3.2 Ab-initio force-constant model
3.3.2 Comparison of phonon frequencies and phonon po-
Figure 3.2: Phonon frequencies along high-symmetry directions obtained from fitted force constants (red line) and obtained from ab-initio force constants (black line) for fcc Ni.
3.3.2 Comparison of phonon frequencies and phonon
3.3 Comparison of the experimental and ab-initio force-constant model
Figure 3.3: Phonon frequencies along high-symmetry directions obtained from fitted force constants (red line) and obtained from ab-initio force constants (black line) for fcc Al.
Figure 3.4: Phonon frequencies along high-symmetry directions obtained from fitted force constants (red line) and obtained from ab-initio force constants (black line) for bcc Fe.
in the whole phonon Brillouin zone. The procedure is described in the following: The set of fitted force constants yields three phonon frequen-cies ωµqfit, µ = 1,2,3 and three polarization vectors efitµq, µ = 1,2,3 for every phonon wavevector q. The same holds for the set of ab-initio force constants which yields ωνqab-initio, ν = 1,2,3 and eab-initioνq , ν = 1,2,3 . It is not clear from the very beginning how to compare the frequencies and polarization vectors. There are in total six possibilities:
1. µ1= 1 withν1= 1,µ2= 2 with ν2= 2,µ3= 3 withν3= 3 2. µ1= 1 withν1= 1,µ2= 2 with ν2= 3,µ3= 3 withν3= 2 3. µ1= 1 withν1= 2,µ2= 2 with ν2= 1,µ3= 3 withν3= 3 4. µ1= 1 withν1= 2,µ2= 2 with ν2= 3,µ3= 3 withν3= 1 5. µ1= 1 withν1= 3,µ2= 2 with ν2= 2,µ3= 3 withν3= 1 6. µ1= 1 withν1= 3,µ2= 2 with ν2= 1,µ3= 3 withν3= 2 . To determine the correct possibility a combination of frequency and eigen-vector argumentation is used.
1. Look at just one special q-point eq where the following condition is fulfilled: All three frequenciesωfitµ
eq are clearly separated forµ= 1,2,3, i.e., not degenerate or near a degeneracy, and all three fre-quencies ωab-initioν
qe are clearly separated for ν = 1,2,3. Then, it makes sense to compare the maximum frequency of the ab-initio force-constant model with the maximum frequency of the fitted force-constant model|ωfitµ
maxeq−ωab-initioν
maxeq |, and consequently the com-parison of the minimum frequencies|ωfitµ
minqe−ωab-initioν
mineq |and middle frequencies|ωfitµ
mideq−ωνab-initio
mideq |makes sense. In principle, it could be possible that this choice is wrong but the calculation showed that this choice is in general correct. It is certainly wrong for q-points at or near a degeneracy.
2. The eigenvectors of neighboring q-points on the same branchµ or ν are almost perfectly parallel for dense q-point grids. The scalar product is almost 1, for example efitµq
1·efitµq
2 ≈1, whereq1 andq2 are neighboring q-points on a dense grid. This is wrong for q-points at or near degeneracies!
3.3 Comparison of the experimental and ab-initio force-constant model
(a) (b)
Figure 3.5: Differences between the calculation with ab-initio force con-stants and the calculation with fitted force concon-stants [31] for fcc Ni. (a) relative phonon frequency deviation in percent (only greater than 2.5%), (b) modulus of the scalar product between the polarization vectors (only less than 0.95). From ref. [26]. Copyright by the American Physical Soci-ety.
3. For regions which are far away from a degeneracy point we can find the right comparison of branches with a combination of step 1. and step 2. For regions near or at a degeneracy point where the criterion of step 1. and step 2. fails, one cannot find the correct comparison. Therefore, these points are omitted for the comparison of eigenvectors but taken into account for the comparison of the frequencies because the frequencies are anyway almost the same even if one had chosen the wrong combination. Note that directly at the degeneracy point any linear combination of eigenvectors is again an eigenvector.
With this procedure at hand the comparison of phonon frequencies and po-larization vectors obtained from ab-initio and from fitted force-constants can be made. Fig. 3.5, fig. 3.6 and fig. 3.7 show the difference of frequen-cies (see (a)) and polarization vectors (see (b)) for fcc Ni, fcc Al and bcc
(a) (b)
Figure 3.6: Differences between the calculation with ab-initio force con-stants and the calculation with fitted force concon-stants [39] for fcc Al. (a) relative phonon frequency deviation in percent (only greater than 4.5%), (b) modulus of the scalar product between the polarization vectors (only less than 0.95). From ref. [26]. Copyright by the American Physical Soci-ety.
3.3 Comparison of the experimental and ab-initio force-constant model
(a) (b)
Figure 3.7: Differences between the calculation with ab-initio force con-stants and the calculation with fitted force concon-stants [40] for bcc Fe. (a) relative phonon frequency deviation in percent (only greater than 7%), (b) modulus of the scalar product between the polarization vectors (only less than 0.995). From ref. [26]. Copyright by the American Physical Society.
Fe, respectively, for a 100×100×100-grid in the first octant of the first phonon Brillouin zone (qx, qy, qz≥0).
The frequency difference is normalized by the maximum frequency in the Brillouin zone, |ωfitµq−ωνqab-initio|/ωmax and is given in percent. The deviation is up to 2.9% for Ni, up to 5.2% for Al and up to 9% for Fe and for all three materials only in small regions of the Brillouin zone whereas in most regions the deviations between the two results are very small.
The deviation of polarization vectors is given as modulus of the scalar product,|efitµq·eab-initioνq |, i.e.,|cosφ|is given whereφis the angle between efitµq and eab-initioνq . There are only small regions of the Brillouin zone for which the scalar products are smaller than 0.95 for Ni and Al and smaller than 0.995 for Fe, and in these regions the scalar products reach values down to 0.6 for Ni and Al and down to 0.98 for Fe. Obviously, the smallest deviations can be found for Fe although the frequency deviations are the largest of the three materials.
In summary, the investigations showed that there are indeed deviations between the frequencies and polarization vectors obtained from the ab-initio force constants and obtained from the fitted force constants. How-ever, the deviations are not large and only small parts of the phonon Brillouin zone are affected. One can conclude that for a calculation for which reliable phonon frequencies and polarization vectors are necessary it is also possible to take force constants which were fitted to phonon frequencies only (without using polarization vectors). Probably, the sym-metry requirements for the unitary transformation of the force-constant matrix is restrictive enough to give good results for both frequencies and polarization vectors. Probably, similar results can be obtained for other three-dimensional metals and it is concluded that time-consuming ab-initio calculations are not necessary to obtain reliable frequencies and polarization vectors in the whole phonon Brillouin zone.
4 Ultrafast demagnetization after laser pulse irradiation
As already explained in the introduction one can distinguish two main ul-trafast demagnetization effects: the all-optical switching and the ulul-trafast demagnetization. They are closely related but only the latter is discussed in this thesis and fig. 1.1 shows a schematic picture of the typical experi-ment. In fig. 1.1 (a) a ferromagnetic material with in-plane magnetization (on a substrate) is irradiated by a fs laser pulse and in (b) one can see a sharp drop of the magnetization by about 40% within about 100 fs.