solid state physics

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Last Updated: Tuesday, 15 August 2023 14:54

Published: Wednesday, 15 May 2019 18:48

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Before we get into complicated stuff, let's start from the simplest buliding bllock, atom. Section 3 of Feymann lectures deals with quantum physics that we deal with when talking about atoms. Feynman has one of the best material on Physics.

http://www.feynmanlectures.caltech.edu/

ATOMS:

Atom is the smallest entity that exists for any material. It has neutrons and protons in the nuclues, and an electron circulating around it in orbit. These would seem like particle, but modern quantum theory states that quantum scale objects have wave-particle duality, and exhibit both wave like and particle like properties. There have been many controversies over this wave-particle nature of objects, but is now generally accepted. Bigger the size of object, it's wave characteristics become so small, that it acts like a particle fr most practical purpose. However, for very small objects, as electrons, wave characteristics are noticeable. Similarly all electromagentic radiation (i.e light, heat), etc are assumed to hold both wave and particle characteristics to help explain many phenomenon which can't be explained by assuming only a particle mode, or only a wave model.

For our purpose, we will treat all electron, proton and neutron as particles, with the understanding that they show wave properties. Electrons have -ve charge, protons have +ve charge, and neutrons are chargeless. All of them have mass, but their mass is too small (neutron/proton have 1800 times greater mass than electron, but still small). So, gravitational forces between these masses is too small. They are held together via electrostatic forces (forces b/w opposite charges).

electron: mass = 9.1 x 10^-31 kg = 0.0005amu, charge = -1.6 x 10^-19 C, (electron lot smaller than neutron/proton, so mostly viewed as wave)

proton: mass  = 1.7 x 10^-27 kg = 1.007amu, charge = +1.6 x 10^-19 C, 

neutron: mass = 1.7 x 10^-27 kg = 1.008amu, charge = 0 C,  (neuton slightly heavier than proton)

In any atom, neutron and proton are in center nucleus, while electron is in circular orbit. Number of protons and electron are equal, resulting in 0 charge for an atom. This is also called "atomic number" of atom (num of electrons).

With any atom, energy is needed to break it apart into electrons and protons, That means some energy was supplied to form an atom (electron and proton joined to form an atom). They were formed right after the "Big Bang" when our known universe originated from pure energy some billions of years ago. The energy was converted to the elementary or sub atomic particles (quarks, gluons, leptons etc...). These condensed to form electrons and protons. Very soon, a proton and an electron met, and formed a hydrogen atom (no neutron, so mass=1amu). The neutron came about in a similar manner, and condensed with neighboring electrons and protons to form a complex mixture of the isotopes of hydrogen and helium (helium has 2 neutrons, 2 protons and 2 electrons, so mass=4amu). From this sprang everything else. That is why we find hydrogen and helium in abundance everywhere in universe, since these were the earliest and rudimentary atoms formed.

We can calculate total energy present in any isolated atom, by calculating the electrostatic energy b/w proton and electron, and kinetic energy of electron. We postulate that this total energy has to be -ve, since we have to supply energy to it, to break it. If total energy was greater than 0, then electron would be free to roam around, w/o being bound to the proton.

Electrostatic potential energy: Two charges at infinite distance are assumed to have 0 potential energy. they have attractive/repulsive forces depending on polarity of charges. Let's assume there are 2 charges +q and -q at infinite distance. They have 0 potential and 0 kinetic energy. They start attracting each other because of attractive force. Since Work=F*distance*cos(theta), they do work W=∫ F*dx = ∫ k*q1*q2/r^2 * dx. Integrating this we get W= - k*q1*q2/distance, which represents the final electrostatic potential energy stored in the system of these 2 charges, as +ve work was done on the +q charge. This work done didn't get lost anywhere, and by conservation of energy has to be stored somewhere. The +q charge will finally have some velocity and acceleration (due to force F acting on it), but the final velocity doesn't account for all of the energy. Since the system had 0 initial energy (at infinite distance and zero velocity), final energy also has to be 0. Since final kinetic energy is +ve, an equivalent -ve energy has to be there, which is what is stored as electrostatic potential energy.

When electron and proton come together from infinity to form an atom, we see that the final kinetic and potential energy is not 0. Again, this is expected, since they formed a bond by giving out energy, and so total energy goes -ve. This means to get this atom to have 0 net energy, we need to apply a +ve energy of that value, and then this atom will break apart into an electron and proton. Let's do this calculation for hydrogen atom.

Bohr's model: Very early, Bohr gave a simplified model of what radii, electron are circulating around nucleus. Ground state is the lowest state available for electron which is at radii R = 5.3 * 10^-11m = 0.53A. This radius can actually be calculated by getting total energy of atom (PE+KE), and then minimize this energy. PE decreases as electron gets closer to proton, but KE increases because of velocity increase (required due to uncertainty priniciple, which states that as electron gets coser, uncertainty in it's position is reduced, which requires corresponding increase in velocity to keep the product above the limit). It will turn out that min energy happens at radius R=0.53A.  Look at Feymann's lectures III-02 on the derivation.

centripetal force for electron to revolve around proton is provided via electrostatic force. F = m*v^2/R = k*q^2/R^2 => m*v^2 = k*e^2/R (e is the charge of an electron)

KE = 1/2*m*v^2 = 1/2*k*e^2/R

PE = -k*q1*q2/R = -k*e^2/R

Total energy = KE+ PE = -1/2*k*e^2/R = -2.2*10^-18J = -13.6eV (NOTE: KE is 1/2 of PE but of opposite sign. If KE was same as PE then net energy would have been 0)

1eV=1.6*10^-19J since Force F = q*E = q*V/d (For electric field E= V/d, where V=potential difference b/w 2 points and d=distance b/w them). Then Work W=F.d=q*V/d*d = q*V = 1.6*10^-19*1V = 1.6*10^-19J = 1eV

So, it will take 13.6eV of energy to ionize an hydrogen atom (i.e remove an electron from ground state).

-13.6eV is the lowest possible energy state. An electron can't get any closer than this to the nucleus. An electron can get into larger radius, but then it's speed will decrease (in accordance with hisenberg uncertainty principle). The total energy in this case will increase (though will still be -ve). It turns out that electron can only be in certain discrete radius (i.e it's not possible for it to be in any arbitrary radius). These distinct radius gives rise to distinct energies. We call these different enery levels. This is what is referred to as quantum physics, as electron energy levels are quantized. Of course the highest energy can be anything infinite, but at that point electron and proton are separate and electron is moving at infinite speeds. For our purpose, highest energy level is 0, which is the state where electron and proton just separated, and electron is at rest (no KE). We call this energy level Einfinite. The lowest energy level (-13.6eV) is called E0 (ground state), next one is called E1 and so on (highere states called as excited states, which they get to by absorbing energy from outside. 2 primary ways to excite an electron are thru absorbing light and thru collisions) . In energy state E100 or higher, electron is so loosely bound to the atom, that any small perturbation or external energy will pull the electron out.

En=-E0/n^2 where n is the energy state = 1,2,3 ...

So, E1=-E0/4 = -3.4eV (i.e if electron was in this state, ionization energ = 3.4eV only)

Periodic table shows atoms with 1 electron, all the way to atoms with 100 or more electrons. Each of these atoms consitututes a different element, and that is all the elements we know about. Interesting that just having a different number of electron changes the material so drastically (i.e aluminum vs silicon). Electrons can be in state 1s, 2s, 2p, 3s, 3p, 4s, etc. s states can have max of 2 electrons, while p states can have max of 6 electrons. 1s states correspond to n=0, 2s/2p states to n=1, 3s/3p to n=2 and so on. For a simple atom like hydrogen which has only 1 electron, all states in same "n" have same energy. i.e all states in 3 (3s,3p,3d) have same energy E2=-13.6/9=-1.5eV, However for atoms that have multiple lectrons, these electrons interact with each other, giving rise to splitting of energy levels within same n. So, 2p states are little higher on energy diagram than 2s states, but still lower than 3s states. Detailed diagrams in II-19 of Feymann's lectures. This is why s states are occupied before p states, as electrons always occupy lowest energy states. One important observation is that from states 3s and beyond, energy states start getting closer to each other (E=E0/n^2), which causes many of these energy states to overlap each other. This results in 3d being at higher energy than 4s states. This is why 4s states are occupied before 3d states, as electrons always occupy lowest energy states. More complicated maths involving schrodinger eqn is needed here.

For helium, which has 2 electrons, E0=-24.6eV (so 24.6eV needed to ionize it, or remove 1 electron). If we want to ionize it further, by removing the last electron, ionization energy=-54.4eV. So any further ionization always requires more energy.

For silicon, which has 14 electrons, ionization energy = 8.1eV. Ionization energy is b/w 5-10eV for most elements except 10 elements in 1st and 2nd row of periodic table (since electrons are further out, so lower energy needed).

Collection of atoms:

Above theory applies to isolated atoms. However, when 2 atoms come close together, there is an interaction between the orbits of their electrons. This interaction causes a splitting of each individual energy level into two slightly different levels. The atoms in almost every crystalline solid are so close together that the energy levels produced after splitting due to interaction between the various orbits of different electrons will be very large and so close together as to form a band.

Consider imaginary formation of a diamond crystal from isolated carbon atoms. Link: http://www.engineeringenotes.com/electrical-engineering/conductivity/energy-bands-in-solids-conductivity-electrical-engineering/33944

Each isolated carbon atom has an electronic structure 1s2 2s2 2p2 in the ground state. Each atom has available two 1s states, two 2s states, six 2p states and higher states. If we consider, N atoms, there will be 2N, 2N and 6N states of type 1s, 2s and 2p respectively. As the interatomic spacing decreases, their energy level split into bands, beginning with the outer shell, i.e., n = 2. As the 2s and 2p bands grow, they merge into a single band composed of a mixture of energy levels. This band of 2s-2p levels contains 8N available states.

As the distance between atoms approaches the equilibrium interatomic spacing of the diamond, this band splits into two bands separated by an energy gap Eg. The upper band (conduction band) contains 4N states, as does the lower band (valence band). Thus, apart from the low lying and tightly bound 1s levels, the diamond crystal has two bands of available energy levels separated by an energy gap Eg wide which contains no allowed energy levels for electrons to occupy. These energy bands are actually made up of many discrete energy levels which are too close together to resolve. Within a band the number of levels is of the order of the number of atoms in the crystal, so although electrons are actually restricted to these energies, they appear to be able to take on a continuum of values. 

The lower 1s band is filled with the 2N electrons which originally resided in the collective 1s states of the isolated atoms. However, there were 4N electrons in the original isolated (n = 2) shell (2N in 2s and 2N in 2p states). These 4N electrons must occupy states in the valence band or the conduction band in the crystal. At 0 K the electrons will occupy the lowest energy states available to them. In the case of diamond crystal, there are exactly 4N states in the valence band available to the 4N electrons. Thus at 0 K, every state in the valence band will be filled, while the conduction band will be completely empty of electrons.

In an insulator and pure semiconductor, lower band is completely filled and the upper band is completely empty. The energy of the forbidden gap is denoted by Eg. The conduction takes place only when the electron in valence band jumps to the conduction band. In other words, the electron in valence band requires energy equal to Eg to jump to the conduction band. When the electron jumps from the valence band to the conduction band, then a vacancy electron called a hole is created in the valence band. Since hole is a deficiency of an electron and hence is positively charged. The forbidden energy gap in an insulator is of the order of 5 to 10 eV. The amount of energy cannot be imparted to the electrons in the valence band and hence the electron cannot jump from the valence to conduction band. Therefore, conduction is not possible in the insulators. The forbidden energy gap in case of semiconductor is usually, of the order of 0.75 to 1 eV. This amount of energy can be easily imparted to the electrons in the valence band by thermal agitation of the crystal lattice. Thus, with the increase in temperature, many electrons from the valence band acquire the required amount of energy to jump to the conduction band and these results in the increase of electron hole pairs. The forbidden energy gap Eg is the energy required to break the covalent bands so as to make the electron free for conduction. In a conductor, this bandgap is of order of 0.01eV, so virtually all electrons in valence band avilable for conduction.