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2

Basic Physics of Semiconductors

Ø Semiconductor materials and their properties

Ø PN-junction diodes

ØReverse Breakdown

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3 SCL SAS NagarMOS - HSJ

Semiconductor Physics

Ø Semiconductor devices serve as heart of microelectronics.

Ø PN junction is the most fundamental semiconductordevice.

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Charge Carriers in Semiconductor

Ø To understand PN junction’s IV characteristics, it isimportant to understand charge carriers’ behavior in solids,how to modify carrier densities, and different mechanismsof charge flow.

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Periodic Table

Ø This abridged table contains elements with three to fivevalence electrons, with Si being the most important.

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Silicon

Ø Si has four valence electrons. Therefore, it can formcovalent bonds with four of its neighbors.

Ø When temperature goes up, electrons in the covalent bondcan become free.

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Electron-Hole Pair Interaction

Ø With free electrons breaking off covalent bonds, holes aregenerated.

Ø Holes can be filled by absorbing other free electrons, soeffectively there is a flow of charge carriers.

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Free Electron Density at a Given Temperature

Ø

E g , or bandgap energy determines how much effort isneeded to break off an electron from its covalent bond.

Ø There exists an exponential relationship between the free-electron density and bandgap energy.

3150

3100

32 / 315

/ 1054.1)600(

/ 1008.1)300(

/ 2

exp102.5

cmelectronsK T n

cmelectronsK T n

cmelectronskT

E T n

i

i

g

i

×==

×==

−×=

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Doping (N type)

Ø Pure Si can be doped with other elements to change itselectrical properties.

Ø For example, if Si is doped with P (phosphorous), then ithas more electrons, or becomes type N (electron).

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Doping (P type)

Ø If Si is doped with B (boron), then it has more holes, orbecomes type P.

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Summary of Charge Carriers

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Electron and Hole Densities

Ø The product of electron and hole densities is ALWAYSequal to the square of intrinsic electron density regardlessof doping levels.

2

innp =

D

i

D

A

i

A

N

n p

N n N

nn

N p

2

2

≈

≈

≈

≈Majority Carriers :

Minority Carriers :

Majority Carriers :

Minority Carriers :

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Example # 1

Ø What is the electron and hole concentration in intrinsicsemiconductor at room temp?

Answer : n = p = ni = 1.5 X 1010 carriers/cm3

This may seem like a lot of carrier. However the no of siliconatoms Nsi in a given volume of crystalline silicon is

Nsi = 50 X 1021 atms/cm3

So there is only one excited electron/hole pair for every 1012

silicon atoms

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Example #2

Ø Suppose is silicon is doped with P having a density Nd of1018 atoms/cm3. Estimate the doped silicon’s holes andelectron concentration.

Ans : n = Nd = 1018 electrons/cm3

P = ni2 / Nd = (1.5 X 1010)2 / 1018

p = 210 holes / cm3

There is one dopant atom for every 50,000 silicon atoms

Assumption : Nsi >> Nd >> ni

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Ni vs temperature

Ø Intrinsic concentration ni varies with temperature T

ni2

= A0 T3

e-EG0/kT

EG0 : energy gap at 0K ( eV)

K ; boltzmann constant

A0 : constant

K = 8.62 X 10-5 eV/k or 1.38 X10-23 J/k

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Forbidden gap energy vs temperature

Ø Forbidden gap energy EG in semiconductor depends upontemperature

EG(T) = 1.21 – 3.6 X 10-4 T

and at room temp(300K) EG = 1.1 eV

Ø For Ge

EG(T) = 0.785 – 32.23 X 10-4 Tand at room temp, EG = 0.72 eV

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First Charge Transportation Mechanism: Drift

Ø The process in which charge particles move because of anelectric field is called drift.

Ø Charge particles will move at a velocity that is proportionalto the electric field.

→→

→→

−=

=

E v

E v

ne

ph

µ

µ

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Mobility vs temperature

Ø Mobility varies with temperature over a range of100 to 400K

µµ ∝∝ T –m

For Si m = 2.5 for electrons

m = 2.7 for holes

For Gem = 1.66 for electrons

m = 2.33 for holes

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Mobility vs electric Field Intensity

Ø Mobility is a function of Electric field intensity.

Ø E < 103 V/cm : µµ is constant

Ø 103 < E < 104 V/cm : µµ ∝∝ 1/√√E

Ø For higher fields : µµ ∝∝ 1/E

Ø Carrier velocity approaches constant value of 107 cms/sec( vsat) at higher electric fields

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Mobility vs electric Field Intensity

Ø Velocity saturation occurs at high electric fieldsØ Optical phonon scattering : at high fields, carriers are

accelerated enough to gain sufficient KE betweencollisions to emit an optical phonon

Ø M V2emit/2 = hωωphonon

Ø Relation between scattering and mobility

– Ionized impurity scattering

– Lattice(phonon) scattering

Ø 1/µµ = 1/ µµ + 1/ µµ

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Current Flow: General Case

Ø Electric current is calculated as the amount of charge in v meters that passes thru a cross-section if the charge travelwith a velocity of v m/s.

qnhW v I

⋅⋅⋅⋅−=

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E pnq

q p E qn E J

qn E J

pn

pntot

nn

)( µµ

µµ

µ

+=

⋅⋅+⋅⋅=⋅⋅=

Current Flow: Drift

Ø Since velocity is equal to µµE, drift characteristic is obtainedby substituting V with µµE in the general current equation.

Ø The total current density consists of both electrons andholes.

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Velocity Saturation

Ø A topic treated in more advanced courses is velocity

saturation.Ø In reality, velocity does not increase linearly with electric

field. It will eventually saturate to a critical value.

E

v

E v

bv

bE

sat

sat

0

0

0

0

1

1

µ

µ

µ

µµ

+=

=

+=

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Second Charge Transportation Mechanism:

Diffusion

Ø Charge particles move from a region of high concentrationto a region of low concentration. It is analogous to anevery day example of an ink droplet in water.

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Current Flow: Diffusion

Ø Diffusion current is proportional to the gradient of charge(dn/dx) along the direction of current flow.

Ø Its total current density consists of both electrons andholes.

dxdnqD J

dx

dn AqD I

nn

n

=

=

)(dxdp D

dxdn Dq J

dx

dpqD J

pntot

p p

−=

−=

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Example: Linear vs. Nonlinear Charge DensityProfile

Ø Linear charge density profile means constant diffusioncurrent, whereas nonlinear charge density profile meansvarying diffusion current.

L

N qD

dx

dnqD J

nnn⋅−==

d d

n

n L

x

L

N qD

dx

dnqD J

−−== exp

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Einstein's Relation

Ø While the underlying physics behind drift and diffusioncurrents are totally different, Einstein’s relation provides amysterious link between the two.

q

kT D=

µ

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Properties of Ge and Si

Property Ge SiAt No 32 14

At Wt 72.6 28.1

Dielectric constant 16 12

EGo, eV at 0K 0.785 1.21

ni (300k) cm-3 2.5 X 1013 1.5 X 1010

Intrinsic resistivity 300k( -cm) 45 230,000

µµn, cm2 /V-sec at 300k 3800 1300

µµp, cm2 /V-sec at 300k 1800 500

Dn, cm2/s = µµn. VT 99 34

Dp, cm2/s = µµp. VT 47 13Atoms/cm3 4.4 X1022 5 X1022

Density g/cm3 5.3 2.3

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PN Junction (Diode)

Ø When N-type and P-type dopants are introduced side-by-side in a semiconductor, a PN junction or a diode is formed.

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Diode’s Three Operation Regions

Ø In order to understand the operation of a diode, it isnecessary to study its three operation regions: equilibrium,reverse bias, and forward bias.

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Current Flow Across Junction: Diffusion

Ø Because each side of the junction contains an excess ofholes or electrons compared to the other side, there existsa large concentration gradient. Therefore, a diffusioncurrent flows across the junction from each side.

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Depletion Region

Ø As free electrons and holes diffuse across the junction, aregion of fixed ions is left behind. This region is known asthe “depletion region.”

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Depletion region

hole diffusionelectron diffusion

p n

hole driftelectron drift

ChargeDensity

Distancex+

-

Electrical

xField

x

PotentialV

ξ

ρ

W2-W1

ψ 0

(a) Current flow.

(b) Charge density.

(c) Electric field.

(d) Electrostaticpotential.

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Band diagram – forward bias

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Current Flow Across Junction: Drift

Ø The fixed ions in depletion region create an electric fieldthat results in a drift current.

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Current Flow Across Junction: Equilibrium

Ø At equilibrium, the drift current flowing in one direction

cancels out the diffusion current flowing in the oppositedirection, creating a net current of zero.

Ø The figure shows the charge profile of the PN junction.

ndiff ndrift

pdiff pdrift

I I

I I

,,

,,

=

=

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Built-in Potential

Ø Because of the electric field across the junction, thereexists a built-in potential. Its derivation is shown above.

∫ ∫ =

−=

n

p

p

p p

x

x p

p p

p

dp DdV

dx

dpqD pE q

2

1

µ

µ

n

p

p

p

p p

p

p D xV xV

dx

dp D

dx

dV p

ln)()(12 µ

µ

=−

−=−

200ln,ln

i

D A

n

p

n

N N

q

kT V

p

p

q

kT V ==

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Diode in Reverse Bias

Ø When the N-type region of a diode is connected to a higherpotential than the P-type region, the diode is under reversebias, which results in wider depletion region and largerbuilt-in electric field across the junction.

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Band diagram – reverse bias

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Reverse Biased Diode’s Application: Voltage-Dependent Capacitor

Ø The PN junction can be viewed as a capacitor. By varyingVR, the depletion width changes, changing its capacitancevalue; therefore, the PN junction is actually a voltage-dependent capacitor.

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Effect of bias on diffusion current

Ø When the diode forward-bias-voltage is

increased, the barrier for electron and holediffusion current decreases linearly. See the banddiagram.

Ø Since the carrier concentration decreasesexponentially with energy in both bands,diffusion current increases exponentially as thebarrier is reduced.

Ø As the reverse-bias-voltage is increased, the

diffusion current decreases rapidly to zero, sincethe fall-off in current is exponential.

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Effect of bias on drift current

Ø When the reverse-bias-voltage is increased, the netelectric field increases, but drift current does not

change. In this case, drift current is limited NOT byHOW FAST carriers are swept across the depletionlayer, but rather HOW OFTEN.

Ø The number of carriers drifting across the depletionlayer is small because the number of minority

carriers that diffuse towards the edge of thedepletion layer is small.

→ To a first approximation, the drift current does not

change with the applied voltage.

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Relation between carrier concentration and vbi

( ) ( ) side-niFside-pFibi E E E E V −+−=

+

=

i

n0

i

p0lnln

n

n

q

kT

n

p

q

kT

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Relationship between carrier concentration and V bi

=

2i

n0p0bi ln

n

n p

q

kT V

=

n0

p0ln

p

p

q

kT n0

n0

2

i p

n

n=because

Therefore, kT

qV

p

p bi

e0n

0p = and kT

qV

n

nbi

e0p

0n =

side-ponconc.electron

side-nonconc.electrone

side-nonconc.hole

side-ponconc.holebi

== kT

qV

Strictly, these concentrations are at the depletion layer edge

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Majority and minority carrier concentration under bias

When an external voltage is applied, the minority carrier concentration

at the edge of the depletion layer will change. If a forward voltage(V A=positive) is applied, the barrier will be lower and carrier injection

(diffusion part) will increase. The minority carrier concentration at the

edge of the depletion layer will increase.

If a reverse voltage (V

A = negative) is applied, the barrier for carrierinjection (diffusion part) will increase, and the minority carrier

concentration at the edge of the depletion layer will decrease.

The drift of minority carriers across the junction does not change

much with applied voltage. Why?

At V A = 0, the carrier injection and the drift of minority carriers cancel

each other such that an equilibrium conc. is maintained.

If “low-level-injection” condition is assumed, then the majority carrier

concentration will not change under any of the above conditions.

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Relationship between carrier concentration and V A

( )

kT

V V q

p p

Abi

e

n

p

−

= since (V bi – V A) is the net voltage (or barrier)when a forwarded voltage is applied.

At low-level injection: pp= pp0; Recall that

then

kT

qV

kT

qV

kT

qV

kT

V V q

p p p p

AAbiAbi

eeee n0p0

)(

p0n ===−−

−

kT

qV

p

p bi

en0

p0

=

kT

qV

p

pA

e

0n

n = kT

qV

n

n A

ep0

p =

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Voltage-Dependent Capacitance

Ø The equations that describe the voltage-dependentcapacitance are shown above.

0

0

0

0

1

2

1

V N N

N N qC

V

V

C C

D A

D Asi

j

R

j

j

+=

+=

ε

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Voltage-Controlled Oscillator

Ø A very important application of a reverse-biased PN

junction is VCO, in which an LC tank is used in anoscillator. By changing VR, we can change C, which alsochanges the oscillation frequency.

LC

f res

1

2

1

π

=

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Diode in Forward Bias

Ø When the N-type region of a diode is at a lower potentialthan the P-type region, the diode is in forward bias.

Ø The depletion width is shortened and the built-in electricfield decreased.

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Minority Carrier Profile in Forward Bias

Ø Under forward bias, minority carriers in each regionincrease due to the lowering of built-in field/potential.Therefore, diffusion currents increase to supply theseminority carriers.

T

F

f p

f n

V

V V

p p

−=

0

,

,

exp

T

e p

en

V

V

p p

0

,

,

exp

=

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Diffusion Current in Forward Bias

Ø Diffusion current will increase in order to supply theincrease in minority carriers. The mathematics are shownabove.

)1(exp

exp 0

−≈∆T

F

T

D p

V

V

V

V

N n )1(exp

exp 0

−≈∆T

F

T

An

V

V

V

V

N p

)(2

p D

p

n A

nis

L N

D

L N

D Aqn I +=)1(exp −=

T

F stot

V

V I I

)1(exp

exp

)1(exp

exp 00

−+−∝T

F

T

D

T

F

T

A

tot

V

V

V V

N

V

V

V V

N I

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Minority Charge Gradient

Ø

Minority charge profile should not be constant along the x-axis; otherwise, there is no concentration gradient and nodiffusion current.

Ø Recombination of the minority carriers with the majoritycarriers accounts for the dropping of minority carriers asthey go deep into the P or N region.

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Forward Bias Condition: Summary

Ø In forward bias, there are large diffusion currents of

minority carriers through the junction. However, as we godeep into the P and N regions, recombination currents fromthe majority carriers dominate. These two currents add upto a constant value.

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IV Characteristic of PN Junction

Ø The current and voltage relationship of a PN junction isexponential in forward bias region, and relatively constantin reverse bias region.

)1(exp −= T

D

S D V

V

I I

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Parallel PN Junctions

Ø Since junction currents are proportional to the junction’scross-section area. Two PN junctions put in parallel areeffectively one PN junction with twice the cross-sectionarea, and hence twice the current.

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Constant-Voltage Diode Model

Ø Diode operates as an open circuit if VD< VD,on and aconstant voltage source of VD,on if VD tends to exceed VD,on.

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Example: Diode Calculations

Ø

This example shows the simplicity provided by a constant-voltage model over an exponential model.

Ø For an exponential model, iterative method is needed tosolve for current, whereas constant-voltage model requiresonly linear equations.

S

X

T X D X X I

I V R I V R I V ln

11+=+=

mA I

mA I

X

X

2.0

2.2

=

=

V V

V V

X

X

1

3

=

= for

for

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Reverse Breakdown

Ø When a large reverse bias voltage is applied, breakdownoccurs and an enormous current flows through the diode.

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Zener vs. Avalanche Breakdown

Ø Zener breakdown is a result of the large electric field insidethe depletion region that breaks electrons or holes off theircovalent bonds.

Ø Avalanche breakdown is a result of electrons or holescolliding with the fixed ions inside the depletion region.

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Diode Implementation of OR Gate

Ø The circuit above shows an example of diode-implementedOR gate.

Ø Vout can only be either VA or VB, not both.

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Input/Output Characteristics

Ø When Vin is less than zero, the diode opens, so Vout = Vin.

Ø When Vin is greater than zero, the diode shorts, so Vout = 0.

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Diode’s Application: Rectifier

Ø A rectifier is a device that passes positive-half cycle of asinusoid and blocks the negative half-cycle or vice versa.

Ø When Vin is greater than 0, diode shorts, so Vout = Vin;however, when Vin is less than 0, diode opens, no currentflows thru R1, Vout = IR1R1 = 0.

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Signal Strength Indicator

Ø The averaged value of a rectifier output can be used as asignal strength indicator for the input, since Vout,avg isproportional to Vp, the input signal’s amplitude.

[ ]π

ωω

ω

pT p

T

p

T

out avgout

V t

V

T

tdt V

T

dt t V

T

V

=−=

∫ =∫ =

2 /

0

2 /

00,

cos1

sin1

)(1

T t T ≤≤2 for

20

T t ≤≤ for 0sin == t V V

pout ω

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Question for the Audience

Can we measure the built-in potential across a diode

by volt meter. If yes, then try in your lab. If NO, then why

Cant we measure, Think!

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Thank You

Shri Sukhdev Singh of VLSI Design

Division made this

Presentation. Thanks to him.

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