Plan for the quantitative solution

Plan for the quantitative solution

n p p

n

n x

D n



 





 

2

0

p n n

p

p x

D p



 





 

0

x ≤ -x _p x ≥ x _n

1) Solve the minority carrier diffusion equations employing boundary conditions

0 )

(   

 p

x

 ¹ 

)

(

2









A i p

p

e

N x n

n ^{ ( ) }



^{ 1} 

D i n

n

e

N x n

p 0

)

(   

 n

x

p

p x L

L x

n

x Ae Be

p ( ) 





n

n x L

L x

p

x Ae Be

n ( ) 





Plan for the quantitative solution

2) Compute the minority carrier current densities in the quasi- neutral regions using

dx n qD d

J





dx p qD d

J

 



x ≤ -x _p x ≥ x _n

 



3) Evaluate the quasi-neutral region solutions for J _n (x) and J _p (x) at the edges of the depletion region and then sum the two edge current densities.

) ( )

(

n

x J x

J

J   

4) Finally, multiply the result by the cross-sectional area of the diode.

AJ

I 

It is convenient to shift the origin of coordinates to the n-edge or the p-edge of the depletion region.

Quantitative solution for ideal diode equation

For the n-type quasi-neutral region,

For the p-type quasi-neutral region,

x n

x x '  

x p

x

x "   

In the new coordinates, the minority carrier diffusion equations are

Quantitative solution for ideal diode equation

n p p

n

n x

D n



 





 

2

0

p n n

p

p x

D p



 





 

0

x ≤ -x _p

x ≥ x _n ^{x '  x  x}

x

x x "   

p n n

p

p x

D p



 





 

0 ' x’ ≥ 0

n p p

n

n x

D n



 





 

2

"

0 x” ≥ 0

In the new coordinates, the boundary conditions are

0 )

'

(   

 p

x

 ¹ 

) 0 '

(

2









n

e

x

p

side -

 n

  0

)

"

(   

 n

x

 ¹ 

) 0

"

(

2









A i p

e

N x n

n

side -

 p

 

Quantitative solution for ideal diode equation

p n n

p

p x

D p



 





 

0 ' x’ ≥ 0

n p p

n

n x

D n



 





 

2

"

0 x” ≥ 0

n

n x L

L x

p

x Ae Be

n ( ) 





p

p x L

L x

n

x Ae Be

p ( ) 





In the new coordinates, the general solutions are

n

n x L

L x

p

x Ae Be

n ( " ) 





p

p x L

L x

n

x Ae Be

p ( ' ) 





where A and B are solution constants, L

 D



L

 D



x’ ≥ 0

x” ≥ 0

Quantitative solution for ideal diode equation

p n n

p

p x

D p



 





  ² ₂

0 ' _{x’ ≥ 0}

As x '   ,  p

( x '   )  Ae

 Be

 Be

According to the boundary conditions,

thus B = 0 0

) '

(   

 p

x

 ¹ 

) 0 '

(

2









D i n

e

N x n

p

side -

 n

 



¹ 

2





D

i

e

N A n

At x '  0 ,  p

( x '  0 )  Ae

 Be

 A

Thus,

p

p x L

L x

n

x Ae Be

p ( ' ) 









D i

n

e e

N x n

p

2

1 )

'

(  



Quantitative solution for ideal diode equation





D i

n

e e

N x n

p

2

1 )

'

(  



dx p qD d

x

J

( )  



In the new coordinates of x’, the hole current density is x

x x '  

) ' '

( dx

p qD d

x

J

 







D i p n p

p

p

e e

N n L q D dx

p qD d

x

J

2

' 1 )

'

(     

dx dx ' 

At the n-side edge (x’ = 0) of the depletion region,

 ¹ 

) 0 '

(

2







p

e

N n L q D x

J

x’ ≥ 0

Quantitative solution for ideal diode equation

As x "   ,

According to the boundary conditions,

thus B = 0

At x " 0 ,

Thus,

n p p

n

n x

D n



 





  ₂

2

"

0 x” ≥ 0

side -

 p

 

 ¹ 

) 0

"

(

2









A i p

e

N x n

n

0 )

"

(   

 n

x

n

n x L

L x

p

x Ae Be

n ( " ) 





n n

n x L x L

L x

p

x Ae Be Be

n ( "   ) 







A Be

Ae x

n

 





 ( " 0 )



¹ 

2





A

i

e

N A n





A i

p

e e

N x n

n

2

1 )

"

(  



Quantitative solution for ideal diode equation

In the new coordinates of x”, the electron current density is





A i

p

e e

N x n

n

2

1 )

"

(  



dx n qD d

x

J



 )

(

x x x "   

dx

dx "   ( " ) "

dx n qD d

x

J











A i n p n

n

n

e e

N n L q D dx

n qD d

x

J

2

" 1 )

"

(   





 ¹ 

) 0

"

(

2







A i n

n n

e

N n L q D x

J

At the p-side edge (x” = 0) of the depletion region, x” ≥ 0

x” ≥ 0

 ¹ 

) 0 '

(

2







D i p

p p

e

N n L q D x

 ¹  J

) 0

"

(

2







A i n

n n

e

N n L q D x

J

Quantitative solution for ideal diode equation

The total current density (J) is given by

 ¹   ¹ 

) 0 ' ( )

0

"

(

2 /

2

  











D i p kT p

qV A

i n

n p

n

A

A

e

N n L q D N e

n L q D x

J x

J J



¹ 

2

2

  





 



 



D i p

p A

i n

n

e

N n L D N

n L q D

The total current (I) is given by I = AJ



¹ 

2

2

  





 



 



D i p

p A

i n

n

e

N n L D N

n L qA D I

) ( )

(

n

x J x

J

J   

The ideal diode equation )

1

( ^/

0 

 I e ^qV

^kT

I _ ^





 



 



D i p

p A

i n

n

N n L D N

n L qA D I

2 2

0

It is also referred to as the Shockley equation.

 ¹ 

0 

 I e ^{V A V ref} I

q V

 kT

At 300 K, kT/q = 26 mV

I I

I  

Ideal diode I-V characteristics )

1

( ^/

0 

 I e ^qV

^kT

I ^{if V}

^{ few kt/q} I  I ₀ e ^qV

^{/ kT}

    ^V _A

kT I q

I  ln ₀ 

ln

 





 



 



D i p

p A

i n

n

N n L D N

n L qA D I

2 2

)

1

( ^/

0 

 I e ^qV

^kT I

Ideal diode I-V: saturation current

The saturation current can vary by many orders of magnitude depending on semiconducting materials and doping

concentrations.

diode Ge

diode

Si 0 ,

,

0 I

I 

Ex) Because of Si ( ) and Ge ( ) n

 10

/ cm

n

 10

/ cm

For p ⁺ n junction,

D i p

p

N n L qA D I

2

0



For pn ⁺ junction,

N n L qA D I

2

0



determined I

 

 



side doped

- lightly the

by

Ideal diode I-V: saturation current

One can neglect the heavily doped side of asymmetrical junctions in computing the depletion width and other

electrostatic variables.

As a general rule, the heavily doped side of an asymmetrical junction can be ignored in determining the electrical

characteristics of the junction.

V _A = 0

Majority hole diffusion I

Minority hole drift I

Majority electron diffusion I

Minority electron I

P

n

E

0 0

/

0 ( e 1 ) I I

I

I  ^qV

^kT   

At V = 0,

Example: two ideal p ⁺ n diodes

Two ideal p ⁺ n step junction diodes at room temperature are identical except that N _D,1 = 10 ¹⁵ /cm ³ and N _D,2 =10 ¹⁶ cm ³ .

Compare their I-V characteristics.

D i p

p

N n L qA D I

2

0



) 1

( ^/

0 

 I e ^qV

^kT

I

Ideal diode I-V: carrier currents J _n & J _p

A reverse-bias plot is essentially identical except all current





A i n p n

n

n

e e

N n L q D dx

n qD d

x

J

2

" 1 )

"

(   









D i p n p

p

p

e e

N n L q D dx

p qD d

x

J

2

' 1 )

'

(     

) ( )

(

n

x J x

J

J    Forward-biased pn junction

x J x

J



 



 

0

Ideal diode: minority carrier concentrations





D i

n

e e

N x n

p

2

1 )

'

(  







A i

p

e e

N x n

n

2

1 )

"

(  



Forward biasing increases the

minority carrier concentrations over their equilibrium values.

Reverse biasing lowers the minority carrier concentrations below their equilibrium values.

Forward

Reverse In each case, after several diffusion

lengths the perturbations effectively

disappear and the minority carrier

concentrations approach their

Announcements

• Next lecture: p. 260 ~ 281