Introduction to Condensed Matter Physics Crystalline Solids - Introduction M.P. Vaughan
Overview
Overview of course Crystal solids Crystal structure Crystal symmetry The reciprocal lattice Band theory X-ray diffraction Phonons Specific heat Elasticity
Overview of crystal solids Ordered and disordered materials Atomic orbitals The hydrogen atom The Aufbau Principle Molecular bonds Hybridised orbitals Covalent bond Ionic bond Polar covalent bond Metallic bond
Ordered and disordered materials
Classifying solids structural Ordered (crystalline) Disordered (glasses) Semidisordered N.B. glass is used as a generic term here (not just the glass you drink from!)
Disordered (amorphous) solids Disordered materials Physics similar to liquids (difficult!) Usually electrically insulating E.G. amorphous SiO 2 (silicon dioxide or silica primary component of every day glass)
Ordered (crystalline) solids Crystalline materials Regularly repeating structure Physics far more tractable E.G. crystalline SiO 2 (quartz), rock salt, diamond, most metals, silicon, germanium...
Electronic classification of crystals insulators (dielectrics) conductors (metals) semiconductors For crystalline materials, the explanation of the electronic properties is quite tractable
Atomic orbitals
The hydrogen atom The hydrogen atom is one of the few real systems for which analytical solutions of the Schrodinger equation exist [1] The solutions are characterised by 3 quantum numbers, n, l and m l A fourth quantum number, m s, must also be added describing the intrinsic angular momentum or spin
Atomic quantum numbers Quantum number Name Meaning Values n principle (radial) energy level 1,2,3,... l azimuthal orbital angular momentum 0,1,2,..., n-1 m l magnetic projection of orbital angular momentum along an axis 0, ±1, ±2,..., ±l, m s spin projection* projection of spin along an axis -s, -(s-1),..., s-1, s *Note that the spin quantum number (giving intrinsic angular momentum), s, of an electron is 1/2
The Pauli Exclusion Principle Each set the four quantum numbers, n, l, m l and m s, characterises a quantum state of the system. In an atom, these are often called atomic orbitals. Since electrons are fermions, they are subject to Pauli s Exclusion Principle, which states that: No two fermions may occupy the same quantum state.
Electron shells l = 0 l = 1 l = 2 m = 0 m = 0 m = -1 m = 1 m = 0 m = -2 m = -1 m = 1 m = 2 n = 1 1s n = 2 2s 2p 2p 2p n = 3 3s 3p 3p 3p 3d 3d 3d 3d 3d... 4s 4p 4p 4p 4d 4d 4d 4d 4d... Note that the number of states in each shell is twice that shown due to spin.
Subshells - notation The notation used is a legacy of the categorisation of spectral lines: s p d f sharp principle diffuse fundamental These correspond to l = 0, 1, 2 and 3. There after, we use g, h, i,.... Each of these denotes a subshell. (Unfortunately!) the following mnemonic is highly effective: sexy people don t fart. (Sorry!)
The Aufbau Principle The Aufbau Principle describes how electronic shells are built up ( aufbau comes from the german aufbauen, meaning building up ). 1s 2s 2p Shells build up in the order 1s 2s 2p 3s 3p 4s 3d... 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 6s... N.B. only works well for first 18 atoms
Hund s Rule When we consider how states fill with electrons, we must take account of spin and apply Hund s Rule: States of the same energy are each filled with one electron of the same spin before electrons of the opposite spin are added
Hund s Rule - example Example: 2p orbitals (1) (4) (2) (5) (3) (6)
Rule of stability A subshell has the highest stability if it is half full or full Stable configurations Unstable configurations readily loses the electron readily accepts an electron
Occupation notation To denote the occupation of a subshell, we use a superscript notation. E.g. 2p 3 means there are 3 electrons in the 2p subshell We may now see how the electronic configurations of the elements of the periodic table are built up
Electronic configurations atomic number element configuration abbreviated notation 1 H 1s 1s 2 He 1s 2 1s 2 3 Li 1s 2 2s 1 [He]2s 1 4 Be 1s 2 2s 2 [He]2s 2 5 B 1s 2 2s 2 2p 1 [He]2s 2 2p 1 6 C 1s 2 2s 2 2p 2 [He]2s 2 2p 2 7 N 1s 2 2s 2 2p 3 [He]2s 2 2p 3 8 O 1s 2 2s 2 2p 4 [He]2s 2 2p 4 9 F 1s 2 2s 2 2p 5 [He]2s 2 2p 5 10 Ne 1s 2 2s 2 2p 6 [He]2s 2 2p 6 11 Na 1s 2 2s 2 2p 6 3s 1 [Ne]3s 1 12 Mg 1s 2 2s 2 2p 6 3s 2 [Ne]3s 2............
Core and valence electrons Note that each of the noble gases, He, Ne, etc., has a full shell Also, each element with a higher atomic number than a given noble gas has the electronic configuration of that gas in common (hence the abbreviated notion) These filled inner shells may be referred to as core electrons The outer most electrons on top of the filled shells are called valence electrons and are responsible for an atom s chemical interactions
Valence configurations I II III IV V VI VII VIII H 1s 1 He - Li 2s 1 Be 2s 2 B 2s 2 2p 1 C 2s 2 2p 2 N 2s 2 2p 3 O 2s 2 2p 4 F 2s 2 2p 5 Ne - Na 3s 1 Mg 3s 2 Al 3s 2 3p 1 Si 3s 2 3p 2 P 3s 2 3p 3 S 3s 2 3p 4 Cl 3s 2 3p 5 Ar - K 4s 1 Ca 4s 2 Sc 4s 2 3d 1... Zn 4s 2 3d 10 Ga 4s 2 3d 10 4p 1 Ge 4s 2 3d 10 4p 2 As 4s 2 3d 10 4p 3 Se 4s 2 3d 10 4p 4 Br 4s 2 3d 10 4p 5 Kr -.................................
Atomic orbitals: n = 1 Images are isosurfaces of constant amplitude 1s orbital 1s orbital (section) red and blue regions indicate a change of sign.
Atomic orbitals: n = 2 s orbital p orbitals
Atomic orbitals: n = 3 s orbital p orbitals d orbitals
Molecular bonds
Hybridised orbitals The atomic orbitals may hybridise with one another to produce molecular bonds between atoms Typically, these hybridised orbitals may be formed from s and p orbitals. Two hybrid orbitals of particular interest are sp 3 hybrid sp 2 hybrid formed from 1 s and 3 p orbitals formed from 1 s and 2 p orbitals
sp 3 hybrid An s and 3 p orbitals may hybridise to produce
sp 3 hybrid Note that the sp3 hybrid has a tetrahedral form
s bond (covalent bond) If other atoms are close enough together, the lobes of the sp 3 hybrids may overlap to produce the s bond, in which 2 electrons are shared. This is the covalent bond. The red shading is electron density.
s* bond As well as the bonding s bond, there is also the antibonding s* bond, which has a nodal plane between the atomic nuclei (region of zero electron density)
Diamond structure For s bonded atoms of the sp 3 type, the atoms form the tetrahedral diamond structure lattice. This will be discussed in more detail later in the section on crystal structure.
Diamond structure Examples of materials with the diamond structure include Diamond (carbon) Crystalline silicon Crystalline germanium Note that these elements all come from group IV of the periodic table
Diamond structure elements Group IV (group 14 in chemistry) III IV V B 2s 2 2p 1 C 2s 2 2p 2 N... 2s 2 2p 3 Al 3s 2 3p 1 Si 3s 2 3p 2 P... 3s 2 3p 3... Ga 4s 2 3d 10 4p 1 Ge 4s 2 3d 10 4p 2 As... 4s 2 3d 10 4p 3...............
The ionic bond Recall the rule of stability Stable configurations Unstable configurations readily loses the electron readily accepts an electron
The ionic bond For s and p orbitals, this means ns np N.B. configuration of outer subshell of noble gases (Ne, Ar, etc.) stable ns np unstable In other words, the orbitals like to fill a complete subshell of 8 atoms (an octet). This is Lewis octet rule of stability.
The ionic bond Consider elements from groups I and VII I VII VIII H 1s 1 He - Li... F 2s 1 2s 2 2p 5 Ne - alkali metals Na... Cl 3s 1 3s 2 3p 5 Ar - halogens K... Br 4s 1 4s 2 3d 10 4p 5 Kr -............
The ionic bond These have the valence electrons in the configurations ns np alkali metals ns np halogens
The ionic bond Thus, if the alkali metal loses its s state electron, the remaining outer subshell will be filled ns np alkali metals ns np halogens By accepting this electron, the halogen then fills its outer shell
The ionic bond The alkali metal (e.g. Na) then becomes positively charged and the halogen ion (e.g. Cl) becomes negatively charged, resulting in an electrostatic force between them. Na + Cl - This is the basis of the ionic bond.
The ionic bond packing rules For the ionic bond, the resulting crystal structure depends on packing rules of the atoms. For simplicity, we shall model the atoms as rigid spheres. We will consider two general categories: Atoms of the same size Two types of atoms of different size
Close packing of equal spheres hexagonal close packing (HCP) face-centred cubic packing (FCC)* *The FCC structure will be discussed later in the section on crystal structure
Close packing of equal spheres Comparison of the HCP and FCC structures HCP FCC
Packing of two different types of sphere Example: NaCl (salt)
Ionicity In practice, all ionic bonds have some covalent character due to the unequal distribution of electron density between them The ionicity of a bond will depend on the difference in electronegativity of the two atoms Electronegativity is a measure of the power of an ion to attract electrons to it Bonds with mixed ionic and covalent character are often termed polar covalent bonds
Polar covalent bonds Consider elements from groups III and V III IV V B 2s 2 2p 1 C 2s 2 2p 2 N... 2s 2 2p 3 group III Al 3s 2 3p 1 Si 3s 2 3p 2 P... 3s 2 3p 3 group V... Ga 4s 2 3d 10 4p 1 Ge 4s 2 3d 10 4p 2 As... 4s 2 3d 10 4p 3...............
Polar covalent bonds Pairs of atoms from these groups have 8 valence bands between them as do the group IV elements. They may therefore form sp3 hybridised orbitals with some ionic character. Two types of polar structure are typically formed: zinc blende like the diamond structure but with two types of atom wurtzite typically, the nitrides (GaN, AlN etc.) These structures will be discussed in more detail later
The metallic bond In the metallic bond, the localised ions are bonded together via their interaction with a sea of delocalised conduction electrons The nature of metals will be discussed in more detail later
Summary
Summary of atomic orbitals Quantum mechanics yields electronic solutions for an atom characterised by 4 atomic numbers These solutions are known as atomic orbitals Using the Exclusion Principle and Hund s rule, the systematic occupation of these atomic orbitals explains the periodic table The atomic orbitals can hybridise to produce bonding states The resulting bonds give rise to crystal structure
References [1] See, for example, L.I. Schiff, Quantum Mechanics, McGraw-Hill
Additional material
The periodic table
Electronegativity Pauling s measure of electronegativity