Physics 322a – 2007  Exam 2 Coverage

 

Exam dates: distributed Monday 11-26-07, due at 4 pm on the following Friday (11-30-07).

This exam is cumulative, but will focus primarily on the material covered since the last exam:

 

Chapters covered: 

Livingston chapters 1, 12

Note: much of what we covered during this period is not in the text, so you will need to review your lecture notes carefully

 

Classes covered: Class 13 (Monday 10-1-07) through Class 30 (Friday 11-16-07), through and including empty lattice model.

 

Assignments covered: 6-9

 

Equation sheet:  You will be allowed 35 equations.

 

Most important topics (the exam is not necessarily limited to these topics, but these are the highlights of what we have done since the last exam):

 

Drude model

Basic assumptions

Definition of

Be ready to derive the expression for the resistivity from the basic assumptions

Drift velocity

Elastic mean free path

Definition

Why the experimental values of it imply that the electron doesn’t collide with nuclei that are part of the regular crystal lattice.

Be familiar with the relation between voltage and electric field

Know the definition of current density

Microscopic version of Ohm’s law                

Mobility

Hall Effect

 

Tight Binding Model

       Understand the basic idea (i.e. that it’s LCAO applied to crystals)

       Understand that the result has the form of a “discretized plane wave”:

       Be able to use the equation for the expectation value of the energy:

       Using the very rough approximation to make rough arguments about the energy

       Why the above equation tells us that the weaker the overlap, the narrower the band

       The band derived from an atomic orbital has 2N states

       Qualitative argument for the shape of the second band in 1D

       Understand qualitatively why higher bands have higher bandwidth

 

 

Metals and Semiconductors

       Why band overlap is needed to create a divalent metal

 

Bloch Theorem

       Understand both versions:  and

       Be able to derive the second version from the first

       Meaning:

       1)  is the quantum mechanical phase accumulated as you go from one point in a unit cell to the equivalent point in a different unit cell which is T away

       2) Bloch states are not momentum eigenstates

       3) Changing k by a reciprocal lattice vector doesn’t change y  (This is analogous to saying that changing an angle by 360o doesn’t change its physical significance.)

       4) The meaningful range of k is the first Brillouin zone.  (This is analogous to saying that the meaningful range of an angle is 0 to 360o.)

       Understand the tradeoff in “wiggliness” between the  part of  and the part when k is changed by a reciprocal lattice vector

 

Brillouin zones

       How the first Brillouin zone is defined

       How higher order Brillouin zones are defined

       Each BZ has the same volume

       Each BZ can “hold” 2 electrons/unit cell

       Any k outside the 1BZ is physically equivalent to one inside the 1BZ: k + G = k1BZ

       Extended, Reduced, and Repeated Zone representations

 

Free electron model

       Born-von Karmann (i.e. periodic) boundary conditions

       “volume” per allowed k

       Density of states in 3D

       Understand in detail why higher bands have higher bandwidth

 

Empty lattice model

       What it is, how to use it to roughly predict band structures

       Why, in 1D, it predicts a band structure which is qualitatively similar to that from tight binding

       Each band can “hold” 2 electrons/unit cell

       Each band occupies a “volume” equal to the 1BZ => One can make a one-to-one association between bands and BZs (i.e. second band corresponds to second BZ, etc.)

       Understand how multiple bands are represented in the reduced zone scheme (i.e. that each k in the 1BZ can be associated either with a state in band 1, or with a state in band 2, etc.)

       Extended, reduced, or repeated zone schemes

 

Fermi level topics (covered in reading and assignments only)

       Contact potential

       Thermionic emission, including Richardson-Dushmann equation

       Photoelectron spectroscopy

 

Fermi surface

       Definition

       Know what the free-electron Fermi surface looks like in 1D, 2D, and 3D, be able to calculate its radius, given the number of electrons per unit cell

       Volume inside Fermi surface is conserved when interactions are turned on

 

Band diagrams

     How to construct them in the empty lattice model

     Understand what the horizontal scale signifies

     Understand why there are multiple curves

     Be able to use a band diagram with the Fermi energy marked to determine where there would be a peak in the optical absorption

     Understand what an “absorption edge” is.  Be ready to answer questions such as, “Why can’t the blue coloration of an object be due to an absorption edge?”

     Be able to use a band diagram with the Fermi energy marked to determine whether a substance is metallic

     Direct (no change in k-needed for lowest energy transitions) and indirect bandgaps

 

Heat capacity of electrons

     Be able to use equation 12.26

 

Fermi-Dirac distribution

    

     Know how variation of T affects the graph of this.