Methods for transmission mode x-ray analysis of a sample by means of apparatuses comprising an x-ray radiation source that provides x-ray radiation for irradiating the sample and a detector for detecting x-ray radiation transmitted through by the sample.
Figure 3-1 shows a schematic with the intensity I0 of incident x-ray and the intensity It
of being transmitted through by the sample.
Figure 3-1: Schematic with the intensity I0 of incident x-ray and the intensity It of being transmitted through by the sample.
A narrow parallel monochromatic x-ray beam of intensity I0passing through a sample of thickness dx will get a reduced intensity It according to the expression :
dIt= I0(x) n dx, (3.1)
where dIt is the change in intensity, I0 is the initial intensity, n is the number of atoms/cm3, is a proportionality constant that re‡ects the total probability of a photon being scattered or absorbed and dx is the incremental thickness of material transversed.
When this equation is integrated, it becomes :
It = I0exp( n x), (3.2)
The number of atoms/cm3 (n) and the proportionality constant ( ) are usually combined to yield the linear attenuation coe¢ cient ( ). Therefore the equation becomes :
It = I0exp( x). (3.3)
where It is the intensity of photons transmitted across some distance x, I0 is the initial intensity of photons, is the linear attenuation coe¢ cient, and x is distance traveled.
The linear attenuation coe¢ cient ( ) describes the fraction of a beam of x-rays or gamma rays that is absorbed or scattered per unit thickness of the absorber. This value basically accounts for the number of atoms in a cubic cm volume of material and the probability of a photon being scattered or absorbed from the nucleus or an electron of one of these atoms. Using the transmitted intensity equation above, linear attenuation coe¢ cients can be used to make a number of calculations. These include :
(1) the intensity of the energy transmitted through a material when the incident x-ray intensity, the material and the material thickness are known.
(2) the intensity of the incident x-ray energy when the transmitted x-ray intensity, material, and material thickness are known.
(3) the thickness of the material when the incident and transmitted intensity, and the material are known.
(4) the material can be determined from the value of when the incident and
trans-Figure 3-2 is an illustration of linear attenuation coe¢ cient ( ) versus wavelength ( ) and we can …nd that is as a function of for every material; however, the extinction coe¢ cient k (the imaginary part of refractive index) is proportional to the linear atten-uation coe¢ cient in accordance with Eq. (3.5). The refractive index is complex, the extinction coe¢ cient introduces a decrease of the amplitude of the waves passing through the material and phase changes between the incident and successively re‡ected waves.
N = 1 i = n ik,
where 1 (n) is the real part of refractive index and (k) is the imaginary part of refractive index. And more speci…cally de…nitions are
= re 2
2 Naf20 and = re 2
2 Naf20
4 , (3.4)
= re 2
Amuf10, (3.5)
where re is classical electron radius, Na is the Avogadro’s number, mu is the atomic mass unit, and f10 and f20 are the anomalous dispersion correction factors. And the detail datas above see Appendix.
Figure 3-2: Schematic progression of the linear absorption coe¢ cient of wavelength.
Hence, we regard the …gure as a relation chart of extinction coe¢ cient and wavelength underlying the x-ray spectrum. At certain energies where the absorption increases dras-tically, and gives rise to an absorption edge. Each such edge occurs when the energy of the incident photons is just su¢ cient to cause excitation of a core electron of the absorb-ing atom to a continuum state, i.e. to produce a photoelectron. Thus, the energies of the absorbed radiation at these edges correspond to the binding energies of electrons in the K, L, M, etc, shells of the absorbing elements. See Figure 3-3, the absorption edges are labelled in the order of increasing energy, K, LI, LII, LIII, MI, . . . , corresponding to the excitation of an electron from the 1s (2s1 / 2), 2s (2s1 / 2), 2p (2p1 / 2), 2p (2p3 / 2), 3s
(2s1 / 2), . . . orbitals (states), respectively. In other words, the absorption e¤ect of
mate-rials would apparently lower and its the key to choose what kind of re‡ector would be in each spetrum.
Figure 3-3: The left side is a simpli…ed illustration of Bohr atomic model and the right side is a comparison of continuum level versus absorption edge overplotted.
We called a light element as spacer (i.e. silicon) that absorbs light only weakly, and a heavy element as absober (i.e. molybdenum) that absorbs light very strongly. However, the reason should be considered is not only the extinction coe¢ cient but easily form an stable and/or sharp interface with other element. At present, the materials of silicon (Si),
combination of spacer layer is called absorber layer, as well as the extinction coe¢ cient is much larger than spacer layer. To choose those materials have large di¤erence of extinction coe¢ cient but small one is to avoid the incident light would be absorbed completely, thus the re‡ectanec can not be improved. Figure 3-4 [30] is all di¤erent types of general x-ray re‡ectors with its suitable energy (wavelength). The illumination wavelength of 13.5 nm was chosen for EUV lithography based on the early development and good performance of molybdenum-silicon MLs in this wavelength region. Mo/Si still remains the most extensively investigated and best understood ML material pair to date, and 13.5 nm is in the wavelength region just longer than the L2, 3 absorption edge of Si (12.4 nm), where Mo/Si achieves its best re‡ective performance. In addition, the
…rst available sources for EUVL were LPP sources, with good conversion e¢ ciency in this wavelength region. Even though the output of the LPP source at 13.5 nm was lower than at 11 nm, the natural width of the Bragg peak of a Mo/Si ML at 13.5 nm is broader than the peak width of a Be-based ML at 11 nm. Hence, the overall integrated re‡ectance is comparable at both 11-nm and 13.5-nm wavelengths. The broader peak width at 13.5 nm also relaxes speci…cations for optic-to-optic wavelength matching. There are othe bene…ts associated with operating at 13.5 versus 11 nm.
Figure 3-4: Theoretical multilayer survey: Maximum achieveable normal incidence
re-‡ectance in the soft x-ray range for the systems shown.