IV. Electronic Structure and Chemical Bonding

IV. Electronic Structure and Chemical Bonding
Hand-Outs: 20 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Chain of H atoms; lattice constant a; 1 H atom per unit cell… N (large) = Periodic Boundary Conditions. Atomic Orbital Basis: 1s AO at each H atom (1 AO/atom) OR +  Symmetry Adapted Linear Combination of Basis Functions (SALCs): (Bloch)

Hand-Outs: 21 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Chain of H atoms; lattice constant a; 1 H atom per unit cell… N (large) = Periodic Boundary Conditions. Hamiltonian (Energy) Matrix: 1 H atom/unit cell = 1 1s AO/unit cell… 11 matrix

Hand-Outs: 20 Hand-Outs: 21 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Chain of H atoms; lattice constant a; 1 H atom per unit cell… N (large) = Periodic Boundary Conditions. Hamiltonian (Energy) Matrix: 1 H atom/unit cell = 1 1s AO/unit cell… 11 matrix Hückel Approximation: Ignore interactions beyond first nearest neighbors “Coulomb” integral = AO Energy “Resonance” integral

Hand-Outs: 21 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Chain of H atoms; lattice constant a; 1 H atom per unit cell… N (large) = Periodic Boundary Conditions. Hamiltonian (Energy) Matrix: 1 H atom/unit cell = 1 1s AO/unit cell… 11 matrix Hückel Approximation: Ignore interactions beyond first nearest neighbors “Coulomb” integral = AO Energy “Resonance” integral (NOTE: E(k) = E(k), so we limit k to 0  k  /a)

Hand-Outs: 21 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Outcomes: Band Structure Density of States Crystal Orbital Overlap Population Bandwidth Antibonding Orbitals Fermi Level for H Chain Bonding Orbitals

Hand-Outs: 21 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Outcomes: Comparison of Band Structure and DOS Curve Band Structure Density of States Crystal Orbital Overlap Population Bandwidth Antibonding Orbitals Fermi Level for H Chain Bonding Orbitals k

Hand-Outs: 22 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Bandwidth Band Center /a

Hand-Outs: 22 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 -Bandwidth -Bandwidth Band Center /a

Hand-Outs: 22 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3

Hand-Outs: 23 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Band Crossings: Band centers vs. Bandwidths p  s > |  |’s p-Band

Hand-Outs: 23 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Band Crossings: Band centers vs. Bandwidths p  s > |  |’s p  s < |  |’s

Hand-Outs: 24 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 1 H atom / unit cell 1 1s AO / unit cell 2 a 2 H atoms / unit cell 2 1s AOs / unit cell 2 H atoms / unit cell 2 1s AOs / unit cell

Hand-Outs: 24 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 1 H atom / unit cell 1 1s AO / unit cell 2 a 2 H atoms / unit cell 2 1s AOs / unit cell   2 H atoms / unit cell 2 1s AOs / unit cell 2 1 2 Energy Matrix (Hamiltonian Matrix):

Hand-Outs: 24 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 2 a 1 = 2   No Distortion 2 1 2 Half-filled Band is unstable with respect to a Peierls Distortion: Electronically-driven

Hand-Outs: 24 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 2 a 1 = 2   2 1 2 “Band Folding”

Hand-Outs: 24 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 Polyacetylene Metallic

Hand-Outs: 24 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 Polyacetylene Metallic Semiconducting

Hand-Outs: 25 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 -Bands 11 valence e 10 valence e

Hand-Outs: 25 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 4 orbitals (BC *) -Bands 11 valence e 10 valence e 10 orbitals (BC , ) 2 orbitals (C 2s)

Hand-Outs: 25 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 YBC -Bands 11 valence e 10 valence e

Hand-Outs: 25 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 ThBC -Bands 11 valence e 10 valence e

Hand-Outs: 26 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 NbI4 High Temperatures Low Temperatures

Hand-Outs: 26 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 NbI4 High Temperatures Low Temperatures (33 valence electrons)

Hand-Outs: 26 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 NbI4 High Temperatures Low Temperatures kF = /2a kF = /2a (33 valence electrons)

Hand-Outs: 27 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Preventing Peierls Distortions (a) Oxidation or Reduction Polyacetylene (2x)+ (Br)2x

Hand-Outs: 27 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Preventing Peierls Distortions (b) Chemical Substitutions

Hand-Outs: 28 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Preventing Peierls Distortions (b) Chemical Substitutions: Charge Density Waves (static or dynamic) Wolfram’s Red Salt: [Pt(NH3)4Br]+ (X) + (Pt3+) Susceptible to a Peierls Distortion Pt 5dz2 Br 4p Br 4s

Hand-Outs: 28 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Preventing Peierls Distortions (b) Chemical Substitutions: Charge Density Waves (static or dynamic) Wolfram’s Red Salt: [Pt(NH3)4Br]+ (X) + (Pt3+) Susceptible to a Peierls Distortion Pt 5dz2 Br 4p Br 4s Pt-Br Bond length alternation does not change the qualitative picture!

Hand-Outs: 28 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Preventing Peierls Distortions (b) Chemical Substitutions: Charge Density Waves (static or dynamic) (Pt4+) (Pt2+) Wolfram’s Red Salt: [Pt(NH3)4Br]+ (X) + (Pt3+) Pt 5dz2 Br 4p Br 4s

Hand-Outs: 27 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Preventing Peierls Distortions (c) Interactions between Chains: Polysulfur nitride (SN)x

Hand-Outs: 27 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Preventing Peierls Distortions (c) Interactions between Chains: Polysulfur nitride (SN)x “Less than 1/2-filled” “More than 1/2-filled”

Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Preventing Peierls Distortions (d) Applying Pressure: Near-neighbor repulsive energy vs. orbital overlap (e) Increasing Temperature: Fermi-Dirac Distribution f(Fermi-Dirac) = [1+exp(EEF)/kT]1 EF

R. Hoffmann, Solids and Surfaces: A Chemist’s View of Bonding in Extended Structures, 1988. Summarizes material published in these review articles: “The meeting of solid state chemistry and physics,” Angewandte Chemie 1987, 99, “The close ties between organometallic chemistry, surface science, and the solid state,” Pure and Applied Chemistry 1986, 58, “A chemical and theoretical way to look at bonding on surfaces,” Reviews of Modern Physics 1988, 60,

Hand-Outs: 29 IV. Electronic Structure and Chemical Bonding Square Lattice J.K. Burdett, Chemical Bonding in Solids, Ch. 3 Real Space: H atoms at lattice points Reciprocal Space: Brillouin Zone ky y kx x (0, /a) (0, 0) (/a, /a) (Only nearest neighbor interactions:  )

Hand-Outs: 29 IV. Electronic Structure and Chemical Bonding Square Lattice J.K. Burdett, Chemical Bonding in Solids, Ch. 3 Wavefunctions M X 

Hand-Outs: 30 IV. Electronic Structure and Chemical Bonding Graphite: -Bands J.K. Burdett, Chemical Bonding in Solids, Ch. 3 y x a2 G (2) (1) a1 a2* K M a1* G: (0, 0) M: (1/2, 0) K: (1/3, 1/3)

Hand-Outs: 30 IV. Electronic Structure and Chemical Bonding Graphite: -Bands J.K. Burdett, Chemical Bonding in Solids, Ch. 3 G K M DOS Curve COOP Curve p-Antibonding “Zero-Gap Semiconductor” p-Bonding

Hand-Outs: 30 IV. Electronic Structure and Chemical Bonding Graphite: -Bands – What do the Wavefunctions Look Like at  (0, 0)? G -Antibonding K M -Bonding

Hand-Outs: 30 IV. Electronic Structure and Chemical Bonding Graphite: -Bands – What do the Wavefunctions Look Like at  (0, 0)? Totally Antibonding G K M Totally Bonding

Hand-Outs: 30 IV. Electronic Structure and Chemical Bonding Graphite: -Bands – What do the Wavefunctions Look Like at M (1/2, 0)? G -Antibonding K M -Bonding

Hand-Outs: 30 IV. Electronic Structure and Chemical Bonding Graphite: -Bands – What do the Wavefunctions Look Like at M (1/2, 0)? G K M

Graphite: -Bands – What is the Advantage of Reciprocal Space? Graphite C6 C13 C24

Hand-Outs: 31 IV. Electronic Structure and Chemical Bonding Graphite: Valence s and p Bands DOS Curve C-C COOP Curve -Bands Optimized C-C Bonding at EF 2pxpy “Poor” Metal 2pz  (“sp2”) 2s M G K M

Hand-Outs: 31 IV. Electronic Structure and Chemical Bonding Boron Nitride: Valence s and p Bands – Electronegativity Effects DOS B-N COOP Nonmetallic “N 2p” B-N Bonding “N 2s” B-N Bonding

Hand-Outs: 32 IV. Electronic Structure and Chemical Bonding MgB2 and AlB2: Valence Bands B: 63 Nets Integrated COHP Mg or Al DOS B-B COHP AlB2 MgB2 Mg or Al 3s, 3p AOs Some Mg-B or Al-B Bonding

Hand-Outs: 32 IV. Electronic Structure and Chemical Bonding MgB2 and AlB2: Energy Bands s Band below EF in AlB2 -Bands at EF in MgB2

Hand-Outs: 33 IV. Electronic Structure and Chemical Bonding Tight-Binding Model: Si (Integrated DOS = # Valence Electrons) (Integrated ICOHP) Si-Si Antibonding “sp3” Si-Si Bonding “sp3” 3s

Hand-Outs: 34 IV. Electronic Structure and Chemical Bonding Tight-Binding Model: Main Group Metals Valence s, p only Free-Electron Metal Nearly Free-Electron Metals Semi-Metals

Hand-Outs: 35 IV. Electronic Structure and Chemical Bonding Atomic Orbital Energies A.Herman, Modelling Simul. Mater. Sci. Eng., 2004, 12, Hartree-Fock Valence Orbital Energies

Hand-Outs: 36 IV. Electronic Structure and Chemical Bonding How are Bands Positioned in the DOS? NaCl Structures (Semimetallic) (Semiconducting) (Insulating) CaO ScN TiC

Hand-Outs: 37 IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3 Re 5d (t2g) (3 orbs.) EF (WO3) O 2p (9 orbs.)

Hand-Outs: 37 IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3 yz  (0, 0, 0)

Hand-Outs: 37 IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3 yz R (1/2, 1/2, 1/2)

Hand-Outs: 37 IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3 Re 5d (t2g) (3 orbs.) EF (WO3) O 2p (9 orbs.)

Hand-Outs: 37 IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3 yz  (0, 0, 0)

Hand-Outs: 37 IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3 yz R (1/2, 1/2, 1/2)

Hand-Outs: 38 IV. Electronic Structure and Chemical Bonding Populating Antibonding States: Distortions Inorg. Chem. 1993, 32, t2g Band d2 d3; d5 d6

Hand-Outs: 39 IV. Electronic Structure and Chemical Bonding NbO: Metal-Metal Bonding J.K. Burdett, Chemical Bonding in Solids, Ch. 4 3 “NbO” per unit cell 33 e Nb-Nb 24 e O 2s + 2p Nb-O

Hand-Outs: 38 IV. Electronic Structure and Chemical Bonding NbO: Metal-Metal Bonding J.K. Burdett, Chemical Bonding in Solids, Ch. 4 NbO in “NaCl-type” 3 “NbO” per unit cell 33 e 11 e Nb-Nb Nb-Nb 24 e 8 e O 2s + 2p Nb-O O 2s + 2p Nb-O

Hand-Outs: 40 IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Electron-Electron Interactions: TB Theory predicts NiO to be a metal – it is an insulator! E = 0 “Higher Potential Energy” Spin-Pairing Energy “Higher Kinetic Energy” Ligand-Field Splitting

Hand-Outs: 40 IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Electron-Electron Interactions: E = 0 “Higher Potential Energy” Spin-Pairing Energy “Higher Kinetic Energy” Ligand-Field Splitting EHS  ELS = 22P = 2(P) High-Spin:  < P Low-Spin:  > P

(Independent Electrons)
Hand-Outs: 40 IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5 H2 Molecule A B Energy ( > 0) EIE = 2() (Independent Electrons) A

Hand-Outs: 40 IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5 H2 Molecule Molecular Orbital Approach (Hund-Mulliken; “Delocalized”) A B MO(1,2) = ½ (A1A2 + A1B2 + B1A2 + B1B2) Energy “Covalent” “Ionic” “Ionic” contribution is too large; Poorly describes H-H dissociation ( > 0) EIE = 2() (Independent Electrons) EMO = 2() + U/2 A

Hand-Outs: 40 IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5 H2 Molecule Valence Bond Approach (Heitler-London; “Localized”) A B VB(1,2) = (A1B2 + B1A2) / 2 Energy “Ionic” contribution is too small; Describes H-H dissociation well ( > 0) EVB = 2 EIE = 2() (Independent Electrons) 0th Order – neglecting 2-electron Coulomb and Exchange Terms A

Hand-Outs: 40 IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Energy If U/ is small: If U/ is large: “Microstates” “Configuration Interaction”

Polytetrafluoroethylene
Metals and insulators Measured resistivities range over more than 30 orders of magnitude Material Resistivity (Ωm) (295K) Resistivity (Ωm) (4K) 10-12 “Pure”Metals Copper 10-5 Semi-Conductors Ge (pure) 5  102 1012 Insulators Diamond 1014 Polytetrafluoroethylene (P.T.F.E) 1020 Potassium 2  10-6 10-10

Metals, insulators & semiconductors?
1020- At low temperatures all materials are insulators or metals. Diamond 1010- Resistivity (Ωm) Germanium Pure metals: resistivity increases rapidly with increasing temperature. 100 - Copper 10-10- 100 200 300 Temperature (K) Semiconductors: resistivity decreases rapidly with increasing temperature. Semiconductors have resistivities intermediate between metals and insulators at room temperature.

Bound States in atoms V(r) r
Electrons in isolated atoms occupy discrete allowed energy levels E0, E1, E2 etc. . The potential energy of an electron a distance r from a positively charge nucleus of charge q is V(r) E2 E1 E0 r Increasing Binding Energy

Bound and “free” states in solids
The 1D potential energy of an electron due to an array of nuclei of charge q separated by a distance R is Where n = 0, +/-1, +/-2 etc. This is shown as the black line in the figure. V(r) E2 E1 E0 V(r) Solid V(r) lower in solid (work function). r + R Nuclear positions

Energy Levels and Bands
In solids the electron states of tightly bound (high binding energy) electrons are very similar to those of the isolated atoms. Lower binding electron states become bands of allowed states. We will find that only partially filled bands conduct Band of allowed energy states. E Electron level similar to that of an isolated atom + + + + + position

Reasonable for “simple metals” (Alkali Li,Na,K,Cs,Rb)
Band Theory The calculation of the allowed electron states in a solid is referred to as band theory or band structure theory. Free electron model: U(r) U(r) Neglect periodic potential & scattering (Pauli) Reasonable for “simple metals” (Alkali Li,Na,K,Cs,Rb)

Energy band theory Solid state N~1023 atoms/cm3 2 atoms 6 atoms

Metal – energy band theory

Band theory ctd. To obtain the full band structure, we need to solve Schrödinger’s equation for the full lattice potential. This cannot be done exactly and various approximation schemes are used. We will introduce two very different models, the nearly free electron and tight binding models. We will continue to treat the electrons as independent, i.e. neglect the electron-electron interaction.

Influence of the lattice periodicity
In the free electron model, the allowed energy states are where for periodic boundary conditions nx , ny and ny positive or negative integers. L- crystal dimension Periodic potential Exact form of potential is complicated Has property V(r+ R) = V(r) where R = m1a + m2b + m3c where m1, m2, m3 are integers and a ,b ,c are the primitive lattice vectors. E

Tight Binding Approximation
Tight Binding Model: construct wavefunction as a linear combination of atomic orbitals of the atoms comprising the crystal. Where f(r) is a wavefunction of the isolated atom rj are the positions of the atom in the crystal.

The tight binding approximation for s states
Solution leads to the E(k) dependence!! 1D: + a Nuclear positions

E(k) for a 3D lattice Simple cubic: nearest neighbour atoms at
So E(k) = - a -2g(coskxa + coskya + coskza) Minimum E(k) = - a -6g for kx=ky=kz=0 Maximum E(k) = - a +6g for kx=ky=kz=+/-p/2 Bandwidth = Emav- Emin = 12g For k << p/a cos(kxx) ~ 1- (kxx)2/2 etc. E(k) ~ constant + (ak)2g/2 c.f. E = (hk)2/me k [111] direction p/a -p/a = 10 g = 1 E(k) Behave like free electrons with “effective mass” h/a2g

Each atomic orbital leads to a band of allowed states in the solid
Gap: no allowed states

Reduced Brillouin zone scheme
The only independent values of k are those in the first Brillouin zone. Discard for |k| > p/a Results of tight binding calculation

The number of states in a band
Independent k-states in the first Brillouin zone, i.e. kx < /a etc. Finite crystal: only discrete k-states allowed Monatomic simple cubic crystal, lattice constant a, and volume V. One allowed k state per volume (2)3/V in k-space. Volume of first BZ is (2/a)3 Total number of allowed k-states in a band is therefore Precisely N allowed k-states i.e. 2N electron states (Pauli) per band This result is true for any lattice: each primitive unit cell contributes exactly one k-state to each band.

Metals and insulators In full band containing 2N electrons all states within the first B. Z. are occupied. The sum of all the k-vectors in the band = 0. A partially filled band can carry current, a filled band cannot Insulators have an even integer number of electrons per primitive unit cell. With an even number of electrons per unit cell can still have metallic behaviour due to band overlap. Overlap in energy need not occur in the same k direction EF Metal due to overlapping bands

EF Full Band Empty Band Energy Gap Part Filled Band Full Band Partially Filled Band Energy Gap EF INSULATOR METAL METAL or SEMICONDUCTOR or SEMI-METAL

Insulator -energy band theory

Covalent bonding Atoms in group III, IV,V,&VI tend to form
Filling factor T. :0.34 F.C.C :0.74

Covalent bonding Crystals: C, Si, Ge
Covalent bond is formed by two electrons, one from each atom, localised in the region between the atoms (spins of electrons are anti-parallel ) Example: Carbon 1S2 2S2 2p2 C C 2D 3D Diamond: tetrahedron, cohesive energy 7.3eV

Covalent Bonding in Silicon
Silicon [Ne]3s23p2 has four electrons in its outermost shell Outer electrons are shared with the surrounding nearest neighbor atoms in a silicon crystalline lattice Sharing results from quantum mechanical bonding – same QM state except for paired, opposite spins (+/- ½ ħ)

diamond

semiconductors

Intrinsic conductivity
ln(s) 1/T

Extrinsic conductivity – n – type semiconductor
ln(s) 1/T

Extrinsic conductivity – p – type semiconductor

Conductivity vs temperature
ln(s) 1/T

Physics 250-06 “Advanced Electronic Structure”
Earlier Electronic Structure Methods Contents: 1. Solution for a Single Atom 2. Solution for Linear Atomic Chain 3. LCAO Method and Tight-Binding Representation 4. Tight-Binding with BandLab

Solving Schrodinger’s equation for solids
Solution of differential equation is required Properties of the potential

Properties of Solution for a Single Atom
Atomic potential (spherically symmetric)

Property of solution for symmetric potential
Discrete set of levels is obtained. While degeneracy with respect to m remains, degeneracy with respect to l is now lifted since V V(r) is different from –Ze2/r . We can label the levels by main quantum number and orbital quantum number. For given l, n=l+1,l+2,... which is the property of the solution. For l=0 we have the states E1s, E2s,E3s,E4s,... For l=1 we have the states E2p, E3p,E4p,E5p,... For l=2 we have the states E3d, E4d,E5d,E6d,... No artificial degeneracy as in the hydrogen atom case: E2s=E2p,E3s=E3p=E3d,E4s=E4p=E4d=E4f… At some point, level Enl becomes above zero, i.e the spectrum changes from discrete to continuous. Note also that this spectrum of levels is a functional of V(r), i.e. it is different for a given atom with a given number of electrons N.

We can interpret different solutions as wave functions with different numbers of nodes.
For example, for l=2, there are solutions of the equations with no nodes, with one node, with two nodes, etc. Number of nodes = n-l-1. If n runs from l+1, l+2, ..., nodes =0,1,2,3...

Periodic table of elements
The one-electron approximation is very useful as it allows to understand what happens if we have many electrons accommodated over different levels. Let us take atom with N electrons. Lets us find all discrete levels E1s<E2s<E2p<... Since electrons are the fermions they obey Fermi-Dirac statistics, i.e. they cannot be more than two electrons (with opposite spins) occupying given non-degenerate level. If level is N-fold degenerate (for example, p level is 3 fold degenerate) then it can accomondate 2N electrons. So we can now fill various atomic shells with electrons E1s2,E2s2E2p6 and so on until we accomodate all N electrons within various slots. Thus we obtain a periodic table. For atom with given N we need to find discrete levels E1s,E2s,E3s,E2p,E3p,... We need to order them from lowest to highest. We need to fill the levels with electrons. In many cases, for a given number N ordering the levels is simple E1s<E2s<E2p<E3s... Therefore atoms of the periodic table have configurations which is easy to obtain. In some cases (since levels depend on Vscf(r) which depends on N) this rule is violated. This for example happens for later 3d elements.

Hunds rules For a given l shell, the electrons will occupy the slots to maximize the total spin For a given l shell, the electrons will maximize the total spin and total angular momentum Hunds rules cannot be understood on the basis of one-electron self-consistent approximation. However, for atomic system, the many body problem for N electrons moving in the Coulomb potential can be solved where En is the set of many body energy levels. Hunds rules follow from this Hamiltonian

Bringing atoms together: what to expect?
Properties of the solutions: Energy Bands Core Levels

Solution for a Linear Atomic Chain
Illustration: Periodic array of potential wells placed at distance L between the wells. Basic property of the potential V(x+l)=V(x), and of the Hamiltonian H(x+l)=H(x) Solving variationally:

We obtain infinite system of equations
If overlap is only between nearest wells, we simplify it to be This is infinite set of equations. At first glance it seems that we cannot solve it. But we can do it indeed.

introduce periodic boundary condition
Since Hamitonian is periodical function, all wells are equivalent for the electron. That would mean that the probability to find the electron in each of the well should be the same. That is possible if where u(x) is a periodical function: introduce periodic boundary condition where n is any integer number and phase is frequently called the wave vector k_n

obeys the periodicity condition automatically
Once we understand the form for the wave function, we already see one problem with the representation Each term in this expansion is not a periodical function, only the entire sum is periodical. The question is if we can construct another linear combinations of those basis functions so that each of the terms obeys the periodicity condition. In other words, we would like to have such combinations of with some coefficients c_n so that the combination obeys the periodicity condition automatically

where n can be any integer.
For this to happen or, the coefficient is equal to where n can be any integer. So we can label those combinations with index k_n which automatically obey the periodicity condition for wave function (Bloch theorem)

What we want is to is to use those linear combinations in finding the solutions
The advantage of this formulation is seen because all off-diagonal matrix elements disappear. This is seen because F(x) is periodical function, therefore Therefore, we automatically obtain the diagonalized Hamiltonian That mean that the spectrum is known to us In other words, we are able to solve the problem completely!

Let us analyze the solutions
We see that we can draw the solutions as a cosine centered around h as a function of wave number k. That is a band. Due to periodicity of cos, it is sufficient to draw it within –pi<kl<pi. We do not forget of course that k are discrete set of numbers However since we consider N to be very large number, we can deal with k as with continuous argument. The wave functions corresponding to each solutions are simply

Summary: Bloch Property for a Local Basis
Differential equation using expansion where is a basis set satisfying Bloch theorem To force the Bloch property we now use instead of plane waves: Linear combinations of local orbitals

Apply Variational Principle
Variational principle leads us to solve matrix eigenvalue problem where is hamiltonian matrix is overlap matrix

LCAO Method Linear Combination of Atomic Orbitals (LCAO)
Tailored to atomic potential to be used in variational principle

Hamiltonian of LCAO Method:
Hoppings between the orbitals

Tight-Binding Parametrization
In LCAO Method, the Hamitonain is parameterized via on site energies of the orbitals and nearest-neighbor hopping integrals between the orbitals. In many situations symmetry plays an important role since for many orbitals hoppings intergrals between them are automatically equal zero.

Thinking of Cubic Harmonics
- unitary transformation

Hoppings between various orbitals
ss,sp,sd hoppings pp,pd hoppings dd hoppings

Illustration, CuO2 plane
It is clear that levels are non-bonding and are all occupied. The same is true for level of copper, and for orbitals. The active degrees of freedom here are orbitals which have hopping rate . Using the active degrees of freedom as the basis the hamiltonian has the form

Let us introduce another basis set of bonding (b),
antibonding (a), and non-bonding (n) orbitals Within this basis set, the Hamiltonian becomes The bands are now seen as the bonding band below and the antibonding band above. The non--bonding band is also present.

MINDLab Software

Understanding s-electron band structure.
Tight-binding parameterization for Na. s level position and hopping rate for s electrons.

Understanding s-d electron band structure.
Tight-binding parameterization for Cu. s level position and hopping rate for s electrons. d level position and narrow d-bands. s-d hybridization

Tight-Binding method Band structure calculation Concept: Develop the wave function of the system into a set of atomic orbitals. sp3 tight-binding model: 4 orbitals/atom: 1 s + 3 p Interactions with the third neighbors. Three center integrals. Spin-orbit coupling. 3rd (12) Diamond structure: 2nd (12) 1st (4) Reference 20

Tight-Binding method 20 different coupling terms for Ge:*
Band structure calculation 20 different coupling terms for Ge:* ESS(000) eV ESS(111) Exx(000) Esx(111) Exx(111) Exy(111) Ess(220) Ess(311) Esx(220) Esx(311) Esx(022) Esx(113) Exx(220) Exx(311) Exx(022) Exx(113) Exy(220) Exy(311) Exy(022) Exy(113) Coupling terms between atomic orbitals are adjusted to give the correct band structure: semi-empirical method. * Y.M. Niquet et al. Phys. Rev. B, 62 (8): , (2000). 21 *Y.M. Niquet et al., Appl. Phys. Lett. 77, 1182 (2000).

Energy dispersion relations
In the bulk: The minimum of the conduction band is the DELTA valleys defined by six degenerated anisotropic bands.  Constant energy surfaces are six ellipsoids -valleys 23

Energy dispersion relations
T=1.36 nm T=2.72 nm T=5.15 nm Energy dispersion relations for the Silicon conduction band calculated with sp3 tight-binding model. The wires are infinite in the [100] x-direction. Direct bandgap semiconductor The minimum of D2 valleys are zone folded, and their positions are in k0=+/ Splitting between D4 subbands 24

Conduction band edge and effective masses
Bandgap increases when the dimensions of cross section decrease m* increases when the dimensions of cross section decrease : 25

Conduction band minima
Three types of conduction band minima: L point: four degenerated valleys (ellipsoidal).  point: single valley (spherical).  directions: six equivalent minima (ellipsoidal). -valleys -valleys 29

Dispersion relations*
T=5.65 nm Ge <100> 2 bulk valleys 4 bulk valleys Single  bulk valley 4  bulk valleys 4 bulk valleys Indirect band-gap. The minimum of CB obtained in kX=/a corresponding to the 4  bulk valleys. Second minimum of CB in kX=0, corresponding to the single  bulk valley (75% of s orbitals). *M. Bescond et al. J. Comp. Electron., accepted (2006). 30

Dispersion relations Ge <100> T=1.13 nm
The four bands at kX=/a are strongly shifted. The minimum of the CB moves to kX=0. The associated state is 50% s ( character) and 50% p ( and  character)  Quantum confinement induces a mix between all the bulk valleys.  These effects can not be reproduced by the effective mass approximation (EMA). 31

Effective masses:  point
Ge <100> (1/m*)=(4 ²/h²)( ²E/ k²) Significant increase compared to bulk value (0.04m0): From 0.071m0 at T=5.65nm to 0.29m0 at T=1.13nm  increase of 70% and 600% respectively.  Other illustration of the mixed valleys discussed earlier in very small nanowires. 32

Effective masses: kX=/a
Ge <100> Small thickness: the four subbands are clearly separated and gives very different effective masses. Larger cross-sections (D>4nm): the effective masses of the four subbands are closer, and an unique effective mass can be calculated: around 0.7m (effective mass: mtrans=0.6m0 for T=5nm) The minimum is not obtained exactly at kX=/a: 33

Band-gap: Ge vs Si Ge <100>
For both materials: the band gap increases by decreasing the thickness T (EMA). EG of Ge increases more rapidly than the one of Si: Si and Ge nanowires have very close band gaps.  Beneficial impact for Ge nano-devices on the leakage current (reduction of band-to-band tunneling). 34

Effective masses: Valence Band
Strong variations with the cross-section: from -0.18m0 to -0.56m0 (70% higher than the mass for the bulk heavy hole). 35

Semi-empirical methods
 Effective Mass Approximation (EMA): Near a band extremum the band structure is approximated by an parabolic function:  (Infinite system at the equilibrium) E(k) k Parabolic approximation of an homogeneous material Parabolic approximation of a finished system of atoms extra

IV. Electronic Structure and Chemical Bonding

Similar presentations

Presentation on theme: "IV. Electronic Structure and Chemical Bonding"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

IV. Electronic Structure and Chemical Bonding

Similar presentations

Presentation on theme: "IV. Electronic Structure and Chemical Bonding"— Presentation transcript:

Similar presentations

About project

Feedback