Z ∞
0
rkdrk∆P0(rk, z, z′;iω) Z 2π
0
dφ e−ikkrkcosφ
= 2π Z ∞
0
rk∆P0(rk, z, z′;iω)J0(kkrk)drk.
This integral involves againJ0(kkrk), the Bessel function of zeroth order and is eval-uated numerically using Simpson’s rule
∆P0(kk, z, z′;iω)≈ ∆rk 3
2
nrkmax 2 −1
X
j=1
(2j∆rk)∆P0(2j∆rk, z, z′;iω)J0(2j∆rkkk)
+ 4
nrkmax 2 −1
X
j=0
((2j+ 1)∆rk)∆P0((2j+ 1)∆rk, z, z′;iω)J0((2j+ 1)∆rkkk) +rkmax∆P0(rkmax, z, z′;iω)J0(rkmaxkk)
. (7.19) It involves the two convergence parameter rkmax and ∆rk, which will be determined in the next section.
7.3.3 Convergence parameters
The numerical evaluation of ∆P0(kk, z, z′;iω) involves several convergence param-eters: First, the convolution over the complex frequency axis, i.e. the evaluation of (7.15), (7.16) and (7.17) includes the grid size ∆ω as well as the maximal fre-quency ωmax. Secondly, for the numerical Fourier transformation (see eq. (7.18)) from ∆P0(rk) to ∆P0(kk) the sufficient maximal values for rk and for the step size
∆rk have to be tested. Similarly to the determination of the numerical parameters for the Fourier transformation of the Green function (see chapter 6), all parameters will be investigated for the potential barrier and the quantum well with the largest positive (V(z) = 0.15) and negative (V(z) =−0.1) values of the potential.
Parameters involved in the convolution over the complex frequency axis The polarization function ∆P0(rk, z, z′;iω) for the potential barrier is displayed in figure
-0.0005 and constant maximal frequency ωmax. Whereas z is situated in the center of the potential, the second spatial coordinate is set to z′ = 0.4 (left) or z′ = 5.0 (right). Although the absolute values of the curves for large distances
|z−z′|(right) are much smaller than for small ones (left), a higher resolution, i.e. a smaller ∆ω is required. For z′ = 0.4 only a marginal deviation of the curves for different values of ∆ω can be observed. Thus, for both positions of z′ an adequate choice is ∆ω = 0.02. Bottom: For the constant parameter
∆ω = 0.02 the maximal frequency ωmax is varied between ωmax = 25 and ωmax= 45. Although the five curves for ∆P0 (left) differ from each other for small values ofrk, the multiplication withrk(right) leads to a reduction of the difference of the curves. Since in the next steprk∆P(rk) will be integrated to obtain ∆P0(kk =0), the maximal frequencyω = 40 suffices for an adequate convergence.
7.3 as a function ofrkfor different values of ∆ωandωmax. As one can see, an adequate convergence is achieved for ∆ω = 0.02 and ωmax = 40. The curves corresponding to different parameters ∆ω (see the two upper plots in figure 7.3) are compared for the constant frequency ω = 0.1 which is the lowest common multiple of the chosen values of ∆ω. In the limit of small frequencies ω (i.e. for frequencies much smaller
7.3 Numerical calculation of ∆P0(kk, z, z′;iω) 89 transformation, for the quantum well for different values ofω. For frequencies in the vicinity of 0, ∆P0(r
k) (orrk∆P0(r
k), respectively,) oscillate strongly and decay much slower to zero than for higher frequencies. Bottom: Determination of the convergence parameters ∆ω (left) and ωmax (right) for the convolution on the complex frequency for the quantum well. Compared to the potential barrier, the parameter ∆ω has to be chosen much smaller, i.e. ∆ω = 0.005 (instead of ∆ω= 0.02 for the potential barrier). As far as the parameterωmax is concerned, an adequate convergence is reached forωmax≈30.
thanωmax) the required convergence parameters ∆ω and ωmaxmerely depend on the frequency at which ∆P0 is compared, since the frequency ω shifts only the limits of the Green function Ghom0 (see section 7.1).
In the plot on the right side of the bottom of figure 7.3 the polarization function ∆P0
is multiplied withrk. This multiplication was made having in mind already the next step, which is the two-dimensional Fourier transformation of ∆P0to reciprocal space.
According to eq. (7.18), ∆P0 is multiplied with rk and the Bessel function J0(rkkk).
Since I am interested in ∆P0(kk = 0) only, the Bessel function simplifies to 1 and the Fourier transformation reduces to an integration over the function rk∆P0(rk).
The function rk∆P0(rk) decays much slower to 0 than ∆P0(rk) itself and reveals
the oscillating behavior of ∆P0(rk) as a function of rk. Of course, this oscillating behavior depends strongly on the complex frequency at which ∆P0 (or rk∆P0(rk), respectively,) is considered. This can be observed in the two plots at the top of figure 7.4, where ∆P0(rk) andrk∆P0(rk) are shown for different frequencies for the quantum well. For the calculations I have used the convergence parameters ∆ω and ωmax for the quantum well which are tested in the two plots at the bottom of figure 7.4. There, rk∆P(rk) is displayed for different parameters ∆ω (left side) andωmax (right side) as a function of rk. As the comparison for the potential barrier has revealed that the convergence is worse for larger |z−z′| (see figure 7.3), one of the z coordinates was chosen to be in the center of the potentialz = 10 and the other atz′ = 5. Whereas in the case of the potential barrier an adequate convergence is reached for ∆ω = 0.02, for the quantum well a smaller step size is required. Although the reduction of ∆ω is very expensive, for all further calculations for the quantum well ∆ω = 0.005 is used. In contrast, the maximal frequency ωmax can be reduced to ωmax ≈30, which is much smaller than the maximal frequency for the potential barrier ωmax ≈40.
Convergence parameters of the two-dimensional Fourier transformation The two-dimensional Fourier transformation of the polarization function ∆P0(rk, z, z′;iω) to the representation ∆P0(kk =0, z, z′;iω) is evaluated numerically using Simpson’s rule according to eq. (7.19). It involves again two numerical parameters, ∆rk and rkmax which are determined in the following. As one can recognize in figure 7.4, it is very important at which frequency the check of the parameters is done: The integrand of the numerical integrationrk∆P0(rk) is most structured and has the slowest decay for frequencies close to the real frequency axis. Since in my implementation I can calculate ∆P0(kk = 0, z, z′;iω) only for frequencies on a grid which is determined by the choice of ∆ω of the convolution on the complex frequency axis, the smallest possible frequency amounts to ω = 0.02 for the potential barrier and to ω = 0.005 for the quantum well. In order to treat both potentials similarly, I will test the convergence for the lowest common multiple ω = 0.02. (Though possible for the quantum well, from now on I will not regard ∆P0(kk = 0, z, z′;iω) for frequencies smaller thanω = 0.02.) The results of ∆P0(kk =0, z, z′;iω) as a function ofzwithz′ centered in the potential are presented in figure 7.5 for different parameters ∆rk and rkmax. For the determination of the parameters I ignore the point at which the two z coordinates are the same, i.e. z = z′ = 0. As I will demonstrate in section 7.3.4, it is very difficult to achieve convergence at this special point and therefore it will be treated in a different way. For all other z coordinates an adequate convergence concerning ∆rk is reached both for the potential barrier and the quantum well for
∆rk = 0.2. In contrast, the maximal rkmax has to be chosen differently for the two potentials. Whereas for the quantum barrier convergence is reached for rkmax≈ 40, for the quantum well a higher rkmax≈50 is required.
7.3 Numerical calculation of ∆P0(kk, z, z′;iω) 91
Figure 7.5:Determination of the convergence parameters ∆rk and rkmax for the potential barrier (top) and the quantum well (bottom). In all four figures the polarization function ∆P0(k
k=0, z, z′ = 0;iω) is presented as a function of zfor constant z′ = 0. On the left side, the parameter ∆rkis varied. An adequate convergence is reached both for the potential barrier and the quantum well for ∆rk = 0.2. In contrast, the maximal rk (right) has to be chosen differently for the two potentials: Whereas rkmax ≈ 40 suffices for the potential barrier, for the quantum well convergence is reached not until rkmax≈50.