Page 41 - Electrical Properties of Materials

24 The electron as a particle

(ii) Using the corrected Hall voltage find the carrier density current, and finally find the transverse field from the condition
in the Ge sample. that the transverse current is zero.]
(iii) Estimate the conductivity of the Ge.
1.9. An electromagnetic wave is incident from Medium 1
(iv) What is the mobility of the carriers?
upon Medium 2 as shown in Figs 1.4 and 1.5. Derive expres-
(v) Is it a p or n type semiconductor?
sions for the reflected and transmitted power. Show that the
1.7. The Hall effect (see Fig. 1.3) is measured in a semi- transmitted electromagnetic power is finite when ω> ω p and
conductor sample in which both electrons and holes are zero when ω< ω p .
present. Under the effect of the magnetic field both carriers [Hint: Solve Maxwell’s equations separately in both media.
are deflected in the same transverse direction. Obviously, no Determine the constants by matching the electric and mag-
electric field can stop simultaneously both the electrons and netic fields at the boundary. The power in the wave (per unit
the holes, hence whatever the Hall voltage there will always surface) is given by the Poynting vector.]
be carrier motion in the transverse direction. Does this mean
that there will be an indefinite accumulation of electrons and 1.10. An electromagnetic wave is incident upon a medium
holes on the surface of one of the boundaries? If not, why not? of width d, as shown in Fig. 1.6. Derive expressions for
the reflected and transmitted power. Calculate the transmit-
1.8. Derive an expression for the Hall coefficient R H [still
ted power for the cases d = 0.25 μmand d = 2.5 μmwhen
defined by eqn (1.20)] when both electrons and holes are 15 –1 15 –1
ω = 6.28 × 10 rad s , ω p =9 × 10 rad s (take = 0 and
present.
μ = μ 0 ).
The experimentally determined Hall coefficient is found to
be negative. Can you conclude that electrons are the dominant 1.11. In a medium containing free charges the total current
charge carriers? density may be written as J total = J –iω E E E ,where J is the
[Hint: Write down the equation of motion (neglect inertia) particle current density, E E E electric field, dielectric constant,
for both holes and electrons in vectorial form. Resolve the and ω frequency of excitation. For convenience, the above ex-
equations in the longitudinal (z-axis in Fig. 1.3) and in the pression is often written in the form J total =–iω ¯ ¯ eqvE E E , defining
transverse (x-axis in Fig. 1.3) directions. Neglect the product thereby an equivalent dielectric tensor ¯ ¯ eqv . Determine ¯ ¯ eqv
of transverse velocity with the magnetic field. Find the trans- for a fully ionized electron–ion plasma to which a constant
verse velocities for electrons and holes. Find the transverse magnetic field B 0 is applied in the z-direction.