물리 전자/김삼동 6-1-1

Potential Difference

Sum the energies from the Fermi level to vacuum

level:

The metal-SC work function difference, φMS, is

defined as;

• Work function difference

g' 'm i ox0 i s Fp

Ee + e + eV = e + e + - e + e

2

φ χ χ χ φ φ

gs

E - e

2φ

ox0eV

meφ

m

'eφ

ieχ eχ'eχ

seφ

gE2≈

Fpeφ

g' 'ox0 s m Fp

E V + = - - + +

2e

→ φ φ χ φ

g' 'ms m Fp

E - + +

2e

φ ≡ φ χ φ

- Metal gates

ms m s - φ ≡ φ φ

물리 전자/김삼동 6-1-2

Potential Difference

eφm’ is the energy difference between Ec (oxide)

and EF of the gate material.

→ EF of the n+ poly-Si = Ec of the n+ poly-Si .

→ Ec (oxide) – Ec = eχ’

- Poly-Si gates • n+ poly Si gate

F cE E=

g' 'ms m Fp

E - + +

2e

φ ≡ φ χ φ

g' 'ms Fp

E - + +

2e

φ = χ χ φ

gms Fp

E - +

2e

φ = φ

물리 전자/김삼동 6-1-3

Potential Difference

eφm’ is the energy difference between Ec (oxide)

and EF of the gate material.

→ EF of the p+ poly-Si = Ev of the n+ poly-Si .

→ Ec (oxide) – Ec = Eg + eχ’

• p+ poly- Si gate F vE E=

g' 'ms g Fp

E E + - + +

2e

φ = χ χ φ

gms Fp

E -

2e

φ = φ

물리 전자/김삼동 6-1-4

Potential Difference

The magnitude of the oxide charge seems, in

general, to be a strong function of the oxidizing

conditions such as oxidizing ambient and

temperature.

→ We assume that an equivalent trapped charge per

unit area, Qss’, is located in the oxide directly

adjacent to the oxide-semiconductor interface.

The presence of this charge also produces potential

difference between the gate and the SC.

• Oxide Charges - Fixed charge : Qss

: Broken or dangling bonds near the Si-SiO2 interface can create (+) fixed charges inside the insulators

'ss

ox

QV = C

∆

물리 전자/김삼동 6-1-5

Potential Difference

Work-function difference: φms

Ec

EV

EF

VG = 0

Ec

EV

EF

VG = φms < 0

Ec

EV

EF

VG = 0

VG = -Qss’/Cox

+

+ + + +

+ Ec

EV

EF

'ss

FB msox

QV 0C

= φ − <

Fixed charge : Qss’

Due to the work-function difference φms, band bends

downward at VG = 0. We need to apply VG = φms (<0)

such that there is no bend bending.

“Flat-band Voltage”

+

+ + + +

+

물리 전자/김삼동 6-1-6

Potential Difference

In NMOS, • Threshold Voltage, VT

The “threshold voltage” is defined as a gate voltage

producing the inversion condition when the surface

potential becomes In NMOS (p-type substrate)

In PMOS (n-type substrate)

Therefore, if the band was originally flat at VG = 0, we

need to apply the following gate voltage to achieve an

inversion point.

2s Fpφ = φ

2s Fnφ = φ

ox s ox FpV V + = V + 2 > 0G = φ φ

oxV > 0

s > 0φ

GV > 0

물리 전자/김삼동 6-1-7

Potential Difference

The charge neutrality is

The oxide voltage can be related to the charge on

the metal and to the oxide charge at this inversion

point.

- Oxide voltage at inversion point, VoxT

Consider charge neutrality in the MOS capacitor

At inversion point, xd → xdT (max) and the inversion

layer starts to form (not created yet)

'mTQ

'sD a dTQ (max) = eN x

dTx

'ssQ

' 'mT sD a dTQ Q = eN x=

''sD a dTmT

oxTox ox ox

Q eN xQV = = C C C

=

1/ 2

s FpdT

a

4 x

e N

ε φ ← =

s a FpoxT

ox

4 eNV =

C

ε φ

물리 전자/김삼동 6-1-8

Potential Difference

⇒ In summary - Correction for the VT is required by using the “flat-

band voltage”

In NMOS (p-type substrate);

In PMOS (n-type substrate);

's a Fp ss

TN Fp ms ox ox

4 eN QV + 2 + - C C

ε φ= φ φ

's d Fn ss

TP Fn ms ox ox

4 eN QV - 2 + - C C

ε φ= − φ φ

VG (+)

P

- - - - - - xdT

VFB

Q’sD: eXdTNa

2φFp

oxide

Metal

- - - - - - Vox

's a,d Fp,n ss

T Fp,n ms ox ox

4 eN QV 2 + - C C

ε φ= ± ± φ φ

Substrate doping: p-type (+), n-type (-)

물리 전자/김삼동 6-1-9

Potential Difference

Both electric field and electric flux are conserved in vacuum In general, electric field is not conserved at the interface between the vacuum & polarized material Electric flux is always conserved !

+ + + + + + + + + +

- - - - - - - - -

Ε

In vacuum

D = εrεo Ε

r oD ε (in SI unit)= ε E

• Electric Flux (Displacement)

+ + + + + + + + + +

- - - - - - - - -

- - - - - - -

+ + + + + +

Εin Dout = εrεo Ε (εr = 1)

Din = εrεo Ε (εr > 1) Εout

≠

oxide SC oxide SC + + + + + + +

ox sD = D 'ox ss sD + Q = D

물리 전자/김삼동 6-1-10

Potential Difference

• Electric field profile

GV =0

FmE

dx

cE

vE

FiEFsE

cE

vE

FiEFsE

FmEFBV

'ssQ

oxE

cE

vE

FiEFsE

'ssQ

FmEG TV >V

dTx

'ssIf Q 0≈

oxsD 0 D 0> → >

FmEFBV

cE

vE

FiEFsE

'ssIf Q 0=

oxsD 0 D 0= → =

'ssIf Q 0≠

oxsD 0 D 0= → <

'ox s ssD = D - Q

'ssIf Q 0≠

ox

's ssD 0 and Q D 0>

→ >

물리 전자/김삼동 6-1-11

C-V Characteristics

- Depletion • Ideal C-V characteristics

- Accumulation

' ' oxox ox r,ox o

ox

C = C = ( = )tε

ε ε ε

p-Si

-

Cox’

- +

Cox’

C’

VG

p-Si

+

- - - - - - - - - - - -

Ionized donor

- +

Cox’

C’

VG

Cox’

CD’ ' ' '

ox D

1 1 1 = + C C C

'D

d

C = x

sε←

ox

ox

ox

'' ox

'ox' dD s

CC = = C1 + t + xC

εε

ε

물리 전자/김삼동 6-1-12

C-V Characteristics

“Flat band capacitance”

: Capacitance in the flat-band condition which

occurs between the accumulation and

depletion conditions.

Note that flat-band capacitance is a function

of tox and Na.

- Inversion

Cox’

CdT’

Inversion layer

xdT p-Si

+

- - - - - - - - - - - - - - - - - - - - - - - -

+

- +

Cox

C’

VG

Cmin’

ox

ox

ox

'min

dTs

C = t + x

εε

ε

ox

ox

ox

'FB

s a

C = kTt + e eN

s

ε

ε ε ε

물리 전자/김삼동 6-1-13

C-V Characteristics

• C-V curves

'oxC

'C

'CFB

'minC

High Frequency

'oxC

Low Frequency

Simplifiedlow frequency model

Accumulation Depletion InversionFBV TV GV

dTx

물리 전자/김삼동 6-1-14

C-V Characteristics

• Frequency Effects

Charge change can occurs - QI: e-’s generates/recombine (thermal)

e- diffuses in & out inversion layer

(long-range diffusion) ⇒ Slow ! Effective !

- QD: depletion width change (xd ↑↓)

(short-range diffusion) ⇒ Fast ! Less effective !

Accumulation

Inversion

High Frequency> 1 MHz

Low Frequency< 100 Hz

0 GV

'C

G

dQC = dV

O S M

QD’

QG’

At LF

dQ’

dQ’

O S M

QD

QG At HF

dQ

dQ

Measurement follows 1/Cox +1/Cd

QI

QI

dxd

Not detected by measurement ~ 1/Cox

물리 전자/김삼동 6-1-15

C-V Characteristics

The flat-band voltage is affected by the fixed

charge.

The VFB shifts to more negative voltages for

a positive fixed charge. Therefore, VT is also

affected by the magnitude and polarity of

fixed charge.

⇒ The C-V characteristics can be used to

determine the Qss’ by measuring the amount

of shift in VT.

• Fixed charge effects

'ss

FBox

QV - Cms= φ

물리 전자/김삼동 6-1-16

C-V Characteristics

- Depletion

- Inversion

• Interface states

: Periodic nature of the SC is abruptly terminated at the

interface (Si-SiO2); many allowed interface states can

exist within the forbidden gap. In general, acceptor levels exist in the upper half and

donor states exist in the lower half. - Accumulation mode

Neutralacceptors

PositivedonorsNeutraldonors

Neutralacceptors

Neutraldonors

Neutralacceptors

Neutraldonors

Negativeacceptor

물리 전자/김삼동 6-1-17

C-V Characteristics

- High frequency C-V curve showing the effects

interface states

⇒ C-V curves become “smeared-out”

“Positive charge effects”

“Negative charge effects”