Overview of Silicon Device Physics Dr. David W. Graham West - - PowerPoint PPT Presentation

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Overview of Silicon Device Physics

• Dr. David W. Graham

West Virginia University

Lane Department of Computer Science and Electrical Engineering

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• Nucleus

Valence Band Energy Bands (Shells) Si has 14 Electrons Silicon is the primary semiconductor used in VLSI systems At T=0K, the highest energy band occupied by an electron is called the valence band. Silicon has 4 outer shell / valence electrons

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• Electrons try to
• ccupy the lowest

energy band possible

• Not every energy

level is a legal state for an electron to

• ccupy
• These legal states

tend to arrange themselves in bands

Allowed Energy States Disallowed Energy States Increasing Electron Energy

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Energy Bands

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• Valence Band

Conduction Band Energy Bandgap

Eg EC EV

Last filled energy band at T=0K First unfilled energy band at T=0K

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• Eg

EC EV Band Diagram Representation Energy plotted as a function of position EC Conduction band Lowest energy state for a free electron EV Valence band Highest energy state for filled outer shells EG Band gap Difference in energy levels between EC and EV No electrons (e-) in the bandgap (only above EC or below EV) EG = 1.12eV in Silicon Increasing electron energy Increasing voltage

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• Silicon has 4 outer shell /

valence electrons Forms into a lattice structure to share electrons

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• EC

EV The valence band is full, and no electrons are free to move about However, at temperatures above T=0K, thermal energy shakes an electron free

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• For T > 0K

Electron shaken free and can cause current to flow e– h+

• Generation – Creation of an electron (e-)

and hole (h+) pair

• h+ is simply a missing electron, which

leaves an excess positive charge (due to an extra proton)

• Recombination – if an e- and an h+ come

in contact, they annihilate each other

• Electrons and holes are called “carriers”

because they are charged particles – when they move, they carry current

• Therefore, semiconductors can conduct

electricity for T > 0K … but not much current (at room temperature (300K), pure silicon has only 1 free electron per 3 trillion atoms)

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• Doping – Adding impurities to the silicon

crystal lattice to increase the number of carriers

• Add a small number of atoms to increase

either the number of electrons or holes

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• Column 4

Elements have 4 electrons in the Valence Shell Column 3 Elements have 3 electrons in the Valence Shell Column 5 Elements have 5 electrons in the Valence Shell

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• Donors
• Add atoms with 5 valence-band

electrons

• ex. Phosphorous (P)
• “Dontates an extra e- that can freely

travel around

• Leaves behind a positively charged

nucleus (cannot move)

• Overall, the crystal is still electrically

neutral

• Called “n-type” material (added

negative carriers)

• ND = the concentration of donor

atoms [atoms/cm3 or cm-3] ~1015-1020cm-3

• e- is free to move about the crystal

(Mobility µn ≈1350cm2/V) +

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• Donors
• Add atoms with 5 valence-band

electrons

• ex. Phosphorous (P)
• “Donates” an extra e- that can freely

travel around

• Leaves behind a positively charged

nucleus (cannot move)

• Overall, the crystal is still electrically

neutral

• Called “n-type” material (added

negative carriers)

• ND = the concentration of donor

atoms [atoms/cm3 or cm-3] ~1015-1020cm-3

• e- is free to move about the crystal

(Mobility µn ≈1350cm2/V) + + + + + + + + + + + + + + + + + – – – – – – – – – – – – – – – – – + + n-Type Material + – + Shorthand Notation Positively charged ion; immobile Negatively charged e-; mobile; Called “majority carrier” Positively charged h+; mobile; Called “minority carrier”

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• –

– h+ Acceptors

• Add atoms with only 3 valence-

band electrons

• ex. Boron (B)
• “Accepts” e– and provides extra h+

to freely travel around

• Leaves behind a negatively

charged nucleus (cannot move)

• Overall, the crystal is still

electrically neutral

• Called “p-type” silicon (added

positive carriers)

• NA = the concentration of acceptor

atoms [atoms/cm3 or cm-3]

• Movement of the hole requires

breaking of a bond! (This is hard, so mobility is low, p ≈ 500cm2/V)

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• Acceptors
• Add atoms with only 3 valence-

band electrons

• ex. Boron (B)
• “Accepts” e– and provides extra h+

to freely travel around

• Leaves behind a negatively

charged nucleus (cannot move)

• Overall, the crystal is still

electrically neutral

• Called “p-type” silicon (added

positive carriers)

• NA = the concentration of acceptor

atoms [atoms/cm3 or cm-3]

• Movement of the hole requires

breaking of a bond! (This is hard, so mobility is low, p ≈ 500cm2/V) – – – – – – – – – – – – – – – – – + + + + + + + + + + + + + + + + + – – p-Type Material Shorthand Notation Negatively charged ion; immobile Positively charged h+; mobile; Called “majority carrier” Negatively charged e-; mobile; Called “minority carrier” – + –

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• f(E)

1 0.5 E Ef

The Fermi Function

• Probability distribution function (PDF)
• The probability that an available state at

an energy E will be occupied by an e- E Energy level of interest Ef Fermi level Halfway point Where f(E) = 0.5 k Boltzmann constant = 1.38×10-23 J/K = 8.617×10-5 eV/K T Absolute temperature (in Kelvins)

( )

( ) kT

E E

f

e E f

−

+ = 1 1

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f(E) 1 0.5 E Ef

~Ef - 4kT ~Ef + 4kT

( )

( ) kT

E E

f

e E f

− −

≈

kT E E

f >>

−

If Then

Boltzmann Distribution

• Describes exponential decrease in the

density of particles in thermal equilibrium with a potential gradient

• Applies to all physical systems
• Atmosphere Exponential distribution of gas molecules
• Electronics Exponential distribution of electrons
• Biology Exponential distribution of ions

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Eg EC EV Band Diagram Representation Energy plotted as a function of position

EC Conduction band Lowest energy state for a free electron Electrons in the conduction band means current can flow EV Valence band Highest energy state for filled outer shells Holes in the valence band means current can flow Ef Fermi Level Shows the likely distribution of electrons EG Band gap Difference in energy levels between EC and EV No electrons (e-) in the bandgap (only above EC or below EV) EG = 1.12eV in Silicon

Ef

f(E) 1 0.5 E

• Virtually all of the

valence-band energy levels are filled with e-

• Virtually no e- in the

conduction band

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Ef is a function of the impurity-doping level

EC EV Ef

f(E) 1 0.5 E

n-Type Material

• High probability of a free e- in the conduction band
• Moving Ef closer to EC (higher doping) increases the number of available

majority carriers

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Ef is a function of the impurity-doping level

EC EV Ef p-Type Material

• Low probability of a free e- in the conduction band
• High probability of h+ in the valence band
• Moving Ef closer to EV (higher doping) increases the number of available

majority carriers

f(E) 1 0.5 E f(E) 1 0.5 E

( )

E f − 1

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• Applies to both electronic systems and

biological systems

• Look at drift and diffusion in silicon
• Assume 1-D motion

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%

Drift → Movement of charged particles in response to an external field (typically an electric field) E E-field applies force F = qE which accelerates the charged particle. However, the particle does not accelerate indefinitely because of collisions with the lattice (velocity saturation) Average velocity <vx> ≈ -µnEx electrons < vx > ≈ µpEx holes µn → electron mobility → empirical proportionality constant between E and velocity µp → hole mobility µn ≈ 3µp

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%

Drift → Movement of charged particles in response to an external field (typically an electric field) E-field applies force F = qE which accelerates the charged particle. However, the particle does not accelerate indefinitely because of collisions with the lattice (velocity saturation) Average velocity <vx> ≈ -µnEx electrons < vx > ≈ -µpEx holes µn → electron mobility → empirical proportionality constant between E and velocity µp → hole mobility µn ≈ 3µp Current Density

qpE J qnE J

p drift p n drift n

µ µ = =

, ,

q = 1.6×10-19 C, carrier density n = number of e- p = number of h+

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Diffusion → Motion of charged particles due to a concentration gradient

• Charged particles move in random directions
• Charged particles tend to move from areas of high concentration to areas of

low concentration (entropy – Second Law of Thermodynamics)

• Net effect is a current flow (carriers moving from areas of high concentration

to areas of low concentration)

( ) ( )

dx x dp qD J dx x dn qD J

p diff p n diff n

− = =

, ,

q = 1.6×10-19 C, carrier density D = Diffusion coefficient n(x) = e- density at position x p(x) = h+ density at position x → The negative sign in Jp,diff is due to moving in the opposite direction from the concentration gradient → The positive sign from Jn,diff is because the negative from the e- cancels out the negative from the concentration gradient

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Einstein Relation

Einstein Relation → Relates D and µ (they are not independent of each other)

q kT D = µ

UT= kT/q → Thermal voltage = 25.86mV at room temperature ≈ 25mV for quick hand approximations → Used in biological and silicon applications

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p-n Junctions (Diodes)

• Fundamental semiconductor device
• In every type of transistor
• Useful circuit elements (one-way valve)
• Light emitting diodes (LEDs)
• Light sensors (imagers)

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+ + + + + + + + + + + + + + + + + + + + + + + + – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – + + + + + + + + + + + + + + + + + + + + + + + + p-type n-type Bring p-type and n-type material into contact

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+ + + + + + + + + + + + + + + + + + + + + + + + – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – + + + + + + + + + + + + + + + + + + + + + + + + p-type n-type

• All the h+ from the p-type side and e- from the n-type side undergo diffusion

→ Move towards the opposite side (less concentration)

• When the carriers get to the other side, they become minority carriers
• Recombination → The minority carriers are quickly annihilated by the large number
• f majority carriers
• All the carriers on both sides of the junction are depleted from the material leaving
• Only charged, stationary particles (within a given region)
• A net electric field

This area is known as the depletion region (depleted of carriers) – – – – – – – – – Depletion Region – – – – – – – – – – – – – – – – – – – – – – – – + + + + + + + + + + + + + + + – – – – – – – – – – – – – – – – – – – – – – – – + + + + + + + + + + + + + + + – – – + + + + + + + + + + + + + + + + + + + + + + + + – – – – – – – – – – – – – – –

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+ + + + + + + + + + + + + + + + + + + + + + + + – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – + + + + + + + + + + + + + + + + + + + + + + + + p-type n-type – – – – – – – – – Depletion Region – – – – – – – – – – – – – – – – – – – – – – – – + + + + + + + + + + + + + + + – – – – – – – – – – – – – – – – – – – – – – – – + + + + + + + + + + + + + + + – – – + + + + + + + + + + + + + + + + + + + + + + + + – – – – – – – – – – – – – – –

ρ(x) qND

• qNA

x

Charge Density The remaining stationary charged particles results in areas with a net charge

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• Areas with opposing charge

densities creates an E-field

• E-field is the integral of the

charge density

• Poisson’s Equation

ε is the permittivity of Silicon

x ρ(x) qND

• qNA

Charge Density E x Electric Field

( )

ε ρ x dx dE =

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• x

ρ(x) qND

• qNA

Charge Density E x Electric Field

( )

x E dx d − = ψ

ψ x Φbi Potential

• E-field sets up a potential

difference

• Potential is the negative of the

integral of the E-field

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• x

ρ(x) qND

• qNA

Charge Density E x Electric Field ψ x Φbi Potential

• Line up the Fermi levels
• Draw a smooth curve to connect

them

EC Ef EV Band Diagram

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(

p n VA

EC Ef EV

p-type n-type

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() *

p n VA

If VA = 0

EC Ef EV EC Ef EV

• Any e- or h+ that wanders into the

depletion region will be swept to the other side via the E-field

• Some e- and h+ have sufficient

energy to diffuse across the depletion region

• If no applied voltage

Idrift = Idiff

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() "#

p n VA

If VA < 0

• Barrier is increased
• No diffusion current occurs (not

sufficient energy to cross the barrier)

• Drift may still occur
• Any generation that occurs inside

the depletion region adds to the drift current

• All current is drift current

Reverse Biased

EC Ef EV

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() +

p n VA

If VA < 0

• Barrier is reduced, so more e-

and h+ may diffuse across

• Increasing VA increases the e-

and h+ that have sufficient energy to cross the boundary in an exponential relationship (Boltzmann Distributions) →Exponential increase in diffusion current

• Drift current remains the same

Forward Biased

EC Ef EV

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(

( )

1 − =

T A nU

V

e I I

Combination of drift and generation Diffusion Drift

q kT UT =

→ Thermal voltage = 25.86mV    = 2 1 n

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(

( )

   − ≈ − = 1 I e I e I I

T A T A

nU V nU V

for VA > 0 for VA < 0

I

• I0

VA

( )

1 1 − = − =

T A T A

nU V nU V

e I I e I I

( )

( )

( )

( ) ( ) ( )

ln ln ln ln ln ln ln I nU V I I e I e I I

T A nU V nU V

T A T A

+ = + = =        

ln(I) ln(I0) VA

nkT q nUT = 1

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'#,!&$

T A nU

V

e I I ≈

Curve Fitting Exponential Data (In MATLAB)

• Given I and V (vectors of data)
• Use the MATLAB functions
• polyfit – function to fit a polynomial (find the coefficients)
• polyval – function to plot a polynomial with given coefficients and x values

[A] = polyfit(V,log(I),1); % polyfit(independent_var,dependent_var,polynomial_order) % A(1) = slope % A(2) = intercept [I_fit] = polyval(A,V); % draws the curve-fit line