Dr. David W. Graham West Virginia University Lane Department of Computer Science and Electrical Engineering 1

Silicon Silicon is the primary semiconductor used in VLSI systems Si has 14 Electrons Energy Bands (Shells) Valence Band Nucleus Silicon has 4 outer shell / valence electrons At T=0K, the highest energy band occupied by an electron is called the valence band.
2

Energy Bands Increasing Electron Energy Energy Bands } Disallowed Energy States } Allowed Energy States • • • Electrons try to occupy the lowest energy band possible Not every energy level is a legal state for an electron to occupy These legal states tend to arrange themselves in bands 3

E C E V Energy Bands E g Energy Bandgap
Conduction Band
First unfilled energy band at T=0K
Valence Band
Last filled energy band at T=0K 4

Band Diagrams E C E V Band Diagram Representation Energy plotted as a function of position E g Increasing electron energy Increasing voltage E C  Conduction band  Lowest energy state for a free electron E V  Valence band  Highest energy state for filled outer shells E G  Band gap  Difference in energy levels between E C and E V  No electrons (e ) in the bandgap (only above E C  E G = 1.12eV in Silicon or below E V ) 5

Intrinsic Semiconductor Silicon has 4 outer shell / valence electrons Forms into a lattice structure to share electrons 6

Intrinsic Silicon The valence band is full, and no electrons are free to move about E C E V However, at temperatures above T=0K, thermal energy shakes an electron free 7

Semiconductor Properties For T > 0K Electron shaken free and can cause current to flow h + e
–
• • • • • 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 + when they move, they carry current come in contact, they annihilate each other Electrons and holes are called “carriers” because they are charged particles – 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) 8

Doping • 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 9

Column 3 Elements have 3 electrons in the Valence Shell Periodic Table Column 4 Elements have 4 electrons in the Valence Shell Column 5 Elements have 5 electrons in the Valence Shell 10

Donors n-Type Material • • • • • • • • 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) N D = the concentration of donor atoms [atoms/cm 3 or cm -3 ] ~10 15 -10 20 cm -3 e is free to move about the crystal (Mobility m n ≈1350cm 2 /V) + 11

Donors n-Type Material • • • • • • • • 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) N D = the concentration of donor atoms [atoms/cm 3 or cm -3 ] ~10 15 -10 20 cm -3 e is free to move about the crystal (Mobility m n ≈1350cm 2 /V) n-Type Material + – + – + – + + – + – + – + – + + – + + – – + – + – – + + – + – + – + – + – + Shorthand Notation Positively charged ion; immobile Negatively charged e-; mobile; Called “majority carrier” Positively charged h+; mobile; Called “minority carrier” 12

Acceptors Make p-Type Material –– 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) N A = the concentration of acceptor atoms [atoms/cm 3 or cm -3 ] Movement of the hole requires breaking of a bond! (This is hard, so mobility is low, μ p ≈ 500cm 2 /V) 13

Acceptors Make p-Type Material p-Type Material – – + + – + – – + – – + + – + – – – + + – – + + – + + – – – + + – + – + – + – Shorthand Notation Negatively charged ion; immobile Positively charged h+; mobile; Called “majority carrier” Negatively charged e-; mobile; Called “minority carrier” • • • • • • • • 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) N A = the concentration of acceptor atoms [atoms/cm 3 or cm -3 ] Movement of the hole requires breaking of a bond! (This is hard, so mobility is low, μ p ≈ 500cm 2 /V) 14

The Fermi Function The Fermi Function • Probability distribution function (PDF) • The probability that an available state at an energy E will be occupied by an e -
f
 1 
e

E
1 
E f

kT
E  Energy level of interest E k f  Fermi level  Halfway point  Where f(E) = 0.5
 Boltzmann constant = 1.38
×10 -23 J/K = 8.617
×10 -5 eV/K T  Absolute temperature (in Kelvins) f(E) 1 0.5
E f E 15

Boltzmann Distribution If
E

E f

kT
Then
f

e
 
E

E f

kT
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 f(E) 1 0.5
~Ef - 4kT E f ~Ef + 4kT E 16

Band Diagrams (Revisited) E C E g E f E V Band Diagram Representation Energy plotted as a function of position E C E V E f E G  Conduction band  Lowest energy state for a free electron  Electrons in the conduction band means current can flow  Valence band  Highest energy state for filled outer shells  Holes in the valence band means current can flow  Fermi Level  Shows the likely distribution of electrons  Band gap  Difference in energy levels between E C and E V  No electrons (e ) in the bandgap (only above E C  E G = 1.12eV in Silicon or below E V ) E 0.5
1 f(E) • • Virtually all of the valence-band energy levels are filled with e Virtually no e in the conduction band 17

Effect of Doping on Fermi Level E f is a function of the impurity-doping level n-Type Material E E C E f E V 0.5
1 f(E) • • High probability of a free e in the conduction band Moving E f closer to E C majority carriers (higher doping) increases the number of available 18

Effect of Doping on Fermi Level E f is a function of the impurity-doping level p-Type Material 1 
f
E C E f E V • • • Low probability of a free e High probability of h + in the conduction band in the valence band Moving E f closer to E V majority carriers (higher doping) increases the number of available 19

Thermal Motion of Charged Particles • • • Applies to both electronic systems and biological systems Look at drift and diffusion in silicon Assume 1-D motion 20

Drift 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
≈ -µ n E x < v x > ≈ µ p E x
electrons holes
µ n µ p
→ electron mobility → empirical proportionality constant between E and velocity → hole mobility
µ n ≈ 3µ p
E 21

Drift 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
≈ -µ n E x < v x > ≈ -µ p E x
electrons holes
µ n µ p
→ electron mobility → empirical proportionality constant between E and velocity → hole mobility
µ n ≈ 3µ p
Current Density
J n
,
drift J p
,
drift
  m
n qnE
m
p qpE q = 1.6×10 -19
C, carrier density n = number of e p = number of h + 22

Diffusion • • • 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)
J n
,
diff

qD n dn
 
dx J p
,
diff
 
qD p dp dx 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 J
p,diff
is due to moving in the opposite direction from the concentration gradient → The positive sign from J
n,diff
is because the negative from the e cancels out the negative from the concentration gradient 23

Einstein Relation Einstein Relation → Relates D and µ (they are not independent of each other)
D
m 
kT q U T = kT/q
→ Thermal voltage = 25.86mV at room temperature ≈ 25mV for quick hand approximations → Used in biological and silicon applications 24

p-n Junctions (Diodes) • • • • • 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) 25

p-n Junctions (Diodes) + – + – + – + – + – + – + – + – + – + – + – + – + – + – + – – + – + – + p-type + – + – + – – + – + – + n-type + – + – + – + – + – + – – + – + – + + – + – + – + – + – + – + – + – + – + – + – + – + – + – + – Bring p-type and n-type material into contact 26

p-n Junctions (Diodes) p-type + + + + + + + + + – + – + – + – + – + – n-type – + – + – + – + – + – + – + – + – + – + – + – + – + – + – + – Depletion Region • • • • 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 of 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) 27

Charge Density p-type + + + + + + + + + – + – + – + – + – + – n-type – + – + – + – + – + – + – + – + – + – + – + – + – + – + – + – Depletion Region  (x) qN D x -qN A The remaining stationary charged particles results in areas with a net charge 28

 (x) qN D -qN A E Electric Field x x • Areas with opposing charge densities creates an E-field • E-field is the integral of the charge density • Poisson’s Equation
dE

dx
  ε is the permittivity of Silicon 29

 (x) qN D -qN A E   bi Potential x x x • E-field sets up a potential difference • Potential is the negative of the integral of the E-field
d

dx
 
E
30

 (x) qN D -qN A E E C E f E V   bi Band Diagram x x x • Line up the Fermi levels • Draw a smooth curve to connect them 31

E C E f E V p-type p-n Junction Band Diagram VA p n n-type 32

E C E f E V p-n Junction – No Applied Bias VA If V
A = 0
p n • 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
I drift = I diff
33

E C E f E V p-n Junction – Reverse Biased VA If V
A < 0
p n Reverse Biased • 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 34

E C E f E V p-n Junction – Forward Biased VA If V
A < 0
p n Forward Biased • Barrier is reduced, so more e and h + may diffuse across • Increasing V
A
and h + increases the e that have sufficient energy to cross the boundary in an exponential relationship (Boltzmann Distributions) →Exponential increase in diffusion current • Drift current remains the same 35

p-n Junction Diode
I

I
0 
e V A nU T
 1  Diffusion Drift Combination of drift and generation
U T

kT q
→ Thermal voltage = 25.86mV
n
 1  2 36

p-n Junction Diode I
I

I
0 
e V A nU T
 1     
I
0
e V A

I
0
nU T
ln(I) for V
A > 0
for V
A < 0
1
nU T

q nkT
-I 0
I

I
0 
e V A nU T
 1 
I I
0 
e V A nU T
 1 V A ln(I 0 ) V A ln  
I I
0 ln    ln  ln 
e V A

e V A nU T nU T
   ln   0 ln 
V A nU T
 ln   0 37

Curve Fitting Exponential Data (In MATLAB) Curve Fitting Exponential Data (In MATLAB)
I

I
0
e V A nU T
• 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 38