Page 117 - Electrical Properties of Materials

Exercises 99

Exercises
6.1. Evaluate the Fermi function for an energy k B T above the (ii) Find a mean value for the temperature coefficient of
Fermi energy. Find the temperature at which there is a 1% resistance.
probability that a state, with an energy 0.5 eV above the Fermi (iii) Find how the heater power varies with temperature, and
energy, will be occupied by an electron. estimate the Stefan–Boltzmann coefficient.
(iv) If the anode voltage is increased to 2.3 kV by how much
6.2. Indicate the main steps in the derivation of the Fermi will the emission current rise?
level and calculate its value for sodium from the data given
Filament temperature (K) 2000 2300 2600
in Exercise 1.4.
Filament current (A) 1.60 1.96 2.30
6.3. Ultraviolet light of 0.2 μm wavelength is incident upon Filament voltage (V) 3.37 5.12 7.40
a metal. Which of the metals listed in Table 6.2 will emit Emission current (mA) 0.131 5.20 91.2
electrons in response to the input light?
6.8. Light from a He–Ne laser (wavelength 632.8 nm) falls on
6.4. Determine the density of occupied states at an energy a photo-emissive cell with a quantum efficiency of 10 –4 (the
k B T above the Fermi level. Find the energy below the Fermi number of electrons emitted per incident photon). If the laser
level which will yield the same density of occupied states. power is 2 mW, and all liberated electrons reach the anode,
how large is the current? Could you estimate the work func-
6.5. Use free-electron theory to determine the Fermi level in tion of the cathode material by varying the anode voltage of
a two-dimensional metal. Take N as the number of electrons the photocell?
per unit area.
6.9. Work out the Fermi level for conduction electrons in
6.6. Show that the average kinetic energy of free electrons, copper. Estimate its specific heat at room temperature; what
following Fermi–Dirac statistics, is (3/5)E F at T =0 K. fraction of it is contributed by the electrons? Check whether
your simple calculation agrees with data on specific heat given
6.7. A tungsten filament is 0.125 mm diameter and has an in a reference book.
effective emitting length of 15 mm. Its temperature is meas- Assume one conduction electron per atom. The atomic
–3
ured with an optical pyrometer at three points, at which also weight of copper is 63.5 and its density 9.4 × 10 kg m .
3
the voltage, current, and saturated emission current to a 5 mm
diameter anode are measured as given in the table below. 6.10. Figure 6.14(a) shows the energy diagram for a
metal–insulator–metal sandwich at thermodynamic equilib-
(i) Check that the data obey the Richardson law, and estimate rium. Take the insulator as representing a high potential
the work function and value of A 0 in eqn (6.37). barrier. The temperature is sufficiently low for all states above

(a) Energy (b) Energy

E +E +eU
0 F 1 eU
E +E E
0 F 1 F 2
E
F 2
E +eU
E 0
0
E=0 E=0
Z (E) Z (E) Z (E) Z (E)
l r l r
Fig. 6.14
Energy against density of states for a metal–insulator–metal tunnel junction.