Optical Properties of Semiconductors 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India http://folk.uio.no/ravi/semi2013
Light Matter Interaction Response to external electric field E 2 Polarizability: Linear approximation: susceptibility c conductivity s dielectric tensor Fourier transform:
Optical absorption in Semiconductors 3 We want to develop a set of equations to describe the absorption of a photon in semiconductor material. The electromagnetic field is a quantized system (with a set of modes, each of which is a harmonic oscillator). In absorption, a photon is absorbed by the crystal and the energy of the electromagnetic field is transferred to the crystal. The initial state in the region of interest in the crystal is E i,while the final state is E f.
What are the assumptions and approximations we must consider? The electromagnetic field is perturbed by the electronic crystal. 4 If the wavelength associated with a mono-energetic field is larger than the perturbing charge (like in an atom or quantum dot), then we can make the dipole approximation and assume there is no position dependence to the field (and solve just using the time-dependent field E(t)=E 0 sin( t)). Otherwise, we assume, Bloch waves. We can assume the intensity of the field is large,so that changes in the photon number in each mode is small. Called semiclassical approximation (which we will make most of the time). This means we neglect the action of the charge back on to the field (back action).
Fermi s Golden Rule 5 In order to determine the probability or amplitude of the absorption we must find the overlap of the initial and final wavefunctions. Instead of single initial and final states in single-particle picture, we have in principle a large density of final states - (k) The probability of absorption or emission will depend on the overlap and energy difference of the initial and final state, and the density of these states.
In quantum structures case 6 Choice of the wavefunctions for the initial and final states Two different kinds of possibilities in quantum structure Transitions between the valence and conduction bands Transitions between the quantum-confined states within a given band, so-called "intersubband transitions E C.B E 2 Barrier QW Barrier Pump Probe E g E 1 Emission E 2 E 1 LH x HH x E g e1-e2 ISBT K HH 1 HH 1 LH 1 V.B LH 1 Resonant optical transition
Energy Light Scattering: Interband transition 7 band structure c k EF E interband transition intraband transition S v k wave vector
Linear optical parameters 8 Complex dielectric tensor: Kramers-Kronig relations Optical conductivity: Complex refractive index: Reflectivity: Absorption coefficient: Loss function:
Intraband Contributions: Metals 9 Dielectric Tensor: Drude-like terms Optical conductivity: Plasma frequency:
Optical Sum rules 10
Form of wavefunctions for "non-excitonic quantum well absorption (quantum well) 11 Start by neglecting any excitonic effects (and other Coulomb effects many particle effects) Treat the initial state as being some electron state corresponding to an electron in the valence band or some lower subband Treat the final state as an electron in the conduction band or a higher subband The absorption process is an interaction between the matter and the electromagnetic field.
General aspects Optical absorption and luminescence occur by transition of electrons and holes between electronic states (bands, tail states, gap states). If electron-phonon coupling is strong enough self-trapping occurs. Choose valence band wavefunction as initial state. Conduction band wavefunction as the final state.
Optical Absorption Absorption coefficient α is defined by I(z) = I o exp {- α z} where I(z) is the flux density if incident light is I o, z is the distance measured from the incident surface. Hence α = - (1/I(z)) di(z)/dz
Tauc law (Tauc plot, A region) The absorption coefficient, α, due to interband transition near the band-gap is well described: αħω = B (ħ ω E g ) 2 ħω is photon energy, E g is optical gap. This Tauc plot defines the optical gap in semiconductors.
Urbach Tail in Absorption 15
Urbach tail (B region) The absorption coefficient at the photon energy below the optical gap (tail absorption) depends exponentially on the photon energy: α(ħ ω) ~ exp (ħ ω/e u ) where E u is called Urbach energy. In addition, optical absorption by defects also appears at energy lower than optical gap (C region). Likewise α is written as another exponential function of photon energy: α(ħω) ~ exp (ħω/e d ), E d belongs to the width of the defect states. C region is rather sensitive to the structural properties of materials.
Photoluminescence Photoluminescence occurs as a result of the transition of electrons and holes from excited states to ground state. After interband excitation, electrons (holes) relax to the bottom (top) of the conduction (valence) band by emitting phonons much more quickly than the radiative transition. In the case of crystalline semiconductors (without defects, there is no localized state) photoluminescence occurs by transition between the bottom of the conduction band and the top of the valence band. k selection rule must be satisfied: k photon = k i k f. (k photon, k i and, k f are the wave numbers of photons, electron of initial and final states.
Direct/indirect transition Since k photon is much smaller than k i and k f, we can rewrite the selection rule: k i = k f. The semiconductors satisfying this condition is called direct-gap semiconductors. c-si is not satisfying k-selection rule (indirect-gap semiconductor). Transition is allowed by either absorption of phonons or their emission.
Microscopic Theory of Linear Optical Properties of Semiconductors 19
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Semic-classical Theory of Interband Transitions 25
Optical Transitions 26
Optical Properties 27
Beer Lambert Law 28
Absorption in Semiconductors : processes 29
Absorption in semiconductors: processes cont. 30
Optical Properties: Semiconductors & Insulators 31
Optical Properties : Impurities 32
Absorption in semiconductors: band-to-band 33
Direct band gap and Indirect band gap 34
Indirect Band Gap 35
Interband absorption above the band gap 36
Dielectric Function and Critical Points in Ge 37
Comparing Direct and Indirect Bandgap Absorption 38
Optical absorptions in Si 39
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RPA Approximation 41
42 One can predict optical properties from DFT calculations
Silicon Optical Absorption 43
Joint Density of States 44
Band edge absorption in direct gap semiconductors 45
External Electric and Magnetic Field Effects 46
Radiative and Non-radiative Recombination 47
Feasible Recombination Processes 48
Interband absorption 49
Interband absorption. 50
Direct versus indirect absorption 51
Silicon band structure 52
Summary of Indirect optical transitions 53
Phonon Assisted Optical Transition 54
Excitonic Effect : Two particle (e-h) interaction 55
Absorption via Excitons 56
Electron-Hole interaction: Excitons 57
Experimental Absorption Edges with exciton 58
Exciton Effect above the bandgap 59
Plasma reflectivity : metals 60
Drude Model 61
Interband transitions in metals 62
Noble Metals : Copper 63
Band structure and DOS in Copper 64
Doped Semiconductors 65
Optical transitions in semiconductors: Impurities 66
Donor absorption in n-type silicon 67
Optical Anisotropy 68
Symmetry of Dielectric Tensor 69 triclinic monoclinic (a,b=90 ) orthorhombic tetragonal, hexagonal cubic
Convergence : Al 70 Interband Im 175 150 125 100 75 50 25 165k 286k 560k 1240k 2456k 3645k 4735k 12.8 12.7 12.6 p 12.5 12.4 12.3 12.2 12.1 12.0 0 1000 2000 3000 4000 5000 k-points in IBZ 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Energy [ev]
N eff [electrons] 5 4 3 2 1 Sumrules : Al 165 k-points 4735 k-points Experiment 71 0 0 10 20 30 40 50 60 70 80 90 100 Energy [ev]
Example: Al 120 100 Loss Function 72 Loss function 80 60 40 20 intraband interband total 0 0 5 10 15 20 Energy [ev]
Sum Rules. 73
Optical Properties of Metals. 74
Optical Properties of Metals: Al and Pd 75
Optical Properties of Metals: Cu and Cd 76
Joint Density of States (JDOS) 77
Dielectric Function (Real and Imaginary parts) 78
Comparison of theory vs. Experiment: ε2(ω) for Ge 79
Absorption Coefficient : α(ε) 80
Index of Refraction: n(ω) 81
Optical Properties: Reflectance & Dielectric Function : Si 82
Optical Properties: Reflectance & Dielectric Function : GaAs 83
Sensitivity of Reflectivity to Surface Contamination 84
Crystalline vs. Amorphous (Exp & Theory) 85
Origin of strong change in absorption 86
87 Band structure of Au: relativistic effects
DOS and Joint DOS for Au: relativistic effect 88
Dielectric function for Au: relativistic effect 89
Optical Spectra : Impact on Solar Cells 90
Optical Spectra : Impact on Solar Cells 91
Current Developments 92 Kohn-Sham theory Gradient Corrections (GGA) LDA + U Exact Exchange (EXX) non-local effects correlation effects band gap problem Generalized Kohn-Sham theory Self-interaction correction (SIC) Non-local exchange / screened exchange Time dependent DFT response to time-dependet perturbation Many-body perturbation theory GW + Bethe-Salpeter equation band gap problem excitonic effects
Theory of Optical Properties 93
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