GENERAL PHYSICS III GENERAL PHYSICS III Optics & Quantum Physics Chapter XXIV Chapter XXIV Crystalline Solids Crystalline Solids §1. Wavefunctions and energy band of electrons §2. Electronic conduction in metals §3. Semiconductors To have a quantum-mechanical treatment we model a crystalline solid as matter in which the atoms have long-range order, that is a recurring (periodical) pattern of atomic positions that extends over many atoms. We will describe the wavefunctions and energy levels of electrons in such periodical atomic structures.  We want to answer the question: Why do some solids conduct currWe want to answer the question: Why do some solids conduct currentent and others donand others don’’t?t? §1. Wavefunctions and energy band of electrons: 1.1 Potential energy of electrons: +e r n = 3 n = 2 n = 1 +e r +e r e )r(U 2   2 2 1 2 rr e rr e )r(U      Atom H Two atoms (molecule H 2 ) +e r n = 3 n = 2 n = 1 Atom Solid Discrete atomic states  band of crystal states  Fill according to Pauli Principle • A simple (1-D) model of the recurring structure is shown below. • An electron interacts not with one, but with many nuclei  every energy level splits into a band of energies 1.2 Wavefunctions 1.2 Wavefunctions and and Energies: Energies: • Atomic ground state: • Molecular states: +e r n = 1  A Bonding state Antibonding state r e )r(U 2   0 a/r e)r(    Now we fill these orbitals with the 2 available electrons (one from each hydrogen atom). Both can go into ‘bonding’, thanks to spin +e r +e  even +e r +e  odd What do these crystal states look like? -- approximately linear combinations of atomic orbitals. Again start with simple atomic state: +e r n = 1  A Bring N atoms together together forming a 1-D crystal (a periodic lattice) (N atomic states N crystal states): Energy band • 1-D periodic lattice: •• SimpleSimple model of a crystalmodel of a crystal withwith covalentcovalent bonding:bonding:  N Highest energy orbital (N-1 nodes)  1 Lowest energy orbital (zero nodes) “in between” states • The “in between" states: Bloch Wavefunction for electron in a solid: The wavevector k has N possible values from k = /L to k = /a. Real Envelope: e ikx Length of crystal, L Lattice spacing, a  n u(x)e(x) xik n n   u(x) depends on the atomic states involved: L/a .1,2,n L n k n   N = L/a states 1s-states u(x) x x 2s-states • An example of Bloch Wavefunctions and the Energy Band: u(x)e(x) xik n n    3  1  2  4  5  6 Energy Lowest energy wavefunction Highest energy wavefunction Closely spaced energy levels in this “1s-band” For N = 6 there are six different superpositions of the atomic states that form the crystal states [...]... scatter more at higher temperatures Here the electrons see a time-varying potential 1.4 Insulators, Semiconductors, Metals: Why do some solids conduct current and others don ’t?  Energy bands and the gaps between them determine the conductivity and other properties of solids    Insulators have a valence band which is full and a large energy gap (few eV)  apply an electric field - no states of higher... decreases) Temperature, T (This graph only shows trends A semiconductor has much higher resistance than a metal.) Summary  Ability to conduct electricity varies enormously between different types of solids I 1 m   1  J  E  Resistivity  is defined by:    2  A    ne   where J = current density and E = applied electric field   Resistivity depends on the scattering time  electrons for . GENERAL PHYSICS III Optics & Quantum Physics Chapter XXIV Chapter XXIV Crystalline Solids Crystalline Solids §1. Wavefunctions and energy band of electrons. We want to answer the question: Why do some solids conduct currWe want to answer the question: Why do some solids conduct currentent and others donand