Physics 551: Solid State Physics F. J. Himpsel

Physics 551: Solid State Physics F. J. Himpsel Background Most of the objects around us are in the solid state. Today s technology relies heavily on new materials, electronics is predominantly solid state. Prominent figures started their career in solid state physics, such as Andrew Grove (former head of Intel, Man of the Year 1997 by Time Magazine). He made his scientific reputation with work on the silicon/sio 2 interface, whose perfection is the key reason for silicon technology. Recent Nobel prizes emphasize both technology (the integrated circuit, the CCD for digital cameras, quantum well lasers for CDs and DVDs, giant magnetoresistance for hard disks) as well as fundamentals (new states of matter in a two-dimensional electron gas, high temperature superconductivity, density functional theory of electrons in solids). Taking a Solid Apart How can we make sense of all the different solids surrounding us? Is it possible to produce a single, unified theory of solids? Typically, we are dealing with a solid containing 10 23 atoms and electrons, which appears to be a daunting task. The trick is to consider ordered (crystalline) solids and set 10 23 to infinity. Disorder can be added later. In a crystal, we have to solve only two problems: 1) Find the arrangement of the few electrons and atoms inside the unit cell of a crystal. 2) Determine the coupling between adjacent unit cells. Electrons are coupled by the overlap of their wave functions, which describes how electrons hop from one cell to the next. The coupling of atoms can be described by interatomic potentials, such as an array of springs. Solid = Atoms + Electrons Atoms: Electrons: Ground state: Crystal structure Excited state: Vibrational motion Ground state: Electron density (all electrons) Excited state: The wave function (one electron) 1

Classification of Solids by Conductivity Conductivity is related to the band structure. Density of States D(E) = number of states per ev and atom. Class Electron density, Density of states D(E) (Band gap E G ) Resistivity (filled bands dark) 1a) Insulators Conduction band LiF (14eV) SiO 2 (9eV) E G = Band gap 1b) Semiconductors GaAs (1.4eV) Si (1.1eV) Ge (0.7eV) 2) Semimetals Graphite 10 20 e /cm 3, 10 4 Ωcm E 3) Metals Aluminum E F = Fermi Level D(E) Energy scales: Band width 10 1 ev, Band gap 10 0 ev, k B T 25meV (room temp.) Metals: Conductivity decreases with increasing temperature. Thermal vibrations scatter electrons. Insulators/Semiconductors: Conductivity increases with increasing temperature. Thermal energy creates charge carriers (electrons, holes). 2

Classification of Solids by Bonding Solid, Molecule What holds a solid together? Separate Atoms Total energy difference between separate atoms and a solid or molecule: ΔU = V + + + V + 2 V + ion ion e e e ion (neighbor) repulsion repulsion attraction An atom is held together by electrostatic attraction between the positive nucleus and the negative electrons. Putting neutral atoms together into a molecule or a solid is more delicate. One has to consider the extra interactions between charges on different atoms. There are two repulsive and two attractive terms. Nevertheless, there is an opportunity for an electron to be attracted by more than one nucleus if the charges are distributed the right way (shown on the left, with V + a factor of 2 larger than either V ++ or V ). Quantum mechanics makes the argument more complicated though. Five generic types of bonds are listed below. The first three are strong bonds between atoms. The last two are weaker bonds between molecules, where most of the electrostatic energy has already been extracted by forming molecules. Types of Bonds Strong Weak 1) Covalent Si Both atoms bind e 2) Ionic NaCl Only one atom binds e 3) Metallic Al Neither atom binds e 4) Hydrogen Hydrophilic Static Dipoles 5) Molecular = Hydrophobic Oscillating Dipoles van der Waals Atom Atom Molecule-Molecule 3

1) Covalent Bond A shared electron pair is bound by two positive ions. Directional bond orbitals. The actual calculation proceeds in two steps. First, we consider the one-electron wave functions in H + 2, then we proceed to the two-electron wave functions in H 2. The bonding and antibonding one-electron wave functions ψ b and ψ a are the sum and difference of atomic wave functions centered on the two atoms. H 2 + ion (1-electron wave function ψ) Wave functions: ψ H 2 + bonding H 2 + anti-bonding ψ b ψ a + + A shared electron is close to both H ions (+), not just one. + + r ψ a antibonding 1 electron levels: H + H Arrows, for the electron spin ψ b bonding H 2 molecule (2-electron wave function Ψ) 2 electron levels: triplet :, ( + ), ψ a ψ b H H ψ b ψ b H H singlet : ( ) The triplet requires one electron in ψ b (bonding) and the other electron in ψ a (antibonding) due to the Pauli principle. For a comparison of 1- and 2-electron energy levels see Fig. 8.3 in the textbook. 4

2-Electron Wave Functions with Spin A 2-electron wave function Ψ 1,2 can be decomposed into products of a spatial wave function Ψ(r 1,r 2 ) and products of spin wave functions S 1,2 : Ψ 1,2 = Ψ(r 1,r 2 ) S 1,2 Ψ(r 1,r 2 ) consists of the 1-electron spatial wave functions: ψ b (r 1 ), ψ b (r 2 ), ψ a (r 1 ), ψ a (r 2 ) S 1,2 describes the spin orientation (up/down) of the two electrons: 1, 2, 1, 2 (Here we use the two molecular wave functions ψ b, ψ a of H + 2. One can also use atomic wave functions located on one or the other H atom, as done in the book.) Since electrons are fermions (spin ½), the two-electron wave function Ψ 1,2 must be antisymmetric with respect to an exchange of the two electrons 1,2: Ψ 1,2 = Ψ 2,1 This condition can be satisfied by making the spatial part symmetric and the spin part antisymmetric, or vice versa. The first option gives the bonding 2-electron wave function: Ψ b (r 1,r 2 ) = ψ b (r 1 ) ψ b (r 2 ) S 1,2 singlet: S 1,2 = [ 1 2 2 1 ]/ 2 The second option gives three possible anti-bonding 2-electron wave functions, because there are three ways to form a symmetric spin wave function: S + 1,2 = 1 2 Ψ a (r 1,r 2 ) = [ψ a (r 1 ) ψ b (r 2 ) ψ a (r 2 ) ψ b (r 1 ) ] S + 1,2 triplet: S + 1,2 = [ 1 2 + 2 1 ]/ 2 S + 1,2 = 1 2 This singlet/triplet pair is characteristic of many two-electron systems, for example in magnetism and in optical excitations of organic semiconductors used in organic LEDs: The ground state is a singlet (S 0 ) with paired spins. Optical transitions (solid arrows) are allowed from singlet to singlet and back (S 0 S 1, S 1 S 0 ). Other types of transitions (dotted) lead to the triplet T 1. From there one can have slow optical transitions back to the ground state (phosphorescence). Singlet and triplet are inverted in this excited state. 5

Tetrahedral semiconductors: Diamond, Si, Ge, GaAs C atom: 1s 2 2s 2 2p 2 (s 2 p 2 ) Diamond crystal: 1s 2 2s 1 2p 3 (sp 3 bonding) Promoting an electron from 2s to 2p in diamond costs energy ( 4eV), but that is more than made up by the energy lowering of the bonding levels. The resulting sp 3 orbitals point from an atom at the center of a tetrahedron to four neighbor atoms at the corners: + + ++ s 1 + p 3 4 sp 3 orbitals (l=0) (l=1) Two useful views for visualizing broken bonds at surfaces: Surface 1 up, 3 down, Orthogonal scissors, (111) Surface (100) Surface Calculated distribution of the outer electrons (blue) in a H-passivated silicon nanocrystal. Left: The bonding electrons sit halfway between the atoms in the bonding orbital: HOMO = Highest Occupied Molecular Orbital Valence Band Maximum = VBM Right: The empty antibonding orbitals are located outside the bond regions: LUMO = Lowest Unoccupied Orbital Conduction Band Minimum = CBM 6

Covalent versus Ionic Wave Functions: Covalent wave function Ψ AB Ionic wave function Ψ A + B A B A + B ψ b - - The two electrons are concentrated between the two atoms. r -- The two electrons are concentrated within one atom. r A linear combination of covalent and ionic wave functions minimizes the energy: Ψ = c 1 Ψ AB + c 2 Ψ + 2 A B c 1 + c 2 2 = 1 This provides a continuous transition from covalent bonding (c 2 = 0) to ionic (c 1 = 0). Even the symmetric H 2 molecule requires a few % of ionic bonding (H + H, H H + ). 2) Ionic Bond Electrons choose the atom with higher binding energy. Attraction between opposite ions. + Na + Cl The total energy is dominated by three terms (neglecting Pauli repulsion, see p. 9): 1) The sum over the electrostatic (Coulomb) energy between all ion pairs i,j (which is called Madelung energy). It is negative (attractive), since it is dominated by the Coulomb energy between the nearest neighbors, which have opposite charges. This energy can be written as a 1/r Coulomb potential (r = bond length) with a geometric pre-factor, the Madelung constant A (not to be confused with the electron affinity A below). Using the experimental r gives the Coulomb energy ϕ i per ion pair: ±e (1) ϕ i = Σ 2 e 2 1 j = A Madelung Energy ϕ i 9 ev (for NaCl) 4πε 0 r ij 4πε 0 r 1.44 ev nm r = 0.28 nm (for NaCl) ±1 p ij = r ij /r A = Σ j = Madelung Constant = 1.748 (for NaCl) p ij Thus, the Coulomb energy in solid NaCl is 1.748 times as large as in a NaCl molecule. 7

The ± signs represent the signs of the charge products q i q j of the ion pairs. The series can be summed to infinity by first summing over adjacent shells of opposite ions. The resulting Madelung constant A depends only on the crystal structure (here NaCl). 2), 3) It costs energy to form positive ions, but part of that energy is gained back by forming negative ions. These energies are the ionization energy and the electron affinity: Na 0 + I = Na + + e defines the ionization energy I +5 ev (for alkali atoms) Cl + A = Cl 0 + e defines the electron affinity A +4 ev (for halogen atoms) In both cases one needs energy to remove an electron (from an atom or ion). Therefore, both I and A both tend to be positive (with some exceptions for A). To form an ionic solid one has to convert neutral atoms to positive and negative ions. That requires switching sides in the second equation. As a result, I and A appear with opposite signs in the energy balance: (2) Na 0 + I = Na + + e (3) Cl 0 + e A = Cl (1)+(2)+(3): U solid U atoms + ϕ i + I A (energy per ion pair) 9 ev + 5 ev 4 ev 8 ev (for alkali halides) Bottom line: The binding energy in ionic crystals is dominated by the Coulomb attraction between the ions (the Madelung energy), as one would expect naively. The energies involved in converting neutral atoms into ions are significant, but they nearly cancel each other. 8

Bond Length The equilibrium bond length r is determined by the minimum of the total energy U as a function of r. That gives the condition: du(r)/dr = 0 For obtaining an accurate derivative, the calculation of U has to be improved. So far, we have mainly considered attractive (negative) contributions to U, but left out the Pauli repulsion. It is significant only at small r. This can be rationalized by going all the way to the limit where the bond length approaches zero. Then the electrons from adjacent atoms have almost the same wave functions. That is forbidden by the Pauli principle. The figure shows the situation for ionic bonding, where we considered already three contributions to the total energy U: 1) Coulomb energy, 2) Ionization energy, 3) Electron Affinity. Now we have to add Pauli Repulsion as 4 th term: r [0.1 nm] The Pauli repulsion increases exponentially at small distances, like the overlap between the wave functions. This positive exponential dominates over the 1/r Coulomb law at small r. At the energy minimum, however, the contribution of the Pauli repulsion to U remains small (red bar). 9