MIT OpenCourseWarehttp://ocw.mit.edu 5.04 Principles of Inorganic Chemistry II �� Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.04, Principles of Inorganic Chemistry II Prof. Daniel G. Nocera Lecture 9: Band Theory in Solids The LCAO method for cyclic systems provides a convenient starting point for the development of the electronic structure of solids. At very large N, as the circumference of the circle approaches ∞, the cyclic problem converges to a linear one, ∞Qualitatively, from a MO energy level perspective, ∞5.04, Principles of Inorganic Chemistry II Lecture 9 Prof. Daniel G. Nocera Page 1 of 10 More quantitatively, in moving from cyclic to linear systems, instead of describing orbital (atom) positions angularly, the position of an atom is described by ma, where m is the number of the atom in the array and a is the distance between atoms. Thus, the θ of the N-cyclic derivation becomes ma, ψk = ∑ eik(ma)φmψ j = ∑ ei jθφmmm Ej = α+ 2β cos 2π j Ek = α+ 2β cos 2π ja multiplied by a/aN Na = α+ 2β coska 2π jwhere k= Na A few words about k. It is: a measure of the number of nodes an index of wavefunction and accordingly symmetry of wavefunction a “quantum number” for a given ψk a measure of length, related to wavelength λ–1 from DeBroglie’s relation, λ= h , therefore k is also a wave vector that measures momentum p Returning to the foregoing discussion, note that k parametrically depends on a. Since a is a lattice parameter of the unit cell, there are as many k’s as there are unit cells in the crystal. In the linear case, the unit cell is the distance between adjacent atoms: there are n atoms ∴ n unit cells or in other terms – there are as many k’s as atoms in the 1-D chain. πLet’s determine the energy values of limits, k = 0 and k = : a at k = 0: ψ 0 = ∑ ei0(ma)φm = ∑φm mm π i ⎜⎜⎛ π a ⎟⎟⎞ (ma) at k = : ψπ = ∑ e ⎝⎠ φm = ∑ eimπφm = ∑ (−1)mφm a n mma 5.04, Principles of Inorganic Chemistry II Lecture 9 Prof. Daniel G. Nocera Page 2 of 10 The energies for these band structures at the limits of k are: E0 = α+ 2β cos(0)a = α+ 2β Eπ =α+ 2β cos(π )a = α− 2β a a Note that k is quantized; so there are a finite number of values between α+2β and α–2β but for a very large number (~1023 atoms) between the limits of k. Thus, the energy is a continuous and smoothly varying function between these limits. ππ πkThe range −≤ k≤ or ≤ is unique because the function repeats itself aa a outside these limits. This unique range of k values is called the Brillouin zone. The ππfirst Brillouin zone is plotted above from 0 to (symmetric reflection from − to aa 0). With a given number of e–s in the solid, the levels will be filled to a certain energy called the Fermi level, which corresponds to a certain value of k (= kF). In the kabove example, there are more electrons than there are orbitals, sokF > . If 2a πeach atomic orbital contributed 1e– to the system, then EF would occur forkF = .2a 5.04, Principles of Inorganic Chemistry II Lecture 9 Prof. Daniel G. Nocera Page 3 of 10 The symmetry of the individual atomic orbitals determines much about band structure. Consider p-orbitals overlapping in a linear array (vs the 1s orbitals of the above treatment). Analyzing limiting forms: at k = 0: ψ0 = φ1 + φ2 + φ3 + φ4 + …NOTE: This is the highest + … energy orbital with N nodes πat k = : ψπ = φ1 −φ2 + φ3 −φ4 + … a a NOTE: Lowest energy + … orbital with N nodes The energy band is opposite of that for the sσ orbital LCAO because the (+) LCAO for a pσ orbital is antibonding. Thus, molecules are easily related to solids via Hückel theory. Not surprisingly, there is a language of chemistry describing the electronic structure of molecules that is related to the language of physics describing the electronic structure of solids. Below are some of the terms that chemists and physicists use to describe similar phenomena in molecules and solids: 5.04, Principles of Inorganic Chemistry II Lecture 9 Prof. Daniel G. Nocera Page 4 of 10 LCAO–MO tight-binding model molecular orbital crystal or band orbital Jahn–Teller distortion Peierls distortionhigh or intermediate spin magnetic low spin non magnetic Band Width or Dispersion What determines the width or dispersion of a band? As for the HOMO-LUMO gap in a molecule, the overlap of neighboring orbitals determines the energy dispersion of a band – the greater the overlap, the greater the dispersion. Note how the band dispersion of a linear chain of H atoms varies as the 1s orbitals of the H atoms are spaced 1, 2, 3 Å apart (E of an isolated H atom is –13.6 eV): 5.04, Principles of Inorganic Chemistry II Lecture 9 Prof. Daniel G. Nocera Page 5 of 10 Density of States The total energy of the system is, normalization constant Etotal = N2π a ∫ − kkFF 2E(k)dk = 2N π a ∫0kF E(k)dk or in other words, it is the area under the curve to kF. Another useful quantity is the number of orbitals between E(k) + dE(k), called the density of states (DOS). For a 1-D system, inversely proportional to slope DOS ⎛⎞−1 of E(k) curve at given k. Since n(E(k))∝ ⎜⎝⎜ ∂∂ E(kk) ⎟⎠⎟ slope at k = 0 and k = π/a approaches 0, n(E(k)) is large A plot of the above equations is, DOS In the above DOS diagram, no energy gap separates the filled and empty bands, i.e. there is a continuous density of states – this property is characteristic of a metal. If an energy gap between filled and empty orbitals is present and it can be thermally surmounted, then it is semiconductor; an energy gap that cannot be surmounted is an insulator. A 1-D Example Arguably the best known 1-D system in inorganic chemistry is K2Pt(CN)4 and its partially oxidized compound (e.g. K2Pt(CN)4Br0.3). 5.04, Principles of Inorganic Chemistry II Lecture 9 Prof. Daniel G. Nocera Page 6 of 10 Normal platinocyanide, K2Pt(CN)4: an insulator, pale yellow compound Partially oxidized platinocyanide, K2Pt(CN)4Br0.3•3H2O: along chain: a conductor (500-1000 Ω–1 cm–1) a copper-colored with metallic luster ⊥ to chain: an insulator (0.05-0.01 Ω–1 cm–1), a yellow compound Note: d(Pt-Pt) = 2.78 Å in Pt metal To explain these disparate properties of the 1-D compounds, consider the molecular subunit Pt(CN)42-: for d8 system valence bonds completely filled … the energy gap between the M-Lσ derived valence band and the M-Lσ* derived conduction band is too large to be thermally populated, thus an insulator 5.04, Principles of Inorganic Chemistry II Lecture 9 Prof. Daniel G. Nocera Page 7 of 10 The dispersion of the bands is due to the different overlaps of the dσ, dπ and dδ orbitals. Band structure (or first Brillouin zones) derived from the frontier MO’s is: For partially oxidized system, the σ bond derived from dz2 should be partially filled and thus metallic, but it is not, partially oxidized K2Pt(CN)4Brx is a semiconductor. To explain this anomaly, consider how the band structure is perturbed upon partial oxidation: 5.04, Principles of Inorganic Chemistry II Lecture 9Prof. Daniel G. Nocera Page 8 of 10 Isolating on the dz2 band in K2Pt(CN)4, the Pt atoms are evenly spaced with lattice dimension a (I). Upon oxidation, the Pt chain can distort to give a lattice dimension 2a (II). In the case of the K2Pt(CN)4Br0.3 •3H2O the distortion is a rotation of Pt subunits and formation of dimers within chain, thus the unit cell dimension is pinned to every other Pt atom. For K2Pt(CN)4 (I): For K2Pt(CN)4Br0.3• 3H2O (II): Thus Brillouin diagram distorts Color code red: for partially occupied dz2 band with no distortion boundary condition for K2Pt(CN)4 boundary condition for Color code aqua marine: for partially occupied dz2 K2Pt(CN)4Br0.3 •3H2O band with distortion or pictorially… 5.04, Principles of Inorganic Chemistry II Lecture 9 Prof. Daniel G. Nocera Page 9 of 10 The Brillouin diagram above is for a distorted K2Pt(CN)4 but filled dz2 orbital. Hence both aqua marine bands are filled and the material that thus possesses the electronic structure shown below is insulating, despite the distortion Upon partial oxidation of K2Pt(CN)4 to produce K2Pt(CN)4Br0.3• 3H2O, the dz2 band is now partially filled. Hence a conductive material results. The conductivity is along the Pt-Pt axis, explaining the electrical anisotropy of the material 5.04, Principles of Inorganic Chemistry II Lecture 9 Prof. Daniel G. Nocera Page 10 of 10