1

• Extrinsic p형반도체가 low-level injection일때 ambipolar diffusion 계수와 ambipolar mobility

계수는상수(constant)인minority carrier (electron)의 parameter로귀착

• Low-level injection 조건의 n형반도체의경우, 비슷한방법으로

0

0

- ,

,

n p n p o

n p n o p

o o

o

n

n p o o

n

n o p o

D D n p D D n n p pD

D n D p D n n D p p

If p type semiconductor p n

If low level injection p p

D D

p p n n

n n p p

, , o o o p pp n n n D D

6. Limits of Extrinsic Doping and Low Injection

2

Low level injection 조건에서의 extrinsic p형반도체

• majority carrier hole의농도는본질적으로일정

• 단위시간당 minority carrier electron이majority carrier hole을만나는확률도기본적으로일정

• 따라서, 1/tnt = 1/tn (여기서 tn은 minority carrier electron의수명) = 일정

Low level injection 조건에서의 extrinsic n형반도체

• minority carrier hole의수명, 1/tpt=1/tp =일정

• minority carrier hole 농도가크게증가

단위시간당 majority carrier electron이 hole을만날확률크게증가

majority carrier 수명크게변화

1 1 : ( ) ( )

nt ptt t

전자 정공가정공전자 과만나재결합할단위시간당확률

3

전자에대해

thermal equilibrium excess

정공에대해

thermal equilibrium excess

excess 전자의생성비율은 excess 정공의생성비율과같아야하므로 g’=gn’=gp’

Low level injection 조건의 p형반도체

Low level injection 조건의 n형반도체

excess majority carriers 움직임은 minority carriers의파라미터에의해결정된다

' '

' ' '

n n no n no n

no no n n n

n

g R g R G g R R

nThermal equilibrium G R g R g R g

t

' '

' ' '

p p po p po p

po po p p p

p

g R g R G g R R

pThermal equilibrium G R g R g R g

t

2

2

no

'

where τ is minority carrier life time under low-level injection

n n

no

n n nnD E g

x x t

t

2

2

po

'

where τ is minority carrier life time under low-level injection

p p

po

p p ppD E g

x x t

t

4

5

• [예제1]

• 반도체가열적평형상태로갈때시간에따른 excess carrier 움직임을결정하라.

• (조건) 1. 무한히큰크기의 homogeneous한 n형반도체

2. 외부에서가해진전계 = 0

3. t=0에서 excess carrier가균일하게발생

4. t > 0 조건에서 g'=0

5. low level injection

[해답]

2

2

2

2

/

/

'

uniform concentration 0

0, ' 0

0

0 (charge neutrality)

po

po

p p

po

po

t

t

p p ppD E g

x x t

p p

x x

p pwhen t g

t

p t p e

n t p e

t

t

t

t

“The excess electrons and holes recombine at the rate determined by the excess minority carrier

hole lifetime in the n-type semiconductor.”

6

• [예제2] excess carrier 농도의공간상분포를결정하라.

• (조건) 1. 무한히넓은 homogeneous p형반도체

2. x=0에서 excess carrier 발생

3. 외부에서인가된전계또는전압 = 0

[해답]

/ /

/

/

0, 0

0 0 0

0 0

n n

n

n

x L x L

x L

x L

n x Ae Be

n B

n A n x n e for x

n x n e for x

2

2

2

2

2 2

2 2 2

2

'

, 0, ' 0 0

, 0

0

0

, minority carrier

n n

no

n

no

n no n

n n no

n n nnD E g

x x t

From assumptions E g for x

nFor steady state

t

n nD

x

d n d nn n

dx D dx L

where L D diffusion length

t

t

t

t

7

excess minority hole concentration in the n-type semiconductor

8

n형반도체에 p가갑자기인가

• 여기서 td=(e/s)는 dielectric relaxation time constant 이며, sub-psec정도의값을가짐

• 일반적으로 excess carrier의수명은 0.1 us 수준

dielectric relaxation time constant ?

• charge neutrality를얼마나빨리회복하느냐의척도

• 순간적으로 bulk로부터전자들을끌어와서 neutralize 시키는정도를나타냄

/

: E=

' : J= E

: J=

J= E=

0 0 dt

Poisson equation

Ohm s law

Continuity equationt

de

t dt

t

e

s

ss

e

s s

e e

7. Dielectric Relaxation Time Constant

9

Excess carrier가반도체에서발생

• 열적평형상태 (ⅹ), Fermi energy 정의 (ⅹ)

• 새로운 Fermi energy 정의필요 quasi-Fermi level

열적평형상태에서

열적비평형상태에서

여기서 EFn과 EFp는 quasi-Fermi 에너지레벨

exp

exp

F Fio i

Fi Fo i

E En n

kT

E Ep n

kT

exp

exp

Fn Fio i

Fi Fp

o i

E En n n

kT

E Ep p n

kT

(a) Nd=1015cm-3, ni=1010cm-3조건에서의열적평형상태

(b) 1013cm-3의 excess 캐리어가존재할때의 quasi-Fermi 레벨

8. Quasi-Fermi Energy Levels

10

Carrier의평균수명은 Shockley-Read-Hall의재결합(recombination) 이론으로부터결정될수있다

forbidden-energy band 에서의 defect states 재결합에영향 tno, tpo 에영향

forbidden 밴드갭안에서의허용된에너지상태 (trap)은 recombination center 의역할

8. Excess Carrier Lifetime

acceptor-type trap 에관련된 4가지기본천이과정 (acceptor-type trap: negatively charged with electron)

• Process 1 : 처음에비어있는 neutral trap에의해 conduction band로부터전자를 capture

• Process 2 : 과정1의반대과정으로처음에 trap을채우고있던전자가 conduction band로다시방출

• Process 3 : 전자를포함하고있는 trap이 valence band로부터정공을 capture (valence band로의전자방출로생각가능)

• Process 4 : 과정3의역과정. neutral trap으로부터 valence band로정공의방출 (또는 valence band로부터전자의 capture)

11

E=Et 에있는 recombination center에의한전자와정공의 recombination rate는다음의식으로주어진다

• 여기서 Cn은전자의 capture rate에비례하는상수이고 Cp는정공의 capture rate에비례하는상수임.

Nt는 trapping center의전체농도를의미함

• 열적평형상태 : np=nopo=ni2 Rp=Rn=0

• Rn, Rp는 excess carrier의 recombination rate 이므로, R=n/t

2

' '

n p t i

n p

n p

C C N np nR R R

C n n C p p

☞ pp. 223~224의수식전개따라가보기

12

Low injection 조건의 n형반도체에서 no >> po, no >> p, no >> n’, no >> p’

no >> n’, no >> p’ trap 에너지레벨이밴드갭의중간근처에있다는의미

• Nt 증가 excess carrier recombination 확률증가 excess carrier lifetime 감소

비슷한방법으로, p형반도체를생각하면 low injection 조건에서

• 결론적으로 extrinsic 반도체에서 excess carrier lifetime은 low injection 조건하에서 minority

carrier lifetime으로귀착됨

Limits of Extrinsic Doping and Injection

1 p t po

po p t

n pR C N p where

C N

t

t t

1no

n tC Nt

capture rate of

minority carrier hole

capture rate of

minority carrier electtron

13

‘반도체가 extrinsic 에서 intrinsic으로갈수록 minority carrier lifetime은길어짐’

Intrinsic 반도체에서의 excess carrier lifetime

2 2

2

' ' ' '

,

2

2

2

2

2

n p t i i

n p po po

i i

i

i no po

i

i

no poi no po

C C N np n np nR

C n n C p p n n p p

Intrinsic semiconductor n n n p n n

n n nR

n n

Low level injection condition n n

n n n nR

n

whe

t t

t t

t t tt t

no pore t t t

Very

14

반도체가갑자기종단되었을때 crystal periodicity는사라짐

• 밴드갭내에허용된전자에너지상태를형성

표면에서의 trap 밀도 > bulk 에서의 trap 밀도

@ : , : s Bs

pos po

p psurface R in bulk R

t t

homogeneous and steady-state condition

: G = R, R = Rs

tpos < tpo ps < pB

9. Surface Effects

15

[Example 6.9]

conditions : n-type semiconductor

pB = 1014 cm-3 and tpo = 10-6 s in the bulk

tpos = 10-7 s at the surface

assume zero applied electric field and let Dp = 10 cm2/s

Determine the steady-state excess carrier concentrations as a function of distance from the

surface of a semiconductor

[Solution]

mDLexp

BBppx

Agpxpx

BeAegxp

sp

g'

pg

dx

pdD

pp

pp

popp

Lx

s

poB

LxLx

po

po

B

po

p

po

pos

Bs

pos

s

po

B

p

pp

t

t

t

t

t

t

t

t

t

6.311010 where 9.0110)(

1091010)0( ,0At

0cm 10')( , As

')(

is solution The

cm 1010

10 is bulk the in rate generation state-steady where

0'

cm 1010

1010

6/14

131314

3-14

//

13-20

6

14

2

2

3-13

6

714