Resumo anterior Aplicações de Raio-X Luz sincrotron Área Analítica Área de Imagem Luz sincrotron Óptica de raio-x, policapilaridade Microscopia de raio-x Laser de raio-x (brandos e duros) Dispoptic 2010

Desde que vimos como são as diferentes formas dos cristais, vejamos como são formados De ligações a bandas Dispoptic 2010

Formação de um sólido Átomos livres Configuração eletrônica dos átomos Aproximação dos átomos Diferentes tipos de forças interatômicas: coulômbica, repulsão, covalente Formação de bandas de energia Formação de sólidos Diferentes tipos de sólidos: metal, isolante, semicondutor Dispoptic 2010

Diferentes tipos de forças interatômicas Eletrostática ~ 20 kJ/mol van der Waals 0.4 – 4 kJ/mol Hidrogênica 12 – 30 kJ/mol Covalente ~ 350 kJ/mol Outras forças fracas ou desprezíveis: magnética e gravitacional Dispoptic 2010

Principais tipos de ligações Argon xstal: http://www.webelements.com/argon/crystal_structure_pdb.html Van der Waals Iônica Metálica Covalente E/kJ/mol r/Å -0.5 +0.5 1 2 3 4 5 repulsão soma atração Sodium xstal: http://www.webelements.com/sodium/crystal_structure_pdb.html Kittel Dispoptic 2010 Carbon xstal; http://www.webelements.com/carbon/crystal_structure_pdb.html

Alguns tipos de ligações Na+ Cl-                                     Ligação Iônica Cl : Cl Ligação covalente não-polar [H : Cl] Ligação covalente polar http://www.chemistry.mcmaster.ca/esam/intro.html Dispoptic 2010

Num sólido iônico Dispoptic 2010

Formação de bandas Dispoptic 2010

Átomo de hidrogênio Dispoptic 2010 http://www.webelements.com/webelements/scholar/elements/hydrogen/electronic.html

Molécula de hidrogênio http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch8/mo.html Valence Bond Model vs. Molecular Orbital Theory Because arguments based on atomic orbitals focus on the bonds formed between valence electrons on an atom, they are often said to involve a valence-bond theory. The valence-bond model can't adequately explain the fact that some molecules contains two equivalent bonds with a bond order between that of a single bond and a double bond. The best it can do is suggest that these molecules are mixtures, or hybrids, of the two Lewis structures that can be written for these molecules. This problem, and many others, can be overcome by using a more sophisticated model of bonding based on molecular orbitals. Molecular orbital theory is more powerful than valence-bond theory because the orbitals reflect the geometry of the molecule to which they are applied. But this power carries a significant cost in terms of the ease with which the model can be visualized. Forming Molecular Orbitals Molecular orbitals are obtained by combining the atomic orbitals on the atoms in the molecule. Consider the H2 molecule, for example. One of the molecular orbitals in this molecule is constructed by adding the mathematical functions for the two 1s atomic orbitals that come together to form this molecule. Another orbital is formed by subtracting one of these functions from the other, as shown in the figure below. One of these orbitals is called a bonding molecular orbital because electrons in this orbital spend most of their time in the region directly between the two nuclei. It is called a sigma ( ) molecular orbital because it looks like an s orbital when viewed along the H-H bond. Electrons placed in the other orbital spend most of their time away from the region between the two nuclei. This orbital is therefore an antibonding, or sigma star ( *), molecular orbital. The bonding molecular orbital concentrates electrons in the region directly between the two nuclei. Placing an electron in this orbital therefore stabilizes the H2 molecule. Since the * antibonding molecular orbital forces the electron to spend most of its time away from the area between the nuclei, placing an electron in this orbital makes the molecule less stable. Electrons are added to molecular orbitals, one at a time, starting with the lowest energy molecular orbital. The two electrons associated with a pair of hydrogen atoms are placed in the lowest energy, or bonding, molecular orbital, as shown in the figure below. This diagram suggests that the energy of an H2 molecule is lower than that of a pair of isolated atoms. As a result, the H2 molecule is more stable than a pair of isolated atoms. Using the Molecular Orbital Model to Explain Why Some Molecules Do Not Exist This molecular orbital model can be used to explain why He2 molecules don't exist. Combining a pair of helium atoms with 1s2 electron configurations would produce a molecule with a pair of electrons in both the bonding and the * antibonding molecular orbitals. The total energy of an He2 molecule would be essentially the same as the energy of a pair of isolated helium atoms, and there would be nothing to hold the helium atoms together to form a molecule. Dispoptic 2010

Distribuição de elétrons e energias de OM Dispoptic 2010

Distribuição de carga homo-heteropolar Dispoptic 2010

Distribuição de carga e distribuição de ligação Dispoptic 2010

Lítio 1s22s Dispoptic 2010

Formação de bandas de energia, número de estados Átomos de Na (1s22s22p63s) Número atômico 11 N átomos (1023 átomos/cm3) 2 átomos 3 átomos Dispoptic 2010

Bandas de energia do Na com N átomos Átomos de Na (1s22s22p63s) Número atômico 11 2(2l+1)elétrons 2 = fator de orientação do spin 2l+1 = número de possíveis orientações do momento angular orbital 2(2l+1)N = capacidade de cada banda para N átomos Dispoptic 2010

Classificação de sólidos Metal Semicondutor Isolante Dispoptic 2010

Zonas de Brillouin http://phycomp.technion.ac.il/~nika/brillouin_zones.html the first Brillouin zone is a uniquely defined primitive cell in reciprocal space Dispoptic 2010

http://phycomp.technion.ac.il/~nika/brillouin_zones.html 1ª 2ª 3ª SC FCC BCC Dispoptic 2010

Em termos de bandas Dispoptic 2010 http://en.wikipedia.org/wiki/Brillouin_zone In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell of the reciprocal lattice in the frequency domain. It is found by the same method as for the Wigner-Seitz cell in the Bravais lattice. The importance of the Brillouin zone stems from the Bloch wave description of waves in a periodic medium, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone. Taking surfaces at the same distance from one element of the lattice and its neighbours, the volume included is the first Brillouin zone. Another definition is as the set of points in k-space that can be reached from the origin without crossing any Bragg plane. Equivalently, this is the Voronoi cell around the origin of the reciprocal lattice. There are also second, third, etc., Brillouin zones, corresponding to a sequence of disjoint regions (all with the same volume) at increasing distances from the origin, but these are used more rarely. As a result, the first Brillouin zone is often called simply the Brillouin zone. (In general, the n-th Brillouin zone consist of the set of points that can be reached from the origin by crossing n − 1 Bragg planes.) A related concept is that of the irreducible Brillouin zone, which is the first Brillouin zone reduced by all of the symmetries in the point group of the lattice. The concept of a Brillouin zone was developed by Leon Brillouin (1889-1969), a French physicist. Dispoptic 2010

Outra representação Dispoptic 2010 Schematic band diagrams for an insulator, a semiconductor, and a metal. Dispoptic 2010

Formação de bandas de energia a partir dos níveis de energia dos átomos constituintes Dispoptic 2010

Exemplo configuração banda de energia do Li Dispoptic 2010

Estrutura de banda de isolante e semicondutor (cristal molecular) Dispoptic 2010

Bandas de energia de níveis permitidos no diamante 1s22s22p2 Dispoptic 2010

Diamante colorido por doadores http://www.webexhibits.org/causesofcolor/11A0.html The band structure of blue and yellow diamonds A pure diamond crystal is translucent, as it is composed only of carbon atoms, each of which has four valence electrons. In a yellow diamond, a few carbon atoms per million have been replaced by nitrogen atoms, each containing five valence electrons. The structure of the diamond crystal does not change significantly, but the extra electrons occupy a donor level. The nitrogen donor level energy in a diamond is large, peaking at about 4 eV. With a concentration of a few nitrogen atoms per million, instead of a clean "spike" donor level energy, the nitrogen donor level energy broadens into a band because of a number of complex factors, including thermal vibrations. This broadened donor energy band results in the difference between donor and conduction bands being as low as 2.2 eV. The most likely transition allows incident light with energy of 4 eV per photon to excite electrons from the donor level to the conduction band. However, it is possible for electrons to be excited to the conduction band with energy of 2.2 eV and upwards. This means that blue and violet light are absorbed from the full spectrum of light normally transmitted, and the resulting color is yellow. Unlike blue boron-doped diamonds, which conduct electricity, nitrogen-doped diamonds remain insulators. This is because nitrogen is a deep impurity: a relatively high energy (2.2 eV, as compared with 0.4 eV for boron) is required to excite electrons from the donor energy level to the conduction band, and only a fraction of the available electrons are freed to carry a charge. An extremely rare green color can result from a higher nitrogen content of about 1 atom per 1000 atoms of carbon. At even higher nitrogen concentrations, the donor level broadens so that all visible light can be absorbed, resulting in a black color. Synthetic blue diamonds are created by adding boron as an impurity. Boron is trivalent, having three valence electrons; where a boron atom replaces a carbon atom in the diamond structure there is one fewer electron than usual. This missing electron, or hole, creates an acceptor energy level above the valence band. The boron acceptor energy is only 0.4 eV, so light of any energy can be absorbed during excitation. The boron acceptor band is broadened, and the absorption tapers off throughout the visible light energies, resulting in stronger absorption at the red end of the spectrum. At a level of one or a few boron atoms for every million carbon atoms, an attractive blue color results. Natural diamonds of this color, such as the Hope Diamond, are rare and highly priced.                                                  Boron has one less electron than carbon, and the presence of a few boron atoms per million carbon atoms in diamond leads to a hole with an energy level within the band gap. This is called an "acceptor" level since it can accept an electron from the full valence band. « Previous Next » webexhibits.org/causesofcolor — Bibliography — About — Credits & Feedback www.exhibits.org/causesofcolor Electrons can be donated to the empty conduction band. The valence band is completely filled. At minute concentrations of nitrogen, the energy required to excite an electron from the donor level to the conduction band is 4 eV, an energy that is greater than the visible light range (left). The diamond will be colorless. At a few nitrogen atoms per million carbon atoms, the donor level is broadened as at the right of this figure, and energies greater than 2.2 eV can excite an electron from the donor level to the conduction band. The absorption of these higher energies (blue and violet light) results in the yellow color of the diamond.   Dispoptic 2010

Teoria de Bandas : duas maneiras Duas aproximações para encontrar as energias dos elétrons associados com os átomos numa rede periódica. 1.- Aproximação de elétron ligado (energia de átomos singulares) Os átomos isolados são juntados para formar um sólido. 2.- Aproximação de elétron livre (não ligado) (E = p2/2m) Elétrons livres modificado por um potencial periódico, i.e. rede de íons. Ambas as aproximações resultam em níveis de energia agrupados com regiões de energia permitida e proibidas. Bandas de energia se sobrepõem em metais. Bandas de energia não se sobrepõem (ou possuem região proibida) para semicondutores. There are two approaches to finding the electron energies associated with atoms in a periodic lattice. Approach #1: “Bound” Electron Approach (single atom energies!) Isolated atoms brought close together to form a solid. Approach #2: “Unbound” or Free Electron Approach (E = p2/2m) Free electrons modified by a periodic potential (i.e. lattice ions). Both approaches result in grouped energy levels with allowed and forbidden energy regions. Energy bands overlap for metals. Energy bands do not overlap (or have a “gap”) for semiconductors Dispoptic 2010 Ver Charles Kittel – Introduction to Solid State Physics

A wide range of energies can cause electrons to be excited from the valence band to the conduction band (absorption; figure shows electronic transitions, A, and corresponding absorption spectrum, B). Dispoptic 2010

Excited electrons will drop from the bottom of the conduction band into the top of the valence band with the emission of light with a very narrow band width (emission; figure shows an electronic transition, A, and corresponding emission spectrum, B) Dispoptic 2010

Diagrama de Banda: Isolante com Egap grande Banda de condução (vazio) T > 0 EC Egap EF EV Banda de valência (cheio) Em T = 0, a banda de valência inferior é preenchida com elétrons e a banda de condução está vazia, conseqüentemente condutividade zero. A energia de Fermi EF está no meio da banda proibida (2-10 eV) entre as bandas de condução e valência. Em T > 0, os elétrons não são termicamente excitados da banda de valência à banda de condução, conseqüentemente também condutividade zero. Dispoptic 2010

Diagrama de Bandas: Função de preenchimento de Fermi-Dirac Probabilidade dos elétrons (férmions) serem encontrados em vários níveis de energia. Em TA, E – EF = 0.05 eV  f(E) = 0.12 E – EF = 7.5 eV  f(E) = 10 –129 Efeito enorme da dependência exponencial T > 0 T >> 0 T = 0 K http://jas.eng.buffalo.edu/education/semicon/fermi/functionAndStates/functionAndStates.html Em T = 0 K, elétrons tem 100% probabilidade de estar abaixo da energia de Fermi EF e 0% probabilidade de estar acima de EF. Em T > 0 K, probabilidade diminui abaixo de EF e aumenta acima de EF, provocando que a função degrau passe a ser mais suave (escorregadia?). Dispoptic 2010 Fermi : http://jas.eng.buffalo.edu/education/semicon/fermi/functionAndStates/functionAndStates.html

Diagrama de Banda: Metal preenchimento da banda. Função de preenchimento Banda de energia a ser preenchida EC,V EC,V EF EF T > 0 T = 0 K Em T = 0, níveis de energia abaixo de EF são preenchidos com elétrons, entretanto todos os níveis acima de EF estão vazios. Os elétrons são livres para se movimentar dentro dos estados vazios da banda de condução com somente um pequeno campo elétrico aplicado E, teremos alta condutividade elétrica. Em T > 0, os elétrons tem uma probabilidade de serem termicamente excitados a partir de níveis abaixo do nível de energia de Fermi para acima. At T = 0, all levels in conduction band below the Fermi energy EF are filled with electrons, while all levels above EF are empty. Electrons are free to move into “empty” states of conduction band with only a small electric field E, leading to high electrical conductivity! At T > 0, electrons have a probability to be thermally “excited” from below the Fermi energy to above it. Dispoptic 2010

Junção pn, diodos, LED’s e diodos lasers Semicondutor tipo p, tipo n Junção pn, circuitos diretos e reversos Equações de transporte LED OLED Diodo laser Dispoptic 2010

Próxima aula será sobre junção pn e emissores de luz Dispoptic 2010