Download Part 3. Free Electrons in Metals

PHYS 353 Solid State Physics
Supplemental Problems
Part 3. Free Electrons in Metals
3-1. The guess for an electron wave function for the free electron gas is of the form
 
( x, y, z )  Aei ( k  r )  Ae
i(k x x  k y y  k z z )
.
(a) Show that this guess is correct by showing that it is a solution to the time-independent
Schrödinger Equation and that the energies of an electron are given by Eq. (3.5), i.e.

 2k 2  2 2

(k x  k y2  k z2 )
2m
2m
(b) Normalize the wave function and show that the amplitude is the reciprocal of the square root of
the volume of the solid cube.
(c) Show that applying the boundary conditions of ( x  L, y  L, z  L)  ( x, y, z ) leads to the
quantization conditions for the components of k stated in Eq. (3.3), i.e.
k x  2p / L
k y  2q / L
k z  2r / L where p, q, and r are integers.
3-2. Recall that the number of conduction electrons with energies between  and +d is given by
n()d  g () f ()d
where the number of allowed electron states per energy interval is
g ( ) 
V
2 3
2 
(2m) 3 / 2 1 / 2
Eq. (3.7)
and the probability that a state is occupied is
1
f () 
e
(  ) / k B T
1
Eq. (3.14)
Consider a solid at absolute zero so that it is in its ground state configuration. It has N conduction
electrons in its volume of V.
(a) What is the probability that a state is occupied if its energy is below the Fermi energy F? What
is the probability if its energy is greater than F?
(b) Using the information from (a), show the Fermi energy at T=0 is given by Eq. (3.9).
(c) If we multiply the number of electrons at energy  by , this will give the energy contribution
from these electrons to the total energy of the solid. If we sum all of these contributions up, we
get the total energy E of the solid. Put mathematically,

E   n()d
0
Show that this total energy at absolute zero is E  3N F / 5 .
3-3. Sodium is monovalent (1 valence electron per atom) with an atomic mass of 23 g/mol and a
density of 968 kg/m3.
(a) Find the atomic density of sodium. Since it is monovalent, this is the conduction electron
density (N/V) in the free electron model. You can do this by noting that this is the same ratio as
NA / Vm where NA is the number of atoms per mole (Avogadro’s number) and Vm is the volume
per mole (the molar volume). (Come on, a little chemistry never hurt anybody.)
(b) Evaluate the Fermi energy and corresponding Fermi wave number at absolute zero for sodium.
Express the energy in electronvolts and the wave number in reciprocal Angstroms.
(c) Xantar, an alien cyborg, gets its energy by absorbing all of the conduction electron energy from
metals. How many Calories does Xantar gain by popping one cubic centimeter of elemental
sodium into its mouth? You can assume that the temperature is near absolute zero. (1 Cal = 1
kcal = 4.186 kJ). (Sodium is so reactive with air and water that you don’t find it as a free
element on Earth but rather in compounds. On Xantar’s home world, however, it is readily
abundant in elemental form and is packaged as a snack food. It is very cold there, also.)
3-4. Consider the equation of motion for conduction electrons, Eq. (3.23).



(a) Show that the function v (t )  C (1  e t /  ) where C is a constant vector is a solution to this

equation when only a dc electric field is present. What must C equal?
(b) Show that the steady-state solution for the drift velocity is the one expressed in Eq. (3.24).
Steady-state means for times much longer than the mean collision time .
3-5. The atomic mass of aluminum is 27 g/mol and its density is about 2700 kg/m3. Its resistivity at
room temperature is 2.828x10-6 -cm. It is in column 13 (old column III) of the periodic table and
so has 3 valence electrons per atom.
(a) Find n, the number of conduction electrons per volume. (This is similar to the calculation in 33(a). Multiply the number of atoms per volume by 3 to estimate the number of conduction
electrons per atom.)
(b) Find aluminum’s Fermi energy.
(c) Find the Fermi velocity of a conduction electron assuming its effective mass is equal to its bare
mass.
(d) Find the mean time between collisions for a conduction electron at room temperature.
(e) Find the mean free path between collisions. Express the distance in nanometers.
3-6. According to Table 3.1, Hall measurements give a value of 3.5 for  1 RH ( N atom / V )e for
aluminum.
(a) Find the Hall coefficient for this aluminum sample in units of m3/C.
(b) Suppose this sample was a block placed in a 500-Gauss magnetic field with 0.5 A of current
flowing through it as shown. How large was the Hall field (in V/m) and how large was the
Hall voltage that was measured? Assume the magnetic field was perpendicular to the current
direction.
(c) In which direction must have the magnetic field pointed if the voltmeter was connected as
shown and the current flowed as shown?
y
x
1 mm
z
i = 0.5 A 1 mm
+
V
_