IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids Experimental Aspects (a) Electrical Conductivity – (thermal or optical) band gaps; (b) Magnetic Susceptibility – localized or itinerant; para- or diamagnetic; (c) Heat Capacity – specific heat due to conduction electrons; lattice; (d) Cohesive Energy – energy required to convert M(s) to M(g); (e) Spectroscopy – XPS, UPS (for example); (f) Phase Changes – under temperature or pressure variations

IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids Experimental Aspects (a) Electrical Conductivity – (thermal or optical) band gaps; (b) Magnetic Susceptibility – localized or itinerant; para- or diamagnetic; (c) Heat Capacity – specific heat due to conduction electrons; lattice; (d) Cohesive Energy – energy required to convert M(s) to M(g); (e) Spectroscopy – XPS, UPS (for example); (f) Phase Changes – under temperature or pressure variations Theoretical Aspects (a) Electronic Density of States (DOS curves) – occupied and unoccupied states; (b) Electron Density – where does electronic charge “build up” in a solid? (c) Analysis of DOS – overlap (bonding) populations, charge partitioning,… (d) Band structure – energy dispersion relations; (e) Equations of State – E(V) curves for various structures; (f) Phonon DOS – vibrational states of crystals; stability of structures (  < 0 ??) (g) “Molecular Dynamics” – phase transitions; crystallization models;

IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids van Arkel-Ketelaar Triangle Average Electronegativity Electronegativity Difference L.C. Allen, J. Am. Chem. Soc. 1992, 114, 1510 Hand-Outs: 1

IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids van Arkel-Ketelaar Triangle Average Electronegativity Electronegativity Difference L.C. Allen, J. Am. Chem. Soc. 1992, 114, 1510  = “Configuration Energy” L.C. Allen et al., JACS, 2000, 122, 2780, 5132 Hand-Outs: 1

IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids van Arkel-Ketelaar Triangle Low valence e  /orbital ratio Low IP(I) Small  High valence e  /orbital ratio High IP(I) Small  Large  Charge transfer from cation to anion Average Electronegativity Electronegativity Difference L.C. Allen, J. Am. Chem. Soc. 1992, 114, 1510 Hand-Outs: 1

IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids van Arkel-Ketelaar Triangle Electrical Conductors Paramagnetic; Itinerant magnetism Soft – malleable, ductile Electrical Insulators Diamagnetic Low boiling points Electrical Insulators; Conducting liquids Diamagnetic; Localized magnetism Brittle Average Electronegativity Electronegativity Difference L.C. Allen, J. Am. Chem. Soc. 1992, 114, 1510 Hand-Outs: 1

IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids van Arkel-Ketelaar Triangle Elect. Semiconductors / Semimetals Diamagnetic “Hard” – Brittle Electrical Semiconductors Diamagnetic “Hard” – Brittle Elect. Semiconductors / Semimetals Diamagnetic “Hard” – Brittle L.C. Allen, J. Am. Chem. Soc. 1992, 114, 1510 Hand-Outs: 1

IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids Schrödinger’s Equation:   {n} = E {n}  {n}  :“Hamiltonian” = Energy operator Kinetic + Potential energy expressions; external fields (electric, magnetic)  {n} :Electronic wavefunctions (complex)  (r) =   * {n}  {n} dV: Charge density (real) E {n} : Electronic energies Temperature: How electronic states are occupied – Maxwell-Boltzmann Distribution:f(E) = exp[  (E  E F )/kT] Fermi-Dirac Distribution:f(E) = [1+exp(  (E  E F )/kT)]  1

IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids Schrödinger’s Equation:   {n} = E {n}  {n} “A solid is a molecule with an infinite number (ca ) of atoms.”

IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids Schrödinger’s Equation:   {n} = E {n}  {n} “A solid is a molecule with an infinite number (ca ) of atoms.” Molecular Solids: on molecular entities (as in gas phase); packing effects?

IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids Schrödinger’s Equation:   {n} = E {n}  {n} “A solid is a molecule with an infinite number (ca ) of atoms.” Molecular Solids: on molecular entities (as in gas phase); packing effects? Extended Solids: how to make the problem tractable? (a) Amorphous (glasses): silicates, phosphates – molecular fragments, tie off ends with simple atoms, e.g., “H”; (b) Quasiperiodic: fragments based on building units, tie off ends with simple atoms, e.g., “H”; (c) Crystalline: unit cells (translational symmetry) – elegant simplification!

IV. Electronic Structure and Chemical Bonding J.K. Burdett, Chemical Bonding in Solids Electronic Structure of Si: Fermi Level Electronic Band StructureElectronic Density of States What can we learn from this information?

IV. Electronic Structure and Chemical Bonding Periodic Functions J.K. Burdett, Chemical Bonding in Solids General, single-valued function, f (r), with total symmetry of Bravais lattice: Plane waves: e i  = cos  + i sin   (r) = K  r,K: units of 1/distance K  t n = 2  N {K m } = Reciprocal Lattice:K m = m 1 a 1 * + m 2 a 2 * + m 3 a 3 * (m 1, m 2, m 3 integers) Therefore, for r = ua 1 + va 2 + wa 3, the general periodic function of the lattice is Hand-Outs: 2

IV. Electronic Structure and Chemical Bonding Group of the Lattice J.K. Burdett, Chemical Bonding in Solids Bravais Lattice: {t n = n 1 a 1 + n 2 a 2 + n 3 a 3 ; n 1, n 2, n 3 integers} (1) Closed under vector addition:t n + t m = t n+m  lattice (2) Identity:t 0 = 0  lattice (3) Vector addition is associative:(t n + t m ) + t p = t n + (t m + t p ) (4) Inverse:t  n =  t n, t n + t  n = 0 ALSO: (5) Vector addition is commutative:t n + t m = t m + t n The (Bravais) Lattice is an “Abelian group”: (a) # classes = # members of the group (b) # members of the group = # irreducible representations (IRs) (c) each IR is one-dimensional (a 1  1 matrix; a complex number, e i  ) (d) Periodic (Born-von Karman) Boundary Conditions: Set N 1 a 1 = identity (like 0), N 2 a 2 = identity, and N 3 a 3 = identity 1  n 1  N 1, 1  n 2  N 2,1  n 3  N 3,Order of {t n } = N = N 1 N 2 N 3 Therefore, N IRs, each labeled k m : Hand-Outs: 2

IV. Electronic Structure and Chemical Bonding Periodic Boundary Conditions: Reciprocal (“k”) Space Real Space (2D) N 1 = 3; N 2 = 3 n 1 = 1, 2, 3; n 2 = 1, 2, 3 a1a1 a2a2 Hand-Outs: 3

IV. Electronic Structure and Chemical Bonding Periodic Boundary Conditions: Reciprocal (“k”) Space Real Space (2D) N 1 = 3; N 2 = 3 n 1 = 1, 2, 3; n 2 = 1, 2, 3 a1a1 a2a2 Lattice Group (9 members) = {a 1 +a 2, 2a 1 +a 2, 3a 1 +a 2,a 1 +2a 2, 2a 1 +2a 2, 3a 1 +2a 2, a 1 +3a 2, 2a 1 +3a 2, 3a 1 +3a 2 } Hand-Outs: 3

IV. Electronic Structure and Chemical Bonding Periodic Boundary Conditions: Reciprocal (“k”) Space Real Space (2D) N 1 = 3; N 2 = 3 4a 2 = 3a 2 + a 2  4a 2  a 2 n 1 = 1, 2, 3; n 2 = 1, 2, 3 a1a1 a2a2 Lattice Group (9 members) = {a 1 +a 2, 2a 1 +a 2, 3a 1 +a 2,a 1 +2a 2, 2a 1 +2a 2, 3a 1 +2a 2, a 1 +3a 2, 2a 1 +3a 2, 3a 1 +3a 2 } Hand-Outs: 3

IV. Electronic Structure and Chemical Bonding Periodic Boundary Conditions: Reciprocal (“k”) Space Real Space (2D) N 1 = 3; N 2 = 3 Lattice Group (9 members) = {a 1 +a 2, 2a 1 +a 2, 3a 1 +a 2,a 1 +2a 2, 2a 1 +2a 2, 3a 1 +2a 2, a 1 +3a 2, 2a 1 +3a 2, 3a 1 +3a 2 } 4a 2 = 3a 2 + a 2  4a 2  a 2 n 1 = 1, 2, 3; n 2 = 1, 2, 3 Reciprocal Space a1a1 a2a2 a1*a1* a2*a2* Hand-Outs: 3

IV. Electronic Structure and Chemical Bonding Periodic Boundary Conditions: Reciprocal (“k”) Space Real Space (2D) N 1 = 3; N 2 = 3 4a 2 = 3a 2 + a 2  4a 2  a 2 n 1 = 1, 2, 3; n 2 = 1, 2, 3 Reciprocal Space a1a1 a2a2 a1*a1* a2*a2* Allowed IRs (9 k-points) k 11 = (1/3)a 1 *+ (1/3)a 2 *; k 12 = (1/3)a 1 *+ (2/3)a 2 *; k 13 = (1/3)a 1 *+ (3/3)a 2 *; k 21 = (2/3)a 1 *+ (1/3)a 2 *; k 22 = (2/3)a 1 *+ (2/3)a 2 *; k 23 = (2/3)a 1 *+ (3/3)a 2 *; k 31 = (3/3)a 1 *+ (3/3)a 2 *; k 32 = (3/3)a 1 *+ (2/3)a 2 *; k 33 = (3/3)a 1 *+ (3/3)a 2 * Lattice Group (9 members) = {a 1 +a 2, 2a 1 +a 2, 3a 1 +a 2,a 1 +2a 2, 2a 1 +2a 2, 3a 1 +2a 2, a 1 +3a 2, 2a 1 +3a 2, 3a 1 +3a 2 } Hand-Outs: 3

IV. Electronic Structure and Chemical Bonding Group of the Lattice: IRs Lattice: {t 1, t 2, t 3, t 4 = identity} Real Space Hand-Outs: 4

IV. Electronic Structure and Chemical Bonding Group of the Lattice: IRs Lattice: {t 1, t 2, t 3, t 4 = identity} IRs: k 1, k 2, k 3, k 4 Real SpaceReciprocal Space Hand-Outs: 4

IV. Electronic Structure and Chemical Bonding Group of the Lattice: IRs Lattice: {t 1, t 2, t 3, t 4 = identity} IRs: k 1, k 2, k 3, k 4 Real SpaceReciprocal Space Hand-Outs: 4 f (x) = General function on 1D Lattice = Basis function of 1D Lattice

IV. Electronic Structure and Chemical Bonding Group of the Lattice: IRs Lattice: {t 1, t 2, t 3, t 4 = identity} IRs: k 1, k 2, k 3, k 4 Basis Function for IR k m : Real SpaceReciprocal Space Hand-Outs: 4 f (x) = General function on 1D Lattice = Basis function of 1D Lattice

IV. Electronic Structure and Chemical Bonding Group of the Lattice: IRs (Character Table) t1t1 t2t2 t3t3 t4t4 Basis Functions (Real / Imaginary)Most General k1k1 i 11 ii 1 Real: Imag: e i  x/2a (Complex conjugate of k 3 ) k2k2 11 1 11 1 Real e i  x/a (Real Representation) k3k3 ii 11 i1 Real: Imag: e 3i  x/2a = e  i  x/2a (Complex conjugate of k 1 ) k4k Real e 2i  x/a = 1 (Totally symmetric rep) Hand-Outs: 4

IV. Electronic Structure and Chemical Bonding Group of the Lattice: Reciprocal Space As the size of the real space lattice increases,N large (ca in each direction)  Reciprocal space becomes continuous set of k-points: … t1t1 t2t2 …tNtN k1k1 :1 k2k2 :1 ::::: kNkN 1111  {k m } is a “quasi”-continuous space;“k m ” = “k-point” or “wavevector” Identity Operation Totally Symmetric Representation Hand-Outs: 4

IV. Electronic Structure and Chemical Bonding Bloch’s Theorem The wavefunctions for electrons, phonons (= lattice vibrations) subjected to periodic potential, i.e., U(r + t) = U(r), take the form  nk (r) = e ik  r u n (r) where u n (r) has the full periodicity of the lattice, i.e., u n (r + t) = u n (r). Note that  nk (r + t) = e ik  t  nk (r) Therefore, for a determination of electronic states or vibrational modes in crystals, we only need to treat the contents of the unit cell (primitive cell)! Hand-Outs: 5

IV. Electronic Structure and Chemical Bonding Bloch’s Theorem The wavefunctions for electrons, phonons (= lattice vibrations) subjected to periodic potential, i.e., U(r + t) = U(r), take the form  nk (r) = e ik  r u n (r) where u n (r) has the full periodicity of the lattice, i.e., u n (r + t) = u n (r). Note that  nk (r + t) = e ik  t  nk (r) Therefore, for a determination of electronic states or vibrational modes in crystals, we only need to treat the contents of the unit cell (primitive cell)! Corollary #1 If K = reciprocal lattice vector, then  nk (r) and  nk+K (r) have the same symmetry properties with respect to translations (same IR!)…  nk (r + t) = e ik  t  nk (r);  nk+K (r + t) = e i(k+K)  t  nk+K (r) = e ik  t  nk+K (r) Hand-Outs: 5

IV. Electronic Structure and Chemical Bonding Bloch’s Theorem The wavefunctions for electrons, phonons (= lattice vibrations) subjected to periodic potential, i.e., U(r + t) = U(r) take the form  nk (r) = e ik  r u n (r) where u n (r) has the full periodicity of the lattice, i.e., u n (r + t) = u n (r). Note that  nk (r + t) = e ik  t  nk (r) Therefore, for a determination of electronic states or vibrational modes in crystals, we only need to treat the contents of the unit cell (primitive cell)! Corollary #1 If K = reciprocal lattice vector, then  nk (r) and  nk+K (r) have the same symmetry properties with respect to translations (same IR!)…  nk (r + t) = e ik  t  nk (r);  nk+K (r + t) = e i(k+K)  t  nk+K (r) = e ik  t  nk+K (r) Hand-Outs: 5

IV. Electronic Structure and Chemical Bonding Bloch’s Theorem The wavefunctions for electrons, phonons (= lattice vibrations) subjected to periodic potential, i.e., U(r + t) = U(r) take the form  nk (r) = e ik  r u n (r) where u n (r) has the full periodicity of the lattice, i.e., u n (r + t) = u n (r). Note that  nk (r + t) = e ik  t  nk (r) Therefore, for a determination of electronic states or vibrational modes in crystals, we only need to treat the contents of the unit cell (primitive cell)! Corollary #2  n,  k (r) is the complex conjugate of  nk (r)…  n,  k (r) = e  ik  r u n (r) =  nk *(r) Hand-Outs: 5

IV. Electronic Structure and Chemical Bonding Brillouin Zones Allowed IRs for the set of lattice translations are confined to one primitive cell in reciprocal space: (first) Brillouin zone {k m : k m = m 1 a 1 * + m 2 a 2 *; 0 < m i  1} a 1 a 2 * a1*a1* a2*a2* Hand-Outs: 6 Consider a 2D Orthorhombic Lattice: a1a1 a2a2

IV. Electronic Structure and Chemical Bonding Brillouin Zones Allowed IRs for the set of lattice translations are confined to one primitive cell in reciprocal space: (first) Brillouin zone {k m : k m = m 1 a 1 * + m 2 a 2 *; 0 < m i  1} {k m : k m =  1 a 1 * +  2 a 2 *;  1/2 <  i  1/2} (First) Brillouin Zone (FBZ) (Wigner-Seitz cell) “Zone Boundary” “Zone Edge” “Zone Center” =  a1*a1* a2*a2* Hand-Outs: 6 a 1 a 2 *