From Last Time…Atoms and MoleculesSimple moleculesSymmetryMolecular SymmetriesMolecule questionPowerPoint PresentationSlide 8Slide 9QuestionSlide 11Quantum state energiesA three-atom moleculeA six-atom moleculeSolidsEnergy levels in a solidEnergy bands in a solidBand structureOccupation of quantum statesSolid sodium (metal)Electrical conductivityMetals, insulators, semiconductorsProperties of some elementsDoped semiconductorsDoping a semiconductorExample: Phosphorus-doped siliconn- and p-type semiconductorsSo what?Semiconductor devicesLight emitting diodeElectrical resistanceResistance questionLife is toughTemperature-dependent resistanceResistanceWhy does temperature matter?Temperature scalesLow temperature propertiesPhy107 Fall 20061From Last Time…Today•Molecules, metals and semiconductors•Important new Quantum Mechanical Concepts–Indistinguishability:–Symmetries of the wavefunction: Symmetric and Antisymmetric–Pauli exclusion principle: only one fermion per state–Spin•Final concepts needed to understand the hydrogen atom and the periodic tablePhy107 Fall 20062Atoms and Molecules•Have talked about atoms–Atoms are a central nucleus with some number of electrons orbiting around it.–Number of orbiting electrons determines what element•Molecules:–One or more atoms bonded together.Phy107 Fall 20063Simple molecules•Water: H2O•Carbon dioxide: CO2•Even these simple molecules can be quite complex.•Many nuclei, many electrons. •However some properties can be determined without worrying too much about the details.Phy107 Fall 20064Symmetry•The symmetries of fermions and bosons were a little subtle.•Symmetries show up in many situations, many times in more direct ways.•Both water and carbon dioxide have spatial symmetries:Phy107 Fall 20065Molecular Symmetries•These symmetries can determine many physical properties.•Can be related to microscopic quantum mechanical properties such as the wavefunction and the probability.•These are easiest to see if we start with a very simple molecule–Two protons and one electron.Phy107 Fall 20066Molecule questionA simple molecule consists of two protons and one electron orbiting around them. This molecule isA. Helium moleculeB. Hydrogen moleculeC. Lithium moleculeH2electronionized hydrogen moleculeElectron must be described as a wave.Use a wavefunction to do this.The square of the wavefunction is the probability of finding the electron.Phy107 Fall 20067A two atom moleculeWhat do we expect for the charge density?If atoms are identical, do we expect more charge on right, left?No reason to expect electron to reside on one atom over the other.One electron orbiting two atomsWhat wavefunction is consistent with electron not preferring one atom over the other?Phy107 Fall 20068•Two ways to superimpose statesTwo possible wavefunctionsElec onleft atomElec onright atom+Elec onleft atomElec onright atom— Symmetric Antisymmetric•In quantum mechanics, we can have BOTH.•Wavefunction is an equal superposition of electron on left atom and electron on right atom.Phy107 Fall 20069•These are obtained by adding or subtraction quantum states on either atom. •Both give symmetric charge density……but details slightly different.Phy107 Fall 200610QuestionWhich state has the lower energy? Symmetric AntisymmetricA. SymmetricB. AntisymmetricC. Both samePhy107 Fall 200611Two-atom moleculeSymmetry of the wavefunctionCompare particle in a box€ λ =LTwo half-wavelengths€ p =hλ=hLmomentum€ λ =2LOne half-wavelength€ p =hλ=h2LmomentumSymmetricAnti-symmetricPhy107 Fall 200612Quantum state energies•Symmetric state is the ‘ground state’.•Antisymmetric state is the excited state.–Wavelength half as large–momentum twice as large–Larger momentum -> larger kinetic energy•Since momentum depends on ‘size of box’ (atomic separation)…•… energy difference increases as atom separation decreases.Atom separationAnti-symmetric stateSymmetric statePhy107 Fall 200613A three-atom molecule-1-0.500.51-4 -2 0 2 4ψPOSITION-1-0.500.51-4 -2 0 2 4ψPOSITION-1-0.500.51-4 -2 0 2 4ψPOSITION-1-0.500.51-4 -2 0 2 4ψPOSITIONEnergy levelsWavefunctionsPhy107 Fall 200614A six-atom moleculeWavefunctionsEnergy levelsCan see different wavelengths for the different molecular statesPhy107 Fall 200615Solids•Solids consist of many atoms bonded together •Many possible ways to combine atomic wavefunctions to get charge density with correct symmetry.•All these quantum states have slightly different energies.•Solid is similar to atom or molecule, except quantum states are extremely close together in energy.Phy107 Fall 200616Energy levels in a solid•Solids consists of ~1024 atoms•Energy levels spaced extremely close together3-atom molecule6-atom molecule1024-atom ‘molecule’Phy107 Fall 200617Energy bands in a solid•This energy region of densely packed quantum states in called an energy band. •Each quantum state on an individual atom (for instance, 1s, 2s, etc) leads to one of these energy bands.•The detailed arrangement of these energy bands is called the band structure.Phy107 Fall 200618Band structureTwo atom moleculeFive atom ‘molecule’Many atom solidAtomic separation Atomic separation Atomic separation•Atomic orbitals interact to form new energy levelsPhy107 Fall 200619Occupation of quantum states•These quantum states are filled with electrons just as atomic states get filled one by one, lowest energy first, just like an atom.•Dramatically more electrons to fill the states!•But since each band arises from an atomic quantum state, •But due to details in which atomic states broaden into bands, sometimes bands overlap and and are not completely full or empty.Phy107 Fall 200620Solid sodium (metal)Sodium atomSodium metal1s2s3s2p3pNa = [Ne]3s1empty1 electron6 electrons2 electrons2 electronsFullFullFull3p3s2p2s1sPartially FullThis band not completely occupiedPhy107 Fall 200621Electrical conductivity•This little detail turns out to dramatically effect the electrical properties of materials.•In particular whether they will carry an electrical currentOnly a partially full band will carry electrical current!Phy107 Fall 200622Metals, insulators, semiconductors•Only partially full bands carry current•Completely full, or completely empty bands, carry no current Metal(at least one partially full band)Insulator(all bands completely full or
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