Review of Optics Absorption in Semiconductors

Review of Optical Properties of Materials

Review of optics

Absorption in semiconductors: qualitative discussion

Derivation of Optical Absorption Coefficient in Direct Semiconductors Photons • When dealing with events on the atomic scale, it is often best to regard light as composed of quasi- particles PHOTONS Photons are Quanta of light Electromagnetic radiation is quantized & occurs in finite "bundles" of energy  Photons – The energy of a single photon in terms of its frequency , or wavelength  is,

Eph = h = (hc)/ Maxwell – Electromagnetic Waves Light as an Electromagnetic Wave

• Light as an electromagnetic wave is characterized by a combination of a time-varying electric field (E) & magnetic field (H) propagating through space. • Maxwell’s equations give the result that both E & H satisfy the same wave equation:

2 2 1    EH,,22  EH ct  

Changes in the fields propagate through space with speed c. Speed of Light, c • Frequency of oscillation, of the fields and their wavelength, o in vacuum are related by; – c = o • In any other medium the speed, v is given by; – v= c/n =  • n = refractive index of the medium •  = wavelength in the medium

• And, n  rr • r = relative magnetic permeability of the medium • r = relative electric permittivity of the medium

The speed of light in a medium is related to the electric and magnetic properties of the medium, and the speed of light can be expressed as Electromagnetic Spectrum

Shorter wavelength

Larger Photon Energy (eV)

Longer wavelength Interaction Between Light & Bulk Material Many different possible processes can occur!

Scattering 3c “Semi-transparent” material 1- Refraction Incident light 2- Transmission 4 3a – Specular reflection 1 3b – Total internal reflection 3a 3b 3c – Diffused reflection 4 Dispersion –where 2 different colors bend differently Refraction, Reflection and Dispersion

Light when it High n

travels in a Small n medium can be absorbed and reemitted by every n1 = refractive index of atom in its path. material 1 n = refractive index of Determined by refractive index; n 2 material 2 Total Internal Reflection

Transmitted (refracted) light

k t t n 2 Evanescent wave n > n 1 2 i  c c  > k i i k r i c TIR Incident Reflected light light (a) (b) (c) Light wave travelling in a more dense medium strikes a less dense medium. Depending on the incidence angle with respect to  c, which is determined by the ratio of the refractive indices, the wave may be transmitted (refracted) or reflected. (a) i < c (b)  i = c (c) i >  c and total internal reflection (TIR).

Optical fibre for communication

What sort of materials do you think are suitable for fibre optics cables? Review of optical processes

Energy levels are everything in quantum mechanics.

Excited level

 E = h Energy Ground level

The atom is vibrating The atom is at least partially in at frequency, . an excited state. Review of optical processes

Before After

Spontaneous emission

Absorption

Stimulated emission • Recall: Semiconductor Bandgaps Eg are usually in the range: 0 < Eg < 3 eV (up to 6 eV if diamond is included) • Also, at equilibrium, at temperature T = 0, the valence band is full & the conduction band is empty. • Now, consider what happens if electromagnetic radiation (“light”) is shined on the material. • In the photon representation of this radiation

If hν  Eg, some electrons can be promoted to the conduction band leaving some holes in the valence band. • Consider various types of spectra associated with this process: • Absorption: – Looks at the number of absorbed photons (intensity) vs. photon frequency ω • Reflection: – Looks at the number of reflected photons (intensity) vs. photon frequency ω • Transmission: – Looks at the number of transmitted photons (intensity) vs. photon frequency ω • Emission: – Looks at the number of emitted photons (intensity) vs. photon frequency ω • Each of these types of spectra are rich, complicated, & varied! • Understanding such spectra gives huge amounts of information about electronic energy bands, vibrational properties, defects, Appearance of insulator, metal and semiconductor

 Appearance in terms of color depends on the interaction between the light with the electronics configuration of the material.  Normally,  High resistivity material: insulator  transparent  High conductivity material: metals  metallic luster and opaque  Semiconductors  colored, opaque or transparent, color depending on the band gap of the material  For semiconductors the energy band diagram can explain the appearance of the material in terms of luster and color. Question: Why is Silicon Black and Shiny? Answer.

 Need to know, the energy gap of Si

 Egap = 1.2eV  Need to know visible light photon energy

 Evis ~ 1.8 – 3.1eV  Evis is larger than Silicon Egap  All visible light will be absorbed  Silicon appears black  Why is Si shiny?  Significant photon absorption occurs in silicon, because there are a significant number of electrons in the conduction band. These electrons are delocalized. They scatter photons. Colors of Semiconductors

Evis= 1.8eV 3.1eV

I B G Y O R

•If Photon Energy, Evis > Egap  Photons will be absorbed

•If Photon Energy, Evis < Egap  Photons will be transmitted

•If Photon Energy is in the range of Egap ;

•Those with higher energy than Egap will be absorbed. •We see the color of the light being transmitted •If all colors are transmitted = White Why is glass transparent?

 Glass is an insulator (huge band gap)  The electrons find it hard to jump across a big energy gap:

Egap >> 5eV

 Egap >> E visible spectrum ~ 3.1- 1.8eV  All colored photons are transmitted, no absorption, hence light transmission – transparent.  Defined transmission and absorption by Lambert’s law:

 I = Io exp (- l)  I = incident beam

 Io = transmitted beam   = total linear absorption coefficient (m-1)   = takes into account the loss of intensity from both scattering centers and absorption centers.   = approaching zero for pure insulator. What happens during photon absorption process?

Photon interacts with the lattice Photon interacts with defects Photon interacts with valance electrons Absorption – an important phenomenon in describing optical properties of semiconductors

 Light, being a form of electromagnetic radiation, interacts with the electronic structure of atoms of a material.  The initial interaction is one of absorption; that is, the electrons of atoms on the surface of a material will absorb the energy of the colliding photons of light and move to the higher-energy states.  The degree of absorption depends, among other things, on the number of free electrons capable of receiving this photon energy. Absorption Process of Semiconductors

 The interaction process is a characteristic of a photon and depends on the energy of the photon  Low-energy photons interact principally by ionization or excitation of the outer orbitals in solids’ atoms.  Light of low-energy photons (< 10 eV) is represented by infrared (IR), visible light, and ultraviolet (UV) in the electromagnetic spectrum.  High-energy protons (> 104 eV) such as x-rays (and gamma rays) scatter mainly elastically and are used for structure determination  The minimum photon energy required to excite and/or ionize the component atoms of a solid is called the absorption edge or threshold. Absorption Process of Semiconductors

Wavelength (m) Vis UV IR -1 Important region: ), cm  vis  ~ g E Absorption coefficient ( Absorption coefficient

Photon energy (eV) Absorption spectrum of a semiconductor. Valance-Conduction Absorption

Process requires the lowest E of photon to initiate electron Conduction band, jumping (excitation) EC • EC-EV = h

• EC-EV = Egap • If h > E then gap h Ephoton transition happens Egap •Electrons in the conduction band and excited.

Valance band, EV Absorption

 Types  Direct and Indirect photon absorption  For all absorption process there must be:  Conservation of energy  Conservation of momentum or the wavevector  The production of e-h pairs is very important for various electronics devices especially the photovoltaic and photodetectors devices.  The absorbed light can be transformed to current in these devices Direct Band Gap

Direct Conservation of E vertical h = E -E = E transition C(min) v (max) gap

K (wave number) h Momentum of Conservation of wavevector photon is negligible Kvmax + photon = kc Interband absorption in indirect gap semiconductors

Indirect-gap semiconductor: highest occupied and lowest unoccupied state have k≠0

Direct transitions possible for k0  strong direct interband absorption

occurs at E > Egap

E gap Other possibility: momentum and energy can be conserved by photon absorption and simultaneous absorption or emission of a phonon:

Indirect transitions possible with

‘assistance of a phonon’

Shown here are optically induced transitions

Egap - during phonon emission a phonon is generated in the process - during phonon absorption a phonon is generated in the process Excitons

Excitons are combined electron-hole states: A free electron and a free hole (empty electronic state in the valence band) exert Coulomb force on each other:

hydrogen-like bound states possible: excitonic states

n=3 E n=2 Eb is the exciton n=1 binding energy = h Coulomb force energy released upon e exciton formation, or Eb energy required for k exciton breakup

Wave functions of electron and hole look similar to free electron and free hole Note: exciton can move through crystal, i.e. not bound to specific atom! Excitonic absorption

Light can excite an electron from the valence band and generate an exciton at energies slightly below the bandgap

 see absorption at Ephot = Egap –Eb (absorption slightly below Egap)

Exciton binding energy on the order of a few meV Thermal energy at room temperature: kT ~ 25 meV  exciton rapidly dissociates at room temperature  absorption lines broaden / disappear for higher temperatures Optical transitions related to dopant atoms

Ga: 3 valence electrons Si: 4 valence electrons As: 5 valence electrons Donor levels

Substitute Si atom with As atom (impurity atom in the Si lattice): weakly bound extra valence electron

Low T

Low T: donors neutral, electron weakly bound low energy light can excite donor electron in to conduciton band

Binding energy Ed similar to kT at room temperature (‘RT’): At room temperature the bound electron is quickly released  impurity mostly ionized at RT : Arsenic is a donor in Si RT At RT such transitions are typically too broad to observe Acceptor levels

Substitute Si atom with Ga atom : empty electronic state just above the Si valence band: at finite temperature, Si valence electron may fill acceptor level  location of unoccupied valence state (hole) can orbit the charged Ga dopant

‘hole’ = available electron state

Binding energy Ea similar to kT at room temperature (‘RT’): At room temperature the hole can leave the dopant, producing a ‘free charge’ Infrared absorption due to dopants

Dopant binding energies low: donor level related absorptions invisible at RT, but observable at low temperatures Example: direct valence band → acceptor level absorption in boron doped Si

Transition at ~40 meV  absorption at 30 m : infrared Dopant related transitions

Possible dopant related transitions:

Typically visible at low T, but not clearly observable at RT Free carrier absorption

At RT, predominant dopant related absorption is free carrier absorption in which a photon excites an electron into a higher lying state Example: p-type semiconductors: filled states in the conduction band:

optical transitions possible at Ephot < Egap !

Free electrons: absorption typically indirect phonon-assisted transition

Free holes can make direct transitions from the heavy-hole band to the light-hole band  holes cause stronger free carrier absorption than electrons Free carrier absorption

Free electron absorption can be described by the Drude model Dopant levels in semiconductors range from ~1014 -1018 /cm3 which is ~108 –104 lower than free electron densities in metals Plasma frequency of doped semiconductors 104 -103 lower than of metals: IR

At frequencies above plasma frequency, εr is complex and  is described by   2  2 2  p   p  p  r '( )  1 , r "( )      2  3  3    2  2 p  ( )  "( )  2  2 c c c p

Electron FCA up for lower energies Free hole absorption less well defined Derivation of Optical Absorption Coefficient in Direct Semiconductors

Chuang Ch. 9 Outline of derivation

•Absorption Coefficient:  ()

() z IzIze(,)(,) o •Examples: lasers, solar cells, etc. •Time-dependent perturbation poly-Si Solar Cells •Fermi’s Golden rule •Direct-gap net absorption rate •Absorption Coefficient & Simplifications Fermi’s Golden Rule

2 2 W H' (( EE )  ( EE )) if  fi f i f i

Ef Ei  

Ei Ef

Absorption Emission Direct-Gap Net Absorption Rate

Ec E 22π 2 R H'1δ(ΕΕ )ff  vc  cv c v v c V kkvc  Assumptions: k kv = kc = k Undoped, low excitation

Ev fv = 1, fc = 0 22k E  22π 2 v 2m* R  H ' δ(ΕΕ ) h abs V   cv c v 22k k EEcg * 2me Absorption Coefficient

R (no. of photons absorbed per second per unit volume) ()  abs P /  (no. of injected photons per second per unit area)

 22π 2 H ' δ(ΕΕ ) 22  cv c v  (/2)ncro A o V k   H '  ψ* H'(r )ψ d 3r •How to find H’cv? cv  c v  1  2  H (r, t)  pˆ  eA  V (r ) 2mo  eA H ' (r )   o eˆ  pˆ Momentum matrix element 2mo 2 πe 2  2 ()  epˆ δ(ΕΕ ) 2  cv c v  ncroo m V k  More Practical Form  222 πeVk 2 2  3 epˆ δ(E ) k ()  2 cv 3 g   ncroo mk V2 2 m r  222 πek 2 2   epˆ δ(E  )d3 k 2 cv 3 g ncroo m 2 2 m r •Using E-k (dispersion) relationship:

2  3/2 πe  2 1 2m 1/ 2 () epˆ r δ(E  )d 222cv  k g k k ncroo m  2     3/2 1 2mr NJk()  22 k 2  2 πe  2 m E  1   ()epˆ N ( E )p 2  o g  1 2 cv J g cv 2  m  ncroo m  e   Conclusions Absorption Coefficient at 5K •Example: InSb •Eg = 0.17eV •Different for 2D,1D,0D •Density of States •Not 100% accurate •Parabolic band approximation

•nr depends on Yu, Cardona: p. 260 wavelength Red: calculation at 300K •Exciton absorption below bandgap