Carrier Action

Chapter 3.

Carrier Action

Sung June Kim g

kimsj@snu.ac.kr

http://helios.snu.ac.kr p

Subject j

1.

반도체의 전계가 가해졌을 때 전자는 어떻게 움직이고 전 자의 이동도와 저항과는 어떤 관계에 있는가

자의 이동도와 저항과는 어떤 관계에 있는가?

캐리어의 확산은 전 에 어떻게 기여하 가

2. excess 캐리어의 확산은 전류에 어떻게 기여하는가?

3.

캐리어의 주입은 어떻게 할 수 있는가? 또, 주입된 캐리어 들은 어떻게 되는가?

1

직접형 및 간접형 반도체

1.

직접형 및 간접형 반도체

2.

재결합 수명

3. quasi-Fermi level 3. quasi Fermi level

4.

확산과 재결합을 동시에 고려했을 때의 전류는?

‰ Drift

‰ Diffusion

‰ Generation Recombination

‰ Generation-Recombination

‰ Equations of State

‰ Drift

‰ Drift

• Definition-Visualization

9 Ch d i l i i l i fi ld

9 Charged-particle motion in response to electric field

9 An electric field tends to accelerate the +q charged holes in the direction of the electric field and the –qq charged electrons in the g opposite direction

9Collisions with impurity atoms and thermally agitated lattice atoms 9 Repeated periods of acceleration and subsequent decelerating 9 Repeated periods of acceleration and subsequent decelerating collisions

9 Measurable quantities are macroscopic observables that reflect the average or overall motion of the carriers

9The drifting motion is actually superimposed upon the always- present thermal motion

present thermal motion.

9 The thermal motion of the carriers is completely random and therefore averages out to zero, does not contribute to current

(a) Motion of carriers within a biased semiconductor bar; (b) drifting hole on a microscopic or atomic scale; (c) carrier drift on a macroscopic scale

• Drift Current

Thermal motion of carrier

• Drift Current

9 I (current) = the charge per unit time crossing a plane oriented normal to the direction of flow

Expanded view of a biased p-type semiconductor bar of cross-sectional

A area A

... All holes this distance back from the normal plane will

d d

v t v −

cross the plane in a time

... All holes in this volume will cross the plane in a time

d

t

v tA t

pv tA ... Holes crossing the plane in a time d

... Charge crossing the plane in a time Charge crossing the plane per nit time

d

t

qpv tA t

A

... Charge crossing the plane per unit time

t

qpv A t

J

_{P drift|}

= qpv

_d

J

The v_d is proportional to E at low electric fields, while at high electric fields v saturates and becomes independent of E

fields v saturates and becomes independent of E

where for holes and for electrons, is the constant of

proportionality between v_d and E at low to moderate electric fields, and is the limiting or saturation velocity

vsat

≅

1

β β

≅

2 μ

0

is the limiting or saturation velocity 9 In the low field limit

v

_d

= μ

₀^E

,

_{d n}

,

_{N drift}| _d

q v μ E qnv

− = − J = −

to drift to drift

9 Unit: cm²/Vs

9 Varies inversely with the amount of scattering (i) Lattice scattering

(ii) Ionized impurity scattering

9Displacement of atoms leads to lattice scattering 9Displacement of atoms leads to lattice scattering

The internal field associated with the stationary array of atoms is already taken into account in m^*

9 , where is the mean free time and is the conductivity effective mass

9 The number of collisions decreases Æ μ varies inversely with

/

*

q m

μ = < > τ < > τ m

^*

τ

< >

The number of collisions decreases Æ μ varies inversely with the amount of scattering

τ

< >

Room temperature carrier mobilities as a function of the dopant

Impedance to motion due to

lattice scattering : Impedance to motion due to ionized impurity scattering : g

-No doping dependence.

-Decreases with decreasing T.

ionized impurity scattering : -Increases with N^A or N^D -Increases with decreasing T.

9 Temperature dependence

9 Low doping limit: Decreasing temperature causes an ever- c eases t dec eas g .

9 Low doping limit: Decreasing temperature causes an ever- decreasing thermal agitation of the atoms, which decreases the lattice scattering

9 Higher doping: Ionized impurities become more effective in deflecting the charged carriers as the temperature and hence the speed of the carriers decreasesp

Temperature dependence of electron mobility in Si for dopings ranging from < 10¹⁴ cm^-3 to 10¹⁸ cm^-3

from < 10¹⁴ cm ³ to 10¹⁸ cm ³.

Temperature dependence of hole mobility in Si for dopings ranging from

< 10¹⁴ cm^-3 to 10¹⁸ cm^-3

< 10¹⁴ cm ³ to 10¹⁸ cm ³.

• Resistivity Resistivity

9 Resistivity (ρ) is defined as the proportionality constant between the electric field and the total current per unit area

Resistivity versus Resistivity versus impurity

concentration at 300 K in Si

• Band Bending

9When exists the band energies become a function of

ε

energies become a function of position

9 If an energy of precisely E_G is added to break an atom-atom bond, the created electron and hole

motionless

the created electron and hole energies would be E_c and E_v,

respectively, and the created carriers ld b ff ti l ti l

would be effectively motionless

9 E-E^c= K.E. of the electrons 9 E^v-E= K.E. of the holes

P E

= E − E f

9The potential energy of a –q charged

c ref

P.E.

E E

c ref

P.E.

= E − E

particle is

P.E. = − qV

9 By definition,

9 In one dimension,

9 Diffusion is a process wherebyDiffusion is a process whereby

particles tend to spread out or redistribute as a result of their random thermal motion,

i ti i l f

migrating on a macroscopic scale from regions of high concentration into region of low concentration

• Diffusion and Total Currents

9 Diffusion Currents: The greater the concentration gradient, the larger the flux

9 Using Fick’s law,

N diff_|

= − qD

N

∇ n

J [ D ] = cm

²

/ s

N diff_|

qD

N

∇ n J

9 Total currents

9 Total particle currents

+ J = J

_N

+ J

_P

J J J

• Relating Diffusion coefficients/Mobilities

9 Einstein relationship

9Consider a nonuniformly doped semiconductor under equilibrium 9Consider a nonuniformly doped semiconductor under equilibrium 9 Constancy of the Fermi Level: nonuniformly doped n-type

semiconductor as an example

dE

_F

/ dx

^{= 0}

= dE

_F

/ dy = dE

_F

/ dz = 0

Electron diffusion current flowing in the +x direction “Built-in”

electric field in the –x direction Æ drift current in the –x direction

9 Einstein relationship:

9 Nondegenerate nonuniformly doped semiconductor Under 9 Nondegenerate, nonuniformly doped semiconductor,Under equilibrium conditions, and focusing on the electrons,E

9 With dE /d 0 9 With dE_F/dx=0,

9 Substituting

k q D kT

n N

= μ

q kT D

p P

= μ

9 Einstein relationship is valid even under nonequilibrium 9 Slightly modified forms result for degenerate materials

p

Slightly modified forms result for degenerate materials

9 When a semiconductor is perturbed Æp an excess or deficit in the carrier concentrations Æ Recombination-generation

9 Recombination: a process whereby electrons and holes (carriers) are annihilated or destroyed

are annihilated or destroyed

9 Generation: a process whereby electrons and holes are created

Band to Band Recombination

• Band-to-Band Recombination

9 The direct annihilation of an electron and a hole Æ the production of a photon (light)

• R-G Center Recombination

9 R-G centers are lattice defects or impurity atoms (Au)

9 The most important property of the R-G centers is the introduction 9 The most important property of the R G centers is the introduction of allowed electronic levels near the center of the band gap (ET)

9Two-step process

9 G ( )

9 R-G center recombination (or indirect recombination) typically releases thermal energy (heat) or, equivalently, produces lattice vibration

Near-midgap energy levels introduced by some common impurities in Si

• Generation Processes

9thermal energy > E_G Æ direct thermal generation 9light with an energy > E_G Æ photogeneration

9light with an energy > E_G Æ photogeneration

9 The thermally assisted generation of carriers with R G centers 9 The thermally assisted generation of carriers with R-G centers

energetic carrier collide with the lattice energetic carrier collide with the lattice 9High -field regions

ε

• Momentum Considerations

9One need be concerned only with the dorminat process

9Crystal momentum in addition to energy must be conserved.

9The momentum of an electron in an energy band can assume only certain quantized values. q

9where k is a parameter proportional to the electron momentum

Si, Ge GaAs

Photons: light Photons: light

Phonons: lattice vibration quanta

Photons, being massless entities, carry very little momentum and a

The thermal energy associated with lattice vibrations (phonons) is very small (in the 10 carry very little momentum, and a

photon-assisted transition is

essentially vertical on the E-k plot

vibrations (phonons) is very small (in the 10- 50 meV range), whereas the phonon

momentum is comparatively large. A phonon- assisted transition is essentially horizontal on assisted transition is essentially horizontal on the E-k plot. The emission of a photon must be accompanied by the emission or

absorption of a phonon absorption of a phonon.

∂t) that must be specified

∂t) that must be specified

9 B-to-B recombination is totally negligible compared to R-G center recombination in Si

9 G f

9Even in direct materials, the R-G center mechanism is often the dominant process.

• Indirect Thermal Recombination-Generation

n₀

, p

₀

…. carrier concentrations when equilibrium conditions prevail

n, p ……. carrier concentrations under arbitrary conditions

, p y Δ

n=n-n₀

… deviations in the carrier concentrations from their equilibrium values

equilibrium values Δ

p=p-p₀

…

N

number of R G centers/cm

³ N_T

………. number of R-G centers/cm

³

9Although the majority carrier concentration remains essentially unperturbed under low-level injection, the minority carrier

concentration can increase by many orders of magnitude.

9 The greater the number of filled R-G centers, the greater the probability of a hole annihilating transition and the faster the rate of probability of a hole annihilating transition and the faster the rate of recombination

9Under equilibrium, essentially all of the R-G centers are filled with electrons because E^F>>E^T

9With , electrons always vastly outnumber holes and

rapidly fill R-G levels that become vacant Æ # of filled centers during rapidly fill R G levels that become vacant Æ # of filled centers during the relaxation N

≅

^T

N

T

t

p ∝

∂

∂

9 The number of hole-annihilating transitions should increase almost linearly with the number of holes

t

R

∂

p c N p

∂ =

y

9 Introducing a proportionality constant, c_p,

c N p

= −

9The recombination and generation rates must precisely balance under equilibrium conditions, or ∂p/∂t|_G = ∂p/∂t|G-equilibrium=- ∂p/∂t|R-equilibrium

p T 0

p c N p

∂ =

_{p T 0}

G

c N p

∂ t

9 The net rate

p T 0

i thermal

R G

R G

( )

p p p

c N p p

t

⁻

t t

−

∂ ∂ ∂

= + = − −

∂ ∂ ∂

p T i thermal

R G

or p for holes in an type material

c N p n

t

⁻

−

∂ = − Δ −

∂

R G

for electrons in a type material

n c N n p

∂ = − Δ −

9 An analogous set of arguments yields

n T

for electrons in a type material

c N Δ n p

∂

n p

9c N and c N must have units of 1/time 9c_pN_T and c_nN_T must have units of 1/time

p n

1 1

N N

τ

_p

= , τ

_n

=

p T n T

c N c N

for holes in an type material

p p

t n

∂ Δ

= − −

∂

^{i thermal} _p

t

⁻R G

τ

−

∂

i th l

for electrons in a type material

n n

t τ p

∂ = − Δ −

∂

^{i thermal} _n

t

⁻R G

τ

−

∂

9 The average excess hole lifetime <t> can be computed 9 The average excess hole lifetime <t> can be computed

: 9

)

(

_p

n

or

t >= τ τ

<

9

τ

_n

( or τ

_p

)

: minority carrier lifetimes

‰ Equations of State

‰ Equations of State

• Continuity Equations

9 C i i h h i b d if diff i i di di

9 Carrier action – whether it be drift, diffusion, indirect or direct thermal recombination, indirect or direct generation, or some other type of carrier action – gives rise to a change in the carrier yp g g

concentrations with time

9Let’s combine all the mechanisms

thermal other processes

drift diff

R G (li h )

n n n n n

t t t t t

∂ ∂ ∂ ∂ ∂

= + + +

∂ ∂

_drift

∂

_diff

∂ ∂

R -G (light, etc.)

p p p p p

∂ ∂ ∂ ∂ ∂

= + + +

∂ ∂

drift

∂

diff

∂

^thermal

∂

other processes

R -G (light, etc.)

t t t t t

∂ ∂ ∂ ∂ ∂

Ny Nz

Nx

J J J

n J

n ∂ + ∂ = ∇

∂ +

∂ =

∂ + 1

) 1 (

Py Pz Px

N diff

drift

J J J J

p p

q J z

y x

q t

t

∂ ∇

∂ +

∂ + + ∂

∂

⋅

∇

∂ =

∂ +

∂ +

∂ =

∂ +

) 1 1 (

) (

P y Pz

Px diff

drift

q J z

y x

q t

p t

p = − ∇ ⋅

+ ∂ + ∂

− ∂

∂ =

∂ + ( )

processes other G

R thermal

N

t

n t

J n q

t n

∂ + ∂

∂ + ∂

⋅

∇

∂ =

∂ 1

other thermal

P

processes G

R

t p t

J p q

t p

q

∂ + ∂

∂ + ∂

⋅

∇

−

∂ =

∂

−

1

processes G

R

t

t q

t ∂ ∂

∂

₋

Mi it C i Diff i E ti

• Minority Carrier Diffusion Equations

9 Simplifying assumptions:

(1) One dimensional (1) One-dimensional

(2) The analysis is limited or restricted to minority carriers

(3) .

(4) The equilibrium minority carrier concentrations are not a (4) The equilibrium minority carrier concentrations are not a

function of position (5) Low level injection

(6) Indirect thermal R-G is the dominant thermal R-G mechanism

0 0

( ),

0 0

( )

n ≠ n x p ≠ p x

(6) Indirect thermal R G is the dominant thermal R G mechanism (7) There are no “other processes”, except possibly

photogeneration

N N

1 1 J

q q x

∇ ⋅ → ∂

J ∂

n

0

n ∂ n n

∂ = + ∂Δ = ∂Δ

x x x x

∂ ∂ ∂ ∂

1

2

n

D ∂ Δ

∇ J

_N

→ D

_{N 2}

q ∇ ⋅ → x

J ∂

9 With low level injection 9 With low level injection,

n n

∂ Δ

∂

^thermal

= −

_n

t

R -G

τ

∂

other L processes

n G

t

∂ =

∂

^{Æ GL}=0 without illumination

processes

9 Th ilib i l t t ti i f ti f ti 9 The equilibrium electron concentration is never a function of time

n

0

n n n

t t t t

∂

∂ ∂Δ ∂Δ

= + =

∂ t ∂ t ∂ t ∂ t

∂ ∂ ∂ ∂

n

2

n n

∂Δ

_p

∂ Δ

_p

Δ

_p

N 2 L

n 2

n n n

D G

t x

p p p

τ

∂Δ ∂ Δ Δ

= − +

∂ ∂

∂Δ ∂ Δ Δ

Minority carrier diffusion equations

n n n

P 2 L

p

p p p

D G

t x τ

∂Δ = ∂ Δ − Δ +

∂ ∂

q

9 Steady state

• Simplifications and Solutions

y

0 0

p n

n p

t t

∂Δ ∂ ∂ t → ⎛ ⎜ ⎝ ⎝ ∂Δ ∂ ∂ t → ⎞ ⎟ ⎠ ⎠

0 0

p n

n p

D

_N

∂ Δ

2

→ 0 ⎛ ⎜ D

_P

∂ Δ

2

→ 0 ⎞ ⎟

D D

x → ⎜ x → ⎟

∂ ⎝ ∂ ⎠

9 No drift current or Æ

E = 0

no further simplification^p 9 No thermal R-G

n ⎛ p ⎞

Δ Δ

0 0

p n

n p

n p

τ τ

⎛ ⎞

Δ Δ

→ ⎜ ⎜ ⎝ → ⎟ ⎟ ⎠

9 No light

0

G

_L

→ 0

G →

9 S l bl 2 A h b l th 0 d f if l 9 Sample problem 2: As shown below, the x=0 end of a uniformly doped semi-infinite bar of silicon with N_D=10¹⁵ cm^-3 is illuminated so as to create Δpp_n0n0=10¹⁰ cm^-3 excess holes at x=0. The wavelength of g the illumination is such that no light penetrates into the interior (x>0) of the bar. Determine Δp_n(x).

9Diffusion and recombination

9 As the diffusing holes move into the bar their numbers are reduced by recombination

9 Under steady state conditions it is reasonable to expect an excess 9 Under steady state conditions it is reasonable to expect an excess distribution of holes near x=0, with Δp_n(x) monotonically decreasing from Δp_n0 at x=0 to Δp_n0= 0 as x Æ ∞

9 E 0 ? Yes

1>Excess hole pile-up is very small

2>The majority carriers redistribute in such a way to partly cancel

≅

) ( Δ p

n max

≅ n

i

j y y p y

the minority carrier charge.

9 Under steady state conditions with G_L=0 for x > 0

2

n n

p 2

p

0 for 0

d p p

D x

dx τ

Δ Δ

− = >

0

Δ p =

n x 0 n x 0 n 0

p

⏐ = +

p

⏐ =

p

Δ = Δ = Δ

p

Δ p = 0

9 Th l l ti 9 The general solution

P P

/ /

n

( )

^{x L x L}

p x Ae

⁻

Be

Δ = +

^where

L

_p

≡ D

_p

τ

_p

exp(x/L_p) Æ ∞ as x Æ ∞

⇒ = B 0

9 With 0 9 With x=0,

A = Δ p

n 0

/ P

n

( )

n0 ^{x L}

solution p x p e

⁻

Δ = Δ ⇐

‰ Supplemental Concepts

‰ Supplemental Concepts

• Diffusion Lengths

9 minority carrier diffusion lengths

P P p

L

_P

≡ D

_{P p}

τ

N N n

L ≡ D τ

9L_P and L_N represent the average distance minority carriers can diffuse into a sea of majority carriers before being annihilated

9 f

9 The average position of the excess minority carriers inside the semiconductor bar is

n n P

0

( )

0

( )

x

^∞

x p x dx

^∞

p x dx L

〈 〉 = ∫ Δ ∫ Δ =

Q i F i L l

• Quasi-Fermi Levels

9 Quasi-Fermi levels are energy levels used to specify the carrier concentrations under nonequilibrium conditionsq

9 Equilibrium conditions prevailed prior to t=0, with n₀=N_D=10¹⁵ cm^-3 and p₀=10⁵ cm^-3

kT E

E

kT E E i

F i

i

e

F

n n

/ ) (

/ ) (

0

−

=

−

9

Δ = p G τ = 10 cm

^{11 -3}

p = p + Δ p ≅ 10 cm

^{11 -3}

n n ≅ n 10

₀ ¹⁵

= 10 cm

^15-3

cm

⁻³

kT E

E i

F

e

i

n

p

₀

=

^{( )/}

9

9 The convenience of being able to deduce the carrier concentrations by inspection from the energy band diagram is extended to

ilib i di i h h h f i F i l l b

10 cm ,

0

10 cm , 10 cm

n L p

p G τ p p p n

Δ = = = + Δ ≅ n ≅ n

₀

10 cm

nonequilibrium conditions through the use of quasi-Fermi levels by introducing F_N and F_P .

Equilibrium conditions Nonequilibrium conditions Equilibrium conditions Nonequilibrium conditions

⎛ ⎞ n

N i

( ) /

N i

or ln

F E kT i

i

n n e F E kT n

n

−

⎛ ⎞

≡ ≡ + ⎜ ⎟

⎝ ⎠

i P

⎛ ⎞

( ) /

P i

or ln

E F kT i

i

p n e F E kT p

n

−

⎛ ⎞

≡ ≡ + ⎜ ⎟

− ⎝ ⎠