MOS FUNDAMENTALS

MOS FUNDAMENTALS

Sung June Kim

kimsj@snu.ac.kr

http://helios.snu.ac.kr

Chapter 16.

CONTENTS

• MOS Structure

• Ideal Structure Assumption

• Effect of an Applied Bias

• Capacitance - Voltage Characteristics

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

Individual energy band diagrams for the metal, insulator, and semiconductor components.

Semiconductor with band bending Semiconductor

E_c

E_F E_i

E_v

( E_c - E_F )_¥

c c

F_M

Metal Insulator

c _i

E_c

E_F

E_v

Ideal Structure Assumption

(1) The metallic gate is sufficiently thick so that it can be considered an equipotential region.

(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.

(6) An ohmic contact between the semiconductor and the metal on the back side.

(7) One-dimensional structure.

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

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

c_i

F

M F¢

M

c

¢

c

E _F E _F

E _v E _c

Effect Of An Applied Bias - Qualitative description

• General observations

–

– With V

G¹ 0, semiconductor Fermi energy is unaffected by the bias and remains invariant as a function of position because of the assumed zero current flow.

G F

F

metal E semiconduc tor qV

E ( ) - ( ) = -

– 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, an upward slope when V

G

>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 more

complex.

• Specific biasing regions

– Accumulation ( V

G < 0 )

·

VG < 0 raises EF(metal) and the hole concentration inside the semiconductor, increases as one approaches the

oxide-semiconductor interface.

·

VG < 0 places negative charges on the gate.

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

( )

[ / ]

exp E E kT

n

p =

_i

-

_F

-

_i

– Depletion ( 0 < V

G < V

T )

·

VG > 0 slightly lowers EF(metal) and the 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 )

Exposed Acceptors

V_G = V_T

+Q

-Q

Electrons

·

As VG is increased positively, the bands at 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.

·

At VG=VT,

A bulk

F i

i

i F i

s

F i

i i

N kT p

E bulk

n E

kT

surface E

n E n

E bulk

E surface

E bulk

E

= ú =

û ù êë

é -

=

úû ù êë

é -

=

-

= -

) exp (

) exp (

] )

( [ 2 ) (

) (

·For VG > VT , ns > NA.

·The surface region :p-type Ù n-type

·In inversion, the depletion width changes little since

- Inversion ( VG > VT )

kT

(surface) E

exp E

n

_s ^{F i}

ú

û ù ê ë

é -

µ

No Bias (V G = 0)

E_C E_F E_V

E_C

E_F E_V

Accumulation (V G < 0)

Depletion ( 0 < V G < V T )

E_C E_F E_V

Inversion ( V G > V T )

E_C E_F E_V

Effect Of An Applied Bias - Quantitative formulation

• Preparatory considerations

( ) [

^{E (bulk) E ( )x}

]

x = q1 _i - _i

f

potential.

surface

the

s

:

f

( ) ( )

[ ]

1 E bulk E surface

q ^{i i}

s

^{= -}

f

( )

[ ]

_F 1 E_i bulk E_F

q -

= f

( / ) ln

N_A n_i q

= kT

For an p-type semiconductor, – Accumulation : – Flat band : – Depletion : – Onset of inversion :

– Inversion :

_{s F}

F s

F s

s s

f f

f f

f f

f f

2 2

2 0

0 0

>

=

<

<

=

<

• Delta-depletion solution

Delta-depletion assumption :

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

– Because the depletion width increases only slightly once the

semiconductor inverts, it is assumed the d - 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 d - function, the electric field and potential are zero for all x > 0.

– Depletion :

2 0

0

0 0

) 2 (

) (

) (

) (

, 0

For

x K W

x qN

x K W

x qN E

K qN K

dx dE

W x

S A S

A

S A S

-

=

-

=

-

@

=

£

£

f e

e

e e

r

^1/2

2

0

0 ú û

ù ê ë

= é

=

=

S A

S S

qN W K

x e f

f

f at ,

with

( )

2 / 1

0

2

2

ú û ù ê ë

= é

_F

A S

T

qN

W K e f

The maximum depletion width,

10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 0.01

0.1 1

T = 300K

N_A or N_D [cm^-3]

W

T

[ m m ]

– Inversion :

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

solution existing at the end of depletion.

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

fixed at their values.

F

s

f

f = 2

Exact solution for the charge density and potential assuming

(a) Accumulation (f

_s

= -6kT/q)

and T K

q

F

= 12 kT / = 300

f

(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 e

OX

is constant in an ideal oxide with no charges,

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

OX Semi

V

G

= D f + D f

ox

ox

e

f = x

_o

D

0 x

s o s ox

Semi ox

K E E K

D D

=

= =

( 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 f f f

f e

f + = + £ £

=

F

s

f

f = 2

voltage) (threshold

F

s A o

o F s

T

K

qN K

V K f

f e

0

2 x 4

+

=

At threshold,

; --- delta-depletion solution, exact solution.

f

s

is a rather rapidly varying function of V

G

when the device is in depletion. However,

when it is accumulated or

inverted, it takes a large change in V

G

to

produce a small change in f

s

.

Capacitance - Voltage Characteristics

High- and low- frequency C-V characteristics.

• CV characteristic is of considerable practical importance.

M O S

A +V_G– +D V_G–

V

_G

: Slowly changed

D V

_G

: AC signal ( high

frequency/low frequency)

• Qualitative theory

– Accumulation

• The state of the system can be changed very rapidly. The majority carrier can equilibrate with a time constant on the order of 10^-10 to 10^-13 sec.

• The small ac signal merely adds or subtracts a charge close to the edges of an insulator.Ù parallel-plate capacitor, C

O.

o 0

x

e capacitanc oxide

G o

o

A K

C acc C

= e

= ( )

) (

M O S

- Q - DQ

Q DQ

C_o

V

G

C Q

D

º D

– Depletion

• Withdrawal of majority carriers

• The charge state can be changed very rapidly.

• The depletion width quasi-statically fluctuates about its dc value Ù two parallel plate capacitors (C

o

and C

s

; oxide and semiconductor capacitance) in series.

• DC bias ­ Ù W ­ Ù C(depl)¯

( )

x

O S

O O S

O

S O

K

W K

C C

C

C depl C

C

+ + =

=

1

M O S

- Q - DQ +Q

+D Q

W

C_o C_S

– Inversion

• W = W

• The charge fluctuation depends on the frequency of the

T

AC signal

– Low frequency :

• Minority carriers can be generated or annihilated in

response to the ac signal. Just as in accumulation, charge is added or subtracted close to the edges of insulator.

M O S

-DQ +D

Q

W_T

( )

_O

LF

inv C

C =

– High frequency :

• The relatively sluggish generation-recombination process can’t supply or eliminate minority carriers in response to the ac signal. The number of minority carriers in the

inversion layer remains fixed and the depletion width fluctuates about the W

T

dc value.Ù Two parallel-plate capacitors in series.

• Since W

T

=constant, C

HF

(inv) = C(depl)

minimum

= constant

( )

x

o s

T o o s

o s HF o

K W K C C

C C inv C

C

+ + =

=

1

M O S

W_T

-DQ +D

Q

– For medium frequency

• A portion of the inversion layer can be created/annihilated.

• C

HF

(inv) < C

MF

(inv) < C

LF

(inv)

M O S

W_T

-DQ +DQ

-DQ’

• Delta-Depletion Analysis (skip)

– Depletion bias

V where

d d

d

e

V V C C

K N K q

V V K

W K

G o

A o

o

o s G o

o s

+

=

º

ú û ù ê ë

é + -

=

1

2

1 1

2

x x

1/ 2

2 _S 0 S A

W K

qN e f

é ù

= ê ú

ë û

0

x 2

s A

G S o S

o s

K qN

V f K K f

= + e

2

2 0 A S

S

qN W f K

= e

2

0 0

x 0

2

s

A A

o G

s o s

qN K qN

W W V

K e ⁺ K K e ^{- =}

2

2 0

x x

s s s

o o G

o o A

K K K

W V

K K qN

æ ö e

= - ± ç ÷ +

è ø

2 0

x 1 1

x

s s o

o G

o A s o

K K K

W V

K qN K

é e ù

= ê+ + - ú

ê ú

ë û

the delta-depletion theory

(x

_o

=0.01mm, N

_A

=10

¹⁷

/cm

³

,T=300K )

0 0.5 1.0 1.5 2.0 -0.5

-1.0

0.2 0.4 0.6 0.8

1.0 Low frequency

High frequency

C/C_o

• Exact Calculation

– Doping dependence

With increased doping, the high-frequency inversion

capacitance increases significantly and the depletion bias

region widens substantially .

• Low frequency characteristics

– Given modern-day MOS-Cs with long carrier lifetimes and low carrier generation rates, even freq. as low as several Hz will yield high-freq. C-V.

– If a low-freq. characteristic is required, the quasi-static technique must be employed.

– The quasi-static displacement current flowing through the device is proportional to the low-freq. capacitance.

C Q

V

I t V

I

V t

I

= D = = = R

D

D

D D / D

(I : displacement current, R : voltage ramp rate)

• High frequency characteristics

– Normal measurement frequency ~ 1MHz.

– In the ramped-measurement, a deep-depletion phenomenon may appear.

• Deep depletion

Note that at even the slowest ramp rates one does not properly plot out the inversion portion of the high-freq. characteristic.

– In accumulation or depletion, only majority carriers are involved ÙCharge configuration rapidly reacts to the changing gate bias.

– In inversion region, minority carriers should be generated

– The generation process is sluggish and has difficulty supplying the minority carriers needed for the structure to equilibrate.

– Deep depletion : the nonequilibrium condition where there is a deficit of minority carriers and a depletion width in excess of W

T Ù C < C

HF(inv)

– Ramp rate ­ Ù C ¯

– The limiting case occurs when the semiconductor is totally devoid of minority carriers - totally deep depleted.

C C

V V

G

=

+

0

1

d

Same form as simple depletion case

Nonequilibrium charge configuration inside an p-type MOS-C

(a) under deep depletion (b) under total deep depletion

Missing electrons W_T

W>W_T

- Q Q

W_T W

+D Q

-DQ