ho #2: elen 384 - mos capacitorspage 1s. saha basic equations for device operation transport...

HO #2: ELEN 384 - MOS Capacitors S. Saha

Basic Equations for Device Operation

Transport equations: The total e- and hole current densities are the sum of the drift due to electric field, and diffusion due to concentration gradient components:

where Jn and Jp are e- and hole current densities, respectively

n and p are e- and hole concentrations, respectively n and p are e- and hole mobilities, respectively.

The total current density is given by: J = Jn + Jp

)(dx

dnDnqJ nnn (1)

(2))(dx

dpDpqJ ppp


Continuity Equations

Continuity equations: based on the conservation of mobile charge, e- and hole continuity equations are given by:

where Gn = e- generation rate

Rn = e- recombination rate

Gp = hole generation rate

Rp = hole recombination rate

pp

p

nnn

GRdx

dJ

qdt

dp

GRdx

dJ

qdt

dn

1

1(3)

(4)


Poisson’s Equation

Poisson’s equation:

where = electric field (x) = space charge density (charge/cm3)

K = dielectric constant

o = permittivity of free space = 8.854x10-14 F/cm

oK

x

dx

d

)(

:form General (5)

oK

x

dx

d

dx

d

dx

d

)(2

2

:as Equation sPoisson' writecan weTherefore,

potential,tic electrosta if Again,

(6)


Poisson’s Equation

Another form of Poisson’s Equation (Gauss’s Law):

where QS = integrated charge density

In silicon, consists of:

mobile charges: e- and holes with concentrations n and p, respectively.

fixed charges: ionized acceptors and donors with concentrations Na

and Nd+, respectively.

)()()()()(

2

2

xNxNxnxpK

q

K

x

dx

dad

oo

:as Equation sPoisson' writecan weTherefore,

(8)

0

)(

K

Q

K

dxx S

o (7)


Carrier Concentration - Boltzmann’s Relations

potential intrinsic level;energy intrinsic

potential Fermi level; Fermi

ionconcentrat carrierintrinsic

where

:by given is ionconcentrat the tor,semiconduc type-p a In

:by given is ionconcentrat the tor,semiconduc type-n an In

q

EE

q

EE

n

enenp

hole

enenn

e-

fii

fff

i

kT

q

ikT

EE

i

kT

q

ikT

EE

i

iffi

fiif

)(

)(

(9)

(10)


Carrier Concentration

type-p for

type-n for

midgap the from potential Fermi of separation where

: writecan we(10) and (9) From

ionconcentrat acceptor

ionconcentrat donor

where

:torsemiconduc type-p a In

:torsemiconduc type-n an In

ab

db

B

i

bifB

a

d

a

ipa

d

ind

NN

NN

n

N

q

kT

N

N

N

nnNp

N

npNn

ln

;

;

2

2

(11)

(12)

(13)


MOS Capacitors - 1. Band Diagrams

Note:• three materials in contact, EF

= constant at equilibrium.

Metal(Al)

Oxide(SiO2)

Semiconductor(p-Si)

E0

qm= 4.1eV

EFM

qox=0.95eV

Eg 8eV

Ec

Ev

EF

E0

Ec

EFEv

qS=4.9eVqS= 4.05eV

Eg=1.12eVMOS structure

p-Si

SiO2

• currents through SiO2 are very small.• holes flow from semiconductor metal on contact.• e flow from metal semiconductor on contact.• bands will bend downwards in silicon at the interface (m < s) where m = metal work function

s = p-type semiconductor work function = S + Eg/2 + B.


MOS Capacitor at Equilibrium: Applied V = 0

• Abrupt transition in Ec and Ev levels at the material interfaces.

• For m < s, band bends downward.

• A typical potential drop ~ 0.4 eV across SiO2. This depends on EF in Si. This potential can be supported because no current flows through SiO2.

• Substantial barriers exist to current flow from:

S M (~ 3.10 eV)

M S (~ 3.15 eV).

Ec

Ev

EF

Al SiO2 p-Si

3.8 eV

3.10 eV3.15 eV

0.4 eVEc

Ev

• Depletion region exists near the surface because EF near the surface is further from Ev than the bulk region.

• An applied voltage, Vfb = m - s = ms can make the band flat.


MOS Capacitor - Flat Band, V = Vfb

• EF is constant in Si because SiO2 prevents any current flow (i.e., Si is at equilibrium).

• Ec and Ev are flat.

• hole concentration is uniform in Si and SiO2 (as shown by flat Ec and Ev).

FLAT-BAND.

• The voltage required to achieve flat band condition is Vfb = ms.

fsgsmms

fsgsmms

E

E

2/

2/

:type)-/Si(n Al/SiOIn

:type)-/Si(p Al/SiOIn

2

2


MOS Capacitor – Accumulation, V < VfB

• EF is still constant in the Si since SiO2 prevents any current flow.

• EF is closer to Ev at the surface.

more holes accumulated near the surface than the bulk.

ACCUMULATION.

qVEFMEc

EF

Ev

Al SiO2 p-Si

ve


MOS Capacitor – Depletion, V > VFB

• EF is still constant in Si (I = 0).

• EF is farther from Ev at the surface.

The surface is depleted of holes.

DEPLETION.

qVEFM

Ec

EFEv

Al SiO2 p-Si

ve

EiB

Qg

Qd


• EF is still constant in Si (I = 0).

• EF is closer to Ec at the surface than it is to Ev.

more e- than holes at the surface.

• If s = 0, the surface becomes intrinsic

• If 0 < s < B, the surface will become slightly n-type:

WEAK INVERSION.

• If s > B, the surface will have an e- concentration > hole concentration in the bulk STRONG INVERSION.

MOS Capacitor – Inversion, V >> VFB

B

s


We now derive the relation among the surface potential, electric field, and charge (Qs) by solving Poisson’s eqn in the surface region of silicon.

For a metal/oxide/p-type silicon system, we define potential at any point x:

2. Electrostatic Potential and Charge Distribution

(x) i(x) i(x ) = amount of band bending at any point x.

where (x = 0) s = surface potential; i(x ) =f = Fermi potential

Thus, (x) < 0 when bands bend upward in accumulation (x) > 0 when bands bend downward in depletion and inversion


Band Bending

)()()()()(

:isEquation sPoisson' The

2

2

xNxNxnxpK

q

K

x

dx

dad

oo

(14)

)()(

and silicon,-pbulk in the (12) Eq. from Also,

2

2

aa

iad

da

ia

NNnxNxN

NNnnNp

qkTveNn

xpnxn

eNenenxp

ttv

x

a

ii

tv

x

atv

x

itv

x

i

Bif

/ where

silicon, type-pa for get we(10) and (9) Eq. From

)(22

)()()(

)()(

)(

(15)

(16)


Band Bending

Thus, the charge density at a depth x in the substrate is given by:

Substituting n(x), p(x), Na, and Nd in Poisson’s equation we can determine surface charge density in silicon substrate.

11

)()()(

)(2)(

tt v

x

a

iv

x

a

ad

eN

neNq

NNxnxpqx

(17)

11)(2)(

02

2tt v

x

a

iv

x

a eN

neN

K

q

dx

d

(18)


Band Bending

Poisson’s equation:

Multiplying both sides of (19) by 2d((x))/dx, we get:

11)(2)(

02

2tt v

x

a

iv

x

a eN

neN

K

q

dx

d

(19)

2

22

2dx

ud

dx

du

dx

du

dx

d

:know We (20)

dx

de

N

neN

K

q

dx

d

dx

d

dx

de

N

neN

K

q

dx

d

dx

d

tt

tt

v

x

a

iv

x

a

v

x

a

iv

x

a

112

112

2

)(2)(

0

2

)(2)(

02

2

or, (21)


Band Bending

Integrate from the bulk ( = 0, d/dx = 0) toward the surface at any point x to get:

At x = 0, = s and E = Es, then from Gauss’s law (7), the total charge per unit area induced in silicon (equal and opposite to the charge on the metal gate is: Qs = K0Es

)(

)]1)(

()1)(

[(2

)(

)]1()1([2

2

)(

2

2)(

0

2

)(

0

)(2)(

00

2

xE

v

xe

N

n

v

xe

K

vqN

dx

d

dx

dxEx

deN

neN

K

q

dx

dd

t

v

x

a

i

t

v

x

ta

x v

x

a

iv

x

adx

d

tt

tt

, at fieldelectric the

(22)

(23)


Band Bending

The charge expression above is valid in all regions of MOS capacitor operation - accumulation, depletion, and inversion. The Eq can, also, be expressed as:

2/12

2

0

0

)]1()1[(2

t

sv

a

i

t

svta

ss

ve

N

n

vevNqK

EKQ

t

s

t

s

(24)

2/1

0

00 )]1()1[(2

t

sv

p

p

t

sv

d

ts v

ep

n

ve

L

KvQ t

s

t

s

(25)

add Nq

kTK LL

20

:as defined length Debye where (26)


Surface Charge Density vs. Surface Potential

Note from (24):• for s < 0: Qs > 0 ACCUMULATION

Qs ~ exp(-s/2vt)

• for s = 0: Qs = 0 FLAT-BAND

• for s > 0: Qs < 0 DEPLETION and

WEAK-INVERSION

Qs ~ SQRT(s)

• fors > 2f

INVERSION

Qs ~ exp(s/2vt)

)2/exp(~ ts v

)2/exp(~ ts v

s~

Plot of (24) showing the variation of the induced surface charge density as a function of surface potential for p-type substrate.


Depletion Approximation

• For the simplicity of analytical integration, we can assume that in the depletion region B < s < 2B, we can approximate (23) as:

• We know: at x = 0, (x) = s; Therefore,

00

21 or, ;

2

K

qN

dx

d

K

qN

dx

d aa (27)

22

0

0 0

12

1)(

2)(

ds

s

as

a

s

W

xx

K

qNx

dxx

K

qNx

d

(28)

(29)


Depletion Approximation

• (29) is a parabolic Eq with the vertex at (x) = 0, x = Wd. Thus, Wd = distance to which band-bending extends = width of the depletion region.

• Then the depletion layer charge density is given by:

• In an MOS system, Wd reaches a maximum, Wd,max at the onset of inversion defined by s 2B = (2kT/q)ln(Na/ni).

a

sd

ds qN

KW

W

xx

0

22

and ;1)( Thus,

(30)

sadad NqKWqNQ 02 (31)

a

iad Nq

nNkTKW

20

max,

)/ln(4 (32)


Strong Inversion

• Beyond strong inversion, the concentration of inversion charge n(x) becomes significant. Therefore, from (23) we get:

• This equation must be solved numerically with boundary condition, = s at x = 0. From the solution of (x), the inversion carriers, n(x) can be calculated from (16).

0.0E+00

2.0E+18

4.0E+18

6.0E+18

8.0E+18

1.0E+19

1.2E+19

0 50 100 150 200

Distance from surface, x (A)

Inv

ers

ion

lay

er

co

nc

en

tra

tio

n (

cm

-3)

N(sub) = 1016 cm-3

s = 0.88 V

s = 0.85 V

t

s

a

i

t

sta ve

N

n

vK

vqN

dx

d

2

2

0

2

• The numerically calculated n(x) Vs. depth plot is shown in Fig. It is seen that inversion charge distribution is extremely close to the surface with an inversion layer width < 50 A.


Strong Inversion

• Generally, inversion-carriers must be treated quantum-mechanically (QM) as a 2-D gas. According to QM model:

– inversion layer carriers occupy discrete energy bands

– peak distribution is 10-30 A away from the surface.

• When the inversion charge density Qi > Qd, we get from (24):

t

s

a

iti

ve

N

nvqKQ

22

02

(33)

itiinv

ti

t

s

a

i

QvKqnQW

nvqKQ

ve

N

nne-

/2)0(/

)0(2

0

0

0

2

thickness, layerinversionClassical

)( :issurface the at ionconcentrat Since,

(34)


3. Low Frequency C – V Characteristics

Assuming silicon at equilibrium. The total capacitance of an MOS structure is: C = d(-Qs)/dVg

where the applied gate bias, Vg of an MOS system is:

Vg = Vfb + Vox + s

or, dVg = dVox + ds

where Vox = voltage drop in the oxide

Thus, the total capacitance equals the oxide capacitance, Cox and the silicon capacitance, Cs connected in series.

sox

s

s

s

ox

s

g

CCC

QQ

V

Q

V

111

or,

(35)


Low Frequency C – V Characteristics

From (24), we get:

From (36), we can calculate C - V characteristics in different regions of MOS operation.

2/1

2

2

2

2

0

)1()1(2

)1(12

)(

t

sv

a

i

t

sv

v

a

iv

d

s

ss

ve

Nn

ve

eNn

e

L

K

d

QdC

t

s

t

s

t

s

t

s

(36)


Low Frequency C – V: Accumulation, 0 << s

In strong accumulation, 0 << s . Then from (36), we can show:

Since 0 << s, for large s, Cs becomes very large.

t

s

t

s

t

s

v

dv

v

ds e

L

K

e

e

L

KC 20

2

0

22

2

(37)

ox

oxsox

CC

CCCC

:onaccumulati inThus,

1111

(38)


Low Frequency C – V: Flat Band, s = 0

For s = 0: the inversion charge term in (36) can be neglected to get:

Expanding the exponential term into power series and retaining the first five terms, we get:

2/10 )]1[(2

t

sv

d

ts v

eL

vKQ t

s

(39)

2

1

2

20

2/1

4

4

3

3

2

20

12

1

3

11

124

1

6

1

2

11

2

t

s

t

s

d

s

t

s

t

s

t

s

t

s

t

s

d

ts

vvL

K

vvvvvL

vKQ

(40)


Low Frequency C – V: Flat Band

2

20

24

1

6

11

211

t

s

t

s

d

ss

vvL

KQ

xx

of expansionBinomial know, We

(41)

1

0

0

1

0)(

K

L

CCC

L

K

sd

sdQC

d

oxfb

dss

band Flat

(42)

Thus, Eq. (42) shows that Cfb is somewhat less than Cox.


Low Frequency C – V: Depletion, B < s < - B

In depletion, the general expression for Cd is given by (36). However, we can derive an approximate expression from depletion approximation using Eq. (30) and (31).

We know that:

Thus, C decreases with the increase in Vg

ds

dd W

K

d

QdC 0)(

0

2

0

21

2

KqN

VC

CC

C

NqK

C

Q

C

QV

a

gox

oxs

sox

sas

ox

ds

ox

sg

:show can we, gEliminatin

(43)


Low Frequency C – V: Strong Inversion

In strong Inversion, s >> . Then from (36), we can show that:

Negative sign indicates that the charge has changed sign.

Since s >> 0, for large s, Cs becomes very large.

ox

oxsox

CC

CCCC

:inversion strongin Thus,

1111

t

s

t

s

t

s

v

a

i

dva

vi

ds e

N

n

L

K

eN

en

L

KC 20

2

0

22

2

(44)


4. Intermediate and High Frequency C - V

• There are plenty of majority carriers which can respond to the A.C. signal.

• The minority carriers, on the other hand, are scarce and have to originate from:

– diffusion from bulk substrate

– generation in the depletion region

– external sources (e.g. n+ diffusion region etc.)

• Thus, the inversion charge can not respond to the applied A.C. signal higher than 100 Hz.

• Therefore, at any high frequency applied signal to Vg:

Qi is assumed to be constant

Qd = Cs


Intermediate and High Frequency C - V

• At any high frequency applied signal to Vg:

– Qd will vary with the signal around its maximum value Qd,max

– Wd will vary with the signal around its maximum value Wd,max.

• Thus, the high frequency capacitance is given by:

a

ia

ox

d

ox

NqK

nNvt

CC

CK

W

CC

0min

min0

max,

)/ln(411

111

(45)


MOS C - V Characteristics


5. Oxide and Interface Charges


5. Oxide and Interface Charges

• Types of origins:

– fixed interface charge, Qf: believed to be uncompensated

Si-Si bonds, on the order of +1010 cm-2.

– interface trapped-charge, Qit: due to un-terminated Si

bonds, on the order of 1010 cm-2.

– Oxide trapped-charge, Qot: due to defects in the SiO2

network, usually negligible in advanced MOSFET devices. Can be guaranteed after a large amount of charges have passed through the SiO2 and broken up the network.

– mobile charge, Qm: sue to alkaline ions (e.g., Na, K), usually

very low concentration (~ 109 cm-2) in advanced technology.


Effects of Qf and Qot on C - V

• The flat-band voltage change due to Qf and Qot is:

• If Qot is small, Qf can be estimated by Vfb.

ox

oxot

oxox

f

ox

oxoxox

ox

ox

oxoxfb

xdx

x

xQ

CC

Q

xdx

x

xx

K

x

xxdxx

KV

0

0

0

1

)(

)(1

(46)


Effects of Qf and Qot on C - V

Figure.


Effects of Qm on C - V

By bias-temperature stress, Qm can be moved from Si-SiO2 interface to the metal-SiO2 interface. The shift in VFB can be used to estimate Qm.

Figure.


Effects of Qit on C - V

• Interface trapped-charge, Qit:

– the traps have energy levels distributed in the band gap

– the charge state of the traps depends on Fermi level.

thus, as the surface potential is swept from flat band towards depletion, the traps will try to pin the surface potential, i.e. it will take more gate voltage to move the surface potential through the traps.


Effects of Qit on C - V

Assume acceptor type traps:

Thus, Cs is in parallel with Cit. Hence, the capacitor model is:

densitytypeacceptor)(

densitytypeacceptor)(

sdit

sait

D

D

its

s

sit

s

ss

s

total

sitsstotal

CC

QQQ

QQQ

Total

))(())((

)()(

(47)

)(

)()(

sit

sdits

aitit

qD

DqDC

(48)


6. Polysilicon Gates

At equilibrium (Vg = 0).

Ec

Ev

EF

n+ poly SiO2 p-Si

Ei

B

Work-function difference:

n+poly with p-type Si: MS = Eg/2 (kT/q)ln(NA/ni) (49)

p+poly with n-type Si: MS = Eg/2 + (kT/q)ln(ND/ni) (50)

At Inversion (Vg >> 0).

Ec

Ev

EF

n+ poly SiO2 p-Si

EiS

VG

Ei

Ev

EF

p

Ec

Vox


Polysilicon Depletion Effect

Note:

• Capacitance at inversion, Cinv does not return to Cox.

• Cinv shows a maximum value, Cmax < Cox.

• Cmax as the polysilicon doping concentration, Np.

• As Np, depletion width

Cmax Coxfor higher Np.

Total capacitance at strong inversion is given by:

1/C = 1/Cox + 1/Csilicon + 1/Cpoly

As VG, Csiliconbut xd(poly) Cpoly.

Low frequencyC V shows a local maximumat a certain VG.


7. Inversion Layer Quantization

• Typically, near the silicon surface, the inversion layer charges are confined to a potential well formed by:– oxide barrier

– bend Si-conduction band at the surface due to the applied gate potential, VG.

• Due to the confinement of inversion layer e (in p-Si): – e- energy levels are grouped in discrete sub-bands of energy, Ej

– each Ej corresponds to a quantized level for e motion in the normal direction.

E2

E1

E0

Edge of EC

Distance from the surfaceBottom ofthe well

E


Inversion Layer Quantization

• Due to Quantum Mechanical (QM) effect, the inversion layer concentration:– peaks below the SiO2/Si interface

0 at the interface determined by the boundary condition of the e- wave function.

• Solve Schrodinger and Poisson Eq self-consistently with the boundary conditions for wave function equal to:– 0 for x < 0 in oxide– 0 at x = .

QM

Classical

Depth

n (c

m-3)


QM Effect on MOS Capacitors

• At high fields, Vth since more band bending is required to populate the lowest sub-band, which is some energy above the bottom of EC.

• Once the inversion layer forms below the surface, a higher Vg over-drive is required to produce a given level of inversion charge density. That is, the effective gate oxide thickness, tOX

eff by:

tOX = (ox/si)z (51)

• Inversion layer quantization can be treated as bandgap widening due to an increase in the effective bandgap by Eg given by:

Here, Eg = EgQM Eg

CL.

(52) niand ndue to QM effect.

kT

ECLi

QMi

g

enn 2

(52)


Home Work 1: Due 0ct 6, 2005

1) An MOS capacitor is built with the structure shown below. The capacitor areas over the N and P regions are the same. The threshold voltages for the N and P substrates are -1 and +1 volts respectively. Sketch the shape of the high frequency (1 MHz) C-V curve that you would expect to measure for this structure. Label as many points as possible and explain.

2) In practical MOS systems, measurements of C - V sometimes show hysteresis effects; that is, the C - V curves look like the sketch in Fig. The sketch refers to measurements made when VG is swept with a very low frequency triangular wave (~ 1 Hz) and the A.C. measurement frequency is of the order of 1 kHz or higher. The sense of the hysteresis on such a curve can be observed experimentally to be either counterclockwise, as shown, or else be clockwise.

N P

SiO2

Al


Home Work 1: Due Oct 6, 2005

The hysteresis sense allows one to differentiate between the two most common cause of non-ideal behavior.

(a) Show this by considering the following non-ideal effects: (i) field-aided movement of positive ions in the insulator and (ii) trapping of free

carriers from the channel in traps at the oxide-Si interface Q it.

(b) Using qualitative reasoning, prepare a table with sketches of the expected C - V plots for n- and p-type substrates; on each sketch indicate the sense of the hysteresis (i.e., clockwise or counterclockwise).



3) MOS capacitors are fabricated on uniformly doped P-substrates with NA = 1x1017 cm3, physical gate oxide thickness TOX = 6.7 nm, and N+ poly-silicon gate doping concentrations Np = 5x1018, 1x1019, and 5x1019 cm3 as shown in the following C – V plots. Here C and Cox are the gate and oxide capacitances, respectively.

(a) Explain the observed variation in C/COX vs. Vg plots for each Np in the strong inversion region.

(b) Explain the observed increase in C/COX for Np = 5x1018 cm3 and Vg 3.5 V.

(c) Describe the possible reasons for C/COX < 1 even for a heavily doped poly with Np = 5x1019 cm3 in the accumulation as well as in the strong inversion regions?



4) A p-type MOS-capacitor with Na = 1x1018 cm-3 and TOX = 3 nm was

fabricated to characterize van Dort’s analytical bandgap widening quantum mechanical (QM) model we discussed in class. Due to inversion layer quantization, the increase in the effective bandgap Eg = Eg

QM EgCL 104

mV. Here EgQM and Eg

CL represent the QM and classical (CL) values of

bandgap (Eg), respectively. Assume Qf = 0, VSUB = 0, and N+ poly gate.

(a) Show that the intrinsic carrier concentration due to QM effect is given by:

(b) Calculate the value of niQM at temperature T = 300oK.

(c) Calculate the value of threshold voltage Vth(CL) using the classical

approach.

(d) Calculate the value of threshold voltage Vth(QM) due to QM effects.

(e) Calculate the shift Vth due to QM effects.

kT

ECLi

QMi

g

enn 2

ho #2: elen 384 - mos capacitorspage 1s. saha basic equations for device operation transport...

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