density) can vary from position to position. Let be the potential along the channel I - V relationships (square-law theory)

G (V_G)

S D (V_D)

Q_N(y) = inversion layer charge

0



VD

 

Let  be the potential along the channel

where the diffusion current is neglected.

For gate voltages above turn-on (V_G ≥ V_T), and drain voltages below pinch-off (0 ≤ V_D ≤ V_Dsat), J_N= q_nnE

Note that current I_D is the same everywhere, but J_N (current density) can vary from position to position.

I

-V

relationships (square-law theory)

Because the current flow within the conducting channel is almost exclusively in the y-direction,

dy n d qμ qμ n

J

J_{N Ny n y n} 







 E

dy d

y

  E 

since

I

-V

relationships (square-law theory)

To find I_D from J_N, we have to integrate the above eqn. within the area of the conducting channel.







 



 ^{( )}

0 )

( 0 0

0

) ( 0

y x

Ny y

x

Ny Z

Z x y

Ny D

c c

c J dxdz dz J dx Z J dx

I







 ^{( )}

0 )

( 0

y x

n y

x

n

c

c qμ ndx

dy Z d dy dx

n d qμ

Z  

Since ,

dy Q d

μ Z

I_{D   n N} 





 ^{( )}

0 ( , ) ( , )

) ) (

( ^{x y} _n

N n

c x y n x y dx

y Q

y q 



I

-V

relationships (square-law theory)

dy Q d

μ Z

I_{D n N} 





Separating variables and integrating,









 ^{n VD} _N

D Q d

L μ I Z

0 

With being position-independent in the long channel MOSFETs,

μn

When V_G exceeds V_T in the MOS capacitor, the equilibrium inversion layer charge is balanced by the charge added to the gate.

N semi

gate Q Q

Q    



Because , ^Q_gate ^C_o^V_G ^C_o^V_G ^V_T  ^Q_N  ^C_o^V_G ^V_T 

ox ox G

ox

o x

K A

C _C _ ⁰

where C_o is the oxide capacitance per unit area

T

G V

V 

...

I

-V

relationships (square-law theory)





 ^{n VD} _N

D Q d

L μ I Z

0 

 _{G T} 

o

N C V V

Q   

In the MOSFET, the bottom-side “plate”

potential (ϕ) varies from zero at the source to V_D at the drain.

Therefore the uniform potential drop (V_G) in the MOS capacitor should be replaced by the potential drop (V_G-ϕ) at an arbitrary point y in the MOSFET.

  



 _{o G T}

N y C V V

Q ( )

Substituting into and integrating,

  



 _{o G T}

N y C V V

Q ( )

  ^D

D

V T

G o

V n

T G

o n

D V V

L μ C d Z

V L V

μ C I Z

0 2

0 ( ) 2 



 



  













 



  

 ( ) 2

2 D D

T G

o n D

V V V L V

μ C I Z

T G

Dsat

D V V V

V  

 and 0

...

I

-V

relationships (square-law theory)

This equation can be applied only below pinch-off.



 



  

 ( ) 2

2 D D

T G

o n D

V V V L V

μ C I Z

T G

Dsat

D V V V

V  

 and 0

...

As pointed out in the qualitative discussion, I_D is approximately constant if V_D exceeds V_Dsat.

V Dsat V

D V

V

D I I

I D_ Dsat  D_ Dsat 

By simply setting



 



  

 ( ) 2

2 Dsat Dsat

T G

o n Dsat

V V V L V

μ C I Z

hence

It should be noted that pinch-off at the drain end of the channel implies Q_N(L)  0 when ϕ(L) = V_D  V_Dsat.

  



 _{o G T}

N y C V V

Q ( )

At the pinch-off,

  ⁰

)

(   _{o G}  _T  _Dsat 

N L C V V V

Q

I

-V

relationships (square-law theory)

  ⁰

)

(   _{o G}  _T  _Dsat 

N L C V V V

Q

Therefore and ^V_Dsat ^V_G ^V_T



 



  

 ( ) 2

2 Dsat Dsat

T G

o n Dsat

V V V L V

μ C

I Z ( )²

2 ^{G T}

o n

Dsat V V

L μ C

I  Z 

Neglecting ’s dependence on V_G, the saturation drain

current varies as the square of the gate voltage above turn-on

μn

 the so called “square-law” dependence.

)2

2 ( ^{G T}

o n

Dsat V V

L μ C

I  Z 



 



  

 ( ) 2

2 D D

T G

o n D

V V V L V

μ C

I Z ... 0 V^D V^Dsat and V^G V^T

T G

Dsat

D V V V

V  and 

...

I

-V

relationships (square-law theory)

n-channel ideal MOSFET

)2

2 ( ^{G T}

o n

Dsat V V

L μ C

I  Z 



 



  

 ( ) 2

2 D D

T G

o p D

V V V L V

μ C I Z

T G

Dsat

D V V V

V  

 and 0

...

T G

Dsat

D V V V

V  and 

...

p-channel ideal MOSFET

T G

Dsat V V

V  

T G

Dsat V V

V  

Experimental determination of threshold voltage and field-effect mobility



 



  

 ( ) 2

2 D D

T G

o n D

V V V L V

μ C I Z

D T G

o n

D V V V

L μ C

I  Z (  )

Very small V_D

Plotting I_D against V_G, a straight line should be obtained, and the V_T and can be determined by the intercept and slope. ^μ_n

L V μ C

slope Z ^{n o D}

Due to the effective mobility being a function of V_G

Due to subthreshold conduction

I

-V

relationships (bulk-charge theory)

  



 _{o G T}

N y C V V

Q ( )

The square-law theory contains a major flaw.

- In the square-law analysis, it is assumed that changes in

gate charge going down the channel were balanced solely by changes in Q_N ( and ). ^Q_gate ^{ Q}_semi ^{ Q}_N - Thus it is implicitly assumed that the depletion width at all

channel points remains fixed at W_T even under V_D ≠ 0 biasing.

p-Si

 Due to V_D ≠ 0 bias, the depletion width widens.

 “Bulk” charge (ionized accepters) must be included in any charge balance.

 ^{ }



 _{o G T}

N y C V V

Q ( ) ^Q_N^{(y)  C}_oBulk-charge theory ^V_G ^V_T ^^qN_A





I

-V

relationships (bulk-charge theory)

The I_D-V_DS relationships in the bulk-charge theory are as below:



























 



 

 



 



 









F D F

D F

W D

D T G

o n D

V V V

V V V L V

μ C I Z



 

4 1 3 1 2

3 4 ) 2

(

2 / 2 3

n-channel ideal MOSFET ... 0 ^V_D ^V_Dsat and ^V_G ^V_T

As in the square-law analysis, the post pinch-off portion of I_D-V_DS curve is approximately modeled by setting

V Dsat V

D V

V

D I I

I D_ Dsat  D_ Dsat 















 



 

 















 



 

 







F W F

W F

T G

W T

G Dsat

V V

V V V

V V

V   1 4

1 4 2

2 / 2 1

where

o T A

W C

W V  qN

negative term added

negative term added

I

-V

relationships (bulk-charge theory)

Because the negative

terms are added in the I_D- V_DS relationships of the bulk-charge theory, I_D and V_Dsat are reduced

compared to the square- law theory.

As N_A (or N_D)  0 and x_ox  0, the bulk-charge theory, mathematically reduces to the square-law theory.

x_ox = 0.1 μm T = 300 K

Charge-sheet and exact-charge theories

Both the square-law and bulk-charge theories suffer from two severe inherent limitations.

1. Q_N was assumed to be zero for V_G ≤ V_T.

 In an actual MOSFET, Q_N becomes small, not zero.

2. The I_D-V_DS relationships from square-law and bulk-charge theories does not self-saturate.

 It is necessary to artificially set .

 As a result, a residual drain current (which is called the subthreshold current) can flow between the source and drain.

V Dsat V

D V

V

D I I

I D_ Dsat  D_ Dsat 

The noted failings are removed in the charge-sheet and exact charge theories (Appendix D).

 The subthreshold current can be calculated.

Charge-sheet and exact-charge theories

Solid line: exact charge theory Dashed line: charge sheet theory

Calculated

subthreshold current (solid or dashed lines)

Announcements

• Next lecture: basic concept for non-ideal MOS p. 645 ~ 683

• Final EXAM: Dec. 15, 13:00 ~ 15:00, 별 232

Everything studied after the Mid-Term EXM

• Homework problem set (due to Dec. 8):

17. 2; 17.18a; 17.20(a-d)