Covalent Crystals - covalent bonding by shared electrons in common orbitals (as in molecules) - covalent bonds lead to the strongest bound crystals, e.g. diamond in the tetrahedral structure determined by the sp 3 hybridization - graphite is another crystalline form of carbon in hexagonal structure (sp 2 hybridization) diamond (C), tetrahedral - strong covalent in-layer bonding but weak (van der Waals) interlayer bonding leading to large anisotropy in crystal structure - due to weak interlayer bonding graphite is used as lubricant (and in pencils) - delocalized electrons lead to electrical conductivity in plane graphite (C), hexagonal phys4.17 Page 1 Graphite, Diamond, Bucky Balls, Nanotubes - under normal conditions (pressure, temperature) graphite is the stable phase of crystalline carbon - at high temperatures and high pressures diamond can form naturally (deep in the earth) or it can be made synthetically dissolving graphite in liquid nickel or cobalt at 1600 K and 60 kbar forming small crystallites - diamond is the hardest existing solid and is used industrially for cutting and grinding, silicon carbide (SiC) has similar properties - carbon also occurs in the form of bucky balls and nanotubes nanotube - generally covalent crystals are hard, have high melting points and are insoluble in ordinary liquids bucky ball phys4.17 Page 2
Van der Waals Bond - the van der Waals force is a weak short range attractive force (~ 1/r 7 ) acting between all atoms and molecules - in absence of covalent, ionic or metallic binding van der Waals forces lead to condensation of gases into liquids and to freezing of liquids into solids - the van der Waals interaction is responsible for effects like friction, adhesion, surface tension, viscosity etc. - polar molecules, e.g. water as in (a), have permanent dipole moment due to a inhomogeneous charge distribution - attractive force between polar ends of molecules, see (b) - polar molecules can attract molecules without permanent dipole moments by inducing polarization (see left) phys4.17 Page 3 Dipole-Dipole Interaction - electric field E of a dipole p at distance r (see J.D. Jackson, Classical Electrodynamics) - dipole moment induced in non-polar molecule - with polarizability α - dipole interaction energy - van der Waals force - strong distance dependence phys4.17 Page 4
Van der Waals in non-polar molecules and in atoms - non-polar molecules have vanishing dipole moments on average - temporal fluctuations in dipole moment due to electron motion in the molecule can mediate van der Waals forces (see figure) - these forces are responsible for condensation in such systems - van der Waals binding energies are in the 1-100 mev range Hydrogen Bonds - hydrogen atoms can mediate the strongest van der Waals interactions - the negative charge of the hydrogen atoms is often concentrated at the binding atom leaving the positively charged nucleus poorly screened - this positive charge can bind to other molecules through electric forces (~ 1/r 2 ) phys4.17 Page 5 Hydrogen Bonds in Water - Hydrogen atoms donate electrons to oxygen to form bonds - orbitals have tetrahedral symmetry - two positively and two negatively charged regions - water molecule can bond to four other water molecules - in liquid these bonds continuously break and reform - in ice water molecules crystallize into structures with only four nearest neighbors (see figure) resulting in the low density observed in ice - hydrogen bonds are also important in biological settings (e.g. DNA replication) phys4.17 Page 6
Metallic Bond - outer (valence) electron in metals are only weakly bound - in a solid these electrons form an electron gas that is relatively free to move in matrix of metal ions - the electron gas provides the bonding in metals, it is also responsible for high thermal and electrical conductivity - as the electrons are delocalized in metals many stable alloys can be formed - in metals conduction electrons are in a continuous energy band - the potential energy of electrons in a solid is decreased with respect to free metallic atoms - the kinetic energy however is increased due to the exclusion principle - the kinetic energy is determined by the Fermi energy (a few ev) phys4.17 Page 7 Electrical Conductivity - the voltage drop V across a conductor is proportional to the electrical current I passing through the conductor and its resistance R - this fact is expressed by Ohm's law - the resistance R is dependent on the dimensions, composition and temperature of the conductor but largely independent of the applied voltage V - Ohm's law follows from the free electron model for metals Derivation of Ohm's Law - consider electrons in a metal as a gas of free particles (matter waves) undergoing scattering with thermally oscillating atoms of the lattice and with defects and impurities - electrons in a perfect single crystal do undergo scattering from the ideal perfect crystal only under specific conditions phys4.17 Page 8
Mean Free Path and Collision Time - mean free path length λ between collisions of an electron with a defect - average time τ between collisions of electron with defect - the collision time τ is nearly independent of the electric field applied to the conductor because of the large Fermi velocity V F ~ 10 6 m/s Drift Velocity - an applied electric field E induces an average directed motion in the electrons with drift velocity v d - find the dependence of electron drift velocity on the electrical current phys4.17 Page 9 Drift Velocity - dependence of current I on electron density n and drift velocity v d - typical drift velocity in a good conductor, e.g. copper (Cu) - drift velocities at typical currents are a lot smaller than the Fermi velocity - find the dependence of the drift velocity on the applied electric field - electron acceleration - electron undergoes collisions, is scattered into random directions and does random walk phys4.17 Page 10
- electron motion in presence of accelerating electrical field - average electron displacement between collisions - with collision time (Poisson distribution) - thus the drift velocity is - resulting in a current - thus Ohm's law is - with resistance R - and resistivity ρ phys4.17 Page 11 Typical Mean Free Path - calculate mean free path in copper from resistivity ρ and Fermi velocity v F - the resisitvity of the metal depends on the concentration of defects and impurities (ρ i ) in the metal (temperature independent) and on the thermal vibrations (ρ t ) of the crystal (obviously temperature dependent) - the total resitivity is the sum of the two - the ratio of the resitivity at high temperatures dominated by scattering from phonons to the one at low temperatures dominated by impurity scattering is called the residual resistance ratio (RRR) and is a measure of how clean a conductor is - in very clean metals the RRR can approach 10 5 phys4.17 Page 12
Band Structure of Solids - electrical conductivity in solids varies over a huge range - good conductors (e.g. copper) - good insulators (e.g. quartz) - how can this variation be explained? - when atoms are brought close to each other their valence electron wave functions overlap and form new electron states of the solid - remember the example of the hydrogen molecule where two energy levels of the orbitals of the constituent atoms at the same energy are combined into two new orbitals (with two split energy levels) phys4.17 Page 13 - the situation in the solid is similar, each valence electron contributes an energy level to the combined electron energy level structure of the solid which then is essentially split into a number of new levels proportional to the number of atoms involved 5 atoms interacting = 5 levels N atoms interacting = N energy levels = a band of allowed electron energies phys4.17 Page 14
Energy Bands in the Metal Sodium (Na) - electron configuration 1s 2 2s 2 2p 6 3s 1, i.e. one valence electron per atom contributed to the metal - equilibrium separation of atoms r = 0.367 nm corresponding to the minimum 3s electron energy - inner shells do not interact, i.e. no splitting or band formation - valence shell electrons interact forming energy bands depending on the inter-atom separation - electrons can take on energies only within the energy bands - energies outside these bands are forbidden - possible electron states are filled up to the Fermi energy ε F, at non-zero temperatures the Fermi-Dirac distribution determines the occupation of higher energy states phys4.17 Page 15