Gavin W Morley Department of Physics University of Warwick

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) Lectures 5 and 6 – Electronic properties: Lectures 5 and 6 – Bandstructure of crystals

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

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 Slide from previous lecture

Metals The Drude Model: Gas of electrons
Electrons sometimes collide with an atomic core All other interactions ignored Slide from previous lecture 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 Slide from previous lecture Arnold Sommerfeld (1868 – 1951)

Metals Sommerfeld The Drude Model
We can draw these permitted states as a grid like this, where the axes are the wavevectors, k, in the x and y directions. The wavevector tells you how many wiggles fit into this metal box. Short wavelengths correspond to a high wavevector k, and a higher momentum and so a higher energy. k is a vector because the electron can have momentum in all three directions x,y and z. The grey dots are the permitted states. We don’t draw this for the wavelengths because the permitted states are not evenly spaced in wavelength: they go 1, ½, 1/3, ¼ and so on. Each of these dots is a quantum state and it can only have one electron in there for spin up and one electron in there for spin down, because of Pauli’s exclusion principle. This red dot shows that the state has been filled, so we can put a few in with the minimum wavevector, before the next electrons must have a bigger wavevector and then soon an even bigger wavevector. We might need to put 10^22 or more electrons in there, so the largest wavevector will be huge, even if we could cool down to zero temperature. The red circle is called the Fermi surface. It doesn’t look like a surface because this schematic is in two dimensions, but for a real 3-dimensional crystal we have to think of this grid in 3D with kz also and then the Fermi surface here in the Sommerfeld model would be a sphere. (Four of Sommerfeld’s PhD students won Nobel prizes: Werner Heisenberg, Wolfgang Pauli, Peter Debye and Hans Bethe). A map of states in k-space, see also page 173, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

Metals Sommerfeld The Drude Model 1 Potential energy (V)
1 Potential energy (V) Here is the Schrodinger equation again. Inside of our box we have assumed that V is zero because the electrons have no interactions: they are free electrons. That simplifies the equation. The solution to this equation is a wave like this with the kinetic energy here of a free particle having momentum h-bar k. The kinetic energy is ½ mv^2 and the momentum is mv. me is the mass of the electron. Now if the boundary condition is that the wavefunction must go to zero at the edge of the box then we get the standing waves I showed on the previous slide. However, we are interested in things like the conductivity of the electrons so we prefer to think of these as travelling waves rather than standing waves. A standing wave is just a sum of two travelling waves going in opposite directions so these views are equivalent for the Schrodinger equation. Drude-Sommerfeld potential Schematics of the potential due to the ions in the crystal, Page 3, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

Metals Sommerfeld The Drude Model
In A-level physics we find out that the kinetic energy is 1/2mv2 and that is all we have here because the momentum is mv and is also h-bar k. We call this kind of graph the dispersion relation, and it will be important later. This free electron is trapped in a box, so the boundary conditions still mean that there are only certain allowed values of k which are equally spaced in k-space. Dispersion relation for a free electron. Page 177, Kittel, Introduction to Solid State Physics, Wiley 1996

Metals fFD The Drude Model vs the Sommerfeld model The Drude Model:
Energy The Drude Model: The Drude Model vs Distribution functions for a typical metal at room temperature, Page 10, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001 the Sommerfeld model Number of electrons fFD Energy Now lets plot things with energy on the x-axis instead of wavevector. In the Drude model the electrons follow Maxwell-Boltzmann statistics from statistical mechanics, so there can be many of them at low energies as shown here. However, we have said that electrons can’t do this because of Pauli exclusion. Electrons are Fermions because they have a spin of ½ and Fermions follow Fermi-Dirac statistics as in this red curve where there can only be one electron for each quantum state. This is why the Drude model fails, and why the Sommerfeld model is better. In the Sommerfeld model, for a typical metal, there are loads of high-energy electrons around <here> and these dominate the properties of the metal. None of these low-energy electrons <here> do much in the Sommerfeld model because the Pauli exclusion principle prevents them from moving to a different state unless they can somehow find this huge amount of energy to get up to an unfilled state <here>. The high-energy electrons <here> are the only ones that can respond to what happens around them which could be that you put a voltage on the metal or warm up one end of it. Both of these give extra energy to the electrons and these high energy electrons are able to take that energy and move up into even higher energy states. When that happens the electrons are conducting electricity or heat respectively. From this we can understand why the Drude model got the wrong answer for the specific heat: in the Drude picture as you heat up the metal all of the electrons can store heat, but here in the Sommerfeld picture these low energy electrons are unable to store heat because they can’t change their quantum state.

Metals the Sommerfeld model Zero temperature T = 0 Finite temperature
Zero temperature T = 0 Finite temperature T << EF/kB The Fermi-Dirac distribution function for a typical metal does not change much even if you warm up from zero temperature <here> in a) to room temperature <here> in b). In the Sommerfeld model, it is only these high-energy electrons close to the Fermi surface that can store heat. At higher temperatures the Fermi-Dirac distribution is more smeared out so there are more electrons that can store the heat. This is why the specific heat increases with temperature. The energy of these high-energy electrons is so important that we refer to this point <here> as the Fermi energy. mu is the chemical potential which is the energy where the probability of occupation is ½. At zero temperature, mu is equal to the Fermi energy, but mu shifts around as the temperature is changed. As you can see <here>, for a typical metal at room temperature, the chemical potential is very close to the value it takes at zero temperature because the Fermi energy is much larger than the room-temperature thermal energy shown <here> as kB T. Now I have another question for you. How fast is an electron moving if it’s one of these high-energy electrons that dominates the properties of a metal? Fermi-Dirac distribution function, Page 9, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

Metals the Sommerfeld model
At any given moment, roughly how quickly does one of the fast electrons travel around in a typical metal at low temperatures? 0 mm s-1 1 mm s-1 7 million mph (1% of c) 200 million mph (30% of c) Officer, I’m so sorry: I’m afraid I wasn’t looking at the speedometer mph is similar to m/s (One meter per second is roughly two miles per hour).

Metals the Sommerfeld model
The Fermi Energy is defined as the chemical potential at zero temperature as we just said, and in the Sommerfeld model this depends only on n : the number of conduction electrons per unit volume. We won’t have time here to derive this equation but you can find it in many books: it’s on page 8 of John Singleton’s book. All of <these> other things on the right hand side of the equation are constants. So sodium metal has one conduction electron per atom and we can look up the density. It’s about 1g/cm^3 so we use the atomic mass and estimate that there are about 10^22 conduction electrons per cm^3. This of course is the same as the number of atoms per cm^3 because each atom gives up one electron. And then the Fermi energy comes out to this huge number: 100 times bigger than room temperature. You can get the Fermi velocity from here because the energy is the kinetic energy which is ½ mv^2. As we said, these energetic electrons have a speed of around 1% of the speed of light in vacuum, even if the metal is cooled close to absolute zero temperature. Fermi-Dirac distribution function, Pages 8&9, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

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 Explains temperature dependence and magnitude of: Electronic specific heat Thermal conductivity (approx.) Electrical conductivity (approx.) But does not explain: Insulators & semiconductors Thermopower Magnetoresistence Hall Effect The Sommerfeld model makes all of these simplifying assumptions, but in spite of this simplicity it agrees with some experiments on metals quite well. The thing which upsets us the most is that it can’t tell us what happens in non-metals. Diamond is an insulator and if we foolishly tried to apply the Sommerfeld model it would just say that a crystal of carbon would be a metal with 4 conduction electrons per atom: very wrong! So we have to improve these assumptions. Assumptions 2 and 3 <here> are that the Sommerfeld model includes collisions between the electrons and the ions, but these are instantaneous and there is no interaction apart from these collisions. Let’s make better assumptions. Arnold Sommerfeld (1868 – 1951)

Metals, Semiconductors & Insulators
Beyond the Sommerfeld Model: Gas of electrons Electrons are in a periodic potential due to the ions Electron-electron interactions ignored Electrons obey the Schrödinger equation and the Pauli exclusion principle 1 Potential energy (V) The next step will be to bring in this assumption number 2. With the Sommerfeld model we were saying that the electrons were trapped in a box, but the potential was flat inside. Then we included the Drude-Sommerfeld assumption of instantaneous collisions with the ionic cores. Now we are going to do better by considering the potential due to the ions: electrons in a crystal move through a periodic potential. We started by assuming that this ionic potential was so strong that the electrons were tightly-bound (the tight binding model). Next we will assume that this periodic potential is really weak, so we have nearly-free electrons. Both cases give us bandstructure so we can be confident that this describes real systems where the strength of the ionic potential is in-between. Drude-Sommerfeld potential real ionic potential Schematics of the potential due to the ions in the crystal, Page 3, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

Bloch’s theorem “Consider a one-electron Hamiltonian with a periodic potential: The eigenstates can be chosen to be a plane wave times a function with the periodicity of the lattice.” 1 Potential energy (V) Bloch demonstrated a very general result about a periodic potential in quantum physics: the eigenstates are a plane wave multiplied by a function with the periodicity of the lattice. This result means that an electron can move in a perfect infinite crystal as a wave forever with no scattering. Any eigenstate is a stationary state: it does not change with time. So all scattering comes from imperfections in the crystal which could be phonons or impurities. This is quite different to the Drude and Sommerfeld models where we assumed that the electrons were scattering with the ionic lattice. Now though, we have a wavevector k which means something a bit different to the k that we had in the Sommerfeld model. In the Sommerfeld model, h-bar k is the momentum of the electron, but the new Bloch wavefunctions are not eigenstates of the momentum operator so we can’t interpret h-bar k as the electron momentum in the Bloch picture. Instead we think of k as a quantum number which characterises a Bloch state. Drude-Sommerfeld potential real ionic potential Bloch’s theorem, Page 16, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

The nearly-free electron model
Let’s start off by thinking about a very weak ionic potential. The electrons are nearly free, so they will behave a lot like the free electrons we discussed yesterday. This means that we can make the approximation that the electrons move through this material as a simple free electron plane wave with a wavevector k. Now let’s apply the Bragg Law, just as you do for X-rays that hit a crystal. Let’s take the simplest case which is one dimensional, with the wave moving directly towards the crystal plane. These dots are the atoms in the crystal with lattice spacing a. The wave reflects off each plane, but only slightly because it is such a weak potential. However, if the wavelength of the electron is 2a then each small reflection adds up giving constructive interference. This means that the electron would be fully reflected back even though the ionic potential is very weak. Except that once it was reflected, it would get Bragg reflected back in the other direction. This means that waves with this wavelength are standing waves rather than the travelling waves we have in the free electron picture. The same thing can happen for certain other wavelengths as long as they are smaller than 2a. The condition for this to happen is that the wavelength should be 2a/n where n is an integer. This means that k=n pi/a so again these features are evenly spaced in k-space and we prefer to think about them in k-space. Drude-Sommerfeld potential weak ionic potential

Here is the dispersion relation I showed earlier for the free electron. With a nearly-free electron things look the same, apart from close to the Bragg condition. The effect of the Bragg condition is that there are these bands of energy that are forbidden, which separate the allowed bands. At point A the electrons spend most of their time close to the positively charged nuclei so this state has a lower energy than it would in the free electron picture. Point B is at a higher energy because then the electrons spend most of their time away from the nuclei. Nearly free electron has bands Dispersion relation for free and nearly-free electrons. Page 177, Kittel, Introduction to Solid State Physics, Wiley 1996

First Brillouin zone The nearly free electron model is good for a typical alkali metal like sodium, because we know that the conduction electrons are nearly free: it is a metal after all! I showed here six filled states because this is a schematic drawing, but if you have a real lump of metal there would be a vast number of states. That is why we call this a band: there are so many states in there that it is very close to being a continuum even though it is made up of individual states. This region of k-space is called the first Brillouin zone. The edge of this zone is where we get Bragg reflection. The first Brillouin zone becomes a 3D shape when we go on to a 3D crystal, instead of this 1D example. (A Brillouin zone is defined as a Wigner-Seitz primitive cell in the reciprocal lattice, where a primitive cell is a minimal-volume unit cell for a crystal). Nearly free electron has bands Dispersion relation for free and nearly-free electrons. Page 177, Kittel, Introduction to Solid State Physics, Wiley 1996

Representing bands Three energy bands of a linear lattice. Page 238, Kittel, Introduction to Solid State Physics, Wiley 1996 The picture I just showed was in what is called the extended zone scheme here. However, k-space is periodic just like real space, so these bands get folded back into the first Brillouin zone. Because of this, the first Brillouin zone actually contains all of the information about the bandstructure, and often people will show that like this in the reduced zone scheme. You can see how this piece here was folded back to make this piece and so on.

Diamond model From the following list, which is the best model of diamond? Drude model Sommerfeld model Nearly-free electron model Tight binding model

Electronic Bandstructure of diamond
Part of the diamond bandstructure looks like that of a free electron W. Saslow, T. K. Bergstresser, and Marvin L. Cohen, Physical Review Letters 16, 354 (1966)

Here is a reminder of the reduced zone scheme. The higher energy bands are folded back into the first Brillouin zone from regions of higher k. Bandstructure in the literature is generally presented like this and it’s good because it’s more densely packed with information. Also it gets the all-important bandgap in the middle of the picture. Kittel page 238 W. Saslow, T. K. Bergstresser, and Marvin L. Cohen, Physical Review Letters 16, 354 (1966)

Heavy-hole band Light-hole band If this looks complex, then just remember that it is describing a crystal with some enormous number of atoms, like It’s an intellectual triumph that we are able to describe something with so many atoms so simply. We still have electrons moving around like waves, and these bandstructure diagrams tell us the allowed wavevectors and their energies. We can still think about a force on one of the electrons as creating an acceleration, using F=ma. However, to do that we can’t use the mass of an electron but the effective mass, m*. This is because the electron does interact with the ionic cores and the other electrons, and these things change the way that electron moves. Many semiconductors, including diamond and silicon have heavy holes and light holes named according to their different effective masses. (The heavy holes tend to dominate the properties near the bandgap because their greater effective mass corresponds to a larger density of states in energy: see Fig 5.1 on page 44 of Singleton’s book.) Effective mass derivation, Page 42, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

Indirect bandgap This is a calculation (actually it used the empirical pseudopotential method (EPM)) and Jon Goss will give you much more detail about modelling and simulations in module 4. This calculation is based on experiments like the optical experiments that Stephen Lynch will tell you about this afternoon. For optical experiments, it is important to notice that there is an indirect bandgap here. The top of the valence band is not directly below the bottom of the conduction band. Instead the bottom of the conduction band is here at a different wavenumber. An optical photon can be absorbed easily as long as it has enough energy to cross this bigger bandgap, or else if the optical photon only provides enough energy to cross this indirect bandgap then a phonon must be involved too to conserve momentum. W. Saslow, T. K. Bergstresser, and Marvin L. Cohen, Physical Review Letters 16, 354 (1966)

This point in the middle, where the k vector is zero has a special name: it is called the gamma point. There are other points also that are labelled here on the x-axis: the X point and the L point. You can see that they are at the edge of the Brillouin zone by comparing with this schematic I showed earlier. Sometimes bandstructure is plotted with just these letters as labels for the x-axis. W. Saslow, T. K. Bergstresser, and Marvin L. Cohen, Physical Review Letters 16, 354 (1966)

This is the shape of the first Brillouin zone for diamond. It looks more complex in 3D, but it is still the reciprocal space shape of one of the primitive unit cells (the Wigner-Seitz cell). The kx, ky and kz axes are labelled. W. Saslow, T. K. Bergstresser, and Marvin L. Cohen, Physical Review Letters 16, 354 (1966)

Bandstructure of Si & diamond
If we compare the bandstructure of silicon and diamond we see a great deal of similarities, because they have the same lattice structure and they are both from group IV. Silicon has a smaller bandgap, so it is a semiconductor rather than an insulator. We still see here this indirect bandgap. There are holes labelled here in the valence band and the conduction band is here. Based on M. Cardona and F. Pollack, Physical Review 142, 530 (1966).) Bandstructure of Si, page 50, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

Any questions? So we have added in a periodic potential and got bands of allowed states. We’re going to move on now, accepting that bands exist so if you have questions about that then now would be a good time to ask.

Effect of an electric field
Maxwell’s first equation gives us the relative permittivity, epsilon_r. Let’s look at what this does to a capacitor. Relative permittivity. Page 271, Kittel, Introduction to Solid State Physics, Wiley 1996

Effect of an electric field - capacitor
+ - We can make a capacitor from two metal plates with vacuum in between, and put some charge onto the two plates. This gives an electric field between the plates. Now if we put some material between the plates instead of vacuum then this will reduce the electric field that results. If we fill that space with a metal then the conduction electrons can move around to completely remove the electric field. However, that’s not much use as a capacitor. Instead let’s put an insulator in there. We call this insulator a dielectric because it partially cancels out the electric field. It does this because the electric field polarizes the medium that we put in. If this medium was water then the water molecules would line up because these molecules are polar: they have an electrically positive end and an electrically negative end. After these molecules have lined up the electric fields they generate are going to cancel out some of the electric field from the capacitor plates. If you put diamond in there, then the electrons that make up the strong sp3 bonds between the carbon atoms will be slightly dragged towards this positive plate, electrically polarizing the diamond. If we make the approximation that the dielectric medium is isotropic and linear then we can just change an equation by replacing 𝜺_𝟎 by 𝜺_𝟎 𝜺_𝒓, where 𝜺_𝒓 is the relative permittivity. This used to be called the dielectric constant. Dielectric properties of insulators, page 533, Ashcroft and Mermin, Solid State Physics, Harcourt 1976.

Effect of an electric field - Coulomb field
Equivalently we can think of the Coublombic electric field from a point charge. In vacuum we would have this expression showing that the electric field is proportional the charge and falls off with distance r as 1/r2. If we have an isotropic linear dielectric medium then we just look up the value of epsilon_r and multiply epsilon_0 by that. Page 240, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985

Dielectric permittivity - static
If you want to look up the dielectric properties of insulators then the textbook to look in is Ashcroft and Mermin. Notice that diamond has a smaller relative permittivity than silicon, because the bonds in silicon are weaker and more easily dragged around by an electric field. See J. C. Phillips, Physical Review Letters 20, 550 (1968) Dielectric constants, page 553, Ashcroft and Mermin, Solid State Physics, Harcourt

Dielectric permittivity - frequency-dependent
+ - The relative permittivity is a function of frequency. If you imagine applying an alternating current to the capacitor plates then the polarized medium in the middle may or may not be able to keep up with the AC electric field depending on how quickly it can change its polarization. This becomes very important for understanding the optical response of dielectric materials because it is the electric field in the electromagnetic radiation which is most responsible for the interaction with matter. This is clear from the fact that the refractive index is related to the relative permittivity like this, and the relative permeability (mu_r) is generally around 1. So we can generally take the approximation that the refractive index is the square-root of epsilon_r. And the refractive index tells you how fast light travels in a material (v), so the refractive index of the vacuum is n=1 and then light travels at c in vacuum, but slower otherwise. Stephen Lynch will tell you more about optics in the next lectures. An electromagnetic wave has an oscillating electric field which tries to move the electrons in a material. This process does work on the electrons as work done=force x distance. The energy for this work comes from the electromagnetic field losing energy. This is dielectric loss. The frequency-dependent dielectric permittivity of diamond is low leading to low dielectric loss, so diamond is used as a window for applications such as where too much dielectric loss would lead to dangerous overheating of the window. This is particularly useful in the infra-red. See: Status report on CVD-diamond window development for high power ECRH, M. Thumm et al, Fusion Engineering and Design 53, 517 (2001) → Dielectric loss Dielectric properties of insulators, page 533, Ashcroft and Mermin, Solid State Physics, Harcourt 1976.

Intrinsic Semiconductor at room temperature
Temperature dependence Energy Intrinsic Semiconductor at room temperature Eg This is a schematic drawing, and we call each of these boxes bands. They can be empty like this white one, full like this gray one, or partially full. An insulator can’t conduct electricity because all of its bands are either full or empty, so the electrons can’t move to a different quantum state. It’s similar to a noble gas atom. This is our schematic at room temperature for an intrinsic semiconductor where some thermal energy has excited electrons over the energy gap. If we cool down semiconductors their resistivity increases. Metal Insulator

Cooling semiconductors down
Energy Intrinsic Semiconductor at room temperature Intrinsic Semiconductor at low temperature Eg At temperatures lower than the bandgap, semiconductors and insulators behave the same: neither one conducts electricity. Metal Insulator

Cooling semiconductors down
Energy Intrinsic Extrinsic for kBT > Eg for Eg > kBT > donor binding energy There are two ways that a semiconductor can get free charge carriers as the temperature is increased: intrinsic and extrinsic. Intrinsic means “from within” and extrinsic means ”from outside”. So the intrinsic conductivity of a semiconductor comes from thermal energy exciting electrons across the bandgap. That means that there are electrons in the conduction band next to free states which can conduct, and also the holes here in the valence band mean that there are electrons here in the valence band next to empty states and we can get conduction here too. Extrinsic conductivity comes from doping the semiconductor: changing the crystal so that there are either extra free electrons as in this diagram, or fewer electrons. Our computers work thanks to the extrinsic doping of silicon.

Semiconductor at room temperature
Intrinsic charge carriers Energy Intrinsic Let’s start with intrinsic conductivity. When electrons are excited across this bandgap they leave behind holes in the valence band. Imaging that just one electron has been excited into the conduction band leaving just one hole. Now to understand what happens in the valence band, it is easier for us to think of this one hole as one positive charge carrier, rather than trying to think about all of the huge number of electrons that are left behind. If we put on a voltage then it can move this single electron to the right. But here when there are loads of electrons and just one missing, it is easier to just follow the hole which is moving to the left, rather than trying to keep track of all of these electrons. When this hole was created it left behind a positive charge coming from the nuclei so the hole looks like a positive charge moving in the opposite direction to the electrons. holes Semiconductor at room temperature

Semiconductor at room temperature
Intrinsic charge carriers Energy Intrinsic Eg The law of mass action tells us roughly how many electrons and hole are present for a given temperature, T. n is the number of electrons in the conduction band per unit volume: “n for negative”. p is the number of holes in the valence band per unit volume with “p for positive”. Eg is the bandgap and when that is bigger than the thermal energy, kBT, the number of electrons and holes is exponentially supressed. W is a constant depending on the details of the valence and conduction band extrema. There is also this dependence on the temperature cubed, but it is the exponential dependence that generally dominates here. For the moment we are assuming that the only charge carriers are these intrinsic charge carriers. That means that n=p and so we can take the square root to get n or p. Semiconductor at room temperature Page 56, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

Intrinsic charge carriers
Ge: Eg = 0.74 eV Si: Eg = 1.17 eV GaAs: Eg = 1.52 eV This plot shows the density of carriers for three important semiconductors as a function of temperature. Even these small differences in the energy gap lead to dramatic differences by orders of magnitude in the number of carriers at room temperature. This comes from the exponential dependence on Eg/kBT. Gallium arsenide is generally more expensive than silicon, but is used for high-frequency chips. Now if you had a crystal of silicon or germanium or GaAs which was 100% pure then this intrinsic behaviour would dominate. However, that is never the case in reality. At room temperature, the conductivity of all of these semiconductors is extrinsic: it is dominated by the impurities that are doped in. Calculated intrinsic carrier densities versus temperature. Page 59, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

Extrinsic charge carriers
Energy Intrinsic Extrinsic (n-type) Extrinsic (p-type) donor impurities acceptor impurities Semiconductor at room temperature Extrinsic conductivity comes from doping the semiconductor: changing the crystal so that there are extra free electrons in this case, or less electrons in this case. If the crystal is doped with extra electrons then we say that it is negatively doped, or n-type. Similarly, p-type material is doped with positive holes. To negatively dope a semiconductor we put in some donor impurities which donate electrons to the conduction band. To make a p-type crystal we put in acceptors which grab spare electrons leaving holes behind. Semiconductor at room temperature Semiconductor at room temperature

Si:P binding energy = 46 meV Page 240, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985 If we look at the periodic table then we can see how to dope semiconductors. The modern computing industry is based on doping silicon with boron and phosphorous. Both of these atoms go in by substituting for a silicon atom to make substitutional dopants without changing the crystal structure. However, boron is missing an electron compared to silicon so it is an acceptor. Phosphorous is a donor. At low temperatures the phosphorous atom recaptures its extra electron, so it looks like a hydrogen atom with one electron bound to one net positive charge. The big difference to a real hydrogen atom is that the binding energy is much less: generally around 5 to 50meV in semiconductors like GaAs and Si as compared to 13.6eV for hydrogen. This means that at room temperature most of the electrons are ionised because kBT295K is around 25meV.

20 ppb To calculate the density of extrinsic electrons and holes for a particular temperature, we can still use the law of mass action, but now instead of setting n=p we use the conservation law for electrons given a certain density of donors (ND) and acceptors (NA). The dashed line is from the intrinsic electrons: it is insignificant at room temperature (see red arrow) but at 500K it quickly starts to dominate. The saturation range is where the donors are all ionised: warming the sample in this range cannot provide any extra free electrons from the donors phosphorous donors per cm^3 is quite low: it’s an impurity level of 20parts per billion. The purest silicon wafers have impurity concentrations of just parts per trillion. 50 years of materials processing have made silicon one of the purest materials we have. To make silicon more insulating without having to work so hard at removing impurities, you can carefully dope it so that the number of acceptors and donors are the same. This is called compensating. The donors freeze out at lower temperature when their binding energy becomes strong enough compared to the thermal energy: they then capture conduction electrons giving us the hydrogenic states I just mentioned. This is why standard silicon chips designed to work at room temperature will stop working at cryogenic temperatures. Dopants in diamond have a larger binding energy (than silicon) so that they are not ionised at room temperature. Very pure diamonds have more impurities than very pure silicon, but the fact that the dopants in diamond are not ionised is what makes diamond an excellent insulator at room temperature. Dopants in diamond have larger binding energies so are not ionised at room temperature Temperature dependence of the electron density in silicon with a net donor density ND-NA=1015 cm-3. Page 61, Singleton

Donor Qubits in Silicon
My group does research on the electrons bound to donors in silicon at low temperatures, using its spin as a quantum bit for future quantum computers. This picture is a substitutional bismuth donor in a silicon lattice and its electronic and nuclear spins make very nice qubits. Picture by Manuel Voegtli

Electron Qubits in diamond
My group also does research using electrons bound to dopants in diamond as qubits. In particular the nitrogen-vacancy centre is great for this. Mark Newton will tell you more about this defect in Module 3 on defects and dopants, and in Module 8 next year you will hear more about their usefulness for Quantum Technologies. One of the reasons for being so keen on these qubits in diamond is that they work at room temperature. When we say that they work, we mean that we can store quantum information on the electron spin of this defect at room temperature. So clearly the electron does not escape from these defects at room temperature, even though it does for silicon. As I said before, this is the reason diamond is an insulator rather than a semiconductor: it’s because the dopants are not ionised at room temperature so there is not much extrinsic conductivity. Picture by Alan Stonebraker

Why is diamond an insulator?
Electron energy Interatomic spacing 2 4 6 So, we explained that diamond and silicon both get bandgaps between the sp3 bonding states and their antibonding states. Then we saw that diamond has a larger bandgap than silicon. However, to understand why diamond is an insulator and silicon is a semiconductor we need to look at the extinsic conductivity: specifically dopants in diamond have larger binding energies than dopants in silicon.

- Binding energies for phosphorous donors: Silicon: 46 meV
Solve Schrödinger’s equation for an electron in a box: Binding energies for phosphorous donors: Silicon: 46 meV Diamond: 500 meV Here is the slide we used to get the energy levels of the actual hydrogen atom. We can use the same physics as long as we change two things: the mass of the electron in the hydrogen atom needs to be replaced by the effective mass, m* and the dielectric permittivity must be changed. For the hydrogen atom we just used the vacuum permittivity, but for an electron bound to a dopant in a semiconductor, the Coulomb potential is screened by the polarizability of the semiconductor. We sort this out by multiplying epsilon_0 in the Coulomb potential by the epsilon_r for the material. This screening means that binding energies of these hydrogenic dopants in semiconductors are smaller than for a hydrogen atom. The important term here is the epsilon_r because it is squared in the expression for the energy levels, and also because the effective masses of silicon and diamond are similar but there is significant difference between the epsilon_r for diamond and silicon. Diamond has slightly larger effective mass than silicon which increases the binding energy a little. The relative permittivity for silicon is up to twice that of diamond, so this is why diamond has larger binding energies for dopants when compared to dopants in silicon. See: Electron effective masses and latice scattering in natural diamond, F. Nava et al, Solid State Communications, 33, 475 (1980). The search for donors in diamond, R. Kalish, Diamond and Related Materials 10, 1749 (2001). - Page 240, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985

Why is diamond an insulator rather than a semiconductor?
a) Wide band-gap means no intrinsic conductivity, deep dopants mean no extrinsic conductivity The binding energy of the dopants is larger: we say that they are deep dopants while silicon has shallow dopants.

But doped diamond and silicon can be metals too
Extrinsic conductivity When we think about doping a semiconductor we have a picture like this where the red line is the energy level of the dopant. If this is a phosphorous dopant in silicon then this energy level is 46meV below the conduction band so it is ionised at room temperature. At low temperature the electron is bound to this level and we have no free charge carriers. If we keep on adding in more and more doping then the conductivity increases until you reach the metal-insulator transition. The big difference here between a metal and a non-metal is that a metal conducts better as you cool it down, while a non-metal conducts worse. One way this can happen is if this impurity state in red becomes so big that it is a tight-binding band. We can picture this as happening when the hydrogenic dopants are so close together that their Bohr radii are overlapping. Semiconductor at room temperature Semiconductor at low temperature

Doped silicon can be a metal
Observed “zero temperature” conductivity versus donor concentration n for Si:P, after T F Rosenbaum et al. Page 285, Kittel, Introduction to Solid State Physics, Wiley 1996 This is experimental data for phosphorous dopants in silicon, which shows that indeed the critical concentration for the transition to a metal occurs when the donor impurity wavefunctions are overlapping.

Doped diamond can be a metal
Charge transport in heavily B- doped polycrystalline diamond films, M. Werner et al Applied Physics Letters 64, 595 (1994) Sample A has 8 x 1021 cm-3 boron Diamond becomes metallic with a high boron doping of ~8x1021 cm-3. Diamond has deeper donors than silicon so they have much smaller Bohr radii. This means that it’s not surprising that a larger density of impurities is needed to make diamond a metal. However, with these large doping levels for metallic diamond we don’t have such a simple picture of the wavefunctions overlapping. Instead the metallic conduction could be due to conduction in the intrinsic diamond bandstructure as well as the impurity bands. See also: Origin of the metallic properties of heavily boron-doped superconducting diamond, T. Yokoya et al, Nature 438, 647 (2005)

So far, for semiconductors and insulators, we have focused on the question of how many free charge carriers there are, but this is not enough to determine the conductivity of a material. The other thing to think about is how easily those charge carriers can move through the material. We describe this with the mobility, which we call mu_e for the electrons. n is the density of electrons and e is their charge. Then we might include the holes too if they are present (heavy holes tend to dominate over light holes because they have a larger density of states in energy). We will always include a subscript on mobilities so as not to get them confused with the chemical potential, mu. Now the Bloch theorem tells us that there will be no scattering in an infinite perfect crystal because the eigenstates are plane waves multiplied by a function with the periodicity of the crystal. In a real crystal thought, there are imperfections: the two main sources of scattering are charged impurities and phonons, but also there are other charge carriers and the boundaries of the crystal. We can use scattering rates like this where tau_c is the time between collisions. Diamond has a high mobility because crystals can be made with a low density of defects. Electrical conductivity of semiconductors. Page 127, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

Diamond has a high mobility because crystals can be made with a low density of defects. These values are at room temperature, and for most of these materials it is thermal phonons which limit the mobility rather than impurities. This means that larger mobilities are reached by cooling down, getting up above 106 cm2/Vs for PbTe and GaAs. GaAs has a higher mobility than silicon which means that GaAs transistors can be faster than silicon transistors. Carrier mobilities at room temperature in cm2/Vs. Page 221, Kittel, Introduction to Solid State Physics, Wiley 1996

10-10 1 1010 1020 Resistivity (ohm-cm) PTFE (Teflon)  > 1018 -cm
(room temperature) Silicon  ~ 104 -cm (room temperature) Pure metal  ~ -cm (1 K) Tin  ~ 10-5 -cm (room temperature) Superconductors  ~ 0 Diamond  ~ 1016 -cm (room temperature) So, this diagram of the resistivities of materials hides a lot of complexity surrounding where the conductivity comes from and the temperature dependence. Our ability to control silicon’s room-temperature resistivity in <this> region is central to building computer chips, along with the possibility of making the silicon both n-type or p-type. Resistivity (ohm-cm)

Diamond properties Diamond has all of these extreme properties and we have covered several of them now. Diamond is an excellent electrical insulator because it has a full band separated from an empty band by a large bandgap, plus the dopant states are deep so they are not ionised at room temperature. The electrons in diamond’s strong bonds are not moved much by applying an electric field: this gives the low dielectric constant. The low dielectric loss follows from this as you will hear from Stephen Lynch. By heavily doping diamond we can make it a good conductor. Metals are good thermal conductors because their free electrons can carry heat around. So how come diamond is a better thermal conductor than metals? The strong bonds in diamond are what carries the heat around in the form of phonons. Martin Kuball will tell you more about this. This afternoon Stephen Lynch will be telling you about optical properties and characterization.

Gavin W Morley Department of Physics University of Warwick

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Gavin W Morley Department of Physics University of Warwick

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