Page 177 - Electrical Properties of Materials

Exercises 159

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8.6. Considering that N e N h = N in a given semiconductor, E A = E g /5. At the temperature of interest E g =20kT and
i
find the ratio N h /N e which yields minimum conductivity. As- E F =5kT. The effective masses of electrons and holes are
23
∗
∗
sume that collision times for electrons and holes are equal and m = 0.12m 0 and m = m 0 .For N A =10 m –3 find
e
h
∗
that m /m = 0.5. (i) the ionized acceptor density,
∗
e
h
8.7. In a certain semiconductor the intrinsic carrier density (ii) the ratio of electron density to hole density,
is N i . When it is doped with a donor impurity N 0 , both the (iii) the hole density,
electron and hole densities change. Plot the relative electron (iv) the electron density,
and hole densities N e /N i and N h /N i as a function of N 0 /N i in (v) the temperature,
the range 0 N 0 /N i 10. Assume that all donor atoms are (vi) the gap energy.
ionized.
8.14. Show that in the low-temperature region the electron
8.8. Consider a sample of intrinsic silicon.
density in an n-type material varies as
(i) Calculate the room temperature resistivity. 1/2 1/2
N e = N c N D exp[–(E g – E D )/2kT].
(ii) Calculate the resistivity at 350 C. +
◦
(iii) If the resistance of this sample of silicon is R find the [Hint: Assume that N e = N and that the donors are only
D
temperature coefficient of resistance at room temperat- lightly ionized in that temperature range, i.e. E F > E D .]
ure defined as (1/R)(dR/dT). How can this be used to 8.15. The Bohr radius for a hydrogen atom is given by
measure temperature? eqn (4.24). On the basis of the model presented in Section 8.3
determine for silicon the radius of an impurity electron’s orbit.
∗
∗
Take E g = 1.1 eV, m = 0.26m 0 ; m = 0.39m 0 ,
h
e
8.16. The conductivity of an n-type semiconductor is σ at an
–1
2
2
–1
μ e = 0.15 m V –1 s , μ h = 0.05 m V –1 s . absolute temperature T 1 . It turns out that at this temperature
the contributions of impurity scattering and lattice scattering
8.9. A sample of gallium arsenide was doped with excess ar- are equal. Assuming that in the range T 1 to 2T 1 the elec-
senic to a level calculated to produce a resistivity of 0.05 m. tron density increases quadratically with absolute temperature,
Owing to the presence of an unknown acceptor impurity determine the ratio σ(2T 1 )/σ(T 1 ).
the actual resistivity was 0.06 m, the sample remaining
8.17. The rate of recombination (equal to the rate of gener-
n-type. What were the concentrations of donors and acceptors
ation) of carriers in an extrinsic semiconductor is given by
present? eqn (8.50). If the minority carrier concentration in an n-type
2
–1 –1
Take μ e = 0.85 m V s and assume that all impurity
semiconductor is above the equilibrium value by an amount
atoms are ionized.
(δN h ) 0 at t = 0, show that this extra density will reduce to zero
8.10. Silicon is to be doped with aluminium to produce according to the relationship,
p-type silicon with resistivity 10 m. By assuming that all δN h =(δN h ) 0 exp(–t/τ p ),
aluminium atoms are ionized and taking the mobility of a
2
–1 –1
hole in silicon to be 0.05 m V s , find the density of where
1
aluminium. τ p = .
αN e
8.11. Estimate what proportion of the aluminium is actually [Hint: The rate of recombination is proportional to the ac-
ionized in Exercise 8.10 at room temperature. The acceptor tual density of carriers, while the rate of generation remains
level for aluminium is 0.057 eV above the valence band. constant.]
8.12. A sample of silicon is doped with indium for which 8.18. Derive the continuity equation for minority carriers in
the electron acceptor level is 0.16 eV above the top of the an n-type semiconductor.
valence band. [Hint: Take account of recombination of excess holes by
introducing the lifetime, τ p .]
(i) What impurity density would cause the Fermi level to
coincide with the impurity level at 300 K? 8.19. Figure 8.16 shows the result of a cyclotron resonance
(ii) What fraction of the acceptor levels is then filled? experiment with Ge. Microwaves of frequency 24 000 MHz
(iii) What are the majority and minority carrier were transmitted through a slice of Ge and the absorption
concentrations? was measured as a function of a steady magnetic field applied
along a particular crystalline axis of the single-crystal speci-
Use data from Exercise 8.8.
men. The ordinate is a linear scale of power absorbed in the
8.13. A certain semiconductor is doped with acceptor type specimen. The total power absorbed is always a very small
impurities of density N A which have an impurity level at fraction of the incident power.