Lecture 10Lecture 10

Lasers & the modern theory of solids

Laser:Light Amplification by Stimulated Emission of Light Amplification by Stimulated Emission of

Radiation

What makes a laser different from a lightbulb?

Laser:Light Amplification by Stimulated Emission of Light Amplification by Stimulated Emission of

Radiation

What makes a laser different from a lightbulb?• coherent photons (i.e., all have the same phase)

high radiation intensities high radiation intensities• all emitted photons have the same wavelength

Rule 1: Only certain optical transitions are allowed, based on conservation of momentum

2/1

Electron orbital angular momentum

2/11 L

mL If we apply a magnetic field along z, Bz then the angular momentum is:

mLz l l f l d b d

1 10 m

Selection rules for electromagnetic radiation absorption and emission (conservation of angular momentum)

1 10, mand

Allowed photon emission processes

Photon emission involves = 1 and m = 0, 1

Rule 2: Photons interact with photons via the following 3 processesg p

Absorption Spontaneous emission

Stimulated emission

The probability of absorption equals the probability of stimulated emissionp y

Absorption Stimulated emission

With a two level system can not achieve “population inversion” – i e more With a two level system, can not achieve population inversion – i.e., more electrons in the excited level E2

Rule 3: Lasers are ‘three-level’ systems

The principle of the LASER. (a) Atoms in the ground state are pumped up to the energy level E3 by incoming photons of energy hν 13 = E3-E1. (b) Atoms at E3 rapidly gy 3 by g p gy 13 3 1 (b) 3 p y

decay to the metastable state at energy level E2 by emitting photons or emitting lattice vibrations. hν32 = E3-E2.

Rule 3: Lasers are ‘three-level’ systems

(c) As the states at E2 are metastable, they quickly become populated and there is a population inversion between E2 and E1. (d) A random photon of energy hν21 = E2-E1 can

initiate stimulated emission. Photons from this stimulated emission can themselves further l l d l h f l d d h hstimulate emissions leading to an avalanche of stimulated emissions and coherent photons

being emitted.

HeNe laser operation

Helium Neon

The principle of operation of the HeNe laser and important HeNe laser energy levels (for 632.8 nm emission).

Schematic of a HeNe laser

HeNe lasers, from 1960 to today1960 1980

Today

Doppler effect: The observed photon frequency depends on whether the Ne atom

Laser output spectrumpp p q y pis moving towards (+vx) or away (‐vx) from the observer

vv xv1

vv xv101

c

vv 102

c

vv 101

vv xv02

Frequency width of the output spectrum is approximately ν2 – ν1

Laser cavity modes: Only certain wavelengths are allowed to exist within the optical cavity L If n is an integer the allowed wavelength λ is

c

Ln

optical cavity L. If n is an integer, the allowed wavelength λ is

From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw‐Hill, 2005)

Ln

2

Laser output spectrum

(a) Doppler-broadened emission versus wavelength characteristics of the lasing medium.(b) Allowed oscillations and their wavelengths within the optical cavity.(c) The output spectrum is determined by satisfying (a) and (b) simultaneously.

Recap So Far

Cl i l l i l & h l d i i lid• Classical electrical & thermal conduction in solids• Diffusion & doping

• Temperature dependence of resistivityp p y• Sensors (i.e., Hall sensors)

• Quantum theory of atomsQuantum theory of atoms•Photon and electron diffraction, the photoelectric effect, blackbody

radiationL•Lasers

• Interacting electrons and atoms solids• Band theory

• Quantum theory of conduction• Foundation for understanding semiconductors & semiconductor Foundation for understanding semiconductors & semiconductor

devices

Bonding & Antibonding orbitals

Two H atoms, both in their 1s state.As they approach, their wavefunctions begin to overlap.

Bonding & Antibonding orbitals

Formation of molecular orbitals - bonding and antibonding ( and *) when two H atomsapproach each other. The two electrons pair their spins and occupy the bonding orbital .

H-H bond: Electron probability distribution

(a) Electron probability distributions for bonding and antibonding orbitals, and *.

(b) Lines representing contours of constant probability (darker lines represent greater relative probability).

Linear combination of atomic orbitals

Two identical atomic orbitals 1s on atoms A and B can be combined linearly in two different ways to generate two

separate molecular orbitals and *p

and * generated from a li bi i f i bi l (LCAO)linear combination of atomic orbitals (LCAO)

Wavefunction around BWavefunction around A

)()( 11 BsAs rr )()( 11 BsAs

)()( 11* BsAs rr

)()( 11 BsAs

Energies using TISE

Energy of and * found using the time-independent Schrodinger equation (TISE) vs. the interatomic separation R.

Energies using TISE antibonding

bonding

Inset: Pictorial representation of “linear combination of atomic orbitals” (LCAO), showing the changes in the electron energy as two isolated H atoms, far left and far right, come together

to form a hydrogen molecule.

He-He bond

antibonding

b dbonding

Two He atoms have four electrons. When He atoms come together, two of the electrons enter the E level and two the E* level, so the overall energy is greater than two isolated He

atoms (since |Eantibonding|>>|Ebonding|).Therefore, He-He does not exist!

Hydrofluoric Acid

Hydrogen

H has one half-empty 1s orbital.

Hydrofluoric Acid

HydrogenFluorine

H (1s) has one half-empty 1s orbital.F (1s2 2s2 2p5) has one half-empty px orbital but full py and pz orbitals.

Hydrofluoric Acid

HydrogenFluorine HF

The overlap between 1s and px produces a bonding orbital and an antibondingorbital. The two electrons fill the bonding orbital and thereby form a covalent b d b t H d Fbond between H and F.

Hydrofluoric Etch of SiO2

Etching SiO2 in Etching SiO2 in integrated circuits &

MEMS

Hydrofluoric Etch of SiO2 - forming CaF2

Etching SiO2 in Etching SiO2 in integrated circuits &

MEMS

Hydrofluoric Etch

Acid-etched cameo glassEtching SiO2 in Acid-etched cameo glassEtching SiO2 in integrated circuits &

MEMS

Hydrofluoric Etch

Acid-etched cameo glassEtching SiO2 in Acid-etched cameo glassEtching SiO2 in integrated circuits &

MEMSDentistry

Raposo, J. Appl. Oral Sci. vol.17 no.2 Mar./Apr. 2009

Three-atom system – Band theory of solids

)()()( 111 CsBsAsa rrr

)()( 11 CsAsb rr

)()()( 111 CsBsAsc rrr

h l l b l f h b l l hThree molecular orbitals from three 1s atomic orbitals overlapping in three different ways.

Three-atom system: three energy levels

The energies of the three molecular orbitals, labeled a, b, and c, in a system with three H atoms.

N-atom system: N energy levels

The formation of 2s energy band from the 2s orbitals when N Li atoms (1s2 2s1) come together The formation of 2s energy band from the 2s orbitals when N Li atoms (1s 2s ) come together to form the Li solid. There are N 2s electrons, but 2N states in the band. The 2s band is

therefore only half full. The atomic 1s orbital is close to the Li nucleus and remains undisturbed in the solid.

Band theory of solids

As Li atoms are brought together from infinity, the atomic orbitals overlap and give rise to bands. Outer orbitals overlap first. The 3s orbitals give rise to the 3s band, 2p orbitals to the 2p

band, etc. The various bands overlap to produce a single band in which the energy is nearly p p g gy ycontinuous.

Note: We can no longer consider the electrons as belonging to specific atoms – they are shared among the entire solid.

Band theory of solids

In a metal the various energy bands overlap to give a single energy band In a metal, the various energy bands overlap to give a single energy band that is only partially full of electrons. There are states with energies up to

the vacuum level, where the electron is free.

Band theory of solids: The Fermi Level

T i l l t b d di f t l All th l l t Typical electron energy band diagram for a metal. All the valence electrons are in an energy band, which they only partially fill. The top of the band is

the vacuum level, where the electron is free from the solid (PE = 0).

Band theory of solids: Work-function redefined

The energy required to excite an electron from the Fermi levelto the vacuum level, that is, to liberate the electron from themetal, is called the work function of the metal.