Download Department of Physical Sciences (Physics)

THE UNIVERSITY OF HULL
Department of Physical Sciences (Physics)
Level 5 Examination
May 2008
Thermal and Solid State Physics
(Module 04203)
Monday 19 May 2008, 09.30 to 11.30
2 hours
Answer TWO questions from Section A and TWO questions from Section B.
Do not open or turn over this exam paper, or start to write anything until
told to by the Invigilator. Starting to write before permitted to do so may
be seen as an attempt to use Unfair Means.
Module 04203
CONTINUED
Page 1 of 6
SECTION A: THERMODYNAMICS
1.
A sample of a diatomic gas, for which the ratio of specific heats  = 1.40,
undergoes a Carnot cycle between reservoirs at temperatures 227oC and
27oC. The initial volume is V = 8.31  10-4 m3 and, during the isothermal
expansion at the higher temperature the volume doubles. Show that
(a) the work done on the isothermal at the higher temperature is 576 J,
[7 marks]
(b) the work done on the isothermal at the lower temperature is -346 J,
[5 marks]
(c) the work done on the two adiabatics cancels.
[4 marks]
Also show that the efficiency of this particular Carnot cycle is 40%.
[4 marks]
[It may be assumed that the gas obeys the equation of state pV = AT, where
A = 1.6628 J/K]
2.
(i) By considering
TdS = dU + pdV,
show that, for a fluid
 U 
 S 
 p 

  T
  p  T    p.
 V T
 V T
 T V
[8 marks]
(ii) State Joule’s Law and show that, for a fluid obeying this law, there exists a
function of volume only, g(V), such that
p.g(V) = T.
[6 marks]
(iii) If an equation of state of the form
pV= f(T)
holds as well as Joule’s Law, find the form of the functions f and g.
[6 marks]
Module 04203
CONTINUED
Page 2 of 6
3.
(i) (a) For a first order phase transition, both the temperature and pressure
remain constant. By considering the Gibbs Free Energy in the form
G = H - TS,
show that, for such a transition
TdS = dH.
[4 marks]
(b) Hence, or otherwise, calculate the entropies of the melting of ice and
vaporisation of water, given that
fusH = +6.008 kJ/mol and vapH = +40.656 kJ/mol.
[6 marks]
(ii) (a) Given that, at 0oC, the heat of fusion of ice is 333.5 J/g, the density of
water is 0.9998 g/cm3, and the density of ice is 0.9168 g/cm3, show that the
change in the freezing point of water per bar pressure is -0.0075 K/bar.
[8 marks]
(b) Hence find the change in pressure necessary to change the freezing point
of water by 1oC.
[2 marks]
It may be assumed that
 fus H
dP

dT T (Vl  Vs )
and that the units of pressure satisfy 105 Pa  1 bar.
Module 04203
CONTINUED
Page 3 of 6
SECTION B: SOLID STATE PHYSICS
4.
(i) The diagram below shows the band alignment for a metal/n-type
semiconductor contact in equilibrium. Sketch equivalent diagrams for small
forward and reverse voltages.
[4 marks]
(ii) Use phenomenological arguments to show that the current-voltage (I-V)
characteristics for the arrangement in (i) are defined by
V
I  I 0 [e
kT
 1]
where
I 0  A. AT 2 e
B

kT
and
A is the area of the contact
A* is the modified Richardson constant
T is the absolute temperature
k = 8.61 × 10-5 eV K-1 is Boltzmann’s constant
B is the barrier height.
[10 marks]
(iii) Sketch the variation of ln[I] against V for both forward and reverse
biases. Mark clearly the quantity I0 and indicate how the presence of
(a) series resistance, and
(b) generation in the depletion region
affects the forward and reverse characteristics respectively.
[6 marks]
Module 04203
CONTINUED
Page 4 of 6
5.
(i) Assuming that electrons in a solid behave like plane waves, show that the
density of electronic states in a solid varies as the square root of the electron
energy.
[10 marks]
(ii) Define what is meant by the effective density of states in the conduction
band of a semiconductor. Using the Boltzmann approximation to Fermi-Dirac
statistics derive an expression for the intrinsic carrier density in terms of the
effective density of states in both the conduction band and the valence band.
[5 marks]
(iii) A typical semiconductor has a band gap of 1.1 eV at room temperature
and an effective density of states in the conduction band of 2.0 × 1019 cm-3. A
donor state lies 40 meV below the conduction band edge. If the density of
donors is 5.0 × 1015 cm-3 calculate
(a) the position of the Fermi level relative to the conduction band edge,
(b) the density of holes if the intrinsic carrier density at room
temperature is 1.0 × 1010 cm-3.
[5 marks]
Assume all donors are ionised.
[The electronic charge is e = 1.6 × 10-19 C and Boltzmann’s constant is
k = 1.38 × 10-23 J K-1]
Module 04203
CONTINUED
Page 5 of 6
6.
(i) Compare and contrast the assumptions of the Sommerfeld model of
electronic conduction with those of its predecessor, the Drude model. Discuss
the principal experimental evidence which led to the abandonment of the
Drude model and show how the Sommerfeld model overcomes the difficulty.
[10 marks]
(ii) Treating the electron as a classical particle within the Drude model, show
that an electron accelerated under the influence of an external electric field E
has a mobility  given by

e.
m
Here  is the mean time between collisions, e is the electronic charge and m*
is the effective mass.
[5 marks]
(iii) Using the expression for mobility in (ii) above show that the conductivity
can be expressed as
n.e 2 .

m
Here n is the density of electrons. Calculate  for moderately doped silicon
(m* = 0.98 m0,   12 S m-1, n  5.0 × 1020 m-3) and hence estimate the
electronic mean free path at a low electric field of 104 V m-1.
[5 marks]
[The electronic charge is e = 1.6 × 10-19 C and the free electron mass is
m0 = 9 × 10-31 kg]
Module 04203
END
Page 6 of 6