PPT Semiconductors: A General Introduction - Seoul National University

Carrier Action

Sung June Kim [email protected] http://nanobio.snu.ac.kr Chapter 3.

1

Subject

1. What relation do mobility of electron and resistivity have?

2. Contribution of diffusion of excess carrier in current 3. Carrier injection and carrier action

1. direct gap semiconductor and indirect semiconductor 2. recombination lifetime

3. quasi-Fermi level

4. How does current change while considering diffusion and

recombination ?

 Drift

 Diffusion

 Generation-Recombination

 Equations of State

3

 Drift

• Definition-Visualization

 Charged-particle motion in response to electric field

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

Collisions with impurity atoms and thermally agitated lattice atoms

 Repeated periods of acceleration and subsequent decelerating collisions

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

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

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

5

• Drift Current

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

(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

Thermal motion of carrier

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

... All holes this distance back from the normal plane will cross the plane in a time

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

d d

d d

v t v

t

v tA t

pv tA



... Holes crossing the plane in a time ... Charge crossing the plane in a time

d

t

qpv tA t

7 P drift d

P drift_

 qpv

d

J

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



sat⁰

. . . 0 0

d 1/ . . .

0 sat

1 v

v

v



 





 

 

 

   

    

   

 

E E

E

E 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

 In the low field limit

v

_d

 

₀

E

P drift_

 q 

p

p

J E

N drift_

 q n 

n

J E  q , v

_d

  

_n

E , J

_{N drift|}

  qnv

_d



1

 



2 

0

9

• Mobility

 Mobility is very important parameter in characterizing transport due to drift

 Unit: cm²/Vs

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

(ii) Ionized impurity scattering

Displacement of atoms leads to lattice scattering

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

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

 The number of collisions decreases   varies inversely with the amount of scattering

/

*

q m

        m

^*

m*



 

Room temperature carrier mobilities as a function of the dopant

11

 Temperature dependence

 Low doping limit: Decreasing temperature causes an ever-

decreasing thermal agitation of the atoms, which decreases the lattice scattering

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

Temperature dependence of electron mobility in Si for dopings ranging

13

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



 J E

 1

  

J E E

=1/: conductivity

drift



N drift_



P drift_

 q ( 

n

n  

p

p )

J J J E

n p

1

( )

q n p

    

n D

1 ... n type semiconductor

 q N

   

p A

1 ... p type semiconductor

 q N

   

Resistivity versus impurity

concentration at 300 K in Si

15

• Band Bending

When exists the band energies become a function of position

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

energies would be E_c and E_v,

respectively, and the created carriers would be effectively motionless

motionless



 E-E^c= K.E. of the electrons

 E^v-E= K.E. of the holes

The potential energy of a –q charged particle is

P.E.   qV

c ref

P.E.

 E  E

c ref

P.E.

 E  E

c ref

1 ( )

V E E

   q 

  dV

E

17

 By definition,

  V E

dV

  dx E

c v i

1 dE 1 dE 1 dE

q dx q dx q dx

  

E

 In one dimension,

 Diffusion

• Definition-Visualization

 Diffusion is a process whereby

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

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

19

• Diffusion and Total Currents

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

 Using Fick’s law,

] /

[# cm

²

s D

F     

Diffusion constant

P diff P

N diff N

qD p qD n





  

   J

J

 Total currents

P P drift P diff p P

N N drift N diff n N

drift diffusion q p qD p

q n qD n





 

 

    

 

    

J J J

J J J

 

E

E

 Total particle currents

N P

 

J J J

s cm

D ] /

[ 

²

+

21

• Relating Diffusion coefficients/Mobilities

 Einstein relationship

Consider a nonuniformly doped semiconductor under equilibrium

 Constancy of the Fermi Level: nonuniformly doped n-type

semiconductor as an example

dE

_F

/ dx dE

^{= 0}



_F

/ dy dE 

_F

/ dz  0

Under thermal equilibrium, there are no charge movements between two materials,

Also, the rate of electron movement from material 2 to material 1 is in proportion to

Under equilibrium condition, these terms are equal. So, f₁(E) = f₂(E) and E_F1 = E_F2

The rate of electron movement from material 1 to material 2 is in proportion to

Fig. Junction of two materials in

equilibrium state. Actually, there is no electron movement, Fermi level should be constant hrough the entire material.

23

 Under equilibrium conditions

N P

0

N P

0

    

J J J J = J

p P

n N

for nonuniform doping 0 and

or for uniform doping 0

q p qD p

q n qD n

p n





 

 

 



 

   



E E E

 Einstein relationship:

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

N drift N diff n N

dn 0

J J q n qD

 dx







 E  

1 dE

i

 q dx

 E  n n e 

_i ^{(EFEi) /kT}

&

= -

Electron diffusion current flowing in the +x direction “Built- in” electric field in the –x direction  drift current in the –x direction

 With dE_F/dx=0,

F i

( ) /

i E E kT i

n dE

dn q

e n

dx kT dx kT

 



  E

 Substituting

n N

( ) ( ) q 0

qn qn D

  kT 

E E

N n

Einstein relationship for electrons D kT

  q

P p

Einstein relationship for holes D kT

  q

 Einstein relationship is valid even under nonequilibrium

 Slightly modified forms result for degenerate materials

25

• Definition-Visualization

 When a semiconductor is perturbed  an excess or deficit in the carrier concentrations  Recombination-generation

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

 Generation: a process whereby electrons and holes are created

• Band-to-Band Recombination

 The direct annihilation of an electron and a hole  the production of a photon (light)

• R-G Center Recombination

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

 The most important property of the R-G centers is the introduction of allowed electronic levels near the center of the band gap (E^T)

Two-step process

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

27

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

• Generation Processes

thermal energy > E_G  direct thermal generation

light with an energy > E_G  photogeneration

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

29

Impact ionization

 e-h is produced as a results of energy released when a highly energetic carrier collide with the lattice

High -field regions



• Momentum Considerations

One need be concerned only with the dorminat process

Crystal momentum in addition to energy must be conserved.

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

where k is a parameter proportional to the electron momentum

31

Si, Ge GaAs

Photons, being massless entities, carry very little momentum, and a photon-assisted transition is

essentially vertical on the E-k plot

The thermal energy associated with lattice 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 the E-k plot. The emission of a photon must be accompanied by the emission or

Photons: light

Phonons: lattice vibration quanta

33

• R-G Statistics

 It is the time rate of change in the carrier concentrations (n/t, p

t) that must be specified

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

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

• Indirect Thermal Recombination-Generation

n

₀

, p

₀

…. carrier concentrations when equilibrium conditions prevail

n, p ……. carrier concentrations under arbitrary conditions

 n=n-n

₀

… deviations in the carrier concentrations from their equilibrium values

 p=p-p

₀

…

N

_T

………. number of R-G centers/cm

³

material type

p a in p

p p

n

material type

n an in

n n

n p

implies injection

level low



















0 0

,

,

35

where n= p=10⁹ cm^-3

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

concentration can increase by many orders of magnitude.

excess carrier

p p

p p

cm N

n p

n n

n n

cm N

n

D i

D



































0 3

6 2

0

0 0

3 14

0

10 /

10

 p

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

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

With , electrons always vastly outnumber holes and

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

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

 Introducing a proportionality constant, c_p,



n

0

p 



T R

t N

p 





37

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

p T 0 G

p c N p t

 



p T 0

i thermalR 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

^

 

     



n T i thermal

R G

for electrons in a type material

n c N n p

t

^

 

     



 The net rate

 An analogous set of arguments yields

um equiliburi G

G

t

p t

p





 





 c_n and c_p are referred to as the capture coefficients

c_pN_T and c_nN_T must have units of 1/time

p n

p T n T

1 1

c N c N

    

i thermalR G p

for holes in an type material

p p

t

^

 n

 

 

   



i thermalR G n

for electrons in a type material

n n

t

^

 p

 

     



These are the excess minority carrier life times.

39

• Continuity Equations

 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

concentrations with time

Let’s combine all the mechanisms

thermal other processes

drift diff R-G (light, etc.)

thermal other processes

drift diff R-G (light, etc.)

n n n n n

t t t t t

p p p p p

t t t t t

    

   

    

        

    

processes other G

thermalR P

processes other G

thermalR N

Pz P Px Py

diff drift

Nz N Nx Ny

diff drift

t p t

J p q

t p

t n t

J n q

t n

q J z

J y

J x

J q

t p t

p

q J z

J y

J x

J q

t n t

n



 



 







 





 



 





 









 

 



 



 

 

 









 

 



 



 



 









1 1

) 1 1 (

) 1

1 (

x

x x

J x

J q t

p

p p

x x x n







 











) (

) 1 (

t p x

t p x

x q

q v p x

J

q 

 

 

 

 









 



  1

t v x



 

• Minority Carrier Diffusion Equations



Simplifying assumptions:

(1) One-dimensional

(2) The analysis is limited or restricted to minority carriers (3) .—no drift current term- so it is called Diffusion Equations.

(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 (7) There are no “other processes”, except possibly

photogeneration

 0 E

0 0

( ),

0 0

( )

n  n x p  p x

N N

1 1 J

q q x

    J 

N n N N

n n

J q n qD qD

x x

  

 

  

E





43

n n 

0

  n



n

0

n n n

x x x x



     

   

2

N N 2

1 n

q D x

     J 

 With low level injection,

thermal n

R-G

n n

t 

 

  

other L processes

n G

t

 



^{ GL}=0 without illumination

 The equilibrium electron concentration is never a function of time

n

0

n n n

t t t t



     

   

2

p p p

N 2 L

n 2

n n n

P 2 L

p

n n n

D G

t x

p p p

D G

t x





   

  

 

      

 

Minority carrier

diffusion equations

45

 No concentration gradient or no diffusion current

2 2

2 ^p

0

2 ⁿ

0

N P

n p

D D

x x

     

   

   

 No drift current or  no further simplification

E  0

 No thermal R-G

0 0

p n

n p

n p

 

 

 

       

 No light

L

0

G 



0 0

p n

n p

t t

           

 Steady state

is set equal to zero. G

_L

L P

n n

P

n

p G

x D p

t

p  

 



 









2 2

L P

n

n

p G

dt p

d  

 



t P

n

n

t p e

p ( )   ( 0 )

^{ /}



Therefore, the solution is

p p p

N 2 L

n 2

n n n

P 2 L

p

n n n

D G

t x

p p p

D G

t x





   

  

 

   

  

 

Sample Problem No.1 : A uniformly donor-doped silicon wafer

maintained at room temperature is suddenly illuminated with light at time t=0. Assuming , , and a light-

induced creation of electrons and holes per

throughout the semiconductor, determine for .

3 15

/

10 cm

N

_D

 

_P

 10

^6

sec

10

17

cm

³

 sec

) (t p

_n

 t  0

Solution :

Step1 – Si, T=300K, , and at all points inside the semiconductor.

Step2 -

with uniform doping, the equilibrium and values are the same

everywhere throughout the semiconductor.

3 15

/

10 cm

N

_D

 G

_L

 10

¹⁷

/ cm

³

 sec

3 15

0 3

10

/ , , 10 /

10 cm N n n N cm

n

_i



_D



_i



_D



3 5

2

0

n / N 10 / cm

p 

_{i D}



Step3 – Prior to t=0, equilibrium conditions prevail and illumination -> increase.

excess carrier number ↑ -> thermal recombination rate ↑

∴ light generation and thermal recombination rates must balance under steady state conditions, we can even state

or if low-level injection prevails.

Step4 –

the boundary condition

 0

 p

_n

p

n



P n

L

p t

G   (   ) /   p

_n

( t   )   p

_n|max

 G

_L



_P

3 15

0 3

11 max

|

G 10 / cm n 10 / cm

p

_n



_{L P}

  

 

L P

n P n

n

p G

x D p

t

p  

 



 









2 2

0

| )

(

₀



 p

_n

t

_t

ordinary differential equation and simplifies to

The general solution is

Applying the boundary condition yields

and

L P

n

n

p G

dt p

d  

 



t P

P L

n

t G Ae

p ( )   

^{ /}



P

G

L

A   

solution )

1 ( )

(  

^/



 p

_n

t G

_L



_P

e

^{t P}

Step5 – A plot of the solution equation is shown below.

Note that, starts from zero at t=0 and eventually saturates at after a few .

) (t p

_n



P

G

L

 

^P

order by the pulse train pictured below.

) 1

( )

(

_{L P} ^{t/ P}

n

t G e

p   

^{ }



The upper equation approximately describes the carrier build-up during a light pulse.

However, the stroboscope light pulses have a duration of compared to a minority carrier lifetime of .

With , one sees only the very first portion of the Fig.

3.25 transient before the light is turned off and the semiconductor is allowed to decay back to equilibrium.

The light-off expression derivation closely parallels the light- on sample problem solution.

Exceptions -> at the beginning of the light-off = at the end of the light-on.

sec 1 

on

 t sec 150 



_P

 1

/

_P



t

on



) (t p

_n



p

n

  p

_n

55

 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 p_n0=10¹⁰ cm^-3 excess holes at x=0. The wavelength of the illumination is such that no light penetrates into the interior (x>0) of the bar. Determine p_n(x).

 Solution: at x=0, p_n(0)= p_n0= 10 cm , and p_n  0 as x  

 The light first creates excess carriers right at x=0

G^L=0 for x>0

Diffusion and recombination

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

 Under steady state conditions it is reasonable to expect an excess distribution of holes near x=0, with p_n(x) monotonically decreasing from p_n0 at x=0 to p_n0= 0 as x  

 Can we ignore the drift current? E 0 ? Yes because 1>Excess hole pile-up is very small , here

2> Also the majority carriers redistribute in such a way to partly cancel the minority carrier charge. Thus use of diffusion eq. is

justified.

 (  p

_{n max}

 n

_i

)

57

 The general solution

P P

/ /

n

( )

^{x L x L}

p x Ae

^

Be

  

^where

L

_p

 D

_{p p}



exp(x/L_p)   as x  

  B 0

 With x=0,

A   p

n0

/ P

n

( )

n0 ^{x L}

solution p x p e

^

   

2

n n

p 2

p

0 for 0

d p p

D x

dx 

 

   

n _x

0

p

_{ }

 

n x 0 n x 0 n0

p

  

p

 

p

    

 So---Under steady state conditions with G_L=0 for x > 0

N N n

L  D 

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

 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

       

 Supplemental Concepts

• Diffusion Lengths

 minority carrier diffusion lengths

P P p

L  D 

59

• Quasi-Fermi Levels



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

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

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

Equilibrium conditions Nonequilibrium conditions

N i

i P

( ) /

N i

( ) /

P i

or ln

or ln

F E kT i

i

E F kT i

n n e F E kT n

n

p n e F E kT p

n





        

 

        

 



11 -3 11 -3 15 -3

10 cm ,

0

10 cm , 10 cm

n L p

p G  p p p n

        n   n

₀

 10

¹⁵

cm

^3