Physics 322a-2007      Solid State Physics    Walter F. Smith    Haverford College

 

Assignment 6

Due: Friday, Oct. 26 at 4 pm (Turn in to envelope outside my office.)

Reading:  Livingston Ch. 12

 

Assigned exercises (except as noted, these are group problems, i.e. you may work on them with other students in small groups)

 

 

Livingston 1-1 (individual problem)  (use the Fermi velocity for silver, 1.396 m/s, instead of the thermal velocity)  Note: The abbreviation “mol” stands for “mole”, i.e. 6.02223 atoms.  The abbreviation “cc” stands for “cubic centimeters”.

 

6A.  For the setup of problem 1.1, calculate the drift velocity.  Express this as a percentage of the Fermi velocity.

 

Livingston 1.4 (Individual problem)

 

6B.  (Adapted from Livingston 1-7) A composite material is made of alternating thin layers of copper (resistivity 1.69-6 W-cm) and a niobium-titanium alloy (resistivity 7-5 W-cm) of equal thickness.  Assume the layers are parallel to the x-y plane.

2 points a.  What is the resistivity of this composite for current flowing in the x-direction?

2 points b.  For current flowing in the x-direction, what fraction of the current is carried by the copper?

1 point c.  For current flowing in the x-direction, how do the electric fields in the two phases compare?

2 points d.  What is the resistivity of this composite for current flowing in the z-direction?

1 point e.  For current flowing in the z-direction, how do the electric fields in the two phases compare?

2 points f.  Under what conditions will the applied electric field and the current be non-parallel for this composite?  Explain.

 

Livingston 1-11 (Individual problem)

 

6C.  Individual problem.  The sheet resistance, R¨ . Consider a square sheet of material, with side length L, thickness d, and electrical resistivity r.  The resistance measured between opposite edges of the sheet is called the sheet resistance, R¨ .  It is also called “R per square”.

a. Find an expression for R¨ in terms of the other quantities above, and show that it is independent of L.  (Hint: this is pretty trivial.)  This property is what makes R  a handy tool for characterizing a thin film of material.

b.  If the sheet is very thin, then the surfaces of the sheet become the dominant sources of elastic scattering.  In this limit, , where vF is the Fermi velocity.  Show that, in this limit, the Drude model gives  R¨ .