Homework Solutions Solid State Physics Wiki

This course is targeted particularly at 3rd year honours students and graduate students. Contents: Properties of crystals; free electron model, band structure; metals, insulators and semi-conductors; phonons; magnetism; selected additional topics in solid-state (e.g. ferroelectrics, elementary transport theory).

Marking Scheme

Assignments 15%
Midterm 35%
Final 50%

whichever is most advantageous.

Textbook

Condensed Matter Physics, Michael P. Mardar.

Bonus Set/Final 2005


What the HELL?? I'd piss myself if I got this as a final...

Maybe that's just me... - Adriano

Argh, I can't believe studying is useless !! Personally I just stopped studying SSP when I had a look at this final… Does anyone has any idea on how to start on ? I guess the temperature gradient is related to the impurity (Si) concentration, but we never studied anything qualitative about that, did we ? - Marianne

"hah. If you thought the 2005 final was bad, go look at the one with carbon nanotube crystals. That's crazy. I can't even find papers on how to do it."

We shouldn't let ourselves use the "Oh, man, this is impossible. Well, at least no one else gets it either..." excuse to not study for it. I've fallen into that trap before, and regretted it. - Adriano

Some ideas:

  • Mention everything we possibly can about semi-conductors
  • Find the Fermi energy (assume GaAs only for now, I guess...)
  • Use the tabulated effective masses to say something about the band structure (I've never actually done this, but I don't think it's too hard)

Introduction

What is Solid State Physics?

Overview of SSP and SSP vs. CMP.

What you need to know

  • Solve Condensed Matter Problems (such as many-body QM, crystal structure, diffraction patterns, ...)
  • Understand advanced literature (on such things as ...)

The Theory of Everything

Theory of Everything known, but not feasible. Systems of atoms can have behavior that can be described very simply (e.g. macroscopic maifestations of quantum behavior such as the Quantum Hall Effect). This is discussed in "More is Different" (it's a great article... read it!).

Water in the TOE picture

The Hamiltonian for water, which formally contains everything we could ever want to know about can be written as:

This on it's own cannot be used to determine the properties of water. Instead, we need to deal with the water molecules in a statistical fashion, using statistical mechanics.

Binding

Hydrogen Molecular Ion

The Hydrogen Molecular Ion (), consists of two protons and one electron. We will take the protons to be fixed, since they are much more massive and move much slower than the electrons (part of the Born-Oppenheimer approximation). Also, we work in 1D, assuming that the electron and the two protons are collinear (not sure why this works... maybe it is simply used because it is simple and demonstrates bonding).

The potential felt by the electron is: . The wavefunction is approximated as being the same as the Hydrogen ground state:

We assume the total wavefunction of the electron is a linear combination of these Hydrogenic wavefunctions: .

Covalent Bonding

To examine covalent bonding, we use the same approach, but extended to N atoms. This is the "Linear Combination of Atomic Orbitals" approach (LCAO).

Hydrogen Molecule and van der Waals

In problem set 1, we derived the van der Waals interaction using second order perturbation theory, and a couple of tricks (!!).

The full Hamiltonian is:

  • Trick 1: Expand in powers of 1/R.
  • Trick 2: In the second order calculation, we arrive at the following expression (?):

Ionic Crystals

Metals

Crystals

He went through some examples...

This section follows Ashcroft & Mermin very closely (Chapter 4).

Bravais Lattice

Note that the choice of the primitive vectors is not unique.

Coordination Number

The number of nearest neighbors of a lattice point (any lattice point, since they are all the same). For example, the simple cubic has 6 nearest neighbors (up, down, left, right, and front, back). Body-centered cubic has coordination number 8 (look at the "center" point; its nearest neighbors are the 8 corners). etc...

Primitive Unit Cell

The parallelipiped (in 2D, parallelogram) generated by the 3 (2) primitive vectors.

Conventional Unit Cell

Another region that fills space, but is larger that the primitive cell. Used for convenience, to emphasize the lattice symmetry (e.g. it is a cube in cubic lattices, etc...)

Miller Indices

Interesting Examples

Helium Crystals

Silicon

Carbon Nanotubes

Diffraction

The Bragg Approach

Consider the Bragg diffraction of Miller planes in the crystal.

The Theory of Everything Approach

Model the incoming rays as plane waves. Compare the phases of the ingoing and outgoing waves.

Example 1 - Linear array of points

Example 2 -

Fermi Energy

Basic idea :

More concretely, the fermi energy of a D-dimensional box (?):

  • Again, the basic question is: "If we have N electrons, and put them into the system, what will be the highest occupied energy?"
  • k-space Volume of a single electron (ignoring spin):

Example 1

From Problem Set 5, which was pretty easy in retrospect (if this is right):

  • Highest occupied energy is where is the number of valence electrons per atom (=1), is Avogadro's, is the molar mass (), and is the density (). The result is 5.5 eV, as expected.
  • The lowest occupied energy is when (I am pretty sure n's can't be 0. Or maybe not all n's can be 0, as in 1,0,0 is ok, but not 0,0,0?). Anyway, , and so:
    since
    Plugging the lengths 4.1 A and 8.2 A into the equation yields and

Unsorted Fermi stuff

Can someone explain what the hell is going on in eqn's (58)-(63) (Sommerfeld stuff) of week 4?

  • Ok, found it: Appendix C of Ashcroft & Mermin. No wonder I didn't get it! Talk about skipping steps!!

Unsorted Topics

Nanowires Project

I donno if this is going to work, but look at one of the references from the paper on SiGe Nanowires (Hicks et al, Phys. Rev. B 47(24) pp. 16631 (1993).) There is a formula for resistance vs. nanowire diameter (conduction in the x dir. confinement in y and z directions). However, it only takes into account the lowest energy state (n=1). It may be a good starting point though to look at conductance vs. nanowire diameter.

where is the width of the nanowire, is the effective mass, F is the Fermi-dirac function (=1 at T=0) and is the mobility of electrons.

Philip Egberts (thanks for the input Philip.)


Here is a very useful reference I found for this project : Properties of Group-IV, III-V and II-VI Semiconductors by Sadao Adachi. An electronic edition can be found on the McGill Library (just search for it and you will be able to access it). Even if this book is not crystal clear, it has the advantage of holding most of the numerical data needed for the project (like the electronic affinity). -Etienne

"Does any one know how to calculate the effective mass in the 111 direction. I had 2 different thoughts, the first was to just square and then add them as in sqrt(mxx^2+myy^2+mzz^2)/sqrt(3)

My second thought was that I should rotate the matrix so that one of the vectors lies in the 111 direction making the eigen value corresponding to that eigen vector. using this method I got (mxx*myy*mzz)^(1/3)


-Jeff"


I found a really sweet paper on solving conductance with the WKB approximation. It's very well done and readable--even for me, which says a lot. "Generalized Formula for the Electric Tunnel Effect Between Similar Electrodes Separated by a Thin Insulating Film," by John G. Simmons. Other papers include:

Properties of amorphous germanium tunnel barries

Tunneling Properties of Barriers in...

Photoexcited Coherent Tunneling in a Double-Barrier Superlattice

Also, has anyone gotten the damn thing to work in 2D? -Andrew Mack

2D?! I have no idea what's going on in 1... -Adriano

Band Structure Project

The project theme is: "What is the band structure of ?" . Use some approximation scheme to determine this.

References

Band Structure

The first thing to clarify would be, what exactly is a band structure? In a finite quantum system with a few electrons, we know that the energy of these electrons is quantized and only allowed to take on specific values. However, in a larger system, with electrons, most of these specific values are so close together as to be considered a continuum. Still, in some of these cases, there are energies which are simply not allowed. Determining the band structure of a solid means finding the allowed electronic energies.

Techniques for Varying x

Pre-averaging

The idea here is to assume that the configuration of Si and Ge molecules is random and that we can multiply any X-Y transition probability by the probability of having 1 X and 1 Y. For example, the probability of having two Si is , whereas having one Si and one Ge is . The other assumption is that the transfer constants of Si-Ge are a weighted average of the transition constants of Si and Ge (not too crude an assumption I hope, both are ~4 in the formulation we are using).

We now take a standard tight-binding matrix that describes both Si and Ge, and put an Si block in the top-left diag, Ge in the bottom-right, a transfer matrix in the off-diags, and the corresponding factors multiplying each (e.g. off-diags are , etc.).

For the form of the matrices, do a Google book search for: physics of semiconductors secular. And look at Sapoval's book. Or, look at the "great page" linked in the reference section above (in turn, they reference "Yu and Cardona - Fundamentals of Semiconductor Physics", which I may have available for anyone who asks).

NOTE: Unfortunately, I am beginning to suspect that this in fact does not work. NOTE2: Instead of putting the matrices in blocks, adding them with their respective probability factors (as Eva suggested) seems to work better (haven't really thought through why...). Behold, an actual result:

Here, x goes from 0 (Si) to 1 (Ge), in steps of .1. I am not sure why there seems to be a point-break at around 225 for the intermediate x values...

NOTE3: The above image was produced with a flat-out mistake in our code (pointed out by Kerry). Here is the much simpler looking result:

Again, x goes from 0 (Si) to 1 (Ge), in steps of 0.1. It'd be nice to see/hear what other people have so far.

Tight-Binding

I am not sure, but I think tight-binding might be the easiest approach here. Just to make things clear (to myself and others) I am going to go through the example he has in the notes, in detail.

1. The general Hamiltonian is , where we have instead of since we are assuming one kind of atom (therefore one kind of potential).
2. The single atom Hamiltonian is the same, without the sum. We call the eigenfunction of this single-atom operator , with eigenvalues . That is,
3. We now make two assumptions. First, no overlap between levels of the one-atom potential (reword this? a bit confusing to me). This means that:
(I assumed they are normalized),
which means that the one-atom Hamiltonian is diagonal:
4. Now we assume that only the tunneling probability between nearest neighbors is significant. That means that the transition probability of going from is nonzero, but the electron cannot go anywhere else. In terms of the matrix of , this means that the off-diagonal entries are some number which we will call , and anything further from the diagonal is zero.
5. As for the diagonal elements, we are essentially asking "What is the energy of the system if the electron stays at this atom?", and the answer is . This is called the "site energy".
6. Putting 4. and 5. together, we have the following equation for the matrix elements of :

Midterm 2004

"Imagine you live in a 2D world. Suppose you have the following crystal X:

All atoms are Carbon, all carbons have 3 neighbors at the same distance 1.4 A. What properties do you expect in your 2D world fo[r] X?"

Lattice

  • Is it a Bravais Lattice? Why or why not?
    • I think that it is a bravais lattice with a basis. The basis will contain 8 atoms at respectively (0,0), (1,1), (1,2.4), (0,3.4), (-1.4,3.4), (-2.4,2.4), (-2.4,1) and (-1.4,0). The bravais lattice will be a square of side 4.8. From this, it should be relatively easy to calculate the diffraction pattern - Dominique
    • Using the middle of the octogon as the center of the basis will lead to a more symmetric structure factor, since atoms would be at (+/- 0.7, +/- 1.7) and (+/- 1.7, +/- 0.7). -Etienne
    • I see a basis of 4 atoms. Basically if you take the 4 adjacent atoms in the octagon you can see that you can create a lattice with just those. This might be easier than the square. For example take the top half of the octagon and then move it around and you'll see that it fits everywhere. -Shreyans
  • Vance: "Would this be a useful four atom basis for the Bravais lattice?"
    • I think so... it is the simplest up to now and it definitely seems like it works -Dominique
  • Having a smaller basis makes it easier to explicitly find the diffraction pattern, since there are less terms to sum.

Diffraction

Jessica

I worked out the structure factor so we can find out which planes interfere constructively, and it isn't that obvious from the answer how you could work it out on paper, but it would be easy to solve with a computer. It is probably more complicated since I used the 8-atom basis, so if someone wants to try it for the 4-atom basis, feel free. Either way the planes should be the same.

Structure Factor Calculation:

Where is the atomic form factor of carbon, G is:

And the ’s are as follows: (Starting at the bottom left and going around the octagon)

The resulting Structure Factor is:

So from this, we can find out for which values we will get:

Where the diffraction planes are given by (v1, v2).

And then from those we can find the diffraction angles using

where

So, did anyone get a simpler answer?

Does anyone know for certain whether the vn's in
G = v1b1 + v2b2
have to be integers? Because it seems like for the 4 atom basis solutions, there are some square roots in the structure factor SG, which mean that in order to get answers like e-in"pi" (where n is some integer), you need some square roots in your v's (as opposed to assignment 3 where they just all had to be even or odd, etc). So, in order to make that into nice planes given by(v1, v2), I think you typically use integers... but maybe that is only true for simple bravais lattices? I've never heard of a (1, root 2) plane though:) Jessica

Jon Buset: In the general reciprocal lattice vector, G, the v's need to be integers. (pg. 36 Kittel, bottom)

Unsorted

  • What are some features we could expect in the diffraction pattern?
    • I get , where the are the 8 basis vectors (should be the same with the Shreyans 4-basis ?).
    • The amplitude squared without the delta function gives us the intensity envelope of the pinpoint peaks. It looks like:

(Too bad we can't have Mathematica with us for the midterm :-/ )

Electronic Properties

Jeff B

I calculated the fermi energy which I found to be 33.53 eV

Then to calculate the Band structure I used the nearly free electron model

you can ballpark the bands becuase you know that there is a parabola located at each reciprocal lattice point. Gaps from because of crossing bands from different locations and the size of the gap is 2 x fourier transform of V. I don't know how to calculate that though. Then find where 33.53 eV lies in the band.

Does this make sense? it seems alright but the only thing is I don't know how to find the Fourier transform of V therefore I don't know how I'm going to determine if the fermi energy lies in a gap, I just guess I just hope that it lies in the middle

-Jeff B

Unsorted

Midterm 2003

"Discuss the hydrogen crystal."

First Thoughts

  • What the hell is the hydrogen crystal?

Some papers on it I found quickly were: Natoli, "Cyrstal Structure of Molecular Hydrogen at High Pressure" Loubeyre, "X-Ray diffraction and equation of state of hydrogen at megabar pressures."

The first thing I would do is open Kittel and look at the table and find that it is hcp and take out the lattice parameters noted. (Table four in my old version--looks like a periodic table.) With the lattice parameters, calculate the diffraction pattern.

I assume this crystal is made by van der waals forces and some calculations could be done on that. To estimate the lattice constants. -Andrew M.

In the front of the CMP text by michael P Marder it gives the crystal lattice configuration for each element. for hydrogen it says it is a hcp: a=3.77, c=6.16 - Jeff B.

Midterm 2001

"Today, a revolutionary new crystal XV-23 was discovered and we want to understand its properties. It is a one-dimensional crystal, which has the following structure, as seen from an AFM:"

NOTE : I just talked to the professor and we don't have to worry about phonons for the midterm (questions 1-3). -Dominique

Question 1

"(4pts) The one-dimensional crystal has two types of atoms: atom A with mass M1 and atom B with mass M2, which can be modeled as having two different spring constants K1 and K2. What is the phonon dispersion relation?"

Answer 1

  • Ok, this is may be a little ridiculous, but I can't get the dispersion relation. Anyone have any luck with this

one?

  • Nevermind, I just needed to refresh my memory. This helped: Semi-classical Phonons
  • I seem to be getting , which is nonsense cause it should reduce to for
  • Ok, I am having better luck following Kittel (page 104, or page 121 in the djvu).

Question 2

"(1pt) How would you measure this dispersion?"

Question 3

"(2pts) Discuss the different dispersion modes." Adriano (or who ever wrote this: Ok, you can fix it... also, do you have an idea how to do number7

Done. As a tip, it is easiest to type the latex stuff directly, and then select all the math parts and click on the math button (the root n). That will automatically put it between the math tags. Also, there is a "Show Preview" so you can check if it's working or not.
I'll have a look at 7 now.

Question 4

"(5pts) The potential the electrons feel is V(x)=Σn Aδ(x-na)+Bδ(x-na-b), where a=9Å and b=3Å. What type of lattice is this? What is the reciprocal lattice? What is the Fourier transform of V(x)?"

This is not a bravais lattice, but a bravais lattice with a basis. The bravais lattice is and the basis have two atoms : one at (0) and one an (3). Note that I work in Angstroms. The reciprocal vector is thus only . The fourier transform of the potential will be -Dominique

Question 5

"(Optional: 5pts) What is the electron’s dispersion relation E(k)? Hint: Do not try optional question 5. before you finished all others as its derivation is lengthy."

Question 6

"(4pts) How would you calculate and measure E(k) (explain how you would proceed to calculate it, without doing the actual calculation)?"

To measure this energy, I would use an angle-resolved photoemission effect. Namely, I would bombard this solid with photons at know energy (and thus momentum) Then, I would detect the electrons emitted by the solid and measure their energy and momentum. Knowing the potential in the solid and using energy and momentum conservation, I would be able to know what was the energy and the momentum of the electron that got kicked out of the solid. Then, by bombarding the solid long enough and with a spread in photons energy, I would be able to get a distribution of the electrons energy as a function of their momentum.

For the calculation, I would slightly modify the Kroning-Penney method. There will be two delta functions delimiting the space in three regions. The two delta functions will be called A and B (for the value of the potential). In the region where there is no potential, the wavefunction will be Then, at each of the two boundary, I will use the following boundary conditions (see eqn 67 in week 4 notes) for and . (1) in the limit (2) When -Dominique

Question 7

"(5pts) Suppose that E(k) can be approximated by E(k)=E0(2-cos(ka)) and that Ef=E0+0.5*E0*(2-√2). What is the average number of electrons per atom?"

Do you think we can use (or is it useful??) - Dominique

Question 8

"(4pts) What is the electrical conductivity of XV-23? Discuss. (There is no single right answer). What would happen if there were impurities in XV-23?"

Formulas

Some quick formulas that might be of use.

Homework 6 - Magneto-oscillations

Question 1

"Imagine that the energy dispersion of the crystal is given by , where the effective masses mi are different in each direction. Calculate the Landau level energies for a magnetic field oriented in directions x, y and z."

Note the typo: it's supposed to be

I missed this lecture so I might be doing something stupid, but anyway: I started from (106)-(108) in his notes, and found that the spacing for the z-direction is:

where

Did anyone else get this?

Question 2

What are the periods of oscillations in 1/B for these three orientations of the magnetic field if the electronic density is and , where is the free electron mass.

I get the following maximal area for the z direction:

News:General Information:

Lecturer:           Reza Sarvari (office EE210)  

Classes:            Saturday-Monday 13:30-15:00    ;  Classroom EE No.310

Texts:       .(S1) Simon, The Oxford Solid State Basics, Oxford University Press , 2013
                 .(K1) Kittel, Introduction to Solid State Physics, John Wiley and Sons, 2004
                 
.(A)    Ashcroft and Mermin, Solid State Physics, 1976
                
.(K2) Kittel and Kroemer, Thermal Physics, W.H. Freeman, and Co, 1980
  
              .(P)   Pierret, Advanced Semiconductor Fundamentals, Prentice Hall, 2002
 
Supplemental:
  Moliton, Solid State Physics for Electronics, 2009
                          Galperin, Introduction to Modern Solid State Physics 
                                 
Syllabus:           

Grades(tentative):  20%    Homework
                             40%    2 Mid-terms
                             40%    Final Exam

Notes: 


            Notes01  

Slides: 

             01  
             02  
             03  

Homeworks/Exams and solutions: 


            homework 1  ;   sol 1
            homework 2  ;   sol 2
            homework 3  ;   sol 3
            homework 4  ;   sol 4
            homework 5  ;   sol 5

Lectures:

Structures of crystals: 
  • Materials: crystalline, poly-crystalline, amorphous 
  • Crystal Structure
  • Basis
  • Lattice: Bravais Lattice vs. Non Bravais Lattice 
  • Unit Cell, Primitive Unit Cell, Wigner-Seitz Unit Cell
  • Crystal symmetry 
  • Miller Indices      


Review of quantum mechanics: 
  • Wave-particle duality 
  • Time-independent Schr�dinger Equation 
  • Probability densities 
  • Operators 
  • Properties of Eigenfunctions and Eigenvalues 
  • Time-Dependent Schr�dinger Equation 
  • Free Particle, Particle in a Box, Quantum Wells, Tunneling




Energy Band Theory:
  • Crystal potential energy structure
  • Bloch theorem
  • Near free-electron model
  • Kronig-Penney model
  • Tight binding model
 


Reciprocal lattice, Brillouin zone, and Effective mass:
  • Brillouin zone and Zone folding
  • Particle motion, Group Velocity, Effective mass
  • Carrier and Electron and Hole Current



Bandstructure in 3D for real materials:
  • E(k) diagram for Si, Ge, GaAs
  • Direct and Indirect Bandgap Semiconductors
  • Constant Energy Surfaces
  • Effective Mass in 3-D

Density of states:
  • Density of states in k-space (periodic/complete reflecting boundary condition)
  • Density of states as a function of energy for free space
  • Specific materials, such as Si, Ge, and GaAs

Phonons:
  • Vibrations in Crystals (Longitudinal/Transverse)
  • Dispersion Relation for Elastic Waves (Brillouin Zones)
  • Two atoms primitive basis: optical phonons and acoustic phonons
  • Quantization of phonon modes

Optical absorption in semiconductors/photonic crystals:
  • Joint density of states
  • Absorption in direct band gap semiconductors
  • Absorption in indirect band gap semiconductors

Binary model systems:
  • Thermal Physics
  • Binary Model Systems
  • Gaussian Approximation to Binomial Coefficients



Temperature and entropy:
  • Probability, Average Values
  • Ensemble Average, Thermal equilibrium
  • Concept of Entropy and Temperature


Laws of thermal dynamics:
  • Entropy and heat flow
  • Laws of Thermodynamics


Helmholtz free energy, chemical potential:
  • Boltzmann factor
  • Partition function
  • Helmholtz Free energy
  • Chemical potential and Diffusive Equilibrium


Applications of chemical potential, semiconductor heterojuction:
  • Chemical Potential and Entropy
  • Chemical potential and potential energy change
  • Gibbs sum and Gibbs factor
  • Fermi-Dirac Distribution Function

Fermi-Dirac distribution:
  • Fermi-Dirac distribution
  • Fermi Gas in three dimensions: ground state and specific heat
  • Classical carrier distribution in semiconductors
  • Donors and Acceptors in Semiconductors
  • Law of mass action
  • Fermi level in intrinsic semiconductors

n-typed and p-typed semiconductors:
  • Extrinsic semiconductors
  • Fermi levels in extrinsic semiconductors
  • Degenerate semiconductors






Bosons, Planck/Bose-Einstein distribution:
  • Example for calculating the Fermi level
  • Dopant ionization consideration
  • PN junction
  • Concept of quasi Fermi level




Application of Bose-Einstein distribution, thermal photon populations::
  • Planck distribution
  • Bose-Einstein distribution
  • Applications of the Planck/Bose-Einstein distribution:
  • Thermal radiation
  • Johnson/Nyquist noise  



Recombination‐Generation:
  • Non‐equilibrium systems
  • Indirect vs. Direct Band gap
  • Recombination generation events
  • Direct Band‐to‐band Recombination
  • Direct Excitonic Recombination
  • Indirect Recombination (Trap‐assisted)
    Carrier Capture Coefficients
    SRH Recombination
  • Auger Recombination (Inverse Impact Ionization)
  • Effective Carrier Lifetime
  • Surface states




Carrier Transport:
  • Scattering mechanisem
    Ionized inpurity scattering
    Phonon scattering
    Matthession rule
  • Electron/Hole mobility
  • High field effects in mobility
  • Hall Effect
  • Diffusion
  • Continuity equation
  • Boltzmann Transport equation  




A:



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