Semi4ExSol6

Semiconductor Physics and Devices: Basic Principles, 4 th edition Chapter 6 By D. A. Neamen Eercise Solutions  !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!  !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!  !!!!!!!! 

Chapter 6 Exercise Solutions

+iii δ  p ( % ×"#

 

−

)

=

"4

% ×"# =

Ex 6.1

δ n  R n′ = τ no

"# "% e −t  $ "#

=

=

"#

−6

"#

+iv δ  p( ∞ )

cm −& s −"

=

% × "#

"4

e

−%

4.*66 × "#

("

=

"%

= "# (" cm −

&

−6

−

"4

$"

)

cm

−&

"# − 6

 R n′

  'or t  = # ,

("

s

+- δ  p( ma )

−

e

−∞

% × "#

)

"4

cm −&

= % ×"#"4 = ( #.#") N d 

/es, lo01in2ection condition is met.  !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!   !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!! 

−"

t  = " µ  s,

 R n′ −& s −" t  =

=

4 µ 

 R n′

"# "% e −" $ "

= &.6) × "# (# cm

−6

"#

Ex 6.4

+a  L n

s,

=

"%

"# e "#

−4 $ "

−6

= ".)& × "#

"*

=

= 4.%4 × "#

"# − 6

"6

= [( (%) ( % ×"# − )]

=

&.%&6 × "#

=

&%.&6  µ  m

or δ n

"# "% e −"# $ "

D nτ no

−&

cm

= "#"% e + x $  L

n

 +  x

≤#

= "# "% e # = "# "% cm −   +ii δ n = "# "% e −&# $ &% .&6 = 4.() × "# "4 &

+i δ n

cm

"$ (

δ n = "#"% e − x $  Ln  +  x ≥ # 

+-

−& s −" t  = "# µ  s,

 R n′

or  L

cm

=

cm −&

−& s −"

 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 

 +iii δ n

= "# "% e −%# $ &%.&6 = (.4& × "# "4

cm −& Ex 6.2 +a "# "4 e − t  $ %# = "# "4 e −" ⇒ t  = %# ns

+- "# "4 e − t  $ %#

= "#

 +iv δ n cm −& +v δ n

"&

 "#"4 −* t  = ( %# × "# ) ln "&  "#

  −     = "."% × "#  

s

or t  = ""% ns  !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!  Ex 6.3

+a δ  p ( t )

=  g ′τ  po (" − e − t  $ τ 

 po

)

cm −&  !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!   !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!  Ex 6.5

δ  p ( x, t )

− t  $ τ  po

 +i δ  p ( # )

= % × "# "4 (" − e −# ) = #

+ii δ  p ("#

−

"4

% ×"#

=

−&

("

−

e

=

e

− t  $ τ  p #

( 4π  D p t ) " $ (

 − ( x − µ  p Ε # t ) (   ep  4 D p t   

 µ  p Ε # t  = ( 4## ) ("## ) ("# − )

−" $

"

&."6 × "#

) "4

cm

= 4 × "# −& c

m or = 4#  µ  m +i  x = (# µ  m.

po

=

= "# "% e −"(# $ &%.&6 = &.&6 × "# "&

+a

= (% × "# (" )("# − )(" − e = % × "# "4 (" − e − t  $ τ 

)

= "# "% e −)% $ &%.&6 = *.#4 × "# "&

δ  p

=

#.&6)) &.%4% × "# − &

 − ( − ( × "# −& ) (  ep    4 × "# − 6 

= &) .")

Semiconductor Physics and Devices: Basic Principles, 4 th edition Chapter 6 By D. A. Neamen Eercise Solutions  !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!    +ii  x = 4# µ  m, 'i5ure %.&,  µ  p ≅ 4## cm ( $1s.

=

+iii  x

 

#.&6))

=

δ  p

&.%4% × "# − &

ep[ #]

σ  = e µ  p N a

"#&.)

=

= 6# µ  m,

δ  p

#.&6))

=

&.%4% × "# − & δ  p

+-  x

=

4# µ 

∈

= =

Ω 1cm −

σ 

"

=

("". ) ().)% × "# −"4 ) ".()

).#* × "#

−"&

s

or τ d  = #.)#* ps  !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 

m −)

Ex 6.7

s,

#.6#6%&

=

  

&).")

=

+i t  = % × "#

δ  p

 − ( ( × "# ) ep   4 × "# −6

(

".() +

7hen τ d 

  −&

= (".6 × "# −"* )( 4##) ( ( ×"# "6 )

(.%#66& × "# − &

= &( .%

 p o

 − ( ( × "# −& ) (  ep    ( × "# − 6 

=  N a − N d  = "#"6 − & × "#"% =

no

=

n i(  p o

 × "#

"%

cm −&

(".% × "# ) = &.("4 ×"# = "#

 × "#

(

4

"%

cm

−&

+a 8n thermal e9uili-rium,

 E  Fi

+ii t  = "# − s,

 

δ  p

#.&6))

=

&.%4% × "# − &

ep[ #]

   × "# "%     = ( #.#(%*) ln  "#   " . % "# ×     = #.&&)#) e

= "#& .)

+iii t  = ( × "# − s,

 

δ  p

#."&%&

=

%.#"& × "# −&

= &.6%

  p   − E  F  = kT ln o       ni  

 − ( − 4 × "# −& ) (  ep    ) × "# −6  +- uasi1'ermi levels,

 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 

 E  Fi − E  Fp

Ex 6.6

+a 'or  N d 

= % × "#"% cm −

&

 in 3aAs,

rom 'i5ure %.&,  µ n

≅ %## cm

(

$1s.

= (".6 × "# −"* )( %## ) (% × "# "% )

σ  = e µ n N d 

= 6 + Ω 1cm − 7hen τ d 

= =

σ 

=

("&.") ().)% ×"# −"4 )

".*& × "#

6 −"&

s

= #."*& ps 'or  N a = ( × "# "6 cm −

or +-

∈

τ d 

&

rom

  ×"#"% + 4 ×"#"4 = ( #.#(%*) ln "#   ".% ×"#

= #.&&*%( e

 E  Fn

"

 

  p + δ  p     = kT  ln o   n   i  

 in silicon,

       

 n + δ n     − E  Fi = kT  ln o   n   i  

 &.("4 × "# 4 + 4 × "#"4 = ( #.#(%*) ln ".% × "#"#  

= #.(6&*% e

       

 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 

Semiconductor Physics and Devices: Basic Principles, 4 th edition Chapter 6 By D. A. Neamen Eercise Solutions  !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!  Ex 6.8 +i 'or  s → ∞ , δ  p ( # ) = #

= "# "% cm −

&

  n1type; n o

 p o

= (.(% × "# δ n

=

τ no

%

+ii 'or  s → # , δ  p ( # )

,

⇒ δ  p ( x )  = constant

cm −&

δ p

=

"#

"4

 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 

cm −& ,

= τ  po = % × "# − s


Test Your n!erstan!in" Solutions

[( n + δ n)(  p + δ  p) − n ] (

 R =

=  g ′τ  p # ,

o

τ  po ( no

o

i

+ δ n + ni ) +τ no (  po + δ  p + ni )

≅ ("# "% + "# "4 )("#"4 ) − (".% × "# "# ) ÷ {( % × "# − )("# "% + "# "4 )

(

TY 6.1 +a p1type; =inority carriers > electrons

+- δ n( t )

  − t      = δ n( # ) ep    τ no  

  7hen

+ (% × "# − )("# "4 )}

δ n( t )

or  (#

− t    − = "# "% ep     cm  % × "# − 6  

&

 R = ".)&×"# cm −& s −"  !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 

 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 

Ex 6.9

TY 6.2 +a p1type; =inority carriers > electrons

+a 'or τ  p # s

 τ  = # ⇒ δ  p s = δ p B   p# s  τ  p #

   = #    

+- δ n( t )

'rom E9uation +6."#*,

δ  p( x ) As  x

=  g ′τ  p # +  Ae + x $ L + Be − x $ L  p

→ ∞,

δ  p

=  g ′τ  p # = "#

"4

p

 

7hen δ  p ( x )

=  g ′τ  p # (" − e

− x $  L p

= #) = # δ  p ( # ) δ  p ( #)  R ′ = = ⇒ R′ = ∞

+- δ  p ( x +c

τ  p # s

 − t  = ("# (# )(% × "# − 6 )" − ep    % × "# − 6 

cm −&

⇒  A = # As  x → #, δ  p = # ⇒  B = − g ′τ  p #

#

  Note: δ  p ( # ) = # is a result o  R .  !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 

′=∞

− x $  L      se    δ  p( x ) =  g ′τ  p # " −  ( D p  L p ) + s        p

+a

+i 'or  s → ∞ ,

=  g ′τ  p # (" − e − x $  L +ii 'or  s → # , δ  p( x ) =  g ′τ  p # δ  p ( x )

+-

p

     

or 

 − t    = % ×"# "4 " − ep      % ×"# − 6    As t  → ∞ , δ  n( ) % "# cm −

δ n( t )

"4 & ∞ = × +c  !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 

TY 6.3

δ n ( x )

 L  p Ex 6.10

   − t    =  g ′τ no " − ep       τ no  

=

   x   = δ  p ( x ) = δ n( # ) ep  −       L  p  

D  pτ  po

= ("# ) ("# −6 ) = &" .6 × "# −4

cm   7hen

δ n( x )

−x   = δ  p( x) = "#"% ep   − 4  cm  &".6 ×"#  

−&

 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!  TY 6.4 n1type; =inority carriers > holes

Semiconductor Physics and Devices: Basic Principles, 4 th edition Chapter 6 By D. A. Neamen Eercise Solutions  !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 

= −eD p

 J diff  

=−

dp

= −eD p

dx

d ( δ  p( x ) ) dx

(".6 ×"# − )("# ) ("# ) ep  − "#    "*

"%

− +&".6 × "# − 4 

 &".6  

7hen hole diusion current density  J diff   = +#.&6* A$cm (
TY 6.6 ?sin5 the results rom 7/? 6.%, 0e ind

δ  p =

ep( − t  τ  p )

( 4π  D t )

 J diff  +holes  

−  

+a +i  x −  µ  p Ε o t 

7hen electron diusion current density  J diff   = −# .&6* A$cm (

=

 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 

".#*& ×"# =

TY 6.5

δ  p =

ep

"$ (

 p

+a

ep( − " %)

[( 4π )("#) ("# ) ] −6

"$ (

⇒ δ  p = &.#

or  δ  p

ep( − % %)

[( 4π )("#) (% ×"# )]

"$ (

⇒ δ  p = "4.

+c

.# × "#

=

ep( − "% %) "$ (

⇒ δ  p = "."%

"$ (

⇒ δ  p = #."(#

[( 4π )("#) ("% ×"# − )] 6

)

−&

+ii  x −  µ  p Ε o t 

 

ep( − (% %)

[( 4π  )("#) ( (% ×"# )]  No0

−&

−

( &)6)("#) ("#

= −.# × "#

or  δ  p

=

)

−&

(#.*

+- +i  x −  µ  p Ε o t 

=  µ  p Ε o t  = ( &)6)("#) t 

7hen +a  x = &).6 µ  m +-  x = "*& µ  m +c  x = %* µ  m +d  x = *6% µ  m  !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 

−6

 − ( − .# × "# −& ) (  δ  p = &.# ep   4("#) ("# − 6 ) 

+d

 x

−6

(#.*

= −&.("× "#

−6

( &)6)("#) ("#

−

 − ( .# × "# −& ) (  δ  p = ⋅ ep  −6 −6 " $ ( ( ) 4 "# "# ( )   [( 4π )("#) ("# )]    − ( .# × "# −& ) (  = &.# ep    4("#) ("# −6 ) 

+- −6

−(

ep( − " %)

− t  τ  po

( 4π  D t )

"$ (

 p

 − ( x − µ  p Ε o t ) (   ⋅ ep 4 D p t   

=

(.64 ×"# =

−(

−

( &)6)("# ) (% ×"#

." × "#

−6

)

−&

 − ( ."× "# −& ) (  δ  p = "4. ep    4("# ) (% × "# − 6 )  or  δ  p

=

"".4

  +ii  x −  µ  p Ε o t  =

".(( ×"#

−(

−

( &)6)("#) (% ×"#

= −." × "#

−6

)

−&

 − ( − ."×"# −& ) (  δ  p = "4. ep    4("#) (% ×"# − 6 )  or 

Semiconductor Physics and Devices: Basic Principles, 4 th edition Chapter 6 By D. A. Neamen Eercise Solutions  !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!  δ    p = "".4

+c +i  x −  µ  p Ε o t  =

6.%# ×"# =

−(

−

( &)6)("# ) ("% ×"#

." × "#

−6

)

−&

 − ( ."× "# −& ) (  δ  p = "."% ep    4("# ) ("% × "# −6 )  or  δ  p

  =

=

".#6

+ii  x −  µ  p Ε o t  %.#) × "#

−(

−

( &)6)("#) ("% ×"#

= −." × "#

−6

)

−&

 − ( − ."× "# −& ) (  δ  p = "."% ep    4("#) ("% ×"# −6 )  or   p δ  

=

".#6

 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!