Capacitance - Voltage Characteristics p g - High- and low

Capacitance - Voltage Characteristics p g

High- and low- frequency C-V characteristics.

• CV characteristic is of considerable practical importance. p

M O S

A

V_G : Slowly changed

∆ V_G : AC signal ( high

+V_G– +∆ V_G– G g ( g

frequency/low frequency)

• Qualitative theory

Q

Accumulation V^G

C Q

∆

≡ ∆

– Accumulation

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

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

O.

p p p

O

Μ O S

G o

o

A K

C acc

C

0

) (

) (

ε

= oxidecapacitance

Μ O S

- Q

Q

∆Q

xo

G o 0

=

Q -∆Q

CC_o

– Depletion

• Withdrawal of majority carriers

• The charge state can be changed very rapidly.

• The depletion width quasi-statically fluctuates

about its dc value 　 two parallel plate capacitors (Co and C

s ; oxide and semiconductor capacitance) in series.

↑ ↑ ↓

• DC bias ↑ W ↑ C(depl)↓

Μ O S

( )

O O S

O

W K

C C

C

C depl C

C =

= +

+Q +∆Q

W

xO S S O

O

K

W C K

C + +

- Q - 1

∆Q

C_o C_S

– Inversion

• W = W

• The charge fluctuation depends on the frequency T

of the 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.

+∆Q

∆Q Q

W_T C_LF

(

inv

)

= C_O

Μ O S

-∆Q

– High frequency :

h l i l l i h i bi i

• The relatively sluggish generation-recombination process can’t supply or eliminate minority carriers

i h i l h b f

in response to the ac signal. The number of

minority carriers in the inversion layer remains

fi d d h d l i id h fl b h

fixed and the depletion width fluctuates about the WT dc value.　 Two parallel-plate capacitors in

i series.

• Since W

T=constant, C

HF(inv) = C(depl)

minimum =

T HF minimum

constant

Μ O S

+∆ ( )

x

T o o s

o s o HF

K W K C C

C C inv C

C

+ + =

=

W_T 1

+∆Q

xo

Ks

-∆Q

– For medium frequencyFor medium frequency

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

created/annihilated.

• C

HF(inv) < C

MF(inv) < C

LF(inv)

Μ O S

W +∆Q

W_T

-∆Q -∆Q’

• Delta-Depletion Analysis (skip)

• Delta-Depletion Analysis (skip)

– Depletion bias

1 / 2

2 _S 0 S

W _{= ⎢}^⎡ K ε φ ^⎤_⎥ φ_S = ^qNA W ²

V V K

W K^s _o ^G ⎥⎤

⎢⎡

− +

= x 1 1

S

qNA φ

⎢ ⎥

⎣ ⎦

2

K N

2 0 S

S

K W

φ ε

V h

δ

K N q

V K

o s o

o

⎥⎦

⎢⎣

x

0

x 2

s A

G S o S

o s

K qN

V φ K K φ

= + ε

V where _δ

ε

C

K N q

A o

o

o

≡ s

2 ^{2 2 sA0 2 os xo s A0 G 0}

K

qN qN

W W V

Kε ⁺ K Kε ^{− =}

2

V C C

G o

+

=

1

2

2 0

x x

s s s

o o G

o o A

K K K

W V

K K qN

⎛ ⎞ ε

= − ± ⎜ ⎟ +

⎝ ⎠

⎡ ⎤

Vδ

+

1 _{2 0}

x 1 1

x

s s o

o G

o A s o

K K K

W V

K qN K

⎡ ε ⎤

= ⎢+ + − ⎥

⎢ ⎥

⎣ ⎦

the delta-depletion theory

(x =0 01µm N_A=10¹⁷/cm³ T=300K ) (x_o=0.01µm, N_A=10 /cm ,T 300K )

1.0 Low frequency

0.8

C/C 0.6

C/C_o

0 2 0.4

High frequency

0 0.5 1.0 1.5 2.0 -0.5

-1.0

0.2

0 0.5 1.0 1.5 2.0 0.5

1.0

• Exact Calculation Exact Calculation

– Doping dependence

With increased doping the high-frequency With increased doping, the high-frequency

inversion capacitance increases significantly and the depletion bias region widens substantially

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 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 m st be emplo ed

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

= ∆ = = = R

∆

∆

∆ ∆ / ∆

V V V t R

∆ ∆ ∆ / ∆

(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

• Deep depletion

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

the inversion portion of the high freq. characteristic.

In accumulation or depletion only majority carriers are – In accumulation or depletion, only majority carriers are

involved 　Charge configuration rapidly reacts to the changing gate bias

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 　 _{C < C}

HF(inv) – Ramp rateRamp rate ↑↑ CC ↓↓

– The limiting case occurs when the semiconductor is

ll d id f i i i ll d d l d

totally devoid of minority carriers - totally deep depleted.

C CC

V _G

=

+

0

1 Same form as simple depletion case + V

1

δ

p p

Non-equilibrium charge configuration inside an p-type MOS-C Non equilibrium charge configuration inside an p type MOS C

+∆ Q

Q

W_T W>W_T

Q

W_T W

Q

- Q

-∆Q

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

Missing electrons

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