The electronic structure of solids We need a picture of the electronic structure of solid that we can use to explain experimental observations and make predictions Why is diamond an insulator? Why is sodium a metal? Why does the conductivity of silicon increase when you heat it? Charge transport in solids The conductivity of a material is determined by three factors: the charge on the charge carriers the number of charge carriers the mobility of the charge carriers σ = n e µ Page 1
Conductivity of common materials ❽Very large variation in conductivity A simple model for metals Consider a metal such as Na, Mg or Al to be essentially a box in which the valence electrons of the metal are confined. The potential within the box is taken to be uniform and much lower than that in the surrounding medium. Treat quantum mechanically as a particle in a box Page 2
The Free Electron Theory You can get solutions of the form ψ n (r) = A sin(πn x x/l) sin(πn y y/l) sin(πn z z/l) these solutions are standing waves Adjust the boundary conditions to get traveling waves ψ k (r) = exp (i k.r), k is the wavevector ε k = (h 2 /2m) (k x2 + k y2 + k z2 ) k can not take all values but in many electron systems it is almost continuous k, the wavevector k is related to the momentum of the electrons in the orbital p =hk k is related to the wavelength of the electron wave k = 2 π / λ Page 3
Solutions to a three dimensional particle in a box problem E = (n 2 x + n 2 y + n 2 z)h 2 / (8ma 2 ) For a large box the energy levels are going to be close together Consider energy levels as forming a continuous band How many energy levels do we have with energy less than some critical value? Number of electrons below E max We can have two electrons per unique combination of nx, ny and nz Set R 2 = n 2 x + n 2 y + n 2 z N = 2 (1/8) (4/3)π R 3 max = (8π/3)(2mE max /h 2 ) 3/2 a 3 Page 4
N(E) - Density of States (DOS) N(E) = 4π(2m/h 2 ) 3/2 E 1/2 At temperatures above 0 K some higher energy states are occupied, f(e) = {1+ exp[e-e F )/kt]} -1 X-ray emission spectra for sodium and aluminum ❽ Spectra strongly resemble simple parabolic density of states predicted by free electron model Na Al Page 5
Temperature dependence of electron distribution ❽ At temperatures above 0K some electron promoted to states with higher k. Electron distribution described by Fermi-Dirac distribution function f ( E) = 1+ exp 1 [( E E ) KT ] F / The wavevector k Classically the kinetic energy of an electron is given by E = p 2 / 2m The free electron model gives the energy of an electron as, E = (n 2 x + n 2 y + n 2 z)h 2 / (8ma 2 ) The momentum p is usually expressed as kh so E = (k 2 x + k 2 y + k 2 z) h 2 /2m Page 6
Electrical conductivity In the absence of an electric field states corresponding to k and -k are equally likely to be populated so there is no overall movement of charge In the presence of an electric field states with the same k but differing k do not necessarily have the same energy. This can lead to charge transport. Electrical conductivity Page 7
Limitations of the free electron model Predicts all materials will be metals! The tight binding approximation Consider a solid to be a large molecule and apply molecular orbital theory Page 8
MOs for evenly spaced H atoms Solids have MOs that are so close in energy they form continuous bands H H 2 H 4 H 9 H n Chains of CH units Consider polyene chains Evenly spaced (CH)H 2 (CH) 2 H 2 (CH) 4 H 2 (CH) 8 H 2 (CH) n H 2 Bond alternation Page 9
Electronic structure of NaCl The band gap ❽The occurrence of groups or bands of orbitals with energy gaps in between them is common Page 10
The origin of band gaps The chemists view atoms in solids have orbitals that overlap to produce large molecular orbitals These molecular orbitals do not occur at all energies The physicists view we need to modify the theory to take into account the periodicity of the structure electron waves can be diffracted by a regular array on ions in a solid Variation of band width and overlap with interatomic distance ❽Pushing atoms closer together increases orbital overlap and increases band widths Calculated for Na using TBA Page 11
Variation of conductivity with pressure ❽ As pressure effects interatomic distances and band widths it can have a profound influence on electronic conductivity What are the coefficients for the orbitals in the bands? Consider a chain of atoms Use LCAO, Ψ(x) = Σ c n ψ n (x) The periodicity of the chain limits the possible solutions for c n c n = exp(ikna) Page 12
Bloch functions for 1D chain Conductivity of solids This approach to the electronic structure of solids naturally introduces electronic states (orbitals) with characteristic momentum p=hk Electrical conductivity can again be related to differing numbers of electrons in states with +k and -k Conductivity is limited by lattice vibrations (phonons) in metals Page 13
Bands in metals, semiconductors and insulators E E E Metal Intrinsic semiconductor Insulator Insulators All bands are fully occupied or empty making it impossible for more electrons to be in states with +k rather than -k Page 14
The band structure of group IV elements Intrinsic and extrinsic semiconductors ❽ In an intrinsic semiconductor the conduction band is populated by thermal excitation of electrons from the valence band ❽ In an extrinsic semiconductor doping is used to produce partially occupied bands Page 15
Band gaps for Group IV Elements Band gaps for inorganic compounds Page 16
Doping semiconductors The addition of very small amounts of dopant can dramatically influence properties P, As added to silicon gives n - type material B, Al,Gagives p -type material The conductivity of doped semiconductors varies less with temperature Extrinsic semiconductors Doping can be used to increase the conductivity of a semiconductor conduction band E E valence band p doping n doping Page 17
Temperature dependence of electron distribution Temperature dependence of conductivity The conductivity of a metal decreases with increasing temperature mobile electrons are scattered by lattice vibrations The conductivity of a semiconductor increases with increasing temperature as more charge carriers become available Page 18
Doped graphites Graphite is a semimetal doping with bromine or potassium improves its conductivity Br 2 E E K E TiO 2, VO 2 and TiS 2 ❽ TiO 2 is an insulator as the d-bands are empty ❽ VO 2 at higher temps is metallic as the d-band is partly filled ❽ TiS 2 is a metal as S 3p and Ti 3d bands overlap Page 19
Li x V 2 O 5 V 2 O 5 has an empty d band and a layered structure Intercalation of Li into the material dopes the V 2 O 5 puts electrons in to the empty d band This improves the solids conductivity VO 2 VO 2 has a rutile like structure chains of edge sharing VO 6 octahedra V(IV) has d 1 electron configuration at low temperatures it displays localized metalmetal bonds and is a semiconductor at high temperatures the structural distortion disappears and it is a metal Page 20
Phase transitions in VO 2 metal d band splits E High T Low T oxygen band Polyacetylene Polyacetylene is a semiconductor because it displays bond alternation without bond alternation it would be a metal It can be doped to make it conducting use oxidizing agents, AsF 5, I 2 etc. to remove electrons from the valence band Page 21
K 2 [Pt(CN) 4 ]Br 0.3. 3H 2 O In KCP the Pt d z 2 orbitals are in an evenly spaced chain (at room temp) forming a single band E Insulator Metal K 2 [Pt(CN) 4 ] Bromine doped Structure of K 2 [Pt(CN) 4 ]Br 0.3. 3H 2 O ❽ 1D chain compound with overlapping d z 2 orbitals Page 22
Peierls distortion On cooling below 150K the conductivity of KCP drops rapidly this is associated with a lattice distortion E Page 23