pn diode characteristics mos capacitor mos transistoree143/fa10/lectures/lec_21.pdf · professor n...

Professor N Cheung, U.C. Berkeley

Lectre 21EE143 F2010

1

Microfabricationcontrols dopantconcentrationdistribution

ND(x) and NA(x)

Electron Concentration n(x)

Hole Concentration p(x)

Electrical resistivity

Sheet Resistance

Fermi level Ef (x) , Electric Field

PN Diode Characteristics

MOS Capacitor

MOS Transistor

Carrier Mobility



2

Electron and Hole Concentrationsfor homogeneous semiconductor at thermal equilibrium

n: electron concentration (cm-3)p : hole concentration (cm-3)ND: donor concentration (cm-3)NA: acceptor concentration (cm-3)

1) Charge neutrality condition: ND + p = NA + n

2) Law of Mass Action : n p = ni2

Note: Carrier concentrations depend onNET dopant concentration (ND - NA) !

Assume completelyionized to form ND

+

and NA-



3

How to find n, p when Na and Nd are known

n- p = Nd - Na (1)pn = ni2 (2)

(i) If Nd -Na > 10 ni : n Nd -Na(ii) If Na - Nd > 10 ni : p Na- Nd

* Either n or p will dominate for typical doping situations



4

Probability of available states at energy E being occupiedf(E) = 1/ [ 1+ exp (E- Ef) / kT]where Ef is the Fermi energy and k = Boltzmann constant=8.617 10-5 eV/K

The Fermi-Dirac Distribution (Fermi Function)

T=0K

0.5

E -Ef

f(E)



5

(2) Probability of available states at energy E NOT being occupied

1- f(E) = 1/ [ 1+ exp (Ef -E) / kT]

Properties of the Fermi-Dirac Distribution

(1) f(E) exp [- (E- Ef) / kT] for (E- Ef) > 3kT Note:At 300K,kT= 0.026eV•This approximation is called Boltzmann approximation

Probability

of electron state

at energy E

will be occupied



6

Fermi Energy (Ei) of Intrinsic Semiconductor

Ec

Ev

Ei

Eg /2

Eg /2



7

Ec

Ev

Ei

Ef(n-type)

Ef(p-type)

q|F|

Let qF Ef - Ei

n = ni exp [(Ef - Ei)/kT]

n = ni exp [qF /kT]

How to find Ef when n(or p) is known

Note: Ef is notvery sensitive tocarrierconcentration

When n doubles,Ef changes bykT/q(ln 2) =18mV



8

Dependence of Fermi Level with Doping Concentration

Ei (EC+EV)/2 Middle of energy gap

When Si is undoped, Ef = Ei ; also n =p = ni



9

Work Function of Materials

Eo

EV

ECq

SEMICONDUCTOR

Ef

Eo

Ef

METAL

Work function= q

Vacuumenergy level

qM is determined

by the metal materialqS is determined

by the semiconductor material,

the dopant type,

and doping concentration



10

Work Function (qM) of MOS Gate MaterialsEo = vacuum energy level Ef = Fermi levelEC = bottom of conduction band EV = top of conductionband

Ef

Al = 4.1 eVTiSi2 = 4.6 eV

Ef

Eo

qM

Eo

qM

Ei

EC

EV

0.56eV

q = 4.15eV

0.56eV

q = 4.15eV (electron affinity)

Eo

Ei

EC

EV

0.56eV

q = 4.15eV

0.56eV

Ef

n+ poly-Si p+ poly-Si

qM



11

Work Function of doped Si substrate

q = 4.15eV

Ef

Eo

qs

Ei

EC

EV

|q F|

0.56eV

0.56eV

* Depends on substrate concentration NB

Ef

Eo

qs

Ei

EC

EV

|qF|0.56eV

0.56eV

q = 4.15eV

n-type Si p-type Si

s (volts) = 4.15 +0.56 - |F| s (volts) = 4.15 +0.56 + |F|

i

BF n

NqkT ln



12

At thermal equilibrium ( i.e., no external perturbation),The Fermi Energy must be constant for all positions

The Fermi Energy at thermal equilibrium

Material A Material B Material C Material DEF

Position x

Electron energy



13

Aftercontactformation

Beforecontactformation

Electron Transfer during contact formation

E

System 1 System 2

EF1

EF2e

System 1 System 2

EF

---

+++

E

System 1 System 2

EF1

EF2

e

System 1 System 2

EF

---

+++

Netnegativecharge

Netpositivecharge



14

Fermi level of the side which has a relatively higher electric potential will have arelatively lower electron energy ( Potential Energy = -q electric potential.)Only difference of the E 's at both sides are important, not the absolute positionof the Fermi levels.

Side 2Side 1

Ef1

f2E

Va > 0

qVa Va

Side 2Side 1

Ef1

f2E

Va < 0

q| |

+ - -+

Potential difference across depletion region= Vbi - Va

Applied Bias and Fermi Level



15

PN junctions

Complete Depletion Approximation used for charges inside depletion region(x) ND

+(x) – NA-(x)

n-Sip-Si

Depletion region

ND+ onlyNA

- only(x) is +(x) is -

ND+ and n

(x) is 0

NA- and p

(x) is 0

E-field

++

++

- -

- -Quasi-neutral

regionQuasi-neutral

region

n-Sip-Si

Depletion region

ND+ onlyNA

- only(x) is +(x) is -

ND+ and n

(x) is 0

NA- and p

(x) is 0

E-field

++

++

- -

- -Quasi-neutral

regionQuasi-neutral

region

http://jas.eng.buffalo.edu/education/pn/pnformation2/pnformation2.html

Thermal Equilibrium



16

x

Electron

Energy E-field+-

EC

EV

21

x

Electron

Energy E-field+ -

EC

EV2

1

Electric potential(2) < (1)

Electric potential(2) > (1)

Energy Band Diagram with E-field

Electron concentration n

kT/)]1()2([qkT/)1(q

kT/)2(q

eee

)1(n)2(n



17

1) Summation of all charges = 0

Electrostatics of Device Charges

x

Semiconductor(x)

xd1xd2

Semiconductor

-

-

x=0

p-typen-type

2) E-field =0 outside depletion regions

2 xd2 = 1 xd1

E = 0E = 0 E 0



18

3) Relationship between E-field and charge density (x)

d [ E(x)] /dx = (x) “Gauss Law”

4) Relationship between E-field and potential

E(x) = - d(x)/dx



19

Example Analysis : n+/ p-Si junction

x

-qNa

xd

x=0

+Q'

n+ Sip-Si

(x)

E (x)

xd

1) Q’ = qNaxd

3) Slope = qNa/s

4) Area under E-field curve= voltage across depletion region= qNaxd

2/2s

Emax =qNaxd/s

Depletionregion

2) E = 0

Depletion region

is very thin and is

approximated as

a thin sheet charge



20

x

-

x=0

-qNA

+qND

(x)

-xp

+xn

x=0

x

1(x)

-xp

-qNA

Q=+qNAxp

x

x=0

2(x)

+qND

+xnQ=-qNAxp

Superposition Principle

If 1(x) E1(x) and V1(x)2(x) E2(x) and V2(x)

then1(x) + 2(x) E1(x) + E2(x) andV1(x) + V2(x)

+



21

x

-

x=0

Slope =- qNA/s

E1(x)

-xp

x=0

x

1(x)

-xp

-qNA

Q=+qNAxp

x

x=0

2(x)

+qND

+xnQ=-qNAxp

+ x

-

x=0

+xn

E2(x)

Slope =+ qND/s



22

x

-

x=0

Slope = - qNA/s

E(x) = E1(x)+ E2(x)

-xp +xn

Slope = + qND/s

Emax = - qNA xp /s= - qND xn /s

Sketch of E(x)



The MOSFET

• No current through the gateto channel

• Current flowing between theSOURCE and DRAIN iscontrolled by the voltage onthe GATE electrode

Desired characteristics:• High ON current• Low OFF current

|GATE VOLTAGE|

CURR

ENT

VT

Metal-Oxide-Semiconductor

Field-EffectTransistor

Substrate

Gate

Source Drain

GATE LENGTH, LOXIDE

THICKNESS, Tox

GATE WIDTH,W



n-channel MOSFET

• Without a gate voltage applied, no current can flowbetween the source and drain regions (2 pn diodesback-to -back)

• Above a certain gate-to-source voltage (thresholdvoltage VT), a conducting layer of mobile electrons isformed at the Si surface beneath the oxide. Theseelectrons can carry current between the source anddrain.

np

oxide insulatorgaten

DSG



The MOSFET as a Controlled Resistor

• When VDS is low, ID increases linearly with VDS

ID

IDS = 0 if VGS < VT

VDS

VGS = 1 V > VT

VGS = 2 V

Inversion charge density Qi(x) = -Cox[ VGS – VT - V(x) ]

where Cox ox / tox.



Pinch-off and Saturation

• As VDS increases, V(x) along channel increases.Inversion-layer charge density Qn at the drain end of the channel isreduced; therefore, ID increases slower than linearly with VDS.

• When VDS reaches VGS VT, the channel is “pinched off” at the drainend, and ID saturates (i.e. it does not increase with further increasesin VDS).

n+n+

S

G

VGS

D

VDS > VGS - VT

VGS - VT+-

pinch-offregion

+–

22 TGSoxnDSAT VVLWCI



NMOS versus PMOS

DS tov V0GSv

TriodeSaturationCut-off

toV

NMOS

DS tov V 0GSv

Triode Saturation Cut-off

toV

PMOS



IDsat Not proportional to

(VGS-Vt)2

pn diode characteristics mos capacitor mos transistoree143/fa10/lectures/lec_21.pdf · professor n...

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