar of cross-sectional area A

vd t … all holes this distance back from the vd-normal plane will cross the plane in a

time t.

vd tA … all holes in this volume will cross the plane in a time t.

pvd tA … holes crossing the plane in a time t.

qpvd tA… charge crossing the plane in a time t.

qpvd A … charge crossing the plane per unit time.

Thus,

I p,drift =qpvd A … hole drift current

J p,drift =qpvd … hole drift current density

Drift: Drift current

+

+ I

E



Drift: Drift current with electric f ield

Drift Current (related with electric field)

From definition, drift current arises in response

to an applied electric field, thus we try to relate

the vd with electric field

From experiments, vd vs E relations

vd =0 E, for E< Esat (0 is mobility)

vd = vsat, for E > Esat (velocity saturation)

Physical explain:

Why saturation (?)

Thus, the drift current is given by

0 p for hole mobility

only valid for low-field

at high-field (E>Emax), then J p,drift =qpvsat ,

one factor limiting speed of MOSFET

E

v d

J p,drift =qpvd =q p pE Si

Si Si

Si

e

E field



Mobility ()

Definition … is a key parameter in characterizing

electron and hole transport due to drift ( = q <>/

m* )

Physical view: mobility is a measure of the ease ofcarrier motion in a crystal

Dominant Mechanisms:

Lattice scattering, involving collisions with

heated lattice atoms mobility increases with decreasing T

no doping dependence

Ionized impurity scattering

mobility decreases with increasing N A or ND

mobility increases with increasing T

Consider all contributions of different scattering:

Drift: Mobil ity

1 =

1

1

2

1

3

1+ + + ...

mobility

mobility

concentration

temperature

T3/2

Impurity scattering Lattice scattering

T-3/2

(Mathiessen’s rule)



Definition: resistivity is a measure of material’s inherent resistance tocurrent flow

J= E=E/

J drift =J p,drift +J n,drift =q(n n+p p)E

Thus, we have

Comments: For nondegenerated semiconductor with ND >>ni

For nondegenerated semiconductor with N A >>ni

Always measure resitivity to determine dopant concentrations

Drift: Resistivity

=1

q(nn+pp)

=1

qNDn

=1

qN Ap



Diffusion: Definition and physical picture

Definition: Diffusion is particle motion result from its spatial variation

Analogy:

Example “perfume” in a closed room

Physical picture:

Microscopic view: redistribute as a result of their random motion

Macroscopic view: migrating on a macroscopic scale from regions

of high particle concentration into regions of low particlesconcentration.

Diffusion +++++

++

+++

Jp,diff

Diffusion -----

--

---

Jn,diff

x x



Diffusion: Diffusion current

Diffusion currentJp,diff = -qDpp

Jn,diff = qDnn

where, Dp is hole diffusion coefficient, Dn is electron diffusion coefficient

Some discussions

Difference in concentration: driving force of diffusion current.

Both hole and electron will diffuse in the -x direction.

Total Current (including diffusion and drift of both electrons and holes)

Jp = Jp,drift + Jp,diff = qpE - qDpp

Jn = Jn,drift + Jn,diff = qnE + qDnnDrift Diffusion

J = Jn + Jp



Diffusion: Einstein relationship

Relating Diffusion Coefficients/Mobilities (Einstein relationship)

Dn/n = kT/q

Dp/p = kT/q

Comments

Derivation is omitted

Valid even under non-equilibrium conditions, however the non-

degenerate restriction still applies

Typically value for Si and Ge at room temperature: D/ ~ 0.026 V

Physical view: both reflect the ease of motion of carriers in crystal

Dn

(cm2/s)

Dp

(cm2/s)

n

(cm2/v-s)

p

(cm2/v-s)

Ge 100 50 3900 1900

Si 35 12.5 1350 480



Recombination and Generation: Recombination

Carrier recombination Definition: Recombination is a process whereby electrons

and holes are annihilated or destroyed

Three main recombination mechanisms

Band-to-band recombination

R-G center recombination

Auger recombination


Ec

EvEv

EcEc

Ev

R-G center recombination Auger recombination

Photon (Light) Thermal energy



Recombination: Band-to-band recombination


Ec

Ev

Photon (Light)


Simplest recombination

process Involves the direct annihilation

of a conduction band e- and

a valence band h+

The excess energy releasedduring the process typically

goes into the production of a

photon (light)



Recombination and Generation: R-G centerrecombination

R-G Center Recombination Involves “third particle”

Takes place only at special locations

within semiconductor known as R-G

centers R-G centers are lattice defects or

special impurity atoms, such as Au in Si

A two-step process

First, one type of carrier, say e-, is

trapped

Then, a hole is attracted to the trapped

electron, or

The e- lose energy again and

annihilating the hole in the valenceband

Typically release thermal energy during the

process or equivalent produce lattice

vibrations

Ev

Ec


Thermal energy

or



Recombination and Generation: Auger recombination

Ec

Ev

Auger recombination

Auger Recombination

Three free carriers interact, two of

the carriers recombine (band-to-band)

The released energy is transferred

into the surviving carrier (the third)

during the collision

The highly energetic carrier will lose

energy in small steps through heat-

producing collision with the

semiconductor lattice



Recombination and Generation: Generation

Ec

Ev

Ec

Ev

Band-to-band generation R-G center generationCarrier generation

via impact ionization

Ec

Ev

Carrier generation

Definition: Generation is a process whereby electrons and

holes are created

Three main generation mechanisms:

Band-to-band generation

R-G center generation

Carrier generation via impact ionization



Recombination and Generation: R-G Statistics

R-G Statistics Take indirect thermal recombination-generation as example

Low-level injection

perturbation

t=0

p0 p< p0

t>0 t

p=0

Ec

ET

Ev

EF

p0 p <<n0

nn0

Low level injection impliesp << n0, nn0, in an n-type material

n << p0, pp0, in a p-type material



R-G Statistics: Hole recombination rate

p/t|R (rate of hole recombination)

Proportional to trap density (NT) and hole

concentration (p)

p/t|R = - cpNTp

Where, cp, capture coefficient, is a

positive constant

Ev

Ec


Thermal energy

or

Note:

(1) Indirect recombination rate is proportional to

the trap states occupied by electrons

(2) Since n-type Si, ET is below EF, so almost all

trap states are filled by electrons(3) Thus, the recombination rate is approximately

proportional to trap density NT

Ec

ET

Ev

EF

p0 p <<n0

nn0



EE5502/Chunxiang Zhu/NUS/Sem I, 2010/11

R-G Statistics: Hole generation rate

p/t|G (rate of hole generation) Depends only on the no. of empty R-

G centers (small for considering case)

no. of empty centers is approximate

constant at its equilibrium value p/t|G= p/t|G-equilibrium

= - p/t|R-equilibrium = cpNTp0

EcET

Ev

EF

p0 p <<n0

nn0

Ec

Ev

Note:

(1) Indirect generation rate is proportional to the trap

states which are empty.

(2) Since n-type Si, ET is below EF, so empty trap states

is very small.

(3) For same n-Si, the empty trap states is also verysmall when there is no injection (in equilibrium).

(4) Thus, generation rate is approximately proportional

to empty trap density in equilibrium




Thus the net rate of change in the hole concentration

due to the R-G center interaction is given by

p/t|R-G = p/t|R + p/t|G= -cpNT(p-p0)

= -cpNTp

Introduction of time constant: p=(cpNT)-1 (discusslater)

R-G center generation/recombination rate under low

level injection

p/t|R-G = -p/p

R-G Statistics: net change rate of holes



R-G statistics: general case

pt

ntR-G =

R-G =ni

2 - np

n(n+n1) + p(p+p1)

where,n1=ni exp[(ET-Ei)/kT]

p1=ni exp[(Ei-ET)/kT]

Comments:

Valid for more general case (arbitrary injection levels

and both carrier types in nondegenerate semiconductor)

n,

p, minority carriers lifetimes

interpreted as average time an excess carrier can

live in a sea of majority carriers

depend on the R-G center concentration (NT)

control of lifetime (introducing of gold to increase NT)

low-level injection (p << n0, n-type material)

pt R-G = - p

p



Equation of States: Continuity Equations

J p(x+ x) J p(x)

x x+ x

Consider only hole current density with one dimension as

example

Jp(x+x): hole current leaving the volume A x

Jp(x): hole current entering the volume A x

p/ t: net increase in hole concentration per unit time,

Relation: p/t is the difference between the hole flux per unit

volume entering and leaving, minus the recombination rate

For x0, we can get continuity equation

For electron minority carrier, similarly

p

t x x+ x

= 1

q

J p(x)-J p(x+ x)

x

- p

p

Rate of

hole buildup

Increase of hole concentra-

tion in A x per unit timeRecombination

rate= -

p(x,t)t = 1q Jpx - pp

pt = -

n(x,t)t = 1q

Jn

x - n

n

nt =



From continuity equation

From diffusion current definition

Combine above 2 equations, we obtain

For electron minority carriers, similarly

Often used in solving transient problems of diffusion with

recombination

consider : what’s assumption used in deriving above results,

what will be the general form of equations?

Equation of States: Minority carrier diffusion equations

p(x,t)t = 1

qJp

x-

pp

pt = -

pJp,diff = -qDp x

2px

- p

p

pt = Dp

2nx

- n

n

nt = Dn

(applying to holes in n-type material)

(applying to electrons in p-type material)



Supplement Concepts: Diffusion Length

In the steady state case, the diffusion equations

simplified as:

where, Lp is minority carrier diffusion length (hole ina p-type material)

definition:

Lp=√Dpp

physical view: represent the average distance

minority carriers can diffusive into a sea of majority

carriers before being annihilated

Example: consider the steady state case of excessholes are injected into a semi-infinite

semiconductor bar at x=0

2p

x2

pDpp

= =pLp

2

<x>=

∫xp(x)dx0

∫p(x)dx0 = Lp

x

p(x)

p0

p0

p(x)=p0+p0exp(-x/Lp)

p(x)=p0exp(-x/L

p)

2px =

pLp

2



Supplement Concepts: Quasi-Fermi Level

Quasi-fermi levels are energy levels used to specify thecarrier concentrations inside semiconductor under non-

equilibrium conditions

Definition:

n = niexp(EFn-Ei)/kT or EFn = Ei+kTln(n/ni)p = niexp(EFp-Ei)/kT or EFp = Ei+kTln(p/ni)

Example:

(a) n0=ND=1015cm-3 and p0=105cm-3

(equilibrium)

(b) p=p0+p=1011cm-3, nn0=1015cm-3

(nonequilbrium, steady state)

Physical view:

JP=p pEFp

namely, a quasi-Fermi level that varies with position

(EFp 0) indicates there is current flowing inside the

semiconductor

EC

EFEi

EV

EC

EFnEi

EV

EFp

(a) equilibrium

(b) non-equilibrium




Summary of Carrier Action

Continuity Equation

Drift

Diffusion

Generation-

recombination

ch1 2 carrier_action

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