Review of Optical Properties of Materials

Transcription

1 Review of Optical Properties of Materials Review of optics Absorption in semiconductors: qualitative discussion Derivation of Optical Absorption Coefficient in Direct Semiconductors

2 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, E ph = h = (hc)/

3 Maxwell Electromagnetic Waves

4 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: 1 c t EH, EH, Changes in the fields propagate through space with speed c.

5 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

6 Electromagnetic Spectrum Shorter wavelength Larger Photon Energy (ev) Longer wavelength

7 Interaction Between Light & Bulk Material Many different possible processes can occur! Incident light 3a 3c Semi-transparent material 1 4 3b Scattering 1- Refraction - Transmission 3a Specular reflection 3b Total internal reflection 3c Diffused reflection 4 Dispersion where different colors bend differently

8 Refraction, Reflection and Dispersion Light when it travels in a medium can be absorbed and reemitted by every atom in its path. Small n Determined by refractive index; n High n n 1 = refractive index of material 1 n = refractive index of material

9 Total Internal Reflection Transmitted (refracted) light t k t n Evanescent wave k i Incident light i i k r n > n 1 c c i > c TIR Reflected 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) S.O. Kasap, Optoelectronics (Prentice Hall)

10 Mechanism and Application of TIR Optical fibre for communication What sort of materials do you think are suitable for fibre optics cables?

11 Review of optical processes Energy levels are everything in quantum mechanics. Excited level Energy E = h Ground level The atom is vibrating at frequency,. The atom is at least partially in an excited state.

12 Review of optical processes Before After Spontaneous emission Absorption Stimulated emission

13 Recall: Semiconductor Bandgaps E g are usually in the range: 0 < E g < 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 E g, some electrons can be promoted to the conduction band leaving some holes in the valence band.

14 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 Understanding such spectra gives rich, complicated, & varied! huge amounts of information about electronic energy bands, vibrational properties, defects,

15 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.

16 Question: Why is Silicon Black and Shiny?

17 Answer. Need to know, the energy gap of Si E gap = 1.eV Need to know visible light photon energy E vis ~ eV E vis is larger than Silicon E gap 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.

18 Colors of Semiconductors E vis = 1.8eV 3.1eV I B G Y O R If Photon Energy, E vis > E gap Photons will be absorbed If Photon Energy, E vis < E gap Photons will be transmitted If Photon Energy is in the range of E gap ; Those with higher energy than E gap will be absorbed. We see the color of the light being transmitted If all colors are transmitted = White

19 Why is glass transparent? Glass is an insulator (huge band gap) The electrons find it hard to jump across a big energy gap: E gap >> 5eV E gap >> E visible spectrum ~ eV All colored photons are transmitted, no absorption, hence light transmission transparent. Defined transmission and absorption by Lambert s law: I = I o exp (- l) I = incident beam I o = 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.

20 What happens during photon absorption process? Photon interacts with the lattice Photon interacts with defects Photon interacts with valance electrons

21 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.

22 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 (> 10 4 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.

23 Absorption Process of Semiconductors Absorption coefficient (), cm -1 UV Important region: Wavelength (m) E g ~ vis Vis IR Photon energy (ev) Absorption spectrum of a semiconductor.

24 Valance-Conduction Absorption Process requires the lowest E of photon to initiate electron jumping (excitation) E C -E V = h Conduction band, E C E C -E V = E gap If h > E gap then transition happens E gap h E photon Electrons in the conduction band and excited. Valance band, E V

25 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

26 Direct Band Gap E Direct vertical transition Conservation of E h = E C(min) -E v (max) = E gap Momentum of photon is negligible K (wave number) Conservation of wavevector K vmax + photon = kc h

27 Interband absorption in indirect gap semiconductors Indirect-gap semiconductor: highest occupied and lowest unoccupied state have k 0 Direct transitions possible for k0 strong direct interband absorption occurs at E > E gap 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 E gap - during phonon emission a phonon is generated in the process - during phonon absorption a phonon is generated in the process

28 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 e n=3 n= n=1 Coulomb force h E b E E b is the exciton binding energy = energy released upon exciton formation, or k energy required for 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!

29 Excitonic absorption Light can excite an electron from the valence band and generate an exciton at energies slightly below the bandgap see absorption at E phot = E gap E b (absorption slightly below E gap ) E E b k Exciton binding energy on the order of a few mev Thermal energy at room temperature: kt ~ 5 mev exciton rapidly dissociates at room temperature absorption lines broaden / disappear for higher temperatures

30 Optical transitions related to dopant atoms Ga: 3 valence electrons Si: 4 valence electrons As: 5 valence electrons

31 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 E d 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

32 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 E a similar to kt at room temperature ( RT ): At room temperature the hole can leave the dopant, producing a free charge

33 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

34 Dopant related transitions Possible dopant related transitions: Typically visible at low T, but not clearly observable at RT

35 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 E phot < E gap! 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

36 Free carrier absorption Free electron absorption can be described by the Drude model Dopant levels in semiconductors range from ~ /cm 3 which is ~ lower than free electron densities in metals Plasma frequency of doped semiconductors lower than of metals: IR 3 3 ) "(, 1 ) '( p p r p r ) "( ) ( p p c c c At frequencies above plasma frequency, ε r is complex and is described by Electron FCA up for lower energies Free hole absorption less well defined

37 Derivation of Optical Absorption Coefficient in Direct Semiconductors Chuang Ch. 9

38 Outline of derivation Absorption Coefficient: ( ) I(, z) I (, z) e o ( ) z Examples: lasers, solar cells, etc. Time-dependent perturbation Fermi s Golden rule poly-si Solar Cells Direct-gap net absorption rate Absorption Coefficient & Simplifications

39 Fermi s Golden Rule ' W i f Hfi ( ( Ef Ei ) ( Ef Ei )) E f E i E i E f Absorption Emission

40 Direct-Gap Net Absorption Rate E c E π R H' δ(ε Ε )f 1f V vc cv c v v c k k k v c Assumptions: k v = k c = k Undoped, low excitation E v f v = 1, f c = 0 E E v c k * mh E g k m * e R abs π H ' cv δ(εc Ε v ) V k

41 ( ) How to find H cv? H ( r, t) H ' ( r ) Absorption Coefficient R abs P / nc A V ( r o o /) 1 m o ea m pˆ ea V ( r ) o o eˆ (no. of photons absorbed per second per unit volume) (no. of injected photons per second per unit area) pˆ r o o H ' cv k π H ' cv δ(ε c Ε v ) ψ * c H' ( r ) ψ πe ( ) eˆ pcv δ(εc Ε v ) nc m V k v d Momentum matrix element 3 r

42 ( ) More Practical Form πe V k nc m V m 3 eˆ p cv δ(e 3 g ) k r o o k r ep δ(e )d k nc m m ( ) πe eˆ pcv NJ ( Eg ) nc m m oeg 1 p cv 1 me r o o πe k 3 ˆ cv 3 g r o o r Using E-k (dispersion) relationship: ( ) nc 3/ πe 1 1/ ˆ mr e p cv k δ(e g k )d k r omo 3/ 1 mr NJ( k) k

43 Conclusions Absorption Coefficient at 5K Example: InSb Eg = 0.17eV Different for D,1D,0D Density of States Yu, Cardona: p. 60 Red: calculation at 300K Not 100% accurate Parabolic band approximation n r depends on wavelength Exciton absorption below bandgap