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Figure 9.7 The two possible scattering events for (left) phonon absorption and (right) phonon emission
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Hence, completing the integration over Kxy:
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Recalling the quadratic for Kxy in the 6-function argument of equation (9.108) then:
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where it is known that as the roots for Kxy are real and distinct, then the argument of the square root function is greater than zero. Furthermore, as it has been specified that a1 > a2, then:
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which also come from the 'sum of the roots= b/a' and 'product of the roots=c/a'. In addition:
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Hence, if equation (9.111) can be manipulated to only contain the roots in the form of these simple constructions, then a reasonably compact expression may be obtained. Consider the term in parentheses in equation (9.111), i.e.
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Using the forms for a1 + a2, a1a2, and a1 a2 in equations (9.115), (9.116), and (9.117), then:
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Substituting back into equation (9.111) gives:
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Thus far, the derivation has followed fairly standard methods; however, it is possible to proceed further with the analytical work and evaluate the integral over the angle o' by using the innovative approach of Hagston, Piorek and Harrison [180]. The remainder of this section is dedicated to this procedure, but if only the result is required, then skip to equation (9.151). For now though, consider this integral over the angle o', where there is clearly a maximum value for 0' which occurs when the argument of the square root function
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in equation (9.112) becomes zero, which is given by ki cos o'max = Labelling as 'I', this is then of the form:
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Putting x = cos2 o', then dx = 2 sin o)' cos o' do', and therefore:
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which is the same as:
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Consider the substitution:
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and so then:
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when x = cos2 omax = b/a; therefore:
i.e. cos a = 1, which implies that the lower limit of the integral becomes TT. In addition, when x = 1, cos a = 1 which implies that the upper limit is 0. Substituting into equation (9.127), then obtain:
Now, the factor can be taken out of the square root, and cancelled with the factor in the numerator; the remainder of the argument then becomes which also cancels with the numerator, and hence:
Writing e = c + d(a + b)/(2a) and / = d(a - b)/(2a), then (for later):
Substituting for e and / gives:
Consider the substitution t = tan (a/2), then:
Making use of the tan half-angle formula (see [38], p. 72):
and changing the integral limits, equation (9.135) then becomes:
Multiplying the top and bottom of the above equation by (1 + t2) then gives the following:
This is a standard form, and given that the coefficient of t2 is greater than zero, the result is then given by the (A = 4(e + f ) ( e f ) > 0) component of equation (2.172) in [23], i.e.
which when evaluated gives:
Recalling the forms for a, 6, c, and d given in equation (9.123) and substituting these into equations (9.134) then gives the following:
Substituting for both (e + f) and (e f) in equation (9.142), and recalling that a = ki2, then:
With this analytical form for the integral over the angle 0', the original equation, i.e. equation (9.121), becomes:
This last equation represents the lifetime of a carrier in an initial subband 'i' with any in-plane wave vector ki, strictly speaking only those ki that satisfy energy conservation can have a lifetime. This information was really lost when the integration over the in-plane phonon wave vector Kxy was performed to remove the second (energy conservation) 6-function. It can be put back in with a Heaviside unit step function:
Recalling that A = Ef Ei = hw then the Heaviside function ensures that there are only finite lifetimes ri when: