1.4 Experimental Results
1.4.2 Absorbed Laser Power Density
Considering a sample, which spacial expansion is described by the vector ~r = (x, y, z), the parameter P(~r, t) in the relation:
P(~r, t) = αabs·F(~r, t)·e−z/Λopt
Λopt ·(1−e−d/Λopt) (1.17) describes the amount of the absorbed power density, when being illuminated by a light pulse with the time dependent pump fluenceF(~r, t) from a laser beam pointing in z direction. The focus on the data analyzed below lies on the time of maximum demagnetization, which is usually reached within around 300 fs after the pump pulse has been absorbed by the sample. The pulse length is 80 fs, and the time between two pulses is 4µs, assuring enough time for temperature relaxation into the initial state between pulses. Therefore, the first part of the analysis will focus on the total energy deposited by a single pulse, neglecting its temporal expansion. Further,
heat diffusion will be neglected in the calculations, because it becomes important only 50 ps after excitation. The spacial expansion of F is already considered by the measurement of the beam profile, 60µm in diameter. The factor αabs is the absorption coefficient given by αabs = (1−R −Tˆ). With R being the amount of light reflected by the material, and ˆT the amount of light transmitted through the material. Considering an angle of incidence of the pump beam around 0◦ to the surface normal, yields inR = 0.68301 for all nickel layers. This can be obtained by applying Fresnel’s equations for the reflection of light from materials with a complex refractive index nC:
R = (nR−1)2+n2I
(nR+ 1)2+n2I. (1.18)
The denominator in equation 1.17 expresses the energy conservation by normalizing the expression. This yields for a sample of the thicknessd:
Z d 0
P(~r)dz =αabsF(~r).
The factor αabs takes into account the energy that is not used for heating the in-vestigated material due to reflection and transmission. The reflected part can be calculated easily, as shown in equation 1.18, and checked experimentally, by mea-suring the intensity of the reflected beam. It appears to be more challenging to estimate the transmission through the layer, especially for the thin nickel layers, with d <Λopt = 14.5 nm. On one hand, the calculation, to be precise, should take care of the effects at the interface between the investigated metal layer, and the substrate, leading to complicated calculations. On the other hand, the transmitted light absorbed by the substrate cannot be measured experimentally. This leavesαabs to be the most critical parameter in the estimation of the absorbed power density, contributing the largest error to the temperature calculated within the 2T model.
A too low estimated ˆT leads to a higher αabs, and thus to an overestimation of the peak temperature, and vice versa.
The transmission ˆT in the absorption factor will be calculated using the exponen-tial decay of the intensity:
using Λopt from section 1.4.1. The factor 1− R takes care of the part reflected from the layer, so that the amount absorbed can be estimated by the integral.
Considering a refractive index of nickel, as given in section 1.4.1 and the refractive index of crystalline silicon n = 3.69 for the wavelength λ = 800 nm, the reflected amount of light at the interface between nickel and silicon according to Fresnel’s equations is around 4 %. That means, the contribution of reflections at the interface between the nickel film and the substrate, contributing to electron excitations can be neglected in this analysis.
1 Ultrafast Spin Dynamics
Figure 1.9: Demagnetization ∆M/M−300 K plotted against the applied pump fluences rescaled to the per nickel layer absorbed power density P/P10 nm,10 mJ/cm2. The open circles represent the calculation assuming the same absorption co-efficient αabs for all nickel thicknesses, whereas the closed circles represent the calculations including thickness dependent absorption coefficients, as de-scribed in equation 1.19. The lines are guides to the eye.
Figure 1.9 shows the applied pump fluences, calculated to the absorbed power density P and rescaled to the power density absorbed by the 10 nm nickel film perturbed by 10 mJ/cm2, P0. The calculation details are given in the appendix A.
The data calculated neglecting the thickness dependence of the transmission coef-ficient ˆT show a linear relation to the measured demagnetization (left open circles), for nickel films of thickness d ≥ 10 nm. Calculating the per layer absorbed power density for the two thinnest films in the analysis, under the assumption of a thick-ness independent ˆT, yields an energy density that is up to five times as high for the 2 nm thick film, as for the others. Surprisingly, those two thinnest nickel layers, d = 2 nm and d = 5 nm do not show higher maximum demagnetization rates with increasing pump fluence, than the thicker films.
Taking into account the thickness dependence of the transmission factors for dif-ferent nickel films, the per layer absorbed power density remains constant for the three thickest nickel films for each fluence. There are no significant changes, al-though for these films the thickness difference is larger, than for the three thinnest films. At the same time the demagnetization ∆M/M300 K increases with decreasing film thickness. Furthermore, the relation between the per nickel layer absorbed en-ergy and the demagnetization stays linear for each of the three thickest films. The data for the three thinnest layers on the other hand, show an increasing per layer absorbed energy with decreasing film thickness not only in comparison to the three
Figure 1.10: The per layer absorbed power density remains constant at all fluences for nickel films thicker than the penetration depth Λopt = 14.5 nm. In nickel films thinner than Λopt, the per layer absorbed power density increases with decreasing film thickness.
thickest layers, but also between each other. That means, the 2 nm thick nickel film absorbs more energy per layer for each fluence, than the 5 nm thick film and the 10 nm film. However, the demagnetization saturates at around 40% and does not increase further with increasing pump fluence.
For clarity, figure 1.10 shows the relation between the pump fluence and the per layer absorbed energy normalized to the energy absorbed by the 10 nm film pumped by a 10 mJ/cm2 laser fluence. The relation for this fluence range and film thickness is linear, as expected from the formulas given in the appendix A. The deviation from linearity will not occur for thicker samples, because the transmission, as defined above does not change significantly for films thicker than Λopt, and also the term exp(−dNi/Λopt) converges to 0 with a decreasing slope. Therefore, the thicker films, starting with dNi = 15 nm absorb approximately the same amount of energy per layer at each fluence. The thinner films on the other hand absorb a higher amount of energy per layer at each fluence, with the highest amount being absorbed by the thinnest nickel film. This suggests that for all nickel films, the maximum electron temperature constantly increases with the pump fluence, and the thinner films reach higher electron temperatures.
1 Ultrafast Spin Dynamics
Figure 1.11: Peak electron (top) and spin (bottom) temperatures (left) and the cor-responding temperatures 20 ps after excitation (right) plotted against the applied pump fluence, for the different nickel film thicknesses.