Midterm is on Thursday, Feb 8. I would like it to include HW 4. HW 4 is pretty short and interested. I would hate for it to get overlooked. We can discuss Tuesday. The ideal think might be to work on it before Tuesday and see if there are any parts where you get stuck.
Notes on the first midterm:
1. a) \( n \approx kT D_c e^{-(E_c-E_f)/kT} \: cm^{-3} = kT D_c e^{-9} = 3.7 \times 10^{17} \: cm^{-3}\)
1. b) \( p \approx kT D_c e^{-(E_f-E_v)/kT} \: cm^{-3} = kT D_c e^{-31} = 1.03 \times 10^{8} \: cm^{-3}\)
1. c) \(4\: eV \times D = 48 \times 10^{22}\) empty states in the conduction band.
4. I think one gets 2,100 nm for the length scale. Is that correct? Is that for a band mass of 1? It is interesting to notice all the parameters this does not depend on! (As well as the ones it does depend on (kT, tau, other tau and m)).
5. a) \(1 eV \times D = 12 \times 10^{22}\) filled states, that is, electrons, in the conduction band. (Because the Fermi energy is 1 eV above the conduction band lower edge. 1/4 of the conduction band is filled.
b) Here you can use the approximation. Why? (Post answer here in comments for extra credit.) \(p \approx kT D_c e^{-(E_f-E_v)/kT} \: cm^{-3} = kT D_c e^{-40} =\) Post number here in comments for extra credit.)
c) \(3 eV \times D = 12 \times 10^{22}\) empty states.
Feel free to comment, discuss and ask questions here. What problems did you like? What problems were you not so fond of...
Wednesday, January 31, 2018
Saturday, January 27, 2018
Homework 3. Due Feb 3
0. Watch the videos on pn junction dynamics. Post a question, comment or correction. Also, you can engage in discussion with other students in the comments.
For the rest of the problems in this assignment, let's imagine a semiconductor (with KT=0.025 eV) with the following density of states:
\(D(E) = 12*10^{22}\, states/(eV cm^{3})\) for 1 eV < E < 5 eV ... (conduction band)
Additionally in the range -4 eV to 0 eV,
\(D(E) = 12*10^{22}\, states/(eV cm^{3})\) for -4 eV < E < 0 eV ... (valence band).
1. Suppose this semiconductor is uniformly doped with 10^18 donors per cm^3 .
a) What would n have to be for that doping?
b) What would \(E_F\) have to be for that doping?
c) What would p, the minority carrier density, be for that doping?
2. Suppose, on the other hand, it is uniformly doped with 10^18 acceptors per cm^3 .
a) What would p be for that doping?
b) What would \(E_F\) be for that doping?
c) What would n, the minority carrier density, be for that doping?
3. (Some of these parts may take a long time to understand and answer)
Consider a p-n junction made using the parts as above (10^18 doping on each side).
a) What is the total amount of band bending you would get?
b) What is the depletion length? (email me this number as soon as you get it so I can gauge your progress. Feel free to include a discussion of unknown or ambiguous quantities.)
c) What is the value of the electric field at the interface? (x=0)
d) What is the value of n(x) at x=0? (this is difficult. (maybe not as difficult as i thought...))
e) Sketch a rough graph of n(x). What is its value at \(x=-x_d\)? What is its value at \(x=+x_d\)?
f) What is the magnitude of the drift current at x=0?
g) What is the magnitude of the diffusion current at x=0?
h) Sketch a rough graph of the derivative of n(x) with respect to x (as a function of x).
4. extra credit. At what value of x does the maximum diffusion current occur. Sketch a graph of the diffusion current as a function of x.
5. extra credit. (warning, this might be messy) In 3c you calculated the depletion length for this particular doping (an actual number in nm or cm). Additionally, so that you can understand the dependencies of \(x_d\) on various parameters, please obtain an expression for the depletion length as a function of like D(Ec), doping, kT and whatever else it depends on. To simplify the calculation a bit, please assume that the density of states is constant within each band, as above, and the D(Ec)=D(Ev); also assume that the doping is equal and opposite on both sides (i.e, \(N_a = N_d\), where \(N_a\) refers to the doping on the left and \(N_d\)is the doping on the right side. Ask questions in the comments here or by email if anything, or everything, here is unclear.
Solution notes
1. a)\( n = 10^{18} electrons/cm^3\)
b) 0.8 eV
c) p=3.8 * 10^7 cm^-3
2. a)\( p = 10^{18} electrons/cm^3\)
b) 0.2 eV
c) n=3.8 * 10^7 cm^-3
3.
a) 0.6 eV
b) \((x_d)^2 = 0.3 volts * 2 \epsilon \epsilon_o/(e N_d) = 25 nm\)
c) Emax= \(2* 0.3 volts/x_d= 2.4 * 10^5 volts/cm\)
d) \( n(0) = 6.2 * 10^{12} electrons/cm^3\), because Ec-Ef=0.5 eV
e) 10^18 and at -xd n=3.8*10^7
h)dn(x)/dx, see plot here
For the rest of the problems in this assignment, let's imagine a semiconductor (with KT=0.025 eV) with the following density of states:
\(D(E) = 12*10^{22}\, states/(eV cm^{3})\) for 1 eV < E < 5 eV ... (conduction band)
Additionally in the range -4 eV to 0 eV,
\(D(E) = 12*10^{22}\, states/(eV cm^{3})\) for -4 eV < E < 0 eV ... (valence band).
1. Suppose this semiconductor is uniformly doped with 10^18 donors per cm^3 .
a) What would n have to be for that doping?
b) What would \(E_F\) have to be for that doping?
c) What would p, the minority carrier density, be for that doping?
2. Suppose, on the other hand, it is uniformly doped with 10^18 acceptors per cm^3 .
a) What would p be for that doping?
b) What would \(E_F\) be for that doping?
c) What would n, the minority carrier density, be for that doping?
3. (Some of these parts may take a long time to understand and answer)
Consider a p-n junction made using the parts as above (10^18 doping on each side).
a) What is the total amount of band bending you would get?
b) What is the depletion length? (email me this number as soon as you get it so I can gauge your progress. Feel free to include a discussion of unknown or ambiguous quantities.)
c) What is the value of the electric field at the interface? (x=0)
d) What is the value of n(x) at x=0? (this is difficult. (maybe not as difficult as i thought...))
e) Sketch a rough graph of n(x). What is its value at \(x=-x_d\)? What is its value at \(x=+x_d\)?
f) What is the magnitude of the drift current at x=0?
g) What is the magnitude of the diffusion current at x=0?
h) Sketch a rough graph of the derivative of n(x) with respect to x (as a function of x).
4. extra credit. At what value of x does the maximum diffusion current occur. Sketch a graph of the diffusion current as a function of x.
5. extra credit. (warning, this might be messy) In 3c you calculated the depletion length for this particular doping (an actual number in nm or cm). Additionally, so that you can understand the dependencies of \(x_d\) on various parameters, please obtain an expression for the depletion length as a function of like D(Ec), doping, kT and whatever else it depends on. To simplify the calculation a bit, please assume that the density of states is constant within each band, as above, and the D(Ec)=D(Ev); also assume that the doping is equal and opposite on both sides (i.e, \(N_a = N_d\), where \(N_a\) refers to the doping on the left and \(N_d\)is the doping on the right side. Ask questions in the comments here or by email if anything, or everything, here is unclear.
Solution notes
1. a)\( n = 10^{18} electrons/cm^3\)
b) 0.8 eV
c) p=3.8 * 10^7 cm^-3
2. a)\( p = 10^{18} electrons/cm^3\)
b) 0.2 eV
c) n=3.8 * 10^7 cm^-3
3.
a) 0.6 eV
b) \((x_d)^2 = 0.3 volts * 2 \epsilon \epsilon_o/(e N_d) = 25 nm\)
c) Emax= \(2* 0.3 volts/x_d= 2.4 * 10^5 volts/cm\)
d) \( n(0) = 6.2 * 10^{12} electrons/cm^3\), because Ec-Ef=0.5 eV
e) 10^18 and at -xd n=3.8*10^7
h)dn(x)/dx, see plot here
Understanding p-n junction dynamics
In p-n junctions there are two opposing currents. One is a drift current associated with an electric field in the interfacial region. The other is a diffusion current associated with the fact that n and p are not the same everywhere, but rather vary as a function of distance from the interface. This video talks about those two types of currents and introduces the scattering rate approximation which we use to simplify each of them.
This second video looks at the nature of the interface where the p-doped material and the n-doped material meet. In includes a discussion of depletion and how that leads to space charge and a non-zero "built in" electric field in the vicinity of the interface. This video derives the x dependence of that electric field in the context of the depletion approximation ansatz.
In the third video we discuss the electric potential, how one can get that from the electric field. One can then use the electric potential to establish the nature of the band-bending that occurs in the vicinity of the interface. From the bent bands one can obtain n(x). One can then calculate the two electron currents that are important near the interface: 1) the drift current which is proportional to n(x) times the electric field, and, 2) the diffusion current which depends on the derivative of n(x) with respect to x.
This second video looks at the nature of the interface where the p-doped material and the n-doped material meet. In includes a discussion of depletion and how that leads to space charge and a non-zero "built in" electric field in the vicinity of the interface. This video derives the x dependence of that electric field in the context of the depletion approximation ansatz.
In the third video we discuss the electric potential, how one can get that from the electric field. One can then use the electric potential to establish the nature of the band-bending that occurs in the vicinity of the interface. From the bent bands one can obtain n(x). One can then calculate the two electron currents that are important near the interface: 1) the drift current which is proportional to n(x) times the electric field, and, 2) the diffusion current which depends on the derivative of n(x) with respect to x.
Tuesday, January 23, 2018
Calculating density of holes
This video discusses a key approximation that enables calculation of the density of holes in the valence band.
Monday, January 22, 2018
Problems for Tuesday, Jan 23
The point of these problems is to develop intuition related to using density of states and the Fermi function together. Also, to learn about the relationship between \(E_F\) and the number of electrons in a conduction band. Watching the video in the neighboring post, Calculating n, may help you with these problems.
Let’s consider a 1 million atom chain. Suppose you are given that there is a band that extends from 1 eV to 5 eV (bandwidth of 4 eV) and that the density of states within this band is constant with a value of 500,000 states per eV.
Let’s consider a 1 million atom chain. Suppose you are given that there is a band that extends from 1 eV to 5 eV (bandwidth of 4 eV) and that the density of states within this band is constant with a value of 500,000 states per eV.
1.a) What is the total number of states in the band?
b) Why does that make sense? What is the correspondence between total number of states in a band and number of atoms?
Calculating n
This video is about calculating n, the number of filled states in the conduction band, from density of states and the Fermi function. In most circumstances relating to semiconductors, one can use an approximation for the Fermi function. This makes things simpler and clearer. This won't work for metals.
terminology:
*Density of states refers to quantum states that may or may not be occupied by an electron. It has units of states/eV.
*The Fermi function tells you the probability that any particular state is occupied. It ranges from 0, no chance, to 1, definitely occupied. It can be viewed as having units of electrons per state.
*occupied states = filled states = electrons. These are interchangeable terms.
terminology:
*Density of states refers to quantum states that may or may not be occupied by an electron. It has units of states/eV.
*The Fermi function tells you the probability that any particular state is occupied. It ranges from 0, no chance, to 1, definitely occupied. It can be viewed as having units of electrons per state.
*occupied states = filled states = electrons. These are interchangeable terms.
Saturday, January 20, 2018
Homework 2 with solutions added Jan 30.
I am hoping you can start this pretty soon, but I realize that is not always possible because you may have other plans or commitments. I like the idea of homework as an opportunity to learn something rather than as a task to be completed by a certain date. So, work on it when you can, and you might not finish it in one sitting, but hopefully it will be a learning experience. I welcome your feedback via email and/or comments here!
Added solution note, Feb 2. What I was looking for in problem 5 was the relationship:
\(n = KT D e^{-(E_c -E_F)/KT}\)
where D is the (constant) Density of States (near the bottom of the conduction band) and K is K Boltzmann.
Consider a semiconductor with the following density of states:
\(D(E) = 10^{22}/\sqrt{4-(E-3)^2}\) for 1 eV < E < 5 eV ... (conduction band)
where E is in eV, the 4 has units of eV squared and the 3 has units of eV.
Additionally in the range -4 eV to 0 eV,
\(D(E) = 10^{22}/\sqrt{4-(E+2)^2}\) for -4 eV < E < 0 eV ... (valence band).
1. a) Sketch D(E) as a function of E. Show that there is a gap, with zero density of states (DOS) between 0 and 1 eV. The top of the valence band is at E=0 eV and the bottom of the conduction band is at 1 eV. (The choice of putting the zero of energy at the top of the valence band is pretty common. The zero of energy is arbitrary.)
Added solution note, Feb 2. What I was looking for in problem 5 was the relationship:
\(n = KT D e^{-(E_c -E_F)/KT}\)
where D is the (constant) Density of States (near the bottom of the conduction band) and K is K Boltzmann.
Consider a semiconductor with the following density of states:
\(D(E) = 10^{22}/\sqrt{4-(E-3)^2}\) for 1 eV < E < 5 eV ... (conduction band)
where E is in eV, the 4 has units of eV squared and the 3 has units of eV.
Additionally in the range -4 eV to 0 eV,
\(D(E) = 10^{22}/\sqrt{4-(E+2)^2}\) for -4 eV < E < 0 eV ... (valence band).
1. a) Sketch D(E) as a function of E. Show that there is a gap, with zero density of states (DOS) between 0 and 1 eV. The top of the valence band is at E=0 eV and the bottom of the conduction band is at 1 eV. (The choice of putting the zero of energy at the top of the valence band is pretty common. The zero of energy is arbitrary.)
Friday, January 19, 2018
Density of States.
Here is a video that shows our derivation of density of states as a function of energy within one particular band. Note that density of states has units of states/eV.
Looking further ahead, at we will use this density of states with the Fermi function to calculate the number of electrons in the conduction band. Note that F(E) has units of electrons per state, so when you multiply F(E) and D(E) together you get units of: electrons per eV. Then when you integrate that as a function of energy (over the band) you get the number of electrons. Thus, integrating f(E)*D(E) gives you the total number of filled states in the band, that is, the number of electrons in the band.
Looking further ahead, at we will use this density of states with the Fermi function to calculate the number of electrons in the conduction band. Note that F(E) has units of electrons per state, so when you multiply F(E) and D(E) together you get units of: electrons per eV. Then when you integrate that as a function of energy (over the band) you get the number of electrons. Thus, integrating f(E)*D(E) gives you the total number of filled states in the band, that is, the number of electrons in the band.
Wednesday, January 17, 2018
Pre-lecture problem (Homework 1c).
Here is a problem that you can do if you like. It is optional, but if you want to email me your answer to the last part (what is it as a fraction of the speed of light, c?) you'll get extra credit. It provides a chance to calculate actual numbers again and it is related to something we will cover tomorrow in class.
5. Consider the expression \(E_{1,k} = E_1 -2 \gamma_1 cos(ak)\). Suppose gamma = 2 eV.
Calculate the group velocity \(v(k) = \frac{1}{\hbar}\frac{\partial E_{1,k} }{\partial k }\). Graph it vs k and evaluate it at its maximum value. I think it is at \(k = \pi/2a\). What is it as a fraction of the speed of light, c?
hint:
5. Consider the expression \(E_{1,k} = E_1 -2 \gamma_1 cos(ak)\). Suppose gamma = 2 eV.
Calculate the group velocity \(v(k) = \frac{1}{\hbar}\frac{\partial E_{1,k} }{\partial k }\). Graph it vs k and evaluate it at its maximum value. I think it is at \(k = \pi/2a\). What is it as a fraction of the speed of light, c?
hint:
Tuesday, January 16, 2018
Physics 155-January Notes
Here are some notes regarding what we could cover over the next couple weeks.
Sunday, January 14, 2018
Homework 1b.
4. Suppose \(y = -cosx\).
a) Take the derivative of y with respect to x. Use a substitution to write that as a function of y.
b) Plot \(\frac{\partial y}{\partial x }\) as a function of y from y =-1 to 1.
c) Plot \(1/|\frac{\partial y}{\partial x }|\) as a function of y from y =-1 to 1.
d) Can you integrate \(1/|\frac{\partial y}{\partial x }|\) as a function of y from y =-1 to 1? If so, what do you get?
PS. Please email what you get for part d. You can add some brief description of the salient features in your email as well if you like.
a) Take the derivative of y with respect to x. Use a substitution to write that as a function of y.
b) Plot \(\frac{\partial y}{\partial x }\) as a function of y from y =-1 to 1.
c) Plot \(1/|\frac{\partial y}{\partial x }|\) as a function of y from y =-1 to 1.
d) Can you integrate \(1/|\frac{\partial y}{\partial x }|\) as a function of y from y =-1 to 1? If so, what do you get?
PS. Please email what you get for part d. You can add some brief description of the salient features in your email as well if you like.
Saturday, January 13, 2018
Square Well Bound States
One can determine the nature and energy of bound state wave functions for a square well by using the wave equation and matching conditions, continuity and smoothness, at the boundaries. This video shows how to do that for a single finite square well. Notice that in the context of finding the wave function parameters and energy, one determines two parameters related to length scales. One is k, which is related to the wavelength of the electron wave function inside the well. The other is q. q is a very important parameter because it tells you how gradually the evanescent part of the electron wave decays outside the well. 1/q is a length scale which tells you how far the evanescent part of the electron wave function extends outside the well. This is important when one is thinking of putting wells next to each other. This length scale helps determine what we mean by close and far.
Homework 1a. Due Tuesday.
This is pretty short and I was going to ask that you try to do it by Tuesday unless someone objects to that. Does that seem okay?
1. Watch the two videos on Boch states and post comments to that post about something related to the videos.
2. Make a wave function graph for \(k=3\pi\)/(4a). Take a picture of your graph and email it to me.
3. This is a "comment question". You can post your thoughts here in the comments. You can say: " I have no idea..." or whatever reflects your thoughts on this problem. We can all work on it together here.
a) Estimate, guess or calculate the energy of the ground state of a square well 0.4 nm wide and 10 eV deep. In units of eV, post your guess, estimate or calculated result in here as a comment. We can perhaps discuss and refine this. (Try using \(\hbar c= 197.3\) eV-nm and mc^2=.511 x 10^6 eV.)
Participate! Have fun.
b) how about the 1st excited state?
I am going to add a comment here to model what I am looking for from you. You might say, for example:
(Spoiler Alert)
1. Watch the two videos on Boch states and post comments to that post about something related to the videos.
2. Make a wave function graph for \(k=3\pi\)/(4a). Take a picture of your graph and email it to me.
3. This is a "comment question". You can post your thoughts here in the comments. You can say: " I have no idea..." or whatever reflects your thoughts on this problem. We can all work on it together here.
a) Estimate, guess or calculate the energy of the ground state of a square well 0.4 nm wide and 10 eV deep. In units of eV, post your guess, estimate or calculated result in here as a comment. We can perhaps discuss and refine this. (Try using \(\hbar c= 197.3\) eV-nm and mc^2=.511 x 10^6 eV.)
Participate! Have fun.
b) how about the 1st excited state?
I am going to add a comment here to model what I am looking for from you. You might say, for example:
(Spoiler Alert)
Crystal (Bloch) states: quantum wave functions and their energies...
This first video reviews what we did in class Thursday. It shows how one can construct quantum wave functions for a crystal from a single atom wave function. (In this case the single atom is a square well, the crystal is 1 dimensional.) k is the crystal quantum number. Within the range \(-\pi/a\) to \(\pi/a \), there are N allowed values of k (where N is the number of atoms in the crystal). For each value of k there is a particular, unique quantum state (which can be occupied by an electron (or not)). These N linearly independent states constitute a band.
The second video shows an expression for the energies of these crystal wave functions (states) as a function of k. This is called an energy band. The energy band centers on the energy of the single atom state and has bandwidth of 4\(\gamma\) associated with the k dependence (often called dispersion) of the band state energies.
1st video: constructing crystal wave functions.
2nd video: energy of crystal wave function states as a function of k.
This 3rd video shows a derivation of the relationship between energy and k for the Bloch state. I do not recommend that you watch this. There is no need, but I made it so I uploaded it anyway.
The second video shows an expression for the energies of these crystal wave functions (states) as a function of k. This is called an energy band. The energy band centers on the energy of the single atom state and has bandwidth of 4\(\gamma\) associated with the k dependence (often called dispersion) of the band state energies.
1st video: constructing crystal wave functions.
2nd video: energy of crystal wave function states as a function of k.
This 3rd video shows a derivation of the relationship between energy and k for the Bloch state. I do not recommend that you watch this. There is no need, but I made it so I uploaded it anyway.
Wednesday, January 10, 2018
Crystal structure: Bravais lattices in 2 dimensions
The image here shows 2 dimensional Bravais lattices. One extra point of interest is: I think there may be a mistake in the generating vectors shown for one of the structures. What do you think?
States of a 4 well system
Here is a problem to think about and comment on here if you like:
1) What is the ground state of a one electron in a 4 well potential? (4 identical, equally spaced wells)
Can you construct 4 plausible, linearly independent wave-functions for an electron in a 4 well potential?
1) What is the ground state of a one electron in a 4 well potential? (4 identical, equally spaced wells)
Can you construct 4 plausible, linearly independent wave-functions for an electron in a 4 well potential?
Tuesday, January 9, 2018
Double finite square well. Tunneling from one well to another.
Here is a reference in case anyone would like to delve more deeply into double square well wave-functions:
Double finite square well
With reference to the above link: why does \(\psi_3\) have a lower energy than \(\psi_4\)?
(You can address this question in the comments here if you like.)
Here is a possibly interesting problem that provides a sense of how quantum mechanics can work:
1. a) Can you use the wave-functions \(\psi_1\) and \(\psi_2\), as shown in the link above, to construct a state in which an electron is initially in the left hand well?
b) How long does it take for the electron to tunnel from one well to the other? What does that depend on?
(These are the same wave functions that we called \(\psi_{1,1}\) and \(\psi_{1,2}\) in class. Assume that these are energy eigenstate wave-functions with energies \(E_{1,1}\) and \(E_{1,2}\) respectively.
Double finite square well
With reference to the above link: why does \(\psi_3\) have a lower energy than \(\psi_4\)?
(You can address this question in the comments here if you like.)
Here is a possibly interesting problem that provides a sense of how quantum mechanics can work:
1. a) Can you use the wave-functions \(\psi_1\) and \(\psi_2\), as shown in the link above, to construct a state in which an electron is initially in the left hand well?
b) How long does it take for the electron to tunnel from one well to the other? What does that depend on?
(These are the same wave functions that we called \(\psi_{1,1}\) and \(\psi_{1,2}\) in class. Assume that these are energy eigenstate wave-functions with energies \(E_{1,1}\) and \(E_{1,2}\) respectively.
Monday, January 8, 2018
Physics 155.
Physics 155: (50% midterms and final. 50% homework, participation and special projects*)
* Special projects typically involve working a problem for this class a presenting your work in a blog post.
0. Crystal structure. Crystals are spatially periodic. There is a basic unit, like an atom or small molecule. A crystal is a collection of many of these basic units are arranged in a regular pattern. A good way to learn about crystals is through examples. Graphene is a 2-dimensional structure, made by arranging a basic unit consisting of 2 carbon atoms at the vertices of a triangular lattice. A one-dimensional chain of atoms is an uncomplicated example.
1. Quantum review. We will look at the quantum eigenstates (wave-functions) of simple atoms and square wells. Going from one square well to several square wells is a good first step toward understanding electron states in crystals.
2. Quantum states of electrons in crystals.
Most of solid state physics deals with quantum states of electrons in crystals. What are they? How are quantum states of electrons in crystals related to states of electrons in atoms? This is a natural starting point for the study of solid state physics. It leads to something called “band theory” which includes states called “Bloch states”. We will explore the nature of “Bloch states”. These provide the quintessential starting point for the understanding of solid state physics.
3. Fermi energy
Fermi statistics and the Pauli exclusion principle play a huge role in the physics of crystals. Most Bloch states of quantum energy below the Fermi energy are occupied by electrons, while most states above the Fermi energy are not occupied. Electrons very near the Fermi energy play a major role in phenomena such as: magnetism and superconductivity as well as characteristics such as conductivity, specific heat and magnetic susceptibility. (Electrons near the Fermi energy in crystals are roughly analogous to valence electrons in atoms.)
4. Semiconductor physics
For semiconductors the Fermi energy is in a gap between two bands. Additionally, it can be manipulated by doping. Semiconductor structures often involve sp2 or sp3 bonding. (This is the same type of bonding that connects carbon atoms graphene as well as in biological systems.)
5. Metal physics.
Metals have a Fermi energy within a band so that there are lots of states very near the Fermi energy. Understanding what happens around the Fermi energy in metals is key to understanding electrical conductivity and superconductivity. Magnetism, superconductivity and other interesting and exotic phenomena. Sometimes a Fermi surface may be unstable and interesting things happen.
6. Disorder.
A crystal is a mathematically perfect thing (at least in our models) but what happens when it is not? How does disorder effect conductivity? This turns out to be pretty interesting. It leads to “localization” and was the subject of a Nobel prize in about 1977. This subject lends itself to learning a matrix formulation of quantum mechanics and can be explored via computer simulations.
7. Magnetism.
A lot of popular literature is confusing with regard to the fundamental nature and origin of magnetism. Magnetism comes from electron-electron repulsion and its relationship to spin via the Pauli exclusion principle.
8. Superconductivity.
Electrons form pairs which act as Bosons, form a Bose condensate and conduct electricity with no resistance. How strange is that? Superconductivity was discovered empirically in about 1908 however there was no real theoretical understanding of superconductivity until about 1958. The difficulty and nature of the theory helped motivate ideas underlying the more is different paradigm.
9. More is different.
Understanding electrons, protons and neutrons and their interactions does not necessarily mean we can understanding the nature and phenomena of systems of many electrons, protons and neutrons. Some phenomena in metals, biological systems and other complex interacting systems seems to elude an understanding starting from the Schrodinger or Dirac equation.
Outline:
Week 1. Brief discussion of crystal structure. Review of key quantum concepts and wave functions including low energy bound states of finite square wells in 1 dimension (1D). Also, hydrogen atom ground and first-excited states and a brief intro to sp2 hybridization and bonding. Thinking about constructing crystal states, as well as states of 2, 3, or 4 wells in 1D. Bloch state construction and quantum energies of Bloch states.
Week 2. More on Bloch states. States of electrons in crystals.
Week 3. Semiconductor theory
Week 4. Semiconductor theory and semiconductor device physics
Week 5. Metal physics
Week 7. Metal physics 2
Week 8. Magnetism
Week 9. Superconductivity
* Special projects typically involve working a problem for this class a presenting your work in a blog post.
0. Crystal structure. Crystals are spatially periodic. There is a basic unit, like an atom or small molecule. A crystal is a collection of many of these basic units are arranged in a regular pattern. A good way to learn about crystals is through examples. Graphene is a 2-dimensional structure, made by arranging a basic unit consisting of 2 carbon atoms at the vertices of a triangular lattice. A one-dimensional chain of atoms is an uncomplicated example.
1. Quantum review. We will look at the quantum eigenstates (wave-functions) of simple atoms and square wells. Going from one square well to several square wells is a good first step toward understanding electron states in crystals.
2. Quantum states of electrons in crystals.
Most of solid state physics deals with quantum states of electrons in crystals. What are they? How are quantum states of electrons in crystals related to states of electrons in atoms? This is a natural starting point for the study of solid state physics. It leads to something called “band theory” which includes states called “Bloch states”. We will explore the nature of “Bloch states”. These provide the quintessential starting point for the understanding of solid state physics.
3. Fermi energy
Fermi statistics and the Pauli exclusion principle play a huge role in the physics of crystals. Most Bloch states of quantum energy below the Fermi energy are occupied by electrons, while most states above the Fermi energy are not occupied. Electrons very near the Fermi energy play a major role in phenomena such as: magnetism and superconductivity as well as characteristics such as conductivity, specific heat and magnetic susceptibility. (Electrons near the Fermi energy in crystals are roughly analogous to valence electrons in atoms.)
4. Semiconductor physics
For semiconductors the Fermi energy is in a gap between two bands. Additionally, it can be manipulated by doping. Semiconductor structures often involve sp2 or sp3 bonding. (This is the same type of bonding that connects carbon atoms graphene as well as in biological systems.)
5. Metal physics.
Metals have a Fermi energy within a band so that there are lots of states very near the Fermi energy. Understanding what happens around the Fermi energy in metals is key to understanding electrical conductivity and superconductivity. Magnetism, superconductivity and other interesting and exotic phenomena. Sometimes a Fermi surface may be unstable and interesting things happen.
6. Disorder.
A crystal is a mathematically perfect thing (at least in our models) but what happens when it is not? How does disorder effect conductivity? This turns out to be pretty interesting. It leads to “localization” and was the subject of a Nobel prize in about 1977. This subject lends itself to learning a matrix formulation of quantum mechanics and can be explored via computer simulations.
7. Magnetism.
A lot of popular literature is confusing with regard to the fundamental nature and origin of magnetism. Magnetism comes from electron-electron repulsion and its relationship to spin via the Pauli exclusion principle.
8. Superconductivity.
Electrons form pairs which act as Bosons, form a Bose condensate and conduct electricity with no resistance. How strange is that? Superconductivity was discovered empirically in about 1908 however there was no real theoretical understanding of superconductivity until about 1958. The difficulty and nature of the theory helped motivate ideas underlying the more is different paradigm.
9. More is different.
Understanding electrons, protons and neutrons and their interactions does not necessarily mean we can understanding the nature and phenomena of systems of many electrons, protons and neutrons. Some phenomena in metals, biological systems and other complex interacting systems seems to elude an understanding starting from the Schrodinger or Dirac equation.
Outline:
Week 1. Brief discussion of crystal structure. Review of key quantum concepts and wave functions including low energy bound states of finite square wells in 1 dimension (1D). Also, hydrogen atom ground and first-excited states and a brief intro to sp2 hybridization and bonding. Thinking about constructing crystal states, as well as states of 2, 3, or 4 wells in 1D. Bloch state construction and quantum energies of Bloch states.
Week 2. More on Bloch states. States of electrons in crystals.
Week 3. Semiconductor theory
Week 4. Semiconductor theory and semiconductor device physics
Week 5. Metal physics
Week 7. Metal physics 2
Week 8. Magnetism
Week 9. Superconductivity
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