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Solution 6.12. The second step (with the hint 'inequalities') is invalid. The rule m<k+l = ra < fc is only valid for integers m and k, and not for real numbers. The mistake made here is an easy one to make because of the overloading of the symbols < and ^ for ordering both real and integer numbers. In this case, the mistake is easily spotted, but in other circumstances it may not be so easy to spot. The moral is: beware of overloaded operators! D Solution 6.13. We have, for all n,

where again the '2' prefactor accounts for spin degeneracy. Therefore:

Thus, by indirect equality, the function / is the ceiling function. Solution 6.14. The defining equation is

where a is the momentum (discussed later). In the rest of this section the equations for calculating Awkj(t) and Av j i (t) are derived. The reference to time, , is omitted for notational convenience. From (3.23), dok _ df0le _ .^ dnet0le dnetOk

and for later:

Indirect equality is used to show that Im I m+n = \ in] We have, for all integers k,

In analogy to both the bulk and quantum-well cases, define the density of states for a one-dimensional wire as:

= = { above definition } arithmetic }

Q J-t-1

As the along-axis dispersion curve can still be described as a parabola, (as given by equation (8.12)), then reuse can be made of equation (2.39), i.e.

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276 = { definition of the floor function }

This, taken together with the result in equation (8.88), then gives:

Solution 7.4. For continued equivalences, pairs of repeated terms cancel each other out. So for continued equivalences, an even number of occurrences of the same term reduces to none, and an odd number of repeated terms reduces to one. D Solution 7.7. p v true { p y ( p = p) { py p =pv { true . Solution 7.10.

(327)

where the energy E is measured upwards from a subband minimum. Therefore, comparing the density of states for bulk (3D), quantum wells (2D) and quantum wires (1D), as summarized in Table 8.2, it can be seen that successive reductions in degrees of freedom for the electron motion, lead to reductions in the functional form of p(E] by factors of E*.

reflexivity of equivalence (5.4) } disjunction distributes over equivalence (7.5) } p reflexivity of equivalence (5.4) }

p(E] p(E]

{ golden rule, p,q := p,p } p = p= pyp { disjunction is idempotent } p =p = p { reflexivity of equivalence (5.4) }

where f'0k is the derivative of the corresponding activation function. Prom (3.27), (3.28), dot dok dnetn. dnet0lc = (1 (3-29)

El B*

Solution 7.12. = p/\(pyq) { golden rule: p,q := p,pvq }

i(^)1 (^)'

From (3.22),

p = pyq = py (pyq)

1(^1)* Ifo)* -i E-*

associativity and idempotence of disjunction }

If there are many (n) confined states within the quantum wire with subband minima Ei, then the density of states at any particular energy is the sum over all the subbands below that point, which can be written as:

d ,!< *

reflexivity of equivalence (5.4) }

Fig. 8.22 gives an example of the ID density of states, for a 60 x 70 A rectangular crosssection GaAs quantum wire surrounded by infinite barriers. In contrast to the bulk and 2D cases displayed earlier in Fig. 2.7, quantum wires show maxima in the density of states at around the subband minima, i.e. at around the point at which charge would be expected to accumulate. Therefore, interband (electron-hole) recombination will have a narrower linewidth than that of the 2D or 3D cases. The situation for quantum dots is quite different. As the particles are confined in all directions, then there are no dispersion curves, and thus the density of states is just dependent upon the number of confined levels. One single isolated dot would therefore offer just two (spin-degenerate) states at the energy of each confined level, and a plot of the density of states versus energy would be a series of s-functions.

(pyq) A(pyr)

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