Gavin W Morley Department of Physics University of Warwick Diamond Science & Technology Centre for Doctoral Training, MSc course Module 2 – Properties and Characterization of Materials Module 2 – (PX904) Lecture 4 – Electronic properties: Lecture 4 – Building a crystal from atoms Philip Martineau: build a crystal from atoms Now: what are the electrons doing in these crystals?

Module 2 – Properties and Characterization of Materials - Overview Diamond properties

Module 2 – Properties and Characterization of Materials - Overview Lectures Lecturer 1-3 Philip Martineau Crystallography 4-6 Gavin Morley Electronic properties 7-8 Stephen Lynch Optical 9 Electronic characterization 10 Richard Beanland Electron microscopy 11-12 Claire Dancer Mechanical 13-14 Martin Kuball Thermal 15-16 Magnetic

Lectures 4 Electronic structure: - Atomic physics - Building crystals from atoms - Tight binding model - Drude model of metals 5 and 6 - Sommerfeld model of metals Bandstructure: - Bloch’s theorem - Nearly free electron model - Semiconductors and insulators - Relative permittivity - Intrinsic and extrinsic conductivity - Metal-insulator transition - Mobility

What explains the Periodic Table? Dmitri Mendeleev (1834 – 1907) Diamond is an insulator Why do metals conduct? Where does this periodic table come from?

What explains the periodicity of the Periodic Table? The Schrödinger equation The Schrödinger equation + the Coulomb potential The Schrödinger equation + the Coulomb potential + electron spin The Schrödinger equation + the Coulomb potential + electron spin + the Pauli exclusion principle What your viewers really want to hear about is how I’ve improved public transport in London Dmitri Mendeleev (1834 – 1907)

Classical physics fails to explain atoms - - - - Nucleus, yes. Electrons do have angular momentum. Not classical orbits as electrons would fall into the middle

Electrons can behave like waves   Electron gun Electron detector Louis de Broglie (1892 – 1987) 1920s: electrons can behave like waves. X-Ray Diffraction with electrons Electrons can behave like waves & can behave like particles crystal

Schrödinger’s equation is a wave equation: Boundary Condition   Schrödinger’s equation is a wave equation: Boundary Condition Erwin Schrödinger (1887 – 1961) wave equation for the wavefunction psi Psi^2: probability of finding the electron KE, PE, total E Normal modes: fundamental, 1st harmonic, 2nd boundary conditions: where the string is held fixed Elastic band video from Acoustics Group, University of Salford, Manchester

Solve Schrödinger’s equation for an electron in a box:   Solve Schrödinger’s equation for an electron in a box: → Discrete energy levels   Erwin Schrödinger (1887 – 1961) electron in a box formed from a potential, V. Wave equation, boundary conditions  normal modes: discrete energy levels 1H 3+1Quantum Numbers Electrons stack because of Pauli’s exclusion principle Page 240, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985

Pauli’s exclusion principle: Two electrons cannot occupy the same quantum state simultaneously Wolfgang Pauli (1900 – 1958) Even without interactions Electrons have spin ½ and are identical Page 308, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985

Solve Schrödinger’s equation for an electron in a box:   Solve Schrödinger’s equation for an electron in a box: → Discrete energy levels   Erwin Schrödinger (1887 – 1961) Stack up electrons with n,l,m_l,m_s Page 240, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985

Number of degenerate eignenfunctions for each l Solve Schrödinger’s equation for electron in Coulomb potential and include spin n 1 2 3 l ml -1,0,+1 -2,-1,0,+1,+2 ms +½,-½ Number of degenerate eignenfunctions for each l 6 10 Subshell name 1s 2s 2p 3s 3p 3d makes the periodic table periodic Page 241, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985

What explains the Periodic Table? 1s, 2s, 3s… 2p, 3p, 4p…, from stacking up electrons into a box. Chemistry is dominated by the outermost electrons Page 330, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985

Schematic of subshell energy levels: The ionization energy of atoms: Difficult to get an electron off noble gas atoms The reason is the energy level spacings Pages 333-336, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985

Any questions so far? Crystals next

An atom Schematic drawing of wavefunction for an electron on a hydrogen atom. Page 245, Kittel, Introduction to Solid State Physics, Wiley 1996   Page 240, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985 The square of the electronic wavefunction shows the probability of finding an electron in that position.

Two atoms Page 240, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985 Here are the wavefunctions for two hydrogen atoms when they are far apart. Underneath I’ve shown the energy levels which are identical. For example each atom has an s level at the same energy. When we bring these atoms together their wavefunctions overlap and their energy levels change. We will still have the same number of energy levels in the combined system but they will have formed a bonding level and an antibonding level. (a) Schematic drawing of wavefunctions for electrons on two hydrogen atoms at large separation. Page 245, Kittel, Introduction to Solid State Physics, Wiley 1996

Building a molecule from atoms …a bond Page 240, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985 (b) shows the covalent chemical bond we get when we bring these identical atoms closer together. The bonding level is the sum of the two wavefunctions, and the antibonding level here in (c) is the difference. The bonding level is lower in energy because of the electrostatics: the Coulombic forces. Where the energy levels before had just one low-lying s-state, the combined system has these two states. So this is H2: we did some chemistry and made a molecule from two atoms. What happens if we have a crystal with a billion atoms? These two energy levels become a billion closely-spaced levels: a band. (b) Ground state wavefunction at closer separation. (c) Excited state wavefunction. Page 245, Kittel, Introduction to Solid State Physics, Wiley 1996

Potentials Atom Molecule Insulating crystal: tight binding model 1 Potential energy (V) After going from an atom to a molecule we can go on to a crystal. There should be more Coulomb potentials for the crystal but then you wouldn’t be able to see them. Schematics of the potential due to the ions in the crystal, Page 3, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

The tight-binding model Atom Molecule Insulating crystal tight binding model With atoms in the 2-electron molecule each atomic energy level became two energy levels. With N atoms (each having one spare electron) in a crystal, each of the single-atom energy levels becomes a band containing N states. Schematics of the potential due to the ions in the crystal, Page 3, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

The tight-binding model With N atoms in a crystal, each of the single-atom energy levels becomes a band containing N states. Each state has some value of the wavevector, k, and then we fill up these states with electrons following Pauli exclusion. Schematic of the formation of tight binding bands as the spacing between atoms is reduced. Page 35, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

The tight-binding model Group IA metal Group IIA metal e.g. sodium e.g. magnesium Atomic separation Atomic separation Energy Energy 3p 3s 2p 3p 3s 2p This diagram is yet another version of the same thing, this time for elements in the first two columns of the periodic table. The vertical dotted line shows a typical atomic spacing. Now remember that any s subshell can take two electrons: spin up and spin down, so a band made from s states can take two electrons per atom. Sodium here on the left has one free conduction electron per atom so this 3s band here is half full. It’s a metal because we have filled states next to empty states. Magnesium here has two valence electrons so if the electronic spacing were larger it would be an insulator because of this bandgap between the full 3s subshell and the empty 3p subshell. Real magnesium is a metal because the 3s and the 3p bands overlap for the actual atomic separation. Schematic of the formation of tight binding bands as the spacing between atoms is increased. Page 36, Singleton

Which of these elements is not in Group IV of the periodic table? C f) N Si Ge Sn Pb Dmitri Mendeleev (1834 – 1907) Some people call it Group 14 instead of Group IV.

Group IV Dmitri Mendeleev (1834 – 1907) It’s particularly surprising as diamond, silicon, germanium and tin all have the same diamond crystal structure.

FCC with two atom basis Here is the diamond crystal structure which Philip described in his lectures. Each carbon atom has two legs and two arms like this as you can see in our logo. The crystal structure is two interlocking face-centred cubic (FCC) lattices. It’s an fcc lattice with a two-atom basis: those two atoms are labelled A and B here, and for diamond they are both carbon atoms, but other important semiconductors, like GaAs, have the same structure different A and B atoms, so for GaAs we have A being Ga and B being As. The primitive unit cell for diamond has two carbon atoms. Diamond crystal structure. Page 37, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

The tight-binding model: diamond Carbon: 1s2 2s2 2p2 Let’s look at the tight-binding description for diamond. On the left and the right are two isolated carbon atoms each with their 2s and 2p states. Remember we can fit a spin up and a spin down electron into each of these energy levels. Each carbon atom has six electrons: 2 are stuck in the 1s state which is not shown here, and then there are 2 electrons in the 2s state and 2 electrons in the 2p state. When these carbon atoms come close to each other the atoms form bonding and antibonding states for both the s and p orbitals. The four lowest states mix to form sp3 hybrids and then there is an energy gap, shown with this green arrow, before you reach the antibonding states. Now let’s see how the electrons stack up. Diamond has two atoms in the primitive unit cell so we take the electrons from two atoms and stack them up in the sp3 hybrid states. (Carbon atom has term symbol 2S+1LJ = 3P0 from Hund’s rules where S is the spin quantum number, L is the orbital quantum number with 0,1,2,3=S,P,D,F. Finally, J is |L±S| ) Schematic of the formation of sp3 hybrid bonding states in diamond. Page 37, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

The tight-binding model: diamond Carbon: 1s2 2s2 2p2 Now all of these hybrid states are full and there is an energy gap up to the antibonding states. So diamond is an insulator. We know that silicon and germanium are semiconductors, and that makes sense too, because the energy gap is just smaller. There is no sharp distinction between a semiconductor and an insulator. But tin is the next element down in group IV and it even has the same crystal structure. So why is tin a metal? Schematic of the formation of sp3 hybrid bonding states in diamond. Page 37, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

The tight-binding model: Group IV Electron energy Element Eg (eV) a (nm) C 5.5 0.356 Si 1.1 0.543 Ge 1.0 0.566 Sn metallic 0.646 4 6 2 4 When the atoms are really spaced out here on the right, we have their s and p atomic orbitals. These spread out as bands and overlap here, and that is the interatomic spacing for tin where they overlap. Silicon and germanium have their atoms closer together so there is a bandgap, and then diamond has its atoms even closer together, with stronger bonds, and into this region with an even bigger bandgap, Eg. Interatomic spacing Schematic of tight-binding band formation in the group IV elements, Page 38, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

Many strong directional bonds Coulomb forces with Pauli exclusion Low-Z atoms are smaller (their electrons are closer to their parent nucleus) Closer atoms are more strongly bound (less screening) Diamond has many strong bonds H-H bond is stronger than C-C, but you can’t make a crystal out of H-H bonds Element Single bond length (Å) Carbon 1.54 Silicon 2.34 Germanium 2.44 Tin 2.80 Lead 2.88 Carbon atoms are small: only 6 nucleons. Jeremy K. Burdett, Chemical Bonding in Solids. New York: Oxford University Press, 1995: 152.           J. J. Gilman, Why silicon is hard, Science 261, 1436 (1993) F. Gao et al., Hardness of Covalent Crystals, Physical Review Letters 91, 015502 (2003).

Bond energy and cohesive energy Bond energy for diatomic molecule at 298 K (kJ/mol) [1] Cohesive energy of crystal at 298 K at 1 atm (kJ/mol) [2] Bond energy of crystal at 298 K at 1 atm = cohesive energy × ½ (kJ/mol) H-H 436.002 ±0.004 O=O 498.340 ±0.2 N≡N 945.33 ±0.59 C-C 607 ±21 for C=C 715 357 Si-Si 450 225 Ge-Ge 376 188 For diamond, see: L. A. Schmid, Physical Review 92, 1373 (1953). B. Holland, H. S. Greenside and M. Schlüter, physica status solidi (b) 126, 511 (1984). X. Jiang et al., Sci. Rep. 3, 1877 (2013). H. Shin et al., The Journal of Chemical Physics 140, 114702 (2014). To convert to the values at 298K from the values in Kittel which are at 0K, add on (3/2) RT = 3.7 kJ/mol for T=298 K and R=8.314 J mol-1 K-1. To get the bond energy from the cohesive energy just divide by 2 as each carbon atom in diamond has four bonds but each bond is shared by two carbon atoms so there are twice as many bonds as carbon atoms. [1] CRC Handbook, Strengths of Chemical Bonds, 57th Edition, 1977 [2] C Kittel, Introduction to Solid State Physics, Wiley 1996, Chapter 3, Table 1

Many strong directional bonds Smaller atoms get closer together can we make a crystal with stronger bonds? BN, BC2N and B are almost as hard as diamond Each boron atom in crystalline boron has five nearest neighbours within the icosahedron. If the bonding were the conventional covalent type then each boron would have donated five electrons. However, boron has only three valence electrons. Jeremy K. Burdett, Chemical Bonding in Solids. New York: Oxford University Press, 1995: 152.           J. J. Gilman, Why silicon is hard, Science 261, 1436 (1993) F. Gao et al., Hardness of Covalent Crystals, Physical Review Letters 91, 015502 (2003).

Hardness (and brittleness) Three things make a covalent crystal hard: - High bond density (electronic density) - Short bond length - High degree of covalent bonding See: F. Gao et al., Hardness of Covalent Crystals, Physical Review Letters 91, 015502 (2003) See Claire Dancer’s lectures (11 and 12 in this module) The covalent bonds in diamond are very directional, so the atoms do not move out of the way if indented, unlike in a metal. Eventually, the crystal must break (with broken bonds) rather than bend, i.e. it is brittle.

Many strong directional bonds hard brittle chemically inert incompressible (i.e. high bulk modulus) High speed of sound See Claire Dancer’s lectures (11 and 12 in this module)

Electronic Bandstructure of diamond Mini-Summary: - Atomic physics  bandstructure …by assuming the electrons in crystals are generally stuck in their atomic potentials - Metals next: we will assume that the electrons are not stuck, and still get bandstructure If you look up the electronic bandstructure of diamond in the literature you get something like this. It might look a bit complex but it’s amazingly simple considering that this describes a crystal containing 10^23 atoms. The really good news is that we can ignore most of these levels most of the time. What matters most is the region around the bandgap here, because we can often ignore all of these full states down here and all of these empty states above. W. Saslow, T. K. Bergstresser, and Marvin L. Cohen, Physical Review Letters 16, 354 (1966)

10-10 1 1010 1020 Resistivity (ohm-cm) PTFE (Teflon)  > 1018 -cm (room temperature) silicon  ~ 104 -cm (room temperature) Pure metal  ~ 10-10 -cm (1 K) Tin  ~ 10-5 -cm (room temperature) Superconductors  ~ 0 diamond  ~ 1016 -cm (room temperature) 10-10 1 1010 1020 Resistivity varies by at least 30 orders of magnitude: a huge range. Teflon is used as the insulator in coaxial cables and has a higher room-temperature resistivity than diamond. Diamond has a resistivity of around 1016 ohm-cm at room temperature. It gets even higher if you cool it down. The distinction between an insulator and a semiconductor is somewhat arbitrary but most people think of the boundary as being near <this> blue mark. Silicon is certainly a semiconductor rather than an insulator. Pure metallic tin has a resistivity of around 10-5 ohm-cm at room temperature, and for copper this is ten times lower. The resistivity of metals goes down when you cool them: the opposite behaviour to what you find for insulators and semiconductors. The resistivity of a pure metal like copper can reach 10-10 ohm-cm at low temperatures, and superconductors have even lower resistivity. Resistivity (ohm-cm)

Bandstructure Energy Eg Metal Insulator Semiconductor Bands explain why some crystals are insulators as you can see here. This insulator has three bands which can take electrons, and two of them are completely full. The electrons down here can’t get up here because of this energy gap labelled Eg, so all of the electrons are stuck where they are, like the electrons in a noble gas atom. The things we have already talked about: the discrete energy levels and Pauli exclusion mean together that electrons in a full band can’t conduct electricity or heat. The Fermi Energy is here in the gap. We call this the bandgap and it can be very important for the properties of non-metals. If this bandgap is large as with diamond then we would call it a wide-bandgap material. It would make more sense to call it a tall bandgap rather than a wide bandgap, but that’s just the terminology. In this metal though, the middle band is half full so we can give some energy to these electrons near the Fermi energy and they can move to a different wavevector and carry a current. And here is a semiconductor: there is no sharp dividing line between a semiconductor and an insulator: one just conducts electricity somewhat more than the other. But where did the bands for a metal come from? Metal Insulator Schematic electron occupancy for allowed energy bands. See page 174, Kittel, Introduction to Solid State Physics, Wiley 1996

What explains the Periodic Table? Dmitri Mendeleev (1834 – 1907) Why do metals conduct?

Metals Which is the most advanced model of metals in the list below? Drude model Sommerfeld model Nearly-free electron model Tight-binding model c) and d) are equally advanced Lady Gaga Paco Rabanne Dolce & Gabbana

Metals Most elements are metals, particularly those on the left of the periodic table Good conductors of electricity & heat Tend to form in crystal structures with at least 8 nearest neighbours (FCC, HCP, BCC) Malleable Schematic model of a crystal of sodium metal. Page 142, Kittel, Introduction to Solid State Physics, Wiley 1996 What properties do metals have? Here is a partial list. We’ve seen already that most pure elements are metals. Metals are good conductors, and they tend to form in crystal structures with a lot of nearest neighbours. And we can bend metals if we’re strong enough. All these things lead us to a model of metals with ionic cores that are like noble gas ions held together by loosely-bound valence electrons. These loosely-bound electrons can conduct electricity and heat. Sodium here gives up one electron per atom and the light circles show the approximate size of the ionic core relative to the space in-between. If you ask me as an experimental physicist to think of a metal then I wouldn’t come up with sodium because I never see it day-to-day. However, sodium is a nice simple example of a metal because it’s not too heavy so the shells are not doing anything strange, and it just has one valence electron. With so many nearest neighbours the bonds between atoms in a metal are not very directional. This makes metals malleable. In contrast an ionic or covalently bonded solid typically has 4 to 6 nearest neighbours. Diamond and silicon have four nearest neighbours and they are not malleable at all: they are brittle ceramics rather than metals.

Metals The Drude Model: Gas of electrons Electrons sometimes collide with an atomic core All other interactions ignored   So how do we describe these conduction electrons? The Drude model is a very basic attempt. It comes from statistical mechanics and does not include quantum mechanics. In a normal gas, the gas atoms bounce off each other, but the electrons don’t do that in this model, and they don’t even interact with each other all. The electrons bounce around only off the ionic cores. This is progress and it does actually agree with some simple experiments, such as the electrical conductivity where we get Ohm’s law: V=IR. The current flowing in a metal is proportional to the voltage applied with some constant which is the resistance R. However, the Drude model fails to explain many other experiments, like the specific heat. This tells us how much heat the electrons can store, and the Drude results is <this>: kB is the Boltzmann constant, and n is the density of conduction electrons. This gives a number which is independent of temperature, whereas in measurements of real metals the specific heat is proportional to temperature: the electrons in real metals can store more heat the warmer they are. Also, this result from Drude is way too big compared to real metals at typical temperatures. Now you might be thinking that the specific heat of metals has not played a big role in your life so far, so who cares if this model spectacularly fails to explain it? Well the underlying reason that it’s interesting is that it shows up a spectacular flaw in our model of reality. To fix that flaw we have to go from classical physics to quantum mechanics. Paul Drude (1863 –1906)

Metals Sommerfeld The Drude Model: Gas of electrons Electrons sometimes collide with an atomic core All other interactions ignored Electrons obey the Schrödinger equation and the Pauli exclusion principle The Sommerfeld model adds quantum mechanics into the Drude model. We keep these first three assumptions but add in this fourth one. Still the electrons’ only interactions are collisions with atomic cores. But now electrons obey the Schrodinger equation so they have wave properties. We already saw that the hydrogen atom gives us an electron trapped in a box which leads to discrete energy levels like a wave on a string. This metal dice is also a box for electrons, it’s just much bigger than a hydrogen atom. So again, because of the boundary conditions, we have discrete energy levels with particular wavelengths. There are permitted wavelengths in the x, y and z directions. An electron must be in a quantum state with x, y and z wavelengths that are permitted for the size of the box. Arnold Sommerfeld (1868 – 1951)

Electronic structure: - Atomic physics - Building crystals from atoms Lectures 4 Electronic structure: - Atomic physics - Building crystals from atoms - Tight binding model - Drude model of metals 5 and 6 - Sommerfeld model of metals Bandstructure: - Bloch’s theorem - Nearly free electron model - Semiconductors and insulators - Relative permittivity - Intrinsic and extrinsic conductivity - Metal-insulator transition - Mobility We found that building a crystal with tightly-bound atoms leads to a bandgap. In the next lecture we will see that we also get bandgaps for crystals in which the ionic potential is really weak: metals.