25 gate voltage and cv - nanohubgatevoltage_and_… · gate voltage and cv 2 1) quick review of...

Lundstrom ECE 305 F15

ECE-305: Spring 2015

Gate Voltage and

Capacitance vs. Voltage

Professor Mark Lundstrom Electrical and Computer Engineering

Purdue University, West Lafayette, IN USA [email protected]

4/3/15

Pierret, Semiconductor Device Fundamentals (SDF) pp. 567-584

1

Gate voltage and CV

2

1)  Quick review of band bending 2)  Surface potential gate voltage 3)  Example 4)  Capacitance vs. voltage


Note: Next week (MOSFET IV), will follow for the most part Lundstrom “Lecture Notes on MOSFETs,” which is posted on the course home page under Week 12.

band banding in an MOS device

3 Fig. 16.6, Semiconductor Device Fundamentals, R.F. Pierret Flat band Accumulation Depletion Inversion

!VG = 0 !VG < 0 0 < !VG <VT !VG > !VT!S = 0 !S < 0 0 < !S < 2!F !S > 2!F

depletion, inversion, and accumulation

4

EC

EV

Ei

EF

Si

q! x( ) ! = 0

W

x

q!F

!F =kBTqln NA

ni

"#$

%&'

!S

!S < 0

0 < !S < 2!F

!S > 2!F

accumulation:

depletion:

inversion:


depletion approximation

5

E x( )

x

What gate voltage produced this surface potential?

P

E S =

qNA

! S"0W

12E SW = !S

E S

W

W = 2! S"0#SqNA

cm

E S =

2qNA!S" s#0

V/cm

QB !S( ) = " 2q# s$0NA!S C/cm2

QB = !qNAW "S( )C/cm2

0 < !S < 2!F

surface electric field and charge

6

E x( )

x

P E S

dEdx

= ! qNA

KS"0

dE = ! qNA

KS"0dx

dE

E W( )

E 0( )

! = " qNA

KS#0dx

W

0

!

E 0( ) =E S =

qNAWKS!0


E ox

KS!0E S = DS = qNAW( ) = "QB

DS = !QB

Dox = !QB

DS = Dox(no charge at interface)


7 Lundstrom ECE 305 F15

QS > 0 C cm2

D =QS

x


8

E x( )

x

P E S


E ox

QB = !qNAW C/cm2

Dox = !QB DS = !QB

W

Gate voltage and CV

9



gate voltage and surface potential

10 10

EC

EV

Ei

EF

Si

metal

!VS

!VOX

EFM

!VG = "VOX +#S

0 < !S < 2!F

!VG = ?Given the surface potential, what is the gate voltage?

relation to gate voltage

11

x

E S

W!xo

metal

!VG = "Vox +#S

!VG

E ox

!Vox = xoE ox

E x( )

Dox = KO!0E ox = "QB

!Vox = xoE ox = xo

"QB

KO#0

Volt drop across a capacitor

12

!Vox

C ! QV

= KO"0Axo

= Fxo

+ + + + + +

- - - - - -

CA= KO!0

xo= F/cm2

Cox =KO!0xo

= F/cm2

relation to gate voltage

13

E x( ) E S

W!xo

metal

!VG

E ox

x

!VG = "QB #S( )Cox

+#S

QB !S( ) = "qNAW !S( )

Cox = KO!0 xo

!Vox = xoE ox = xo

"QB

KO#0

14

E x( )

x

P

E S =

qNA

! S"0W

12E SW = !S

E S

W

W = 2! S"0#SqNA

cm

E S =

2qNA!S" s#0

V/cm

QB = ! 2q" s#0NA$S C/cm2

QB = !qNAW "S( )C/cm2

0 < !S < 2!F

!VG = "QB #S( )Cox

+#S

recap (depletion)

MOS electrostatics: inversion

15

EC

EV

Ei

EF

Si

! x( ) ! 0( )

x

!F!S " 2!F !F

WT

WT =2KS!0qNA

2"F#

$%

&

'(

1/2 Maximum depletion region depth

delta-depletion approximation

16

!

x

metal

!xo

WT

! = "qNA

QB = !qNAWT

Qn

WT =2! S"0 2#FqNA

delta-depletion approximation

17

E x( )

x

P

WT

E S

E 0+( ) = ! QB

KS"0

E 0( ) = ! QS

KS"0


MOS electrostatics: inversion

18

EC

EV

Ei

EF

Si

! x( ) ! 0( )

x

!F

!S " 2!F

!F

WT

WT =2KS!0qNA

2"F#

$%

&

'(

1/2

!VG = "QB 2#F( ) +Qn

Cox

+ 2#F

!VT = "QB 2#F( )Cox

+ 2#F

Qn = !Cox "VG ! "VT( )

!VG = " QS

Cox

+ 2#F


Gate voltage and CV

19



20

example

source drain

SiO

2

silicon

S G DAssume n+ poly Si gate with Φms = -1.0 V

NA = 1018 cm-3 channel doping

xo = 1.5 nm

What is VT?

Electric -field in oxide at VG = 1V?

21

example (cont.)

!VG = "QS #S( )Cox

+ #S

!VT = "QB 2#F( )Cox

+ 2#F

VT = !ms "QB 2!F( )Cox

+ 2!F

!F =kBTqln NA

ni

"#$

%&'

Cox = KO!0 xo

QB = ! 2q" s#0NA2$F

QB = !qNAW 2"F( )

!ms = "1.00 V

!F = 0.48 V

Cox = 2.36 !10"6 F/cm2

QB = !5.71"10!7 C/cm2

!ms = "1.00 VVT = 0.20 V

22

example (cont): electric field in the oxide

Qn = !Cox VG !VT( )

E ox = ! QS

Ko"0= !

Qn +QB 2#F( )Ko"0

Qn = !1.89 "10!6 C/cm2

E ox = 7.1!106 V/cm

Qn

!q= 1.2 "1013 /cm2

VG = 1V

VT = 0.2 V

23

example (cont): volt drop in the oxide at VG = 1 V

!Vox =E oxxo E ox = 7.1!106 V/cm !Vox = 1.07V

!VS = "s !s = 2!F !F = 0.48 V !s = 0.96

VG = !Vox + !VS = 1.07 + 0.96 = 2.03

VG = 1V

Gate voltage and CV

24



MOS capacitor

25

p-Si

vS sin!t

+ VG

+

-

-

VG + vS sin!t

~

MOS capacitor in depletion

26

VG

p-Si

W !S( ) W VG( )

27

MOS capacitor in depletion

xo

W !S( )

KO

KS

C = ?

Gate

Undepleted P-type semiconductor

28

a simpler problem

xo

W !S( )

KO

KS CS =KS!0W "S( )

Cox =KO!0xo

1C

=1Cox

+1CS

C =CSCox

CS + Cox

C =Cox

1+ Cox CS

C = Cox

1+KOW !S( )KSxo

result

xo

W !S( )

! ox

! Si

C = Cox

1+KOW !S( )KSxo

VG

Cox

CS

!S

30

s.s. gate capacitance vs. d.c. gate bias

C

VG!

C = Cox

1+KOW !S( )KSxo

accumulation depletion

inversion

VT!

flat band

Cox

31

s.s. gate capacitance vs. d.c. gate bias

C

VG!C = Cox

1+KOW !S( )KSxo

accumulation

depletion

inversion

VT!

flat band

Cox

32

capacitance vs. gate voltage

C

VG!C = Cox

1+KOW !S( )KSxo


inversion

VT!

flat band

Cox

33

high frequency vs. low frequency

C

VG!C = Cox

1+KOW !S( )KSxo


inversion

VT!

flat band

Cox

high frequency

low frequency

34

high frequency vs. low frequency

p-Si

n+-Si n+-Si

MOS capacitor

25 gate voltage and cv - nanohubgatevoltage_and_… · gate voltage and cv 2 1) quick review of...

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