Chapter 16-2. MOS electrostatic: Quantitative analysis

Chapter 16-2. MOS electrostatic:

Quantitative analysis

In this class, we will

• Derive analytical expressions for the charge density, electric field and the electrostatic potential.

• Expression for the depletion layer width

• Describe delta depletion solution

• Derive gate voltage relationship

– Gate voltage required to obtain inversion (V_T)

Electrostatic variables in the ideal MOS

Let’s establish analytical relationship for the charge density (ρ), the electric field (E), and the electrostatic potential in an ideal MOS capacitor.

Because the metal is an equipotential region, charge

appearing near the M-O interface resides only a few Angstrom into the metal and may be modeled as a δ-function.

Because there is no charge in the oxide, the magnitude of the charge in the metal is simply equal to the sum of the charges inside the semiconductor.

Due to the absence in the oxide, the E-field is constant in the oxide.  linear function of position.

Solving for the electrostatic variables in the ideal MOS

Electrostatic potential,  (x)

Define a new term, (x) taken to be the potential inside the semiconductor at a given position x.

*The symbol  instead of V used in MOS work to avoid confusion with externally applied voltage (V).

)]

( (bulk)

[ )

( E_i E_i x

x  q1 



(surface)]

(bulk)

[ _{i i}

S E E

q 

 1



] (bulk)

[ _{i F}

F E E

q 

 1



Electrostatic potential at x

Surface potential (x = 0)

| _F |, reference voltage related to doping concentration.

Electrostatic parameters

_S is positive if the

band bends downward

(surface)]

(bulk)

[ _{i i}

S E E

q 

 1



] (bulk)

[ _{i F}

F E E

q 

 1



)]

( (bulk)

[ )

( E_i E_i x

x  q1 

 q_S

] (bulk)

[ _{i F}

F E E

q 

 1



 

A D

D kT

E E

ie N N N

n

n_bulk  ^{F i(bulk) /}  ... if 

 

D A

A /kT

E (bulk) E

i

bulk n e N ... if N N

p  ^{i  F}  

Combining the equations for each type,













 



 



 







type -

n ...

type -

p ...

F

D i A

n N q

kT

n N q

kT

ln ln



Electrostatic parameters

Therefore,

Under flat band conditions,

_S = 0 and  (x) = constant

E_i(surface) – E_F = E_F – E_i(bulk)

or E_i(surface) – E_i(bulk) = 2[E_F – E_i(bulk)]

q_S

At the depletion-inversion transition point (V_G = V_T),



  2

Surface potential, 

(surface)]

(bulk)

[ _{i i}

S E E

q 

 1



Electrostatic parameters

For a p-type semiconductor (_F > 0),

For a n-type semiconductor (_F < 0),

Exercise 16.2

Consider the following _F and _S parameters. Indicate

whether the semiconductor is p-type or n-type, specify the biasing condition, and draw the energy band diagram at the biasing condition. _F = 12 kT/q; _S = 12 kT/q

_F = +12 kT/q means that E_i – E_F in the semiconductor is 12 kT (a

positive value); So, p-type.

E_C

E_i E_F E_V

12kT

_S=12 kT/q means E_i (bulk) – E_i(surface) = 12 kT; i.e. the

band bends downward near the surface.  depletion of holes

(surface)]

(bulk)

[ _{i i}

S E E

q 

 1



] (bulk)

[ _{i F}

F E E

q 

 1



_F =  9 kT/q; _S = 18 kT/q

E_C

E_F E_i

-9kT

Exercise 16.2

] (bulk)

[ _{i F}

F E E

q 

 1



(surface)]

(bulk)

[ _{i i}

S E E

q 

 1



here _F =  9 kT/q means [E_i(bulk) – E_F] =  9 kT; i.e., E_i is below E_F. Thus the semiconductor is n-type.

_S=  18 kT/q means that E_i (bulk) – E_i(surface) = –18 kT; So band bends upwards near the surface.

The surface is “inverted” since the surface has the same number of holes as the bulk has electrons.

Delta-depletion approximation

Consider p-type MOS in an accumulation condition.

Charge on metal = Q_M

Charge on semiconductor =  (Q_M )

Accumulation of holes

x

M O S

V_G < 0

p-Si

Assume that the free carrier

concentration at O-S interface is a

-function.

The accumulation charges are mobile holes, and appear close to the surface and fall-off rapidly as x increases.

E

E = 0 for all x > 0 x

_s = 0 for all accumulation biases

Consider p-type MOS in a depletion condition (applying a positive V_G such that _s < 2_F).

Using the depletion approximation, the charge density in the depletion region is

) 0

( )

(p n N N qN x W

q   _D  _A   _A  

 ...



Charges in Si are immobile ions - results in depletion layer.

Then the Poisson’s equation reduces to

) 0

(

0 0

W K x

qN K

dx d

S A S



 



 ...



 E 

Integrating with the boundary condition (E = 0 at x = W) yields





A

(x) dx

ε K d qN

0 0

E E





ε K x qN

S

A 



0

)

E(  0  x W

Delta-depletion approximation

From the definition of E-field,

dx d

 E 





ε K

qN dx

d

S

A 









0

 E

W x 

 0 

Separating variables and integrating from an arbitrary point x to W ( = 0 at x = W),

 





S A

(x) W x dx

ε K d qN

0 0





 

2 0

)

( W x

ε K x qN

S

A 



Quadratic function

Delta-depletion approximation

Applying  = _s at x = 0,

 

2 0

)

( W x

ε K x qN

S

A 

  _{ 0  x W}

2

2 0 W

ε K qN

S A s 



and therefore ^s

A S

qN ε W  ²K ⁰ 

Note that for an n-type MOS, N in the preceding equations The depletion width at the deletion-inversion transition point (_s = 2_F) is the maximum attainable equilibrium depletion width (W_T)

) 2 2 ₀ (

F A

S

T qN

ε

W ^ K 

Delta-depletion approximation













type -

n ...

type -

p ...

F

) / ln(

) / ln(

i D

i A

n q N

kT

n q N

kT



) 2 2 ₀ (

F A

S

T qN

ε

W  K 

Maximum equilibrium depletion width

Charges in Si are immobile ions - results in depletion layer.

Depletion of holes

Q_M w

|q N_A A W| = |Q_M| () (+)

When surface potential is _s ,

E

x

M O S

V_G > 0

p-Si For a p-type MOS in a depletion

condition





ε K

x)  qN^A  E(

s A S

qN ε W  ²K ⁰ 

ε W K

qN

S A S

0

E 

0

) (

ε K

qN dx

x d

S

 A

E 

Delta-depletion approximation

M O S

V_G >> 0

p-Si For a p-type MOS in an

inversion condition

Depletion of holes

w_T Q_M

Inversion electrons:

-function-like

Once inversion charges

appear, they remain close to the surface since they are mobile.

The -function-like charge added in inversion precisely balances the charge added to the metal.

 The charge in the depletion region and E-field (x > 0) remain

 

E

T S

A

S W

ε K

qN

0

E 

)

(x qN

dE  _A

Delta-depletion approximation

Gate voltage relationship (V

versus 

)

The applied V_G is dropped partly across the oxide and partly across the semiconductor.

V_G= _ox +_semi

Consider a p-type MOS.

Because  = 0 in the bulk,

_semi = (x = 0)  (bulk) = _S

In the ideal insulator without any carriers and charge centers,

 0 dx dE_ox

and E   constant

dx d _ox

ox



Integrating across the oxide of which thickness is x₀,

ox(x 0) 0 0











^



 

V_G > 0

M O S

p-Si

_Semi

_ox

-x₀ 0 x

Gate voltage relationship (V

versus 

)

ox ox

 x

E

 

According to the well-known boundary condition on the E-field normal to an interface between two dissimilar materials,

S - interface O

S -

O Q

D

D_semi  _ox)  (

where D = εE is the dielectric displacement and Q_O-S is the charge density (charge/unit area) located at the interface.

Since Q_O-S = 0 in the ideal MOS,



semi x

ox

D

D

S S ox

O

K

K E  E

where K_O and K_S are dielectric constants of the oxide and semiconductor, respectively (K_O = 3.9 for SiO₂ and K_S = 11.8 for Si  ).

E

 3 E

Gate voltage relationship (V

versus 

)

Combining all equations,

S O

S

ox x

K

K ₀E



 V_G= _ox +_semi _semi = _S

S O

S S

G x

K

V   K ₀E

s A S

qN ε W  ²K ⁰ 





ε K x qN

S

A 



0

) E(

Employing the delta-depletion solution,

A combination of the two equations gives at x = 0. _s

S A

S K ε

qN 

0

 2 E

Therefore _s

S A O

S S

G K ε

x qN K

V  K 

0 0

 2

  0 S  2F

Gate voltage relationship (V

versus 

)

The exact solution can be found in Appendix B.

_s = 2_F for all inversion biases

_s = 0 for all

accumulation biases

s S

A O

S S

G K ε

x qN K

V  K 

0 0

 2



F

S 

 2

0  



The V_G-_s relationship obtain using the delta-depletion approximation can be used only for the depletion bias

) 2 2 (

2

0

0 F

S A O

S F

T K ε

x qN K

V    K 

V

- 

relationship: Alternative method

V_G= _Semi + _ox

_ox= Q_M/C_ox = Q_s/C_ox where C_ox is oxide capacitance and Q_s is the depletion layer charge in semiconductor

Q_s = q A N_AW C_ox = _ox A / x_ox

Consider a p-type MOS

 

s Si

ox A ox

s Si Si

ox A ox

s Si G

Si ox A

ox Si ox

ox ox A

2 /

 



 



 



 









 

 





x qN qN W

x V

qN W x x

A

W N

A q

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