Optical Properties of Solid from DFT

Transcription

1 Optical Properties of Solid from DFT 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India & Center for Materials Science and Nanotechnology, University of Oslo, Norway

2 Light Matter Interaction Response to external electric field E 2 Polarizability: Linear approximation: Electric susceptibility c conductivity s Displacement Field D is defined as dielectric tensor Fourier transform:

3 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.

4 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).

5 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.

6 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

7 Energy Light Scattering: Interband transition 7 band structure c k EF E interband transition intraband transition S v k wave vector

8 Linear optical parameters 8 Complex dielectric tensor: Kramers-Kronig relations Optical conductivity: Complex refractive index: Reflectivity: Absorption coefficient: Loss function:

9 Intraband Contributions: Metals 9 Dielectric Tensor: Drude-like terms Optical conductivity: Plasma frequency:

10 Optical Sum rules 10

11 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.

12 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.

13 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

14 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.

15 Urbach Tail in Absorption 15

16 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.

17 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.

18 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.

19 Microscopic Theory of Linear Optical Properties of Semiconductors 19

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25 Semic-classical Theory of Interband Transitions 25

26 Optical Transitions 26

27 Optical Properties 27

28 Beer Lambert Law 28

29 Absorption in Semiconductors : processes 29

30 Absorption in semiconductors: processes cont. 30

31 Optical Properties: Semiconductors & Insulators 31

32 Optical Properties : Impurities 32

33 Absorption in semiconductors: band-to-band 33

34 Direct band gap and Indirect band gap 34

35 Indirect Band Gap 35

36 Interband absorption above the band gap 36

37 Dielectric Function and Critical Points in Ge 37

38 Comparing Direct and Indirect Bandgap Absorption 38

39 Optical absorptions in Si 39

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41 RPA Approximation 41

42 42 One can predict optical properties from DFT calculations

43 Silicon Optical Absorption 43

44 Joint Density of States 44

45 Band edge absorption in direct gap semiconductors 45

46 External Electric and Magnetic Field Effects 46

47 Radiative and Non-radiative Recombination 47

48 Feasible Recombination Processes 48

49 Interband absorption 49

50 Interband absorption. 50

51 Direct versus indirect absorption 51

52 Silicon band structure 52

53 Summary of Indirect optical transitions 53

54 Phonon Assisted Optical Transition 54

55 Excitonic Effect : Two particle (e-h) interaction 55

56 Absorption via Excitons 56

57 Electron-Hole interaction: Excitons 57

58 Experimental Absorption Edges with exciton 58

59 Exciton Effect above the bandgap 59

60 Plasma reflectivity : metals 60

61 Drude Model 61

62 Interband transitions in metals 62

63 Noble Metals : Copper 63

64 Band structure and DOS in Copper 64

65 Doped Semiconductors 65

66 Optical transitions in semiconductors: Impurities 66

67 Donor absorption in n-type silicon 67

68 Optical Anisotropy 68

69 Symmetry of Dielectric Tensor 69 triclinic monoclinic (a,b=90 ) orthorhombic tetragonal, hexagonal cubic

70 Convergence : Al 70 Interband Im k 286k 560k 1240k 2456k 3645k 4735k p k-points in IBZ Energy [ev]

71 N eff [electrons] Sumrules : Al 165 k-points 4735 k-points Experiment Energy [ev]

72 Example: Al Loss Function 72 Loss function intraband interband total Energy [ev]

73 73 Role of plasmons on Optical Properties Plasmons play a large role in the optical properties of metals. Light of frequencies below the plasma frequency is reflected, because the electrons in the metal screen the electric field of the light. Light of frequencies above the plasma frequency is transmitted, because the electrons cannot respond fast enough to screen it. In most metals, the plasma frequency is in the ultraviolet, making them shiny (reflective) in the visible range. Some metals, such as copper and gold, have electronic interband transitions in the visible range, whereby specific light energies (colors) are absorbed, yielding their distinct color

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76 Sum Rules. 76

77 Optical Properties of Metals. 77

78 Optical Properties of Metals: Al and Pd 78

79 Optical Properties of Metals: Cu and Cd 79

80 Joint Density of States (JDOS) 80

81 Dielectric Function (Real and Imaginary parts) 81

82 Comparison of theory vs. Experiment: ε2(ω) for Ge 82

83 Absorption Coefficient : α(ε) 83

84 Index of Refraction: n(ω) 84

85 Optical Properties: Reflectance & Dielectric Function : Si 85

86 Optical Properties: Reflectance & Dielectric Function : GaAs 86

87 Sensitivity of Reflectivity to Surface Contamination 87

88 Crystalline vs. Amorphous (Exp & Theory) 88

89 Origin of strong change in absorption 89

90 90 Band structure of Au: relativistic effects

91 DOS and Joint DOS for Au: relativistic effect 91

92 Dielectric function for Au: relativistic effect 92

93 Reflectivity: Theory vs. Experiment for Pt 93

94 Optical Spectra : Impact on Solar Cells 94

95 Optical Spectra : Impact on Solar Cells 95

96 Current Developments 96 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

97 Theory of Optical Properties 97

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