Lectures Solid state materials

Slides:



Advertisements
Similar presentations
The one that is lower in energy is called the bonding orbital, The one higher in energy is called an antibonding orbital. These two.
Advertisements

© 2013 Eric Pop, UIUCECE 340: Semiconductor Electronics ECE 340 Lecture 3 Crystals and Lattices Online reference:
Energy Band View of Semiconductors Conductors, semiconductors, insulators: Why is it that when individual atoms get close together to form a solid – such.
CHAPTER 3 Introduction to the Quantum Theory of Solids
P461 - Solids1 Solids - types MOLECULAR. Set of single atoms or molecules bound to adjacent due to weak electric force between neutral objects (van der.
P461 - Solids1 Solids - types MOLECULAR. Set of single atoms or molecules bound to adjacent due to weak electric force between neutral objects (van der.
Lectures Molecular Bonding Theories 1) Lewis structures and octet rule
Lecture 15 Molecular Bonding Theories 1) Molecular Orbital Theory Considers all electrons in the field of all atoms constituting a polyatomic species,
Drawing the structure of polymer chains
An amazing thing about the universe - It works in a way that sometimes when things come together, they stick… Sections Sections H H H H.
Electronic Structure of Solids
Project topics due today. Next HW due in one week
SEMICONDUCTOR PHYSICS. BAND THEORY OF SOLIDS  Ge and Si are pure semiconductors  Electronic configuration of Si is  1S 2, 2S 2, 2P 6, 3S 2, 3P 2.
Formic acid: HCOOH Acetone.
Chemistry 100 Chapter 9 Molecular Geometry and Bonding Theories.
IV. Electronic Structure and Chemical Bonding
1 SEMICONDUCTORS Semiconductor atomic structure. 2 SEMICONDUCTORS We are going to look at the basic structure of an atom and compare and contrast the.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) These PowerPoint color diagrams can only be used by.
ECE 4339 L. Trombetta ECE 4339: Physical Principles of Solid State Devices Len Trombetta Summer 2007 Chapter 2: Carrier Modeling Goal: To understand what.
Chemistry 100 Chapter 9 Molecular Geometry and Bonding Theories.
Ψ 1 = c 1 φ 1 + c 2 φ 2 ψ 2 = c 1 φ 1 - c 2 φ 2 Molecular Orbital Theory LCAO-MO = linear combination of atomic orbitals Add and subtract amplitudes of.
Of bonds and bands How to understand MO theory for extended solids?
EEE 3394 Electronic Materials
Chemistry 100 Chapter 9 Molecular Geometry and Bonding Theories.
Molecular Orbital Energy Diagrams (16.7) MO energy diagrams are useful in that they show how atomic orbitals from different atoms may combine to molecular.
Chapter 10 Chemical Bonding II. Valence Bond Theory Valence Bond Theory: A quantum mechanical model which shows how electron pairs are shared in a covalent.
Atomic QM to Molecular QM ( ) Solution of SE for molecules is more complicated due to much larger number of electrons and multiple nuclei – SE.
BASIC ELECTRONICS Module 1 Introduction to Semiconductors
Energy of 3 π bonding orbitals lower than energy of 2p (isolated) orbitals on C from which they come. π antibonding are higher than isolated 2p. Find experimentally.
The Tightbinding Bandstructure Theory
Molecular Orbital Theory Molecular Orbitals Just as atomic orbitals belong to a particular atom, a molecular orbital belongs to molecules as a whole.
Why do bonds form? Energy of two separate H atoms Lennard-Jones potential energy diagram for the hydrogen molecule. Forces involved: We understand that.
Chemical Bonding II: Molecular Geometry and Hybridization of Atomic Orbitals Chapter 10 Copyright © The McGraw-Hill Companies, Inc. Permission required.
Molecular Orbital Theory
Last Time The# of allowed k states (dots) is equal to the number of primitive cells in the crystal.
CHEMISTRY 2000 Topic #1: Bonding – What Holds Atoms Together? Spring 2008 Dr. Susan Lait.
Topic #1: Bonding – What Holds Atoms Together?
Molecular Geometry and Bonding Theories
BASICS OF SEMICONDUCTOR
المملكة العربية السعودية وزارة التعليم العالي - جامعة أم القرى كلية الهندسة و العمارة الإسلامية قسم الهندسة الكهربائية ELECTRONIC DEVICES K INGDOM.
Drawing the structure of polymer chains
Origin of energy band formation:
Molecular Orbital Theory Molecular orbital theory describes covalent bonds in terms of molecular orbitals, which result from interaction of the.
Introduction to Semiconductors CSE251. Atomic Theory Consists of Electron, proton, neutron Electron revolve around nucleus in specific orbitals/shells.
Kronig-Penney model and Free electron (or empty lattice) band structure Outline: Last class: Bloch theorem, energy bands and band gaps – result of conduction.
Chemical Bonding And Intermolecular Forces. Chemical Bonds Forces of attraction that hold atoms or groups of atoms together and allow them to function.
Introduction to Molecular Orbital Theory.
Question on Van der Waals Interactions
Electrical Engineering Materials
Metallic Solids Metallic bond: The valence electrons are loosely bound. Free valence electrons may be shared by the lattice. The common structures for.
Solids: Conductors, Insulators and Semiconductors
Band Theory of Electronic Structure in Solids
Introduction to Semiconductor Material and Devices.
Molecular Orbital Theory
Schrödinger's Cat A cat is placed in an airtight box with an oxygen supply and with a glass vial containing cyanide gas to be released if a radiation detector.
SEMICONDUCTORS Semiconductors Semiconductor devices
Insulators, Semiconductors, Metals
Tightbinding (LCAO) Approach to Bandstructure Theory
Band Theory of Solids So far we have neglected the lattice of positively charged ions Moreover, we have ignored the Coulomb repulsion between the electrons.
Band Theory The other approach to band theory solves the Schrodinger equation using a periodic potential to represent the Coulomb attraction of the positive.
Physics 342 Lecture 28 Band Theory of Electronic Structure in Solids
Energy of 3 π bonding orbitals lower than energy of 2p (isolated)
Solids and semiconductors
Chapter 12 – Solids and Modern Materials
Band Theory of Solids 1.
Molecular Orbital Theory
Spin quantum number – ms
Energy Band 7 In free electron model, electrons occupy positive energy levels from E=0 to higher values of energy. They are valence electron so called.
Bonding & Molecular Structure:
Energy Band View of Semiconductors
Presentation transcript:

Lectures 21-22 Solid state materials Lectures 21-22 Solid state materials. Electronic structure and conductivity 1) Band theory The electronic structure of solids can also be described by MO theory. A solid can be considered as a supermolecule. One mole of atoms (NA), each with X orbitals in the valence shell contributes X moles of atomic orbitals producing X moles of MO’s. Consider qualitatively bonding between N metal atoms of ns1 configuration (Li, Na etc) arranged in a chain; N = 2, 4, NA. Assume that X=1 for simplicity. In the case of N ~ NA atoms they form not bonds but bands. The band appearing in the bonding region is called valence band. The antibonding region is called conduction band. In the case of metals the valence and conduction bands are immediately adjacent.

2) Band theory. Insulators, semiconductors, conductors If we apply now an electrostatic potential to a conductor, the population of the energy levels will tend to change and electrons will be able to flow using empty adjacent conduction band. In the case of insulators and semiconductors, the energy gap between the valence and conduction bands is more or less significant; electrons cannot easily get into the conduction band and cannot move along the sample; thermal or photo-energy is needed to bring some electrons to the conduction band. Bandgap, eV Conductivity, W-1cm-1 C 6.02 < 10-18 Si 1.09 5·10-6 Ge 0.72 0.02 Sn 0.07 104

3) Crystal Orbital theory The band structure of a crystalline material of virtually any complexity can be found through the application of the MO theory for solid state materials (Crystal Orbital theory). One of the ways to model a real (finite size) crystal is by using cyclic boundary conditions assuming that a chain of bound atoms forms a very large ring. It turns out that the energy levels in a cyclic molecule composed of N hydrogen atoms look as shown below.

4) Crystal orbitals (Bloch functions) If we have N hydrogen atoms with atomic wave functions fm (m = 1 … N) related by symmetry and spaced at distance a, we can get N MO’s yn (n = -N/2, …, 0, …, N/2) which are called Bloch functions. For the n-th crystal orbital, yn, we will have: When n changes from 0 to N/2, variable k = 2pn/(aN)) (wave vector) changes from 0 to p/a and the type of the MO changes from the completely bonding y0 to the completely antibonding yN/2: Energy levels of the resulting set of MO’s (band structure) can be described with help of continuous functions E and density of states dn/dE (DOS)

5) Bonding in solids: Crystal Orbital Overlap Population A common way to analyze bonding in solids is by calculating and analyzing the crystal orbital overlap population (COOP). COOP is defined in the same way as the bond order is defined in MO theory of molecules. For any two atoms i and j COOP(i-j) = S2cicjSij (Sij is the overlap integral for two atomic wavefunctions; summation should be performed for all pairs of overlapping orbitals of atoms i and j). A negative value of COOP means antibonding situation while a positive value is characteristic for bonding. For the chain of hydrogen atoms the lower half of the band is bonding while the upper half is antibonding (see diagram on the right).

6) Simplified picture of bonding in crystalline metals Using crystal orbital theory we can rationalize the well-known fact that the metals with highest melting points are those belonging to 6th and 7th groups (see diagram below).

7) The Peierls distortion When working with highly symmetrical structures one has to be cautious. Highly symmetrical structures with not completely filled degenerate or near-degenerate levels are a subject to distortions which lower the symmetry and the energy of the system (Peierls distortion). Diagrams on the left and in the center show how we can form bands for polymeric dihydrogen (s-MO) with twice larger four-atomic unit 2a and then distort the polymer to produce an array of dihydrogen molecules (the diagram on the right). Similarly an infinite polyene -HC=HC-HC=HC-… polyacetylene will have alternating HC-HC and HC=HC bonds due to the Peierls distortion. Because of the large band gap it will behave not as a conductor but as an semiconductor.

8) Band structure of one dimensional polymers: a stack of PtII square planar complexes In some cases one dimensional consideration is sufficient for a satisfactory analysis of band structure of solids – one dimensional polymers. For example, we can get a satisfactory description of bonding and conductivity of K2[Pt(CN)4Clx] (x = 0 … 0.3) using just one-dimensional model of crystal. The complexes K2[Pt(CN)4Clx] (x = 0 … 0.3) have Pt(CN)4 – squares stacked one above another with Pt-Pt separation of 3.3 (x = 0) or 2.7-3.3 Ǻ (0 < x < 0.3). Purely PtII complex (x = 0 in the formula above) is an insulator while oxidized cyanoplatinates are low-dimensional conductors.

9) Forming bands: Principles To predict a qualitative band structure of stacked [Pt(CN)4]2-, we will consider [PtH4]2- as a model. We will need for this analysis a MO diagram of PtL4. Each of the monomer’s MOs generates a band when we form a polymer. We can analyze all MO’s one by one and then combine all bands together. To get an idea about bands width use the rule which states that better orbital overlap will produce a wider band (s>p>d):

10) How bands behave To learn, how the “frontier” bands will run (“up” or “down”) let’s write corresponding Bloch functions for frontier orbitals, pz, and all d-orbitals, for k = 0 and k = p/a.

11) Band structure of a stacked [PtH4]2- The predicted band structure of a stacked [PtH4]2- in the center match well a calculated diagram on the right. With band structure or DOS diagram in hands we can answer the questions: 1) why oxidized K2[Pt(CN)4Clx] (x>0) is a conductor and 2) why Pt-Pt distance shortens as x increases.

12) Bonding and conductivity in stacked [PtH4]2- Conductivity. The Fermi level of stacked [PtH4]2- is on the top of the z2-band since the monomer HOMO is dz2 orbital. The conduction band is pz-band which is almost 3 eV higher in energy. When the z2-band is completely filled (case of PtIIL4), no conductivity is expected / observed. For partially oxidized materials z2–band is filled only partially and we expect and observe conductivity. Bonding. In solids like in molecules if bonding and antibonding MO’s are completely filled, the net bonding is zero. For partially oxidized materials K2[Pt(CN)4Clx] (x = 0 … 0.3) z2–band is partially empty and we observe s(dz2-dz2) bonding between Pt atoms.