EEE 3394 Electronic Materials Chris Ferekides Fall 2014 Week 8

Pauli Exclusion Principle and Hund’s Rule NO two electrons can have the same set of quantum numbers … i.e. if one electron in ψ1,0,0,1/2 then a second electron in the same system will have … ψ1,0,0,-1/2 Electrons in the same orbital “like” to have “parallel" spins …

Some possible states of the carbon atom, not in any particular order Some possible states of the carbon atom, not in any particular order. Arrange in the order of increasing energy …

LASERS u h E What does LASER stand for ? What is unique about LASER light ? E 1 2 h u ( a ) A b s o r p t i n S e m c l d I N O U T

LASERS What is a condition necessary for lasing ? “population inversion” Pumping: exciting e’s to a higher energy level Metastable state – “high” energy state where e’s can exist relatively longer before they can be “stimulated” to lower their energy and emit a photon E 1 3 2 h u C o e r n t p s O U T M a b l ( ) c d

H2 Molecule Two H atoms … Pauli Exclusion Principle What happens in the H2 molecule ? The 1s level splits into 2x …

Energy Bands What happens when we bring together N atoms … to form a solid ? The splitting of N levels leads to a large number of closely spaced 2N energy levels … that are so close together they can be considered as “continuous energy bands” From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

Metals – Energy Bands In metals these bands ovelap … with many ending up above vacuum … i.e. free e’s Inner/core levels also split … but spacing even close and completely filled! Of no interest right now … I n t e r a o m i c S p ( R ) E 2 s 3 1 = ¥ l d A T h V u L v F U M P Y g y

Metals – Energy Bands From now on … we will use “bands” even though we know that there are discrete energy levels … but they are VERY closely spaced that can be viewed as bands! 2 s p 3 O v e r l a i n g y b d E c t o V u m 1 S A = F

Metals – Energy Bands Energy distance from vacuum level to Fermi level is called the work function … recall from photo-electric effect … E l e c t r o n g y V a u m L v F B i s d h ­ 2 . 5 7 4 p b Where is the reference ?

Under an Applied E-Field x F B V ( ) - e n r g y b a d i m l c t o s p v Visualize momentum and energy changes … Energy band diagram … - under the E-field … we have a “distributed” V - which means that the electron Energy varies - electrons will try to move to lower energy states … E p x L a t i c e s r n g ­ v > - F O D = ¢ b l o m y

Under an Applied E-field Therefore … in the absence of an external force … random e motion … leads to ZERO current … (Net momentum ??) Under an applied E-field … the e-momentum is non-zero … and we have net electron flow i.e. current. As electrons gain energy they start “moving” … during this movement they lose energy via collisions with lattice In order for electrons to move “up” or down in energy … there must be an empty energy “state” for them to move into E = p^2 / 2m

Semiconductors – Si Si 14 electrons

Semiconductors – Energy Bands In semiconductors the interaction between N atoms … and the overlap of their orbitals … leads to similar “bands” i.e. MANY closely spaced discrete levels BUT these do not overlap … they are separated by a forbidden gap … the Energy Gap … 2 s p 3 O v e r l a i n g y b d E c t o V u m 1 S A = F S i A T O M y B h b C N D U I V L E n e r g a p , 3 s R Y Si 14 electrons

Semiconductors – Energy Bands At 0K “Top” band: Conduction Band - empty “Bottom” band: Valence Band - filled Energy space between C and V bands: Energy Gap (bandgap) Electrons can move into an empty state in the C-band … leaving behind an empty state - HOLE

Energy Bands – Overview Energy Band Model: - Top band EC: Conduction Band (contains “free” electrons) - “Bottom” Band EV: Valence Band (that’s where we find “Holes) - Space between EC and EV known as the ENERGY GAP or Bandgap EG (no electron states, for “pure” material); Note: at 0ºK the valence band is completely filled and the conduction band completely empty. Conductors – Insulators - Semiconductors

Effective Mass EFFECTIVE MASS Electrons in vacuum: mo Under an applied electric field: Electrons (conduction band electrons) in a semiconductor under an applied electric field: above equation does not describe motion of electrons in a semiconductor; collisions between electrons and atoms decelerate the carriers. electrons (in a crystal) are also affected by complex crystalline fields. QUANTUM MECHANICS must be used to describe the motion of carriers in a crystal. Quantum mechanical treatment leads to a simplified equation similar to the one above with mo replaced by m*: effective mass. NOTE: both conceptually and mathematically, electrons and holes can be treated as classical particles, as long as the carrier effective mass replaces the particle mass in mathematical relationships.

Density of States There is a specific “seating” arrangement for the electrons in the energy bands … There is a limited number of states and they have a certain distribution … … much like a seating arrangement in a stadium … or an auditorium Density of states g(E): the number of available electronic states (?) per unit volume per unit energy ! GET used to dealing with “per unit volume” units! … charge carriers … electronic states etc. IF one needs to calculate the TOTAL number of states from energy E1 to energy E2 … then, must integrate g(E) from E1 to E2

Density of States in a Metal Density of states g(E): the number of available electronic states (?) per unit volume per unit energy !

Statistics - Boltzmann Boltzman Probability Function … Is the probability of finding a particle at an energy E… … “good” in situations where there is NO restriction that only a number of particles can occupy certain states etc. … “good” where high density of states i.e. MANY electronic states and few electrons (particles) … “good” for classical particles … for semiconductors it is “valid” in the conduction band where the number of electrons << number of states

Statistics - Fermi Fermi Probability Function … Is the probability of finding a particle at an energy E… with “reference” EF the Fermi Energy … “good” in situations where states can be occupied with one particle only … approximated by Boltzman for energies “far from” EF

Boltzmann vs. Fermi Dirac 1 / 2 f ( ) T = > µ e x p ( ­ E / k T ) N 1 2

How Would you Calculate n ?

Calculating n: Electron Concentration Density of states g(E): the number of available electronic states Fermi Function f(E): probability a state is occupied by an electron … therefore the product of g and f … gives us ?? E 1 / 2 f ( ) T K g = A n F + dE ò ¥

Calculating n: Electron Concentration n is fixed … and is a function of T and EF … therefore ??

T-Dependence of the Fermi Energy … to calculate the average energy of the electrons … … and the results are ! …

T-Dependence of the Fermi Energy … the temperature dependence is weak and for all practical purposes … … and since electrons are “free” in a metal … then … therefore the electron average speed is …