Anyway, these pairs of electrons are bosons and a superfluid bose condensate can form that is able to transport current (charge) with no resistance. Superconductivity is an unusual and completely unexpected phenomenon. The idea that "more is different" in physics are partly motivated by the emergence of the superconductivity and the difficulty of explaining it in a reductionist manner.
Sunday, May 31, 2015
Superconductivity
A number of people have indicated they would like to learn about superconductivity. Perhaps on Monday we can discuss that. In superconductor there electrons near the fermi surface pair with each other. This is pretty surprising. (Electrons typically repel each other. Why would electrons pair?)
Tuesday, May 26, 2015
Ferromagnetism and anti-ferromagnetism.
The relationship between ferromagnetism and what is called "anti-ferromagmetism" is not simple. Ferromagnetism occurs in metals and is driven by the tendency of electrons to, in some circumstances, use spin alignment in order to avoid each other. In this way the overall coulomb energy associated with electron-electron repulsion can be diminished.
Antiferromagnetism, on the other hand, usually involves electrons that are confined (that is, electrons in non-metallic systems). Confinement has increased the kinetic energy of these electrons. The electrons seek a way to lower their confinement-related kinetic energy. Communicating through their spins, the electrons find a way to lower their KE through an inherently collective, cooperative behavior involving spin alignment. In an antiferromagnet, each spin tends to point in the opposite way of its nearest neighbors. Antiferromagnetism is not just the alternating of classical spins. It is a sophisticated many-electron quantum collective state.
The names, FM and AFM, tend to be rather misleading. The magnetism part makes sense, however, ferro is just from an ancient greek name for a material in which magnetism was observed long ago. It tells us nothing about the underlying origin. One might instead call it magnetism from spontaneous spin alignment, or just "spontaneous magnetism".
More importantly, anti-ferromagnetism is not really the opposite of ferromagnetism. It is its own collective phenomenon with its own origins. These are very different and probably even more intriguing and subtle than those of ferromagnetism. Anti-ferromagetism involves a collective state with broken symmetry which arises to enable electrons to lower their quantum kinetic energy.
Antiferromagnetism, on the other hand, usually involves electrons that are confined (that is, electrons in non-metallic systems). Confinement has increased the kinetic energy of these electrons. The electrons seek a way to lower their confinement-related kinetic energy. Communicating through their spins, the electrons find a way to lower their KE through an inherently collective, cooperative behavior involving spin alignment. In an antiferromagnet, each spin tends to point in the opposite way of its nearest neighbors. Antiferromagnetism is not just the alternating of classical spins. It is a sophisticated many-electron quantum collective state.
The names, FM and AFM, tend to be rather misleading. The magnetism part makes sense, however, ferro is just from an ancient greek name for a material in which magnetism was observed long ago. It tells us nothing about the underlying origin. One might instead call it magnetism from spontaneous spin alignment, or just "spontaneous magnetism".
More importantly, anti-ferromagnetism is not really the opposite of ferromagnetism. It is its own collective phenomenon with its own origins. These are very different and probably even more intriguing and subtle than those of ferromagnetism. Anti-ferromagetism involves a collective state with broken symmetry which arises to enable electrons to lower their quantum kinetic energy.
Friday, May 22, 2015
Ferromagnetism and HW 8. With Solutions.
This assignment is optional. You can get extra-credit for it or you can not turn it in.
...we will look at the origins of magnetism, particularly ferromagnetism in metals. We will look at the role of the Fermi energy, Pauli exclusion principle, the fermion nature of electrons. Electron-electron repulsion plays a key role. Why is that? How do electron spins get involved? What causes spins to align?
HW 8.
1. Band energy: The total band energy can be defined as the sum of the energies of all electrons in single-electron-states. That is, it is the sum of the energies of the occupied single-electron states. Considering a band of bandwidth B for which we make the simplifying assumption that the density of states, \(D_c\), does not depend on energy.
a) What is the total band energy in terms of E_f and E_c?
b) What is the total band energy of we set \(E_c = 0\) ?
c) What is the total band energy in terms of n?
d) What is the relationship between n and Ef?
e) How many total states are there on the band
Suppose that just, for the fun of it, we divide the density of states into two parts: one associated with spin-up single-electron states and the other associated with spin-down single-electron states.
2. On what basis does one choose the spatial direction (axis) with respect to which spin up and spin down are defined?
3. a) What is the density of states in the spin up band? (in terms of \(D_c\))
b) What is the relationship between \(n_\uparrow\) and \(E_f^{\uparrow}\)?
c) What is the relationship between \(n_\downarrow\) and \(E_f^{\downarrow}\)?
d) Suppose for a fixed value of n, we allow \(n_\uparrow\) and \(n_\downarrow\) to vary. What is the band energy of all n electrons as a function of \(n_\downarrow\) and \(n_\downarrow\)?
e) Express this band energy as a function of \(n_\uparrow - n_\downarrow\).
f) Plot this band energy as a function of \(n_\uparrow - n_\downarrow\). What is the domain of this graph? What configuration(s) of the system have the lowest energy?
4. For most of this quarter we have been using single-electron states (filling them with many electrons), and we have been ignoring the coulomb repulsion between electrons. Sometimes the electron-electron (e-e) doesn't make much difference, but sometimes it makes a lot of difference. A fairly simple model from the e-e interaction if to write: \(V(n_\uparrow, n_\downarrow) = U n_\uparrow n_\downarrow /n\).
a) Plot this as a function of \(n_\uparrow - n_\downarrow\).
b) What is its value at \(n_\uparrow - n_\downarrow = 0 \)?
c) What is its value at \(n_\uparrow - n_\downarrow = n \)?
5. In the Stoner model one combines the coulomb interaction energy associated with electron-electron repulsion with the band energy associated with the filled one-electron states.
a) Do this. For a given value of U, for what values of bandwidth the system will have a ground state with spin alignment (a ferromagnetic ground state).
b) For the case where the system is spin-aligned (magnetic), explain the nature of the origin of the magnetism? What is the driving mechanism that leads to the ferromagnetic state?
6. On can add a entropy related term to this model and explore why magnetism disappears at higher temperature. Generally, \( S= k ln(\Omega)\). For small B, perhaps one can approximate \(ln(\Omega) \) by \(ln(\Omega) \approx \frac{2}{3} n - \frac{1}{2n} (n_\uparrow- n_\downarrow)^2 - \frac{1}{8n^3} (n_\uparrow- n_\downarrow)^4 \).
a) For U=2 eV and B = 1 eV, find the ground state of the system as a function of temperature, T. You can assume the the ground state is the state with the lowest free energy, that is, the lowest value of F= E-TS.
b) Plot the ground state value of \((n_\uparrow- n_\downarrow) \) as a function of kT.
c) extra credit: basically one can think of \(\Omega\) as the number of possible states of the system with a given value of n and \(n_\uparrow\). What is that? That is, what is the value of \(Omega\) in terms of \(n_\uparrow\) and \(n_\downarrow\).
7. another optional special project: Calculate the magnetic susceptibility of the system from problem 6 as a function of temperature. Do this only in the temperature range above where it becomes magnetic, and show that it diverges as you approach the transition temperature from above. What is the form of that divergence? What sort of "power law"?
8. Optional special project: This project I think will show the how electrons with align spins tend to avoid each other. Consider an infinite square well with two electrons in it. Let's try making two electron states from products of the one electron states. Let's call the one electrons states \(\psi_n\) where \(\psi_n(x) = \sqrt{2} sin (n \pi x)\) where x is in nm and the well extends from x = 0 to x= 1 nm.
a) Consider a two-electron state in which the 11th and 12th states are occupied. Is \(\psi_{11}(x_1) \psi_{12}(x_2) \) an appropriate two electron state? Why or why not?
b) Suppose both electrons have the same spin (\(\uparrow \uparrow\) ). What is the appropriate two electron (spatial) state for this case?
c) On the other hand suppose that the 11th and 12th states are occupied, but the spin state is \(\frac{1}{\sqrt{2}} [ \uparrow \downarrow - \downarrow \uparrow ] \), then what is the two electron (spatial) state?
d) How are these (spatial and spin) states different? Why? Discuss here if you like. Don't wait too long. Do it now.
The expectation value of \( |x_1 - x_2| \) can tell us how close the electrons tend to be in these two-electron wave states.
e) for the state from b), calculate the expectation value \( |x_1 - x_2| \).
f) for the state from c), calculate the expectation value \( |x_1 - x_2| \).
g) are they different? by how much? what do you infer from this?
========
Solution notes link:
https://drive.google.com/file/d/0B_GIlXrjJVn4V0g5V0NERFVlU3M/view?usp=sharing
...we will look at the origins of magnetism, particularly ferromagnetism in metals. We will look at the role of the Fermi energy, Pauli exclusion principle, the fermion nature of electrons. Electron-electron repulsion plays a key role. Why is that? How do electron spins get involved? What causes spins to align?
HW 8.
1. Band energy: The total band energy can be defined as the sum of the energies of all electrons in single-electron-states. That is, it is the sum of the energies of the occupied single-electron states. Considering a band of bandwidth B for which we make the simplifying assumption that the density of states, \(D_c\), does not depend on energy.
a) What is the total band energy in terms of E_f and E_c?
b) What is the total band energy of we set \(E_c = 0\) ?
c) What is the total band energy in terms of n?
d) What is the relationship between n and Ef?
e) How many total states are there on the band
Suppose that just, for the fun of it, we divide the density of states into two parts: one associated with spin-up single-electron states and the other associated with spin-down single-electron states.
2. On what basis does one choose the spatial direction (axis) with respect to which spin up and spin down are defined?
3. a) What is the density of states in the spin up band? (in terms of \(D_c\))
b) What is the relationship between \(n_\uparrow\) and \(E_f^{\uparrow}\)?
c) What is the relationship between \(n_\downarrow\) and \(E_f^{\downarrow}\)?
d) Suppose for a fixed value of n, we allow \(n_\uparrow\) and \(n_\downarrow\) to vary. What is the band energy of all n electrons as a function of \(n_\downarrow\) and \(n_\downarrow\)?
e) Express this band energy as a function of \(n_\uparrow - n_\downarrow\).
f) Plot this band energy as a function of \(n_\uparrow - n_\downarrow\). What is the domain of this graph? What configuration(s) of the system have the lowest energy?
4. For most of this quarter we have been using single-electron states (filling them with many electrons), and we have been ignoring the coulomb repulsion between electrons. Sometimes the electron-electron (e-e) doesn't make much difference, but sometimes it makes a lot of difference. A fairly simple model from the e-e interaction if to write: \(V(n_\uparrow, n_\downarrow) = U n_\uparrow n_\downarrow /n\).
a) Plot this as a function of \(n_\uparrow - n_\downarrow\).
b) What is its value at \(n_\uparrow - n_\downarrow = 0 \)?
c) What is its value at \(n_\uparrow - n_\downarrow = n \)?
d) What simple symmetry does this function have?
e) What configuration of the system would be favored by this coulomb interaction?
5. In the Stoner model one combines the coulomb interaction energy associated with electron-electron repulsion with the band energy associated with the filled one-electron states.
a) Do this. For a given value of U, for what values of bandwidth the system will have a ground state with spin alignment (a ferromagnetic ground state).
b) For the case where the system is spin-aligned (magnetic), explain the nature of the origin of the magnetism? What is the driving mechanism that leads to the ferromagnetic state?
6. On can add a entropy related term to this model and explore why magnetism disappears at higher temperature. Generally, \( S= k ln(\Omega)\). For small B, perhaps one can approximate \(ln(\Omega) \) by \(ln(\Omega) \approx \frac{2}{3} n - \frac{1}{2n} (n_\uparrow- n_\downarrow)^2 - \frac{1}{8n^3} (n_\uparrow- n_\downarrow)^4 \).
a) For U=2 eV and B = 1 eV, find the ground state of the system as a function of temperature, T. You can assume the the ground state is the state with the lowest free energy, that is, the lowest value of F= E-TS.
b) Plot the ground state value of \((n_\uparrow- n_\downarrow) \) as a function of kT.
c) extra credit: basically one can think of \(\Omega\) as the number of possible states of the system with a given value of n and \(n_\uparrow\). What is that? That is, what is the value of \(Omega\) in terms of \(n_\uparrow\) and \(n_\downarrow\).
7. another optional special project: Calculate the magnetic susceptibility of the system from problem 6 as a function of temperature. Do this only in the temperature range above where it becomes magnetic, and show that it diverges as you approach the transition temperature from above. What is the form of that divergence? What sort of "power law"?
8. Optional special project: This project I think will show the how electrons with align spins tend to avoid each other. Consider an infinite square well with two electrons in it. Let's try making two electron states from products of the one electron states. Let's call the one electrons states \(\psi_n\) where \(\psi_n(x) = \sqrt{2} sin (n \pi x)\) where x is in nm and the well extends from x = 0 to x= 1 nm.
a) Consider a two-electron state in which the 11th and 12th states are occupied. Is \(\psi_{11}(x_1) \psi_{12}(x_2) \) an appropriate two electron state? Why or why not?
b) Suppose both electrons have the same spin (\(\uparrow \uparrow\) ). What is the appropriate two electron (spatial) state for this case?
c) On the other hand suppose that the 11th and 12th states are occupied, but the spin state is \(\frac{1}{\sqrt{2}} [ \uparrow \downarrow - \downarrow \uparrow ] \), then what is the two electron (spatial) state?
d) How are these (spatial and spin) states different? Why? Discuss here if you like. Don't wait too long. Do it now.
The expectation value of \( |x_1 - x_2| \) can tell us how close the electrons tend to be in these two-electron wave states.
e) for the state from b), calculate the expectation value \( |x_1 - x_2| \).
f) for the state from c), calculate the expectation value \( |x_1 - x_2| \).
g) are they different? by how much? what do you infer from this?
========
Solution notes link:
https://drive.google.com/file/d/0B_GIlXrjJVn4V0g5V0NERFVlU3M/view?usp=sharing
Wednesday, May 20, 2015
Illuminated Junction With A Capacitor
Daniel and I have been messing around with the differential equation (HW6 P.5) Zack gave us in class. Instead of attempting to solve it analytically we plugged it into Mathematica and got a solution and some cool graphs.
This is the equation Zack turned into a DE
\(I = I_{ill} + I_o(1-e^\frac{-eVa}{kT})\)
where \(I = \frac{dQ}{dt}\) and \(Va = \frac{Q}{C}\) our \(Va\) is negative and \(C = 10 Farads\)
So this becomes
\(\frac{dQ}{dt} = I_{ill} + I_o(1-e^\frac{eQ(t)}{CkT})\) and the exponential becomes \(e^{4Q(t)}\)
This is the code for Wolfram Mathematica:
DSolve[{Q'[t] == 12.8*10^-3 + 3*10^-11*(1 - Exp[4 Q[t]]), Q[0] == 0}, Q, {t}]
Then it gives you this solution:
\(Q(t) = -0.25 Ln[2.34375*10^{-9} + 7.8125*10^{-10} * e^{20.9701 - 0.0512 t}]\)
Here is the solution graphed
(Time is in seconds and Charge is in coulombs)
It takes around 400 seconds to charge
Here we used a value of \(I_o = 3x10^{-7}A\)
Notice the cap doesn't fully charge and steady state is achieved faster.
Here is also a link to some work Jimmy Layne did on this. It looks like he got an analytic solution for the time dependence.
Extra Special Bonus Pictures (not related to things above)
This is a three dimensional graph of E vs K where the energy surface is displayed
\(E(k_x,k_y)\)
\(E(k_x,k_y,k_z)\) Contour plot (Surfaces of constant energy)
This picture is interesting anyone have some cool thoughts about it?
Tuesday, May 19, 2015
Homework 6 solution notes.
Homework 6 included some problems that illustrate some of the issues involved in converting the energy from electron excitation due to absorbed photons to useable energy. This includes a highly non-linear I-V relationship for the n-p junction and the circuit it is part of.
In problem 7 the current is pretty much constant up to R=16 Ohms and, because of that, the power generated in the resistor is essentially linear in R in that range. At R=32 Ohm there is just a hint of non-linearity. (We also know already from our work with the capacitance that the current drops to zero just before Va = -.5 volts.) At R= 64 Ohm, I think that the current will have dropped a lot. Picturing the graph of I vs V is very helpful here I believe.
Can some people comment on whether my numbers in the table agree with what you got. Please discuss what you learned here. Also, what did you find for the R, I, V values that give you the highest power generated?
https://drive.google.com/file/d/0B_GIlXrjJVn4MVV2VmVnaVI0eFU/view?usp=sharing
In problem 7 the current is pretty much constant up to R=16 Ohms and, because of that, the power generated in the resistor is essentially linear in R in that range. At R=32 Ohm there is just a hint of non-linearity. (We also know already from our work with the capacitance that the current drops to zero just before Va = -.5 volts.) At R= 64 Ohm, I think that the current will have dropped a lot. Picturing the graph of I vs V is very helpful here I believe.
Can some people comment on whether my numbers in the table agree with what you got. Please discuss what you learned here. Also, what did you find for the R, I, V values that give you the highest power generated?
https://drive.google.com/file/d/0B_GIlXrjJVn4MVV2VmVnaVI0eFU/view?usp=sharing
Sunday, May 17, 2015
HW 7.
Some people asked for extra time and I think it would be okay to just turn this in Wednesday in class.
For any of these problems, assume a scattering rate of \(\tau = 10^{-13} s\) unless otherwise specified.
1. a) Show that you can combine two 1D Bloch states to obtain a standing wave.
b) How are their k values related?
2. Suppose the E vs k relation for the partially filled band of a metal is:
\(E(k) = E_o - (B/2) cos(ak)\) where k ranges from \(-\pi/a\) to \(+\pi/a\) and a = 0.157 nm.
a) What is the relationship between the density of atoms, N, and the integral of g(k) dk from \(-\pi/a\) to \(+\pi/a\) ?
b) If the density of electrons is N, what is \(k_f\)?
c) If the density of electrons is N/2, what is \(k_f\)?
3. (Same system) When an electric field is applied, the Fermi boundaries (in k space) shift.
a) For an electric field of 10 Volts/cm applied, by what amount is the Fermi boundary shifted if the density of electron is N? Illustrate.
b) For an electric field of 10 Volts/cm applied, by what amount is the Fermi boundary shifted if the density of electron is N/2?
4. When the Fermi boundary is shifted, electrons can be viewed as belonging to two classes: those that have an occupied partner state with which they can form a standing wave state and those that do not. What percentage electrons fall into each class for:
a) n=N
b) n=N/2
5. Some people say that the current density of metal is proportional to: the density of electrons in states that have no partner, multiplied by the fermi velocity. (Let's use a bandwidth of 6 eV for this problem.)
a) Argue for or against that as a good approximation. What other factor do you need?
b) What is the fermi velocity for a half-filled band (n=N)?
c) What is the fermi velocity for a quarter-filled band (n=N/2)?
d) How does that compare with: the thermal velocity of electron in a semiconductor with an effective mass of 1? the average speed of an electron in a semiconductor conduction band with an electric field of 10 V/cm applied?
Fermi Boundaries:
A Fermi boundary in k-space is the set of points that separate a regions of occupied states from a region of unoccupied states. In 1D it is just two points -- one at kf, the other at -kf. In order to explore the concept of a Fermi boundary we therefore need to go up to at least 2D.
6. Consider a two-dimensional (2D) metal for which the E vs k relationship for the conduction band is: \(E(k) = E_o - b cos(ak_x) - b cos(ak_y)\) where kx and ky each range from \(-\pi/a\) to \(+\pi/a\) and a = 0.157 nm. This range of k includes all the states of the band (2N states), however, they are not all filled.
a) What is the bandwidth of this band?
b-e) Figure out and illustrate the Fermi boundary for the cases where the band is roughly: 1/8, 1/4, 1/2 and 3/4 filled, respectively.*
f) Which one of these is simplest? Which one of this is not like the others? Describe and discuss your results and illustrations.
g) To what 2D lattice structure do you think this E vs k relationship corresponds? How come?
* If doing this for a specified filling fraction (1/8, 1/4...) is too difficult, then maybe we could use a sequence of fermi energy values instead. Would that be more manageable? What do you think? What might be a reasonable selection of values of Ef that meet our goal of exploring?
7. One can make this anisotropic in the following way:
\(E(k) = E_o - b cos(a_xk_x) - \frac{b}{4} cos(a_yk_y)\)
a) Figure out and illustrate the Fermi boundary for the case where the fermi energy is \(E_o -b/8\).
b) Figure out and illustrate the Fermi boundary for the case where the fermi energy is \(E_o -b/4\).
c) Figure out and illustrate the Fermi boundary for the case where the fermi energy is ... your choice.
d) What would you guess is a lattice structure to which that this might correspond??
6.2 (problem added on Wednesday) I think it might be interesting to also examine the fermi boundaries for an "upside down" E vs k relationship. That is, suppose:
\(E(k) = E_o + b cos(ak_x) + b cos(ak_y)\) where kx and ky each range from \(-\pi/a\) to \(+\pi/a\) and a = 0.157 nm.
a-d) Figure out and illustrate the Fermi boundary for the cases where the band is roughly: 1/8, 1/4, 1/2 and 3/4 filled, respectively.
e) Discuss
8. The previous problems are for a simple structure, but it is not a very realistic or stable structure (because the atoms are not very efficiently (closely) packed). A more realistic 2D metal could have an E vs k relationship for the conduction band of:
\(E(k) = E_o - b cos(ak_x) - b cos(\frac{1}{2}ak_x + \frac{\sqrt{3}}{2}a k_y) - b cos(\frac{-1}{2}ak_x + \frac{\sqrt{3}}{2}a k_y) \).
a) Explore the nature of the Fermi boundaries for this band for a some particular examples of filling (corresponding to different values of Ef). What structure would you guess that this could be the and structure for? (educated guess --based on symmetry considerations perhaps)
For any of these problems, assume a scattering rate of \(\tau = 10^{-13} s\) unless otherwise specified.
1. a) Show that you can combine two 1D Bloch states to obtain a standing wave.
b) How are their k values related?
2. Suppose the E vs k relation for the partially filled band of a metal is:
\(E(k) = E_o - (B/2) cos(ak)\) where k ranges from \(-\pi/a\) to \(+\pi/a\) and a = 0.157 nm.
a) What is the relationship between the density of atoms, N, and the integral of g(k) dk from \(-\pi/a\) to \(+\pi/a\) ?
b) If the density of electrons is N, what is \(k_f\)?
c) If the density of electrons is N/2, what is \(k_f\)?
3. (Same system) When an electric field is applied, the Fermi boundaries (in k space) shift.
a) For an electric field of 10 Volts/cm applied, by what amount is the Fermi boundary shifted if the density of electron is N? Illustrate.
b) For an electric field of 10 Volts/cm applied, by what amount is the Fermi boundary shifted if the density of electron is N/2?
4. When the Fermi boundary is shifted, electrons can be viewed as belonging to two classes: those that have an occupied partner state with which they can form a standing wave state and those that do not. What percentage electrons fall into each class for:
a) n=N
b) n=N/2
5. Some people say that the current density of metal is proportional to: the density of electrons in states that have no partner, multiplied by the fermi velocity. (Let's use a bandwidth of 6 eV for this problem.)
a) Argue for or against that as a good approximation. What other factor do you need?
b) What is the fermi velocity for a half-filled band (n=N)?
c) What is the fermi velocity for a quarter-filled band (n=N/2)?
d) How does that compare with: the thermal velocity of electron in a semiconductor with an effective mass of 1? the average speed of an electron in a semiconductor conduction band with an electric field of 10 V/cm applied?
Fermi Boundaries:
A Fermi boundary in k-space is the set of points that separate a regions of occupied states from a region of unoccupied states. In 1D it is just two points -- one at kf, the other at -kf. In order to explore the concept of a Fermi boundary we therefore need to go up to at least 2D.
6. Consider a two-dimensional (2D) metal for which the E vs k relationship for the conduction band is: \(E(k) = E_o - b cos(ak_x) - b cos(ak_y)\) where kx and ky each range from \(-\pi/a\) to \(+\pi/a\) and a = 0.157 nm. This range of k includes all the states of the band (2N states), however, they are not all filled.
a) What is the bandwidth of this band?
b-e) Figure out and illustrate the Fermi boundary for the cases where the band is roughly: 1/8, 1/4, 1/2 and 3/4 filled, respectively.*
f) Which one of these is simplest? Which one of this is not like the others? Describe and discuss your results and illustrations.
g) To what 2D lattice structure do you think this E vs k relationship corresponds? How come?
* If doing this for a specified filling fraction (1/8, 1/4...) is too difficult, then maybe we could use a sequence of fermi energy values instead. Would that be more manageable? What do you think? What might be a reasonable selection of values of Ef that meet our goal of exploring?
7. One can make this anisotropic in the following way:
\(E(k) = E_o - b cos(a_xk_x) - \frac{b}{4} cos(a_yk_y)\)
a) Figure out and illustrate the Fermi boundary for the case where the fermi energy is \(E_o -b/8\).
b) Figure out and illustrate the Fermi boundary for the case where the fermi energy is \(E_o -b/4\).
c) Figure out and illustrate the Fermi boundary for the case where the fermi energy is ... your choice.
d) What would you guess is a lattice structure to which that this might correspond??
6.2 (problem added on Wednesday) I think it might be interesting to also examine the fermi boundaries for an "upside down" E vs k relationship. That is, suppose:
\(E(k) = E_o + b cos(ak_x) + b cos(ak_y)\) where kx and ky each range from \(-\pi/a\) to \(+\pi/a\) and a = 0.157 nm.
a-d) Figure out and illustrate the Fermi boundary for the cases where the band is roughly: 1/8, 1/4, 1/2 and 3/4 filled, respectively.
e) Discuss
8. The previous problems are for a simple structure, but it is not a very realistic or stable structure (because the atoms are not very efficiently (closely) packed). A more realistic 2D metal could have an E vs k relationship for the conduction band of:
\(E(k) = E_o - b cos(ak_x) - b cos(\frac{1}{2}ak_x + \frac{\sqrt{3}}{2}a k_y) - b cos(\frac{-1}{2}ak_x + \frac{\sqrt{3}}{2}a k_y) \).
a) Explore the nature of the Fermi boundaries for this band for a some particular examples of filling (corresponding to different values of Ef). What structure would you guess that this could be the and structure for? (educated guess --based on symmetry considerations perhaps)
Monday, May 11, 2015
Hi everyone,
These are some of the responses that I gave for the midterm. There may be some mistakes or misunderstandings but it would be helpful and interesting to see how other students may have solved these problems. Also, could please correct me on any mistakes I may have made, this way if other students made similar mistakes we can all benefit.
Homework 6
If you like solar cells, try to make sure you are able to spend a lot of time on problems 6 and 7. Those are the problems that really help you understand solar cells. Especially problem 7.
Extra credit: Can some of you solve problem 1 soon and post what you get here? That might be helpful in contextualizing some of the later problems of this homework.
For these problems, assume a symmetric n-p junction doped to \(10^{17} cm^{-3}\) on either side. [and to a semiconductor for which \(E_g = 1 eV, \quad kT=.025 eV\) and \(D_c = D_V = 12 \times 10^{21} \frac{states}{eV*cm^3}, \quad B_c = B_V = 3 eV\)].
1. In the ideal junction approximation, I believe that we found that for a junction connected to a battery of voltage V_a, the current density will have the form:
\(J(V_a) = J_o (1 - e^{-eV_a/kT})\)
Based on the calculation we did of the diffusion current at x_d, what is a reasonable estimate for the value of J_o? What are the units of J_o?
2. (LED question) With a battery connected to the junction (in series),
a) how much current do you get through the junction for an area of 1 cm^2 and \(V_a = -0.4 V\)?
b) If 50% of the electrons involved in that current flow emit a photon, how much power is that? In what units would you like that power to be expressed?
c) how much radiated power (in photons) do you get from the junction if \(V_a = -0.5 V\)?
Extra credit: Can some of you solve problem 1 soon and post what you get here? That might be helpful in contextualizing some of the later problems of this homework.
For these problems, assume a symmetric n-p junction doped to \(10^{17} cm^{-3}\) on either side. [and to a semiconductor for which \(E_g = 1 eV, \quad kT=.025 eV\) and \(D_c = D_V = 12 \times 10^{21} \frac{states}{eV*cm^3}, \quad B_c = B_V = 3 eV\)].
1. In the ideal junction approximation, I believe that we found that for a junction connected to a battery of voltage V_a, the current density will have the form:
\(J(V_a) = J_o (1 - e^{-eV_a/kT})\)
Based on the calculation we did of the diffusion current at x_d, what is a reasonable estimate for the value of J_o? What are the units of J_o?
2. (LED question) With a battery connected to the junction (in series),
a) how much current do you get through the junction for an area of 1 cm^2 and \(V_a = -0.4 V\)?
b) If 50% of the electrons involved in that current flow emit a photon, how much power is that? In what units would you like that power to be expressed?
c) how much radiated power (in photons) do you get from the junction if \(V_a = -0.5 V\)?
Saturday, May 9, 2015
Midterm answers, solutions and notes.
Who would be willing to post answers and explanations to midterm problems here? Go for it. If you got something right, or see how to do it now, and would be willing to share your results, thoughts and questions here, please do. Thanks. -Zack
PS. Here are my solutions & notes.
https://drive.google.com/file/d/0B_GIlXrjJVn4ZHV3N0J5eTlfRnM/view?usp=sharing
PS. Here are my solutions & notes.
https://drive.google.com/file/d/0B_GIlXrjJVn4ZHV3N0J5eTlfRnM/view?usp=sharing
Wednesday, May 6, 2015
Midterm on Friday. see added notes.
Our midterm is this Friday, May 8. (This is a repost/top post.)
In this class you are allowed, even encouraged, to bring a calculator. (Phones, however, are not allowed as per general department standards and practice.)
You will be asked to share your understanding of np junctions, fermi energy, carrier density and the origin of bands in crystals. Also drift current and diffusion current, mobility, effective mass and its relationship to bandwidth...
Some things that will help you:
Understand some basic definitions: n, p, bandwidth, bandgap, density of states...
A natural intuitive feel for and understanding how n and p are influenced by the relationship between the fermi energy and the band edge.
An understanding of the approximations we use (sometimes) to evaluate n and p. Good judgement regarding when some approximations are appropriate and when they are not. Confidence to decide what approximations to use. (Failure to make approximations can hurt you. It is often not okay to use the exact expression for something when an approximation can be made.)
A sense of energy scales and their importance and relationships. e.g. kT, bandwidth, E_gap, etc
An understanding of diffusion current.
An understanding of drift current.
Understand what a band represents. How many states are in a band?... the relationship between bandwidth, density of states and effective mass.
Added notes: The ability to actually calculate numbers, for current as well as n and p, will be tested.
Also, a deep understanding of the nature of an unbiased and biased np junction (negative on the n side) and the assumptions of the ideal junction approximations. Understand the concepts behind the assumptions, they are ideal, so not perfect, but in what way are they sort of reasonable? what thinking lies behind them? What are their consequences??
How do these equations look?
In this class you are allowed, even encouraged, to bring a calculator. (Phones, however, are not allowed as per general department standards and practice.)
You will be asked to share your understanding of np junctions, fermi energy, carrier density and the origin of bands in crystals. Also drift current and diffusion current, mobility, effective mass and its relationship to bandwidth...
Some things that will help you:
Understand some basic definitions: n, p, bandwidth, bandgap, density of states...
A natural intuitive feel for and understanding how n and p are influenced by the relationship between the fermi energy and the band edge.
An understanding of the approximations we use (sometimes) to evaluate n and p. Good judgement regarding when some approximations are appropriate and when they are not. Confidence to decide what approximations to use. (Failure to make approximations can hurt you. It is often not okay to use the exact expression for something when an approximation can be made.)
A sense of energy scales and their importance and relationships. e.g. kT, bandwidth, E_gap, etc
An understanding of diffusion current.
An understanding of drift current.
Understand what a band represents. How many states are in a band?... the relationship between bandwidth, density of states and effective mass.
Added notes: The ability to actually calculate numbers, for current as well as n and p, will be tested.
Also, a deep understanding of the nature of an unbiased and biased np junction (negative on the n side) and the assumptions of the ideal junction approximations. Understand the concepts behind the assumptions, they are ideal, so not perfect, but in what way are they sort of reasonable? what thinking lies behind them? What are their consequences??
How do these equations look?
Video on enhanced value of \(n(x_d\)).
The enhancement of n(x) at \(x_d\) in a biased n-p junction is the key thing that leads to a exponential current-voltage (I-V) relationship. This video illustrates and discusses the origin and nature of that enhancement. (Questions and comments are welcome.)
Tuesday, May 5, 2015
Video on np junction physics
This video looks at the nature of np junctions. It includes both the static case (no bias, no applied voltage, no steady-state current flow), and the case of a bias that can produce flow of electrons from the n-side to the p-side of the junction. (An applied voltage that is negative on the n-side will tend much electrons over toward the p-side.)
Interestingly, the key thing aspect of the current flow is diffusion of electrons from the n-side to the p-side, and, at the same time, diffusion of hole from the p side to the n side.
The video is long, but it can be divided into parts.
The first 12 minutes or so (00:00 to 12:00) has a discussion of the nature of the static junction (no applied voltage, no current flowing).
The rest deals with a biased junction with electrons flowing from the n to p side.
12:00-17:30 Big picture discussion.
17:30-24:00 The enhanced value of n(x) at x= x_d. The ideal junction approximation and its consequences.
23:40 Minority carried enhancement...
24:00 - 44:00 Calculation of enhanced n(x) for x greater than x_d. Assuming a specific enhancement at x =x_d, what does that imply about n(x) for x greater than x_d?
44:00-50:30 Discussion of diffusion region (x>x_d) length scale. How rapidly does n(x) regress to its equilibrium value?
Please post lots of comments and questions here!
Interestingly, the key thing aspect of the current flow is diffusion of electrons from the n-side to the p-side, and, at the same time, diffusion of hole from the p side to the n side.
The video is long, but it can be divided into parts.
The first 12 minutes or so (00:00 to 12:00) has a discussion of the nature of the static junction (no applied voltage, no current flowing).
The rest deals with a biased junction with electrons flowing from the n to p side.
12:00-17:30 Big picture discussion.
17:30-24:00 The enhanced value of n(x) at x= x_d. The ideal junction approximation and its consequences.
23:40 Minority carried enhancement...
24:00 - 44:00 Calculation of enhanced n(x) for x greater than x_d. Assuming a specific enhancement at x =x_d, what does that imply about n(x) for x greater than x_d?
44:00-50:30 Discussion of diffusion region (x>x_d) length scale. How rapidly does n(x) regress to its equilibrium value?
Please post lots of comments and questions here!
Monday, May 4, 2015
Carrier Density Enhancement Factor for Biased NP Junctions
When Chris and I were working on homework set 5, we noticed a discrepancy between the enhancement factor that we both agreed upon and that which was discussed online and in class. It was established online that the enhancement factor was about 54, or e^4 for the problem in which the applied voltage was -.1v. However, we believe that the exponent here is off by a factor of 2, and instead should be e^8. You can see our work for this in the picture provided, but note that the sign convention is that which Zack provided in the comments on HW 5.
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