MOS FUNDAMENTALS

Chapter 16.

MOS FUNDAMENTALS

Sung June Kimg kimsj@snu.ac.kr http://helios.snu.ac.krp

Contents

‰ MOS Structure

‰ Ideal Structure Assumption

‰ Eff t f A li d Bi

‰ Effect of an Applied Bias

‰ Capacitance - Voltage Characteristics

2

MOS Structure

Metal - Oxide (SiO

2

) - Semiconductor (Si)

– The most common field plate (gate) materials are heavily doped polycrystalline silicon.

– The silicon-side terminal is called the back or substrate contact – The silicon-side terminal is called the back or substrate contact.

– The more general designation: metal-insulator-semiconductor (MIS)

Individual energy band diagrams for the metal insulator and Individual energy band diagrams for the metal, insulator, and

semiconductor components.

χ

( E - E_F )

χ χ Φ_M

χ _i

E_c

E_c

E E_i

( E_c E_F )_∞

E_F

E_F E_v

Semiconductor with b d b di

Semiconductor

Metal Insulator

E_v

band bending

Ideal Structure Assumption Ideal Structure Assumption

(1) The metallic gate is sufficiently thick so that it can be considered (1) The metallic gate is sufficiently thick so that it can be considered

an equipotential region.

( ) h id i f i l

(2) The oxide is a perfect insulator.

(3) No charge centers located in the oxide or at the interface.

(4) Uniformly doped.

(5) The semiconductor is sufficiently thick so that a field-free

region(“bulk”) is encountered before reaching the back contact.

region( bulk ) is encountered before reaching the back contact.

(6) An ohmic contact between the semiconductor and the metal on th b k id

the back side.

(7) One-dimensional structure.

(8) No work function difference between metal and semiconductor.

χ_i

Φ Φ′ χ

′ χ

Μ Φ

Μ

′

Ε _F Ε _F

Ε _c

F Ε _v

Energy band diagram of an ideal MOS structure with no bias.

Effect Of An Applied Bias Effect Of An Applied Bias

- Qualitative description

• General observations

–

– With V

G≠ 0, semiconductor Fermi energy is unaffected by the bias and

G F

F

metal E semiconduc tor qV

E ( ) − ( ) = −

G

remains invariant as a function of position because of the assumed zero current flow.

– Since the barrier heights are fixed quantities, the movement of the metal Fermi level leads to a band bending.

• In the metal, no bend-bending.

• In the oxide and semiconductor In the oxide and semiconductor, an upward slope when V

G

>0

d d l h V 0

a downward slope when V

G

< 0.

• With no oxide charges, the Poisson's equation yields a constant slope in the oxide.

• Band bending in the semiconductor is somewhat g

more complex.

• Specific biasing regions g g

– Accumulation ( V

G < 0 )

• VG < 0 raises EF(metal) and the hole concentration inside the semiconductor,

i h th

( )

[

^/

]

exp E E kT

n

p = _i − _F − _i increases as one approaches the oxide-semiconductor interface.

VG 0 l i h h

• VG < 0 places negative charges on the gate.

To maintain a balance of charge, positively charged holes must be drawn toward the Si charged holes must be drawn toward the Si- SiO2 interface.

– Depletion ( 0 < V

G < V

T )

p (

G T )

• VG > 0 slightly lowers EF(metal) and the hole concentration is decreased (depleted) hole concentration is decreased (depleted) in the vicinity of the Si-SiO2 interface.

• VG > 0 places positive charges on the gate, which in turn repels holes from the interface and exposes the negatively

charged acceptor sites.

– Onset of Inversion (V

G = V

T )

• As VG is increased positively, the bands at the Si surface will bend down more and the the Si surface will bend down more and the electron concentration at the surface (ns) will increase from less than ni when E i

(surface) > EF, to ni when Ei (surface) = EF, to greater than ni when Ei (surface) < EF.

Exposed

A t

V_G = V_{T •} At VG=VT,

F i

i

i bulk E surface E bulk E E ( )− ( )= 2[ ( )− ]

Acceptors

+Q

i F

i s

F i

i i

kT

surface E

n E n

E bulk

E surface

E bulk

E

⎥⎦⎤

⎢⎣⎡ −

= ( )

exp

] )

( [ 2 ) (

) (

-Q

Electrons

A bulk

F i

i p N

kT

E bulk

n E

kT

=

⎥⎦ =

⎢⎣ ⎤

⎡ −

=

⎦

⎣

) exp (

kT ⎦

⎣

−

Inversion ( VG > VT ) Inversion ( VG VT )

•For VG > VT , ns > NA.

•The surface region :p-typeThe surface region :p type n-typen type

•In inversion, the depletion width changes little since

little since

)

( ⎤

⎡ E

_F

− E

_i

( surface )

exp ⎥⎦ ⎤

⎢⎣ ⎡

∝ kT

surface E

n

_s

E

^{F i}

No Bias (V (

_GG

= 0) )

E_C E_F E_F E_V

Accumulation (V < 0) Accumulation (V

_G

< 0)

E_C E_F E_V

Depletion ( 0 < V p (

_GG

< V

_TT

) )

E_C EE_F E_V

Inversion ( V (

_GG

> V

_TT

) )

E_C E_F E_V

Effect Of An Applied Bias pp - Quantitative formulation

• Preparatory considerations ( )x = 1

[

E ⁽bulk⁾ E ^{( )}x

]

φ( )

[

E ⁽bulk⁾ E ^{( )}x

]

x = q _i − _i

φ

potential.

surface

the

s:

φ

( ) ( )

[ ]

1 E bulk E surface

q ^{i i}

s ^{= −}

φ

( )

[ ]

_F 1 E_i bulk E_F

q −

=

φ q

(

^/

)

ln

N_A n_i q

=kT q

F i d

For an p-type semiconductor,

– Accumulation :

– Flat band :

s

s

φ φ

0 0

=

<

Flat band :

– Depletion :

O t f i i

F s

s

φ φ

φ φ

φ

2

2 0 < <

– Onset of inversion :

– Inversion :

_{s F}

F s

φ φ

φ φ

2 2

>

=

• Delta-depletion solution

Delta-depletion assumption :

– The functional form of the accumulation charge & the inversion charge : δ - function.

– Because the depletion width increases only slightly once the p y g y semiconductor inverts, it is assumed the δ - function of charge added in inversion precisely balances the charge added to the gate.

– The actual depletion charge is replaced with a squared-off distribution

– Accumulation :

• Because of the assumed δ - function, the electric field and potential are zero for all x > 0. p

– Depletion :

, 0

For

N dE

W x ≤

≤

2 / 1

0

⎤

⎡

=

= φ_S x

φ at ,

with

0 0

N

K qN K

dx dE

S A S

−

≅

= ε ε

ρ ^1/2

2 0

⎥⎦

⎢ ⎤

⎣

= ⎡ _S

A S

qN

W K ε φ

0

) (

) (

N

x K W

x qN E

S

A −

= ε

2 /

2 ⎤1

⎡ K ε

The maximum depletion width,

2 0

) 2 (

)

( W x

K x qN

S

A −

= ε

φ 2 ⁰

( )

2

⎥⎦

⎢ ⎤

⎣

= ⎡ _F

A S

T qN

W K ε φ

1

T = 3 0 0 K

0 . 1

W T [ μm]W

0 . 0 1

1 0^{1 5} 1 0^{1 6} 1 0^{1 7} 1 0^{1 8}

0 . 0 1

N _A o r N _D [ c m ^{- 3}]

– Inversion :

• The solution is established by merely adding The solution is established by merely adding a δ - function of surface charge to the

solution existing at the end of depletion.

• The depletion charge the x > 0 electric field

• The depletion charge, the x > 0 electric field, and the x > 0 potential remain

fixed at their values.

F

s φ

φ = 2

Exact solution for the charge density and potential Exact solution for the charge density and potential

assuming

(a) Accumulation (f = 6kT/q)

andT K q

F =12kT/ = 300

φ

(a) Accumulation (f

_s

= -6kT/q)

(b) Middle of depletion (f = f = 12kT/q) (b) Middle of depletion (f

s

= f

F

= 12kT/q)

(c) Onset of inversion (f

s

= 2f

F

= 24kT/q)

(d) Deep into inversion

•

Gate voltage relationship (delta-depletion solution)

Because ε is constant in an ideal oxide with no charges,

OX Semi

VG = Δφ + Δφ

OX g

Since there is no charges at the interface, (in the depletion region)

ox

ox ε

φ = x_o Δ

g , ( p g )

0 x

Semi ox

K D

D = =

s o s

ox E

K E = K

(

0 2

)

2 x

x

0

F S

S s

A o

o s S

s o o s S

G K

qN K

E K K

V K φ φ φ

φ ε

φ + = + ≤ ≤

=

F

s

φ

φ = 2 At threshold,

4

2 K

^s

x

^o

qN

^A

V 2 φ + φ

0

F s

A o

o s

F

T

K K

V φ

φ + ε

=

φ

s

is a rather rapidly varying

s

function of V

G

when the device is in depletion. However, p ,

when it is accumulated or

inverted it takes a large change inverted, it takes a large change in V

G

to

produce a small change in φ produce a small change in φ

s

.

; delta-depletion solution, exact solution.