Two-dimensional lattice

Transcription

1 1 Two-dimensional lattice a 1 *, k x k x = 0, k y = 0 X M a 2, y a 1, x Γ X a 2 *, k y k x = 0.5 a 1 *, k y = 0 k x = 0, k y = 0.5 a 2 * Γ k x = 0.5 a 1 *, k y = 0.5 a 2 * X X M k x = 0.25 a 1 *, k y = 0 k x = 0, k y = 0.25 a 2 * k x = 0.25 a 1 *, k y = 0.25 a 2 *

2 2 Crystal orbitals Ψ( x, y) = cr sχ c r, s, r, s ( x, y) ( irk a isk a) r, s = exp x + y

3 3 Two-dimensional lattice k = (k x,k y ) the wave vector of electron showing the direction and length of the wave λ ( 2 2 k ) 1/ 2 x k = 2π / k = 2π / + y For square array with N atoms in each direction (k x,k y ) = (2 /Na) (p,q), p, q are integers - /a = k x, k y < + /a E(k) = α + 2β{cos(k x a + cos(k y a)}

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10 (c) Corresponding density-of states curves. 10 (a) Molecular orbitals of C 60. (b) Band structure of K 3 C 60.

11 Total density of states for NbO.The Fermi level corresponds to a d 3 electron count. 11

12 Band theory diagrams diamond (C), silicon (Si), germanium (Ge), Gray tin (Sn) 12

13 13 Band Structure of Insulators and Semiconductors > 6eV = 3eV

14 14 Density of state = dn/de n = number of states (a) (b) Density of states in (a) metal, (b) semimetal (e.g. graphite).

15 Fermi level- the highest occupied orbital at T= 0 (a) (b) Fermi distribution (a) at T= 0, and (b) at T> 0. The population decays exponentially at energies well above the Fermi level. Population, P 1 = ( E ) / kt where, µ = chemical potential e µ + 1 When E= µ, P= 1/2 15

16 (a) population (b) Fermi distribution at T> 0 for (a) Intrinsic semiconductor, (b) Fermi distribution and the band gap 16

17 Extrinsic semiconductor: (a) n-type, e.g. P doped Si (b) p-type, e.g. Ga doped Si. 17

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21 Electrical Properties Control of the electrical properties of solids by the location and filling of their energy bands. (a)typical metal with a partially filled bands. (b) Metal generated by the overlap of filled and empty bands. (c) Small gap between filled and empty bands. As the temperature increases, then the upper band may become thermally populated, a semiconductor. (d) Similar to (c) except that population of the upper band arises through photoexcitation. (e) A large gap between the two bands, an insulator. (f) Two bands just touch, a semimetal. 21

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26 The Photoelectric Effect Albert Einstein considered electromagnetic energy to be bundled into little packets called photons. Energy of photon = E = hv Where, h = Planck constant ( 6.62 x J s ) v = frequency (Hz) of the radiation Photons of light hit surface electrons and transfer their energy hv = B.E. + K.E. hv e - (K.E.) The energized electrons overcome their attraction and escape from the surface Photoelectron spectroscopy detects the kinetic energy of the electron escaped from the surface. 26

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28 28 Photoelectron spectroscopy - a single photon in/ electron out process X-ray Photoelectron Spectroscopy (XPS) - using soft x-ray ( ev) radiation to examine core-levels. Ultraviolet Photoelectron Spectroscopy (UPS) - using vacuum UV (10-45 ev) radiation to examine valence levels.

29 He(I) UPS spectrum of HCl gas. 1. Loss of a bonding electron decreases the bond order, increasing the bond length in the resulting cation compared to the parent molecule. 2. Loss of a nonbonding electron has no effect on bond order or bond length. 3. Loss of an antibonding electron increases the bond order, decreasing the bond length of the cation compared to the parent molecule. 29

30 30 E Peak shift- charging effect Broadening- molecular solid bonding and relaxation effects.

31 Metal oxides Eg Decrease in overlapping of the d-orbitals 31

32 Overlapping of d-orbitals of early transition metal elements in the Oxide structures Energy level diagram of early transition metal elements in the 32

33 metallic 33 Metal sulfides semiconducto

34 Density-functional studies of tungsten trioxide, tungsten bronzes, and related systems Physics, 2005, vol. 1 34

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36 FIG. 8: Band structure diagrams of (a) cubic WO 3 and (b) NaWO 3. 36

37 FIG. 7: Calculated density of states for cubic tungsten bronzes, MWO3, near the Fermi level: (a) WO3, (b) HWO3, (c) LiWO3, (d) NaWO3, (e) KWO3, (f) RbWO3, (g) CsWO3. The Fermi level is indicated in each case. 37

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39 39 Electrochromic material - color change by applying electric field semiconducting metallic Figure 1: The color of Na x WO 3 with different x values (degree of reduction of W).

40 40 The measurement of absorption edge and band gap properties of novel nanocomposite materials T. Nguyena, A. R. Hind, Varian Australia crystalline phases of TiO 2 - anatase, rutile, brookite layered titanates - K 2 Ti 3 O 7, K 2 Ti 4 O 9

41 41 Diffuse reflectance spectra of nanocomposite materials: (a) TiO 2, (b) K 2 Ti 4 O 9, (c) (C 3 H 7 NH 3 ) 2 Ti 4 O 9, (d) C 6 H 12 (NH 3 ) 2 Ti 4 O 9 and (e) (Fe 3 (CH 3 COO) 7 OH)Ti 4 O 9. Absorption edges and band gap energies of nanocomposites and precursors.

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44 44 Titanium(IV) Oxide - properties high refractive index (n > 2.5, comparing to 1.45 of SiO 2 ) - pigments - photonic crystals n-type semiconductor (Eg ~ 3.2 ev) - photocatalysts reducible center - catalysts or catalyst supports

45 Photocatalysis over a Semiconductor Oxide such as TiO 2 Band gap of TiO 2 ~ 3.2 ev Amy Linsebigler et al., Chem. Rev., 95, 735,

46 46 O 2 (ads) TiO hν 2 h + + e - O -. H + 2. OOH H 2 O O 2.-. OH hν H 2 O 2 + H. - OOH + O2 Scheme II Possible pathways for formation of hydroxy radical.

47 Electronic Band Structure of Titania Semiconductor Nanosheets Revealed by Electrochemical and Photoelectrochemical Studies J. AM. CHEM. SOC. 2004, 126,

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49 49 p- n junction Excess excess electron hole No current flows (reverse bias) Current flows (forward bias)

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51 51 Photovoltaic Cell A photovoltaic cell, or solar cell, is a semiconductor device that converts light to electricity. The cell consists of a thin layer of p-type semiconductor, such as Si doped with Al, in contact with an n-type semiconductor, such as Si doped with P. The p-type semiconductor in the solar cell must be very thin - about 1 x 10-4 cm (1 µm). This is to reduce the tendency for conduction electrons produced by sunlight to be captured by positive holes and immobilized in covalent bonds.

52 Photovoltaic Cell 52

53 53 Simple PV Systems PV with Battery Storage

54 PV Connected to Utilities This electric vehicle recharging station in southern Florida is powered by a grid-connected PV array mounted on the roof. When no vehicles need charging, power from the modules is transferred to the utility line. (Photo: University of South Florida) 54

55 Applications and Uses PV cells and modules are very reliable in space and on the earth. The Hubble space telescope (pictured here) and virtually all communications satellites are powered by photovoltaic technology. 55

56 56 Solar Spectrum Visible Large Area Pulsed Solar Simulator

57 In search of better efficient Semiconductors Light emitting diodes (LED) made of indium gallium nitride (Eg = 0.7 ~ 3.4 ev) held clues to the potential new solar cell material by W. Walukiewicz at Berkerly 57

58 A newly established indium gallium nitride system of alloys (In 1-x Ga x N) covers the full solar spectrum 58