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Lecture 4: Equilibrium Carrier StatisticsECE5590: Nanoscale Devices and circuitsMostafizur Rahmanrahmanmo@umkc.edu11ECE5590 Fall 2015 MRRecapECE5590 Fall 2015 MR2

ECE 663So far, we saw how to calculate bands for solids

Kronig-Penny was a simple example

Real bandstructures more complex

Often look like free electrons with effective mass m*

Recap3OutlineDensity of StatesFermi Dirac StatisticsDopingConclusions

ECE5590 Fall 2015 MR4Electrons and holes are called carriers because they are charged particles when they move, they carry current

Equilibrium is the condition that prevails when the semiconductor is left unperturbed from external stimuli (i.e., voltage, light, etc)

Carrier populations depend on Number of available energy states (density of states)statistical distribution of energies (Fermi-Dirac function)

ECE 663Equilibrium Carrier Statistics5Density of StatesECE5590 Fall 2015 MR6

How many states are there in a band? N

How many are occupied upto E?Or How many states per unit Energy? (DOS)Density of StatesECE5590 Fall 2015 MR7dEk

dEdkxxxxxxxxxE/a

2/NaDensity of StatesECE5590 Fall 2015 MR8

dEk

dEdkxxxxxxxxxEPi/aECE 663dEk

dE

dkxxxxxxxxxEkFor 1D parabolic bands, DOS peaks at edgesGraphical RepresentationAs you go up, DOS tapers downECE5590 Fall 2015 MR91D-DOSECE5590 Fall 2015 MR10

M. Alam, ECE 606 Purdue3D-DOSECE5590 Fall 2015 MR11

3-D DOSECE5590 Fall 2015 MR12

DOS for Real Semiconductor (GaAs)ECE5590 Fall 2015 MR13

M. Alam, ECE 606 PurdueDOS for Real Semiconductor (GaAs)ECE5590 Fall 2015 MR14

DOS for Real CrystalsECE5590 Fall 2015 MR15

M. Alam, ECE 606 PurdueOutlineDensity of StatesFermi Dirac StatisticsDopingConclusions

ECE5590 Fall 2015 MR16What is Fermi-Dirac Function?Find number of carriers in CB/VB - need to knowNumber of available energy states (g(E))Probability that a given state is occupied (f(E))

ECE5590 Fall 2015 MR17

ECE 663Fermi-Dirac FunctionFermi-Dirac function derived from statistical mechanics of free particles with three assumptions:Pauli Exclusion Principle each allowed state can accommodate only one electronThe total number of electrons is fixed N=NiThe total energy is fixed ETOT = EiNi 18The 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 interestEfFermi levelHalfway pointWhere f(E) = 0.5kBoltzmann constant=1.3810-23 J/K=8.61710-5 eV/KTAbsolute temperature (in Kelvins)

19The Fermi function at three different temperatures.ECE5590 Fall 2015 MR20

http://ecee.colorado.edu/~bart/book/book/chapter2/ch2_5.htmThe Fermi-Dirac distribution function, also called Fermi function, provides the probability of occupancy of energy levels by Fermions. Fermions are half-integer spin particles, which obey the Pauli exclusion principle. The Pauli exclusion principle postulates that only one Fermion can occupy a single quantum state. Therefore, as Fermions are added toan energy band, they will fill the available states in an energy band just like water fills a bucket. The states with the lowest energy are filled first, followed by the next higher ones. At absolute zero temperature (T= 0 K), the energy levels are all filled up to a maximum energy, which we call the Fermi level. No states above the Fermi level are filled. At higher temperature, one finds that the transition between completely filled states and completely empty states is gradual rather than abrupt.Electrons are Fermions. Therefore, the Fermi function provides the probability that an energy level at energy,E, in thermal equilibrium with a large system, is occupied by an electron. The system is characterized by its temperature,T, and its Fermi energy,EF. The Fermi function is given by:The Fermi function has a value of one for energies, which are more than a few timeskTbelow the Fermi energy. It equals 1/2 if the energy equals the Fermi energy and decreases exponentially for energies which are a few timeskTlarger than the Fermi energy. While atT= 0 K the Fermi function equals a step function, the transition is more gradual at finite temperatures and more so at higher temperatures.ECE5590 Fall 2015 MR20ECE 663Carrier ConcentrationsNumber of electrons in conduction band

And number of holes in valence band

El. DensityStateDensityOccupancyper state21Carrier ConcentrationECE5590 Fall 2015 MR22

M. Alam, ECE 606 PurdueFermi Dirac StatisticsECE5590 Fall 2015 MR23

ECE5590 Fall 2015 MR24

Boltzmann DistributionFermi dirac integral of order half can be approximated by exp()ECE5590 Fall 2015 MR25

Effective Density of StatesECE5590 Fall 2015 MR26

OutlineDensity of StatesFermi Dirac StatisticsDopingConclusions

ECE5590 Fall 2015 MR27Intrinsic Semiconductor

Silicon has 4 outer shell / valence electrons

Forms into a lattice structure to share electrons28Intrinsic Silicon

ECEVThe valence band is full, and no electrons are free to move about

However, at temperatures above T=0K, thermal energy shakes an electron free29Semiconductor Properties

For T > 0KElectron shaken free and can cause current to floweh+Generation Creation of an electron (e-) and hole (h+) pairh+ 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 otherTherefore, 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)

30DopingDoping Adding impurities to the silicon crystal lattice to increase the number of carriersAdd a small number of atoms to increase either the number of electrons or holes31Periodic Table

Column 4 Elements have 4 electrons in the Valence ShellColumn 3 Elements have 3 electrons in the Valence ShellColumn 5 Elements have 5 electrons in the Valence Shell32

Donors n-Type MaterialDonates an extra e- that can freely travel aroundLeaves behind a positively charged nucleus (cannot move)Overall, the crystal is still electrically neutralCalled n-type material (added negative carriers)Mobility?+33Acceptors Make p-Type Material

h+Add atoms with only 3 valence-band electrons ex. Boron (B)Accepts e and provides extra h+ to freely travel aroundLeaves behind a negatively charged nucleus (cannot move)Overall, the crystal is still electrically neutralCalled p-type silicon (added positive carriers)Mobility?34Band Diagrams (Revisited)EgECEVEC 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

Virtually all of the valence-band energy levels are filled with e-Virtually no e- in the conduction band35Effect of Doping on Fermi LevelEf is a function of the impurity-doping levelECEVEf

High probability of a free e- in the conduction bandMoving Ef closer to EC (higher doping) increases the number of available majority carriers36Effect of Doping on Fermi LevelEf is a function of the impurity-doping levelECEVEfLow probability of a free e- in the conduction bandHigh probability of h+ in the valence bandMoving Ef closer to EV (higher doping) increases the number of available majority carriers

Dr. Ethan Farquhar, university of tennassee37Revisiting Boltzmann Distribution Fermi level within 3KT: degenerateECE5590 Fall 2015 MR38

Intrinsic Semiconductor (Law of Mass Action)ECE5590 Fall 2015 MR39

n = p = ni

ECE 663Charge Neutrality

Poissons Equation relating charge density to electric fieldIn equilibrium, E=0 and =0

Charge NeutralityRelationship--40SummaryDOS allows us to know possible states

The Fermi-Dirac distribution helps us fill these states

For non-degenerate semiconductors, we get simple formulae for n and p at equilibrium in terms of Ei and EF, with EF determined by doping

Next-> non equilibrium states

Suggested Reading: Chapter 4 (Advanced Semiconductor Fundamentals)

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