Energy Band Diagrams

Overview of Silicon Semiconductor Device Physics

Dr. David W. Graham

1 Silicon

Silicon is the primary semiconductor used in VLSI systems

Si has 14 Electrons

Energy Bands (Shells) Valence Band

Nucleus

At T=0K, the highest energy band occupied by Silicon has 4 outer shell / an electron is valence electrons called the valence band. 2 Energy Bands

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

Energy Bands

3 Energy Bands

EC Conduction Band First unfilled energy band at T=0K E Energy g Bandgap

EV Valence Band Last filled energy band at T=0K

4 Band Diagrams

Increasing electron energy EC

EV Increasing voltage Band Diagram Representation Energy plotted as a function of position

EC Æ Conduction band Æ Lowest energy state for a free electron

EV Æ Valence band Æ Highest energy state for filled outer shells

EG Æ Band gap Æ Difference in energy levels between EC and EV - Æ No electrons (e ) in the bandgap (only above EC or below EV) Æ EG = 1.12eV in Silicon

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

However, at temperatures above T=0K, thermal energy shakes an electron free

7 Semiconductor Properties

For T > 0K Electron shaken free and can • Generation – Creation of an electron (e-) cause current to flow 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

h+ e– • 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)

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 Periodic Table

Column 3 Column 4 Elements have 3 Elements have 4 electrons in the electrons in the Valence Shell 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) • “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] ~10 15-10 20cm-3

• e- is free to move about the crystal 2 (Mobility μn ≈1350cm /V)

11 Donors n-Type Material

Donors n-Type Material • 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 Shorthand Notation • Called “n-type” material (added + Positively charged ion; immobile negative carriers) – Negatively charged e-; mobile;

• ND = the concentration of donor Called “majority carrier” atoms [atoms/cm3 or cm-3] + Positively charged h+; mobile; ~10 15-10 20cm-3 Called “minority carrier”

• e- is free to move about the crystal 2 (Mobility μn ≈1350cm /V)

12 Acceptors Make p-Type Material

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 h+ 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, 2 so mobility is low, μp ≈ 500cm /V)

13 Acceptors Make p-Type Material

p-Type Material 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 Shorthand Notation electrically neutral – Negatively charged ion; immobile • Called “p-type” silicon (added + Positively charged h+; mobile; positive carriers) Called “majority carrier” • N = the concentration of acceptor – Negatively charged e-; mobile; A atoms [atoms/cm3 or cm-3] Called “minority carrier” • Movement of the hole requires

breaking of a bond! (This is hard, 2 so mobility is low, μp ≈ 500cm /V)

14 The Fermi Function

The Fermi Function • Probability distribution function (PDF) • The probability that an available state at f(E) an energy E will be occupied by an e- 1 1 ()Ef = ()− f kTEE 1+ e 0.5

E Æ Energy level of interest E Æ Fermi level f E Æ Halfway point f E Æ Where f(E) = 0.5 k Æ Boltzmann constant = 1.38×10-23 J/K = 8.617×10-5 eV/K T Æ Absolute temperature (in Kelvins)

15 Boltzmann Distribution

>>− kTEE If f f(E) Then 1 ()≈ eEf ( −− f ) kTEE 0.5 Boltzmann Distribution • Describes exponential decrease in the density of particles in thermal equilibrium with a potential gradient Ef E • Applies to all physical systems • Atmosphere Æ Exponential distribution of gas molecules ~E -4kT ~E +4kT • Electronics Æ Exponential distribution of electrons f f • Biology Æ Exponential distribution of ions

16 Band Diagrams (Revisited)

EC Eg Ef

EV Band Diagram Representation 0.5 1 f(E) Energy plotted as a function of position

EC Æ Conduction band Æ Lowest energy state for a free electron Æ Electrons in the conduction band means current can flow

EV Æ Valence band • Virtually all of the Æ Highest energy state for filled outer shells valence-band energy Æ Holes in the valence band means current can flow levels are filled with e- - Ef Æ Fermi Level • Virtually no e in the Æ Shows the likely distribution of electrons conduction band

EG Æ Band gap Æ Difference in energy levels between EC and EV - Æ No electrons (e ) in the bandgap (only above EC or below EV) Æ EG = 1.12eV in Silicon

17 Effect of Doping on Fermi Level

Ef is a function of the impurity-doping level

n-Type Material

EC Ef

EV 0.5 1 f(E)

• High probability of a free e- in the conduction band • Moving E closer to E (higher doping) increases the number of available f C majority carriers

18 Effect of Doping on Fermi Level

Ef is a function of the impurity-doping level

p-Type Material 1− (Ef ) E EC

Ef EV 0.5 1 f(E)

• Low probability of a free e- in the conduction band • High probability of h+ in the valence band

• Moving E closer to E (higher doping) increases the number of available f V majority carriers

19 Equilibrium Carrier Concentrations

n = # of e- in a material + p = # of h in a material

- ni = # of e in an intrinsic (undoped) material

Intrinsic silicon • Undoped silicon • Fermi level • Halfway between Ev and Ec • Location at “Ei”

EC Eg Ef

EV 0.5 1 f(E)

20 Equilibrium Carrier Concentrations

Non-degenerate Silicon • Silicon that is not too heavily doped • Ef not too close to Ev or Ec Assuming non-degenerate silicon

( − if ) kTEE = ienn

()− fi kTEE = ienp Multiplying together

2 = nnp i

21 Charge Neutrality Relationship

• For uniformly doped semiconductor • Assuming total ionization of dopant atoms

− + − NNnp AD = 0

# of carriers # of ions

Total Charge = 0 Electrically Neutral

22 Calculating Carrier Concentrations

• Based upon “fixed” quantities

• N A, ND, ni are fixed (given specific dopings for a material) • n, p can change (but we can find their equilibrium values) 1 ⎡ 2 ⎤ 2 − AD ⎛ − NNNN AD ⎞ 2 n = + ⎢⎜ ⎟ + ni ⎥ 2 ⎣⎢⎝ 2 ⎠ ⎦⎥ 1 ⎡ 2 ⎤ 2 − DA ⎛ − NNNN DA ⎞ 2 p = + ⎢⎜ ⎟ + ni ⎥ 2 ⎣⎢⎝ 2 ⎠ ⎦⎥ n 2 = i n 23 Common Special Cases in Silicon

1. Intrinsic semiconductor (NA = 0, ND = 0) 2. Heavily one-sided doping 3. Symmetric doping

24 Intrinsic Semiconductor (NA=0, ND=0)

Carrier concentrations are given by

= nn i

= np i

== npn i

25 Heavily One-Sided Doping

>>≈− nNNN iDAD

>>≈− nNNN iADA This is the typical case for most semiconductor applications

If D >> , >> nNNN iDA (Nondegenerate, Total Ionization)

Then ≈ Nn D n 2 p ≈ i N D

If A >> , >> nNNN iAD (Nondegenerate, Total Ionization)

Then ≈ Np A n 2 n ≈ i N A

26 Symmetric Doping

Doped semiconductor where ni >> |ND-NA|

• Increasing temperature increases the number of intrinsic carriers • All semiconductors become intrinsic at sufficiently high temperatures

≈ ≈ npn i

27 Determination of Ef in Doped Semiconductor

⎛ N ⎞ ⎜ D ⎟ if =− kTEE ln⎜ ⎟ for D >> , >> nNNN iDA ⎝ ni ⎠ ⎛ N ⎞ ⎜ A ⎟ for fi =− kTEE ln⎜ ⎟ A >> , >> nNNN iAD ⎝ ni ⎠

Also, for typical semiconductors (heavily one-sided doping)

⎛ n ⎞ ⎛ p ⎞ ⎜ ⎟ ⎜ ⎟ [units eV] if =− kTEE ln⎜ ⎟ −= kT ln⎜ ⎟ ⎝ ni ⎠ ⎝ ni ⎠

28 Thermal Motion of Charged Particles

• Look at drift and diffusion in silicon • Assume 1-D motion • Applies to both electronic systems and biological systems

29 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

≈ -µnEx electrons < vx > ≈ µpEx holes

µ → electron mobility n → empirical proportionality constant between E and velocity

µp → hole mobility E µ ≈ 3µ µ↓ as T↑ n p

30 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 Current Density indefinitely because of collisions with the lattice (velocity saturation) J ,driftn = μnqnE Average velocity ≈ -µnEx electrons J ,driftp = μ pqpE < vx > ≈ µpEx holes

µ → electron mobility q = 1.6×10-19 C, carrier density n → empirical proportionality constant n = number of e-

between E and velocity p = number of h+

µp → hole mobility

µ ≈ 3µ µ↓ as T↑ n p

31 Resistivity

• Closely related to carrier drift • Proportionality constant between electric field and the total particle current flow 1 ρ = where q ×= 10602.1 −19 C ()+ μμpn pnq n-Type Semiconductor p-Type Semiconductor 1 1 ρ = ρ = μ Nq Dn μ Nq Ap • Therefore, all semiconductor material is a resistor – Could be parasitic (unwanted) – Could be intentional (with proper doping)

• Typically, p-type material is more resistive than n-type material for a given amount of doping

• Doping levels are often calculated/verified from resistivity measurements

32 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)

(xdn ) q = 1.6×10-19 C, carrier density ,diffn = qDJ n dx D = Diffusion coefficient

()xdp n(x) = e- density at position x + ,diffp −= qDJ p p(x) = h density at position x dx

→ 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

33 Total Current Densities

Summation of both drift and diffusion

= ,driftnn + JJJ ,diffn ()xdn μ += qDqnE (1 Dimension) n n dx (3 Dimensions) μn n∇+= nqDqnE

,driftpp += JJJ ,diffp ()xdp μ −= qDqpE (1 Dimension) p p dx

μ p p∇−= pqDqpE (3 Dimensions)

Total current flow

= + JJJ pn

34 Einstein Relation

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

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

35 Changes in Carrier Numbers

Primary “other” causes for changes in carrier concentration • Photogeneration (light shining on semiconductor) • Recombination-generation

Photogeneration ∂n ∂p = = G Photogeneration rate ∂t ∂t L light light

Creates same # of e- and h+

36 Changes in Carrier Numbers

Indirect Thermal Recombination-Generation ∂p Δ− p = + h in n-type material n0, p0 Æ equilibrium carrier concentrations ∂t −GR τ p n, p Æ carrier concentrations under arbitrary conditions

∂n Δ− n Δn, Δp Æ change in # of e- or h+ from = e- in p-type material equilibrium conditions ∂t −GR τ n

Assumes low-level injection

<<Δ 0 , ≈ nnnp 0 in type-n material

<<Δ 0 , ≈ pppn 0 in type-p material

37 Minority Carrier Properties

Minority Carriers • e- in p-type material + • h in n-type material Minority Carrier Lifetimes

• τn Æ The time before minority carrier electrons undergo recombination in p-type material

• τp Æ The time before minority carrier holes undergo recombination in n-type material

Diffusion Lengths • How far minority carriers will make it into “enemy territory” if they are injected into that material

- n = DL τ nn for minority carrier e in p-type material

+ p = DL τ pp for minority carrier h in n-type material

38 Equations of State

• Putting it all together • Carrier concentrations with respect to time (all processes)

• Spatial and time continuity equations for carrier concentrations

∂n ∂n ∂n ∂n ∂n = + + + ∂t ∂t ∂t ∂t ∂t other drift diff −GR light)( 1 ∂n ∂n J n +⋅∇= + q ∂t ∂t other 14243 −GR light)( Current to Related to Current

∂p ∂p ∂p ∂p ∂p = + + + ∂t ∂t ∂t ∂t ∂t other drift diff −GR light)( 1 ∂p ∂p J p +⋅∇−= + q ∂t ∂t other 142 43 −GR light)( Current to Related to Current

39 Equations of State

Minority Carrier Equations • Continuity equations for the special case of minority carriers • Assumes low-level injection

2 Δ∂ np Δ∂ np Δnp = Dn 2 − + GL ∂t ∂x τ n

Light generation

Indirect thermal recombination

∂n 1 ∂Jn J, assuming no E-field qDn and also →⋅∇ DJ nn ∂x q ∂x

2 Δ∂ pn Δ∂ p Δpnn = Dn 2 − + GL ∂t ∂x τ p

np, pn Æ minority carriers in “other” type of material